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PREFACE.
1 FEAR I cannot defend the title
1
of this volume
as very accurately describing its contents. The title
“Lessons Introductory”applied well enough to the
former edition, which grew out of Lectures given
some years ago to my Class, and which was not
intended to do more than to supply students with
the preliminary information necessary to enable them
to read with advantage the original memoirs whence
my materials were derived. That edition has,how
ever, been for a long time out of print, and in re
printing it now I found so many additions necessaryto bring it up to the present state of science, that
the book has become doubled in siz e, and might
fairly assume the less modest title of a“Treatise”
on the subjects with which it deals. Neither does
the name“Modern Higher Algebra”
very preciselydefine the nature of these subjects . The Theory of
Elimination,and that of Determinants cannot be
said to be very modern ; and I do not meddle with
some parts of Higher Algebra , for which much has
been done in modern times ; as,for instance
,the
v i PREFACE.
Theory of Numbers, or the General Theory of the
Resolution of Equations. But it is no great abuse
of language to give, in a special sense, the name
“Modern Higher Algebra”to that which forms the
principal subject of this volume— the Theory of
Linear Transformations. Since Mr. Cayley’s dis
covery of Invariants, quite a new department of
Algebra has been created ; and there'
is no part of
Mathematics in which an able mathematician, who
had turned his attention to other subjects some
twenty years ago , would find more difficulty in
reading a memoir of the present day, and would
more feel the want of an elementary guide to informhim of the meaning of the terms employed, and to
establish the truth of the theorems assumed to be
known.
With respect to the use of new words I have
tried to steer a middle course. In this part of
Algebra combinations of ideas require to be fre
quently spoken of which were not of important use
in the older Algebra . This has made it necessaryto employ some new words, in order to avoid an
intolerable amount of circumlocution. But feelingthat every strange term makes the science more
repulsive to a beginner,I have generally preferred
the use of a periphrasis to the introduction of a neW
word which I was not likely often to have occasionto employ. Students who may be disappointed by
PREFACE. v ii
not finding in this volume the explanation of some
words which occur in modern algebraical memoirs,
'
will be likely to find the desired information in the
Glossary added to Mr. Sylvester’s paper (Philosolnlzteal
Transactions,1853
, p.
The first four or five Lessons in this volume were
printed a year or two ago : it having been at that
time my intention to publish separately the Lessons
on Determinants as a manual for the use of Students.
At the time when these Lessons were written I had
not met Baltz er’s Trea tise on Determinants
,a work
remarkable for the rigorous and scientific manner in
which its principles are evolved. But I most re
gretted not to have met with it earlier, on account
o f his careful indication of the original authorities
for the several theorems. Although very sensible of
the value of these historical notices, I have , in the
text of these Lessons,too often omitted to assign
the theorems to their original authors, because my
knowledge not,having been obtained by any recent
“
coursewo f study, I did not find it easy to name the
sources whence I had derived it, nor had I mathe‘
matical learning enough to be able to tell whether
these sources were the originals. I have now tried
to supply the references omitted in the text by
adding a few historical notes ; following Baltz er’s
guidance as far as it would serve me. Where I
have had only my own reading to trust to , it is
v iii PREFACE.
only too likely that I have in several cases fa iled
to trace theorems back to their first discoverers ,
and I must ask the indulgence of any liv ing authors
to whom I have in this way unwittingly done
injustice.
I have to thank my friends Dr. Hart,Mr. Traill,
and Mr. Burnside, for help given me at various times
in the revision of the proofs of this work, though,in justice to these gentlemen
,I must add that there
is a cons iderable part of it which was printed under
circumstances where I could not have the benefit
of their assistance,and for the errors in which they
'
are not responsible. I have already intimated that
my obligations to Messrs . Cayley and Sylvester are
not merely those which every one must owe who
writes on a branch of Algebra which they have done
so much to create. I was in constant correspondence
with them at the time when some of their most
important discoveries were made,and I owe my
knowledge of these discoveries as much to their
letters as to any printed papers. I must also ex
press my thanks to M. Hermite for his obliging
readiness to remove by letter difficulties which cc
curred to me in my study of his published memoirs.
TRINITY COLLEGE, DUBLIN,October 16th, 1866.
C ONTENTS.
LESSON 1.
DETERMINANTs—PRELIMINARY ILLUSTRATIONS AND DEFINITIONS.Rule of signs
LESSON II.
REDUCTION AND CALCULATION OF DETERMINANTS.Minors [called by Jacobi partial determinants]Examples of reduction
Product of difierences of n quantities expressed as a determinantReduction of bordered Hessians
LESSON III.
MULTIPLICA ’I‘ION or DETERMINANTS.The theorem stated as one of linear transformationExtension of the theorem
Product of squares of diiferences Of n quantifies
Relation connectingmutual distances of points on a circle or sphereOf five points in space
LESSON IV.
MINOR AND RECIPROCAL DETERMINANTS.Solution of a.system of linear equationsReciprocal systems [called by Cauchy acyoz
'
nt systems]Minors of reciprocal system expressed in terms of those of the originalMinors of a determinant which vanishes
X CONTENTS.
LESSON V.
SYMMETRICAL AND SK EW SYMMETRICAL DETERMINANTS.Difierentials of a determinant with respect to its coefficientsIf a determinan t vanishes, the same bordered is a perfect squareSkew symmetric determinants of odd degree vanish
Of even degree are perfect squares [Cayley]Nature of the square root [see Jacobi , Crelle, II. 354 ; XX IX. 236]Orthogonal substitutions
LESSON VI.
SYMMETRICAL DETERMINANTS.Equation of secular inequali ties has always real rootsSylvester’
s prodf
Another new proofBorchardt
’s proof
Sylvester’s expressions for Sturm’
s functions In terms of the Ioots
LESSON VII.
SYMMETRIC FUNCTIONS.Newton’
s formulae for sums of powers of rootsRules forweight and order of a symmetric flmctionRule for sum of powers of difierences of rootsDifferential equation of function of diiferences
Symmetric functions of homogeneous equationsDifiere
‘
ntial equation where binomial coeficients are used
Serret’s notation
Brioschi ’s expression for the operation '
In terms of the roots
LESSON VIII.
ORDER AND WEIGHT OF ELIMINANTS.Elimination by symmetric functionsOrder and weight of resultant of two equationsSymmetric functions of common values of system of two equationsExtension of principles to any number of equations
LESSON IX.
EXPRESSION OF ELIM INANTS As DETERMINANTS.Elimination by process for greatest commonmeasureEuler
’s method
Conditions that two equations should have two common factorsSylvester’
s dialytic methodBez out
’s method
Cayley’5 statement of it
Jacob i ans defined
Expression by determinants, in particular cases, of resultant of three equationsCayley
’s method of expressing resultants as quotients of determinants
CONTENTS. xi
LESSON X.
DETERMINATION OF COMMON ROOTS.Expression of roots common to a.system Of equations by the differentials of the
resultantEquations connecting these difierentia ls when the resultant vanishesExpressions by the minors of Bez out
’s matrix
General expression for difierentials of resultants with respect to any quantities
General conditi ons that a systemmay have two common roots
Order and weight of discriminantsDiscriminant expressed in terms of the rootsDiscriminant of product of two ormore functionsDiscriminant 15 of form coq> 4 1
24;Formation of discriminants by the difierential equationMethod of finding the equal roots when the discriminant vanishesExtension to any number of variablesDiscriminant of a quadratic function
LESSON XII.
LINEAR TRANSFORMATIONS.Invariance of discriminantsNumber of independent invari antsInvariants of systems of quanticsCovariantsEvery invariant of a covariant is an invariant of the originalInvariants of emanante are covariants
m2 m &c. absolutely unaltered by transformation
Evectants
Evectant of discriminant of a quantic whose discriminant vanishes
LESSON XIII.
FORMATION OF INVARIANTS AND COVARIANTS.Method by symmetric functionsInvariants, th e. which vanish when two or more roots are equalMutual diiferentiation of covariants and contravariantsDifferential coefficients substituted for the variables in a. contravariant
covariantsEvery binary quantic has an invariant of even degreeCubinvariant of a quarticEvery quantic of odd order has invariantof the 4th orderWeight of an invariant of givenorder
Binary quantics of even degree cannot have invariants of odd order
X l l CONTENTS.
Skew invariantsDetermination of numbers of independent invariants by the difierential equationCoefficients of covariants determined by the difierential equationSource of product of two covariants is product of their sourcesExtension to any number of variables
LESSON XIV.
SYMBOLICAL REPRESENTATION or INVARIANTS AND COVARIANTS.Formation of derivative symbols
Order of derivative in coefficients and in the variablesTable of invariants of the third orderHermite’
s law of reciprocityDerivative symbols for ternary quanticsSymbols for evectants
Method of Aronhold and ClebschClebsch
’s proof that every invariant can be thus expressed symbolically
LESSON XV.
CANONICAL FORMS.Generality of a form determined by its number of constantsReduction of a quadratic function to a sum of squaresPrinciple that the number of negative squares is unaffected by real substitutionReduction of cubic to its canonical form
Discriminant of a cubic and its Hessian diifer only 111 srgn
General reduction of quantic of odd degreeMethods of forming canoniz antCondition that quantic of order 271 should be reducible to a sum of n, 2271
th
powers [called by Sylvester cata lecticam]Canonical forms for quantics of even order
Canonical forms for sexticFor ternary and quaternary cubic
LESSON XVI.
SYSTEMS or QUANTICS.Combinants defined, differential equation satisfied by themNumber of double points in an involutionFactor common to two quantics i s also factor in JacobianOrder of condition that u Icv may have cubic factorNature of discriminant of JacobianDiscriminant of discriminant of u 761:
Proof tha t resultant is a combinantDiscriminant with respect to cc , y, of a function of u, v
Discriminant of discriminant of u Iw for ternary quanticsTact-invariant of two curvesTact-invariant of complex curves
Of It 1 , k-ary quantics
Of two quaternary quanticsOeculants
rAOn
115
116
116
118
119
CONTENTS. xur
LESSON XVII.
APPLICATIONS To B INARY QuANTIcs.Invari ants when said to be distinctNumber of independent covariantsTHE QUADRATIO
Resultant of two quadraticsCondition that three should form a system in involutionTHE CUBIC
Geometric meaning of covari ant cubicSquare of this cubic expressed In terms of the other covari antsSolution of cubic
SYSTEM OF CUBIC AND QUADRATICSYSTEM or Two CUB ICS
Resultant of the systemCondition that u M may have a cubic factorMode of deal ing wi th equations whi ch contain a.superfluous variableInvariants of invariants of u M are combinanteTHE QUARTIO
Catalecticants
Every invariant of a quartic 1s a rational function of S and T
Discriminant of a quarticThe same derived from theory of two cubicsRelation between covariants of a cubic derived from that of Invariants of a
quarticConditions that a quartic should have two square factorsReduction of quartic to its canoni cal formGeneral solution of quarticCriteria for real and Imaginary rootsThe quartic can be brought to its canonical form by real substitutionsSextic covariant of quarticInvariants of system of quartic and its Hessian
SYSTEM or Two QUARTICSTheir resultantCondition that u Xv should be perfect squareCondi tion that u Xv should have cubic factorSpecial form when both quartics are sum of two fourth powersTHE QUINTICCondition that two quartics should be capable of bemg got by differentiation
from the same quinticDiscriminant of quinticIts fundamental invariantsConditions for two pairs of equal rootsAll invariants of a quantic vanish if more than half its roots be all equalHermite ’
s canonical form
Hermi te’s skew invariant
Covariants of quinticSign of di scrim in ant of any quantic determines whether it has an odd or even
number of pairs of '
Imaginary rootsCriteria furnished by Sturm’
s theorem for a quinticIf roots all real , canonizant has imaginary factorsInvariantive expression of criteria for real rootsSylvester’
s criteria
xiv CONTENTS.
Conditions involving a.constant varyingwithin certain limitsSextic resolvent of a quinticHarley and Cockle’
s resolventExpressi on of invariants in terms of rootsTHE SEXTIC— its invariantsCondi tions for cubic factoror for twosquare factorsThe discriminantSimplerinvariant of tenth i
order
The skew invariant expressed in terms of other invariantsFunctions likely to alford criteria for real roots
LESSON XVHI .
ON THE ORDER or RESTRICTED SYSTEMS or EQUATIONS.Order and weight of systems definedRestricted systemsDeterminant system,
7c rows, 70 1 columns
Order and weight of.condi tions that two equations Should have two common
roots
System of condi tions that three equations Should have a common rootIntersection of quantics having common curvesCase of distinct curvesProblem to find the number of quadrics which pass through five points and
touch four planesRank of curve represented by a system of lo rows, k 1 columns
Systemo f krows, k 2 columns
System of conditions that two equations should have three common rootsSystem,
of quantics having a surface common
Having common two surfaces .
System of conditions that three ternary quantics should have two common
points 235
Rule when the constants in systems Of equations are connected by relations
LESSON XIX.
APPLICATIONS or SYMBOLIOAL METHODS.When afigure disappears from a derivatIve symbol, U is a factorResult of substi tuting in reciprocal , differentials for the variablesSymbols for derivatives of derivativesSymboli cal expression for resultant of quadratic and equationof nth degreeFormula for discriminant of quarticClebsch
’s investigation of resultant of quadratic and n-tic
APPENDIX.
Value of skew invariant of a sextic
CONTENTS. XV
NOTES.History of determinantsCommutanteHessians
Symmetric functionsEliminationDiscriminantsLinear transformationsOn the number of invari ants or covariants Of a binary quanticCanonical forms
Comb inants
Applications to binary quanticsThe quintic
Every Skew invariant vanishes when alternate terms in a quanticwanting
Hermite’s forme-types
The Tschirnhausen transformationOn the order of systems of equations
Mr. S. Roberts’ methods of investigationBez outiants
Tables of resultantsHirsch and Cayley’
s tables of symmetric functionsMr.M. Roberts’ tables of sums of powers of differencesIndex
ERRATA.
p. 27, line 11,for Emmy , A, read 2 .0mm, A.
p. 140, lines 2 and 3 from bottom,f or 1 ,read I .
p. 168, line 16,for “invariants,” read “covari ants.
p. 194, note, line 3 from bottom,f or is of the recurring form,
” is capablebeing linearly transformed to the recurring form.
”
p. 197, line 17, the word “not ” has been omitted.
p. 200, note, line 11 from bottom,f or D,read JD .
p. 211, lines 4 and 5, interchange the words positive and negative.
p. 211, line 3 frombottom,fo r A3 100B2 , read A2 IOOB.
LESSONS ON HIGHER ALGEBRA.
LESSON I.
DETERMINANTS.-PRELM INARY ILLUSTRATIONS.
1. IF we are given or homogeneous equations ofthefirst degreebetween at variables
,we can eliminate the variables
,and Obtain a
result involving the coefficients only, which is called the deter
minant of those equations. We shall,in wh t follows, give ,
mics
for the formation of these determinants, and?Shall state some of
their principal properties ; but we think that the general theorywill be better understood if we first give illustrations of its appiication to the simplest examples.
Let us commence, then, with two equations between two
variablesup b
ly=
)
Q,
ape bzy 0.
The variables are eliminated by adding the first equationmulti
plied by b to the second multiplied by when we get
a b — a 6 = 0, the left-hand member of which Is the determinant
required. The ordinary notation for this determinant 13
We shall,however, often, for brev ity, write to express this
determinant, leaving the reader to supply the termwith the negative sign ; and in this notation it is Obv ious that (a b (a bThe coefficients a
, ,bu &c., which enter into the expression of a
determinant, are called the sonatituents of that determinant, and
the products a are called the elements of the determinant .
B
2 DETERMINANTS.
2. It can be verified at once that we should have obtained
the same result if we had eliminated the variables between
the equationsblw+ bzy
= 0.
In other words
27 2 17 2
or the value of the determinant is not altered if we swrite the
horiz ontal rows vertically, and vice versé.
3 . If we are given two homogeneous equations-b etweenthree variables
,
blx + bzy+ bez
= 0 ;
these equations are sufficient to determine the mutual ratios of
w, y, z . Thus
,by eliminating y and a: alternately, we can
express a: and ygr terms of e ; when we find
(a 162)w (afibB)z (a lb2)y (G86!)z '
In other worde '
rn,"
y, z , are propOrtional respectively to (a zbs),(ESQ), (a lbe). Substituting these values in the original equations,we obtain the identical relations
“I (d ebs) a
2 (0 861) as (d l ) O
7bl (a flb
fl) 62 (a sbl) 6
8 (d l ) 0
relations which are verified at once by Writing them at full
length,as for instance
a, (uzbel
age) a
, (<nl a b at, (a b a
,b,) O.
The notation.a ,, a” a
s
b, ,by, b
(where the number of columns is greater,
than the number of
rows)is used to express the three determinants ,
which can be
obtained by suppressing in turn each one of the columns ; v iz .
the three determinants of which we have been speaking, (cabs),
(d ab):
4. Let us now proceed to a system of three equations
8
a1w+ bly+ c
lz = O
,
Then,if we multiply the first by (a zbs), the second by (ask), the.
third by and add ; the coeficients of a: and 3; will vanish
DETERMINANTS. 3
in virtue of the identical relations of Art. 3, and the deter
minant required is
61
02
08 (d l );
or,writing at full length
,
a b csagbl.
c a b — c a b ” d b l- gaps“ , 1 2l 2 8 1 8 2 2 3 1
It may also be written in either of the forms,
a.(5203) a
.are"
o. b.(ca ) b
.(ca ) 6.(as)
This determinant Is expressed by the notation“15 bi , 0
1
def 627. ca
as, be? cs .
though we shall Often‘
use forit the abbreviation (0 16303)It is useful to observe that
but
For by analogy of notation,
(a 2bscl)= “a (6801) a
3 (b,cg) a1
which is the same as
while
a1 (6802) a
, (bael) a,
which is the same as (a lbaca).
5 . We should have Obtained the same result of elimination ifwe had eliminated between the three equations
a ltr day O
,blw b
zy 65a 0, e
lm e
zy 632 0.
For if we proceed on the same system as before,multiplying the
first equation by the second by (ages), and the third by(agba), and add, then the coefficients of y and z vanish
,and the
determinant is obtained In the form
a, (6203) 6
1 (czas) O] (d ebs)?
which, expanded, is found to be identical with Hence
an bl) 61 an “a,
as
as?be)
02
bu6)be
as?be,
63
”I,
027
as
or the determinant is not altered by writing the horiz ontal rowsvertically, and vice verse) a property which will be proved:tobe true of every determinant.
4 DETERMINANTS.
6. Using the notation
an
as?
as)
“4
61)be?be?bs
oncan
09)
04
to denote the system of determinants obtained by omitting in
turn each one of the columns ; these four determinants are con
nected by the relations
at (d ebtors) a
s (asbACr) “3 (6546102) “
407
bl (0 116804) 6
: (6186401) be (“Abroad 6
4 (a ibacs) 0,
01 0126304) 0
2 (6136401) 03
04 (6216208) 0°
These relations may be either verified by actual expansion of the
determinants,or else may be proved by a method analogous to
that used in Art. 3 . Take the three equations
an: azy a
sz a
4w 0
,
b,a: b
aa = 0
,
on: cg Caz c
,w = O.
Then (as in Art. 5)we can eliminate y and a by multiplyingthe equations by (agba), respectively, and adding
,
when we get
In like manner, multiplying by (bag), (aah)respectively,we get (“abscri .7/ w 0'
And in like manner,
(61536102)z w 0.
Now, attending to the remarks about signs (Art. theseequations are equivalent to
(a rbiter!) -(a zbSOA)w? (651320931: (asb4cl)740? (61162092 (“4brcalw'
or at, y, z , w are respectively proportional to (a b c (a b c
substituting which values in thisbiiginal squations
,we obtain the identities already written.
7. Ifnow we have to eliminate between the four equa tionsale: by 0
12 d
lw 0
,
bzy+ 0
2z + d
2w= 0
,
ago: c
sz d
sw 0
,
we have only to multiply the first by the second by
(a sbyzl), the third by the fourth by (a lbgca), and add,
6 DETERMINANTS.
9 . A cyclic interchange of sufixes a lters the sign when the
number of terms in the p roduct is even ; but not so when the
number of terms is odd. Thus a2b,,being got from a
lb2by one
interchange o f suffixes,has a different sign ; but a
,b,clhas the
same sign with albzc9from which it is derived by a double permu
tation. For,changing the suffixes of a and b
,a1b2c8b ecome‘s
a b c and changing the suffixes of b and c,this again become
'
s2 1 87
aebscl. In like manner a
2b304d1has an opposite sign to a
lbzcacl“
being derived from it by a triple permutation, viz . through thesteps a
eblcsdn a
zbscgln
10. We are now in a position to replace our former definition
of a determinant by another,which we make the foundation of
the subsequent theory. In fact,since a determinant is only a
function of its constituents a, ,b, ,cl ,&c.
,and does not contain
the variables x, g, z , &c.
,it is obviously preferable to give a
definition which does not introduce any mention of equationsbetween these quantities x
, g, z .
*Let there be n.
2 quantities arrayed in a square of 72 columns
and n rows,then the sum with proper signs (as expla ined, Art. 9)
of all possible products of n constituents,one constituent being
taken from each horiz ontal and each vertical row is called the
determinant of these quantities and is said to be of the nmorder.
Constituents are said to be conjuga te to each other, when the
place which each occupies in the horiz ontal rows is the same as
that which the other occupies in the vertical rows. A deter
minant is said to be symmetr ica l when the conjugate constituents
are equal to each other ; for example,a,h, g
k,5, f
.97 j; c
sometimes given in the form that the sign of an element is when the totalnumber of displacements as compared with the order in the leading term is even,and vice versd.
We might have commenced with this defini tion of a determinant, the preceding
articles being unnecessary to the scientific development of the theory. We havethought, however, that the illustrations there given would make the general theorymore intelligible ; and also that the importance of the study of determinants wouldmore clearly appear, when it had been shown that every elimination of the variablesfrom a.system of equations of the first degree, and every solution of such a system,
gives rise to determinants, such systems of equations being of constant occurrence inevery department of pure and applied mathematics.
DETERMINANTS. 7
11. In these first lessons,as in the previous examples, we
write all the constituents in the same row with the same letter,
and those in'
the same column,
with the same suffix . The -i
most'
common notation however is to~
write the constituents of a determinant with a double suffix
,one suffix denoting the row
,and
the other thecolumn,to which the constituent .belongs. Thus
the determinant of thethird order would be written
7
or else 2
where in the sum the suffixes are interchanged in all possible
ways. The preceding notation is occasionally modified by the
omission of the letter a,and the determinant is written
(17 (17 (17 3)(2 ; (2 1 (2 , 3)
(37 (37 (353)Lastly, Mr. Sylvester has suggested What he calls an umbra l
notation. Consider,for example, the determinant
a a,ba
,ca
,cla
as, be, as, anan My or, 017
a d,b8
,08,038
the constituents of which are a a,ba
, &c., and where a,b,c,&c.
are not quantities,but as it were shadows of quantities ; that
is to say, they have no meaning separately, except in combi
nation with one of the other class of umbrae a, B, y, &c. Thus
,
for example, if a, B, r
‘
y, 8 represent the suffixes 1,2,3,4,the
constituents in the notation we have ourselves employed, are
all formed by combining one of the letters a,b,c,cl with one
of the figures 1,2,3,4. Now the determinant above written
is written by Mr. Sylvester more compactly
a,b,0,cl,
“2 77 87
which denotes the sum of all possible products of the form
obtained by giving the terms in.
the second line.
every possible permutation, and changing sign according to theordinary rule with every permutation.
LESSON I I.
REDUCTION AND CALCULATION OF DETERMINANTS.
12. WE have in the last Lesson given the rule for the forma
tion of determinants, and exemplified some of their properties in
particular cases. We shall in this Lesson prove these propertiesin general
,together with some others
,which are most frequently
used in the reduction and calculation of determinants.
The va lue of a determinant is not a ltered if the vertica l rows
be written horiz onta lly, and vice versa‘
(see Arts. 2,
This follows immediately from the law of formation (Art.which is perfectly symmetrical with respect to the columns and
rows. One of the principal advantages of the notation withdouble suffixes is that it , exhibits most distinctly the symmetrywhich exists between the horiz ontal and vertical lines.
13 . If any two rows (or two columns)be interchanged, the signof the determinant is a ltered.
For the effect of the change is evidently a single permutationof two of the letters (or of two of the suffixes), which by thelaw of formation causes a change of sign.
14. If two rows (or if two columns)be identica l, the determinant vanishes .
For these two rows being interchanged,we ought (Art. 13)
to have a change of sign : but the interchange of two identical
lines can produce no change in the value of the determinant.
Its value,then
,does not alter when its sign is changed ; that is
to say, it is 0.
This theorem also follows immediately from the definition of
a determinant, as the result of elimination between n linear
equations. For that elimination is performed by solving for thevariables from n 1 of the equations
,and substituting the values
so found in the nth
. But if this nt" equation be the same as one
of the others, it must vanish identically when these values are
substituted in it.
REDUCTION AND CALCULATION OF DETERMINANTS. 9
15 . If every consti tuent in any row (or in any column)bemultip lied by the same factor, then the determinant is multip lied
by tha tfactor.This follows at once from the fact that every term in the ex
pansion of the determinant contains as a factor,one
,and but one
,
constituent belonging to,the same row or to the same column.
Thus, for example , since every element of the determinant
anbi )
01
“27621
02
as)be?
03
contains either a, ,a, ,or a
n,the determinant can be written in the
form alA
1+ a
2A
2+ aaA s (where neither nor A8contains any
constituent from the a column); and if a, ,a2 ,asbe each multi
plied by the same factor'
h,the determinant will be multiplied
by that factor.
COR. If the constituents in one row or column differ onlyby a constant multiplier from those in another row or column,the determinant vanishes. Thus
as)
a” “a
h b, ,b, ,b3
0 (Art.
02)
027
Ca
16. If in any determinant we erase any number of rows
and the same number of columns,the determinant formed with
the remaining rows and columns is called a minor of the givendeterminant. The minors formed by erasing one row and one
column may be called first minors ; those formed by erasing tworows and two columns, second minors, and so on.
We have,in the last article
,observed that if the constituents
of one c olumn of a determinant be a, ,
a, ,
as,&c.
,the deter
minant may be written in the form
And it is ev ident that A ,is the minor obtained by erasing the
line and column which contain a, ,&c. For every element of
the determinant which contains a1can contain no other con
stituent from the column a or the line and almust be
multiplied by all possible combinations of products of n 1
constituents,taken one from each of the other rows and
columns. But the aggregate of these form the minor A,.
C
10 REDUCTION AND CALCULATION or DETERMINANTS.
Compare Art. 7. In like manner the determinant may be
written where B1is the minor formed
by erasing the row and column which contain b,.
If a ll the constituents but one vanish in any row or column
of a determinant of the nmorder
,i ts ca lcula tion is reduced to the
ca lcula tion of a determinant of the n order. For,ev idently,
if a, ,
as,&c. all vanish
,the determinant re
duces to the single term alA
l ; and AIis a determinant having
one row and one column less than the given determinant.
18. If every constituent in any row (or in any column)be re
solvable into the sum of two others,the determinant i s resolvable
into the sum of two others .
This follows from the principle used in Art. 16. Thus, if inthe Example there given, we write a
1a1for a
lb1 B, for b
cl y, for cl then the determinant becomes
(a l + al)A l
+ (61 +Bl)B 1 (01 71)O!
{a 1A 1blBl
cl01} {a lA l BIB ]
Thus we haveQt
anas?
as
a a27
as
an
G2 ,
as
l),+ 18 17 62 ? be b
l ,be?be 18 1 ? be) 63
01
02,
Ca
047
ca?
cs 71? 0
2)08
In like manner,if the terms in any one column were each
the sum of any number of others,the determinant could be
resolved into the same number of o thers.
19. If again,in the preceding, the terms in the second
column were also each the sum of others (if, for instance, wewere to write for a
, ,a2
012 ; for b, , b, 3 2 ; for c
2,c2
theneach of the determinants on the right-hand side of the last
equation could be resolved into the sum of others ; and we see,without difficulty, that
(“I al lb2+B27 68) (a l 2
03) (a lb208) (a lB208).
And if each of the constituents in .the first column could be re
solved into the sum of m others,and each of those of the second
into the sum of n others,then the determinant could be resolved
REDUCTION AND CALCULATION OF DETERMINANTS. 11
into the sum of mn others. For we should first,as in the last
Article; resolve the determinant into the sum of m others,by
taking, instead of the first column,each one of the m partial
columns ; and then, in like manner, resolve each of these into nothers, by dealing similarlywith the second column. And so
,in
general, if‘
each of the constituents of a determinant consist of
the sum of a number of terms,so that each of the columns can
be resolved into the sum of a number of partial columns (thefirst into m partial columns
,the second, into n
,the third into
p , then the determinant is equal to the sum of all the deter
minants which can be formed by taking, instead of each column,
one of its part ial columns ; and the number of such determinantswill be the product of the numbers m,
n, p , &c.
20. If the constituents of one row or column are resp ectivelyequa l to the sum of the corresp onding constituents of other rows
or columns,multip lied resp ectively by constant factors, the deter
minant vanishes. For in this case the determinant can be re
solved into the sum of others which separately vanish. Thuska
s
' l' zas?
as)
as
kan)
as?
as
law as)
as
kbslbs?be?be
kbz ?be?be
lbs,67be
[cos
02,
ca
7602 ,
02,
ca
1cs)
027
08
But the last two determinants vanish (Cor., Art.
21. A determinant is not a ltered if we add to ea ch constituent
of any row or column the corresp onding constituents of any of the
other rows or columns multip lied respectively by constantfactors.Thus
ai + kas+ lasl as) as an
as?
as
has-Flam a
)as
62,b,
51 ,62 ,b,
kb, + lbs, 6 , b,
61 + kce + lcs7 0
,ca
01 ,
cs?
03
kca+ lcsf 0
2,63
But the last determinant vanishes (Art. The following
examples will show how the principlesjust explained are applied
to simplify the calculation of determinants.
The beginner will be careful to observe that though the determinant is not
altered if we substi tute in the first row a l lea, 1c , for a” &c., yet if wemake the
same substitution in the second row for “a; &c.we multiply the determindnt by lo ;and if for a, , &c.wemultiply it by I.
12 REDUCTION AND CALCULATION OF DETERMINANTS.
Ex. 1. Let it be required to calculate the following determinant9, 1s, 17, 4 1, 1, 1, 4 1, 1, 1, 1
13, 2s, 33, s 2,4, 1, s 2, 4, 1, 1
30, 40, 34, 13 4, 1, 2, 13 4, 1, 2, 6
24, 37, 4s, 11 2, 4, 2, 11 2,4, 2, 3
The second determinant is deri ved from the first b y subtracting from the constituentsof the first, second, and third columns, twice, three times, and feurtinies, the corre
sponding constituents of the last column. The third determinant is deri ved fromthe second by subtracting the sum of the first three columns from the last. Whenever we have, as now, a determinant for which all the constituents of one row are
equal, we can get by subtraction one for which all the constituents but one of one
row vanish, and so reduce the cal culation t o that of a determinant of lower order
(Art. Thus'
subtracting the first column from each of those following, the determinant last written becomes
1, 0, o, 0
2, 2, - 1, 1
a - a - a 2
2, 2, 0, 1
The third of these follows from the one preceding by subtracting double the lastcolumn from the first, when we have a determinant of only the second order, whose
value is —8 - 7 z 15.
Ex. 2. The calculation of the following determinant is necessary (Solid Geometry,p. 175)5, 10, 11, 0 5, 10, 11, 0
10, 11, 12, 4,
32, 35, 34, 0
11, 12, 11, 2 11, 12, 11, 2
o, 4, 2, - e 1, a, 3, 0
a - a i a - a l
10 32, 7, 1 27, 9, 0
1, 1, s - 39, 17, 09“
The first transformation is made by subtracting double the third row fromthe second,and adding-( the sum of the second and third to the fourth. In the next step it willbe observed that since the sign of the term a lbzcfl ; is opposite to tha t ofwhen c, is the only constituent of the las t column which does not vanish, the determinant becomes c, In the next step, we add the second and third columns,we take out the factor 6 common to the second column, and the sign common tothe second row. We then subtrac t the first row from the second, and eight timesthe first row from the last, and the remainder is obvious.Ex . 3.
972 (Solid Geometry, p.Ex. 4. 25, 16, 23, 5
28, 19, 15, 9
5, 5 , 9, 5 i 194400 (Solid Geometry, p.
14 REDUCTION AND CALCULATION OF DETERMINANTS.
Ex. 8. (b c)2, a2,
a.2
622 (c a)21 62
cg, 02, (a b)2 2abc (a. b
Ex . 9. l , 1, 1
sin a, sinfi, 51117
00 8 0 . 00 86. 00 8 7 4 sia a 3)sinew 7)emi t! 7)
Ex. 10. coma coats—v), cesay — a)’
COSHa -t fl),mi ld -Hi), Shitty-t a) =2 sin1} (a
Ex. 11. 811!fi, sin 7
cosB, cosy
sin a cos a , sinfl cosfi, siny 005 7
2 sini (a 3)Gimme 7)sinfl a 7){sin (a 5) £4t 7) Isin('y + a)} o
Ex. 12. Many of these examples may be applied to the calculan'
on of areas of
tri angles, it being remembered that the double area of the triangle formed by threepoints is
y’
, y y
and by three lines ax+ by+ c, &c. is a , b, c f
a'
,b'
, c’
a"
,b"
, c" divided by (ab’ —cb
’
)(see Conic Sections, p. For example
, the area of the triangle formed by thecentres of curvature of three points on a parabola is (the co-ordinates of a centre of
4a“
curvature being 517 3x’
,
7 )
1, 1, 1
316
w, a" ’s
17, (3/ (11 —
y’
Liz’
s"
In like manner may be investigated the area of the triangle formed by three normals,or any other three lines connected with the curve.Ex. 13. 0
,c, b 0
, c, b, d
c, 0, a c , 0, a , e
b,a, 0 =2abc ; b
, a , fd, e, j; 0 =a2d2+ bzez + 07 2— 2abde— 2beef — 2cdcf .
Ex. 14. Prove 0, 1: 17 1 0
, x, Z
170; 2
2, x
702 z ! y
1, z2, 0, as? y, z, 0, a:
1; 3
21 0 ”
1 ‘0 ’ 0
— w)(2 + z —3l)(“PHI— z)
Ex. 15. a, A, X2 A, &c
A, b, X, A, &c.
h,A, c, 7k, &c.
A, A, A, d, &c.
&c.
1
REDUCTION AND CALCULATION OF DETERMINANTS. 15
where all the constituents are equal except those in the principal diagonal, is«p (A) 7Liii ; where ct (7k)is the continued product (a. A)(b A), &c.
Ex. 16. Let u be a homogeneous function of the with order in any number of
variables ; and let a” 142, u,”&c. denote its differential coeficients with regard tothe variables w” 35 2, ms, 810. and
, in like manner, let um, um, um denote the difierential coefficients of fax, 810. Then, by Euler’
s theorem of homogeneous functions ,we have
em mm, uzwz 14391, &c (n 1)u1 mlun maul, &c., 610.
We shall hereafter speak at length of the determinant (called the Hessian)formed with the second difierenti al coefficients, whose rows are u m um a”,
&c.
an , an , u m, &c.; &c. At present our object is to show how to reduce a class of
determinants of frequent occurrence, viz . those which are formed by bordering the
matrix of the Hessian, either with the first differential coefiid ents, or with other
“In “In, “13, ”I ; “I
“211 “222 "231 “27 “2
“311 “32 2 “337 “37 “a
v1. “2; ”a: 0. 0
a ” (12, a s, 0, 0
In this example we only take three variables, but the processes which we shall
employ are applicable to the case of any number of vari ables. In this example thedeterminant formed by the first three rows and columns is the Hessian which weshall call H. We shall denote the determinant written above by the abbreviation
(Z whilewe use (Z)(Z)to denote the determinants of four rows, formedby bordering the matrix of the Hessian wi th a single row and column, either both23
’s or both a
’s, or one u and the other a . We also write a las, dgz z ' i' use , a .
If now we mul tiply the first column of the above written determinant by 401, the
second by $ 2 , the third by $ 3, and subtract from n 1 times the fourth column,
the first three terms vanish, the fourth becomes —nu, and the fifth - a . Again,
multiply the fourth row by n 1, and sub tract in like manner the first, second, andthird rows multiplied by $ 1, 3 2.ma res pectively, and the first three terms vanish, thefourth remains unchanged, and the last becomes a . Thus then (n— 1)
° times thedeterminant originallywritten is proved to be equal to
0, a ]
0, “2
0, “a
n (n 1)u, a
But now since (Art. 15)a determinant which has only two terms d“ (15 of the fourthrow, which .do not vanish, is expressible in the form ds ; the above determinant may be resolved into the sum of two others, and wefind that the originally
In like manner it is proved that He . Or, again, i f there be four
16 MULTIPLICATION OF DETERMINANTS.
variables, and if the matrix of the Hessian be triply bordered, .we prove in the
same way thatu a n a 1
(a
)2
(z)(n ag)_n— l
u (ap) (n — l)
2 B“a
Whenu is of the second degree, it is to be noted that, in the case of three variables,
(2)expresses the condition that the line (1 should touch the conic n ; and (2p)[3
expresses the condition that the intersection of the lines -a , 13 should be on the conic.
In likemanner for four variables (2)(2fl)(2gg)respectively are the conditions
that a plane should touch, that the intersection of two planes should touch, and thatthe intersection of three planes should be on the quadric a . The equation then
CZ”)0 expremes that the polar plane of a point passes through one or other
of the two points where the line afimeets the quadric, But points having this
at the points where 443meets the quadric, and the transformation for gives theequation of the tangent cone where a meets the quadric.
LESSON III.
MULTIPLICATION orDETERMINANTS.
22. WE shall in this lesson show that the product of twodeterminants may be expressed as a. determinant.
Theproduct of two determinants is the determinant whose constituents are the sums of the products of the constituents in any row
qf one by the corresponding consti tuents in any row of the other.For example, the product of the determinants (0 164011)and
(093m)is“tarbIBl “l' 0171) a
ideblfie
01721 “
Ian
c{ Ya
anal628 1 0
271) atlas6232 0
2727 “ads6538 0
2711
aadr‘l' 655 1 + 0
3711 “tau
"i biz/82 “We “ear l
“ 6888+ 0
8 o
The proofs which we shall give for this particular case will
apply equally in general. Since the constituents of the deter
minant just written are each the sum of three terms, the determinant . can (by Art. 19)be resolved into the sum of the 27
determinants,obtained by taking any one partial column of the
MULTIPLICATION or DETERMINANTS. 17
first,second
,and third column. We need not write down the
whole 27 but give two or three specimen terms
a lal , a las, a las a lali 61627 0178 “1a17
01727 6189
“sax?
“20‘s:
dean 623 27 0273 d
ean 0
272? 6263 &c’
“ear?
“nae?
“sac
“Ban68327 Caryn “
Ban 0
3727 6868
Now it will be observed that in all these determinants everycolumn has a common factor
,which (Art. 15)may be taken out
as a multiplier of the entire determinant. The specimen termsalready given may therefore be written in the form
an an “
1an 617 cl a
n 01)bl
aiazas
as)
as)
“a
al 'Ba'ys a
s?627
ca
“17283 a
s,02,be
as)
as,
as
as?be?
on
as:
on,be
But the first of these determinants vanishes,since two columns
are the same ; the second is the determinant (a lbgcn); and thethird (Art. 13) is (a lbzos). In like manner
,every other
partial determinant will vanish which has two columns the
same ; and it will be found that every determinant which doesnot vanish will be (a lbzos), while the factors which multiply itwill be the elements of the determinant (a1It would have been equally possible to break up the deter
minant into a series of terms,every one of which would have
been the determinant (a l multiplied by one of the elements
of
23. On account of the importance of this theorem,we give
another proof, founded on our first definition of a determinant.
The determinant which we examined in the last Article is
the result of elimination between the equations
(a la1+ 6161+ 0171)w (d ial -i. 6132+ c
iv2)y (a la3+ 6163+ 6173)z 0
,
(a2a1+ 6261+ w (a2a2+ 6262+ 0272)y (a2a3+ 6268+ 0
278)z 0,
(d ear 635 1 0
371)w 6332+ G
aye)y (asas ' l' bsBs' l' 0378)z 0‘
Now ifwe write
Ii
a,sc+ a
gy use =X
,
fiiw+fifl +firz = Y,
w no as Z,
18 MULTIPLICATION or DETERMINANTS.
the three preceding equations may be written
alX -i b
1Y+ e
lz = O
,
a2X +n + c
22 = 0
,
a3X + bsY+ c
°Z = 0
,
from which eliminating X,Y,Z,we see at once that (a b c)1 2 8
must be a factor in the result. But also a system of values of
at, y, z can be found to satisfy the three given equations
,provided
that a system can be found to satisfy simultaneously the equations X O
,Y 0
,Z 0. Hence (0113 278) 0
,which is the
condition that the latter should be possible,is also a factor in the
result. And since we can see without difficulty that the degreeof the result in the coefficients is exactly the same as that of theproduct of these quantities
,the result is (a lbfica)(al 2W8).
It appears from the present Article that the theorem con
cerning the multiplication of determinants can be expressed inthe following form
,in which we shall frequently employ it
If a system of equations
alX + blY+c = O
,a2X + b2Y+ 02Z = O, a
aX+ b8Y+ O
BZ = O
be transformed by the substitutions
X = alm+ a
2y+ a z Y= fl1x +B2y+Bsz ,
then the determinant of the transformed system will be equa l to
(0 15203) the determinant of the origina l system,multip lied by
which we shall call the modulus of transforma tion.
24. The theorems of the last Articles may be extended as
follows : We might have two sets of constituents,the number
of rows being different from the number of columns ; forexample
“17bu0; Ba 71
as.)be?
02 1
32? 72
and from these we could form,in the same manner as
Articles,the determinant
a la l btfil
0171, a a 6
23 1 0271
alazb1'82
01727 “
ads6262 0
272 1
whose value we purpose to investigate.
MULTIPLICATION OF DETERMINANTS. 19
Now,first
,let the number of columns be greater than the
number of rows,as in the example just written ; so that each
constituent of the new determinant is the sum of a number of
terms greater than the number of rows : then proceeding as in
Art. 22,the value of the determinant is
ala
fl“sax
dial ,6261
a las? a sa z a las) bafiz
(65152) (a 0 (3 172)
That is to say, the new determinant is the sum of the pr
oducts ofeverypossible determinant which can be formed out of the one set
of constituents by the corresponding determinant formed out of the
other set of consti tuents.
25 . But in the second place, let the number of rows exceed
the number of columns. Thus, from the two sets of con
stituents,
an bl
be
as?be
let us form the determinant
a a 6161) “
sat62617 “
sax6361
a1a26132, a
2a26262 , b
882
a las 31“ads62637 “
ads63183
Then whenwe proceed to break this up into partial determinantsin the manner already explained
,it will be found impossible to
form any partial determinant which shall not have two columns
the same. The determinant therefore wi ll vanish identica lly. Or
this may be seen immediately by adding a column of cyphers toeach matrix and then multiplying ; whenwe get the determinantlast written as the product of two factors each equal cypher.
26. A useful particular case of Art. 22 is, that the square ofa determinant is a symmetrica l determinant (see Art. Thus
the square of (4 15203)is
“1
2 bl
201
2
,ala26162
0102,
a las 6163
clot!ala
fi6162
01027
“2
262
2 62
2
,“2a84g6263
0203
a la3+ b b + 0
108,
aeds+ b 6 + 0 0 a: b: c
s
”
20 MULTIPLICATION or DETERMINANTS.
Again,it appears by Art. 24 that the sum of the squares of the
determinants (al Y (blog)2
(clam)2 is the determinant
a,
”b,
’a la
2+ b b + 0
102
a a 6163+ 0
102,
a: b,.
Ex. 1. If a , , b, , c, ; a», 62 , c, be the direction-confines of two lines in space, and 8
their inclination to each other, cos 6 am, and the identity last provedgives sin? 9 (a tba)2 (b,cz)
2 (cla2)2'
Ex. 2. In the theory of equations it is important to express the product of thesquares of the difierenees of the roots ; now the product of the difierences of n
quantities has been expressed as a determinant (Ex. 5, p. and if we form the
square of this determinant we obtain
80, 3 1, 32 " J o-1
3 1, 82 , n o : 8”
82, 88, 84 n o t
Sal-1, SM P H SM
where s,, denotes the sum of thep“ powers of the quantities a , 3, &c.
Ex. 3. In likemanner it is proved by Art. 24 that the determinantso, 3 1 2
3” 32
2 (a 3)
3m 3 b ”a
81: sss 33 E (a B)2 (p (732; 38 ! 34
We thus form a series of determinants, the last of which is the product of thesquares of the differences of a
, B, &c. ; all similar determinants beyond this vanishing identically by Art . 25. This series of determinants is of great importance in thetheory of algebraic equations.Ex. 4. Let the origin be taken at the centre of the circle circumscribing a triangle,
whose radius is R ; and let M be the area of the triangle, thense'
, y’
,R
a"
, yl l
’ R
mm , 31m,R
5 l/ y
Mul tiply these determinants according to the rule,and the first term w
’2y” R’
vanishes ; the second 93's,”
y’
y'R2 a} { (m
‘x
”
)2 (y’
y"
)2} go
”,where
c is a side of the triangle. Hence then
4111 2122 i { azbzcfl
whence R 01—60as is well known.
4M
Ex. 5. The same process may be used to find an expression for the radius of thesphere circumscribing a tetrahedron. Starting with the expression for the volumeof the tetrahedron
22 MULTIPLICATION or DETERMINANTS.
Ex. 8. Tofind a relation connecting the mutual distances of three points on a line,four points on a plane
, or five points in space. We prefix a unit and cyphers to thetwo determinants which we multiplied in the last example, thus
1, 0, 0, 0 0, 0, 0, 1
z ”y”,
2y'
, 1 1, x’
, y'
, m'2
&c. &c.
We have now got five rows and only four columns, therefore the product formed as
inArt. 25, will vanish identically. But this is the determinanto, 1, 1
,I,
1
1, 0, (12V, (13V. (14V
1, (21V, 0, (23V, (24?1, (32V, 0, (34V1, (41V, (42V, (43V, 0 0,
which is the relation required. If we erase the outside row and columnwe have therelation connecting three points on a line ; and if we add another row 1, &c.,we get the relation connecting the mutual distanws of five points in space. Wemight proceed to calculate these determinants by subtracting the second column fromeach ot these succeeding, and then the first row fi-om those succeeding, whenwe get
2 (23V,
(23V, 2 (13V
(24V, (34V, 2 0.
Now the determinants might have been obtained directly in this reduced but un
symmetrical form by taking the origin at the point and farming, as in Art.25,with the constituents x’
y’
,m”y
"
, &c., the determinant which vanishes identically,$12
yrz
’ ml
x/ I
g/yu
, “I III
gl
yfll
zI
z /I+ y
l
yfl
’EII2 ++ y
l l z, w
l lzIII+ y
II
yHI
zldl ’
yyl l
, ml lzIII
yl
yfll
’ zl l 'z l
lz
which it will readily be seen is equivalent to that last written.
Ex. 9. To find the relation connecting the arcs which join four points on a sphere.Take the origin at the centre of the sphere , and form with the direction-cosines ofthe radii vectores to each point, cos a ’
, ecap‘,cos-y
'
; cos a”
, ao.a determinant which
1, cos ab, cosac, cosad
cosba , 1, ocebe, cosbd
cosca , 005 06, 1, cosca
cosda , cos cfi, cosdc, 1
Ifwe eubstitute for each cosine, cosab, 1
of the sphere to be infinite, we derive from the determinant of this article, that of
EL IO. It «13 00 h, 5 — 79 The deter
terminant is one of like form vfith A2 instead of A, the first line being A A”,H, G, cc., where
MINOR AND RECIPROCAL DETERMINANTS. 23
and the expanded determinant equated to cypher gives LN M1 2 N 0,where
L a2 + bz + 02 +
M (50 ~ f i)z(M (ab hz)2 2 (af 2 (by hf ? 2 (ch
and N is the square of the original determinant with A in it 0 ; L,M
, N are thenall essentially positive quantities. In like manner it (X)be formed similarly from
any symmetrical determinant, (7k)4) 7k)equated to nothing, gives an equation forA2, whose signs are alternately positive and negative, which therefore by Dee Cartes’
s
rule cannot have a negative root. The above constitutes Mr. Sylvester’s proof tha t
the roots of the equation (p (X) 0 are all real. It is evident, from what has beenjust said, that no root can be of the form BJ( and in order to see that no root
can be of the form a BJ( it is only necessary to write a a a'
, b a b'
,
c a z c' when the case is reduced to the preceding.
LESSON IV.
MINOR AND RECIPROCAL DETERMINANTS.
27. We have seen (Art. 16)that the minors of any deter
minant are connected with the corresponding constituents bythe relation
alA
lazA
2aaAs&c. A
,
and these minors are connected with the other constituents bythe identical relations
61A
,62A
,68AB&c. 0
,
0214
203As&c. 0
,&c.
For since the determinant is equal to and sinceAn A2 ,
&c.,do not contain a
l ,a”&c., therefore
is what the determinant would become if we were to make in ita = b
z ,&c.; but the determinant would then have two2
columns identical, and would therefore vanish (Art.
28. We can now briefly write the solution of a system of
equationsaim o
le &c. f,
azm b
2y 022: &c. 17,
asw b
ay csz &c. é
'
,&c.
,
multiply the first byA t,the second by A 2,
&c., and add,coefficients of y, z , &c.
,will vanish identically, While
24 MINOR AND RECIPROCAL DETERMINANTS.
coefficient of a: will be “114
1 + which is the determinant formed out of the coefficients on the left-hand side of
the equation,which we shall call A . Thus we get
Aw A } A27; A
8§+ &c.,
A31 1315+ B2
7; BBC+ &c.,
As 015 0
277 &c.
29. The recip roca l of a given determinant is the determinantwhose constituents are the minors corresponding to each con
stituent of the given one. Thus the reciprocal of (a lbgcs)is“A B
1701
A B2,02
A
where A1 ,B
1 ,&c.
,have the meaning already explained. If we
call this reciprocal A '
,and multiply it by the original deter
minant A ,by the rule of Art. 22 we get
alA
lblBl
cl01,
a2A
lbZBl
0201,
aBA
I6831
0301
alA
iZbl
'B2
01027
azA
a s zCa027
asA
absBa
cs02
al
'ASblBS
Cl03 ,
“2A36238
C2087
“BADbBB
flcs08
But (Art. 27)alA
1 + 613 1+ cl0,A,
all /1
2cl0,0,&c.
This determinant,therefore
,reduces to
A,o,0
0,A,0
0,0,A
Hence (a lbgcs)(4 3 203) therefore
And in general,A
'
A A”
; therefore A'
A“
.
30. If we take the second system of equations in Art. 28,
and solve these back again for E, a), &c., in terms of A40, A31, &c.,we get
A'
f alAac b
,Ay o
lA z &c.
,
where an b” 01are the minors of the reciprocal determinant.
But these values for E, n, C, &c. must be identical with the ex
pressions originally given ; hence remembering that A '
A“
,
we get, by, comparison of coefficients,
b,= A
H b clM
c &c.,
MINOR AND RECIPROCAL DETERMINANTS. 25
which express,in terms of the original coefficients
,the first
minors of the reciprocal determinant.
31. We have seen that,considering any one column a of a
determinant, every element contains as a factor a constituentfrom that column
,and therefore the determinant can be written
in the form EapAp. In like manner
,considering any two
columns a,b of the determinant
,it can be written in the form
2 where the sum 2 (apbq)is intended to express allpossible determinants which can be formed by taking two rowsof the given two columns.For every element of the determinant contains as factors
,a
constituent from the column a,and another from the column b ;
and any term aphqcrdn &c., must, by the rule of signs
,be aecom
panied by another,
&c. Hence we see that the formof the determinant is A
” ; and, by the same reasoning as
inArt. 16,we see that the multiplier A“ is the minor formed by
omitting the two rows and columns in which 6,occur.
In like manner,considering any 1) columns of the deter
minant,it can be expressed as the sum of all possible deter
minants that can be formed by taking anyp rows of the selectedcolumns
,and multiplying the minor formed with them,
by. the
comp lementa l minor ; that is to say, the minor formed by erasingthese rows and columns. For example
,
(a. (a.b4)(czdses) (Cabal
(01d46.) (a. (a (Odes)
(d ebs)(c d e (cabs)(c d e1 2 4 1 2 8
The sign of each term in the above 18 determined without difficulty by the rule of signs (Art.It is ev ident, as in Art. 27, that if we write in the above a
c for every I), the sum 2 (caches)must vanish identically,since it is what the determinant would become if the 6 Column
were equal to the 6 column.
32. The theorem of Art. 30 may be extended as follows
Any minor of the order p which can be formed out of the inverse
constituents A1,Bu&c.
,is equa l to the comp lementary of the cor
26 MINOR AND RECIPROCAL DETERMINANTS.
responding minor of the origina l determinant, multiplied by the
(p 1)th
p ower of tha t determinant.
For example,in the case where the original determinant 18
of the fifth order,
it4il A (c d e la (A nBeas) A (d4e6)’ 853 °
The method inwhich this is proved in general will be sufficientlyunderstood from the proof of the first Example. We have
A3: A15+ A 2
7; + A 3§+ Ap A v
Ag B27) B
SC+ B40) s .
Therefore
AB2x AA
23/ (AR){5 (ABE2) (A4
132)a) (A v.
But we can get another expression for x in terms of the samefive quantities
, y, E,L’
,w,v. For
,consider the original equations,
1; asw h
sy caz d
sw e
sa,
w ape 6g c4z d
4w e
4u,
v abs: 6
531 csz d
sw c
an,
and eliminate z,w,a,when we get
(a lcsd4e5)a: (blc3d4e6)y (c d e l’g‘
(c d e (c d e a)3 4 5 4 5 1 6 1 8
and since (a ,csd4e5)is by definition = B , ,comparing these equa
tions with those got already, we find A (csd4e5), &c.
Q .E.D.
Ex. 1. If a determinant vanish, its minors A 1, A, , &c. are respectively proportionalto B " 8 2 , &c. For we have just proved that 11l A zB l A0, where 0 is the
second minor obtained by suppressing the first two rows and columns . If thenA =0, we have A l : : B 1 s , &c.
Ex. 2. A particular example of the above,which is of frequent occurrence, is
obtained by applying these principles to the determinant considered, Ex.16, p. 15 . We
2
thus find,using the notation of that example (
a
)(g)(
a
) A (G 3)(see SolidGeometry, p. a 6 fl a {3
*LE SSON V.
SYMMETRICAL AND SK EW SYMMETRICAL DETERMINANTS.
33 . IN this lesson it is convenient to employ the doublesuffix notation
,and to write the constituents a“, “m &c. ; and
we therefore begin by expressing in this notation some of the
results already obtained. We denote the constituents of the
reciprocal determinant. by an ,
an ,&c.
,where
,if a
"be any con
stituent of the original,a
”is the minor obtained by erasing the
row and column which conta in that constituent. The equationsof Art. 27 may then be written
a a + a a + a a71 72 V2 VS 73
amour.2 amen.3 &c. 0,
s m that is to say, the sum
of the products away, (where we give every value to s from
1 to n)is 0,when r and r
'
are different,and A when r = r
'
.
Since any constituent a”enters into the determinant only in
the first degree,it is obv ious that the factor ca
n ,which multiplies
it,is the differential coefficient of the determinant taken with
respect to a" ; similarly, that the second minor (Art. 3 1)which
multiplies the product of two constituents am ,
a”is the second
differential coefficient of the determinant taken with respect tothese two constituents, &c.If any of the constituents be functions of any variable x
,
the entire differential of the determinant, with regard to thatdada
11 12vari able
,1s ev idently a
dc doc
or more briefly A,2 a a
Ex. If a l , &c. denote the first differentials of u,v,&c. with respect to a: 5 a2,
the second differentials, &c., provea,v w a , v w
“2: ”2; we “3 ;”Us; w3
The differential is the sum of the nine products of the minor obtained by suppressingeach term into the differential of that term,
vi z .
(“nut “120 1 c rawl) “22772 0 2370 2) (“anus “32173 a sawa)~
But the first three terms denote the result of changing in the gi ven determinant thePortions marked thus may be omitted on a first readi ng.
28 SYMMETRICAL AND SK EW SYMMETRICAL DETERMINANTS.
first row into a“ 10 1, and therefore vanish ; the second three terms vanish as
denoting the result"
of changing the second row into uz , oz , 102 ; and there only
remain the last three terms which denote the result of changing the last row into
as, ms. The same proof evidently appl ies to the similar determinant of the
n‘“
order formed with n functions.
24. The determinant is sa id to be symmetrical (Art. 10)when every two conjugate constituents are equal (a , ,
In
this case it is to be observed that the corresponding minors willalso be equal (a for it easily appears that the determinant got by suppressing the r
mrow and the 3
“ column,differs
only by an interchange of rows for columns,from that -got by
suppressing the s"1
row and the rtltl column. It appears from
the last article that if any constituent a"were given as any
function of its conjugate a the differential coefficient of thed
determmant,Wi th regard to a
” ,would be a
n + a"
a"
In theda
present case then where an
a" ,
a" ,the differential cooth
cient of the determinant,with regard to i s 2a
". The diffe
rential coefficient, however, with respect to one of the terms inthe leading diagonal a
" , remains as before a" ,since such a term
has no conjugate distinct from itself.
35 . If,as before
,a
”denote tlie
'
first minor of any determinant answering to any constituent a
" ,and if B”, denote the
first minor of the determinant an answering to any constituenta ,“ which will, of course
,be a second minor of the original
determinant,then this last may be written
A a a z ira rkaifia i
where we are to give i every value except r,and 71: every value
except s . For any element of the determinant which does notconta in the constituent a
"must contain some other constituent
from the rmrow and some other from the column ; that is to
say, must contain a product such as arka‘, where i and k are
two numbers different from r and s respectively. But as we
have already seen the aggregate of all the terms which multiplya is a
n ; and the coefficient of (by Art. 3 1)differs onlyin sign from that of a
ha“; that is to say, differs only in sign
from the coefficient of a”, in a Therefore is the value
of the coefficient in question.
30 SYMMETRICAL AND s w SYMMETRICAL DETERMINANTS.
and if a “, &c. be negative the determinant is a positive square.What has been just proved may be stated a little differently.We may consider the bordered determinant as the original determinant ; of which that obta ined by suppressing the row and
column containing 7» is a first minor,and a
s,obtained by sup
pressing the next outside row and column is a second minor.And what we have proved with respect to any symmetricaldeterminant wanting the last term a
n" ,is that if the first minor
obtained by erasing the outside row and column vanish, thenthe determinant itself and the second minor, similarly obtained
,
must have opposite signs. And this will be equally true if am,does not vanish. For in the expansion of the determinant, amis multiplied by the first minor
,which vanishes by hypothesis,
and therefore the presence or absence of am, does not affect thetruth of the result.
37. A shew symmetric determinant is one in which everyterm is equal to its conjugate with its sign changed. The
terms in the leading diagonal,being each its own conjugate
,
must in this case vanish ; otherwise each could not be equal toitself with sign changed.A skew symmetrica l determinant of odd degree vanishes. For
if we multiply each row by 1 ; in other words,if we change
the sign of every term,it is easy to see that we get the
same result as if we were to read the columns of the originaldeterminant as rows and vice versa
‘. Thus then
,a skew
symmetrical determinant is not altered when multiplied byand therefore when n is odd
,such a determinant must
vanish.It is easy to see that the minor or
"differs by the sign of
every term from the minor a" ,
and therefore a"
Hence a"
ah , when n is odd
,and is equal with contrary
sign when n is even. a"is itself a skew symmetric deter
minant and therefore vanishes when the original determinantis of even degree.The differential coefficient of the determinant
,with regard
to any constituent being is a, ,
When
therefore n is even it i s 2a"and when n is odd it vanishes.
SYMMETRICAL AND s w SYMMETRICAL DETERMINANTS. 31
38. Every shew symmetrica l determinant of even degree is
a perfect square.We have seen (Art. 35)that any determinant is
am an/n B
reamarn rs)
and in the present case am,
vanishes,as does also a
n”,which is
a skew symmetric of odd degree ; On this account thereforewe have , as in Art. 36
, and therefore exactly as
in that article,the determinant is shown to be
{a W3 W3 22) dam/(3 38) &C-l
z
The determinant is therefore a perfect square if B” , 22are
perfect squares,but these are skew symmetries of the order
n — 2. Hence the theorem of this article is true for determinants of order n if true for those of order n 2
,and so on.
But it is ev idently true for a determinant of the second order
0’ which is = a Hence it is generally true.
_ a12’
0
39. We have seen that the square root of the determinantcontains one term where BW LM contains noterms with either of the suffixes n _ 1 or n. But taking any
two of the rema ining suffixes,such as n — 3
,n —2
,we see that
conta ins a term where ‘Yn-sm-scon
tains no term with any of the four suffixes of which account hasalready been taken. Proceeding in this manner we see thatthe square root will be the sum of a number of terms suchas each of which is the product of %n con
stituents,and in which no suffix is repeated twice. The form
however obta ined in the last article am WK?does not show what sign is to be affixed to each term. Thusif the method of the last article be applied to the skew sym
metric of the fourth order,its square root appears to be
away ; awafi i “
14923 5 but it has not been shown which signswe
are to choose. This however will appear from the followingconsiderations : If in the given determinant we interchangeany two suffixes since this amounts to a transposition of
the first and second row,and also of the first and second column,
the determinant is not altered. Its square root then must be
32 SYMMETRICAL AND s w SYMMETRICAL DETERMINANTS.
a function,such that if we interchange any two suffixes it will
remain unaltered,or at most change sign. But that it will
change sign is evident on considering any term awaw &c.
which,if we interchange the suffixes 1 and 2 becomes amaw
&c. ; that is to say, changes sign, since a am. It followsthen
,in the particular example just considered, that the signs
of the terms are ama34 a lga24
aua” ; for if we give the second
term a positive sign,the interchange of 2 and 3
,which alters
the sign of the last term would leave the first two unchanged.And generally the rule is
,that the square root is the sum of
all possible terms derived from by interchange of
the suffixes 2,
where,as in determinants, we change sign
with every permutation.
40. We can reduce to the calculation of skew symmetricdeterminants
,the calculation of what Mr. Cayley calls a skew
determinant,vi z . where
,though the conjugate terms are equal
with opposite signs,a“ a“, yet the leading terms a“, a”, &c.
do not vanish. We shall suppose,for simplicity, that these
leading terms all have a common value x. We prefix the
following lemma : If in any determinant we denote by D,the
result of making all the leading terms = O,by D, what the
minor corresponding to a n becomes when the leading terms areall made = 0
,by Du what the second minor corresponding to
becomes when the leading terms vanish,&c. ; then the
given determinant, expanded as far as the leading terms areconcerned
,is
A D Z ai iD, Sufi
akkDi,‘ a
where in the first sum i is given every value from 1 to n,where
in the second sum i,k are any binary combinations of these
numbers,&c.
For the part of the determinant which contains no leadingterm is ev idently D. Since the terms which contain a are
where A IS the corre sponding minor,the terms which
contain a“ and no other leading term are got by making theleading terms 0 in A &c.
11a22
l Q 0
u ,
41. If this lemma be applied to the case of the skew determinant defined in the last article
,all the terms D“, &c.
SYMMETRICAL AND SK EW SYMMETRICAL DETERMINANTS. 33
are skew symmetric determinants ; of which, those of odd order
vanish , while those of even order are perfect squares. The
term is N",and the determinant is
A A”
AMZ D
,AMED
4&c.
,
where D2,D, ,&c. denote skew symmetrical determinants of the
second,fourth
,&c. orders formed from the original in the
manner explained in the last art icle.Ex. 1. h , an, a“,
“217 A» “as
an , an, X where a21 am 610. M h (an? an,2
Ex. 2. The similar skew determinant of the fourth order expanded isM M “18
2 ‘l' “142 “l‘ “23
2 “242
0 342)4” (amass “tease “l" “140 29
2
42. Mr. Cayley (see Crelle, Vol. xxx11., p . 119)has appliedthe theory of skew determinants to that of orthogonal substitutions
,of which we shall here give some account. It is known
(see Solid Geometry, p. 10)that when we transform from one
set of three rectangular axes to another,if a
,b,c,&c. be the
direction-cosines of the new axes, we have
Z = a"
ac+ b y + c a ;
that we have X’+ Y
2.+
whence a”
a"
a’”
1,&c.
,ab a
’
h’
a"
b"
0,&c.;
that also we have
y= bX + b'
Y+ b"
Z , c"
Z,
and that we have the determinant formed bya,b,c ; a
'
,b'
,c'
;
It is also useful (in studying the theory of rotation for example)instead of using nine quantities a , b, c, &c. connected by six
relations, to express all in terms of three independent variables.Now all this may be generaliz ed as follows : If we have a
function of any number of variables,it can be transformed by
a linear substitution by writing
a: auX+ a
l zY+ a
,,Z + &c., y a
ux -Fa”
Y+ a, ,Z + &c.
,&c.
,
and the substitution is called orthogonal if we have
which implies the equationsa + a a a + 01 0. &c
11 12‘21 as
34 SYMMETRICAL AND SK EW SYMMETRICAL DETERMINANTS.
Thus the n2 quantities “m &c. are connected by §n (n+ 1)relations and there are only Qn (n 1)of them independent.We have then conversely
Y= a az + a y+ a &c.,
equations which are immediately verified by substituting on the
right-hand side of the equations for ac, y, z , &c. their values.
And hence,the equation X 2 Y2
&c. x”
y" &c. gives us
the new system of relations
a + a a a + a al l 21 12 22
Lastly, forming by the ordinary rule for multiplication of
determinants,the square of the determinant formed with the
n" quantities “m &c.
,every term of the square vanishes except
the leading terms which are all 1. The value of the squareis therefore = 1. Thus the theorems which we know to be
true in the case of determinants of the third order are generally true
,and it only remains to show how to express the n2
quantities in terms of&n (n 1)independent quantities.
43 . Let us suppose that we have a skew determinant of the
(n— l) order,6 b &c. where by, b“, and b“ 1 ;111 127
and let us suppose that we form with these constituents twodifferent sets of linear substitutions
,v iz .
m= b gu m+ 5 r+ &c.,X= b g+ 6 5+ &c.,
y= b § + bz zn+ b Y= b
e = h E+ bszn+ b Z = b f +~ bzsn+ b
from adding which equations we have , in v irtue of the givenrelations between b
u ,bu ,&c.
,
33+ X : 25, 3, Y= &c.
If new the first set of equations be solved for E, n, &c. in termsof x
, y, &c.,we find
,by Art. 28
,
AE= 3 T +Bmy+ 6
(where B“, 13m &c. are minors of the determinant in questionand putting for 25, x + X ,
&c.,these equations give
AX = (w A)w fees use &c.,
A Y: 23 12-77 A)y 2B82z &c.
,&c.
,
SYMMETRICAL AND SK EW SYMMETRICAL DETERMINANTS. 35
which express X,Y,&c. in terms .of w
, y, &c. But if we had
solved from the second set of equations 5, n, &c. in terms of
X and Y,we should have got
A§=BHX +B 622Y+ B Z + &c.
whence,as before
,
Aw= (2,3 A)X + Y+ 2B
Ay 2,8 2,X + (2B A)Y+ 2,8 23Z + &c.
Thus,then
,if we write
— A — A 2BA A
we have x, y, &c. connected with X,
Y,&c. by the relations
w auX + a Y+ &c., y a
21X + a
22Y+ &c., &c.
,
X and amy &c.
,Y= amcc a
ny &c.,&c.
We have then x, y, &c.
,X,Y,&c. connected by an orthogonal
substitution,for if we substitute in the value of cc
,the values
of X,Y,&c. given by the second set of equations
,in order that
our results may be consistent, we must have
a a a + a a + a a &e11 21 12 22 18 23
Thus then we have seen that taking arbitrarily the %n (n 1)quantities
,613,&c.
,we are able toexpress in terms of these
the coefficients of a general orthogonal transformation of the
nth
order.
Ex. 1. To form an orthogonal trans formation of the second order. Wri te
A z
I1’
A, 1 1 V,
then [in fizz 1, em h , fin A, and our transformation is(1 + X
2)m — A2)X + 2AY, (1 + 7t
2)y Y,
(1 A2)X : (1— A2)a: my, (1 2M: (1 y.
Ex. 2 . To form an orthogonal transformation of the third order. Write
A 11 V; l“
v,I,
X
F"“ A, 1
the constituents of the reciprocal system are
u + lku, 1 + u2,
X-l-pw
fl + 7\u, l + u2
‘
36 SYMMETRICAL DETERMINANTS.
consequently the coefficients of the orthogonal substitution hence derived are1 +A2 —p
2 — v", “Kr —
fl),
2 (7\n— u), — X2 — v2,
sa ws). 2 (; w H us—Ah a”.
where each term is to be divided by 1 h? a?
*LESSON VI.
SYMMETRICAL DETERMINANTS.
44. IF we add the quantity A. to each of the leading termsof a symmetrical determinant, and equate the result to 0, wehave an equation of considerable import ance in analysis.
‘
l’ We
have already given one proof (Mr. Sylvester’s)that the roots
of this equation are all real (Ex . 10,p. and we purpose in
this Lesson to give another proof byM.Borchardt (seeLiouville,Vol. XII.
,p. 50)chiefly because the principles involved in this
proof are worth knowing for their own sakes. First,however,
we may remark that a simple proof may be obtained by theapplication of a principle proved in Art. 36. Take the de
terminantan
A,
are,
“i s,
&c’
a”
A,
am, &c.
at“, a”,
a“ 7L, &c.
&c.
and form from it a minor, as in Art. 36, by erasing theline and column : form from this again another minor
The geometric meaning of these coefficients may be stated as follows : Write
h z a tmgrfl, p =b tan§9, v =c tan } 0, then the new axes may be derived from the
old by rotating the system through an angle 0 round an axis whose direction-cosine;are a , b, c. The theory of orthogonal substitutions was first investigated by Euler,(Nov. Comm.Petrop ., Vol. xv., p. 75 , and Vol. xx ., p. 217)who gave formula for thetransformation as far as the fourth order. The quantities A, n, v, in the case of the
third order, were introduced by Rodrigues, Liouvi lle, Vol. v.,p. 405. The general
theory, explained above, connecting linear transformations with skew determinantswas given by Cayley, Crelle, Vol. xxxn , p. 119.1' It occurs in the determination of the secular inequalities of the planets (seeLaplace, Mecanique Celeste, Part 1 Book Art.
38 SYMMETRICAL DETERMINANTS.
only by positive square multipliers from the following. The
first two (namely, the function itself, and its derived) are of
course, (x — B)(tn &c., S he - B)(ac &c.; and
the remaining ones are
2 (at—Bf lm—al)(90 -5 2 (OP -
v)"
07
where we take the product of any k factors of the given equation
,and multiplying by the product of the squares of the diffe
renees of all the roots not contained in these factors, form the
corresponding symmetric function. We commence by provingthis theorem.
*
46. In the first place, let U be the function, V its first
derived,R
2,&c. the series of Sturm’s rema inders : then it is
easy to see that any one of them can be expressed in the form
A V B U. For,from the fundamental equa tions
U: Q1V’ Ry)
V: Qs Ra)Ra QaRe
B4?&c'
)
we have
R. e.V U.
B.= o.R. V= (s o. 1)V
RF (4230. 1)Rg Q3V= (Q1 Q2 Q3 _ Q, Q.)V 1)U:
and so on. We have then in general‘l‘R, = A V— EU
,where
,
since all the Q’
s are of the first degree in x,it is easy to see that
A is of the degree k— l,and B of the degree k
— 2,while B,
is of the degree n k.
I suppose that Mr. Sylvester must have originally divined the form of thesefunctions from the characteristic property of Sturm’
s functions, viz . that if theequation has two equal roots (1 3, every one of them must become divisible byz a . Consequently, if we express any one of these functions as the sum of a
number of products (or a)(a: B), &c., every product which does not include
either z — a or z — fimust be divisible by («r and it is evident in this waythat the theorem ought to be true. The method of verification here employed doesnot difl er essentially from Sturm’
s proof, I/iotwi lle, Vol.VII., p. 356.1' The theory of continued fractions which we are virtually applying here shows
that if we have R} , AEV BkU; Rm, 2 Ak flv Bkfl U, then AkBkfl AIM-13 k is
constant and 1. In fact, since Qn RH , we haveAb fl M k A le—I; Bt u (2k B ic-1,
whence AbBb fl AmBk Z Ag. ,Bg AkB IH ,
and by taking the values in the first two equations above, namely, where k 2,
and k 3, we see that the constant value 1.
SYMMETRICAL DETERMINANTS. 39
But‘
now this property would suffice to determine R2,R
8,&c.
directly. Thus,if in the equation 113
2= Q1V U we assume
Ql aw b,where a.and b are unknown constants, the condition
that the coefficients of the two highest powers of x on the righthand side of the equation must vanish (since R2
is only of the
degree 72 — 2)is sufficient to determine a and 6. And so in
general,if in the function A V— BU we write for A the most
general function of the k— lSt
degree containing 70 constants,
and for B the most general function of the k degree,con
taining 70 — 1 constants,we appear to have in all 2lo 1 constants
at our disposal,and have in reality one less
,since one of the
coefficients may by div ision be made We have then justconstants enough to be able to make the first 270 — 2 terms of
the equation v anish,or to reduce it from the degree n+ k — 2
to the degree n k. The problem,then
,to form a function of
the degree n 7c,and expressible in the form A V— B U
,where
A and B are of the degrees k 1,7c — 2
,is perfectly definite,
and admits but of one solution. If,then
,we have ascerta ined
that any function R,is expressible in the form A V B U
,where
A and B are of the right degree,we can infer that B
,must be
identical with the corresponding Sturm ’s remainder,or at least
only differ from it bya constant multiplier. It is in this waythat we shall identify with Sturm
’s remainders the expressionsin terms of the roots
,Art. 45 .
47. Let us now,to fix the ideas
,take any one of these
functions, suppose2 (a — B)
2
(B (ar- d l (w 3)(56 &0
and we shall prove that it is of the form A V— BU,where in
the example chosen A is to be of the second degree,and B of
the first in m. Now we can immediately see what we are to
assume for the form of A,by making w a on both sides of the
equation. The right—hand side of the equation will then become A (a (a r
y)(a 8)(ar— s), &c. since U vanishes ; and
the left-hand side will become
2 (u a)(cc- 8)(oz — e), &c.
Just as the six constants in the most general equation of a conic are only
equival ent to five independent constants, and only enable us to make the curve
satisfy five conditions.
40 SYMMETRICAL DETERMINANTS.
It follows,then
,that the supposition as a must reduce A to the
form 2 (B— vy)
”
(a B)(a — ry), and it is at once suggested thatwe ought to take for A the symmetric function
2 (B — B)(ac—v)
And in like manner, in the general case, we are to take for Athe symmetric function of the product of k 1 factors of theoriginal equationmultiplied by the product of the squares of thedifferences of all the roots which enter into these factors. It
will not be necessary to our purpose actually to determine thecoefficients in B , which we shall therefore write down in itsmost general form. Let us then write down
— 'Y)2
(v- a)i
(m — a)(96 — 3)
x E (x (av—q), &c. + (am+ b)(x — a)(oz — B), &c.,
which we are to prove is an identical equation. Now,since an
equation of the pmdegree can only have p roots
,if such an
equation is satisfied by more than p values of ac,it must be an
identical equation, or one in which the coefficients of the severalpowers of w separately vanish. But the equation we havewritten down is satisfied for each of the n values m=a
,&c.
,
no matter what the values of a and 6mayb e. And if we sub
stitute any other two values of x,then
,by solving for a and b
from the equations so obtained,we can determine a and b so
that the equation may be satisfied for these two values. It is,
therefore,satisfied for n 2 values of as
,and since it is only an
equation of the n 1“degree, it must be an identical equation.
And the corresponding equation in general,which is of the
n k 1 degree,is satisfied immediately for any of the n values
as : a, &c. ; while B being of the k 1 degree we can determinethe k constants which occur in its general expression
,so that
the equation may be satisfied for k other values ; the equationis,therefore
,an identical equation.
48. We have new proved that the functions written inArt. 45 being of the form A V BU are either identical withSturm’s rema inders, or only differ from them by constant factors,It remains to find out the value of these factors
,which is an
essential matter, since it is on the signs of the functions that
SYMMETRICAL DETERMINANTS. 41
everything turns. Calling Sturm’s remainders,
as before,
R2 ,R8,&c.
,let Mr. Sylvester
’s forms (Art. 45)be T2, T8 , &c.,
then we have proved tha t the latter are of the form T2= 7t
2R2 ,
&c.,and we want to determine a
s,&c. We can
at once determine by comparing the coefficients of the
highest powers of a: on both sides of the identity T,=A 2V— B
2U;
for as”does not occur in T
a,while in V the coefficient of w"" is n
,
and the coefficient of a: is also n in A2 ,which E (a: a) hence
Ez= u
z. But the equation T
2= A
2V B
2U must be identical
with the equation R2= Q, V U multiplied by we have
,
therefore,
To determine in general 7th it is to be observed that sinceany equation T
k= A
kV E
kU is 7x, times the corresponding
equation for and since in the latter case it was proved
(note, p. 38)that A ,B,+1
1,the corresponding quan
t ity for Tk, T,“ must Now from the equations
Tk= A
lcV' T
lc+1= A
we have
A TtA4Tk-n (Ae /r-H
AmBk)U: MM“U'
Now,comparing the coefficients of the highest powers of a: on
both sides of the equation, and observing that the highest power .
does not occur in Ak fl ,we have the product of the leading
coefficients of A and T, MN,+1. But if we write
2 (a By=I92a 2 (a V)(13 WY=ps1 &c.,
we have,on inspection of the values in Arts. 45
,47
,the
leading coefficient in Te=p2, in 72
—4
103, &c., and in A 2= n
,in
As=19m in A ,
=p e, &c. Hence
t+1
2 2 2 1792
"3732
p , Pa =7t87» 4, 194 whence NF ?X4
102 ,&c.
2
The important matter then is, that these coefficients are all
positive squares,and
,therefore
,as in using Sturm’s theorem
we are only concerned with the signs of the functions, we mayomit them altogether.
49. When we want to know the total number of imaginaryroots of an equation
,it i s well known that . we are only con
G
42 SYMMETRICAL DETERMINANTS.
cerned with the coefficients of the highest powers of a: in
Sturm’s functions,there being as many pa irs of imaginary roots
as there are variations in the signs of these leading terms.
And since the signs of the leading terms of T2 ,T8 ,&c. are the
same as those of &c.,it follows that an equation has as
many pa irs of imaginary roots as there are variations in the
series of signs of 1 , n, 2.(a 2 (a B)”
(B —y)
”
(7 &c.
This theorem may be stated in a different form by means of
Ex . 3,p. 20
,and we learn that an equation has as many pairs
of imaginary roots as there are variations in the signs of 1, so,and the series of determinants
30)
st ,
3
311se?
38
837
3a? 4
834 i91
387
the last in the series being the discriminant ; and the conditionthat the roots of an equation should be all real is simply thatevery one of these determinants should be positive.
50. We return now,from this digression on Sturm’s theo
rem,to M. Borchardt
’s proof
,of which we commenced to give
an account,Art. 45 ; and it is ev ident that in order to apply
the test just obtained,to prove the reality of the roots of the
equation got by expanding the determinant of Art. 44,it will
be first necessary to form the sums of the powers of the roots ofthat equation. For the sake of brev ity, we confine our proofto the determinant of the third order
,it being understood that
precisely the same process applies in general ; and,for con
venience we change the sign of 7» which will not affect thequestion as to the rea li ty of its values. Then it appears immediately, on expanding the determinant
,that 3
.“11 + a
22+ a
88,
since the determinant is of the form (a l l a22
a”) &c.
And in the general case .s1is equal to the sum of the leading
terms. We can calculate 32as follows : The determinant. may
be supposed to have been derived by eliminating x, y, z be
tween the equations
hx= aux + amy
' l' a 13z 7 Ry a2 1x+aazy+ a zaz 7 7tz =a
81w+“
3231+assz '
SYMMETRICAL DETERMINANTS. 43
Multiply all these equations by a , and substitute on the righthand side for Ma
,Ry, kz their-values, when we get
hyw=(a
9+a
9+a
2
)m+Ka
y a28a18)w+
xaz =(a31a l [+a 82a 19+a ssa la)m+ a
98a28)y+
from which eliminating w, y, z , we have a determinant of form
exactly similar to that which we are discussing,and which
may be written6 7x
”
,b
6217
baa
baa
bu ,
I)
Then,of course
,in like manner
,
82bl !
622
638
a2
“22
2a83
22a
12
22a
The same process applies in general and enables us from s,to
compute 8,+1
. Thus suppose we have got the system of equations
Nam=duw+dmy+dxsz $ N
a
y=d
2 1x+d223/ J
azz)Nuz =d
aiw+d
ssy+dssz ’
from which we could deduce,as above
,3,d
”dz,d
” ; then
multiplying both sides by 7t , and substituting for 7m, &c. theirvalues
,we get
xfi lw (d! la l l El
i sa
”!dJSaIB)x (d11a 2l d
lfiaafi
di sa
flb)y
(du d s! dudes
di sarm)z ,
(dz la‘l ld
flBGJQ
dfial s)x (dil lafil
d22a92
dflaa
fifl)y
(dfiaal
dasass
dudes)z ’
Mme (J a + cl a + d33am)w+ (d a + d a + d88a”)ya: i t as 12 a: 21 82 22
(deta in daadaa
dudes)2 )
Whence 894-1 Cluau
dama ge d
esdes
24126112
2dasasa
2d81aal '
51. We shall now show, by the help of these values fors” &c. and of the principle established Art. 24
,that every
one of the determinants at the end of Art. 49 can be expressedas the sum of a number of squares
,and is therefore essentially
positivefi“ Thus write down the set of constituents
1?1117
0?0,0,
0)
an ,
“sci “as, “
291asx)
aw.
M. K ummer first found out by actual trial that the discriminant of the cubicwhich determines the axes of a surface of the second degree is resolvable into a sum
44 SYMMETRIC FUNCTIONS.
then it is easy to see that81 is the determinant formed
1, a
from this by the method of Art. 24,and which expresses
the sum of all possible squares of determinants which can be
formed by taking any two of the nine columns written above.
The dete81 is thus seen to be resolvable into the
sum of the squares
(a — a + (a 88 — a, 6 (a
and is therefore essentially positive. Aga in,if we write down
“11?
“227
ass)
“as?
ast)
an)
“as?
“an
“12
bu )baa)baa?
baa?banbi s?best591)bi s 7
where b &c. have the meaning already explained,it will be
307
Sn 32
easily seen from the values we have found that 317
32,
38is the
827
88 ,
S4
determinant which,in like manner
,is equal to the sum of the
squares of all possible determinants which can be formed out of
the above matrix. And so in like manner in general.
LE S SON VII.
SYMMETRIC FUNCTIONS.
52. WE assume the reader to be acqua inted with the theoryof the symmetric functions of roots of equations as usually givenin works on the Theory of Equations. Thus we suppose himto be acqua inted with Newton’
s formula for calculating the sumsof the powers of the roots of the equation
x”
p lw“+p a
acH
&c. O,
Viz ’ 81_P1
07
32
—p‘8 1 2172 0
)33
“
P182
3p, O)&c‘
)
of squares. (Crelle, Vol. xx v 1., p. The general theory gi ven here is due, aswe have said, to M. Borchardt.
46 SYMMETRIC FUNCTIONS.
of a beyond the first,it is plain that the order of any symmetric
function 2 0323 979
(where m is supposed to be greater than por 9)must be at least m. For of course, unless there are at
least m factors,each containing a
,a
’” cannot appear in the product. But
,conversely, any symmetric function, whose order is
m,will contain some terms involving For if 91, g, , 931 &c.
be the sum,sum of products in pairs, in threes
,&c. of B, 7, 8
&c we have 10. at + a ,saws a &C and the
coefficient of the highest power of a in such a term as
will be and,conversely, the multiplier gl
'
gz’
gs‘can only
arise from the term p z
'
pa’
pj. It therefore-lcannot be made tovanish by the addition of other“terms:J It fbllows their thatthe order of any symmetric function 2 aMfip iy is equal to thegreatest of the numbers m
, p , g ; for we have proved that itcannot be less than that number, and that it cannot be greater,since functions of a hig powersof a than a
m.
Ml ild s.
By the help of then
just proved we can writedown the literal part of any symmetric function, and it onlyremains to determine the coefficients. Thus if it were requiredto form 2 u
”
(6 ty)3, we see on inspection that this is a functionwhose weight is four
,and that it is
'
of the second order ; thatis to say, there cannot be more than two factors in any term.
The only terms then that can enter into such a function are
179201, 1922
,and the calculation would be complete if we knew
with what coefficients these terms are to be affected.
55 . Symmetric functions of the differences of the roots ofequations* being those with which we shall have most to deal,it may not be amiss to give a theorem by which the sum of
any powers of the differences can be expressed in terms ofthe sums of the powers of the roots of the given equation.
Expanding (cc— a)”by the binomial theorem
,and adding the
similar expansions for (ac &c.,we have at once
£7(ac a)”
30mm
mslwm"
5m (m 1)szw'H &c.
Now if we substitute a for a: in E (a: a)” it becomes
(a (a similarly if we substitute [3 for a: it
Such functions have been called critica l functions.
SYMMETRIC FUNCTIONS. 47
becomes (B— oc)+ (B— fy) and so on ; and if we add
the results of all these substitutions,if m be odd
,the sum
vanishes,since the terms (oz (B— u)
m cancel e ach other.If m be even
,the result is 22 (a But when the same
substitutions are made on the right-hand side of the equationlast written
,and the results added together
,we
i
get
1 171-1
If m be odd,the last term will be — s
mso,which will cancel the
first term,and
,in like manner
,all the other terms will destroy
each other. But if m be even,the last term will be identical
with the first,and so on
,and the equation will be divisible by
two. Thus then when m is even,we have
ms 8 %m (m 1)s m—z&c.
é (a 308m
mslsm_2
“5m (m 1)Sasm—Q&c'
)
where the coefficients are those of the binomial until we cometo the middle term with which we stop
,and which must be
div ided by two . Thus
2 (u B)4= s
os4— 4s s 2 (Ot—B)
8sose
1532345 108 3
2
,&c.
56. Any function of the differences , will of course be un
changed if we increase or diminish all the roots by the same
quantities,as,for instance
,if we substitute x for a: in the
given equation. It then becomes
xn
a: 1)M91 4 5" (n ”7”lx
{pa -i (72 — 2)Mag i w
Now any function (I) of the coefficients Pu &c. will,when
we alterPIinto 391 + 8p”p ,
into 112 + 8392 , &c., become
4, {06
1
1—5820‘ 810 2 &c. &c.
If then,in any function of Pu 102, &c., we substitute p l
—Hfit
for Pu p , (n 1)hp ,71271 (n 1)a
”for 102, &c. and arrange the
result according to the powers of 7t, it becomes
dd) dcb dd) 2— 2 a‘f’ “i" 7"61191
(n 1)p l dp.(n ”92 dp s
c l c a? a
t
But since we have seen that any function of the differences is
unchanged by the substitution, no matter how small 7» be, it is
48 SYMMETRIC FUNCTIONS.
necessary.
that any function of the differences, when expressedin terms of the coefficients
,should satisfy the differential equation
mg (n Up ,- 2)p ,di n dr. dp.
Ex. 1. Let it be required to form 2 (a fi)2. We know that its order‘and weightare both 2. It must therefore be of the form Ap 2 Bp l
z. Applying the differential
equation, we have { (n 1)A 2nB } p , 0, whence B is proportional to n 1 and
A to 2a . The function then can only differ by a factor from (n 1)p l z 272102.
The factor may be shewn to be unity by supposing a 1 and all the otherroots =O, when p , 1
, p 2 0, and the value just wri tten reduces to n 1 as it
ought to do.
Ex . 2. To form for a cubic the product of the squares of the differences
(a (fl (y a)2. This is a function of the order 4 and weight 6. It musttherefore be of the form
441732 “l“ BPaPzPI Bps
s EPzzp i
z'
d d dOperating Wi th 3
£ 1
2p , + 172 i t becomes
(2A 3B)P3172 (23 90)P3P12 ‘
i‘ (B GB SE)172
2191 (C 4E)P219 1.»
and as this is to vanish identically, we must have 0 4E, B ISE
, A
D 4E, or the function can only differ by a fac tor from171
2192
2 18101172173 417m " 27p 82o
The factor may be shown to be unity by supposing 7 and consequently11a to be 0.
57. We shall in future usually employ homogeneous equa
tions. Thus writing 5for ac,and clearing of fractions
,the
equation we have used becomesas"
fl aw“
? M axwa
’m i p y 0
We give w”a coefficient for the sake of symmetry ; and We find
it convenient to give the terms the same coefficients as in thebinomial theorem and so write the equation
-1 Aaom
”na
,x""
y i n (n 1)a zzc y‘
“ nam xy any
”0.
One advantage Of using the binomial coefficients is,that thus
all functions of the differences of the roots will,when expressed
in terms of the coefficients, be such that the sum Of the numericalcoefficients will be nothing. For we get the sum of the nu
merical coefficients by making ao a,
a2&c. 1 but on this
supposition all the roots Of the original equation become equal,and all the differences vanish.When we speak of a symmetric function Of the roots of the
homogeneous equation, we understand that the equation having
SYMMETRIC FUNCTIONS. 49
been div ided by coy
"
,the corresponding symmetric function has
a a asbeen formed of the coeffiCIents
a
l
j,&c. of the equati on In
0 0
and that it has been cleared of fractions by multiplying by thehighest power of or
0in any denominator. In this way, every
symmetric function will be a homogeneous function of the
coefficients ao,a.”&c. ; for before it was cleared of fractions
,it was
a homogeneous function of the degree 0,and it remains homo
geneous when every term is multiplied by the same quantity.
Or we may state the theory Of the symmetric functions of theroots of the homogeneous equation
,without first transforming it
to an equation inx
If one of the roots of the latter equation
be a,then it is ev ident that the homogeneous equation is satisfied
by any system Of values x'
, y'
for which we have x’
ay'
,since
it is manifest that we are only concerned with the ratio w’
y'
.
And since the equation div ided by y" is resolvable into factors
,
so the homogeneous equation is plainly reducible to a productof factors (y
'
x yx'
)(y"
x gm"
)(y x &c. Actually multiplying and comparing with the original equation
,we get
I H " I I H H )
a0=yy y na %n(n 1)a = 2wx y ,
&C .
an
+ 53 90 9: &c.,na &c.
By making all the y’s 1
,these expressions become the ordinary
expressions for the coefficients of an equation in terms of its
roots x '
,x"
,&c. And conversely, any symmetric function ex
pressed in the ordinary way in terms of the roots w’
,w"
,may
be reduced to the other form,by imagining each x
'
divided bythe corresponding y
'
,and then the whole multiplied by such
a power of y'
y"
,&c. as will clear it of fractions. Thus the
sum of the squares of the differences 2 (a:'
becomes
2 (cc'
y"
&c. And generally any functions of
the differences will consist of the sum of products of deter
minants of the form (x'
y"
(ac'
ym
&c. by powersof y
'
, y &c.
58. The differential equation which we have given for
functions of the differences of the roots,requires to be modified
when the equation has been written with binomial coefficients.H
50 sYMMETRIC FUNCTIONS.
Thus,if in the equation a
oao"
na lm’“
y &c. 0,we write a: a
for w,the new a becomes a ka
o,a becomes a 2a 7t + a
°7x“,
a becomes a + 37ta + 3Va -I—7t"a0
and any function 4:ofthe coefficients Is altered by this substitution into
d¢ dd; dd;
a? da.
.d—
as
Any function then of the differences,since it remains unaltered
when for a: we substitute a: A,must satisfy the equation
0145
daz
In like manner any function which remains unaltered, when fory we substitute ?y 7x
,must satisfy the equation
lci,%t-l- (n —2)
Functions of the latter k ind are functions of the differences ofthe reciprocals Of the roots
,and in the homogeneous notation
consist of products Of determinants Of the form w'
y"
yw'
,&c.
by powers of a"
,as"
,&c. Functions of the determiriants
w’
y"
y'
m"
alone,and not multiplied by any powers of the w
’s
or the y’s,will satisfy both the differential equations.
59. It is to be Observed,that the condition
«a a deb
dal
1d d,
da,
is not only necessary but sufficient,in order that gb should be
unaltered by the transformation 90+ 7» for x. We have seenthat the coefficient of 7t in the transformed equation thenvanishes
,and the coefficient of 7c
“ is without difficulty foundto be
dd) d_ _d> 01
—4) d_d_
dag
“161715
+ 6622 + 860 .'
l+l
_
2 d—‘
a,
where m the latter symbol the ao,an&c.which appear explicitly,
are not to be differentia’tatl.But it will be seen that this is
preciselyl
— + 3a
+ 810
SYMMETRIC FUNCTIONS. 5 1
For when we operate with the symbol on itself, the result willbe the sum of the terms got by differentiating the a.
” as,&c.
which appear explicitly, together with the result on the supposition that these a
na, ,&c. are constant. Thus then
,the coeffi
l C d 0
cient of h”vani shes
,since a‘,
—g+ &c. Is supposed to vani shl
identically. SO in like manner,for the coefficients of the other
powers of h .
Ex. To form for the cubic cow“ 3a lw2y 3a29:_z/2 aay
s, the function
2 (s xix/x)”(w e mas/z)
” (ms/t xxx/s)”
This can be derived fromEx . 2,p. 48, or else directly as follows. The function is to
be of the order 4 and weight 6. It must therefore be of the formAssesses. Baaazfz i a. Caaa iaxa. Da zaz aza o Ew an
d d deratewith a — + 2a and weOP 0
tia lI da ,
“2dc ,
get
(B 6A)aaazaoao-I (30+ 23 )050 1a 10 o+ (2E 6D+ 3B)azaza lao (4E+ 30) 0.
Equating separately to Othe coefl‘icient of each term,and taking A z l , we find
B = -6,
E -3.
d60. M. Serret wri tes the operatIon a
o&c. in a compact
form,which is sometimes convenient. If we imagine a fictitious
variable C, of which the coefficients ao, an &e. are such func
tions, that«a da
2 ,«a
dc d: dc
then evidentlyC
g? if
3as
MIn like manner na l da + &c. may be written in the compact
0
d
is, where n is a variable,of which c
o,an &c. are sup
posed to be such functions that
da da
3770
1 ,-l)a &c.
61. It is worth while to add an observation of M. Brioschi
as to the meaning of the first of these operations when expressed
in terms of the roots. Let there be any function of the coeffi
52 SYMMETRIC FUNCTIONS.
cients Pu p , , &c.,let us express these coefficients in terms of
the roots,and examine the differential lief the function with
respect to any root. Now if g” 92, &c. denote the sum,sum
of products in pa irs,&c. of all the roots
,omitting any one root
a,then ev idently
dr. dp. div.da 7 da g} , da
—gea &c i
d d d d
d“ dp 1
g‘dPsgs dp a
We have of course corresponding expressions for the differentials with respect to the other roo ts
,and if now we add
will ev idently be n. That of1
all together,the coefficient of
d?a; will be (n 1)p ] ; for since §1 =p ,
a,when we
(
add,the
coefficient In question will be np , (a B «y &c. 1)Pr
In like manner the coeffi cient ofE
(n 2)p”,and we have
8
d d d d
It Is ev ident now why, when the operation on the right-handside of this equation is applied to any function of the differenees of the roots
,the result vanishes
,since when the equi
dvalent operat ion
a&c. i s applied to any difference, the result
vanishes.In the same manner
,if the equation had been written with
binomial coefficients,we should have had
da92
70: fin (n 1) 1i a &c.,
whence,exactly as before
,
d= a
o
54 ORDER AND WEIGHT OF ELIMINAN'
I‘S.
system of equa tions, whose vanishing expresses tha t they can be
sa tisfied by a common system of va lues of the variables.
63 . We have now to show how elimination can be per
formed,and what is the nature of the results arrived at. We
commence with two equations written in the non-homogeneousform
mmm
—p x or
ac”
glx 9290 &c,t= 0
,or x]x (ac)= 0.
The eliminant of these equationsA i-s,2s we have seen
,the condi
tion that they should have a common root. If this be the case,
some one of the roots of the first equation must satisfy the
second. Let the roots of the first equation be a, B, ry, &c. ;
and let us substitute these values successively in the secondequation
,then some one of the results vr (a), &c. must
vanish,and therefore the product of all must be sure to vanish.
But this product is a symmetric function of the roots of the first
equation,and therefore can be expressed in terms of its coetfi
cients,in which state it is the eliminant required. The rule
then for elimination by this method is to take the m factors
“I”(a) an
gloat—1+ gza
nfl &c‘7
= B"-
Q,B + 9 3
a}, (v) a"
&0-5&0
to multiply all together,and then substitute for the symmetric
functions &c.,their values in terms of the coefficients of
the first equation.
Ex. To eliminate z between p ix p , 0, $ 2 qlay Q2 0. Multiplying91“ 92)(fie 913 we get
azfi2 qi afi(a fl) 92 £3
2) qfiafi qIQZ (a fl) 92
2;
and then substituting a [3 cf) p z , a2 32 p ,
” 2122, we haveP2
2 “
19 117291 42 (P12 2132)4“
P2912
Q22,
(192 a)?(m 91)(mag —
p zqt),
whi ch is the eliminant required.
64. We obtain in this , way the same result (or at leastresults differing only in sign), whether we substitute the roots
of the firSt equation in the second,or those of the second in the
first. In other words,if a
'
, B’
,ry'
,&c. be the roots of the
ORDER AND WEIGHT OF ELIMINANTS. 5 5
second equation, the eliminant may be written at pleasure as
the continued product of am,&c.
,or as the
product of 11r (a), «14m, &c. For remembering that(f) (w) (x a)(so B)(a: y), &c.
,the first form is
(a'
a)(a'
B)(a'
a &c. (B'
a)(B'— B)(3
'
w ), &c.,and the second is
(a a'
)(a B'
)(a &c. (B a'
)(B —m(B n.&c.
In either case we get the product of all possible differencesbetween a root of the first equation and a root of the second ;and the two products can at most differ in sign.
65 . If the equations had been given in the homogeneousform
,with or without binomial '
coefficients,
aowm
ma1xm’ 1
y 13m (m 1) &c. 0,
boot"
nblx
’H
y + 312n (n 1) az
w
y2+ &c. = O
,
we can reduce them to the preceding form by dividing them re
bspecti vely by by
"
,when we have 19 , Til l bi
b—1
,&c.
0
We substitute then these values for P1 , 91 , &c. in the result
obta ined by the method of the last article,and then clear of
fractions by multiplying by the highest power of ac,
or b0 ,in \
.
any denominator. Thus the eliminant of a0x2
+ 2a locy+ a2yfl
box2
+ 2blwy + b2y
2
,obtained in this manner from the result of
Ex .,Art. 63
,is
(ar-0b2 4250) 4 (a ,bo — a
0bl)(a b — a
2bl).
It is ev ident thus that the eliminant is a lways a homogeneous
function of the coeficients of ei ther equa tion. For before wecleared of fractions
,it was ev idently a homogeneous function
of the degree 0,and it remains homogeneous when every term
is multiplied by the same quantity.
The same thing may be seen by applying to the equationsdirectly the process of Art. 63 . Let the values which satisfythe first equation be x
’
y'
,ce
”
y"
,&c. ; then, if the equations have
a common factor,some one of these values must satisfy the
second equation. We must then multiply together
(bowm
nblm'M
y'
&c.)(box nblm'm ' 1
y"
&c.)
56 ORDER AND WEIGHT or ELIMINANTS.
and then substitute for the symmetric functions &c.their values in terms of the coefficients of the first equation
,as
found by Art. 57.
66. The eliminant of two equa tions of the m and n ordersresp ectively, is of the n
thorder in the coeficients of the first equa
tion,and qf the m
min the coeficients of the second.
For it may be written either as the product of m factorsxlr (a), «1: (B), &c.
,each containing the coefficients of the second
equation in the first degree,or else as the product of n factors
d) &c.,each containing in the first degree the coeffi
cients of the first equation. Or confining our attention to theform ( t argets), &c. we can see that this form
,which obviously
conta ins the coefficients of the second equation in the degreean, contains those of the first in the degree n
,since the sym
metric functions which occur in' it may contain the and no
higher,power of any root (Art.
67. The weight of the eliminant is mn ; that is to say,the sum of the suffixes in every term is constant and =mn.
For if each of the roots a, B a
'
, B'
,&c. be multiplied by the
same factor 7x,then since each of the nan differences a a
'
(seeArt.
'
64)is multiplied by this factor a,the eliminant will be
multiplied by X”
. But the roots of the two equations will bemultiplied by 7x if for 1717 q1 we substitute Np” kg ; for 102 , q, ;X‘p z , 7&
2
q2 ; &c. We see then that if we make this substitutionin the eliminant
,the effect will be that every term will be
multiplied by N“
; or, in other words,the sum of the suffixes
in every term will be i nn. The same thing may also be seen
to follow from the principle of Art. 5 3 . In that)the sum of
the index of every term and the suffix of the correspondingcoefficient is n ; that is to say, alr (:zc)consists of the sum of a
number of terms,each of the form q .x
‘. If then we
'
i ake any
terms at random in each of the factors 44 a), &c.,the
corresponding term in the product will be &c.,
and if we combine with this all other terms in which the same co
efficients of the second equation occur,we get
&c. The sum of the suffixes of the q’s is n'
i+n -j+n
or since there are m factors,the sum is mn
ORDER AND WEIGHT or M inimum. 57
But, by Art. 53 , the sum of the suffixes of the p
’s in the ex
pression for &c. is i j+ k+ &c. Therefore the sumof both sets of suffixes is mn
,which was to be proved.
The result at which we have arrived may be otherwise statedthus If ql conta in any new var iable z in the first degreeif p g, q2 conta in it in the second and lower degrees ; if 93 in
the third,and so on wi ll in genera l conta in
this variable in the m
It is ev ident that the results of this and of the last article areequally true if the equations had been written in the homo
geneous form aowm+ &c., because the suffixes in the two forms
mutually correspond. And again,from symmetry it follows that
the result of this article would be equally true if the equationshad been written in the form a
maim " H
y &c.,where the
suffix of any coefficient corresponds to the power of a: whichit multiplies
,instead of to the power of y.
68. Since the eliminant is a function of the differencesbetween a root of one equation a nd a root of the other
,it
will be unaltered if the roots of each equation be olncreased bythe same quantity?)that is to say, if we substitute x + 7t for a:
in each equation. It follows then,as in Art. 56, that the climi
nant must satisfy the differential equation
32+ (m—(m — 2)
+ nd—
g+ (n
or,as in Art. 58, if the equations had been written with
binomial coefficients, we haVedA dA
a; dot—
1t
+&c.+ 6 + 25lA
69. Given two homogeneous equa tions between three variables,
of themmand n
tndegrees respectively, the number of systems of
Or again thus : if in, the eliminant we substi tute for each coefficient p a ,the term
a“ which it multiplies in the original equation, every term of the eliminant will bedivisible by am". Or, in the homogeneous form, it we substitute for each coefficienta , the term a
ah/"W ,
which it multiplies, every term of the eliminant will be divisibleby am y
m".
58 ORDER AND WEIGHT OF ELIMINANTS.
va lues of the variables which can befbund to sa tisfy simultaneouslythe two equa tions is mn.
*
Let the two equations arranged according to powers ofm,be
ax”
(by as)w (dyg
eye fez
)a: + &c.= O,
a'
x”
(b'
y c'
z)a: (d'
yz
e'
yz f'
z"
)a)”
&c. 0.
If now we eliminate a; between these equations,since the co
efficient of m’“is a homogeneous function of y and z of the first
degree, that of ad” is a similar function of the second degree
,
and so ou,— it follows from the last Article that the eliminant
will be a homogeneous function of y and z of the mnmdegree.
It follows then tha t mn values of y and zTcan be found whichwill make the eliminant 0. If we substitute any one of thesein the given equations
,they will now have a common root when
solved for a: (since their eliminant vanishes) and this value of a',
combined with the values of y and z already found, gives one
system of values satisfying the given equations. So we plainlyhave in all mn such systems of values. We shall
,in Lesson X.
,
give a method by which , when two equations have a commonroot
,that common root can immediately be found.
Ex. To find the co-ordinates of the four points of intersection of the two conics
as:2 by2
or2 2fye 2gzm 2hay 0
,a
’a:2 b
’
y2 o
’s2 2f
’
yz 2g'z a: 2h
’my 0.
Arrange the e uations according to the powers of m, and eliminate that variable ;then, by Art. 5 )the result is
312 2 (af
'
)W (M’
)
4 y (as’
)3 ]WW) 2 (f W 2Us?)2231 0,
where, as in Lesson I., we have written (ab’
)for ab'
a’b. This equation, solved for
y : z,determines the values corresponding to the four points of intersection. Having
found these , by substituting any one of them. in both equations, and finding theircommon root, we obtain the corresponding .value of a: : z . We might have at once
got the four values of a: : z by eliminating y between the equations, but substitutionin the equations is necessary in order to find which value of y corresponds to eachvalue of a . By making 2 1
,what has been sai d is translated into the language of
ordinary Cartesian co-ordinates .
These equations may be considered as representing two curves of the mmand
nm degrees respectively ; the geometrical interpretation of the proposition of this
Article being, that two such curves intersect in ma points. The equations are re
duced to ordinary Cartesian equations by making z 1.
1 The reader will remember tha t when we use homogeneous equations, the ratio
of the variables is all with which we are concerned. Thus here, z“may be taken arbitrarily, the corresponding value of y being determined by the equation iny z .
ORDER AND WEIGHT OF ELIMINANTS. 59
70. Any symmetric functions of the mn va lues which simul
taneously sa tisfy the two equa tions can be exp ressed in terms ofthe coeficients of those equa tions .
In order to be more easily understood,we first consider
non-homogeneous equations in two variables. Then it is plainenough that -we can so express symmetric functions involvingeither variable alone. For eliminating y, we have an equationin x
,in terms of whose coefficients can be expressed all sym
metric functions of the mn values of a: which satisfy both equations. Similarly for y. Thus
,for example
,in the case of two
conics ; my ,&c.
,being the co-ordinates of their points of intersec
tion,we see at once how to express such symmetric functions as
xl+ w + x + w
uu7 y2
r+ y
2+ y
2
+ y2
uu7&c
"
and the only thing requiring explanation is how to express symmetric functions into which both variables enter, such as
way:
wi ly/I “
any“:xi i i /yum
To do this,we introduce a new variable
,t= 7tw
,uy, and by the
help of this assumed equation eliminate both a: and y from the
given equations. Thus y is immediately eliminated by substi
tuting in both its value derived from t= 7ta3+ uy, and then wehave two equations of the mtn
and nmdegrees in w
,the eliminant
of which will be of the mn degree in t,and its roots will be
obv iously Mal -t uy, ,7m &c.
,where say ,
any“
are the
values of a: and y common to the two equations. The coeffi
cients of this equation in t will of course involve 7t and n. We
next form the sum of the lam powers of the roots of this equa
tion in t,which must plainly be (Me +uy
The coefficient,then
,of M in this sum will be En
": the coeffi
cient of hw y. gives us and so on.
Little need be sa id in order to translate the above into the
language of homogeneous equations. We see at once how to
form symmetric functions involv ing two variables only, such as
Eynuem z w ,for these are found
,as explained, Art. 57, from the
homogeneous equation obtained on eliminating the rema iningvariable ; the only thing requiring explanat ion is how to form
symmetric functions involv ing all these variables, and this is
done precisely as above , by substituting t= 7tx ,u«y.
60 ORDER AND wmen'r OF ELIMINANTS.
Ex. To form the symmetric functions of the cc-ordinates of the four pointscommon to two conics. The equation in the last Example gi ves at once
M Hz/"s un: 4 (as
’
) (ab'
)z 4 (ah’
)(bh'
)
and from symmetry, (bc'
)2 4 (bf
2 (Milan /2m) 4 (af'
) (ah’
) (“y'
l (ck'
) 2 (as’
) &c
To take an Example of a function involving three variables, let us form2
,
zz mz2m),
which corresponds to 2 when the equations are written in the non-homogeneousform.
By the preceding theory we are to eliminate between the given equations, andt M: fly ; and the required function will be half the coefficient of hp. in
2 If the resul t of elimination be
At‘ (BK tsz (B ltz EM; Fa?)fiz z &c.
2 (BX 24 (D7\2 EM;
2 (x, B C AE.
By actual elimination
A (ab'
)z 4 (ah’
) B 4 W’
)(ah’
) (W)W’
) 2 (W)(W).C 4 (af
'
l (as’
)(W) (ah'
)(by’
l 2 (ah’
)
E 4 (W) W)(ah’
) 2 (cf )(kf’
) 2 (hi )(ky'
) 4W’
)
71. To form the eliminant of three homogeneous equa tions in
three variables, of the m”
,nth
,andp
mdegrees respectively.
The vanishing of the eliminant is the condition that a systemof values of x , y, z can be found to satisfy all three equations.*When this
,then
,is the case
,if we solve for any two of the
equations,and substitute successively in the remaining one the
values so found for to, y, z , some one of these sets of values must
satisfy that equation, and therefore the product of all the resultsof substitution must vanish. Let
,then
,x'
, y'
,z'
; y"
,z"
,&c.
be the system of values which satisfy the last two equations,and
which (Art. 69)are np in number : we substitute these valuesin the first
,and multiply together the np results <j> (cc y
'
,z
'
, y"
,z &c. The product will plainly involve only sym
metric functions of a'
, y'
,z'
,&c. which (Art. 70)can all be
expressed 111 terms of the coefficients of the last two equations ;and
,when they are so expressed
,it is the eliminant required.
If the three equations represent curves , the vanishing of the eliminant is the
condition that all three curves should pass through a common point.
62 EXPRESSION OF ELIMINANTS AS DETERMINANTS.
first will be y'
,The eliminant, then, which is the
product of np factors of the form 4) (a'
, y’
,z'
)will be multipliedby N
M”. If then any term in the eliminant be a
kbpm, &c.,where the suffix corresponds to the power of e
,which the
coefficient multiplies,since the alteration of a , into Ma“ b
,
into Nb” &c.,multiplies the term by N
”
,we must have
Q .E.D.
74. It is proved in like manner that three equations are ingeneral satisfied by mnp common values ; that any symmetricfunction of these values can be
“
expressed in terms of their coefficients ; and that we can form the eliminant of four equationsby solv ing from any three of them
,substituting successively in
the fourth each of the systems of values so found,forming the
product of the results of substitution,and then
,by the method
of symmetric functions, expressing the product in terms of thecoefficients of the equations. In this way we can form the
eliminant of any number of equations ; and we have the following general theorems : The eliminant qf 7c equa tions in k— l
indep endent variables is a homogeneous function qf the coeficients
of each equa tion, whose order is equa l to thep roduct of the degrees
of a ll the rema ining equa tions . If each coeflicient in a ll the
equa tions be afi'
ected with a sufiw equa l to the power of any one
variable which i t multiplies, then the sum of the sufimes in everyterm of the eliminant wi ll be equa l to the product Q)
"the degrees
qf a ll the equa tions. And,again
,if we are given 70 equations
in h: variables,the number of systems of common va lues of the
variables,which can befound to sa tisfy a ll the equations, will be
equa l to the product of the orders of the equations.
LESSON IX.
EXPRESSION OF ELIMINANTS AS DETERMINANTS.
75. THE method of elimination by symmetric functions is,in a theoretical point of v iew
,perhaps preferable to any other,
it being universally applicable to equations in any number of
ExPRESSION OF ELIMINANTS As DETERMINANTS 63
variables ; yet as it is not very expeditious in practice,and does
not yield its results in the most convenient form,we shall in
this Lesson give an account of some other methods of elimi
nation. The following i s the method which most Obviouslypresents itself. It is in substance identical with what is calledelimination by the process of finding the greatest commonmeasure. We have already seen that the eliminant of two
linear equations ax b 0,
u'
m b'
0 is the determinantab
'
ba'= 0. If now we have two quadratic equations
multiplying the first by a'
,the second by a
,and subtracting,
we get
(a b'
)w (ao'
) O ;
and,again
,multiplying the first by c
’
,the second by c
,sub
tracting,and div iding by cc
,we get
(ad)a: (bc'
) 0.
The problem is now reduced to elimination between two linear
equations,and the result is
(ac'
)2
(be)(bc'
) o.
76. So,again
,ifwe have two cubic equations
aw3
+ bw2+ ca3+ d = o,
we multiply the first by a’
,the second by a
,and subtract ;
and also multiply the first by d'
,the second by d, subtract and
divide by so. The problem is thus reduced to elimination be
tween the two quadratics
(ab'
)as”
(ad)a: (ad’
) 0, (ad
'
)(192
(bd’
)ac (cd’
) 0.
By the last article, the result is
(ab'
) (ad) (ab'
)(M il (db'
) 0
Now it is to be observed that the equation
(ab'
)(cd’
) (a o'
)(db'
) (ad'
)(bc'
) 0
is identically true. Consequently when we multiply put,the
preceding result becomes divisible by and the reduced
result is
2 (ad'
)(ab'
)(cd'
) (ad'
)(ad)(bd'
)
(acr (cd'
) (bdreb'
1 (ab'
)(bc'
)(cd'
) o.
64 EXPRESSION or ELm rNANTs As DETERMINANTS.
The reason that in this process the irrelevant factor (ad'
)is introduced is that, if and therefore a to a
'
in the sameratio as d to d
’
,we must get results differing only by a factor
,
if from the first equation multiplied by a'
we subtract the secondequation multiplied by a
,or if from the first equationmultiplied
by d'
,we subtract the second equation multiplied by d. Thus,
on the supposition 0,even though the original two cubics
have not a common factor,the two quadratics to which we re
duced them would have a common factor. In general then,
when we eliminate by this process,irrelevant factors are intro
duced,and therefore other methods are preferable.
77. Euler’s ill ethod. If two equations of the m“1
and nm
degrees respectively have a common factor of the first degree,
we must obtain identical results,whether we multiply the first
equation by the rema ining n — l factors of the second, or the
second by the remaining m 1 factors of the first. If thenwemultiply the first by an arbitrary function of the (n degree,which
,of course
,introduces n arbitrary constants ; if wemultiply
the second by an arbitrary function of the (m 1)tlhdegree
,intro
ducing thus m constants ; and if we then equate,term by term,
the two equations of the (m n 1)mdegree so formed
,we shall
have m + n equations,from which we can eliminate the m+n
introduced constants,which all ehter into those equations only
in the first degree ; and we shall thus obtain, in the form of
a determinant, the eliminant of the two given equations.
Ex.
We are to equate, term by term,
(Ae + By)(aa’ + buy + cy’)and (A
'z + B
'
y)
Aa A'
a’
o,
Ab+ B a — A'h’
Ac + Bb
Be B'
e o,
eliminating B , A'
, B’
, the result is the determinant
a , o, a"b,a,
c, by c"o o,
EXPRESSION OF ELIMINANTS AS DETERMINANTS. 65
78. This method may be ex tended to find the conditions thatthe equations should have two common factors. In this case iti s ev ident
,in like manner
,that we shall obta in the same result
whether we multiply the first by the rema ining 92 2 factors ofthe second
,or the second by the remaining 772 2 factors of the
first. As before,then
,we multiply the first by an arbitrary
function of the 92 — 2 degree (introducing n 1 constants), andthe second by an arbitrary function of the m — 2 degree ; andequating
,term by term,
the two equations of the m + a 2 de
gree so found,we have m n 1 equations
,from any m n 2
of which,eliminating the m + n — 2 introduced constants
,we
obtain m + n 1 conditions,equivalent
,of course
,to only two
independent conditions.Ex. To find the conditions that
am“ easy2 dy
a 0, d z " b’xzy (
fatty
2 d' 3 0,
should have two common factors. Equating(Aw By)(m 9 bwzy easy
2 dy3) (A
'x B
’
y)(ala s b
'wzy c
’
aty2 d
'
y3),
we have Aa A’a
’0,
Ab Ba A’b’
B’a,
’O,
Ac Bb A’o
’B
’b
’z 0,
Ad Bo A’d
’B
’c
’0,
Bd B'd'0,
from which, eliminating A,B,A
’
, B'
, we have the system of determinants [for the
notation used, see Art.
79. Sylvester’s dia lytz
'
c method. This method is identical in
its results with Euler’s,but simpler in its application, and more
easily capable of being extended. Multiply the equation of the
m“1degree by as
“,
w" ’ 3
y2
,&c. ; and the second equation
by azw ‘
,svm’ g
y, wm ' s
yz
,&c.
,and we thus get on+ 72 equations
,
from which we can eliminate linearly the m + n quantitie s90mm "
,wm n' s
yz
,&c.
,considered as independent un
knowns. Thus in the case of two quadratics, multiply bothby a: and by y, and we get the equations
aws
bxz
y exp/2
0,
awz
y buoy”
cy"
0
afar8b
’
w"
y dang/2
O,
d affy b'
wyz
c'
y”
0,
66 EXPRESSION or ELIMINANTS as DETERMINANTS.
from which,eliminating x
3
,my, my
”
, y“,we get the same deter
minant as before
7 7
In general,it is evident by this method, that the eliminant
is expressed as a determinant of which n rows contain the coefficients of the first equation
,and m rows contain the coefficients
of the second. Thus we obtain the rule already stated for theorder of the eliminant in the coefficients of each equation.
80. Bez out’s metkod. This process also expresses the elimi
nant in the form of a determinant ; but one which can be more
rapidly calculated than the preceding. The general methodwill
,perhaps
,be better understood if we apply it first to the
particular case of the two equations of the fourth degree
d x4
+ cx"
y”+ Ch g/
3
+ey4= 0
,al
so“b'
xa
y c'
w'
fly"d'
wy”
e'
y‘O.
Multiplying the first by a'
,the second by a
,and subtracting
,
the first term in each is eliminated,and the result
,being divisible
by y, gives
(ab'
)w”
(ao’
)w“
y (ad’
)mg”
(ae'
)y“0.
Again,multiply the first by a
'
w b'
y, and the second by am by,
and the two first terms in each are eliminated,and the result
,
being div ided by y”
,gives
(ad)w”
(bc'
” (Mi lmy”
(be'
) 0
Next,multiply the first by a a: b
'
wy c’
y”
; and the second byax
”bony cg
”
; subtract , and div ide by y”
; when we get
W)o f 11ae’
) (Mnas“
? me2+wer o.
Lastly, multiply the first by a'
ac”b'
wz
y day2d'
y”; the second
by aw“ba
f
y 1» easy
"dy
“; subtract, and div ide by y
‘; when we
etg(ae
'
)w”
(be'
)we+ (ca'
1my 0
From the four equations thus formed we can eliminate linearly
EXPRESSION or ELIMINANTS AS DETERMINANTS. 67
the four quantities,as
”
,w
‘
iy, my”
,
'
y”
,and obtain for our result the
determinant
(a (ad) (ad'
)
(ad), (ad'
) (ae'
)+ 1507) (be
(ae'
(M). (be (of ). (ed)
(0637 (dd)
81. The process here employed is so ev idently applicable toany two equations
,both of the 72
“degree
,that it is unnecessary
to make a formal statement of the general proof. On inspectionof the determinant obtained in the last article
,the law of its
formation is apparent,and we can at once write down the deter
minant which is the eliminant between two equations of the
fifth degree by simply continuing the series of terms, writing an
(af’
)after every &c. Thus the eliminant is
(ab'
)7 (ad) 7 (ad'
) 7 (ad) 7
ga
y), (ad
'
) (bc'
) (d e'
)a
7
l
) 7
(ae'
)A“7 (df
'
)
(ef'
)
It appears hence that in the eliminant every term must contain a or a
'
; as was ev ident beforehand,since if both of these
were 0,the equations would ev idently have the common
factor y 0.
It appears also that those terms which contain a or a’
only in
the first degree are (ab'
)multiplied by the eliminant of the equations got by making a and a
'= o in the given equations. For
every element in the determinant written above must contain a
constituent from the first row,and also one from the first column ;
but as all the constituents of the first row or column contain a or
a'
,the only terms which contain a and a
’
in only the first degree,
are (ab'
)multiplied by the corresponding minor ; and this, whena and a
’
are made O,is the nex t lower eliminant.
82. It only remains to show that the process here employedis applicable when the equations are of d ifferent dimensions ;
68 EXPRESSION or ELIMINANTS AS DETERMINANTS.
and,as before
,we commence with a particular example
,viz .
,
the equations
acc‘hw
“y ca
f
y”dwy
"ey
‘O,
ai
m”h
’
wy c'
y"0.
Multiply the first by a'
,the second by am
”
,and subtract
,when
we have(ba
'
) (ca'
)w”
y (da'
)my”
(ea'
)y"0.
In like manner,multiply the first by a
'
x b'
y, and the second by(can by)x
”
,and we get
(cd'
)ms
(Ja i l 9031”
31°0
This process can be carried no further ; but if we join to thetwo equations just obta ined the two equations got by multiplying the second of the original equations by w and by y, we havefour equations from which to eliminate (139
,w
’
y, my”
, y”.
And in general,when the degrees of the equations are nu
equal,m being the greater
,it will be found that the process of
Art. 80 gives us 77 equations of the (m degree,each of
these equations being of the first order in the coefficients of eachequation : to which we are to add the m n equations found bymultiplying the second equation by Q
M "
,wM "
y, &c., and we
can then eliminate the m quantities mm",wm' 2
y, &c., from the m
equations we have formed. Every row of the determinant contains the coefficients of the second equation, but only n rowscontain the coefficients of the first. The eliminant is
,therefore,
as it ought to be,of the degree in the coefficients of the first,
and of the mulin those of the second equation.
83 . Cayley’s sta tement of Bez out
’s method. If two equations
47 (w, y), 11» (at , y), have a common root , then it must be possibleto satisfy any equation of the form (17+ 7Wr = o7 independentlyof any particular value of X. Take then the equation
4’ 19)‘f’ (357 ff) ‘f’ y’
)“If 31)= 07
which,
'
if (f) and a!» have a common factor, can be satisfied inde
pendently of any particular values of w’
and y'
. We may in the
first place div ide it by wy’ —ym
'
,which is obv iously a factor : then
equate to 0 the coefficients of the several powers of w’
, y'
; and
then eliminate the powers of ac and y as if theywere independent
70 EXPRESSION OF ELIMINANTS AS DETERMINANTS.
Jx = U,nu V
lae W
,nw
,from which it appears at once that if
u,v,w vanish
,J will vanish too. Again, differentiating the
equation just found, we have
J—i-az zlL—
m
, c
% + 7m + nw
$50
2—3But
,remembering (Art. 27)that
we see that the supposition u 0,v 0
,w 0 (in consequence of
a:
which J is also 0)makes if, also to vanish.
86. We can now express as a determinant the eliminant ofthree equations
,each of the second degree. For their Jacobian
is of the third degree,and therefore its differentials are of the
second. We have thus three new equations of the seconddegree
,which will be also satisfied by any system of values
common to the given equations. From the six equations,
wall 6” C”
we can eliminate the six uantities7 dx ’ dy
’ dz ’ q
a”
, y“,z”
, ya , eat, my, and so form the determinant required.Aga in
,if the equations are all of the third degree
,J is of the
sixth,and its differentials of the fifth
,and if we multiply each
of the three given equations by as”
, y”
,z”
, ye, z ar, my, we obta in
then,u,v,
eighteen equations,which
,combined with the three differentials
of the Jacobian,enable us to eliminate dialytically the twenty
One quantities,
w‘y, &c.
,which enter into an equation of the
fifth degree. This process,however
,cannot, without modifi
cation,be extended further.
87 Mr. Sylvester has shown that the eliminant can alwaysbe expressed as a determinant when the three equations are of
the same degree. Let us take,for an example
,three equa
tions of the fourth degree. Multiply each by the six termsmy, y
”
,&c.)of an equation of the second
’
degree [or gene
The beginner may omit the rest of this Lesson.
EXPRESSION OF ELIMINANTS As DETERMINANTS. 71
rally by the %n (n 1)terms of an equation of the degree (nWe thus form eighteen [gn (n equations. But since theseequations
,being new of the sixth [2n— 2] degree, consist of
twenty-eight [n (277 l)] terms ; we require ten ad
ditional equations to enable us to eliminate dialytically all the
powers of the variables. These equations are formed as followsThe first of the three given equations can be written in the
form Am‘+ By+ Oz ; the second and third
,in the form
A"
w + B"
y + 0"
z ;
and the determinant (AB’
Q which 18 of the sixth degree inthe variables must Obviously be ,
satisfied by any values whichsatisfy all the given equations. We should form two similardeterminants by decomposing the equations into the form
Ay‘+ Bx Oz
,A s
‘Ba: Cy.. So again
,we might decompose
the equations into the forms A ac3+ By
2+ Oa , O
'
z,
A"
a: 0"
z (for every term not div isible by x”or y
2must
be divisible by z); and then we obtain another determinantwhich will be satisfied when the equations vanish
together. There are six determinants of this form got by interchanging a
, y, and z in the rule for decomposing the equations.Lastly, decomposing into the form Am
2+ By
2
+ Oz”
,&c. we
get a single determinant, which , added to the nine equationsalready found, makes the ten required. In general
,we decem
pose the equations into the form Ax“+ By
fl+ Oz such thata ,
8 + «y :- n+ 2 , and form the determinant AB ’
0"
and it canbe very easily proved that the number of integer solutions ofthe equation a + .3 17 71 + 2 is %n (n+ exactly the numberrequired.
88. When the degrees of the equations are different, it isnot possible to form a determinant in this way, which shall givethe eliminant clear of extraneous factors. The reason why
such factors are introduced, and the method by which they are
to be got rid of,will be uii derstood from the following theory,
due to Mr. Cayley : Let us take for simplicity three equations,u,77,w,all of the second degree. If we attempt to eliminate
dialytically by multiplying each by a'
, y, a , we get nine equations
,which are not sufficient to eliminate the ten quantities
a"
,fly, &c. Again
,if we multiply each equation by the six
72 EXPRESSION OF ELIMINANTS As DETERMINANTS.
quantities,w
”
,my, y
”
,&c.
,we have eighteen equations, which are
more than sufficient to eliminate the fifteen quantities a‘, ma
y, &c.
If we take at pleasure anyfifteen Of these equations, and form
their determinant,we shall indeed have the eliminant, but it will
be multiplied by an extraneous factor ; since the determinant is ofthe fifteenth degree in the coefficients, while the eliminant is onlyof the twelfth (Art. 72 , mn np +pm 12
,when m n= p
The reason of this is,that the eighteen equations we have formed
are not independent,but are connected by three linear relations.
In fact,if we write the identity uv = vu
,and then replace the
first u by its value, as;2by
2and in like manner
,with the
v on the right-hand side of the equation,we get
In like manner,from the identit ies vw=wv
,wu= uw
,we get two
other identical relations connecting the quantities m’a, y
2u,w
”v,
wzw,&c. The question then comes to this : “If there be m+p
linear equations in m variables,but these equations connected by
p linear relations so as to be equivalent only to m independentequations
,how to express most simply the condition that all
the equations can be made to vanish together.” In the presentcase m 3 .
89. Let us,for simplicity, take an example with numbers
not quite so large,for instance
,m= 3
, p: 1. That is to say,
let us consider four equations,3,t,u,v,where s a lso by c
lz,
&c.,these equations not being independent
,but
satisfying the relation,D
13 D
2t D
au 0. Now I say,
in the first place,that if we form the determinant (a lb2ea)of any
three of these equations,3,t,u,this must containD
4as a factor.
For if D4= O
,we shall have 3
,t,a connected by a linear rela
tion, so that any values which satisfied bo th 3 and t should satisfyi t also ; and therefore the supposition D
4= O would cause the
determinant (a lbzcs) to vanish. And in the second place,I
say that we get the same result (or, at least,one differing
only in sign)whether we div ide (0 16203)by D4or (a lbzc4)by
D8. For (Art. 15) (a 1b204)is the same as the determinantwhose first row is “
17bn c
, ,the second
,a b c and the third
,27 27 27
D4a‘, D4
b‘, D405 but we may substitute for its value
EXPRESSION OF ELIMINANTS AS DETERMINANTS. 73
—D,al
—D2a2— D
sa3 ,
and in like manner for D4b4 ,D
404. The
determinant would then (Art. 18)be resolvable into the sum
Of three others ; but two of these would vanish,hav ing two
rows the same,and there would remain D
4 (a 1b2c4)= D3 (a 1b2c3).
It follows then that the eliminant of the'
system may be ex
pressed in any of the equivalent forms obta ined by forming thedeterminant (a lbzcs)of any three Of the equations
,and dividing
by the remaining constant D4.
Suppose now that we had five equations 3,t,u,v,w,con
useted by two linear relationsE
13 E
2t E
su E
47) E
520 0. Eliminating w from these
relations,we have (B l
E5)s +
and we see, precisely as before
,that the supposition (D4
Eb)= 0
would cause the determinant (a lbgcs)to vanish ; and that we
get the same result whether we div ide (a 1b2cs)by (D4Es)or
div ide the determinant Of any other three of the equations bythe complemental determinant answering to (D4
E5). This
reasoning may be extended to any number of equations con
nected by any number of relations,and we are led to the
following general rule for finding the eliminant of the systemin its simplest form . Write down the constants in the m +pequations
,and complete them into a square "
form by addingthe constants in the p relations : thus
8 ; “17bi )
01D E
1
t!
“27be?
02D
27E
2
u ; as?be,
03
D37E3
0 ; “47647
04D
47E4
w)
as)657
05D
57E5
then the eliminant in its most reduced form is the determinantof any m rows of the left-hand or equation columns
,div ided
by the determinant got by erasing these rows in the right-handcolumns.Thus
,then
,in the example of the last Article
,we take
the determinant of any fifteen Of the equations,and
,div iding
it by a determinant formed with three of the relation rows,Obtain the eliminant ; which is of the twelfth degree, as it
ought to be.
74 EXPRESSION OF ELIMINANTS AS DETERMINANTS.
90. And,in general
,given three equations of the m
and pmdegrees, we form a number
_Of equations Of the degree
m n+p 2,by multiplying the first equation by all the terms
am”
,and so on. We should in this manner have
sews 1)
equations. But the number Of terms, &c.
,to be elimi
nated from the equations formed,is &(m n p 1)(on n+p),
or,in general
,less than the number of equations. But again
,
if we consider the identity a t:= eu,which is of the degree m n
,
and multiply it by the several terms at“,&c.
,we get Mp 1)10
identical relations between the system of equations we haveformed and in like manner &(n 1)n Hm 1)m other identities ; and the number of identities subtracted fromthe numberof equations leaves exactly the number of variables to be elimi
nated,and gives the eliminant in the right degree.
91. If we had four equations in four variables,we should pro
ceed in like manner,and it would be found then that the case
would arise Of our hav ing m+ n linear equations in m variables,
these equations not being independent,but connected by n+p
relations ; these latter relations aga in not being independent,but connected by p other relations. And in order to find the
reduced eliminant of such a system,we should diwde the deter
minant of any m of the equations by a quantity which is itselfthe quotient Of two determinants. I think it needless to go intofurther details
,but I thought it necessary to explain so much of
the theory, the above being,as far as. I know
,the only general
theory of the expression Of eliminants as determinants ; sincewhenever
,in the application of the dialytic method
,any of the
equations is multiplied by terms exceeding its own degree,we
shall be sure to have a number of equations greater than the
number of quantities which we want to eliminate.
( 75 )
LESSON X.
DETERMINATION OF COMMON ROOTS.
92. WHEN the eliminant of any number of equationsvanishes, these equations can be satisfied by a common systemof values, and we purpose in this Lesson to show how thatsystem of values can be found without actually solving theequations. The method is the same whatever he the numberOf the variables ; but for greater simplicity we commence withthe system of two equations, gb O
,«In: 0
,where
c]: cc
1p= hna“
bflw"
Let us suppose that some root of the second equation,w= a
satisfies the first,and therefore thatR the eliminant of the system
vanishes. Now in (,bwemay alter the coefficients (“m into am+A
into aw
AW “&c. and the transformed equation
ammm+ a
m_ M + &c =
will obv iously still be satisfied by the value provided onlythat the increments Am, &c. are connected by the singlerelation
Ama
’”A &c.
.
0,
since the remaining part of the equation, by hypothesis, vanishesfor x= a. The transformed equation then has a root commonwith «p, and therefore the eliminant between 3b and that transformed equation v anishes. But this eliminant is obtained fromR,the eliminant of 47 and by altering in it a Into a
m+Am, &c.
The eliminant so transformed is
dR
da da771 ill—1
We have B 0 by hypothesis ; and since the increments A'
&c.
may be as small as we please, the terms conta ining the firstpowers of these increments must vanish separately. We have
&c. &c. O.
76 DETERMINATION OF COMMON ROOTS.
then A(if A
diR
&c. 0. This relation must be iden
tical with the relation Amc‘Q—i-A &c. 0
,which we have
seen is the only relation that the increments need satisfy. Itfollows then that the several differential coefficients are pro
’
portional to am
,of
“,&c.
,and therefore that a can be found
by taking the quotient of any two consecutive differential coefficients.
COR. 1. If a,be any two coefficients in (I), we must have,
dR dR dR dBwhen R= O
, da,daM da
,
s1nce the quotIent either
of the first by the second, or of the third by the fourth will 01"
It follows dR dB dRvanishes when R= 0
,and
da,
therefore must contain R as a factor ; or,in other words
,
dR dB (113 dB
da,da
,da
,da
,
COR. 2. It is ev ident,by parity of reasoning
,that the difi
'
e
rential coefficients of the eliminant,with regard to the several
coefficients in are proportional to a”
,a“
,&c. ; and hence, as
dR dR dB dR
d“?da
y-tTb
sdb
a-k
’
contains R as a factor if we havep“
4 g r s.
in the last corollary, that, when R 0
dB dB dR dBor that
da,db
,da
,db,
contaIns B as a factor when we have
p + g= r + a
COR. 3 . Or,aga in
,if we substitute in the second equation
the values of a"
of" 1&c. iven above we have7 7g
7
dB dB
da da&c. 0
’
when R = 0. But the left-hand side Of this equation cannotconta in It as a factor
,for it obv iously contains the coefficients
of d: in a degree less by one than that in which R containsthem. It must therefore vanish identically.
93 . The results of the preceding article may be confirmedby calculating the actual values of the differential coefficientsof R. We know (Art. 63)that R : 4) (01)gt d) (y)&c. But
78 DETERMINATION OF COMMON ROOTS.
are a degree lower than the d iesel-minim in the coefficients ofeach equation ; whereas the values found by differentiating the
Q—d-isem i-nant are a degree lower than it only In the coefficientsOf one of the equations. For example , the common value
which satisfies the pa ir Of equations
b I
(ail Eli—c
w) whereas by the
preceding method it is given in the less simple form
is by this method found to be
a’
(bc'
) c’
(ab'
) 2a'
(ao'
) b'
(ah'
)
All these values are equal in v irtue of the relation which issupposed to be satisfied (ab
'
)
95 . If we substitute in any of the equations used in the lastdR
artICle,the values for a: &c.
,thi s equatIOn must beda
m_I
satisfied when R : 0,and therefore the result of substitution
must be divisible by R. In other words,if a
" ,a, ,&c. be the
constituents of any of the lines of the determinant of Art. 80,
we must have af l d
dR
dill)
+ &c. div isible by R. But if
we examine what a &c. are,we see that 01 Is the determinant
(ambM ), &c., and thus that the function a, 1
contains
the b coefficients In a degree one higher than R while its weightexceeds that of R by n — r + 1. Consequently the remainingfactor must be b
, ” Hmultiplied by a numerical coefficient. TO
determine this coefficient we suppose all the terms of \]r to
vanish except bnfi fl
. Now it follows at once from the method ofelimination by symmetric functions that if a)» consist of factorsV,17V
,&c., the eliminant of 4) and 111‘ is the product of the
eliminants of (15, V; W; &c. For if V be (w— a)(w &c.,
and W b e (x — B'
)&c., the eliminant of (I) and V is¢(a)(NB)&c., that of <1) and W is &c.
,and the
product Of all these is the eliminant of <1) and 11AAgain
,if‘f’ reduce to the single term b
ax
‘yfl
,since the elimi
nant of and a: is aoand of 47 and y is am, the eliminant of
DETERMINATION OF COMMON ROOTS. 79
c]: and 1? will be The only one then of the series of
termsdiR
&c., which will not vanish when all the coefficients
of except b“ are made to vanish,will be 45
,and this will
dao
be But in the case we are considering,it will
777-1
be found that the term by which if0
and hence that in general, when a n r 1,
01dB
rI dam-1
Ex. In order to make what has been said more intelli gible, we repeat the prooffor the particular case of the two cubics (134? 0123
2 a le: ac, bzx’ baa
” be,thenwe have the system of equations (Art. 80)
(asbz)“32 (Gabi) a: (4 360) 07
(Gabi)x2
2 (0 260) 07
(Gabe)“32 (“2b0) w (“Ibo 0.
Substituting then, suppose in the second equation ; the following quantity must bedivisible by R,
is multiplied will be baao,
R
da01—2
dR dR dR(d ebt)5 ;
{ (asbo) (a2b1)} dd ;(0250)E
.
But, considering the order and weight of the function in question, it is seen at once
that.the remaining factor must be 62 multiplied by a numerical coefficient. To deter
mine that coeficient, let b, , 6, all vanish, then the quantity we are discussingR
reduces to be (a,g? a0371-0) But R,on the same supposIti on, reduces to
and therefore the functionwe are calcul ating at most differs in sign from 26213.
96. There is no difficulty in applying the method of
Arts. 92,93 to the case of any number Of variables. For
greater clearness we confine ourselves to three variables,but
the same proof applies word for word to any number of variables. Let there be three equations 0
,xlr = 0, x= 0, where
&c.,and let the values w’
y’
z'
satisfy all the equations ; then they will still satisfy them if
in 47we alter “mom .
yinto a
n ,“Ammo, a a ’fi,
oy A g
’g’y,&c.
prov ided only that Ammw'm
-l &c.
But,as in Art. 92, the equation must also be satisfied
dBA
clR
”314 We, 7 971405 13 , 7m’ 0, 0
and comparing these two equations,we see that the value of
80 DETERMINATION OF COMMON ROOTs.
each term x'a
y’Bz
'7 must be proportional to the differential of
the eliminant with respect to the coefficient which multiplies it.We obta in the values of w
'
, y'
,z'
,by taking the ratios of the
differentials of ,R with respect to the coefficients of any termswhich are in the ratio of ac
, y, z . And this may be verifiedas in Art. 93 . For let the common roots of X substitutedin (I), give results &c. Then R &c. And
dRwlx
ytpz
17
¢u
¢m&c. w
rrd
yu
s W¢
f
¢u
&c. &c.da d
’s ,
and if we suppose to v anish,the value of this differential
coefficient reduces to its first term,and it is seen as before
,that
the differential with regard to each coefficient is proportionalto the term which that coefficient multiplies. The same corollaries may be drawn as in Art. 92 .
97. And more generally, in like manner,if the coeffi
cients of 4) be functions of any quantities a , b, c,&c. which
do not enter into «it , x, it is proved by the same method,that
(ZR dB dd) C} ? where in the latter differentials as, y, z , are
supposed to have the values x'
, y'
,a'
,which satisfy all the
equations. For either,as in Art. 93
,we have when
d 'R d¢ H " IdB d¢ H ”I
0
da £75 4) <1) &c.
5 4: d) &c.,or agam
,as 111 Art. 92
if a,b,c be varied
,so that the same system of values continues
to satisfy 96, we have
dcb d¢74586”
while,because in this case
,the eliminant of the transformed d)
and of the other equations continues to vanish,we have
dR dR dR
dd3a +
fiSb ' l-m80
and these two equations must be identical.
98. The formulae become more complicated if we take thedifferentials of the eliminant with respect to quantities a , b, &c.which enter into all the equations. As before
,if we give these
DETERMINATION OF COMMON ROOTS. 81
quantities variations, consistent with the supposition that theeliminant still vanishes
,we have
dB dR dR
da8a +
dbSb+
dc
Now in the former case,where a
,b,c,&c. only entered into
one of the equations,a change in these quantities produced no
change in the value of the common roots,since the coefficients
remained constant in the other equations,whose system of
common roots was therefore fixed and determinate. But thiswill now no longer be the case
,and the common -roots of
the transformed equations may be different from those of the
original system. Let the new system of common roots be
y'
+ Sy’
,z
’
+ Se'
, &c.,then the variations are connected
by the relations
d i”8a +dd’
555 +dac
d
d
¢8a +dd»
8x'
+dd dx dy
If there are is such equations,there will be k 1 independent
variables we may therefore,between these lo relations climi
nate the k— l variations 5g'
,&c.
,and so arrive at a re
lation between the variations 8a,86
,&c. only ; the coefficients
R dR
dd db
Ex. 1. Let there be two equations and one variable. The final relation then is
(d4’ d‘” d‘”d—t”)6aa + (
”l—‘l’ d‘p d‘”
ab + &c. z o,
of which must be severally proportional to
da da: da da: db da: db dz
dR dRand the several coefficients are proportlonal to dd
&c. If the equations hadbeen given in the homogeneous form, we might have taken a: as constant, and sub
stituteddd) W,
for _d_? dip
in the preceding formula. This makes no change,dy
’dy dz dz
because it was proved, Art. 85, that the common root satisfies the Jacobian, ormakesdd) dlll d4>
dx dy dy dz
If the equations had been given as homogeneous functions of la variables, still
since their ratio is all '
we are concerned with, we may suppose any of the variables zto be the same in all the equations and may suppose da' 0.
I
82 DETERMINATION OF COMMON RooTs.
Ex. 2. If there are three equations, the coefficient of ou isdef) dx
da’da
’da
¢l 7 ‘I’ I: XI
¢2J ‘f’ zr X2
where (p l , ¢2 denote the differential coefficients of «p with respect to a: and y, &c.
99. If a system of equations is satisfied ‘
by two commonsystems of
‘
values,not only will the eliminant R vanish
,but
also the differential of R with respect ' to every coefficient ineither equation. For evidently the values of the differentials
,
given Art. 93,all vanish if both (Ma)and or
,in
Ar t. 96,if cp
'
,both = 0. In this case the actual values of
the two common roots can be expressed by a quadratic equat ion in terms of the second differentials Of R. The following
,
though for brevity, stated only for the case of two equations,
applies word for word in general. We have (see Art. 93)
j“, aw e M8 &c. p ”¢(a)c» (8)&c.
which,when (a), d) (B) 0
,reduces to the single term
a”fi”¢(v)d) (3)&c
In like manner,in the same case
,
d2R
43W) &0 MS)&0
If then we solve the quadratic ipA ,u.,
d‘R d
eR
dag
z ,u.da
yd”
?
the roots will give the ratios a”: a
If the equations have three common systems of values, allthe second differentials of R vanish
,and the common
l
roots are
found by proceeding to the third , differential coefficients andsolving a cubic equation.
LESSON X I.
DISCRIMINANTS.
100. BEFORE entering on the subject of discriminants,we
shall explain some terms and symbols which we shall frequentlyfind it convenient to employ. In ordinary algebra we are whollyconcerned with equa tions , the object usually being to find the
values of x which will make a given function = 0. In whatfollows we have little to do with equations
,the most frequent
subject of investigation being that on which we enter in the next
Lesson : namely, the discovery of those properties of a functionwhich are unaltered by linear transformations. It is convenient
,
then,to have a word to denote the function itself
,without being
obliged to speak of the equation got by putting the function = Oa word
,for , example , to denote without being
obliged to speak of the quadratic equationWe shall, after Mr. Cayley, use the term quantic to denote a
homogeneous function in general ; using the words quadriccubic
,quartic
,quintic
,to denote quantics of the 2nd
,3rd
,
4th,5th
,of
“degrees. And we distinguish quantics into binary,
ternary, quaternary, n-ary, according as they contain 2,3,4,
72 variables. Thus,by a binary cubic, we mean a function
such as aw3
+ bx2
y+ cwy2
+ dy3
; by a ternary quadric, such a s
aw”
by2
oz”+
~
2fyz 2gz x 2hwy, &c. Mr. Cayley usesthe abbrev iation (a , b, 0
, fi x, y)
3to denote the quantic
aw3
+ 3bx2
y + 3cwy + dy9
,in which
,as is usually most con
venient,the terms are affected
‘
with the same numerical coefficients as in the expansion of So the ternary quadricwritten above would be expressed (a , b, c
, f , 9 , 11133, y, z)2.
When the terms are not thus affected with numerical coeffi
cients,he puts an arrow-head on the parenthesis
,writing
,for
instance (a , b, c,di m, y)
“to denote aw
3
+ bx2
y + cxy2
+ dy3.
When it is not necessary to mention the coefficients, the quanticof the 92 degree i s written (on, (w, y, &c.
84 DISCRIMINANTS.
101. If a quantic in k variables be differentiated with respectto each of the variables
,the eliminant of the k differentials is
called the discriminant of the given quantic.If n be the degree of the quantic, i ts discriminant is a homo
geneous junction of i ts coeflicients , and of the order lz (uFor the discriminant is the eliminant of k equations of the
degree,and (Art. 74)must contain the coefficients of
each of these equations in a degree equal to the product of thedegrees of all the rest
,that is (n And since each of
these equations conta ins the coefficients of the original quanticin the first degree
,the discriminant contains them in the
k (n degree. Thus,then
,the discriminant of a binary
quantic is of the degree 2 (n 1) of a ternary, is of the degree3 (n &c.
102. If in the origina l quantic every coefioient multiplying the
first p ower of one of the variables x,be ejected with a sufiw 1,
every term multiplying the second p ower by a sufix 2,and so on
then the sum of the sufixes in each term of the discriminant isconstant and = n (n It was proved (Art. 74) that ifevery coefficient in a system of equations were affected with a
suffix corresponding to the power of as which it multiplies, thenthe sum of the suffixes in every term of their eliminant would beequal to the product of the degrees of those equations
,viz .
,
= mnp &c. Now suppose,that in the first of these equations
the suffix of instead of being 0,was l ; that of m
lwas l+ 1,
and so on ; it is ev ident that the effect would be to increase thesum of the suffixes by l for every coefficient of the first equationwhich enters into the eliminant ; and since (Art. 74)every termcontains up &c. coefficients of the first equation ; the total sumof suffixes is mnp &c. + lnp &c.
: (m + l)np 810. Now,in the
present example,it is evident that every coefficient in the h— l
differentials [72 ,U,” multiplies the same power of a: as
it did in the original quantic U. But in the remaining differential, every coefficient multiplies a power of at one lessthan in U
,and the coefficient multiplying any term in this
We write , as before, U” U2, U3 , &c. to denote the di fferential coefficients of Uwith respect to z , y, z , &c.
86 DISCRIMINANTS.
105. To exp ress the discriminant in terms qf the va luesany] , w2y2, &c., which make the quantic vanish.
Let U (my. ex.)(we.w e.)(xys aw.)&0 ~ (see Art
then
U. y.(ma —
yr .)(ivy. an.)8 m a te .waxed —ex.)&0 8m;
and the result of the substitution in Ul of any root 9 o 1of U is
y‘(w1y2 —y1x2)(wlys yla's)&c. Similarly, the result of sub
stituting aszy2 is (e gyl (my, yzws)&c. If
,then, all the
results of substitution are multiplied together,the product is
i l/Iyzys 850 “ (x i-ya y1x2)2
(x 1y3 _ y1x8)2
(x zys ‘
ysxay&c°
This, then, is the eliminant of U and U1 ,and if we divide it
by ac,which is = yly2y3 &c.
,we shall have the discriminant
(say, ylw2)2
(x lya y1w3)2
&c. If we make in it all the y’s = 1
,
we get the theorem in the well-known form that the discriminantis equal to the product of the squares of all the differences ofany two roots of the equation. We shall
,for simplicity, refer
to the theorem in the latter form.
106. ~ The discr iminant of thep roduct qf two quantics is equa lto the p roduct of their discriminants multip lied by the square oftheir eliminant. For the product of the squares of differences ofall the roots evidently consists of the product of the squares ofdifferences of two roots both belonging to the same quantic
,
multiplied by the square of the product of all differences betweena root of one and a root of the other
,and this latter product is
the eliminant (Art. A s a particular case of this,the dis
criminant of (w a)4) (w)is the discriminant of 4; (w)multipliedby the square of <1) (a). For if ,3 , 7, &c. be the roots of 4) (m),then ' (a — B)
”
(ac — 0y)2&c. is equal to the square of
(a (a — fy)&c. which is (f) (a), multiplied by the product ofthe squares of all differences not conta ining a.
107. The discriminant of (ac, ani on, y)
”is of the
form and) where «if is the discriminant of the equa tion of
the (n degree (no, am i ss
, y)’H
. For we evidently
get the same result whether we put any term a,
O in the
discriminant ; or whether we put an= O in the quantic, and thenform the discriminant. But if we make a
nO in the quantic, we
DISCRIMINANTS. 87
get a: multiplied by the (n written above,and (Art. 106)
its discriminant will then be the discriminant of that (n— l)ic
multiplied by the square of the result of making in it m= o ;that is
,by the square of a
m ]. In like manner we see that the
discriminant is of the form ao¢> afwlrfi
"
108. The discriminant being ' a function of the determi
nants xlya w
,y, , &c. must satisfy the two differential equations (Art.
dA dA dA
d a,
dA dA dAa, 33
0 a 825;&C._
or,if the original equation had been written with binomial
coefficients,
dA dA da
d Ua—
a
“0 33
—
1
Ex . To form the discriminant of (ac, a , , a2 , which we suppose arrangedaccording to the powers of d o. We know (Art . 107) that the absolute term is
a lzD
, where D is the discriminant of (a l , y)’H
. The discriminant then is
&c. 0.
da lzD ao¢ ao
’df &c. ; operating on this with a , E2a, d
ih 8a, dja;
&c.,we
0 I.
may equate separately to z ero the coeflicient of each power of a , . Thus then thepart independent of a, is
dor, remembering that (a, a?! 2as 75
-
2
&c.)D 0, we. have
d d— 4a,,D -l-a , (ca d—a-2 + 2a4 d
-as
and the discriminant isd
(a ,z mag)D a lao (as (Ta &c.)D 211; &c.
2 i n
In like manner, from the coefiicient of a 1 we can determine it, but the result is not
simple enough to seem worth writing down.
This theorem was first published by Joachimsthal I had , however, previouslybeen led by simple geometrical considerations to the following theorem in which itis ind pded. If a , contain .
a factor z , and if ao contain 23 as a. factor, the discriminantwill be divisible by e”. If a2 contain z as a factor, if a , contain z ’
, and a , contain-z“,
the di scriminant will in general be divisible by z“. In like manner, if as'
contain z ;
a z , z” a " z
"; and ac, a
‘, the discriminant will be divisible by z " , &c.
88 DISCRIMINANTS.
109. If the discriminant of a binary quantic vanishes,the
quantic has equal roots,and the actual values of these roots can
be found by a process similar to that employed in Lesson X.
Let U a2w" ' 2+ &c. be a quantic whose discrimi
nant vanishes,and hav ing therefore a square factor (fr
Then ev idently V, where
V= A090 A
l
'
s: Azx &c.
will also be d iv isible by scia,provided that A A &c. be any
quantities satisfying the condition
Aoa"
A1a A
za &c. O.
In this case then we shall have U+ 7xV div isible by w— a.
Let it (as — a){(w— a)(Mac) It follows then,from
A rt. 1_QZ , that the discriminant of U+ 7tV is the discriminantof the quantity within the brackets
,multiplied by the square
of the result of substituting a for a inside the brackets. But
this result is Mir We have proved then that in the casesupposed
,the discriminant of U+ 7t V is div isible by X
".
But since U+ 7tV is derived from U by altering aointd
ao+ M4
O,&c.
,the discriminant of U+ 7\V is derived from the
discriminant of U by a like substitution,and is therefore
dA dA
da0
1 dc?l
By hypothesis A 0. But the discriminant will not be divisibleby X
‘unless the coefficient of 7t vanish. Now the relation thus
obtained between AO,A
1,&c. must be identical with the relation
Ana"
A lan" 1&c. 0
,which we have already seen is the only
relation that need be satisfied by A O,A
1 ,&c. in order that the
discriminant of U K Vmay be divisible by We must havetherefore the quantities a
"
, at"
,a“
,&c. respectively propor
tional to 330
, 22, 32, &c. Div iding any one of these terms
by that consecutive , we get an expression for a . We may statethis result : When the discriminant vanishes
,the severa l dif e
rentia l coeg'
z’
oients of the discriminant with respect to ao,a], due.
a re p rop ortiona l to the difirentia l coeficients of the quantic with
respect to the same quantities.
&c.)
90 mscnmmsnrs.
tiale U, ,U, ; the resulting equations of the (n 2)
mdegree
,when
expressed in terms of the roots are 2 (a B)”
(se 'y)(m 8)2 91 (“J- 7) 2 9901 where 917are the sum
,sum of products in pairs, &c. of all the roots
except a and Bi l“ The discriminant is then,by Bez out
’s
method,expressed as a determinant
,whose constituents are
E (oz Be. (a l”
,2 9. (at &c
2 91 01 2 91i
(a — B)2
7Effie (Of &c.
,
2 92 (a f fi)7 z gigs 01 (a _ B):&c ’)&c'
And when the given equation has two roots equal,the first
minors of this determinant will, by Art. 94,be in the ratio
1,a,a
”
,&c. A somewhat simpler series, possessing the same
PrOPerty, i? 2 (B '7 (3 3 a (fi-v)
”
ffr (3Ea
“ &c.
112. The following proof of the theorem of Art. 109 isapplicable to the case of a quantic in any number of variables.For simplicity, we confine ourselves to the case of two independent variables, the method, which is that of Art. 98, beingequally applicable in general. Let the coefficients in U befunctions of any quantities a , b, &c.
,and let variations be given
to these quantities consistent with the supposition that the discriminant st ill vanishes, and therefore such that
dA— 8a +
db
ASb -l-dtc.
And if the effect of this change in a,b,&c. is to alter the
singular roots from to, 31 into w+ 8x , y+ 8g since these new
values satisfy U1 , U, , U, , &c.,we must have
dUHSa + —
U—U8y=
0
0,
dU,
dU dU dU
da8a +
d_
h“Sb-Hirec + ' 8w+
The first of these functions of degree n 2 is one of the series to which weare led by Sturm
’s process ; but any of the others may be substituted for it, and will
give rise to another series possessing the same property.
DISCRIMINANTS. 91.
Multiply these equations by w, y, z respectively and add ;
then since nU= wU,+ yU + z U
,the coefficient of 80. will be
91(Ll—I and since
dU-d v
’dU d v‘
da ’dc; dy doc dz ’
will be (n 1)U, , which will vanish, since U, i s satisfied for thesingular roots. We get therefore
dU dU
dd8aa .+ -
d—
686+ &C = 0,
and therefore the differentials of A with respect to a,b,&c. are
proportional to the differentials of U with respect to the samequantities
,it being understood that the a
'
, y, z which occur inthe latter differentials are the singular roots.
,the coefficient of 8x
113 . The theorem proved for binary quantics (Art. 107 maybe extended to quantics in general. Let a be the coefficientof the highest power of any of the variables
,6,c,d,&c. those
of the terms involving the next highest power ; then the dis“
criminant is of the form
“9 (ff), X7 ‘f’ 3 &C 'Iba 07 d)&c.)2
Thus,for a ternary quantic to which , for greater simplicity,
we confine ourselves,if a be the coefficient of z
"
; I), 0 those of
s,z
”"
y, then if in the discriminant we make a = o,the re
ma ining part will be of the form b2
¢+ bcxlr + 02
x. To provethis : first
,let U be any quantic whose discriminant vanishes,
V any other satisfied by the singular roots of U,then I
say that the discriminant of U+ 7t V will be div isible byFor let U a z
”
bz"“w &c.
,V= A z
"
B z"“w then
the coefficient of 71. in the discriminant of U will be
A6
3, and (Art. 112) dz
; &c. are proportional
to z"
an“
x,&c. The coefficient of 7x is therefore proportional
to the result of substituting the singular roots inV,and there
fore‘vanishes.Now in the case we are considering
,the supposition of
a = 0,
a= o must make the discriminant vanish,since
then all the differentials vanish for the singular roots
y= 0. Any other quantic V will vanish for the same values,
prov ided only A = O. The general form of the discriminant
92 LINEAR TRANSFORMATIONS.
then must be such that if we substitute for b, b+ 7\B ; for c,
&c.,and then make a
,b,c = 0
,the result must be
divisible by or,in other words, if we put for 6, RB ; for c,
&c.,and then make a = 0
,the result is divisible by a
“,
which was the thing to be.
proved.
114 . Concerning discrim inants in general, it only remainsto notice that the discriminant of a quadratic function in anynumber of v ariables is immediately expressed as a symmetricaldeterminant. And conversely, from any symmetrical determinant
,we may form a quadratic function which shall have
that determinant for its discriminant. The simplest notationfor the coefficients of a quadratic function is to use a doublesuffix
,writing the coefficients of w
”
, y“
,&c.
,as”,&c.
,
and those of my, acz , &c.,am, am ; a ,2 and a
, 1being identical in
this notation. The discriminant is then obviously the sym
metrical determinant
a117
“127 137
“an
ass) 287
LESSON X II.
LINEAR TRANSFORMATIONS.
115 . Invariants. The discriminant of a binary quanticbeing a function of the differences of the roots is ev identlyunaltered when all the roots are increased or diminished bythe same quantity. Now the substitution of w+ 7\. for w is a
particular case of the general linear transforma tion, where, , in
a homogeneous function, we substitute for each variable a linear
function of the variables ; as for example, in the case of a
binary quantic where we substitute for x,M:+ ,
uy, and for
y, It will‘ illustrate the nature of the enquiries inwhich we shall presently engage if we examine the effect of
94 LINEAR TRANSFORMATIONS.
theorem is true for the discriminant . of a quantic in any numberof variables.What I have called Modern Algebra may be said to have
taken its origin from a paper in the Cambridge Ma thema tical
Journa l,for Nov . 1841
,where Dr. Boole established the prin
ciples just stated and made some important applications of
them . Subsequently Mr. Cayley proposed to himself the problem to determine a p riori what functions of the coefficientsof a given equation possess this property of invariance , viz .
that when the equation i s linearly transformed, the same function of the new coefficients is equal to the given functionmultiplied by a quantity independent Of the coefficients. The
result of his investigations was to discover that this propertyof invariance is not peculiar to discriminants and to bringto light other important functions
, (some Of them involvingthe variables as well as the coefficients)whose relations tothe given equation are unaffected by linear transformation.
In expla ining this theory, even where,for brevity, we write
only three variables,the reader is to understand that the
processes are all applicable in exactly the same wav to any
number of variables.
117. We suppose then that the variables in any homo
geneous quantic in k variables are transformed by the sub
stitution
y= X2X + u2Y+ V2Z + &c.,
z X3X + a, Y+ V
SZ + &c., &c.
,
and we denote by A the modulus of transforma tion ; namely,the determinant
,whose constituents are the coefficients of
transformation, p.” v
, ,&c.
, v2,&c.”Ate.
Now it is evidently not possible in general so to choose thecoefficients a
, , p , ,&c.
,that a certa in given function
shall assume , by transformation,another given form a
’
X”&c.
In fact,if we make the substitution in aw
n
+ &c., and thenequate coefficients, we Obta in
,as in Art. 115
,a series of equa
tions a '
the number Of which will be equal to thenumber of terms in the general function of the n
thdegree in
LINEAR TRANSFORMATIONS. 95
71: variables. And to satisfy these equations we have only at
our disposal the la“ constants &c.,a number which will
in general be less than the number of equations to be satisfied.*
It follows then that when a function is capable of
being transformed into there will be relations connecting the coefficients a
,b,&c.
,a'
,b'
,&c. In fact we have
only to eliminate the k”
constants from any 78 4-1 of the
equations and we obta in a series of relationsconnecting a
,a
’
,&c.
,which will be equivalent to as many
independent relations as the excess over k“ of the number of
equations. Thus in the case of a binary quantic, the numberof terms in a homogeneous function of the n
tndegree is n+ 1.
If then,in any quantic aw
"
+&c., we substitute for w
,x,x+ ,
u,Y,
and for y, 71.,X agY, and if we then equate coefficients withwe have n + 1 equations connecting a
,a'
,7g, &c.,
from which,if we eliminate the four. quantities a
n ,u, ,we
get a system equivalent to n — 3 independent relations betweena,b,a'
,b'
,&c. It will appear in the sequel that these relations
can be thrown into the form 4) (a , b, &c.)= <I> (a'
,b
’
, or,
in other words,that there are functions of the coefficients
a,b,&c. which are equal to the same functions of the trans
formed coefficients. The process indicated in this article isnot that which we shall actually employ in order to find suchfunctions
,but it gives an apriori explana tion of the ex istence
of such functions, and it shows what .number of such functions,independent of each other, we may expect to find.
118. Any function of the coefficients of a quantic is calledan invariant, if when the quantic is linearly transformed, thesame function of the new coefficients is equal to the old function
The number of terms in the general equation of the a“ degree homogeneous
1)cases where this number is not greater than are first ; when n 2
, when it becomesi h (k a number necessarily less than B ,
Is being an integer ; and secondly, the
case k 2, n 3 , when both numbers have the same value 4. That is to say, the
only cases where a given function can be made by transformation to assume any,
assigned form,are first, the case of a quadratic function in any number of variables ;
and secondly, the case of a cubic function homogeneous in two variables.
96 LINEAR TRANSFORMATIONS.
multiplied by some power of the modulus of transformation ;that is to say, when we have
(1)(a'
,b'
,c'
,&c.) AP
<I> (a , b, c,
Such a function is said to be an absolute invar iant when
that is to say, when the function is absolutely unaltered by
transformation even though A be not 1. If a quantic havetwo
“
ordinary invariants, it is easy to deduce from them an
absolute invariant. For if it have an invariant qb, which when
transformed becomes multiplied by A”,and another which
when transformed becomes multiplied by A"
,then ev idently the
gm power of (1) div ided by the p
m power Of qr will be a functionwhich will be absolutely unchanged by transformation.
It follows,from what ’
has been just said,that a binary
quadratic or cubic can have no invariant but the discrimi
nant,which we saw (Art. 116)is an invariant. For if there
were a second,we could
'
from the two deduce a relation
¢(a , b, b'
,But we see from Art. 117 that
there can be no relation connecting a,b,&c. with a
'
,b'
,&c.
,
since,with the help of t he four constants 7g, &e.
‘
at our dis
posal,we can transform a given quadratic or cubic
,so that the
coefficients of the transformed equation may have any valueswe please. In the same manner we see that a quantic of thesecond order in any number of variables can have no invariantbut the discriminant.
119 . In the same manner as we have invariants of a singlequantic we may have invariants Of a system of quantics. Let
there be any number Of simultaneous equationsa'
os"
+ &c. = 0,&c.
,and if when the variables in all are trans
formed by the same substitution,these become AX "
+
&c.,then any function Of the coefficients is an
invariant if the same function of the new coefficients is equalto the old function multiplied by a power of the modulus oftransformation ; that is to say, if
<1) (A ,B,A
'
,B
'
,A &c.) AM; (a , b, a
'
,b'
,a
The simplest example of such invariants is the case of a
system of linear equations. The determinant of such a system
is an invariant of the system. This is ev ident at once from
98 LINEAR TRANSFORMATIONS.
a'
g; + b'
we get an invariant of the system of two
quantics ant a'
az”
+ &c. We may repeat the same operation and thus get another invariant of the system,
or we may
operate with a"
3
6—2 1) 52+ &c., and thus get an invariant of
a system of three quantics ; and so on. This latter processgives us the invariants which we should find by substitutingfor a
,a + ka
'
+ la"
,&c., and taking the coefficients of the pro
ducts Of-every power Of k and Z. In the same manner we get
invariants of a system of any number of quantics.
120. Cova riants. A covariant is a function involving not
only the coefficients of a quantic,but also the variables
,and
such that when the quantic is linearly transformed,the same
function of the new variables and coeficients shall be equalto the old function multiplied by some power of the modulusof transformation ; that is to say, if aw
”&c. when transformed
becomes a function e will be a covariant* if it issuch that
d) (A ,B,&c.
,X,Y,&c.) A
pr/1 (a , b, &c., a
'
, y,
Every invariant of a covariant is an'
invariant of the original
quantic. This follows at once from the definitions. Let the
quantic be and the covariant which are
supposed to become by transformationNow an invariant of the covariant is a '
function of its coefficientssuch that
it (A'
,B
'
,&c.) A
’
d) (a'
b
But A'
,B
'
,&c. by definition can only differ by a power of
the modulus from being the same functions of A,B,&c. that
a'
,b'
,&c. are of a
,b,&c. Hence when the functions are both
In the geometry of curves and surfaces, all transformations of coordinates are
efiected by linear sub stitutions. An invariant of a ternary or quaternary quantic isa function of the coefficients, whose vanishing expresses some property of the curve
or surface independent of the axes to whi ch it is retened, as, for'
instance, that thecurve or surface should have a double point. A covariant will denote another curveor surface, the locus of a point whose relation to the given curve is independent ofthe choice of axes . Hence the geometrical importance of the theory of invariants
and covariants.
LINEAR TRANSFORMATIONS. 99
expressed in terms of the coefficients Of the original quantic andits transformed
,we have
‘f’ (A )B,&c.) Aq
1k(a , byor the function is an invariant. Similarly, a covariant of a
covariant is a covariant of the original quantic.
121. We shall in this and the next article establish principles which lead to an important series of covariants.If in any quantic u we substitute x km
’
for x, y ky
'
for y,&c.
,where w
’
y'
z'
are cogredient to xyz , then the coeffi
cients Of the several powers of k,which are all of the form
(w’
dill:+ y
'
g; u,have been called the first
,second
,third
,
&c. enzanants* ~
0f the quantic. Now ea ch of these emanants is
a cova riant of the quantic. We ev idently get the same resultwhether in any quantic we write x + hx
'
for m,&c.
,and then
transform as,w
’
,&c. by linear substitutions
,or whether we make
the substitutions first and then wri te X hX'
for X,&c. For
plainlyA,X + M.Y+ v
,Z + k (A,X
'
,u.,I”
v,Z
’
)— 7t
, (15 +
If then u becomes by transformation U, we have proved thatthe result of writing 90 + km
'
for x,&c. in u
,must be the same
as the result of writing X + c'
for X,&c. in U
,and since It
is indeterminate,the coefficients of It must be equal on both
sides Of the equation ; or
da dux'
&c.= X &c. &c. .E.D.
122. If we regard any emanant as a function of cc'
, y'
,(fee.
trea ting x, y, &c. as constants ; then any of its invariants wi ll be
a covariant of the origina l quantic when x, y, &c. are considered
as variables.
We have just seen that x &c. becomes X
when we substitute for a'
,7»,X
'
a,Y &c.
,and for x
,
In geometry emanants denote the polar curves or surfaces of a point with regardto a curve or surface.
100 LINEAR TRANSFORMATIONS.
AX p , Y+ &c. It is evidently a matter Of indifferencewhether the substitutions for m
’
,&c. and for ac
, &c. are
simultaneous or successive. If then on transforming a"
,&c.
alone,a: —
u
becomes aX"+ &c., then a
,&c. will be
dx9
such functions of as,&c. as when x
, y, &c. are transformed willd"U
becomep ,
&c. Now an Invari ant of the gi ven emanant
considered as a function only of w’
, y'
,&c. is by definition such
a function of its coefficients as differs only by a power of the
modulus from the corresponding function of the transforrfied
coefficients a b &c. But since,as we have seen
,a, &c. become
ZXUF ’&c. when as
,&c. are transformed
,it follows that the given
invariant will be a function of &c.,which when x
,&c. are
dpa
dxp 7
transformed will differ only by a power of the modulus from
the corresponding function of gXU,&c. It is therefore by
definition a covariant of the quantic.Thus then
,for example
,since we have proved (Art. 115)
that if the binary quantic -i-ogz becomes by trans
formation AX22EXY 0Y2
,then
A O— B"= A
2
(ao — b2
);
it follows now,by considering the second emanant (a
'
$+yi
u
of a quantic Of any degree, that
an] an]
(era
)(d
zu d u
(d
’a
dX”dY”
dXdY ill—
x dy'
dmd' “
y
a theorem of.which other demonstrations will be given.
123 . In general, if we take the second emanant of“
a quanticin any number of variables
,and form its discriminant, this will
be a covariant which is . called the Hessian of the quantic. It
was noticed (Art. 114)that the discriminant of every quadraticfunction may be written as a determinant. Thus then if, as
we have done elsewhere,we use the suffixes l , 2 , &c. to de
note differentiation with respect to x, y, &c., so that, for
102 LINEAR TRANSFORMATIONS.
terms of the old. Stated thus,it is evident that the relation
between the two substitutions is reciprocal. Solving for f, gin terms of E
,H,Z,we get (Art. 28)
AC: LSE MH N
SZ,
where L1 ,M,
&c. are the minors Obta ined by striking out fromthe matrix of the determinant Na z i/8 (the modulus of transformation)the line and column conta ining F’
n &c.
Sets of variables ac, y, z ; E, v, supposed to be transformed
according to the different rules here explained,are said to be
contragredz’
ent to each other. In what follows,variables sup
posed to be contragredient to x, y, z are denoted by Greek
letters,the letters a
,B, 7 being usually employed in subsequent
lessons. We proceed to explain two of the most importantcases in which the inverse substitution is employed.
126. When a function of x, y, z
,&c. is transformed by
linear substitutions to a function of X,Y,Z,&c.
,then the
difi'
erential coefficients, with respect to the new variables,are
linear functions of those with respect to the old,but are ex
pressed in terms of them by the inverse substitution. We have
d’
d dx+d dy
+ édz
dX dx dX dy dX dz dX
But from the expressions for w, y, &c. in terms of X,
Y,&c.
,
we haveOla: dy dz
dX”
fl dx M
d d d dHence thendX 5
A, 35
+ A8(Te
&c.
d d d dSimi larly dY
=,u1 a ;
a, 25p , 32
&c.,&c.
Thus then,according to the definition given in the last article,
d d d
dzc ’ dy’ dz ’
ac, y, z , &c. ; that is to say, when the latter are linearly transformed, the former will be linearly transformed also
,but accord
ing to a different rule, viz . the rule explained in the last article.
the operating symbols &c. are confiragredz’
ent to
LINEAR TRANSFORMATIONS. 103
If,as before
,a”a
s,&c. denote the differential coefficients of u
,
and U] ,
&c. those of the transformed function U,we have
just proved that
U, Na
, Mu, Mus, U2 “a“: p ans, &c.
Consequently, if “n a
2,naall vanish
,U1,U8,U; must all vanish
likewise. Now we know that u] ,uz ,aaall vanish together only
when the discriminant of the system vanishes ; if then the dis
criminant of the original system vanishes,we see now that the
discriminant of the transformed system must vanish likewise,
and therefore that the latter conta ins the former as a factor,
as has been already stated (Art.
127. In plane geometry, if m, y, e be the trilinear co-ordinatesof any point, and as? yr, at : 0 be the equat ion of any line,5, v, Cmay be called the tangential co-ordinates of that line
(see (Ionics , Art. Now,if the equation be transformed to
any new system of axes by the substitution w= 7t1X + &c.
,the
new equation of the line becomes
E(A1X + a ,Y+ v
iZ ) 070t ”2
17+ V2Z ) { (NBX+ aaY+ s ),
so that if the new equation of the right line be writtenwe have
ME 7m M,H‘=
a ? m a t, Z 136+m v
s?
In other words,when the co—ordina tes of a p oint are transformed
by a linear substitution, the tangentia l co-ordina tes of a line a re
transformed by the inverse substitution. In like manner,in the
geometry of three dimensions, the tangential co-ordinates of anyplane are contragredient to the co-ordinates of any point. Whenwe transform to new axes
,all co-ordinates myaw, &c.
expressing different points, are cogredient : that is to say, all
must be transformed by the same substitution w= >t1X + &c.
,
&c. But the tangential co-ordinates of everyplane will be transformed by the inverse substitution, which hasbeen just explained.We shall make frequent use of the principle stated in this
article,that
where m, y, z being supposed to be changed by the substitution,
104 LINEAR TRANSFORMATIONS.
x alX + a ,
Y+ &c.,
’g‘
, n, C are supposed to b e changed by theinverse subst itution 5 A
154» az t; MC, &c. In other words,in
the case supposed,avg 3m rat is a function absolutely unaltered
by transformation.
128. If a function am“
+ &c. becomes by transformationAX &c.
,then any function involving the coefficients and those
variables which are supposed to be transformed by the inversesubstitution
,is said to be a contravariant if it is such that it
differs only by a power Of ° tlle modulus from the correspondingfunction of the transformed coefficients and variables : that isto say, if
9b (A ,B,&c.
, E, H ,&c.) N d) (a , b, &c., E, n,
Such functions for instance constantly present themselves ingeometry. If we have an equation expressing the conditionthat a line or plane should have to a given curve or surface arelation independent of the axes to which it is referred ; as, forexample
,the condition that the line or plane Should touch the
curve or surface ; then, when we transform to new axes,it is
obv iously indifferent whether we transform the given relationby substituting for the old coefficients their values in terms of
the new,or whether we
'
derive the condition from the trans
formed equation of the curve by the same rule as that by whichit was originally formed. In this way it is seen that such a con
dition is of such a kind that ¢(a , b, g, &c.)differs only by a
factor from gb (A ,B, E,
129. Besides covariants and contravariants there are also
functions involving both sets of variables,and such as to differ
only by a power of the modulus from the corresponding transformed functions : i .e. such that
q; (A ,X, E
, A’
d:
Mr. Sylvester uses the name concomitant as a general wordto include all functions whose relations to the quantic are nu
altered by linear transformation, and he calls the functions nowunder consideration mixed concomitants . I do not choose tointroduce a name on my own responsibility ; otherwise 1 shouldbe inclined to call them divariants. The simplest function of
106 LINEAR TRANSFORMATIONS.
If we perform the operation 8”
db &c. upon any covariant,0
we obtain a mixed concomitant,for it is proved in the same way
that the result,which will evidehtly be a function involving
variables of both kinds,will be transformed into a function of
similar form.
Ex . 1. We know that ao— b2 is an invariant of aa 2+ 2bmy+ cy2; hence oE’
is a contravariant of the same system.
Ex. 2. Similarly abc 2fgh af‘2 by
2ch”, being the discriminant, and therefore
an invari ant of as:2 by2
oez 2fye 2gz a: 2kccy,
(bC ~ f2)52 + (ca —
92)n2 + (ab { 2 4-2 (ah — af )"K+ 2 (hf — by){ 5+ 2029 En
is a contravariant of the same quantic. Geometrically, as is well known, it expressesthe tangential equation of the conic represented by the given quantic.
Ex. 3 . Given a system of two ternary quadrics as:2 &c.,a'
r2 &c., then since
(bc f 2) &c. is an invariant of the system (Art. 119) operating with 52 die &c.,
we find that(bc
'b
’
c £2 (ca
'c'a 2gg
'
)"2 (ab
’a
’b 2hh
'
)Z2
2 af’
aif)n§ + 2 (kf’
+ k'
f (Ad — b’
y)Z5 2 ch’o'h)En
is a contravariant of the system. We might have equally found this contravariant
dby operating with a
’
db&c. on the contravari ant of the last article. Geometrically,
it expresses the condition that a line should be cut harmonically by two conics.
131 . When the discriminant of a quantic vanishes,it has
a set of singular roots w’
y'
z’
[geometrically the co-ordinates ofthe double point on the curve or surface represented by the
quantic] ; and in this case the first evectant will be a perfectn"1 power of Since we have seen that the
evectant is a function unaltered by transformation, it is sufficientto see what it becomes in any particular case. Now if the
discriminant vanishes,the quantic can be so transformed that
the new coefficients of ai”
,w
’H
y, w'Hz shall vanish ; that is to
say, so that the singular root shall be y= 0, a = o, [geome
trically, so that the po int yz shall be the double point]. Now
it was proved (Art. 113)that the form of the discriminant is
“on afqb a b if bfx
Ev idently then, not only will this vanish when ac,anb1vanish ;
but also its differentials will vanish with respect to everycoefficient except a
c. The evectant then reduces itself to
FORMATION OF INVARIANTS AND OOVARIANTS. 107
if, multiplied by the perfect n
k
power E”
; which is what0
(m'
f+ becomes when y'
and z'= 0
,and w
'= 1. Thus
then, if the discriminant of a ternary quadric vanish, the quadricrepresents two lines : the contravariant
(bc—f’
)E”
(ca g“
)n”&c.
becomes a perfect square and if we identify it withwe get w
’
y’
z'
the co-ordinates of the intersection of the pair of
lines. If a quantic have two sets of singular roots,all the first
differentials of the discriminant vanish,and its second evectant
becomes a perfect n“ power of
z'
t)(w E+e n+ z C),where a: y z are the two sets of singular roots. And
so on.
LESSON X I I I.
FORMATION OF INVARIANTS AND COVARIANTS.
132. HAVING now shown wha t is meant by invariants, &c.we go on to expla in the methods by which such functions canbe formed. Three of these methods will be explained in thisLesson ; and a fourth in the next Lesson.
Symmetric functions. The following method is only appli
cable to binary quantics. Any symmetric function of the difi'
e
renees of the roots is an invariant, provided that each root enters
into the expression the same number of times.* It is evident that
If in the equation the hi ghest power of a: is written with a coeficient an, we
have to divide by that coefl'
icient in order to obtain the expression for the sum, &c.
of the roots ; and all symmetric functions of the roots are fractions containing powersof a. in the denominator. When we say that a symmetric fimction of the roots is
a.power of a, as will clear it of fractions ; or, what comes to the same thing, if weform the symmetric function on the supposition that the coefficient of a“ is 1, thatwe make it homogeneous by multiplyingfiach term by whatever power of a° may
be necessary.
108 FORMATION OF INVARIANTS AND COVARIANTS.
an invariant must be a function Of the differences Of the roots,
since it is to be unaltered when for a: we substituteNow the most general linear transformation is ev idently equi
Mc
Na i f: By this
change the d ifference between any two roots a,8,becomes
our m»)(a 6)(Na p
’
)(N3 M'
)difi
'
erences may, when transformed, differ only by a factor fromits former value
,it is necessary that the denominator should be
the same for every term ; and therefore the function must bea product of differences, in which each '
root occurs the same
number Of times. Thus for a biquadratic,2 (at— B)
”
(fy— S)”is
an invariant,because
,when we transform
,all the terms of
which the sum is made up have the same denominator. But
Ski — B)” is not an invariant
,the denominator for the term
(cc— B) being and for the term (7being (My M)
"
(N8
valent to an alteration of each root a into
In order then that any function Of the
'
133 . Or perhaps the same thing may be more simply stated
by writing the equation in the homogeneous form. We saw
(Art. 116)that if we change a: into Mc+ ,u.y, y into Nw+ ply,
the quantity wlyz
— wzy1 becomes (M4! Mu.)(wlyg my ), and
consequently, that any function of the determinants aclyfi
— sihyt
&c. is an invariant. Now (Art. 57)any function Of the rootsexpressed in the ordinary way, is changed to the homogeneous
form by writing for a , B, &c. E ,
002
, &c., and then multiplyingl 2
by such a power of the product of all the y’s as will clear it
of fractions. If any function of the differences in which all the
roots do not equally occur,be treated in this way , powers of
the y’s will remain after the multiplication
,and the functionwill
not be an invariant. Thus, for a biquadratic 2 (a be
comes Eyaz
y‘,
2
(x ,yz but the function 2 (cc— ,B)(fy—SY,
in the expression for which all the roots occur, becomes2 (wg ,
— wg s)2 which being a function Of the de
terminante only is an invariant.It is proved in like manner, that any symmetric function
formed of differences of roots and differences between w'
and
110 FORMATION OF INVARIANTS AND COVARIANTS.
the reciprocal substitution. Now we have shown (Art. 126)
that the differential symbols di ’622, &c. are so transformed.
We may, therefore, in any contravariant substitute these differential symbols for E, n, &c., and we shall obtain an operatingsymbol unaltered by transformation, and which, therefore, ifapplied either to the quantic itself or to any of its covariants
,
will give a covariant if any of the variables remain after differentiation ; and if not
,an invariant. Similarly, if applied to a
mixed concomitant,it will give either a contravariant or a new
mixed concomitant,according as the variables are or are not
removed by differentiation. Or again,in any contravariant ih
stead of substituting for f, n, &c., 5; 622, &c. and so Obtaining
an Operating symbol, we may substitute where U is
either the quantic itself or any of its covariants,and so Obtain
a new covariant. The relation between the sets of variablesx, y, z , &c., f, n, C, &c. being reciprocal, we may, in like manner,
substitute in any covariant, for ac, y, z , &c., 575 , (in,
(fig,when we get an Operative symbol which when applied to anycontravariant will give either a new contravariant or an in
variant.Thus then
,if we are given any covariant and contravariant,
by substituting in one of them,differential symbols and operating
on the other we obtain a new contravariant or covariant ; whichagain may be combined with one of the two given at first
,so
as to generate another ; and so on.
136. In the case of a binary quantic this method may bestated more simply. The formulae for direct transformationbeing
m= X1X + I’ fi
Y’ y= 7t2X + p 2Y,
those for the reciprocal transformation are (Art. 125)x16 X
27)? H ”i f ”2
77,
whence A5 XzH,An mg NH ,
which may be written
A E) +M. E)
FORMATION OF INVARIANTs AND CovARIANTs. 111
Thus we see that, with the exception of the constant factorA , n and —Eare transformed by exactly the same rules asa: and y ; and it may be said that y and — 5c are contragredientto a: and y. Thus then in binary quantics
,covariants and
contravariants are not essentially distinct, and we have only inany covariant to write n and ‘
g‘for a; and y when we have
a contravariant ; or vice versct. In fact,suppose that by trans
formation any homogeneous function whatever (w, y)becomes<b (X,
Y); the formulae just given show that — f)will
become (H,E), where p is the degree of the function in
a: and y. If then (03, y)is a covariant ; that is to say, a
function which becomes by transformation one differing only bya power Of A from a function of like form inX and Y; evidently
5)will by transformation become one differing only bya power of A from one of like form in E and H ; that is to say,it will be a contravariant. For example
,the contravariant
,
noticed (Art. 130, Ex. cf” by the substitution
just mentioned,becomes the original quantic.
Instead then of saying that the differential symbols are
contragredient to a: and y, we may say that they are cogredi‘entto y and — :r ; and if either in the quantic itself or any Of its
covariants we write6251 , th
i
sfor a: and y, we get a differential
symbol which may be used to generate new covariants in the
manner expla ined in the last article. Or we may substitutedu du
dy’ dx
following examples will sufficiently illustrate this method
for ac and y, and so get a new covariant. The
Ex. 1. To find an invariant of a quadratic, or of a system of two quadratics.Suppose that by transformation as:2 2bxy cy
2 becomes AX2 2EXY CY2, then
since we have seen that Ag,“ A
d—i are transformed by the same rules as z and y,
it follows that the Operative symbolA2 (a (g, 2573, g)becomes by transformation (AIt thenwe operate on the given quadratic itself, we get
(ac be): 4 (A0 B ),
which shows that as b” is an invariant ; or if we operate on a'w’ 2h
'wy o
’
gz and
its transformed, we get2A2 (ao
'ca
’2bb
’
) 2 (A0'
d’
edXdY (1X ?)
1 12 FORMATION OF INVARIANTS AND COVARIANTS.
which shows that ac’ca
’
2bb’is an invari ant. We might also infer that
a (be: cy)2 2b (bx cy)(as: by) c (as: by)2
is a covari ant but this is only the quantic itself multiplied by ac 62.
Ex. 2. Every binary quantic of even degreehas an invariant of the second order inthed d
coeffi cients. We have only to substi tute, as Just explained, Ey ’d_
a:for a: and y, and
operate on the quantic itself. Thus for the quartic (a , b, c, d, el m, we find thatcc 4bd 3c2 is an invariant ; or for the general quantic (ao, ”4323 ,we find that a oa“ .} n (n 1)azam 2 &c. is aninvariant ; where the coefiiciente are those of the binomi al, but the middle term is divided by two.If we apply this method to a quantic of odd degree ; as, for example, if we operate
da d“ d” d32 2 8 fi-i -dy ,
w1th d, 8 , 2 ,
+ 3b, , 2
ad ,
will be found that the result vanishes identically. We thus find however that asystem of two cubics has the invari ant (cd
’c'd) 3 (bc
’b'
c). Or, in general , thata system of two quantics of Odd degree, aoai" &c., box
" &c., has the invariant
(“obs a nbo) n (aql 6Lu-b ) ’l‘n (n 1) -2 “vi-abs) &c.,
which vanishes when the two quantics are identical .
137. When,by the method just explained
,we have found
an invariant of a quantic of any degree, we have immediatelyby the method of Art. 122
,a covariant of any quantic of higher
degree. Thus knowing that a c — b2 is an invariant of a quad
ratic,by forming that invariant of the quadratic emanant of
al gal d zu d2a
as dy" (W
of any quantic above the second degree. In like manner,from
the invariant Of a quartic ae 4c we infer that for everyquantic above the fourth degree
,
d‘a d
‘a
4d‘a d
‘a d
‘a
dab“a’
y4
deasdy ( lad y
8 (de f tly2is a covariant
,&c. In this way we see that a quantic in
general has a series Of covariants,Of the second order in the
coefficients,and Of the orders 2 (n 2 (n 2 (n &c.
in the variables. These covariants may be combined with theoriginal quantic and with each other
,so as to lead to new co
variants or invariants.
2
any quantic, we learn that is a covariant
Ex . 1. A quartic has an invariant of the third orderin the coefficients. We knowthat its Hessian
(as:2 2bxy cg?)(er
z 2dary eyz) (bar;2 c y
or (ac b?)m‘ 2 (cd 60)$ 33] (as 2bd 3c?)mfg? 2 (be cd)my“ (cs d?)y
‘,
114 FORMATION OF INVARIANTs AND CovARIANTs.
Art. 53 that the weight, or sum Of the suffixes in every termi s constant .Again
,the invariant must remain unaltered
,if we change
m into y, and y into x,a linear transformation
,the modulus of
which is - 1 . The effect of this substitution is the same as iffor every coefficient a“ we substitute a
w l. Hence the sum of
a number of suffixes
whence 2 = n6. Q.E.D.
COR. nand 9 cannot both be Odd since their product is aneven number ; or
,a binary quantic of odd degree cannot have
an invariant of odd order.
140. The principle just established enables us to write downimmediately the literal part of any invariant whose order is
given. For the order being given,the weight is given also.
Thus if it were required to form for a quartic an invariant ofthe third order in the coefficients, the weight must be 6, andthe terms of the invariant must be
jala4a2a0Baa a Ca a a .Da a a Ea
zaaaz ,4 l 1 8 8 0 8 2 l
where the coefficients A,B,&c. remain to be determined. The
reader will Observe that there are as many terms in this invariant as the ways ia which the number 6 can be expressedas the sum of three numbers from 0 to 4 inclusive ; and gener
rally that there may be as many terms in any invariant as theways in which its weight gnd can be expressed as the sum of
6 numbers from 0 to n inclusive.
We determine the coefficients from the consideration thatsince an invariant is to be unaltered by the substitution either
of x + >t for w, or y+ 7t. for y, ev idently, as in Art. 58,every
invariant must satisfy the two differential equations
d] d] d] all all
da,
‘da,
das
+ &c
° O,na
1a0
+ (n
it being supposed that the original equation has been writtenwith binomial coefficients. In practice only one of these equations need be used ; for the second is derived from the first bychanging each coefficient a ,,
into and
. It is sufficient then to
FORMATION OF INVARIANTs AND OovARIANTs.
use one of the equations,prov ided we. take care that the func
tion we form is symmetrical with regard to a: and y ; that i sto say, which does not change (or at most changes sign)* whenwe change a “ into a
w ”. And this condition will always be
fulfilled if we take care that the weight of the invariant isthat which has been just assigned. Thus then
,in the example
chosen for an illustration,if we Operate on A a
4a2ao+ &c. with
we get
(2B + 2A)a a a4 1 0 3 2 0
= 0,8 1 l 2 2 1
whence if we take A = 1,the other coefficients are found to be
B = 1,D = 2
,0 = 1
,E = 1
,and the invariant is
2a3a2a1
a4a1a1asaaa0
a2azaz.a d a
4 3 0
141. In seeking to determine an invariant of given order
by the method just explained,we have a certain number of
unknown coefficients A ,B,C,&c. to determine
,and we do so
by the help of a certain number of conditions formed by means
of the differential equation. Now ev idently if the number Of
these conditions were greater than the number of unknowncoefficients
,the formation of the invariant would in general
be impossible ; if they were equal we could form one invariant ;if the number of conditions were less, we could form more
than one invariant of the given order. We have just seen
that the number Of terms in the invariant,which is one more
than the number of unknown coefficients, is equal to the numberof ways in which its weight And can be written, as the sum
of 9 numbers,none being greater than n. But the effect Of
the Operation ao
i + &c. is ev idently to diminish the Weighta]
by one ; the number of conditions to be fulfilled is therefore
equal to the number of ways in which %n0 1 can be expressed
When we change a: into y and y into a this is a transformation whose modulus0,1 or 1. Any invari ant therefore whi ch when transformed becomes mul
1,0
tiplied by an odd power of the modulus of transformation will change signwhenweinterchange a: and ‘
y. Such invariants are called s/cew invari ants.
is
116 FORMATION OF INVARIANTs AND CovARIANTs.
as the sum Of 9 numbers,none exceeding n. Thus
,in the
example of Art. 140,the number of conditions used to deter
mine A,B,&c. was equal to the number of ways in which
5 can be expressed as the sum of three numbers from O to 4
inclusive. To find then generally whether an invariant of abinary quantic of the order 9 can be formed
,and whether
there can be more than one,we must compare the number
of ways in which the numbers 5n9, én9 can be expressedas the sum Of 9 numbers from 0 to n inclusive. On this principle is founded Mr. Cayley
’s investigation of the number of
invariants of a binary quantic, of which we give an accountin an Appendix.
142. Similar reasoning applies to covariants. A covariant,
like the original quantic,must remain unaltered
,when we
change a uto pm, and at the same time every coefficient a ,into p
’aa. If
”
then the coefficient Of anv power of as,we in
the covariant be a abflcy , &c. it is obv ious,as before
,that
p. a + must be constant for every term ; and we maycall this number the weight of the covariant.Again
,in order that the covariant may not change when
we alter a; into y and y into ac,we must have
14 + a + ,8 + ry
—,u)j
—(n
where p is the degree of the covariant in a: and y ; whence if9 be the order of the covariant in the coefficients, we haveimmediately its weight (n9 Thus if it were requiredto form a quadratic covariant to a cubic
,of the second order
in the'
coefficients,n= 3
,9 = p
= 2, and the weight is 4. We
have then for the terms multiplying a and theseterms must be a
za0and a la l. In like manner the terms mul
tiplying my must be “3am a
2a1,
and those multiplying y’ must
be aaal ,aza2. In this manner we can determine the literal part
of a covariant of any order. The coefficients are determinedas follows
143 . From the definition of a covariant it follows that wemust get the same result whether in it we change a: into as Ry,or whether we make the same change in the original quantic
118 FORMATION or mvsmsm s AND COVARIANTS.
we get, as may be easily seen,
y d”u
1 2+ &c.
Hence if we know any relat ion connecting any functions of thedifferences A
0,B
o,00,&c. the same relation will connect the
covariants derived from these functions.
Ex. 1. To find the quadratic covariant of a cubic. We have seen (Art. 142)thatd
A, is of the form azao-i-Ba la l. Operate on this with and we
1
get (2 23)(soc l 0, whence B 1 and A. a za, a la l. Operate then withd d d
3al d7 °
+ 2avza
+ th dzand we have Operate With the same
on A” and we have A2 a las agar The covariant therefore is
(“rue - ana l)“52 + (aoa ,Ex. 2. To find a cubic covariant of a.cubic of the third order in the coeficients.Here 1; =3, 6=8, +p) 6. The sum then of the sumxes of the coefficient
of will be 3 ; and this coeflicient must b e of the form Aaaaoao Bagels, Ca,a,al.d d d
eratewith a — + 2a and we etOp 0dd ]
1 da,“2 da,
’ g
(3A B)a zaoao (23 30)a la lao,whence if we take A 1, we have B 3 , 0 2
, or A, aaaoa, 3:12am
Operate on this three times successively with 3a 1 i 2a2j a,
i and we haveda
0 da (l a,
the remaining coefli cients and the covariant is (see Art. 138)3024 100 2a'la la 1)”3 3 (“aa i ao 20125524 0 4
“ az a la l)may
4“ 3 (2a8a l a l “2054 1 aa“z“o)$ 192 (sasa2a‘1 20 2“s “ran“o)y’
144. We have seen that a quantic has as many covariantsof the degree 10 in the variables and of the order 0 in the cc
efficients,as functions A
Owhose weight is (n9—
p)can be
found to satisfy the equation(24
79 0. And
,as in Art. 141, we
see that this number is equal to the difference of the ways inwhich the numberswe p)an d a(710—
p) 1 can be expressedas the sum of 0 numbers from O to n inclusive. It may be t e
marked that p cannot be odd unless both n and 0 are odd.
Hence only quantics of odd degree can have covariants of odddegree in the coefficients, and these must also be of odd degree
in the variables.
145. The results arrived at (Art. 143)may be stated a little
differently. The operation y; performed on any quantic is
FORMATION OF INVARIANTS AND COVARIANTS. 119
equivalent to a certain operation performed by difl'
erentia
ting with respect to the coefficients. Thus,for the quantic
al ,a2
we get the same result whether we operated
on i t Wi thi
or Wi th ao537
1did
&c. .Thi s latter opera2
tion then may be written(5
and the property already
proved for a,covariant may
‘
be wr itten that we have for it
yé—i 0. In other words,that we get the same
result whether we operate on i the covariant with yé or with
fl i + &c. In his Memoirs on Qu’
antics,Mr. Cayleyda
1da
2
has started with this property as his definition of a covarianta definition which includes invariants also
,since for them we
have yg 0,and therefore also
146. It can be proved, in like manner, that quantics in any
number of variables satisfy differential equations which may be
written y diw
d zgm &c. Thus,for the
quantic (a , b, c,f, g, hi m, y, we have
61 d d d d cl d d1,1c
a
fi-i g
c
-
Z—
f-i—fi
db ’e
d;a
zg+ 7zzf+ 2g dc ,
and every covariant must satisfy these two equations. Wh ileevery invariant must satisfy the two equations
d] all all dIa
ZZ—
h+g 26
0,
a
zq+ k 29
670
as may easily be proved from the consideration that the invariantrema ins unaltered if we substitute for w, w Ky or a: p z .
( 120 )
LESSON XIV.
SYMBOLICAL REPRESENTATION OF IN'
VARIANTS AND COVARIANTS.
147. IT remains to expla in a fourth method ,of finding ihvariants and covariants
,given byMr. Cayley in 1846 (Cambridge
and Dublin Ma thema tica l Journa l,vol. I.
,p . 104
,and Crelle
,
vol.XXX .) which not only enables us to arrive at such functions,
but also affords the basis Of a regular calculus by means of whichthey may be compared and identified.
Let x1,y1 ; be any two cogredient sets of variables ; then
0 d d d dit has been proved (Arts. 126 ; 116: 135)that dx
ldil l dm$z dy,d d d
transformed by the reciprocal substi tution ; thatalso
,doc
,da,
i s an invariant symbol of Operation ; and that if we operatewith any power Of this symbol on any function of y] , ya,we shall Obta in a covariant of that function. We shall use ford d d d
the abbreviation 12 .
dwldy2 dw
?dyl
Suppose now that we are given any two binary quanticsU,V,We can at once form covariants of this system of two
quantics. For we have only to write the variables in U withthe suffix those in V with the suffix and then Operateon the product UV with any power of the symbol 12 when theresult must be an invariant or covariant. Thus if
U are.
” m s . ca"V= as
.
"CW,
and if we operate on UV with which,written at full
length,is
(P d2 cl‘dz '
2(P d2
dmf dag
dz.
fly.”
da dy. da ds/Jthe result is ac’
+ ca' which is thus proved to be an inva
riant of the system Of equations. In general,it is obvious that
the differentials marked with the suffix (1)only apply to U, andthose ' with the suffix (2)only to V; and it is unnecessary to
122 SYMBOLICAL REPRESENTATION or
follows,when we use such symbols as 12
"
&c. without addingany subject Of operation
,we mean to express derivatives of a
single function U. We take for the subject Operated on,the
product of two or more quantics U] ,U” &c., where the variables
w1, y, ; 511
2 , y, ; &c. are written in each respectively, instead of
a: and y ; and we suppose that after differentiation all the
suffixes are omitted,and that the variables
,if any remain
,are
all made equal to x and y.
149. From the omi ssion of the suffixes after differentiation,
it follows at once that it cannot make any difference what
figures had been originally used,and that 12“ and 34
"
are
expressions for the same thing. In the use of this method wehave constantly to employ transformations depending on this
obvious principle. Thus,we can show that when n is Odd
,12
”
applied to a single function vanishes identically. For,from
what has been said 12 = 2 1"
, but 12 and 2 1 have oppositesigns
,as appears immediately on writing at full length the
symbol for which 12 is an abbrev iation. It follows then that12
”must vanish when n is Odd. Thus
,in the expansion of
given at the end of Art. 147,ifwe make U V
,it will obviously
vanish identically. The series 122,12
4
,12
6&c. gives the series
of invariants and covariants which we have already found
(p. It is easy to see that,when n is even
,12
”applied to
a,a2. " Ix , y)
” gives- na
lan
where the last coefficient must be divided by two, as is evidentfrom the manner of formation.
150. The results of the preceding Articles are extendedwithout difficulty to any number Of functions. We may takeany number of quantics U, V, l/V, &c.
,and
,writing the variables
in the first with the suffix those in the second with the suffixin the third with the suffix and so on
,we may Operate
on their product with the product of any number of symbols2319
,3 1
7
,&c. ; where, as before, 23 is an abbreviation
d d d ddue
,dx
After the differentIationwe suppress
INVARIANTS AND COVARIANTS. 123
the suffixes,and we thus get a covariant of the given system of
quantics,which will be an invariant if it happens that 110 power
of a: and y appear after differentiation. Any number Of the
quantics U,V,V”, &c.
,may be identical ; and in the case with
which we shall be most frequently concerned, v iz .,where we
wish to form derivatives of a single quantic,the subject Operated
on is U,U;U, &c.
,where U; and U
2only differ by having the
variables written with different suffixes.It is ev ident that in this method the degree of the derivative
in the coefficients will be always equal to the number of differentfigures in the symbol for the derivative. For if all the functionswere distinct
,the derivative would evidently contain a coefficient
from every one of the quantics U,V, W,
&c. ; and it will be stilltrue
,when
,
U,V,W are supposed identical
,that the degree in
the coefiicients is equal to the number of factors in the productU1UzUS&c. which we Operate on. Thus
,the derivatives con
‘isidered in the last Article being all of the form 12? are all of the
second degree in the coefficients.Aga in
,if it were required to find the degree of the derivative
in a: and y. Suppose, in the first place,that the
’ quantics weredistinct, U being of the degree 12
,V of the degree n’
,W of the
degree n",and so on and suppose that in the Operating symbol
the figure 1 occurs a times ; 2 , 6 times ; and so on ; then, sinceU is differentiated a times ; V, B times, &c., the result is Of the
degree (n a) (n’
,8) (n"
'y) &c. When the quanticsare identical, if there are p factors in the product U
,
which we operate on,the degree of the result in a: and y will be
7219 (01+ ,8 7 While aga in
,if there be r factors such
as 12 in the operating symbol, it is obv ious thata -l-BIt is clear that if we wish to obta in an invariant
,we must have
151. To illustrate the above principles,we make an ex
amination of all possible invariants of the third degree in thecoefficients. Since the symbol for these can only conta in threefigures
,its most general form is while
,in order
that it should yield an invariant,we must have
124 SYMBOLICAL REPRESENTATION or
whence a = B= fy. The general form,then, that we have to
examine i s Aga in,if a be odd
,this derivative
vanishes identically ; for, as in Art. 149,by interchanging the
figures 1 and 2,we have but these
have Opposite signs. It follows,then
,that all invariants of the
third order are included in the formula where a
i s even. Thus,
is an invariant Of a quartic,since
the differentials rise to the fourth degree ; is an
invariant Of an octav ic ; of a quantic Of the twelfthdegree
,and so on ; only quantics whose degree is of the form
4m hav ing invariants of the third order in the coefficients. If
we wish actually to calculate one of these,suppose
I write,for brev ity, $1 , 711, &c.
,instead of
d
d
w&c. Then
1
, da.’
we have actually to multiply out
(51772In the result we omit all the suffixes
,and replace 5
“by
4Uda
" &c.,
or, when we operate on a quartic, by a
othe coefficient of on“.
There are many ways which a little practice suggests forabridging the work of this expans ion
,but we do not think it
worth\
while to give up the space necessary to explain them ;and we merely give the results of the expans ion of the three
invariants just referred to. 3 1"
yields the invariant of
a quartic already Obtained (Art. 137 Ex . 1 andArt.
a4a,ao
2a8a2a1
a,al
aoaa
gives
aa (a,a0 4a
,,al3a
2az) a
74a
bao
l 2adral8a
,a,)
-a6 (3a 6a0 8a
,,al
22a,az24a
aas)4 a
6 (24a a — 36a
And 3 1“gives
— 10a8a3)+ a 6a a + 3Oa6a 1
aw (15aaao 54a
,al
24a6ag
15Oa5a,l 35a
,a,)
a,
10a9ao3Oa
8a,
l 50a,a2430a
6a3270a
5a,)
as
l 35uaa2
270a,a,,
495a,,ad540a
5a6)
4 a7
54oa7a‘ 720a
6a6) 280a
6a6aQ.
126 SYMBOLICAL REPRESENTATION or
(at (B— oy)”
(y a)”and its powers ; and the discriminant
is of the fourth order in the coefficients. Hence onlyquantics ofthe degree 4m have cubic invariants whose general type is
It will be proved that quartics have two in
dependent invariants,one of the second
,and one of the third
degree,in the coefiicients ; and
,of course
,any power of one
multiplied by any power of the other is an invariant. Hence,
quartics have as many invariants of the pmorder as the equation
2m+ 3y =p admits of integer solutions : this is,therefore
,the
number of invariants of the fourth order which a quantic of thepmdegree can possess.
154. Hermite has proved that his theorem applies also tocovariants of any given degree in w and y ; that ‘is to say, thatan nio possesses as many such covariants of the p
t"order in the
coefficients as a pic has of the order in the coefficients. For,
consider any symbol, 12K .23M.34V &c.,where there are p figures
,
and the figure 1 occurs 0. times,2 occurs b times
,and so on ;
then we have proved that the degree of this covariant in a: andyi s (n But we may form the symmetricfunction
2 (a Br (3 r)“(a' (w at)”
(w B)“ &0
which has been .proved (Art. 133)to be a covariant of thequantic of the p degree
,whose roots are a
, B, &c. Everyroot enters into its expression in the degree n
,which is there
fore the order of the covariant in the coefficients, and itobv iously contains a: and y in the same degree as before
,viz .
(n —b)+ &c. Thus,for example
,the only covariants
which a quadratic has are some power of the quantic itselfmultiplied by some power of its discriminant, the general type ofwhich is
(at (a’ — a)“(a’
the order of which in the coefficients is 2p g, and in a: and y is2g. Hence we infer that every quantic of the degree 2p + qhas a covariant of the second degree in the coefficients
,and of
the degree 2g in a: and y, the general symbol for such covariantsbeing When g : 1
,we obtain the theorem given (p.
that every quantic of odd degree has a quadratic covariant.
INVARIANTS AND covmmnrs. 127
155. This notation affords a complete calculus , by means ofwhich invariants or covariants can be transformed
,and the
identity of different expressions ascerta ined. Postponing to a
subsequent Lesson the explanation of this,we go on to show
how the same notation is to be applied to express derivatives ofquantics in three or more variables. If <1}
,s wsysz a, be
cogredient sets of variables,then
,by the rule for multiplication
of determinants,the determinant
wt (yaz s blaz e) w
e (yaz i yl z s) ms (yi z a _yaz i)
i s an invariant,which by transformation, becomes a similar
function multiplied by the modulus of transformation. And if in
d1) dx
,
for y, , d—d
and so on,we obtain
an invariantive symbol of operation,which we shall write 123.
When,then
,we wish to obtain invariants or covariants of
any function U;we have only to operate on the product
with the product of any number of symbols123“ 235
" &c.,and after differentiation suppress all the
suffixes. Thus,for example
,let U
1 ,U2 ,U,be ternary quadrics,
and let the coefficients in UIbe a
,b,0,2f , 29, 2h, as at p. 83
,
then 123“expanded_is
— 2f:ffl
)+ b (cI
aII
+ 2f (g _ bl
q blI
gI
)
+ 2k((f igII
Hf" I
_ cI
kII — c
Il
kI
);
which, when we suppose the three quantics Uh to be
identical, or a = a'= a
"
&c. reduces to six t imes
abc 2fgh af’by
”or .
If in ,the above we replace a
,the coefficient of at”
,by
the above we write for a:
2. ‘dw? &c.
we get the expanmon of 1232as applied to any ternary quantic.
This covariant is called the Hessian of the quantic.It is seen
,as at Art. 149, that odd powers of the symbol 123
vanish when it is applied to a single quantic. We give as a
further example the expansion of 1234applied to the quartic
,
ax“by
‘as44 (a zx
s
y asxsz by z b
lysso 0
1s c
zz
’
y)
6 (dyfiz2
ez"a:2
fxz
yz
) 12xyz (la: my me).
128 SYMBOLICAL REPRESENTATION or
Then 1234 is
abo 4 (abso, be a ca,b,) 3 (ad
z be”
Cf 4 (a 2b80,12 (a2 'nd a
amd b
,
'ne bale e
,mf + c
, (f )12 (lb,c, mc
,a,
naabs) 12 (dl
‘em
"+fn
2
) 6def 12177272.
156. We can express in the same manner functions containing contragredient variables ; for if 01, B, y be any variables con
tragredient to x, y, z , and therefore cogredient withdi ’02,g,
1t follows,as before
,that the determinant
d d_d d
) (d d d d
‘
dy, dz , dyz dz , dz,dx
,dz
2den
,
7doc
,d_
y, dx,dy,)
(which we shall denote by the abbreviation 0112)is an invariantive symbol of operation. Thus
,if U
, ,U,be two different
quadrics,0c12
2 is the c ontravariant called gb (Combs, p.which expanded is
a?
(bI
cII
+ b”
fiz
72
(aI
bH
+aII
bI—2k
l
k"
)267 + g
Il
kI
“f”
aI
y‘I
) 27“(hffl
kI
ZfI
b'
y"
bII
gI
)2aB (f
I
gII
+f7
gl
el
k"
cII
kI
)
and which , when the two quadrics are identical,becomes the
equation of the polar reciprocal of the quadric.In like manner, the quantic contravariant to a quartic
,which
I have called S (Higher Plane Curves, p. may be written
symbolically a 12‘,and the quantic T in the same place may be
wri tten 011220123
20131
2. In any of these we have only to replace
the coefficient of any power of at,x
”by$7, to obtain a symbol
which will yield a mixed concomitant when applied to a quanticof higher dimensions. Thus is
U deU
(d U
dz2
dydz
which,when applied to a quadri c
,is a contravariant
,but, when
applied to a quantic of higher order,contains both to
, y, z , as
well as the contragredient a , B, y, and therefore,is a mixed
concomitant.
130 SYMBOLICAL REPRESENTATION OF
being the discriminant of a cubic,its evectant
, got by omittingthe factors which contain 4
,is
dU dU dU
doc dy dz
for 01, ,8 ,
«y, we also get a covariant,and the symbol for it is
obtained from that for the contravariant by writing a difirent
If in a contravariant of any quantic we substitute
new figure in place of every01. Thus,from a34
”we get
and so on.
158. In the explanation of symbolical methods which hasbeen hitherto given, I have followed the notation and courseof proceeding originally made use of by Mr. C
‘
ayley. I wishnow to explain some modifications of notation introduced byMM. Aronhold and Clebsch
,who have employed these sym
bolical methods with great success,but who perhaps have
scarcely sufficiently recogniz ed the substantial identity of
their methods with those prev iously given by Mr. Cayley.
The variables are denoted x, ,w, ,m, ,&c.
,while the coefficients
are denoted by suffixes corresponding to the variables whichthey multiply. Thus the ternary cubic, the ternary quartic, &c.may be briefly denoted Eanmxfl k
mlwm,&c.
,where the
numbers i,It,I,m are to receive in succession all the values
1,2,3 . It will be observed tha t in this notation am : a am,
so that when we form the sums indicated we obtain a quanticwritten with the numerical coefficients of the binomial theorem.
Thus when we form the sum the three termsa mac a a x x x a mas s: are identical
,as in like manner
112 1 1 2) 121 1 2 17 211 2 1 1
are the s1x terms
a128x1w2w87
“maxex ixa? a281w2w8x1 ’
a '
312wax1w2)
aaz l w l I
so that the sum written at length would be“111x1x1x1
“222x2x2w2
“839588373333
3a1 12w1w1w2
6a123w1x2m3
’
And so in like manner in general. Now M. Aronhold uses,as an abbrev iated expression for the quantic in general,
where after expansion we are to sub
stitute for the products &c. the coefficients am. Thusthe ternary cubic given above may be written in the abbre
v iated form the terms
3a 1 &c.
INVARIANTS AND COVARIANTS. 131
in the expansion of the cube being replaced by&c. The quantic might equally have been writtenb8w3)3
, &c.,it being understood
that we are in like manner to substitute for &c. the
coefficients am ,&c. Now the rule given by M. Aronhold
for the formation of invariants is to to take a number of determinants
,whose constituents are the symbols a
, ,a, ,as ; b, , b, , &c .
,
to multiply all together,and after multiplication to substitute
for the symbols bmbnbp ,the coefficients am,
am p,
&c.
Thus M. Aronhold first discovered a fundamental invar1ant of
a ternary cubic by forming the four determinants 2 + a b o1 2 87
2 i b c d Ei o d a 2 i d d b multiplying all together and1 2 37
.
1 2 3 1 1 2 3 )0 ,
then performlng the substitutions already indicated. Th1s IS
the same invariant ” which,in Mr. Cayley
’s notation
,would be
designated as
159. . In order to see the substantial identity of the two
methods,it is sufficient to observe that by the theorem of homo
geneous functions any quantic u of the order differs only
d dn
by a numer1cal mult1pl1er from (or, dab,
Jeri
d—
x)u,so
that if we write it the symbols a, ,
a2 ,
a,
differ only by a numerical constant from the differential sym
bols &c. And we ev idently get the same results whether
with Mr Cayley we form determinants whose constituents are
d d
dx,
’ dzo,
’ dw,
’
“17
“2?aa'
And the artifice made use of by both is the same.
If we multiply' together a number of differential
”
symbols
7.g)(di-7+ ,u.
di Z
y&c. and operate on U
,it is evident
the result will be a linear function of differentials of U of an
order equal to the number of factors multiplied'
together ; and
that in this way we can never get any power higher than thefirst of any differential coefficient. When then it is required
to express symbolically a function involv ing powers of the
differential coefficients, the artifice used by Mr. Cayley was
or with Aronhold,whose constituents are
132 REPREsENTATmN or 1NVAR1ANTs AND COVARIANTS.
to write the function first with different sets of variables and
form such a function as (dc
; + x d_d
(cl—G
a
l
e,u.
of?) and
after differentia tion to make the variables identical. So in likemanner Aronhold in his symbolic multiplication uses differentsymbols which have the same meaning after the multiplication has been performed. By multiplying together symbols
a” a,,&c.
,we can only get a term such as a
,“ of the firstdegree only in the coefficients. When then we want to ex
press symbolically functions of the coefficients of higher order
than the first,the artifice is used of multiplying together diffe
rent sets of symbols an a” a , ; b" b, , b,, &c., the products63,-aka" o
,oke,,
&c. all equally denoting the coefficient a
160. Hav ing then so fully expla ined Mr. Cayley’s symbolical
method,it is unnecessary to expla in at greater length one which
only differs from it by notation. Reserv ing,then
,to a sub
sequent lesson illustrations of the applications of these methods,
we only add here Clebsch ’s proof that every invariant can be
expressed in the manner indicated. If we are given any in
variant d; (a , b, o, &c.)of the coefficients of a single quantic,it
has been proved, Art. 119 , that by performing on it the opera
tion a'
j—a -l-b'
dbi we obtain a simultaneous invariant of a
pair of quantics,a'
,b'
,c' being the coefficients of the second,
answering respectively to a,b,o in the first. In like manner
,if
d dwe operate on th eW1th a
0171+b
d_
bwe get an invariant of
a system of three quantics,and we can repeat this process as long
as there remains coefficients a , b, c, &c. in the invariant. Thus,then, when we are given any invariant of a quantic whose orderin the coefficients is p , we can derive from it an invariant of
a system of p quantics which shall be of the first degree in thecoefficients of each quantic. And we can fall back at any time
on the original invariant by supposing the p quantics to be
identical , For the operation a ; b; + &c. performed on theinvariant could only have effected it with a numerical multiplier.Again, we are at liberty to suppose all the new quantics thus
134 CAN'
ONICAL FORMS.
quantics in general what the reader is familiar with in the caseof ternary and quaternary quantics ; since
,when we wish to
study the properties of a curve or surface,every geometer is
familiar with the advantage of choosing such axes as shallreduce the equation of this curve or surface to its simplest formfi“The simplest form, then, to which a quantic can, without loss ofgenerality, be reduced is called the canonica l form of the quantic. We can
,by merely counting the constants
,ascertain
whether any proposed simple form is sufficiently general to betaken as the canonical form of a quantic
,for if the proposed form
does not, either explicitly or implicitly, contain as many con
stants as the given quantic in its most general form,it will not
be possible always to reduce the general to the proposed form.1‘
Thus, a binary cubic may be reduced to the form X°+ Y
'
; for
the latter form,being equivalent to con
tains implicitly four constants, and therefore is as general as(a , b, c
, d , So, in like manner,a ternary cubic in its
most general form conta ins ten constants ; but the form
.X°+ 'contains also ten constants
,since
,in
addition to the m which appears explicitly, X,Y,Z,implicitly
It must be owned however that as in the progress of analysis greater facility isgained in dealing with quantics in their most general form, the advantage diminishesof reducing them to simplerforms.f It is not true, however, conversely that a form which contains the proper number
of constants is necessarily one to which the general equation may be reduced. Forwhen we endeavour by comparison of coefficients to identify such a form with thegeneral equation, although the number of equations is equal to the number of
quantities to be determined, it may happen that the constants enter into the equationsin such a.way that all the equations cannot be satisfied. Thus
(ft
is a form containing five constants, and yet is not one to which the general.
equationof a ternary quadric can be reduced ; since the constants enter the equationm such away that though we have more than enough to make the coefficients of a:andyand theabsolute term identical with those in any proposed equation, we have no means ofidentifying the coeficients of a”
, my and y”. A more important example is
where z , u, 'v are linear functions. In the case of a ternary quantic this formcontains
implicitly fourteen independent constants, and themfore seems to be one to whi ch thequart ic in general can be reduced . But Clebsch has shown that a conditionmust befulfilled in order that a quartic should be reducible to this form, namely, the
vanishing of a certain invariant.
cauomcan roams. 135
involve three constants each. This latter,then
,may be taken
as the canonical form of a ternary cubic, and, in fact, the most
important advances that have been recently made in the -theoryof curves of the third degree are owing to the use of the equationin this simple and manageable form.
162. The .quadratic function (a , 6, 01'
s, y)
”can be reduced in
an infinity of ways to the form as” since the latter form im
plicitly contains four constants , and the former only three. In
like manner the ternary quadric which conta ins six constants canbe reduced in an infinity of ways to the form x
”
y”
z“,since
this last conta ins implicitly nine constants ; and in general aquadratic form in any number of variables can be reduced in an
infinity of ways to a sum of squares. It is worth observ ing,
however,that though a quadratic form can be reduced in an
infinity of ways to a sum of squares, yet the number of positive
and negative squares in this sum is fixed. Thus, if a binaryquadric can be reduced to the form w
”+ y
’
,it cannot also be re
duced to the form 'we— v
”
,since we cannot have as
”
y” identical
with 149
the factors on the one side of the identity beingimaginary, and those on the other being real. In like manner
,
for ternary quadrics we cannot have 502+ y
”242+
since we should thus have a)”
u’+ 0
2
+ w”
,or,in other
words,
(lx my (l'
w My (Z x m”
y n”a)2
,
and if we make w and y 0,one side of the identity would
vanish,and the other would -reduce itself to the sum of four
positive squares which could not be 0. And the same argument applies in general.
163. We commence by showing that, as has been juststated
,a cubic may always be reduced to the sum of two cubes.
To do this is in fact to solve the equation, since when the
quantic is brought to the form X 3 Y“,it can immediately be
resolved into its linear factors. Now,if the cubic (a , b, c, (11413, y)
3
become by transformation (A ,B,0, DIX, then, since
(Art. .
122)the Hessian dy) is a cc
variant,it will by the definition of a covariant, be transformed
136 cas ement, roams.
into a similar function of A,B,0,D,X,Y. That is to say,
we must have
(ac b”
)w”
(ad be)my (bd c“)y
”
(A O— B”
) —B 0)K Y+ (EB —C’
)Y‘.
Now, if in the transformed cubic
,B and 0 vanish
,the Hessian
takes the form ADXY; and we see at once that we are to ta kefor X and Y the two factors into which the Hessian may bebroken up. When we have found X and Y
,we compare the
given cubic with AX s -Z)Y3
,and determine A and D by
comparison of coefficients.
Ex. To reduce 4x 3 18a: 17 to the formAX ’ DY’. TheHessian is
(4a: 3)(6m 17)or 15,whose linear factors are a: 3
, 3m 1. Comparing then the given cubicwithA (x D (3w+
we have A 27D 4,27A D 17, whence 7280 91
,728A 465, or A is to D
in the ratio of 5 to 1. The given cubic then only differs by a factor (viz .8)from5 (a:
and it.is obvious that the roots of the cubic are givenby the equation
=0.
164. It is evident that every cubic cannot be brought byrea l transformation to the form AX”
DY for this last formhas one real factor and two imaginary ; and therefore cannotbe identical with a cubic whose three factors are real. The
discriminant of the Hessian
4 (ac be
)(bd — c”
) (ctd m bcyI
is,with sign changed
,the same as that of the cubic, When
the discriminant of the cubic is positive,the Hessian has two
real factors,and the cubic one real factor and two imaginary.
When it is negative,the Hessian has two imaginary factors,
and the cubic three real. When it vanishes,both Hessian and
cubic have two equal factors,and it can be directly verified that
the Hessian of X”Y is X ’fi“
It is to be observed, that a quantic cannot always be reducedto its canonical form. The impossibility of the reduction indi
In general, when a b inary quantic has a square factor, this will also ba a squarefactor in its Hessian, as may be verified at once by forming the Hessianof
138 CANomen , FORMS.
Now it will be observed that the coefficients of a in everycolumn are proportional to lm
,m
“. Consequently
,if we re
solved this determinant,as in Art. 22
,into partial determinants
,
every such determinant which contained two of the u columnswould vanish as hav ing two columns the same. And so
,in like
manner,would any which contained two 1) or two 10 columns.
The determinant then will be new multiplied by a numericalfactor.*
When, then, the determinant written in the beginning ofthis Article has been found
,by solv ing a cubic equation
,to be
the product of the factors (x + 7ty) we knowthat a
,v,w can only differ from these by numerical coefficients,
and we may put
(a , b: c, d) ehfIx)”5 A (33+ Ky)
5
+ B (x flylb
' i' ”yls
i
and then A,B,0 are got from solv ing any of the systems of
simple equations got by equating three coeflicients on both sidesof the above identity.The determinant used in this Article is a covariant
,which is
called the canoniz ant of the given quantic.
166. The canoniz antmay be written in another,and perhaps
simpler form,namely
,
98
) yzx) yx
”
, ‘ we
a,
b,
c,
d
b,
c,
d,
e
c,
d)
e, f
This last is the form in which we should have been led to it ifwe had followed the course that naturally presented itself, andsought directly to determine the six quantities
,A,B,0, 7t, pl, V,
by solving the six equations got on comparison of coefficients ofthe identity last written in Art. 165 . For the development
of the solution in this form,to which we cannot afford the
necessary space here,we refer to Mr. Sylvester’
s Paper (Philo
sophica lMaga z ine, November, Meanwhile,the identity
of the determinant in this Article with that in the last has been
Viz . (hn'
(z'
l”m'
)2 (l"m M y.
CANONICAL FORMS. 139
shown by Mr. Cayley as follows. have,by multiplication
of determinants (Art.
ya
)“
yzwa
170101 -0
a,
61 $
7 $9 1 01 0
b; 0
1 07w: fl ) O
07
d; f 0
107x, y
319
7O
7 07
0
ax + by, bx + cy, cx + dy
0,bx + cy, cx + dy, dx + ey
0,
cx + dy, dx + ey, ex +fywhich
,div iding both sides of the equation by y
”
,gives the
identity required.
0
167. We have still to mention another way of forming thecanoniz ant. Let this sought covariant be (A ,
B,(J,Di x , y)
3
,
where we want to determine A,B,0,D ; then (Art. 136)
A,B,0,D i will also ield a covariant. But if this
dy dx y
operation is applied to where x + 7ty is a factor in
(A ,B,0,Di x , y), the result must vanish
,since one of the
factors in the operating symbol 15d d
Since,then
,the
given quantic is by hypothesis the sum of three terms of theform (x the result of applying to the g1ven quantic theoperating symbol just written must vanish. Thus
,then
, We
hav e
A (d) eh fl w’ y)2 — B (c, d: eIx,m2 G ib 0 y)
D (a , by 01 377or
,equating separately to O the coefficients of x
”
,xy, f ,
we
haveAd — Bc + Cb — Da = 0
,
A e — Bd+ 00 — Db = 0,
Af — Be + Cal — Dc = O,
whence (Art. 28)A is proportional to the determinant gota,b,c
suppressing the column A or b,
c,d and so for B
,C,D,
c,d,
140 CANONICAL FORMS.
which values give for the canoniz ant the form stated in the lastArticle.
168. We proceed now to quantics of even degreeSince this quantic conta ins 2n+ 1 terms, if we equate it to a
sum of 77 powers Of the degree 277,we have one equation more
to satisfy than we have constants at our disposal. On the otherhand
,if we add another 2nm power
,we have one constant too
many,and the quantic can be reduced to this form in an infinity
of ways. It is easy,however
,to determine the condition that
the given quantic should be reducible to the sum of 77,
powers. Thus,for example
,the conditions that a quartic
should be reducible to the sum of two fourth powers,and that
a sextic should be reducible to the sum of three sixth powers,
are respectively the determinantsa,b,c,d
b,c,d,e
07d;97 f
d:67 f: 9
and so on. For in the case of the quartic the constituents ofthe determinant are the several fourth differentials of thequantic
,and
,expressing these in terms of a and v precisely as
in Art. 165,it is easy to see that the determinant must vanish,
when the quartic can be reduced to the form u“
Similarlyfor the rest. This determinant expanded in the case of the
quartic is the invariant already noticed (see Art. 137, Ex.
ace 2bcd adz
eb2
03
169. When this condition is not fulfilled,the quantic is re
duced to the sum of n powers,together with an additional term.
Thus,the canonical form for a quartic is u
4
+ v4
+ 67tu2772. We
shall commence with the reduction of the general quartic to itscanonical form the method which we shall use is not the easiestfor this case, but is that which shows most readily how the reduction is to be effected in general. Let the product, then, ofa,v,which we seek to determine, be (A ,
B,01x , and let us
operate with (A,B,
11 both sides of the identity
(a , b, c, d, eIx , y)‘
u“
77‘
67tu2v2.
142 CANONICAL FORMS.
V is a covariant of this latter function such that when Vuvw&c.
is operated on by (A0,A
,(fix)the result is propor.
tional to the product uvw&c. Suppose, for the moment, thatwe had found a function V to fulfil this condition, then, proceeding exactly as in the last Article
,and operating with the
differential symbol last written on the identity got by equatingthe quantic to its canonical form
,we get the system of equations
Anan (n 1)A a
2
A a — 77A,a,l 177 (n 1)A g
a0 11—1
A a — nA a
114-1
+ 177, (77 1)A 2an
&c.= 7\.A &c.,0 71—2 1 w—l
whence,eliminating A
0,A A &c.
,we get the determinant
( bu
— x,a'
fl f' l ’an“ ,
o o o o c o o a o o o o am‘
1a" Z
X,a"+1,
an,
“1?
and hav ing found 7» by equating to 0 this determinant expanded
(a remarkable equation,all the coefficients of which will be
invariants), the equations last written enable us to determine thevalues of A
0,Au&c.
,corresponding to any of the 77+ 1 values
of h .
171 . To apply this to the‘case of the sextic,the canonical
form here is u“v“w" Vuvw
,where
,if 77v be
(AO,A
17A
27 yr?V is the evectant of the discriminant of this last quantic
,and
whose value is written at full length (Art. Now it will
The determinant above written may be otherwise obtained as follows. Leta’, g/ be cogredient to z , y, and let us form the function
—m~.
which (Arts. 121, we have proved to be linearly transformed into a function of
similar form. Take the a 1 coefficients of the several powers as", zfl-ly, &c., and
from these eliminate linearly the n 1 quantities sum“, &c., and we obtain the
determinant in question.
CANONICAL FORMS. 143
afford an excellent example of the use of canonical forms if weshow that in any cubic the result of the operation
a"
(ac) an as) a
performed on the product of the cubic and the evectant justmentioned, will be proportional to the cubic itself. For it is
sufficient to prov e this,for the case when the cubic is reduced to
the canonical form in which case the evectant will ben:a
gsas appears at once by putting b c O
,and a = d = 1 in
the value given,p. 113. The product
,then
,of cubic and cvcc
8
tant will be x“ which,if operated on by
c%* cil
x“ gives a
result manifestly proportional to x8
y”. And the theorem now
prov ed being independent of linear transformation,if true for
any form of the cubic,is true in general. The canonical form
,
then,being assumed as above
,we proceed exactly as in the last
Art icle,and we solve for 71. from the equation
“11
a
1as,
“8 3 K , “
4?
“47
which,when expanded
,will be found to contain only even
powers of a. If we suppose uvw reduced as above to itscanonical form x”
y‘, the three factors of which are
“ 31, H wy, w+ w”
y,
where a) is a cube root of unity, then it is evident from the
above that the corresponding canonical form for the sextic is
A (90 3 (50+ ms)“ 0 (w My)
" I? (936 —y6
)
It can be proved (see Lesson XVI I.)that if u, v, w be the factorsof the cubic
,then the factors of the evectant used above are
a v,v w
,w u
,so that the canonical form of the sextic
may also be written
u°+ (u — v)(v— w)(w— a).
172. In the case of the octav ic the canonical form is“8+ v
sw8+ z
8+ xu
fivgw
flz8
3
144 SYSTEMS OF QUANTICS.
for if we operate on u"v”w2.e2with a symbol formed according to
the same method as in the preceding Articles,the result will be
proportional to uvwz . This will be proved in a subsequentLesson.
A s for higher canonical forms we content ourselves with againmentioning that for a ternary cubic
,v iz .
and that given by Mr. Sylvester for a quaternary cubic,
LESSON XVI.
SYSTEMS OF QUANTICS.
173 . IT still remains to explain a few properties of systemsof quantics
,to which we
“devote this Lesson. An invariantof a system of quantics is called a combinant if it is unaltered(except by a constant multiplier)not only when the variablesare linearly transformed
,but also when for any Of the quantics
is substituted a linear function Of the quantics. Thus the elimi
nant of a system of quantics u,
'v,
'
w is a combinant. For
evidently the result of substituting the common roots of ex ina + 7w+ aw is the same as that of substituting them in u ; andthe eliminant Of a Nu+ aw, 77, w is the same as the eliminantof new. In addition to the differential equations satisfied byordinary invariants
,combinants must evidently also satisfy the
equation7
a d] bdl+
It follows from this that in the case of two quantics a combinantis a function of the determinants &c. ; in the
case of three,of the determinants &c. ; and will accord
ingly vanish identically, if any two Of the quantics becomeidentical. If we substitute for u
,v ; Xu + p v, 7t
'
u + a'
v,every
one of the determinants (ab'
)will be multiplied by (ha —Na);and therefore the combinant will be multiplied by a power
146 SYSTEMS OF QUANTICS.
the 2 (n 1)double points of the involution determined bvU.W
'
175. If u and 7) base a commonfactor, this will appear as a
squarefactor in their Jacobian. First let it be Observed, that
since 7i a : xa, yum 7m= xv
, yv2, then if we write J foruzol ,we shall have n (a v2 ru
g) xJ,72 (we)l va
l) -yJ.
Differentiating the first of these equations with regard to y, andthe second with regard to x
,we get
pus,
xJ,, n (m; yJ;.
It follows from the equations we have written, that any value aof x which makes both a and o vanish
,will make not only J
vanish but also its differentials J” J” and therefore x a mustbe a square factor in J.
Or more directly thus: let u se, where B= lx+m3j;the“14
.14> 734mu2=W ? v
.= All mtlr Bit} ;
and ufig +73 1(Me. ¢nlf l
whence (71 1) u,v,)
(n 1)B”
as.) B(193 my)(self. Ma)(int , fl ash)
It follows from what has been said,that the discriminant of the
Jacobian of u,7) must conta in R their resultant as a factor ;
since whenever R vanishes,the Jacobian has two equal roots.
Thus in the case of two quadratics,
b:01377 7 (a
'
,
the Jacobian is (ab’
)x”
(ao’
)xy (bc'
)y’
,
whose discriminant is (ab’
)(bc'
) which is the eliminant ofthe two quadratics. In the case of quantics of higher order,the discriminant of the Jacobianwill, in addition to the resultant,contain another factor, the nature of which will appear fromthe following articles.
In like manner, for a temary quantic, theJaeobian of U, K W is the locus of the
double points of all curves of the system 17+ kV-l IW which have double point-l.And so in like manner for quantics with any number of variables.
SYSTEMS OF QUANTICS. 147
176. It has been. said that we can always determine k,so
that u+ he shall have a square factor. But since two conditionsmust be fulfilled
,in order that u + he may have a cube factor,
k cannot be determined so that this '
shall be the case unless acertain relation connect the coefficients of u and v. This condi
tion wi ll be of the order 3 (n 2)in the coefioz’
ents both of u and v.
If (as— oi)abe a factor in u + kv+ lw, w a will be a factor
in the three second differential coefficients,or w= a will satisfy
the equationsu +kv Zw
u= 0
,wi z+ ~ho Zw = O
,
'
u22+ hvm+ he”
: 0,
whence eliminating k and l,a: a will satisfy the equation
“1 17
”117wn
”127
”127
“221
”as?was
O’
If then we use the word treble-po int in a sense analogousto that in which we used the word double-point (Art.we see that the equation which has been just written
,gives
the treble points of the system u + kv+ lw ; and since the
equation is of the degree 3 (n there may be 3 (n 2)suchtreble points. But we could find the number of treble po intsotherwise. Suppose we have formed the condition that u -l—hv
should admit of a treble po int,and that this condition is of the
order p in the coefficients of a . If in this condition we sub
stitute for each coefficient (a)of u , a la"
,we get .an equation
of'
the degree 10 in l ; and therefore p values of I will befound to satisfy it. In other words
_p quantics of the systemu + 7w lw will have a treble point. It follows then from what
has just been proved that p = 3 (ri And the same argument proves that the condition in question is of the order3 (n 2)in the coefficients of v.
This condition is evidently a combinant for if it is possibleto give such a value to h
,as that u + /cv shall have a cubic
factor,it must be possible to determine k, so that (u + av)+ kv
shall have a cube factor.
177. If u he have a cube factor (m then the Ja cobianof u and 1)will contain the square factor (a: For the two
d ifferentials u,kv
l ,u,kv
9will ev idently contain this square
148 srsrsms or QUANTICS.
factor, and therefore it will appear also in the Jacobian whichmay be written (a t he.)'os (”a lav
a) If then S= 0 be thecondition that u + 701)may hav e a cube factor
,S will be a factor
in the discriminant of the Jacobian, since if S= 0 the Jacobianhas two equal roots
,and therefore its discriminant vanishes.
If R be the resultant,the discriminant of the Jacobian can
only differ by a numerical factor from RS. For since theJacobian is of the degree 2 (n— l), its discriminant is of thedegree 2 {2 (n 1) 1} in its coefficients
,which are of the first
order in the coefficients of both a and Now R is of the ordern in each set of coefficients, 8 of the order 3 (n Boththese are factors in the discriminant and it can have no othersince
n+ 3 (n {2 (n— l)—l } .
178. If we form the discriminant of a he,this considered
as a function of 70 will have a square factor whenever a and a
have a common factor. In fact (Art. 107)the discriminant ofu + kv will be of the form dr. But if aand e have a common factor, we can linearly transform a and 1)
so that this factor shall be y, that is to say, so that both a
and a'
shall vanish. The discriminant will therefore have thesquare factor and since the form of the discriminantis not affected by a linear transformation of the variables
,it
always has a square factor in the case supposed.It follows that if we form the discriminant of u+kv, and.
then the discriminant of this aga in considered as a function ofk,the latter will contain as a factor R the resultant of u and v.
For it has been proved that when R= O, the function of Ithas two equal roots, and therefore its discriminant vanishes.For example , the discriminant of a quadratic ao—h
”
,becomes
by the substitution of a + ha ' for a,&c.
(ac b”
) 70 (ao'
ca'
2bb'
) k”(a c
whose discriminant is
(ac h“)(a
'
o' b 4 (ao
'
ca'
But th is is only ' a form in which, as Dr. Boole has shewn, theresultant of the two quadratics (a , b, me, (a ', h’
,01 371 l
can be written. This form,all the component parts of which
150 SYSTEMS OF QUANTICS.
3 (n and it will be proved in a subsequent lesson that Tis of the order 2 (n —2)(n and
2 (n —1)(2n — 2)(77.
180. It was stated (Art. 173)that every combinant of u, v
becomes multiplied by a power of (MIX Np.)when we sub
stitute h a + av, for a,o . It will be useful to prove
otherwise that the eliminant of u,7; has this property. First
,
let it be observed that if we have any number of quantics,
one of which is the product of several others,a,v,ww
’
w"
,their
resultant is the product of the resultants (uvw),For when we substitute the common roots of u
,v in the last
and multiply the results,we evidently get the product of the
results of making the same substitution in w,w
'
,w
"
. Again,
the resultant of u,v,low is the resultant of u
,v,w multiplied by
km”
since the coefficients of w enter into the resultant in the
degree m e. If new R (u, v)denote the resultant of u c whichare supposed to be both of the same degree n
,we have
pf"
R (7m m), Na ,u'
v) R (Mr/
u Na p'
v)*
R { (M’
X'
s)u, Nu u’
v} (M1 W if”)
(MU Nlt lfl
ll'mR (u ,
whence R (7m av, Na pfv) (Ml Ma)"
R (u,By the same method it can be proved that the eliminant of7m+ pv + vw
,Nu + it
"
v —l-vw, l s (hp/v timesthat of a
,v,w,and so on.
181. If U,V be functions of the orders m and 92 respectively
in u,c,which are t hemselves functions of w,
'
y of the order p ,and if D be the result of eliminating u
,between U, V; then
the result of eliminating x, y between U, V will be DP times
the math power of the resultant of u
,v. For U may be re
solved into the factors a — av,a — Bv, &c., and V into u — a
’
o,
a — B'
v,&c. And
,Art. 180
,the resultant of U
,V will be the
product of all the separate resultants u av,u a
'
o . But one ofthese is (a There are mn such resultants. When
The resultant of u he, 21, beingthe same as the resultant of a , 11, Art. 173,
we next subtract fl. times the second quantic from the first.
SYSTEMS or QUANTICS. 151
therefore we multiply all together,we get the ma
“ power of
R (u , v)multiplied by the pm power of (a &c.
But this last is the eliminant of U,V with respect to a
,v.
182. Similarly,let it. be required to find the discriminant
,
with respect to w, y, of U
,where U is a function of u
,v.
First,let it be remarked (see Ar t. 106)that the discriminant
of the product of two binary quantics a , 'v is the product of thediscriminants of u and v multiplied by the square of the ir re
sultant.
If then U (u am)(a Be), &c . the discriminant of U willbe the product of the discrim inants of a — ao
,u ,Bv, &c. by
the square of the product of all the separate resultants u av,
u —Bv. But,as before
,any of these will be (a — B)"R (u ,
'v).
If then m be the degree of U considered as a. function of a,0 ;
there will be 5m (m 1)separate resultants,and the square of
the product of all will be (cc— B)”
(a &c. x
But (a (a &c. is the discriminant of U considered as
a function of a , v. If then we call this A , we have provedthat the product of the squares of the separate resultants isAPR
m‘m' ” Let us now consider the product of the discriminantsof u — av
,u — Bc, &c. ; this is the result of eliminating a be
tween the discr iminant of u cm,which is a quantic of the
order 2 (p 1)in a,and the quantic of the m‘h order got by
substituting u = av in .U. Or this product has been otherwiserepresented by Mr. Sylvester. If “
0160be the coefficients of
a ’ in u,c,then (Art. 104)the resultant of a — av
,16,— av
,will
be ao
ab0times the discrim inant of u av. But
E(u, v)E(u— ao,u,
— ao,u,c cw a v- uv
l).
Now (Art. 175)p (u,v where J is ulvz
— azvl ,
and
E (u —ow, y)is a
o— ab
o. It appears thus that the discriminant
of u — ao differs only by a numerical factor from the resultant
of (a — ao,J)div ided by R (u, v). The product then of all the
discriminants will be the resultant of J and the product u ow,
u Bo, &c. ; in other words, the resultant of U, J div idedby the m
mpower of E (u, Thus we have Mr. Sylvester’
s
result (Comp tes Rendus, LVIII., 1078)that the discriminant of
U with respect to my is A"R (u, (U, J). But it will
152 SYSTEMS or QUANTICS.
be observed that the result expressed thus is not in its mostreduced form since R (U, J)contains the factor R (u,
183 . It is not my purpose in this volume to enter with anydetail into the theory of ternary and quaternary quantics. For
as all this theory has a geometrical meaning,it will be naturally
treated of in geometrical treatises ; for instance, in my Geometry of Three Dimensions
,and in the New Edition
, whichI hope to publish of my Higher Plane Curves. It will beproper however here to shew what corresponds in
,the case ofternary and quaternary quantics to the theory just explainedfor systems of binary quantics. Let , then u and v be two
ternary quantics,and let us suppose that we have formed the
discriminant of a + hv. Then this discriminant considered as
a function of lo will have a square factor ; in the first place,
if the curves represented by u and v touch each other. For
we have seen (Art. 1 13) that if the equation of a curve beits discriminant is of the form
a 9 + bodr + c2
x. The discriminant then of u +kv will beof the form 0+ But if we take forthe point my, a point common to u and v
,both a and a
'
willvanish ; and if we take the line y for the common tangent
,both
6 and b’
vanish ; and the discriminant will be of the formand therefore will always have a square factor in
the case supposed.
184. Again,the discriminant will have a square factor
if u have either a cusp or two double points. The discriminantA of a ternary quantic is the condition that a
na, ,u,should
hav e a common system of v alues. If,however
,we have either
two double points, or a cusp,an
as,
aswill have two systems
of commonvalues,distinct or coincident
,and therefore (Art. 99)
not only will A vanish,but .also its differentials with respect
to all the coefficients of a . The discriminant then of u +hv,
1 kc'
a’A b
’
a’A '
ll th'
be ing in genera A (y +m— 4 0. W1 1n 1s
case be divisible by h”. And as in Art. 179
,it will be divisible
by (k a)2 if the curve a av have either a cusp or two double
points. Let then R= 0 be the condition that u and v should
154 SYSTEMS or QUANTICS.
that the tangents are identical. If then we eliminate thevariables between u
,v,
and the determinant above written.
which is of the order m + n — 2 in the variables and of thefirst order in the coefficients of each curve
,the result will be
R multiplied by the result of elimination between. u,v and
ax + &c. Now the complete resultant is of the order run in
a, ,8 , ry ; it contains the coefficients of u in the order
n (m + r¢
and in like manner those of v in the degree m (2u+mThe resultant of u
,v,ax &c. conta ins a
, ,8,«y in the degree r m,
the coefficients of u in the degree n,and those of v in the
degree m. Hence R contains the coefficients of u in the degreen (2m+ n — 3) and those of v in the degree m (2n+mWhen we and n are equal we get the number already stated3n (nThis result may be otherwise found as follows : The order
of the tact-invariant in the coefficients of v is evidently thesame as the number of curves of the form u + hv +uw whichcan be drawn to touch u
,where w is another curve of the same
order as v . But the point of contact with u is easily seen tobe a point on the Jacobian of v
,v,w. And this being a curve
of the order (m 2n 3)meets u in m (m 2n 3)points.*
186. The theorem given,Art. 106
,for the discriminant of
the product of two binary quantics cannot be extended to
ternary quantics ; for the discriminant of the product of twoWill
,in this case
,vanish identically. In fact
,the discriminant
is the condition that a curve should have a double point ; anda curve made up of two others must have double points ;namely
,the intersections of the component curves. Or
, without any geometrical considerations
,the discriminant of uv is
the condition that values of the variables can be found to
satisfy simultaneously the differentials uv;+ vv, ,uv
2+ vu
2,&c.
But'
these will all be satisfied by any values which satisfy
I reserve for a geometrical treatise an account of the modifications which thetheorems of this chapter receive when u and v are not the most general quanticsof their degree, but denote curves having double points.
SYSTEMS or QUANTICS. 155
simultaneously u and v ; and such values can always be foundwhen there are more than two variables.But the theorem of Art. 106 may directly be extended to
tact-invariants. The condition that u should touch a compoundcurve vw. w ill evidently be fulfilled if u touch either v or w
,
or go through an intersection of either. For an intersectioncounts
,as has been said
,as a double point on the complex
curve ; and a line going through a double point of a curve isto be considered doubly as a tangent. Hence if T (u, v)denotethe tact-invariant of u, v, we have
T (u , vw)= T (u , v)T (u , w){R (u v
where R (u , v, w)is the resultant of u,v,w. And the result
may be verified by comparing the order in wh ich the coefficientsof u
,v,or w occur in these invariants. Thus for the coefficient
of u,we have
187. What was said, Art. 185,as to the tact-invariant of two
ternary quantics may be extended almost word for word to
that of a system of three quaternary quantics, or of k 1,k—ary
quantics. Thus the tact-invariant of three quaternary quanticsexpresses the condition
'
that two of the mnp points of the sur
faces represented by them coincide ; or,in other words, that
at any point of intersection the three tangent planes to the
surfaces have a common line. And it is found , as in Art. 185 ,
by dividing the resultant of v, v, w, and
“17
“27
“37
“4
”17
”a?
”37
v4
wl ’we)we)
704
a , 13 7 7? 8
by the resultant of u, v, w and In this way it
is seen that the tact-invariant contains the coefficients of u in
the degree np (2m + rz p a result which is confirmed byobserving that this is the number of
“points in which the inter
section of vw meets the Jacobian of u, u'
,v,to . See Geometry
of Tbree Dimensions , p . 438. And, in like manner, in general
the tact-invariant of lo 1,k—ary quantics u
,v,w,&c. contains
156 SYSTEMS or QUANTICS.
the coefficients of n in the degree found by multiplying theproduct of n
, p , &c. by 2m + n+ p + &c.—k.
188. In order to complete the subj ect we give here thetheory of the tact-invariant of two quaternaryalthough we shall have to use principles which are to be
established in a subsequent Lesson. The same argument as
that used before shews that the order, in the coefficients of U,
of the tact-invariant of U,W is the same as the number of
quantics of the system U+ K V which can touch W; U and Vbeing of the same order. But comparing the coefficientsof thetangent planes of U+ 7~V and W we see that the point ofcontact must satisfy the system
,equivalent to two conditions
,
UnU2:
U.
Vn VsVs)V.
”717WeW
3 ,W.= 0
It will be shewn in the Lesson on the order of systems of
equations, that this system is of the order
+ v + p v+ v7t + 7tmwhere p , v are the orders of U
1:VI,VV
, ,&c.
,that is
,in the
present case m 1,m 1
,n 1. The order is then
n2+ 2mn+ 3m
”— 4n— 8m+ 6.
And the required order of the tact-invariant is n times thisnumber ; since the point of contact is got by combining withthese conditions W: O
,which it must also, satisfy.
189 . There is now not the least difficulty in forming thegeneral theory of the class of invariants we have been con
sidering, to which Mr. Sylvester proposes to give the nameof osculants. Let there be quantics
,U,V,W,&c. in k
variables ; then the osculant is the condition that for the same
system of values which satisfy U,V,&c. the tangential quantics
mU,’
yU,’
&c.,&c.
,shall be connected by an identical relation
See Geometry of Three Dimensions, p. 439.
158 APPLICATIONS TO BINARY QUANTICS.
k 2 with U, that of the system of k 2 with V, and the squareof the resultant of all the quantics.It would
,next in order
,be proper to discuss the form of the
discriminant of the Jacobian of XU+ ,a V+ VVV, where U, V, Ware ternary quantics
,and that of the discriminant of the dis
criminant with respect to 7x, p , v. This theory
,when U
,V,W
are quadrics,will be given in Lesson XVIII. In other cases
the latter discriminant vanishes identically. I do not know thegeneral theory of the former.
LESSON XVII.
APPLICATIONS TO BINARY QUANTICS.
190. HAVING now explained the most essential parts of thegeneral theory
,we wish in this lesson to illustrate its application
by examining in some of the simplest cases the different invariants and covariants which a quantic may possess. The
method of Art. 117 shows that a binary quantic has in generaln — 2 independent invariants. For it was there proved, thatthere may be n — 3 absolute invariants ; and since we see
,as in
Art. 118,that from any two ordinary invariants an absolute
invariant can be deduced,the number of ordinary invariants
is at most one more. If we have two invari ants S and T ofthe same degree then S a T (where a is any numerical factor)wi ll of course be also an invariant. We do not consider invariants included under this form as new invariants
,nor as
essentially distinct from the invariants S and T. Nor again,
if S were of the second degree,and T of the third
,would
S8+ uT
2be sa id to be a. new invariant essentially distinct from
S and T. But if another invariant were not rationally ex
pressible in terms of S and T,but only connected with them by
an equation such as R”: S
3
+ a T”
,then we speak of R as a
new invariant distinct from S and T,though not independent
of them. Thus then,though the number of indep endent in
variants of a quantic is, as has been said,n — 2
,the number
of distinct invariants is unlimited after we passlthe sixth degree.
APPLICATIONS To BINARY QUANTICS. 159
If we have a system of two quantics,of the degrees m and n
respectively ; proceeding still by the method of Art. 117,the
number of equations to be satisfied being m + n + 2 , and the
number of constants at our disposal being still only 4,we find
that there are m n 2 absolute,and m n 1 ordinary
,inde
pendent invariants of the system. That is to say, there will bein general three new independent invariants of the system in
addition to the vi — 2 and n — 2 independent invariants of thequantics considered separately. If one of the quantics be eitherlinear or a quadratic
,there will be only two new independent
invariants. This is because the number n — 2 does not expressthe number of independent invariants of a linear or quadraticquantic
,that number being in both cases one more ; that is to
say, 0 in the one case and 1 in the other. The number of otherinvariants will therefore be one less than in the general case.The invariants of any quantic and of the linear system
mf yr; &c. (see Art. may be regarded as contravariantsof the given quantic ; and in binary quantics contravariants may
be reduced to covariants by changing f and 77 into y'
and -w.
It has been shewn then that a binary quantic has in addition toits invariants
,only two independent covariants ; or
,since we
may take the quantic itself for one of these covariants,that all
covariants may be expressed (though not necessarily expressedrationally) in terms of the quantic
,its invariants
,and one
covariant.We now proceed to enumerate the fundamental invariants
and covariants of all the most simple systems.
191. The quadra tic. We have already stated the principalpoints in the theory of the quadratic form (a , b, 0190, It
has but one independent invariant (Art. v iz . the discri
minant ao — b Any other invariant must be a power of this
(ao We have already showed_(Art. 153)that it follows
by Hermite’s law of reciprocity
,that only quantics of even
degree have invariants of the second order whose symbol is
12 If we make y 1 in the quantic,it denotes geometrically
a system of two points on the ax is of x,and the vanishing of
the discriminant expresses the condition that these points shouldcoincide (Art.
160 APPLICATIONS TO BINARY QUANTICS.
In like manner the system of two quadratics
(a , b, 01 331 (a'
,
has the invariant 122 or ad a'
o — 2bb'
. When each quantic istaken to represent a pair of points in the manner just stated
,
the vanishing of this invariant expresses the condition (seeConics , p. 291)that the four points should form a harmonicsystem,
the two points represented by each quantic being con
jugate to each other. We have also proved (Art. 174)that the
covariant 12 (or the Jacobian of the system)represents geometrically the foci of the system in involution determined by the
four points.The eliminant of the system may be written in either of
the forms(ao
'
ca'
)z
4 (ba'
b'
a)(bc’
b'
c),
or (ac'
ca'
2bb’
)2
4 (ac b”
)(a c
Lastly,given a system of three quadratics
(a , by al so , ”2
1 (a abac Ix ,
the vanishing of the determinanta b c
a b c'
If
a,b
,c
expresses thecondition that the three pairs of points representedby the quadratics shall form a system in involution (Sonics,p .
192. l e cubic. We come next to the concomitants ofthe cub ic
(a , by 07 d a”3°
It has but one invariant (Ar t. v iz . the discriminantazd2 4a o
”4db
a311
202
6abcd.
The Hessian 122 is
(ao — b”
)x + (ad — bc)xy + (dd — c )y2
,
which may also be written as a determinantb, c
c,
d
162 APPLICATIONS TO BINARY QUANTICS.
193. When we wish to establish any relation between the
preceding covariants and invariants, we use the canonical forms,
which are,for U: ax
3
+ dy3
,the discriminant D= a
2d
”
; the
Hessian H : adxy ; and the cubicovariant, J= ad (arcs
Thus we can prove that the discriminant of J is the cube ofthe discriminant of U, for the discriminant of J in its canonicalform is as 6
. Aga in,we have been led
,by Art. 190, to foresee
that J is not independent of U and H ; and we can easilyestablish
,by the help of the canonical form
,the relation con
necting them,due to Mr. Cayley, v iz .
P DU2 — 4H”.
Mr. Cayley has used this equation to solve the cubic U,
or,in other words
,to resolve it into its linear factors. For
,
since J"— DU"’is a p erfect cube
,we_are led to infer that the
factors J i UVD will also be perfect cubes,and
,in fact
,the
canonical form shows that they are 2a“dai:
8and 2ad
2
y”. Now
,
since ma + yd’is one of _the factors of the canonical form
,it
immediately follows that the factor in general is proportional to
(UVD + J)+ (UVD — J)a linear function which ev idently vanishes on the suppositionU 0.
Ex. Let us take the same example as at p. 136, viz ., U : 9m:2 1890 17.
Here we have D 1600, J : 90w2y 63omy2 67031
3, whence
UJD
and the factors are 3a: y + (w+ 3y)
194. System of cubic and quadra tic. Let these be
(“a b: 0, (A :B)01271 f/l
2'
The simplest invariant of the system is got by combining thequadric with the Hessian of the cubic
,and forming the inter
mediate invariant of this system of two quadratics,when we have
I = A (bd c?
)— B (ad — bc)+ 0 (ac
The resultant of the system,formed by the method of Arts.
63 or 82,is
R it"O’a 6abB 0 .
2GaoO’
(2B2A O) ad (GABO 8B
“)
9b2A C
218bcABO + 6bd4£1 (2B
2A 0)
902 20 6c A
"dzA
".
APPLICATIONS To B INARY QUANTICS. 163
These two may be taken as the fundamental invariants of the
system. In comparing other invariants with these it is con
venient to take A and O= O,which is equivalent to taking for
a: and y the two factors of the quadratic. I then reduces toB (be ad)and R to 8B
3ad. The Jacobian of the system is
(Ab—Ba) —Bb Ca).v’
y+ (Ad+Bc 2 Cb)wy”+(Bd Cc)y
“.
But it has,besides
,linear covariants. Thus
,if we substitute
differential symbols in the quadratic and operate on the cub ic,
We getL,= (a O
and if we operate in like manner with this on the quadratic,
we get
L, {aB O 6 (23
1+A C) ScAB dA
’
} a:
— dAB } y,wh ich expressed in terms of the roots at, B, 17 of the cubic and
a'
, B'
of the .quadric,are
2 (a' — a)(3 - 5)(Go 3 (a
' — d l (KY—fl )(a' —v)(00— 3 )
The resultant of L,and L when we make A and is
proportional to B abe. If
,then
,A denote the discriminant
A O B” of the quadratic
,we see that the resultant of L
, ,L, ,
expressed in terms of the fundamental invariants,is R 8A ] .
There are other linear covariants got by writing for a,b,&c.
the a,b,&c. of the cubi—covariant (as in Art. 138)of the given
cubic.
If we eliminate between the linear covariant L,and the
quadratic,we get the same result as if we eliminate between
L,and L
2. But, if we eliminate between L,
and the cubic,we
get an invariant distinct from those already given. If we make
A and 0 = 0,this invariant is B s
(ac°—db
8
), which we see is
not reducible to.
the prev ious forms. But its square can withoutdifficulty be reduced to those forms. For
,since the discriminant
of the cubic is
D a"d’
4m“4db
a3b
sew
18abcd,
sixteen times the square of the invariant now under consideration is
B°
(D a’d" 3b
gc"
18abcd)’64B
°ad
ao"
,
164 APPLICATIONS TO BINARY QUANTICS.
and we have only to write in this for Bbc,I Bad
,for B ’
ad,
fil l, and for B“,
A,when the whole is expressed in terms
of the fundamental invariants.
195 . System of two cubics. A system of two cubics(a , b, 0 , fi x
, (a'
,b'
,c'
,d'
Ix, y)”has for its simplest invariant
(see Art. 136, Ex . 2)(ad'
)— 3 (bc'
) which is a combinant,and
which we shall refer to as the invariant P. The properties ofthis system may be studied most conveniently by throwing theequations into the form Aus+ Bv
°+ Ow
”
, C'
w"
,a.
form to which the two cubics can be reduced in an infinity ofways. For the cubics conta in four constants each
,or eight in
all. And the form just written conta ins six constants explicitly ;and u
,v,10 contains implicitly a constant each
,since u stands for
m+ 7ty, &c. The second form then is‘equivalent to one withnine constants
,that is to say, one constant more than is necessary
to enable us to identify it with the general form.
Any three binary quantics of the first degree are obv iouslyconnected by an identical relation of the form an Bv «yw 0.
We shall suppose the constants a, B, ry to have been incorporated
w ith u,v,w,so as to write the two cubics
,Au
a+ Bv
°+ Cw
“,
A'
u’ B
'
v8O
'
w”
,where u v w 0.
Putting for w its value,and writing the cubics
(A) 070707B “
.CI“
;70
3
: (A ; Ol
a“ O
i
:B
and forming the invariant P of the system,we find it to be
(AB'
) (B C'
)The resultant of the system is found by solving between
the equations Au“ Be
“Ow
”0,A
'
u8 B '
os 0
'
w"
0,when
we get u”: v
“: w
°= (AB'
) and,substituting
in the identity u 'v w 0,the r esultant is
(As'
)é was?" 0
,
(B O'
) 27 (AB'
)(BO'
)
Now,if we denote the two cubics by U and V
,it has been
proved, Art. 176, that there is an invariant,which we shall call
Q, of the third order in the coefficients of each cubic,which
expresses the condition of its being possible to determine 7t sothat U XV shall be a perfect cube. Now this invariant is
166 APPLICATIONS To BINARY QUANTICS.
value of (ad'
) 3 We shall for the sake of establishing auseful general principle
,giv e another way of making the same
calculation. We know that we may substitute in any binary0 d d 0 l U 0quanti c7g
, da:for a: and y, and so obta in an Invariantive
operative symbol. Now when th is change is made in a functionexpressed in terms of x
, y, z , where z is (a: y), z will becomed d
And when the Operation is performed on a function
similarly expressed,since its differential with respect to a: will
d d dzbe3} dz dx ’
or In Virtue of the relati on between at , y, z ,
d
div
substitute in any covariant for x, y, z respectively
d d d d d d
dy dz ’ dz dx ’ dx dy’
and so obtain an operative symbol which we may apply to anycovariant expressed in terms of w
, y, z , without first reducingit to a function of two variables. Thus
,in the present case
,we
find the invariant P by Operating on O'
z“,with
d d3
d d d d8
A bs; dz)+ B (i z da:
and the result only differs by a numerical factor fromas we found before.
In like manner we find that,in the symbolical notation, 12
as applied to a function expressed in terms of m, y, z , denotes
(d d d d
div,dy, dzr:
2dy, dy, dz , dy, dz dz
,doc
,dz
adx
,
d
dz ’we see that the rule may be expressed, that we may
198. The Jacobian of the system
Ax“By
“Oz
“,A
'
x”B
'
y“0’z
may be found by means of the formula last given, or directly asfollows. In v irtue of the relation connecting z with a: and
y, the differentials of Uwith respect to x and y are proportionalto Aw —0z
2
,By
2 — Oz"
; and U, V, U2V, ,is
(AB’
)coz
y”
(B 0 y"z2
(OA’
)zed
”.
APPLICATIONS TO BINARY QUANTICS. 167
This is a. biquadratic, for which we may calculate the two invariants noticed in p. 112
,v iz .
S= ac 4bd T= ace 2bcd ad”
eb”
c”.
Putting in for z2
,and multiplying the Jacob ian by
six to avoid fractions,We get
3 : 6 (BC'
Ld = 3 iB 0'
)7c (B O
'
) (OA'
) (143 )=P,
whence S= 3P ”
,T= 54Q— P
3. We shall presently show that
the discriminant of a biquadratic is S“27T
2. The discriminant
of the Jacobian therefore,is proportional to Q (P
827Q), which
agrees with Art. 177
199. If in general we form any invariant of U+ 7tV, andthen form any invariant of this again considered as a functionof 7t , the result will be a combinant of the system U
,V; that
is to say, it will not be altered if we substitute lUJI mV,l’
U+ m’
V for U, V. For,by this
ysubstitution
,we get the
corresponding invariant of U V,which is
equivalent to a linear transformation of 7t , -by which the
invariants of the function of A will not be altered. If then,
in the case of two cubics,the discriminant of U+ 7tV be
A 4E). 607k"
AIDA“EM
,and if it be required to calculate
the invariants of this b iquadratic, we may without loss ofgenerality, take instead of U and V two quantics of the systemU+ 7\.V which have square factors
,taking a: and y for these
factors ; and so write U For this
system we have P = ad -3bc, Q = b
”c"
(ad bc); the resultantP
“27Q being a
"d"'
(ad 9bc). Now,for this form
,the biquad
ratic is (d‘d26abcd 3b
"c"
) or multiplying by six to av oid fractions A = E = O
,B = 6ac
”
,D = 6db
’
,
0 = a’dg 6abcd 3b
2 = P”
12bfc’. Hence,
S= 3 0”— 4BD = 3P (P
3 — 24Q);T= 2B CD - 0 8 = (P
G
whence the discriminant of the biquadratic 8 9 — 27T 2 is pro
portional to Q)which agrees with Art. 179. We
infer from Art. 117,by counting the constants, that a system
of two cubics cannot have more thanfive independent invariants ;
168 APPLICATIONS To BINARY QUANTICS.
and we see here that P and Q can be expressed,though not
rationally,in terms of the five A
,B,0,D,E.
200. We may also conveniently use the form ax3+ 3bm
2
y,in examining the relations of invariants which are
not combinants ; but it will be necessary to verify by moregeneral forms
,that the relations found to ex ist in this particular
case are true in general. The invariants are all of even orderin the coefficients of the system ; and there is no invariant of thesecond order but the combinant P. We may form severalinvariants of the fourth order
,as follows. Form the Hessian
of U+ 7tV, which will be
(a)B7 vi a), y)2+ 7“(a
'
, Bl
) (a a 13 1”YKw,W,
where the absolute term and the coefficient of A”are the Hessians
of U and Vrespectively ; and the coeficient,of 7» is an interme
diate covariant. Now,we
,
may form the quadratic invariant of anyone
,or any pa ir of these invariants, 4ary 2 (my
'
087) ,BB’
,
&c., and these will all,be invariants of the system. If we
remember that the discriminant of the Hessian is the same asthat of the quantic, and therefore identify
(B NB'
7u2
3 )”
4 (a M’ W "
)(7 Wwith (A ,
B, 0, D, E1 1, we identify all these invariants with
A,B,D
,E except the two 2 (afy
"
B”4a
’ry'
,which
we shall call J,K,and which we find to be connected by the
relation 2J+ K : 60.
In the form arc“
3b9c”
y A (3cxy1a
the Hessian is
b l ac)$2
A (ad be)mg (l bd Age”
)y”
;
and we find J 2bzc"
,K =
1a2d2 6abcd b
’c”
; and, using thevalue 60 = a
2d"— 6abcd 3b
202
,we have GO= P 2
+ 6J, K =P2+4J,
relations which are without difficulty verified in general.Lastly
,if we form the cubicovariant
,as in Art. 138, of
U+ 7\V, the coefficients of the powers of A give us four co
v ariants of the same kind,v iz . those of U and V
,and two ’
intermediate covariants. And if we form the combinant P of
each pair of these cub ics,we get six invariants Of the sixth
order in the coefficients. Four of these will,however
,be only
PA,PB
,PD,
PE. The remaining pa ir, viz . the combinante
170 APPLICATIONS To BINARY QUANTIes.
which does not affect the invariant, it is sufficient to prove thetheorem for that form. Now,
in the first place, we say'
thatany invariant which vanishes when m 0
,will also vanishwhen
m i 1 . For the form x4
y“is changed
,by the transformation
of w+y and w y for a: and y, into the formand by the change of ac+ y N/( x —
y V( 1)for a: and y,into x
46x
2
y2
+ y4 Thus
,then
,we see that if m be a factor
in any invariant, m2
1 will also be a factor; or the invariantwill be div isible by m m
"
,that is to say by T.
Let us take now any invariant expressed in terms of thegeneral coefficients a
,b,c,d,e ; and we say that if it does not
vanish when we make b= 0,e = 0
,d = o
,the part remaining
must be a power of ac. For it ev idently must be a symmetricalfunction of a and e ; and it cannot be of any form such as
a"
e"
,since the weight of every term must be the same. Let
this part remaining then be ake"
,and let us subtract from the
given invariant (ae 4bd or S" : the rema inder,then
,will
ev idently vanish when we make b= 0,c= 0
,d = o ; or, in the
case of the canonical form for which b and d are always 0,it
w ill vanish when m 0. By what has been proved then it mustbe div isible by T : that is to say, we have proved that the invariant is of the form S"+ T4). But
,by the same argument
,
we prove that qt is of the form Sk'
+ Tap ; and so on : so that,by repeating the process
,the invariant can altogether be ex
pressed rationally as a function of S and T.
203 . To exp ress tbc discriminant in terms of S and T. It
has been already remarked (Art. 107)that the discriminant Of
a quantic must vanish,if the first two coefficients a and b vanish ;
for,in that case
,the quantic has a square factor being divisible
by y". On the other hand it is also true that any invariant
which vanishes when a and b are made 0,must contain the
discriminant as a factor. Such an invariant,in fact
,would
vanish whenever the quantic had any square factor (rc ay)for
,by linear transformation, the quantic could be brought to
a form in which this factor was taken for y, and in whichtherefore the coefficients a and b= O. But an invariant whichvanishes whenever any two roots of the quantic are equal, mustwhen expressed in terms of the roots contain as a factor the
APPLICATIONS TO BINARY QUANTICS. 171
difference between every two roots : that is to say, must containthe discriminant as a factor.It is easy now, bymeans of S and T
,to construct an ih
variant which shall vanish when we make a and b 0. For on
this supposition 8 becomes 30” and T becomes — e°
; therefore183
27T‘vanishes. Now this invariant of the sixth order in
the coefficients is of the same order as that which we know(Art. 101)the discriminant to be. It must therefore be thediscriminant itself, and not the product of the discriminant byany other invariant. The discriminant is therefore
(ac 45d 27 (ace 250d _ ad”
ebs
We can in various ways v erify this result. For instance,it
appears, from Art. that the discriminant of the canonicalform a 6mx
2
y’+y
‘is . the square of the discriminant of the
quadratic cc" 6mmy+y’
; that is to say, is (1 But
(1 (1 3m”
)a
27 (m m8
)”
204. We may also derive this result from the form P 327Q
of the resultant of a system of two cubics given,Art. 195 . The
combinant P of the system U2 ,
am3 3bw”
y 3cwy’dy
"
,bas
e3cx
2
y 3dryg
eys
,
is (ae— bd) 3 (bd that is to say, is no other than the
invariant S of the biquadratic. And Q, if calculated,will be
found to be a perfect square,namely
,the square of the invariant
T. We can give an a p riori reason why Q should in this casebe a perfect square. For Q : 0 expresses the condition that forsome value of a, U, + a ll, shall be a perfect cube. If this beso,we can (Art. 126)linearly transform so that it shall be the
differential with regard to a: wh ich shall be the perfect cubesuppose (ze+ Then the quantic itself must be of the form
and in this case we have also 0,— 7tU, a per
feet cube. Thus Q 0 expresses the condition that the quartic
We may also see this directly, thus : The redultmt of
is the it“ power of ab
’ 'b, since the substitution of each root of the first equationin the second gives ab' a
’b. Now the discriminant of as:4 Gca fl
y2 cy
4 is the
resultant of as:3 Bossy“, 3cw
’y of . If we sub stitute a: 0 in the second , and y 0
in the first, we get results 6, a , respectively, and the resultant of am” 3og2, 3cac
2eyz
is (ac 902V. The discriminant is therefore ae (ae
172 APPLICATIONS To BINARY QUANTICS.
shall be the sum of two fourth powers ; and,in this case
,
AU,+ 14 0; can in two ways be made a perfect cube.We can also easily apply the theory of a system of two
cubics by writing the quartic under a form more general thanthe canonical form
,v iz . Oz
“
,where
In this case,then
,we have a =A + O, e= B + 0,
and we easily calculate T= AB O. But
if we equate to nothing the two differentials,viz . Aw
sCz
s
,
BysOz
”
, we get as“, y
”
,z3respectively proportional to BC’
,
CA,AB ; and, substituting in we get the dis
criminant in the form
(B O)+ (0A)+ (AB)or (B 0 + 0A + AB)
’or 8
3 — 27T"= O
205 . From the expression just given for the discriminant ofa quartic in terms of S and T
,can be derived the relation
(Art. 193)which connects the covariants of a cubic.
If we multiply two quantics together,the i nvariants of the
compound quantic will be invariants of the system formed bythe two components. If then we multiply a quantic by an?+ 7
the invariants of the compound will (Art. 130)be contravariants
of the original quantic ; and if we then change if and 77into y and
as,will be covariants of it. If we apply this process to a cubic
,
the coefficients of the quartic so formed will be“A,
‘l‘659 “ ax), 1} 1“ (0231 309 97 d” ;
and the invariants S and T Of this quartic are found to be thecovariants QB ,
of the cubic. But the discriminant of theproduct of any quantic by a t? yn, by Art. 106, becomes, whentreated thus
,the discriminant of U multiplied by U
". Express
ing then the discriminant of the compound quartic in terms ofits 8 and T
,we get the relation connecting the H
,J,and dis
criminant of the cubic.
206. The Hessian of the quartic is the evectant of T,and is
(ac b”
)w"
2 (ad be)wa
y (ae 2bd 3c”
)tidy”
2 (be ed)my”
(cc d”)y
‘,
which, for the canonical form,is
m (a:4
y“) (1 3m
”
)ze’
y".
174 APPLICATIONS TO BINARY QUANTICS.
207. We have already shown (Art. 169)how to reduce aquartic to its canonical form
,a problem in which is included
that of the solution of the equation,since when it has been
reduced to the form it can be solved like aquadratic. The reduction may also be effected by means of
the values given for S and T. Imagine the variables transformed by a linear transformation whose modulus is unity
,
and so that the new b and d Shall vanish : then we haveS= ae+ 3c
z
,T= ace e
’
; and the new 0 is given by the equation 4e
3 — Se+ T : 0. We get the a: and y which occur inthe canonical form from the equations
h a s Ba u er,H = acx + (ae so
”
)new oerwhence 0U H : (9c ac)$
2
312
Our process then, is to solve for c from the cubic just given ;then with one of these values of c to form cU— H whichwill be found to be a perfect square. Taking the square rootand breaking it up into its factors we find the new a; and y, and
consequently know the transformation by means of wh ich the
given quartic can be brought to the canonical form. Having
got it to the form we can of course,if we
please,make the coefficients of a " and y
4 unity by writing$2and y
2for x
2 M(a), and y2
V(e).
Ex . Solve the equationz ‘+ 8:r3y
We have here 8 2 216,T : 756, and our cubic is 403 2160 756, of which
c 3 is a root. The Hessian is
H 614 6ow3y 72:192y2 24xy
3
3U H 9 (m‘ 4z 3y 12:7:2y2 32a:y
8 64y‘) 9 (a:2 2:1:y 8y
2)2.
The variables then of the canonical form are X x 2y, Y a: 4y, which give
63: 4X 6y X Y ; whence substituting in the given quartic the canonical
form is found to be 3X 2X 2Y2 Y4. The roots then are given by the equations— l)=x - 4tl
208. Mr. Cayley has given the root of the quartic in a more
symmetrical form. Let e, ,02,03be the roots of the cubic of the
last article, and it has been shown that H 0,U,H c
,U,
H e3U are respectively perfect squares ; the square roots being
of course of the second degree in a: and y. But further
(oz — ca)(H (os — c
,)(H — ceUfi-l (c, (H— c
nUfi,
APPLICATIONS TO BINARY QUANTICS. 175
is also a perfect square,and its root is one of the factors of the
quartic. It is only necessary to prove that this quantity is a
perfect square , for it evidently vanishes when U 0. We workwith the canonical form, taking for simplicity a and e 1. Now
if we solve the equation 4z8
z (1 e c"
0,we find the
three roots to be e, i} (e+ and the three cor
responding values ofH eU are
(1 9c“)70
’
s”
, i (se 1)(r i (ao 1)(w’
Now in order that any quantity of the formmy+ 3 (a
f
a r/2
) v (w’
may be a perfect square, we must obviously hav e a24
which is verified when
a’ = 1 9c
”
, (3e (3c+ 'y”: g, (3c+ (3c
Ex. If this method be applied to the last example, the other values of c are
1} 3 i 9 .j( and the squares of the linear factors of the quartic are given
in the form—2wy 83/
z li i {1—J( 4( —Z2J(
i i {1 J( [ll J( $3
{10 2 J( my {2 10J( 3)l If]
209. It remains to distinguish the cases in which the transformation to the canonical form is made by a real or byimaginary substitution. The discriminant of the canonical formis,as we have seen (p. 171, note), cc (ae and since the
Sign of the discriminant is unaffected by linear transformation,we see that whenever the discriminant is positive
,a and e of
the canonical form have like signs ; and when the di scriminantis negative
,unlike signs. Now the form aw
‘Goa-
”
y2
ey‘
ev idently resolves itself into two factors of the form,either
(ao’+ or (a
’ — Ry”
)(af
P792
); that is to say, the'
quartic has either four imaginary roots or four real roots. On
the contrary,if a and e have opposite signs, the two factors are of
the form (7172 Ly“)(a:
2or the quartic has two real and two
imaginary roots. Hence then when the discriminant is negative ; that is to say, when S
”is less than 27T"
,the quartic has two
real roots and two imaginary ; and when the discriminant ispositive
,it has either four real or four imaginary rootsfi" Now
The signs of the invariants do not enable‘
we to distingui sh the case of four
real roots from that of four imaginary ; but the application of Sturm's theorem
176 APPLICATIONS TO B INARY QUANTICS.
the discriminant of the equation 403Se+ T= 0 is 27T
’
therefore (Art. 164)when S 3 is less than 27T’
,the equation in c
has one root real and two imaginary ; in the other case it hasthree real roots. The transformation therefore can only be effectedin one real way, when the quartic has two real and two
imaginary roots. It is easy to see that if a and e have likesigns
,in which case the equation can be brought to the form
a?
(Smash/2
+ y‘,it can by two other linear transformations be
brought to the same form for write a: y and a: y for a: and y,and we have (1 3m)as
“6 (1 m)in?
“(1 3m)y
‘. Write
ac+ y t/( l), and x —y V( l) for a: and y, and we have
(1+ 3m)oz."l 6 (m 1) 3771)y
“. Thus then when the
quartic has four real or four imaginary roots,though there are
three real values for 0,one of these corresponds to imaginary
values of a: and y ; and there are only two real ways of
making the transformation.
The same thing may be also seen thus. Imagine the quarticto have been resolved into two real quadratic factors
(a , b, “I“, y)
2
7 (a'
,a'
i
then these two factors U, Vcan by simultaneous transformation bebrought to the formAX “
B I”,A
’
X’B
’
Y”
,where X 2
and Y”
are the values of XU+ V corresponding to the two values ofgiven by the equation
(ao 79+ 2bb
'
)7t + (a c — b )= O.
In order that the values of A. should be real,we must have
the eliminant of the two quadratics positive,or
(a a'
)(a B'
)(B a'
)03 — 3 )
positive. Thus then,when the quantic has four real roots
,if
we take for a and B the two greatest roots, and for a'
and B’
the
two least, or again,if we take for a and
,3 the two extremeroots
,and for a
’
and ,8'
the two mean roots,we get real values
for 7t . In the remaining case we get imaginary values. If
shews that (the discriminant being positive), when the roots are all real, both the
quantities b2 an and 3aT 2 (b2 ac)S are positive, while if either is negative
the four roots are imaginary. (Cayley, Quarterly Jaw-m l, vol. IV., p.
178 APPLICATIONS TO BINARY QUANTICS.
The last is a perfect square,because
,as we have just mentioned
,
instead of six cases where aU+ GfiH has a square factor,we
have three cases where it has two square factors.Hermite has noticed that if we call G the function of a
, B,at3
9 lS'aiB
254718
3
,then the values just given for the S and T
of aU 6,8H are respectively the Hessian and the cubicovariant
of G. The discriminant of G differs only by a numericalfactor from the discriminant of U.
The covariants of aU+ G,BE are also covariants of U. Its
Hessian is(aBS-l U+ (a
‘”H,
which is the Jacobian,with respect to a
, ,3,of G and aU+ G,BE.
Since J is a combinant of the system U,H,the J of aU+ 6311
will be the same,multiplied however by the numerical factor G.
The Hessian of J is S2
U2 — 36TUH + 12SH
2
; which is theresultant of a U+ 6,8H and the Hessian of G. Mr. Cayleyhas thrown this into the form
(3 01
2TH)
1
5,- (Sa mmy,
showing that it is a perfect square when the discriminant Of Uvanishes.
212. It may he remarked here , that, by the principleestablished Art. 137, theorems concerning invariants and co
variants of a quantic give at once theorems concerning thecovariants of quantics of higher degree. Thus
,it hav ing been
just proved that the Hessian of the Hessian of a quartic is ofthe form a TU+BSH,
we can infer that the same is true of theHessian of the Hessian of any quantic.
‘
For if we form the
Hessian of uni t21a
“12
2
;this involves the second
,third
,and fourth
differentials of u. But,by the equations (n
&c.,we can express the second and third differentials in terms
of the fourth, and so write the Hessian as a function of fourthdifferentials only, and of the a; and g which we have introduced,and which, it will be found, enter in the fourth degree. It willthen be a covariant of the quartic emanant. Now every co
variant of a quartic is a function of U and H (Art. and
when the covariant is of the fourth degree it must be a linear
APPLICATIONS To BINARY QuANTIcs. 179
function of these quantities. This covariant then will be ofthe form aTU+BSH,
when S and T are invariants of the
quartic emanant,but covariants, as in Art. 137, of the higher
quantic.
213 . System of two quartics . We purpose now to enumeratethe combinants of a system of two quartics. If we form the
S and wh ich we may write
Sh.”
2 7W. S'
pf’
,TV twp, t
'
Me’T
the invariants of this quadratic and cubic W ill be combinants ofthe system U, V (Art.Thus the discriminant of the quadratic is without difficulty
found to be
16 (bd)”12 (ac
'
)(cc'
) 48 (bc'
)(cd’
) 8 (ab'
)(de'
) 8 (ad’
)(be’
)wh ich we shall refer to as the comb inant A .
Combinants of the same order in the coefficients may be
found by other processes. Thus the Jacobian of U and V is
(ah'
)as“
3 (ao'
)$5
3] {3 (ad'
) 6 mfg/9
8 wa
ys
{3 6 (“I'
llw”
? 3 90215
(de'
)if :
and their combinantiv e covariant 128 is
3 (Ml 50”
2 (56m9031 3 (“I'
ll31”
Now the discriminant of the latter covariant, and the quadraticinvariant, as at p. 112 , .Of the former will be combinants of the
same order as A ,but will not be identical with it. If we
write the new combinant B ,
(bd)”
(ad) (cd'
) (ab'
)(de'
) (ad'
)(007) (bc'
)
then the quadratic invariant of the Jacobian will be found to beA 4SE
,and the discriminant of the other covariant to be
A 12B .
It has beenmentioned that the eliminant of two quadrics may be written either
(ac b?) (ac'
ca'
2bb’
)A); (a'c
’b")“
2,
or of th e Jacob ian (ab’
) (ao'
)my (bc’
)y”
but it may be added that the.
former expression is linearly transformed into the latterby taking A6 ab
'
, ha M,for w and y. Similar transformations may be used to
facilitate the following calculations.
180 APPLICATIONS TO BINARY QUANTIcs.
214. In writing these combinants in terms of the determinants &c.
,I find it convenient to use the abbreviations
(ab'
) a, (de
’
) a'
, (ad'
) B, (be'
) B’
, (and) A, (cc
'
) N,
(cd'
) We have then
A I2A)!
48pp'
'y”
1682
Saa'
B =kk’
[All By.’
8” ad .
The resultant of U, V, found by expanding the determinantof p. 67
,is
R 1296A’A
’”3456 (apx
’1152 (crea
m
727’n
'
57678706 9216aa'
pp’
967 (sec mat)15368 a
'
Bp)+3072aa'
(a nB'
p)‘
«y‘48aa
'fy"
256aa'eys 5mm”
256 (aB'°
a'
fl”
) 4096aa'
8’.
In terms of the preceding combinante can be e xpressed thecombinant which we have called T (p. but which, in orderto avoid confusion
,we shall now call 0 ; and which expresses the
condition that a quartic of the system U AV can have twosquare factors. Such a combinant must vanish if V reduce tothe single term In such a case a
,a'
, ,B, y, 8 all vanish ;
and we have A l 27u>J 48mt'
,B = 7t7t
’
pp'
,R 12967
09 0
;
hence we see that (A 48B)” R is a combinant which vanishes
on this supposition. And since it is of the same order that wehave seen
,Art. 179
,that T must be
,it is identical with it.
Using the values already given for A,B,R,we find that
(A 48B)” R 128 0
,where
C= 1 - 36d a'
yy'
ISTSM'
WWI 37 (aB'
N a'
BX) 243”
(515 +B'
s)
68 (B’A
’
H? ) 68 a'
BA) 2 (oi/8"
a'
B’
) aa'
y’
4d’a"
2d a’
y8 478”
1601068”
Again,if we form the invariant which we called I (Art. 194)
of the quadratic and cubic of Art. 213,it will be found that
(A 48B)(A 16B) If : — 128I.
215 . The other combinants of the system which we shallnotice are D, the resultant of the cubic and quadratic, and E
182 APPLICATIONS To BINARY QUANTICS.
To save room, we write ae l,M : m
,cc
'
n,can?
2c'
eb‘I=p
’
We find then
A (l 4m)2
1271 (l 4m),B = m —
p2
+ n (l— m),R l8 (l 16m) 96l
2
p2
72nl"
(l+ 8m)+ 1296Pn’.
I Pm”
41m8
+192
(l2
21m 8m”
) 6104
n (l3
61m216m
”
) 12712
(l2lm
n2
{9lm2
(l 4m) 6172
(2 l‘7lm— 4m
"
) 27104
1612 (l
E : 12m4
2 lm'z
p2
(l 2772) (l 27702
17“
2nlm"
(l+ 2m)2
4p“
2711) (l 18 'np21m
2n2
(l+ 2m)4
367121m
2
(l 2m) 672174
(1+ 2m) 672""
p2
(l 2m)“
277120“
47as
(l 2m)3
108n3hn
2.
By the help of these values we can v erify the equation
16B3 — AB 2
2IB + E= D.
NOW if we take for the fundamental invariants of the system,
the coefficients of the cubic and quadratic of Art. 2 13,the com
binants A ,D,E,I are explicitly giv en as functions of these
quantities ; and the present equation gives B in terms of thesame quantities, and shows that all the combinante we haveused can be expressed in terms of the same fundamental invariants.The Jacobian, with this form
,wants the extreme terms.
There is no difficulty therefore in calculating its discriminantand thus verifying the theorem of Art. 177.
217. I have also sometimes found it convenient to supposeeach quartic to be the sum of two fourth powers
,so that for
each the invariant T vani shes. Let the quartics bewhere u is a
lx + B,g/, &c. We use (12)to denote
053 2
" and we use the abbreviations
(12)(34) L, (13) M
, (14)(23) N ;
where it will be observed that we have identically L+M+N=0,
for mx)
in the quartic and then operating on itself. If we Operate in
Now the invariant S is got by substituting£22,
APPLICATIONS To BINARY QUANTICS. 183
this way with u upon it the result vanishes ; but if we operateon v the result is
'
We find then at once that the S of
7tU [AV isfl ab (l 7m.{aa
'
(1 p’a'
b'
The combinant then which we have called A is
{d a'
ah'
ba'
66'
In the same case B is found to be aba'
b’
LgMN .
The invariant T is found by operating with the Hessian of
a quartic on itself. But here the Hessian of U is ab
We find then that the T of AU+ p V, is
{aba’
abb’
Mt“
{a'
b'
a d'
b’
b
Hence,we have immediately
E {ea/N“
D= {af'aft’N
a Hami ll ”
2MNa”a'
b’
2MN b’a '
b’
2MNa’2ab 2MN b
'2ab
2M 2N
2aba
'
b'
(M2N
2
I aba’
b'
L"
[aza
’iN
(M”+N
22L”
) b’a'
b’
a"ab b
"ab
2M 2N
”
(M2N
’I4L
”
)
by the help of which values we can verify the equation alreadyObtained.
218. The Quintic. In studying the quintic we constantlyuse the canonical form ax
5
+ by5+ oz
“, (where 96 + to
which it has been shown (Art. 165)that the general equationmay be reduced. Differentiating with regard to w and ysuccessively
,we have u
,ax
‘— cz4
,u2= Inf — os
“. It is evident
that the resultant of these two will be the discriminant of thequintic
,and that the combinante of this system will be invariants
of the quintic. These invariants are then immediately foundfrom the expressions in the last article
,where we must write
for a and b,a and — c
,for a
’
and b'
,b and c. We have
184 APPLICATIONS To BINARY QUANTICS.
and therefore M : O ; (13) 1, (12) 1
, (34) 1, (14) 1
,
(23) 1. We observe then at once that B vanishes. We can
see by counting constants,that any two cub ics can be brought
by linear transformation to be the two differentials of a singlequartic ; but two quartics cannot be similarly brought to be thedifferentials of a single quintic
,unless the condition B = 0 be
fulfilled. The combinant A in like manner becomes
6202
cw a’b”
2abc (a b c).
This,which we shall call J
,is the simplest invariant of the
quintic,and it may be Obtained in other ways. The quintic
,it
will be observed,has two covariants of the second order in the
coefficients,
v iz . the Hessian 12 5,which for the canonical
form is bcy°z3
+ ca z’x
”+ abx
s
y”
,and a covariant quadratic
the S of the quartic emanant,which in the same case is
bcyz + cazw+ abwy. If the quintic be written in the generalform (a , b, c, d, 6 ,f l at , y)
5
,these covariants are respectively
(a0 (138+ 3 (cd be)m
b
y+ 3 (ae+bd 2c (af+ 7be 80d)wa
y”
+ 3 (bf + ce— 2d"
)w’
y‘+ 3 (cf 016)my” (df
(as 46d 302
)on"
(af 3be 2cd)any (bf 406 3d‘)y
“.
Now we get invariants differing only by a numerical factor,whether we form the discriminant of the latter covariant, or the
quadratic invariant 12° of the Hessian. In either way we
obtain the general value of J, v iz .
a'
ff2
10abef + 4aedf + 16ace“
12ad’e 166
ӢZf
Qbee"
12602
766cde 486da
486°e 32c
’d”.
219. The discriminant of the quintic may be obta ined eitherfrom the theory of two quartics, or by direct elimination
between the two differentials ax‘
cz“,by
‘- cz‘. When these
vanish together, we may take abe as the common value of1 _l. l.
am‘,by
‘,cz
“; whence as y= z = Substituting
in w y z 0,we get the discriminant in the form
3. .l .
(50) = 0,
or {b c c"’
a.2
a2bz2abc (a b c)}
2128a
’bsc”
(be ca ab)=0.
Thus then we are led to the form for the discriminant J 12SK,
186 APPLICATIONS To BINARY QUANTICS.
R aff‘
20a‘ffsbe 120a
s 80d 160 (ay
”ce°
a’
ffsbgd)
360 (aff2d2e aff
sbc
z
) 640 (ori
ffdes
256 (aae5bff
“)
10a’
ff"b2e2
1640a’
fzbecd 320 (a
’
f be’b af
ibsed)
l 44oai 2
(d ese) 4080 (a
t
f bew
1920 (a’be
‘d b
‘ecf
’
) 2640a’
ff2c2d
’4480 (affe
2de
’
2560 10080 (affcdae af
’bc
sd)
5760 (a’ccl
2e8bscedf
’
) 3456 (a’dff + aj
2o"
)2160 (a
’zcl‘e2
180af b”c3
14920af b”e”cd
7200 (abfie‘c b
‘e”df) 960af (b
zed
3be
ecs
)
600 (ab263d2+b
ee’c
‘
ff) 28480af bec2d 2 16000 b
secd’
ff)1] 520af (bcd
4c‘de) 7200 (abe
"’
cd8bzecs
a’
f)6400 (ac
‘es69d? ) 5 120afe
f’al8 3200 (ae
268d9
3375b“e4
9000b°e30d 4000 (b
ae‘id 3 0
26303
)
The discriminant may also be expressed as follows Let
A a'
ff234af be 76afcd 32ace 32b
’af 12aed’
12bc’
ff225b
ze"
820becd 480 (ba’a
086) 32OCfd
"
;
B 3a’
ff222afbe 12afcd 64 (ace
2b2df)
36 (aed2boff) 45b
"e"20becd
C affe 2af bd 9abe"9afc
"32acde 18aal
86bof
15666”
Iobed
D 3a2df 2a
”’e29af bc abed 18ac
ge 12acd
”6bf
’
f15b
"’
ec 10b2d"
;
and let C '
,D’
be the functions complemental to 0 and I), (where
all these functions vanish if three roots be equal), then threetimes the discriminant is
AB + 6400“64DD
'
.
220. Quintics‘have also an invariant of the twelfth degree
,
which may be most simply defined as the discriminant of thecanoniz ant. For the canonical form for which the canoniz antis abcxyz , the discriminant is And
,in general, the
discriminant is - L,where the following is the value of L as
calculated byM. Faa de Bruno. To save space in printing we
APPLICATIONS TO B INARY QUANTICS. 187
omit the complementary terms. Thus (a2c2de2f
s
)stands for
a2c2de2f
2a3b2cd
2
f2.
L a2e2d2
f2— 2 (a
2c2de2f
s
) (a2c2e‘
ff2
) 6 (a‘cd
s
efs
) 16 (a2ca7262f
2
)14 (a
2ca762f ) 4 4 (a
2d2f
‘2
) 11 (a2a74e2f
2
) 10 (a2d2ef/ )
3 (a2d2ee
) 4a2b2cde
2
f°
2 (a2b2ce
'
ff2
) 6 (a2b2d2ef
2
)16 (a
ab2d2e2f
2
) 14 (a2b2de
‘ff ) 4 (a
2b2c7
) 50a2bc
2d23f
‘2
82 (asbc
2de2f
2
) 32 (a2bc
2e’
ff) 36 (asbcd
‘
ffs
) 30 (aabcd
3e2 2
)30 (a
abcd
2e
‘
ff ) 24 (a2bcdee) 28 (a
sd ef
2
) 50 (a2bd
2e2f )
22 (asbd
ses
) 16 22a2c2d2f
850 (a
2csd2e2f
2
)16 (a
scsdeff ) 16 (a
scsee
) 54 (aac2d2e
2
) 46 (a2c2d2e2f)
60 (a202d2e'2
) 6 (a2cd
‘ff
2
) 70 (ascd
se'
ff ) 56 (a2cd
2e2
)
18 (ac7cf ) 14 (a
sd662
) a2b2e‘
ff2+ 132a
2b26de
2
f250 -
a2bsce5
f )14 (a
2b2dse2f
2
) 60 (a2b2d2e‘
ff) 30 (a2bade
e
) 168a2b2c2a’2e2 2
48 4 (a2b2c262
) 48 (a2b2cd
2
ef2
) 2 (a2b2cd
362f)
6 (w as? ) 62 (a2b2d‘ff
2
) 90 (Curr ey) 39 (a
2b2d2e2
)112 82a
2bc
2d26f
2170 (a
2bc
2d2e’
ff ) 104 (a2b0
3d6
5
)
108 (atbe
tda
f?
) 42 (a2bc
2d2e2 — 242
294 (a bcdses
) 72 (a bd'
ff ) 78 (a2bd
7e2
) 164 (a2cba762f
'
)
24 (6120565
) 63a2c2d
‘
ff2
394 (a2c‘2dse2
194 (a2c2d2e2
)
324 (a2csd2
8f ) 440 (a2csd
‘2es
) 78 (a2c2d2f) 428 (a
2c2d6e
2
)
180 (a2cd
se) 27 (a
2d12
) 18ab5e2f 38ab
2cdc
2
f + 36 (ab‘cee
)
204 (ab‘dseff ) 102 (ab
2d2e2
) 308 (ab202de
'2
) 42ab262d26
’
ff
674 (abacd
2c2f) 590 (ab
scd
se2
) 128 (ab2de
ef) 138 (absd
fies
)
4 (ab2o2e
‘2
) 652 (ab2c2d2e2
) 714ab2csdae2f + 498 (ab
2c2d5
ef'
)
1246 (ab2c2d2e2
) 224 (ab2cdff) 516 (ab
2cd
262
) 48 (ab2dee)
136 (abcsde
2
) 1078abc2d2ef 206 (abc
‘262
) 342 (abcsd‘ff )
804 (abcsd2e2
) 506 (abc2d7e) 90 (abcd
2
) 16 (ac7c2
)220 (ao
° 262
) 106acsd‘ff — 392 (ac
sd‘e2
) 222 (ao2 2
c) 40 (acsda
)276673“ 234b
20d6
232 (b
tdse‘) 713b
2c2d2e2
246
4 866b2c
’2d262550 (b
sc2d262
) 56 (bscd
7e)-l 4 (b
sa2
)
139b202d4e2
354 (b2csdee) 83 (b
2c2d2
) 330bcsd5e
72 1606 6
.
188 APPLICATIONS To BINARY QUANTICS.
On inspecting this invariant it will be seen that it vanishes ifb,c,d all vanish. Consequently the form aw
5
+ 5exy2
+fy°
,to
which Mr. Jerrard has shown that the quintic can be broughtby a non-linear transformation
,is one to which no quintic can
be brought by linear substitution unless L 0.
221 . We take J,K,L as the fundamental invariants of the
quintic, and we proceed to show how all its other invariants canbe expressed in terms of these. In the first place
,it will be
observed that the interchange either of a' and y, or of ac and z,
is a linear transformation whose modulus is 1. Hence,if any
invariant is such that when transformed.
it is multiplied by an
even power of the modulus of transformation, it must, for thecanonical form
,be unaltered by any interchange of a , b, c ; that
is to say, it must be a symmetric function of these quantities.If the invariant is multiplied by an odd power of the modulus,it must
,for the canonical form
,be such as to change sign when
any two of the quant ities a,b,c are interchanged ; it must
therefore be of the form (a b)(b c)(c a)multiplied by a
symmetric function of a,b,c. Now an invariant is in trans
formation multiplied by a power of the modulus equal to itsweight. And (Art. the weight of an invariant of thequintic whose order is n
,is an. A quintic cannot have an in
variant of odd order in the coefficients. If the order is a
multiple of 4 the weight is an even number,and the sign of the
invariant is unaltered by the interchange of a: and y. If theorder be not div isible by 4, the invariant is what we have calledskew
,that is to say, such as to change sign when a: and y are
interchanged . Let us first examine the former kind, which, wehave seen
,must
,for the canonical form he symmetric functions
of a,b,6. Now
,since J (be ca ab)
24abc (a b c),
K a2b2c2
(be ca ab), L a2b‘c‘, (from which we infer
E — JL)= a2b
fic5
it follows that if we are
given any quintic, and transform it to the canonical form by a
The reader must be careful to observe that though , in the case of the canonical
form,a 5b5c5 (a b c), for example, is div isible by we have no right to infer
that in general H is divisible by L, unless in cases where the quotient abc (a b 0)has been also proved to be an invariant.
190 APPLICATIONS TO BINARY QUANTICS.
the possibility of the existence of skew invariants had not beenrecognised . I took the trouble to calculate this invariant
,and
the result is printed (s'
IOSOp l n'
ca l Transactions,1858
,p .
but as it consists of nearly nine hundred terms I cannot affordroom for it here. The leading terms are a
’dff
ea°cff
7
; in this,as in ev ery skew invariant
,the complementary terms hav ing
opposite signs, and the symmetrical terms vanishing. By the
argument used in the last article,it is proved that every skew
invariant of a quintic must be the product of this invariant'
I
by a rational function of J,K,L.
223 . The square of I being of the thirty-sixth degree can
be expressed rationally in terms of J,K,L (Art. The
actual expression is easily found.By forming the discriminant of the cubic (Art. 221)
— L,
Lz
Lsz
we obtain the product of the squares of the differences of a b c
in terms of J,K
,L,and thus have
PL=E K2+ 18HKL 27L4 — 4K 9L2
or putting for H its value 71; (K
2JL), and dividing byL, we
have
161”JK
“8LK "
2.72LK 2
72JK L2 432L”JsL”
.
224. I do not purpose to enter into detail as to the diii 'ere'
nt
covariants* of the quintic. The most remarkable,after those
already mentioned,are the linear covariants. If we operate
twice with the quadrat ic covariant bcyz + ca zw+ abxy on the
quintic itself,the result will evidently be of the first degree
,
and for the canonical form will be abc (bcw+ cay+ abz). If weeliminate between this covariant and the canoniz ant we getHermite ’s invariant I ; and if between this linear covariant andthe quintic itself we get I (J
2 — 3K Thus,then
,if I vanish
,
the quintic is immediately soluble,one of the roots being given
by the linear covariant, as also if J23K .
The actual values in general of some of the simplest are given by Mr. Cayley,Phi losophica l Transactions, CXLVI. 125.
APPLICATIONS To BINARY QUANTICS. 191
By operating with the linear covari ant on the quadratic,we
get another linear covariant of the seventh order,v iz .
abc {ac (a c2
a"bz
ab2
abc y aa2bzb""c2
abc2
azbc)
z (b"’oz
02a2
agbc
It has been already proved that we can express all the covariantsin terms of any two together with the invariants. AccordinglyM. Hermite has used the transformation Of taking the two linearcovariants for a: and y, when all the coefficients in the trans
formed qu intic are found to be invariants. The actual valueshowever are not simple
,and I have not found any advantage
in the use of this form of the equation. The reduction to thisform will be impossible in the particular case where these twolinear covariants are identical
,which will be when their re
sultant JK 9L vanishes. By forming the Jacobian of thequadratic covariant and any other covariant
,we obtain a co
variant of the same degree in the variables as the latter,and of
an order two higher in the coefficients. Thus,from the canoniz ant
in this way we get another cubic covariant,
abc {bc (y’z ya
”
) ca (22a: aw
”) ab (x
2
y boys
»,which might also have been got by operating with the canoniz anton the Hessian. Quintics hav ing linear covariants of every oddorder above the third, it follows by the principle of reciprocitythat all quantics of odd order abov e the third have linear co
variants Of the fifth degree In the coefficients.
225. It ought to have been stated earlier that the Sign of
the discriminant of any quantic enables us at once to determineWhether it has an even or odd number of pai rs of imaginaryroots. Imagine the quantic resolved into its real quadraticfactors
,then (Art. 106)the discriminant of the quantic is equal ,
to the product of the discriminants of all the quadratics,mul
tiplied by the square Of the product of the resultants of everypair of factors. These resultants are all real
,and their squares
positive,therefore
,in considering the sign of the discriminant,
we need only attend to the discriminants of the quadratic factors.But the square of the difference of the roots of a quadratic ispositive when the roots are real
,and negative when they are
imaginary. It follows then that the product of the squares of
192 APPLICATIONS To BINARY QUANTICS.
the differences of the roots of any quantic is positive when ithas an even number of pairs of imaginary roots
,and negative
when it has an odd number. We have been accustomed towrite the discriminant giv ing the positive sign to the productof the two extreme terms. This will have the same sign as the
product of the squares‘
of differences of the roots when the orderof the quantic is of the form 4m or 4m+ 1, and the oppositeSign when the order is of the form 4m + 2 or 4m+ 3 . We see
then, in the case of the quintic,that if the discriminant be
positive, there will be either four imaginary roots or none ; andif the discriminant be negative
,there will be two imaginary
roots. It remains then further to distinguish the cases whenall the roots are real
,and where only one is so.
226. In order to discrim inate between the remaining cases,
there are various ways in which we may proceed. The
following* are,in their simplest forms
,the criteria furnished
by Sturm’s theorem. Let J be the invariant as before
,and
H : b”
ac, 8 : ae 4bd 30
2
, T= a ce+l
2bccl ad”eb
2o“,
M : a“e”
a”df+ 3abcf 3abde 4acd 2 4ac
"e 2bff
5b2ce abtd
’8bo”d ao
“,
then the leading terms in the Sturmian functions are proportionalto a
,a,H,55 8 + 9a T,
— HJ+ 12b'M+ 4S
s216T‘
, the lastof course being
.the discriminant ; and the conditions furnished
by Sturm’s theorem to discriminate the cases of four and no
imaginary roots,are that when all the roots are real the three
quantities H,5HS 9a T
,HJ+ &c., must all be positive.
227. We may apply these conditions to the canonical form
(0 a)a? 5cw‘3/ Ic
s
y”
IOca’y“5cwy
‘(o b)y
‘,
in which case the equality of all but two of ,the coefficients
renders the direct calculation also easy. We easily find thenthat the constants are c — a
,c —a
,a c
,and the fourth
being essentially negative we need not proceed further,and we
These values are given by Mr. M. Roberts, Quarter ly Jowrnal, vol. IV. p. 176.The read er who may use Mr. Cayley’
s tables of Sturmian functions (Philosop hical
Transactions, vol.OXLVII.p.735)must be cautioned that the fourth and fifth functions
are there givenwith wrong signs.
194 APPLICATIONS To BINARY QUANTICS.
that the J, K ,L of every qu intic with real coefficients are such
that the quantity G is essentially positive ; where G isJE
“8LK °
2J’LK ”
72JL2K 432La J
”L“
For G has been shown to be the perfect square of a realfunction of the coefficients of the general quintic
, vi z . a’dff
“ &c.
this being the eliminant of the quintic and its canoniz ant,and
therefore necessarily real. We may in the above substitute forK its value in the discriminant from the equation J —I2SK =D,and so write G
e 4 (73+ (6J
°
\
J ’D"
(4J°
61.2°J°L JD (J
82“L)(J
‘
If now, to assist our conceptions,we take J, D, L for the co
ordinates* of a po int in space ; then G : 0 represents a surface ;and points on one side of it making G positive answer to realquintics, while po ints on the other side making G negative
'
l'
answer to quintics with imaginary coefiicients.
229. Now, in the next place, we say that if the coefficientsin an equation be made to vary continuously
,the passage from
real to imaginary roots must take place through equal roots.For
,let any quantic (Mm)become by a small change of co
efiicients 95 (as) (so), (where e is infinitesimal), and let a bea real root Of the first
, a + li a root of the second ; then wehav e whence
,since we have
which gives a real value for b. The con
secutive root a b . is therefore also real. But if (a)vanishesas well as 4: (a), the lowest term in the expansion of <1) (a + h)Will b e h”
,and the value of 71. may possibly be imaginary.
When therefore the original quantic has equal roots, the cor
responding roots of the consecutive quantic may be imaginary.It follows then that if we represent systems of values of
J,D,L,by points in space
,in the manner indicated in the last
article, two points will correspond to quintics having the same
Mr. Sylvester takes L in the usual direction of a' , J of y, and D of z .
1' Points for which G 0 answer to real quintics, and it is easy to see that in
this case the equation is of the recurring form. For we have proved that when G 0
two of the coefficients of the canonical form are equal. The equation is therefore ofthe form O.
APPLICATIONS TO BINARY QUANTICS. 195
number of real roots,prov ided that we can pass from one to
the other without crossing either the plane D or the surface G.
If po ints lie on opposite sides of the plane D,we ev idently
cannot pass from one to the other without hav ing at an intervening point D= 0, at which point a change
'
In the characterof the roots might take place. If two points
,both fulfilling
the condition G positive,be separated by sheets of the surface
G, we can not pass continuously from one of the correspondingqu intics to the other ; because when on crossing the surface ,we have G negative
,the corresponding quintic has imaginary
coefficients. But when two points are not separated in one of
these ways, we can pass continuously from one to the other,
without the occurrence of any change in the character of thecorresponding quintics.Now Mr. Sylvester’
s method consists in shewing, by a dis
cussion of the surface G,that all points fulfill ing the condition
G positive,* may be,
d istributed into three blocks separated fromeach other either by the plane D Or the surface G. And sincethere may ev idently be quintics of three kinds, viz . hav ing four,two, or no imaginary roots, the points in the three blocks mustcorrespond respectively to these three classes. I have notspace for the. elaborate investigation of the surface G, by whichMr. Sylvester establishes this ; but the following is sufficientto enable the reader to conv ince himself of the truth of his
conclusions.
230. One of the three blocks we may dispose of at once,
viz . points on the negative side of the plane D,which we have
seen (Art. 225)correspond to quintics hav ing two imaginaryroots. Next with regard to points for which D is positive.We have seen, in the last article, that a change in the characterof the roots only takes place when D 0 ; our attention istherefore directed to the section of G by the plane D. We seeat once
,by making D=O in the value of G (Art. that
the remainder has a square factor, and consequently that thesurface G touches D along the curve J 3 — 2
“L,and cuts it
along J Now,if a surface merely cut a plane, the
Mr. Sylvester calls these facultative points.
196 APPLICATIONS TO BINARY QUANTICS.
line of section is no line of separation between points on thesame side of the surface. If
,for example
,we put a cup on a
table, there is free communication: between all the points insidethe cup and between all those outside it. But if a plane toucha surface, as, for instance, if we place a cylinder on a table
,
then while there is still free communication between the pointsinside the cylinder, the line of contact acts as a boundary line,cutting off communication as far as it extends
,between points
outside the cylinder on» each side of the boundary.NowMr.Sylvester’
s assertion is, that ifwe take the negativequadrant, viz . that for which both J and L are negative
,and
if we draw in the plane of any, the curve J”2“L ; then all
facultative points in that quadrant, lying above the space lncluded between the curve and the axis L 0
,form a block
completely separated from the rest, and correspond to the caseof five real roots.
231. In order to see the character of the surface,I form the
discriminant of G considered as a function of K ,which I find
to be - L"
(J+ Consequently,when both J and L are
negative, the discriminant is negative, and the equation inK hasonly two real roots: To every system of values, therefore, ofJ and L correspond two values of K
, and consequently twovalues of D, and the surface is one of two sheets. Now I saythat it is the space between these sheets for which G is positive.In fact
,since G is it may be resolved into its factors
J (D — a)(D B){ (D and since J is supposed to be
negative in the space under consideration, D must evidently beintermediate between-a andB in order that G should be positive..Now the last term of the equation being (J
"
if J be nearly equal to will be of opposite sign to L, orin the present case will be positive. And the coefiicient ofD‘
being negative, we see that on both sides of the line J 2“L;
the values of D are,one positive and the other negative, that
is to say, the two sheets of the surface are one above and theother below the plane D. But I say it is the upper sheet whichtouches D along J —2
“L. This may be seen immediately by.
looking at the sign of the penultimate term in the equationfor D, by which we see that when the last term vanishes, thc
198 APPLICATIONS TO BINABY QUANTICS.
points between the cylinder and this surface would be non
facultative, and therefore irrelevant to the question. Mr.
Sylvester has thus seen that we may substitute for the criterion2nL— J
3
,2“L— J
"
+ ,aJD,provided that the second represent
a surface not meeting G above the plane D. And on investigating within what limits y. must be taken, in order to fulfilthis condition, he finds that may be any number between1 and — 2 .
He ava ils himself of this to give criteria expressed as symmetrical functions of the roots. In the first place
3 (a (B a)“(3 a)
“
is an invariant (Art. and being of the same order and
weight as J can only differ from it by a numerical factor :which factor must be negative
,since this function is essentially
positive ; and J we have seen is essentially negative when the.
roots are all real. And secondly,the symmetric function
2 (01— 3) — a)2
(e— a)4
(e
(the relation of which to the other may be seen by writing itin the form D2
E (a B)"
cl)
" 2
(«y or)“ 2
(8 where D isthe discriminant), is also an invariant
,and of the twelfth order.
It must therefore be of the form aJ3
+ BJD + fyL. Now,by
using the quintic'*ac(sf — a
"
)((a’
the symmetric functionmay easily be calculated and identified with the invariants ; andthe result is that its value is proportional to 2uL— J
8
+ gJD.
Since then the numerical multiplier of JD is within the pre
scribed limits,it may be used as a criterion
,and Mr. Sylvester’
s
result is that the two symmetrical functions mentioned are suchthat not only are both positive
,as is ev ident
,if the roots are
all real,but also if both are positive
,and D positive, the roots
must be all real. It ought to be possible to verify this directlyby examining the form of these functions in the case of an
equation with four imaginary roots.
It is to be observed that though this form may be safely used in this case, itcannot always be safelyused. For when a linear factor of a quintic is also a factorin the Jambian of the remaining quartic, a relationmust exist between the invariants,which, however, I have not taken the trouble to calculate, it being obviousthat it isof too high a degree inJ, D, L to affect the present question.
APPLICATIONS To BINARY QUANTICS. 199
233. I have also tried to verify these results by examiningthe invariants of the product of a linear factor and a quartic
,
(aw+ By)(w‘+ 6mm?
”
y“) these being necessarily covariants
of the quartic (Art. The coefficients of the quintic are
then 5a ,“
,8 , 3ma, 3m,8 , a , and I find for the J of the quintic,
48 (SSE 3 TU), or 48 times
(5m 27m”
)(on4
84
) (8 18m2
547724
)
Now the roots of the quartic are all real when m is negative,and when 9m” is greater than 1. On inspection of the valuegiven for J
,we see that when m is negative every termb ut
one is negative. G iving then m its smallest negative valueJ is negative
,v iz . 144 (a
‘”and J is afar
-tiori negativefor every greater negative value of m. Or we may see the
same thing by supposing B= O, when we hav e only to look at
the coefficient of the highest power of a in 88H — 3TU,
'
whichis 8
'
(b2
ac)S 3 Ta . But now if we call the three Sturmianconstants A ,
B,G: viz .
A = bz — ac
,B = 2SA + 3Ta , 0 = S
's
the value given for J becomes BAS B,which is e ssentially
negative when the roots are all real.The invariant L, according to my calculation, is
54
'
(SSH 3TU)“
6400 (SS
27T2
)(4113SHU+ TU
”)
150 (Sa
27T?
)U2
(88H + 15TU) 4050Ute
“(2SE 3TU),
whence .2“L -J
8 differs only by a positive constant multiplierfrom
128 (SM 27T”
)(433
317U+ TU“)
3 (SS - a7ze.v
”a
(8811 + 15TU) 3TU).
Writing 02CI 3abc 26
3
,the coefficient of the highest
power of a in this is128 09
28165
283
45a2GB 54a.
2GSA .
All the terms of this but one are positive when ‘
the roots are
all real, but as there is one negative term, it is not Obv ious onthe face of the formula , that the whole will be positive whenthe roots are all real. Still less that if this formula be positiveand Jn egative, the roots are necessarily all real. Therefore,although no doubt Mr. Sylvester’
s rule may be tested by the
200 A PPLICATIONS To BINARY QUANTICS.
process here indicated,to do so requ ires a closer examination of
this formula than I am able to give.*
234. It does not enter into the plan of these Lessons to givean account of the researches to wh ich the problem of resolv ingthe quint ic has given rise.T The following however finds a
place here on account of its connection with the theory of in
variants. Lagrange,as is well known
,made the solution of
a quintic to depend on the solution of a sextic ; and it caneasily be proved that functions of five letters can be formedcapable of six values by transformation of letters. Let 12345
denote any cyclic function of the roots of a quintic such,for
example,as the product
(a 62
03
where ev idently 2345 1 and 15432 would denote the same as
12345 ; then it can easily be seen that there can be writtendown in all twelve such cyclic functions. But further thesedistribute themselves into pairs ; and
'
by so grouping them wecan form a function capable of only six values ; for instance,12345 13524
,12435 14523
,13245 125 34, 13425 14532 ,
The actual formation of the
The verification, however, is easy in the particular case a: (m4 Smash/2
We have then J : 48m (5 L 12m (5 211L J 3 proportional tom (1 9m?)(50 45m2 648m‘ Thus
, when m is negative and 9m2 1,
we have J and L negative and 211L J 5 positive. The latter is positive for imaginary
roots only when m is positive, but in this case J is positive. The imaginary roots
must therefore be detected by one criterion or other.
The discussion of the invariantive characteristics of the reality of the roots of a
quintic was originally commenced by M. Hermite in his classical“ paper in the
Cambridge and Dub linMathematica l Jowrna l for 1854,and has been resumed by him
in his valuable memoir presented to the French Academy this year. His result,
translated into the notation we have used, is that the roots are all real when, the dis
criminant being positive, we have also positive K , 21'L—J 3+D, and K (JL+K 2) 18L2.
It seems to me that this result is superseded by the greater simplicity of Mr.Sylvester's
criteria. I should have wished however to give an account of M.Hermi te ’s method,
as well as of the many other important additions he has made to the theory of quintics.But having neglected to read his papers as they came out in the Comp tes Remhw,I find, to my great regret, on taking up his memoir now, that I have not time to
make myself master of it without delaying indefinitely the publication of this volume.I hope to find a place further on for some
_account of his application of the theory of
invariants to the Tschirnh ausen transformation of equations.1' Among the most remarkable of recent discoveries in this subject is the appli
cation to it of the theory of elliptic functions by M.Hermite and M.K ronecker.
202 APPLICATIONS TO BINARY QUANTICS.
235 . M. Hermite has studied in detail the expression of the
invariants in terms of the roots. He uses the equation trans
formed so as to want the first and last terms ; that is to say,so that one root is 0 and another infinite ; and the calculationis thus reduced to forming symmetric functions of the rootsof a cub ic. I had been led independently to try the sametransformation on the problem discussed in the last article
,but
found that, even when thus simplified,the problem remained
a difficult one. It would be necessary to form for a cubic thesextic whose roots are the six values of
(B—v)(at— v)
”
Bgffl a
and then to identify the result with combinations of the formsassumed by the invariants of the quintic when a and f vanish.
Of M. Hermite ’s results I only give his expression for his owninvariant I. Let
F = (at (Gt— E)(3 - 7) (a Wis - 3)(B - E),0 = (a - B)(at— f
r)(6 — 5) (at - 3)(OP - 3)(B-v)i
H = (at— B)(a 8)(e v) (Oi - v)(at— e)(S—B);
the continued product of these is symmetrical with respect toall the roots except a ; and if we multiply this product by thesimilar products obtained for the other four roots we get I.
236. The Sextt'
c. The theory of the sextic has as yet beenbut little studied. It has four independent invariants
,which we
shall call A,B,0,D,of the orders 2
,4,6,10 respectively ;
and a fifth skew invariant,which we shall call E
,of the fifteenth
order,whose square is a rational and integral function of the
other four. The first,A,is the invariant formed by the
method of p. 112 , which for the general sextic isag 66f + l 503 — 10d
".
I have given (Arts. 170, 171)the canonical form of the sextic ;but I believe it will be found in practice not less convenient touse (as in Art. 217)the more general form
an“
owe dz“.
To this we should be led by the theory of two quintics, whichcannot be more simply expressed than as each the sum of four
APPLICATIONS TO BINARY QUANTICS. 203
fifth powers. For the form just given,the invari ant A is
, byproceeding as in Art. 217
,found to be
ab ac ad be bd cd
wh ich we may write BabThe Hessian of the sextic 12
2 is of the eighth degree, thegeneral coefficients being ac b
"
,4 (ad be), 6ae 4bd
4af + 1666 2oad,
14bf + 5cc &c. ; and for mycanonical form is Eabu
“v4 The sextic has another co
variant of the second order in the coefficients,viz . the S of the
emanant quartic,which is of the fourth order in the variables,
the general coefficients being
as 4bd 302
,2af 6be 4cd
,ag 9ce 8d
”
,&c.
,
this for the canonical form being To these covariants we may add the covariant sextic
,of the third order
,
which is the T of the quartic emanant,and whose general
coefficients are
a ce 2bcd ad”eb
22aqf 2ade 2b
’
ff+ 2bce 2bd2
2c2d,
acg + 2adf Sac”— bg
— 2bcf + 4bde+ 202e—3cd
”
,
2ady 2def 2bcg 4bdf 2be”
2cff+ Gods &c.
and which for the canonical form is Babe q o’.
237. We take for the invariant B, that which has beencalled by Mr. Sylvester the ca ta lectacant
,which expresses the
condition that the sextic should be reducible to the sum of threesixth powers
,and is (Art. 168)the determinant
a,b,c,d
b,
c,d,e
01d:e, f
d)e, f; .9
This expanded is
aceg— acf
2 — ad’
g+ 2adqf aeH— b eg+ bf
°+ 2bcdg 2beef
2bdff+ 2bde2
c”
g 202df+ a
ge”
3cd”e d
“.
If now we form the quadrinvariant of the Hessian, we find itproportional to A”
300B if that of the covariant S, we find
I have thought it unnecessary to add the termswhich may bewritten down fromsymmetry.
204 APPLICATIONS TO BINARY QUANTICS.
A 23GB ; and if we operate on the sextic with the covariant T
,
we get B . Applying this last process then to the canonicalform,
we get, for the value of B,
abcd (23) (41)(13V(24V;which vanishes, as it ought, if any of the quantities a
,b,c,d
vani shes, or if any two of the four functions u,v,w,z become
identical.
238. We take for the form of the fundamental sextinvariant0,that which involves no power higher than the second of the
leading coefiicient a,and which for the general form is
a2d2
g"
6a2defg 4a
2df
94d
2es
y 3a”e'
ff2
6abcdg 18abcqfq12abqf
”12abdffg l Sabde
’
g Gabeff+ 4ac”
g”
24d ozaz
g
18ac”dfg 30ac
“ef
254acd
2eg 12acd”
f2
42acdeffl 2ace
“20ad
‘g 24ad
s
ef 8ad2e°
4b3dg
”l 2b
°
efgSb
’
I
’
f8
3b”c“g230b
zce
”
g 24bgcef
"12b
2d2eg 24b
2dff
"
60b2de
’
ff 27b"e4
+
1
6bc’
ffg 42bo2dey 60bc
"df
“30bc
’e’
ff24bod
°
y 84bcd’ef —i 66bcde
824bd
‘
ff 2~i bd"e2
120469
27072 —80
”e“— 240
2d
’
ff - 8d“.
In terms of these ' the other invariants of the sixth ordercan be expressed. Thus
,the cubinvariant of the covariant
quartic is A 8108AB 54 0 the cubinvariant of the Hessian
is 3A3
100AB + 27500 and the quadrinvariant of the sexticcovariant is 2AB 0. The last-named invariant can be easilycalculated in the case of the canonical form. We have tooperate with Eabc on itself. Now if weoperate with 13 0
222on u
'
hfw"the result is proportional to
MN,where M and N have the same meaning as in Art.
2 17 ; and if with uzvzw2on itself the result is
Hence we get for the invariant in question
2 a2bze2
(23? 2abcd 2 ab MW“.
239. If a , b, c all vanish, the invariants A ,B, 0 become
respectively 10d”
,d‘,- 8d“
. Hence, when the sextic has as
factor a perfect cube, the conditions must be fulfilled A 2 = 100B,
206 APPLICATIONS To BINARY QUANTICS.
1222500a2bc
’
ef’
g
506250658602 5
330000a*bcd
”
g°
1590000a’bcd
2
efq”
330000a2bcdff
3
g
750000a2bcde
° 2
3 172500a”bcde
’
y’
g
1537500a"bcdef
4
375000a2bce
‘
ffg2250000.bee
s 8
780000a2bdffq
"
1350000a2bd
°ez
gg
2190000a”bd
3e
”
g
1200000a’bd
’
ff‘
4650000azbd
26’
ffg2550000a
2bd
26‘ff"
1500000a’bde
s
g
900000aebde
4 2
150000a"c“eg
a
250000a208d2
g8
750000a208dq
"
37500a20°df
“g
1062500a“cse3
g2
2821875a208eff
2
g
126562505284
4
1350000a202dw
3562500a202dzefq
2
4725000a202d2
ef2
9
2062500a202dff
“
4875000a2cwyg
'
1875000a202es
g
1125000a2026ff
‘
3750000a2d
‘e9
2
300000a2cd
4
f2
9
9750000a’cd
8effg
5250000a20d
8e
8
3750000a’cd
”e‘g
2250000a’cd
2e‘ff
”
l OOOOOOa” “g“
3000000a’d56jg
1600000a’d5
f"
1250000a’d4e3
g
9375able'g4
7500ab“dfg
3
9375ab4egg
8
25500ab‘ef
z
gz
11520a6‘f4
9
97500ab302
f"g8
412500ab”cde_q
8
616500ab°cd
”
g”
1222500abl’ceffg
2
2197800abscef
’
iq864000ab
3
q50000ab
3da
g°
330000ab8dzefg
”
83200ab3d
’
fff’
g
37500ab‘i’de
‘1q2
5 11500ab°deff
z
g
288000ab8def
‘
202500abse‘
ffq121500ab
se’
ff’
412500ab"d
’eg
‘1
23250ab20?f
‘g
’
675000ab202dz
g”
3172500ab202defg
”
5 11500ab cdfa
g
2821875ab20268
92
7633125abzcze’
ff’
g
3442500ab202ef
4
219OOOQab20d
‘ffg
’
4725000ab’cd
"e’I
6030000ab’cd
‘tf g
3360000ab"cd
’
ff‘
— 15337500ab*cdeyg
8392500ab’cdeff
”
5062500ab”ce
“g
3037500ab”ce
‘2
300000ab’d‘cg
“
900000ab2d
se’
fq480000ab
”d
”
ef°
375000ab2d264
g
225000ab’d ’
es
f”
90000041566n375000abc
‘efg
202500abcff°
g
4650000ab03dffq
2
4875000abe°de
° 2
15337500abcsd6f
2
9
7087500abc°df
‘
843750abc°ef7fq
506250abcaeff
a
9750000abcéd
8
qq"
900000abc2d
’
ffz
g
24750000ab02dzeffg
— 13350000abc”d26
8
9375000abczde
‘o
‘g
5625000abc'2de
‘ff"
sooooooa cdiq”
9000000abcd‘efg
4800000abcd'
ff8
3750000abcd3e°
g
225OOOOab<3d8eff
2
250000ac"
g8
1500000610‘5df‘q
2
1875000acbez
g2
APPLICATIONS TO BINARY QUANTICS.
50625OQa05ef
z
g
2278125ac’5
f“
3750000ac“d"eg
2
375000ac4d
’
72
9
9375000ac4de
2
fg5062500ac
4def
8
35 15625ac‘e‘g
2 109375ac“e’
ff2
1250000ac”d4
g2
3750000070s q2000000a0
8d fy
°
1562500ac3 2
68
9
937500ae”d
”e’
ff2
3 125b°
g*
3750065
q°
187500b5deg
3
2400002>5
c2
92
5062509 6792
864000b5ef
"
g
331776bff5
187500b4cs’eg
8
11250b4c'
ff2
g2
375000b"’cd
”
g8
1537500b4cdefg
2
2238000640d 9
1265625640638
92
344250054067
2
9
15552006‘c6f
4
20625OOb4d26'
fq2
,
3360000b4dze
2
g
1843200b4dff
“
7087500b4de
’
ffg38880005
4616
2 3
22781256469
13668756467
2
50000069o3dg
°
225000b°cs
efg2
121500b30ff
s
g
2550000b802dffg
”
26250005302d6
2 2
8392500bsc2def”
g
3888000bsc2df
‘
506250bscze’
ffg3037505
80281
”
525000063cd
3
eg’
480000b3cdff
2
9
13350000bscd
26ffg
207
720000053cd
2ef
°
5062500bscde
4
g
30375006”cde
3 2
1600000b8d5
g2
480000069014
4;2560000b
3d
‘
ff“
2000000b8d8e9
g
120000068d3e2
f”
1500006205 8
9000006204dfg
2
30375006204
4 ?1.9; 66875b
204 4
2250000b2d’d
’eg
2
225000b208d2
f2
9
5625000b2csde
’
fq3037500b
203def
s
2 109375b203e‘g
1265625b2c9e3
f”
75OOOOb2c2d4
g2
2250000b202d3
efg12000006
”c”d3
f“
937500b262dzes
g
562500b202dze’
ff2
241. Instead of the discriminant A we may use anotherinvariant D
,in which no higher power than the fourth of the
extreme coefficients a, 9 appears
,and which does not contain
the product a‘g‘. The quantity multiplying a 4 in D is (eg _ f
2
)8
5
and the relation connecting A and D is
A A5
3751193 G25A
20 + 3 125D.
The value of D is
“468
98
3s‘afh
tiff)"6
12a”bde
"
ga
24a°beff
9
g 6a302e’
iq’l
24al’bdef
2
'q2
12af’bef5
Gascgef
2
g”
12a8bdf
4
g (1308
94
3ascff
4
g12a
8be
'
ff9’
12a302dfq
86a
scd
2eg
a
208 APPLICATIONS To BINARY QUANTICS.
42a”cdf
’
f"
g2
420a’bcd
’efg
2
24a"ede
”
fg2
72a“bcd
°
f”
q
108asedef
s
g 156a”bede
s
gSB
60ascdf
5828a
2bcde
2
/z
g
27aflce
‘
tg2
348a2bcdef
‘
114asce
’
ff2
g 48a been60a
sce
’
ff4
36a2bce
9
f9
5a"d
“
g8
246a bdffg2
84a.”dsefg
2372abdse2g
”
66a8 2
eg
g2
816a2bd
3ef
’
g
228a°d2e7
2
9 480azbd
i’
f”t
120a8d2ef
*1536a bd
’d‘fg
288a°de
4
fg 888a2bd
262 3
IGOasde
l
ffa
480azbde
s
g
288a bde‘ 2
48a”e'
ff2
3a2b902
g4 ”
92
24a2b20dfq
”66a
acsda
gs
6412192062
93
156azc°defg
48a2bzcef
2
92
12a208df
”
g
24a2b2cf
4
g 270a‘2cf’ea
g2
4261.262d269
s
804a2036'
ff2
g
9Oa2b2d
'
ff”
g2
405a2c
’
ef60cz
2b2a’e
’
fffq2
372612020379
2
1800."
d48a
2b2df
51428a
202d2ef
”
g
660a”e2d
'
ff4
Gazbzeff
2
g 1200a202deffg
660a202de
'
f’
fa
12a"bc
‘fffq
3429a
20'zefi
g
24a2b0
2deg
°225a
“c2e“f2
6061.2be
zdf
z
g2
957u2ed
*ey
"’
96azbe
zez
fg2
3 l 2a2bc
ief
a
g 2664a20d
3e'
ffg162a
2bc2 5
1440a20d
3
ef°
84a2bcd
3
g8
972a20d
2e‘g
540a2cd
2e‘ff
2
258a2d°
g2
816a2d5
¢g432a
2d'
if2
324a2d2e2
g
180a2 ‘
e2 2
3ab‘eg
4
12ab4dfg
2
3ab262
g2
24ab‘af
2
92
12abff2
g
24ab202fg
2
108abscdeg
2
180ab20df
2
g2
312ab2ee2fg
2
564abscef
“g
216ab2
qf' 6
72ab2d2
efg2
168ab2d2f
“g
12ab2de
2
92
84ab2de
2
f2
g
144ab2def
2
108ab26B 2
114ab2e2eg
2
6ab2c2f
2
g2
'
228ab2c2d
2
g"
828ab c defy2
84ab2e2df
”
g
804ab2c2e2
92
1890ab2c2e2
1f2
g
648ab2c2
ef2
816ab2cd2fg
2
1428ab2cd
2e2
g2
2148ab2cd
2
ef2
g
1320ab2cd2f
‘
4188ab cde‘ffg
2.772ab
2cde
2 3
1458ab2ce°
g
1296ab2ceff
2
84ab2d‘eg
2
240ab2d‘f
2
g
24Oab2d2e2fg
240ab2d26f
3
12ab2d2e2
g
288ab2d2e2 2
288abe2dg
s
48abe2
efg2
1536ab02d2fg
2
1200abc2e2
g2
4188abesdef
2
g
2268abc2df
‘
12OOabeue’
ffgI296abc
262 2
2664abc2d°eg
2
24Oabe2d2
f2
g
5736abc2d2effg
3048abe2 2
af2
3252abc2de
‘g
2340abe2de
2 2
324abe2eff
816abcds
g2
l 572abcd‘ejg
1224abcdff2
1656abcd2e2
g
1296abedae2f
2
432abcd2eff
168abd‘ff'
g
192abdbe2
g
144abd°ef
2
144abd‘e‘1f
80ac"
g3
480ae°dfg
2
210 APPLICATIONS To BINARY QUANTICS.
242. Mr. C ayley’s theory of the number of invariants of a
quantic, shows that, in addition to the four invariants alreadymentioned, sextics hav e a skew invariant E of the fifteenthdegree. I do not know whether In the same manner as the skewinvariant of a quintic is the resultant of two covariants
,this
invariant also can be got by operating with one of two knowncovariants on the other. The invariant however can be formedby means of the differential equation. The highest power of awhich occurs in it is a“
,and the factor multiplying it is
(0192
2f"
)
The expression for E in terms of the other invariants maybe got from the following considerations. If in the sextic b, d,fvanish, E necessarily vanishes. For
,since the weight of E is
forty-five (Art. the weight of some one of the constituentcoefficients in each term must be expressed by an odd number ;and when we make in the equation all the terms vanish whoseweight is odd
,E vanishes. E = 0 is therefore the condition
that the roots of the sextic should form a system in involution.
If then we make 6,d,f = 0 in A ,
B,(J,D,and eliminate a , c, e,g
from the results,the relation thus obtained between A ,
B,0,D
must be satisfied when E vanishes,and must therefore contain
it as a factor.If we write ag= 7t , ee= p , as
°+gc
a= v,the values of the
invariants got by making b, d,f ; 0, may be written
A = 7t + l 5p , B = 7tp +p2 — v,
0 = 24m.2
4 (A+ 3p.)v,'
A 7k {X2
1507t,u. 1875p
250014
2.
Eliminating v in the first place,the last two equations become
0 = 4a (h —p.)
23s)B , A=7t (7
9+ 350M4 1375p
fi
Then eliminating p. by the help of the first equation,we get
1024X2—11527t
2A (132A2—108003 )X+3375 0+2700AB —4A
2=0,
N (25670232OAR. 55A
245OGE)
2A 0.
I have partly calculated the value of E. If I find leisure to complete the
calculation 1 will give the result in an appendix. Ls}
APPLICATIONS TO BINARY QUANTICS. 211
The resultant of these two equations is of the thirtieth degreein the coefficients ;
‘
and therefore from what we have seen can
only differ by a constant multiplier from E 2.
243 . By Art. 225 , when the discriminant is os1t1v9 thesextic has either six or two real roots ; and when it is negative ,has either four or none. We can readily anticipate th at thediscussion of this expression for E is likely to lead to the sameresults in affording criteria for further distinguishing these cases
,
as the corresponding discussion of the expression G in the caseof the quintic. Analogy also leads us to expect that what willbe important to examine will be the result of making A = O
in the expression for E. Now,although the calculation of the
general expression for E may be a little laborious,that part
of it which is independent of A is easily obta ined. It willev idently be the product of 3375 0 + 2700AB — 4A
2by the
square of the resultant of the cubic and of the quadratic
320A).+.55A2
45003 .
And aga in,analogy leads us to believe that the first of these
factors is not important in the question of the criteria for realroots
,and that it is the square factor alone which needs to be
attended to.
The result I find is that, writing for convenience B'
for l OOB ,
0'
for 125 0,the quantity squared differs only by a constant
multiplier from
4A°19A
4B
’
49A2B
'24A
2O
'
80B + 52AB'
0'-40
Analogy then leads me to suppose that‘the criteria for thenumber of real roots of a sextic depend on the Signs of thisquantity, and of A
2 2; 100B 2,A
B125 O’
,
(A8
300AB + 250 5 (A2
which,as we saw
,vanish when three roots are all equal.
( 212 )
LESSON XVIII.
ON THE ORDER OF RESTRICTED SYSTEMS OF EQUATIONS."I
244 . THE problems discussed in this lesson are purely algebraical
,and in the investigation of them I do not make use
of any geometrical principles. But I find it convenient to
borrow one or two terms from geometry,because we can thus
avoid circumlocution,and also can more readily see how to
extend to quantics in general,theorems already known for
ternary and quaternary quantics.We saw (Art. 74)that if we are given k equations in [of
independent variables,the number of systems of common values
of the variables which can be found to satisfy all the equations,
will be equal to the product of the orders of the equations.Now
,in the geometry of two and three dimensions respectively
,
the system of values w= a, y=
,b ; or ze= a
, y= b, z = c,denotes
a point. I find it convenient therefore to use the word point”
in general instead of “system of values of the variables and
the theorem already stated may be enunciated. “A systemof k equations in k variables of degrees I
,m,n, p , g, &c. respec
tively, represents lmnpg &c. p oints,” by which we mean that
so many “systems of values of the variables” can be found
to satisfy all the equations. This number lmnpg &c. will becalled the order of the system of equations.
245. If we have a system of k 1 equations in k independent variables, we have not data enough to determine any
In this place in the former edition followed a Lesson on the application of
this theory to ternary quantics. This would now require much development inorder to bring it up to the present state of the subject ; and as the results all admit of
geometrical interpretation, I have thought thatthey would be placed more properly ina geometrical treatise. Instead, I have inserted here what has in substance already
been published as an appendix to my work on the Geometry of Three Dimensions,
but which seems properly to belong to the algebraical'
theory explained in this
1 If as is usual‘we employ homogeneous equations, the number of variables will
of course be k 1.
214 ORDER OF RESTRICTED SYSTEMS or EQUATIONS.
coeficients of the equations in the degrees a, p , v
, p respec
tively, this quantity will enter into the resultant in the degreea r pnrl vrlm plmn.
We shall use the word ‘order’to denote the degrees l, m,
n,r,
in which the equations conta in the variables which are to be
eliminated, and weight to denote the degrees a, p , v, p in whichthey Contain the quantity not eliminated : and the result justwritten may be stated
,that the weight of the resultant
,or the
weight of the system,is equal to the sum of the weights of each
equation multiplied by the order of the system formed by the
rema ining equations.And this is still true
,if we break the given system up into
partial systems. Thus,the first two equations form a system
whose order is lm and weight Xm+ p l, and the second twoequations a system whose order is ar and weight w + pn ; andthe value just given for the weight of the entire system, is
m'
(Mn p l) lm (w pn),
that is,it is the sum of the weights of each component system
multiplied by the order of the other. The advantage of so
stating the matter will appear presently.
247. What has been hitherto said in this lesson,is but a
re-statement in other words of principles already laid down inthe Lesson onElimination : but my object has been to makemere intelligible the Object of investigations, on which we shallnowenter as to the order and weight of systems of a somewhatd ifferent kind. We have seen that k equations in k variablesrepresent lmap &c. points. But now we may combine withthese 70 equations an additional equation
,wh ich is satisfied for
some of the points but not for others of them. We have then a
system of k+ 1 equations representing points, that is to say, all
satisfied by a number of systems of common values of the
variables, that number being new however generally smallerthan the product of the degrees of any 7c of the equations.Cases are of constant occurrence where a number of points canbe expressed in no other way than that here described. A simplegeometrical example will suffice. Consider p points in a planewhere p is a prime number, and where the points do not lie in a
ORDER OF RESTRICTED SYSTEMS OF EQUATIONS. 215
right line,then these points cannot be represented as the com
plete intersection of any two curves, and if we have any two
curves going through the points,their intersection includes not
only these points but others besides. To define the pointscompletely
,we must add a third curve going through the p
given points,but not through the rema ining points of intersec
tion of the first two curv es. The points are then completelydefined as the only po ints common to all three curves. Our
Object then is,in some important cases where a system of
points is defined by more than k equations, to lay down rulesfor ascertaining the order Of the system ; that is to say, how
many systems of common values satisfy all the equations.In like manner a system of k 1 equations is satisfied by an
infinity Of common values. But it may happen that we can
write down an additional equation satisfied by part Of this seriesof common values
,but not by the rema ining part. In such
a case,the system Of k 1 equations denotes a complex curve
,
and it requires the system Of la equations to define that part ofit for which all the equations are satisfied. It will be the
Obj ect of this lesson to ascertain the order and weight of whatwe may call restricted systems : that is to say, where to a
number Of equations sufficient to define points,curves
,&c.
,is
added one or more others which exclude from considerationthose values Of the variables which satisfy the first set of
equations,but do not satisfy the additional equations.
248. The simplest example of such a system is the set of
determinantsu U w
u,
'v,w
'
or, at full length,'vw
'
wu’
uv'
By writing these equations in the form
it is ev ident that in general values of the variables which satisfytwo of the equations must satisfy the third. But there is an
216 ORDER OF RESTRICTED SYSTEMS OF EQUATIONS.
exception for the case of values which make either u and u'
,
'v and v’
,or w and w
'
0. In any of these cases it is easy to seethat two of the equationswill be satisfied
,but not the third. And
now it is easy to see how to calculate the order of the systemcommon to all three. Let the orders Of u
'
and 'v'
,of 'v and
,v'
,
of w and w'
,be I
,m,n respectively ; then the orders of the first
two equations are m + n,91 + Z, and Of the system formed by
them is (m + n) But in this system will be includedvalues which satisfy both 10 and w
'
,these values not satisfying
the third equation. Excluding then this system,the order of
which is n”
,the order Of the system common to the three
determinants is mn+ nl+ lm.
In like manner suppose we have a system with three rowsand four columns
,
w,w,w
,w
Let us write at full length the determinants formed by, the
omission of the third and fourth columns
to (vw’
v'
w) 'v (wu
'
w'
u) w (uv’
u'
v) 0,
u (um'
v’
w) v (wu’
w'
u) w (nu'
u’
v) 0,
then these two equations are obv iously satisfied for all valueswhich satisfy the three vw
''v'
w,wu
'
w'
u,
But
these values will not satisfy the other determinants of the
given system. From (l+ m + n)2
,then
,which is the order of
the system formed by the two equations written at length,we
must subtract mn+ nl+ lm which has just been found to bethe order of the system special to these two equations
,and the
remainder nl+ lm is the order of the systemcommon to all the determ inants. Hav ing thus determined theorder Of a system with three rows and four columns
,we have
,
in like manner, thence derived the order Of a system with fourrows and five columns. Proceeding thus step by step I arriveby induction at a general formula which I establish by showingthat if it is true for a system with It rows
,it is true for a system
with k 1.
218 ORDER or RESTRICTED SYSTEMS OF EQUATIONS.
have generally either a or a = 0. When all the rows are Of
the same order we have a, ,8 , &c. all = O
,
'
and therefore p , gboth = 0
,and the order of the system is Q.
250. We may proceed in like manner to calculate theweight of the system of determinants considered in the lastarticle. B eginning again with the simplest case
,let us suppose
that the sy v w is to be combined with one or
'v,w
’
more other equations and the variables eliminated. Now the
result of elimination between uv'
u'
v,uw
’
u’
w,and any other
equations will contain as a factor the resultant Of u,u'
,and
the other equations. If we reject this factor we get the sameresult as if we had eliminated between nv
'
u'
v,vw
'— vw
'
and
the other . equations,and then rejected the factor got by climi
nating between C,C'
,and the other equations. To illustrate
the method employed , let us suppose that u,u'
; v,v'
; w, w'
respectively conta in any quantity not eliminated in the degrees7x, p , v ; and that we are to combine with the determinants of
the given system another equation R 0,whose order is r
,and
containing the uneliminated quantity in the degree p. Thisquantity thenwill enter into the resultant ofR
,uv
' —u’
v,uw
'
u’
w
in the degree
p + r 772)(7x+ v) (l+ n)But the resultant Of R u u will contain the same quantity inthe degree
pl2
When then this factor is rej ected from the former result, theremainder is
The order then Of the system of three determinants is thequantity multiplying p, and the weight is the quantity multiplying r.
251. F inding in this way the weight of a system with tworows and three columns
,we find
,step by step
,as in Art. 249,
the weight of a system with k rows and 7c+ 1 columns. The
result is that if the orders Of the several functions be as written
ORDER OF RESTR ICTED SYSTEMS or EQUATIONS. 219
in Art. 249,and if their weights (that is to say, the degrees
in which they contain the variable not eliminated)be a’
+ a’
b'
++ aa, &c., a &c.,then the formula for the weight is
derived from that for the order,by performing on it the
operation “1562+ b'
g; That is to
say, the weight is2 (ab
'
)+pP'
p'
P+ 22 (and) 2
which may also be written
(d a'
)— 2 (ad);
this being the result of performing the operation a'
ga +&c.
on Q+pP +p —g.
252. If we form the condition that the two equations
a t’”
bf“
of" 2
&c. 0,a'
t"
b'
t’H
o'
tM
. &c. 0,
should have a common root,we Obta in a Single equation
,namely
the resultant Of the equations. But if we form the conditionsthat they Should have two common roots
,we Obta in (Art. 78)
not two equations,but a whole system
,no doubt equivalent to
two conditions, yet such that two equations of the system
would not adequately define the conditions in question. Now we
may suppose that t is a parameter eliminated,and that a
,b,&c.
conta in variables,and we may propose to investigate the order
of the system of conditions in question. Now,Art. 78
,these
conditions are the determinants of the systema,b,c,d
a,b,c
a,b
where the first line is repeated 72 1 times,and the second m 1
times ; there are m+ n — 2 rows and m + n 1 columns. The
problem is then a particular case Of that of Art. 249. We
suppose the degrees of the functions introduced to be equidifferent : that is to say, if the degrees Of a , a
'
be 7x, ,u, we
220 ORDER OF RESTRICTED SYSTEMS or EQUATIONS.
suppose those of b,b'
to be 7t + a,
of c,c' to be 7t + 2 a
p + 2a , &c. We may write the formula of Art. 249 in the somewhat more convenient form Q +pP + z} p
”where s
,is the
sum Of the squares of a, B, &c. To apply it to the present
case we may take for the quantities a,b,c,&c. O
,a,2a
,&c. ;
and for the quantities a, B, «y, &c. Of Art. 249 , A, A— a
,7t —2a
,&c.
P is then the sum of m n — 2 terms Of the series a,2a
,3a
,&c.,
and is therefore,if we write m n k
,1)(k 2)a . In the
same case Q is the sum of products in pairs of these quantities,
and is therefore
(k— 3)(lo— 2)(k 1)(3k— 4)a
Again p is the sum of n 1 terms of the series 7t,X a
,A 2a
,
&c., and of m 1 terms Of the series,u, p
— a, p
— 2a,&c.
We have then
_p= (n — l)(n 1)(m
In like manner E2is the sum Of the squares of the same quan
t ities,and is
(n — l)n.2 — l)(n— 2)— p.a (m— 1)(m— 2)
(n 1)(ac— 2)(212 — 3) (m 1)(on— 2)(2m— 3)
Collecting all the terms, the order of the required system isfound to be
£42 (fl — 1)(n 1)M+ %n (u — 1)(2m l)(2n
{mm(m 1)(n 1)a”.
If the two equations considered are of the same degree,that
is to say, if m n,we may write A y.=p, M4. q, and the
order becomes
i n (n 1lall— (w 1)9
If all the functions a,b,&c. are of the first degree
,writing
7t =,u, 1
,and a=0
,in the preceding formula
,the order is found
to be &(m + n 1)(m+ n
253 . If the degrees in which the uneliminated variables occurin any terms be denoted by the accented letters corresponding
222 ORDER or RESTRICTED SYSTEMS or EQUATIONS.
255 . It is a particular case of the preceding to find the
order and weight Of the system of conditions that an equationmay have three equal roots ; because these
conditions are found by expressing that the three second differentials may have a common factor. Writing in the precedingfor l
,m,and n
,n — 2 ; for p , 7t + a ; and for v
,7t + 2a ; we find
,
for the order of the system,
3 (n n (n 1)(n 2)a“;
and in like manner for its weight
6 (n 2)MU 3n (n 2)(a'
h. ax) 2n (n 1)(n 2)d a'
.
Again,to find the order and weight of the system of condi
tions that the same equation may have two distinct pairs of
equal roots ; we form first,by Art. 249
,the order and weight
of the system of conditions that the two first differentialsmay have two common factors. We
subtract then the order and weight of the system found in thefirst part of this article. The result is that the order is
2 (n 2)(n (n 1)(n— 2)(n— 3)a ,
and the weight is
4 (n 2)(n 3)Mt'
+ 2n (n — 2)(n— 3)(a'
7t + d7x'
)
+ n ('
n 1)(n 2)(n— 3)d a’
.
Before proceeding further in investigating the order Of othersystems
,it is necessary to discuss a different problem,
and I com
mence by explaining the use of one or two other terms whichI borrow from geometry.
256. Intersection of quantics hav'
z'
ng common curves. TWO
systems of quantics are sa id to intersect if they have one or
more points” common,that is to say, if they are both capable
of being satisfied by the same system of values of the variables.A “
surface” is said to conta in a“curve” if every system of
values which satisfies the k 1 equations constituting the curve,
satisfies also the la 2 equations constituting the surface. Thus,
in the case of four variables,three equations U= 0
,V: 0
, W=Oconstitute a curve
,and the two equations U O
, V= 0 con
stitute a surface which evidently contains that curve.
ORDER OF RESTRICTED SYSTEMS OF EQUATIONS. 223
Now a system of lo quantics in 7c variables,in general
,as we
have seen,intersect in a definite number Of points
,that number
being the product of the orders Of the quantics. But it mayhappen that they may have an infinity of points common
,
these po ints forming a curve” in the sense in which we have
already defined that word. Besides that curve they will haveordinarily a finite number Of points common
,which it is our
Object now to determine . Let us take,for example , to fix the
ideas,the cas e Of four independent variables ; and suppose that
we have four equations of the form
U = Au +Ev + Ow = O,
V = A'
u + E'
v + C'
m = O,
Z = A u + B 11+ 0 w= 0.
We suppose the degrees of U,V, I/V, Z to be l, m,
n,p ; of u , v, w
to be A, ,u.,v ; and A ,
B ; A'
,B
'
,&c. are therefore functions of
the degrees l — A, Z— p ; m —jt , m ,u,&c. Now
,ev idently
,
these equations will be all satisfied by every system Of valueswhich make u 0
,v 0
,w 0 and these equations not being
sufficient to determine “points,will be satisfied by an infinity
of values Of variables. In other words,the four quantics U
,V,
I/V, Z have a common curve now. And yet U, V, W,Z may
be satisfied by a number of v alues which do not make wall = O. It is our Obj ect to determine this latter number ; andour problem is,When a system of quantics has a common curve
,
it is required to find how many of their lmnp &c. points Ofintersect ion are absorbed by that curve , and in how many pointsthey intersect not on that curve.
257. Let us first consider the curve formed by 7c — 1 Of the
quantics ; for instance , in the example we have chosen forillustration
,the curve UVVV. Now ev idently a portion of this
curve is the curve uvw,but there are besides an infinity Of
po ints satisfying UVW which do not satisfy u,v,w. We speak
then Of the curve UVVV,as a complex curve consisting of the
curv e wow and a complementary curve. Now the order of a
complex curve is always equal to the sum of the orders of its
components. For by definition the order Of the complex c urve
224 ORDER OF RESTRICTED SYSTEMS OF EQUATIONS.
UVW is the number of points obtained by combining with theequations of the system an additional one of the first degreethat order being in the present case lmn. And evidently, sinceof those lmn points MW lie on the curve now
,there must be
lmn on the complementary curve.The two curves intersect in points whose number 7? is easily
obtained. For ev idently all points which satisfy the threeequations
and which do not satisfy u,
'v,w ; must satisfy the determinant
A B,0
A B'
0'
A,B
,C
the degree of which is l+ m+ n —/J.
— v. The intersectionof this new quantic with now gives all the points in whichuvwmeets the complementary curve. we have therefore
—7t —p—v).
258. To find now the number of po ints common to UVWZ ,we have to consider the points in which the curve UVWmeets Z ; and it is required to find how many of these are noton the curve now. But since new is itself a part of the curveUVW,
it is ev ident that the points required are containedamong the p (lmn— 7tp.v)points in which the complementarycurve meets Z . And from these points must be excluded the1?points in which the complementary curve meets uvw. Usingthen the value given in the last article for i
,we find
,for the
number which we seek to determine,lmnr
We shall,state this result thus
,that if k quantics of orders
1,m,n, p , &c. have common a curve of order a ; then the
number of points which they will hav e common in addition tothis curve is less than the product of the orders of the quanticsby B, where B is a constant dependingonly on the nature of the curve and not involv ing the ordersof the quantics. We shall call this constant the rank of thecurve. We have seen that when the curve is given as the
226 ORDER OF RESTRICTED SYSTEMS OF EQUATIONS.
components ; and the rank Of the complex is equal to the sum
of their ranks increased by double the number of points commonto every pair of curves.
260. We give,as an illustration of the application of these
principles,
‘
the problem to determine how many surfaces of thesecond degree can be described through five po ints to touchfour planes. Let S; T, U, V, W be five surfaces passingthrough the five points
,then any other will be of the formand the condition that this should
touch a plane will be a. cubic function of the five quantitiesor, B, y, 8, e. We are given four such equations
,and it is
required to find how many systems of values can be got tosatisfy them all . If the four equations had no common “
curves,
”
the number of their common points” would be 34 or 81. But
the existence of common curves may be seen in this way. The
condition that a surface of the second order should touch a
plane vanishes identically when the surface consists of two planes.Let us take then for S and T two pairs of planes passingthrough the five given points
,S= (I23) T : (123)
then,ev idently
,the condition that 018 +BT+ ¢yU+ 8V+ eW
should touch any plane whatever, must be satisfied by the sup
position vy= 0, e= 0. This “
curve,
”then
,which is of
the first degree,will be common to all four quantics. And
,if
if we call this the line (123) it is ev ident,by parity Of
reasoning, that the quantics have common ten such lines
(124) &c. Now if,as before
,we take 8 as the system of
two planes (123) T= (123) and take U : (145)then
,while the line (123)(45)is denoted by «y: 0
,8 : 0
,a= o
,
the line (145)(23)is denoted by B= 0, e= 0 ; and thesetwo lines intersect, being both satisfied by the common values
B= 0, a= o. And,in like manner, (123)(45)is
intersected by (245) (345) Thus then the ten lineshave fifteen points of mutual intersection The rank of asinglecurve of the first degree being got by making 7t = p = v= 1 in
the formula M W is three. Hence the rank of the
entire system is ten times three increased by twice fifteen or
is 60. And the number Of points which satisfy the fourquantics is 81 10 60 or is 21.
ORDER or RESTRICTED SYSTEMS or EQUATIONS. 227
261. We have shown,Art. 249
,how to determine the order
of a system of determinants,the number of rows and columns
in whose matrix difi'
er by one. We shall now show how,when
such a system represents a curve,to determine the rank of the
curve. Commence, as before, with the simple case
2 1
and we see that the intersection of uv’ —u
’
v,uw
' — u’
w is a
complex curve,consisting of the curve uu’
and of the curvewith which we are concerned. And knowing the order and
rank of uu’
,we find the order and rank of the other curve.
K nowing these aga in,we Obtain
,in like manner, the charac
teristics of the curve represented by a system with four columnsand three rows
,and so on, step by step
,until we arrive
,by in
duction,at the result that the general expression for the rank is
(p2 —q+pP+
where p, g, P, Q have the same meaning as in Art. 249, and
R,r are the sums Of the products in threes of the quantities
a, b, &c., a , B, &c. For we have seen
,Art. 258
,that the ranks
of two complementary curves which together make up theintersection of two quantics whose orders are
,u,v,are con
nected by the equation B B’
(a a'
) And we saw
(Art. 249).that the intersection of two quantics whose ordersare p + P a
, p + P b,is made up of the curve whose order
is a =— p’ —g+pP + Q, and of the curve Of lower order for which
a’ =P
”Q+pP+ g where we have written
p'
a + h, g'= ab. Now to find
,by the formula of this article
assumed true, the rank of the lower system,we are to write for
the newR,-the old r ;
'
and for the new r,
R1 270 + (p
'” P (p'°- 2M )~
We have,then
,after a little reduction
,
B'= g)(2P+p) Q (3P+ 2p)+R+ r
-p'
-r 9
To this value of B'
add (at a'
)(p. v) that is to say,
is”—P’
2 (Q s)+p'
(PM) (2P+ 2p
228 ORDER OF RESTRICTED SYSTEMS or EQUATIONS.
and the result is the value already given for B. As then the
formula can easily be verified for the case Of two rows, it istrue generally.
262. Let us next consider such a system as
a b c d,
e, f
a b c d'
e'
f’
a b c d e, f
a,b
,c
,d
r em
t f
where the number Of columns exceeds the number of rows bytwo; and let us examine the order Of the restricted systemcommon to all these determinants. Now any three of these
have common the curve
d,d
'
,d d
e,
e!
e"
e!"
f, f 7 f 1 fIf then I
,m,n be the order of the three determinants ; a , B,
the order and weight Of this curve. the determinants intersectin points not on this curve
,in number
lmn
But if we represent the orders of the several functions in the
same way as in Art. 249,it is easy to see that the degrees of
the three determinants are P+p —b— c,P+_p c— a
,P+p
- a —b ;
so that if we write a b c=p'
,be ca ab g
'
,abc r
'
,we
are to substitute for bmn,
— 2p'
(PM ? (19 (P 9and for l+m+ n, 3P+ 3p
— 2p'
. The order and rank of thecurve
,a and B, are found from the formulas of Arts. 249
,261
,
by writing for P, Q, and R, p , g, and r ; and for p , g, re
spectively
P—p
'
, Q—PP'
H " a'
,R —
p’
Q+ (p —9
’
)P (p 210 9
We find thus,as before
,
a =P"
Q+rP+ q—p
’
(2P+10) Q (3P+ + R+ r
—p
’
(3P+r)+214 (0 9)+10”(P+10)+22
'
(PM)-10q - r'
230 ORDER OF RESTRICTED SYSTEMS or EQUATIONS.
In the case where we have a 0,7t y. 1
,this reduces to
(m+ n— 2)(m+ n— 3)(m+ n
The weight of the system,found by the same process as
before,is
4n (n 1)(n— 2)7Cw+ §m (m 1)(m— 2)n’
n'
(2 —1)(n—2)(m - 1)(m—2)(n
(m 1)n (n 1)(n 2)(MUa 57006)
(n 1)m (m 1)(m 2)(22 a W )
i (m 2)(n 2){2mn m n} (An'
a h’
na hna'
)
{%n (n 1)(n 2)m (m 2) fin (n 1)(n (03W 2aoc
'
7t)
Em (m 1)(m 2)n (n 2) m(m l)(m (W}m (an— l)(m— 2)n (n 1)(n— 2)e
2
u'
.
264. The nex t problem we inv estigate is when a systemof quantics have a
“surface” common
,to find how many of
their points Of intersection are absorbed by the common surface.We mean by the order and rank of a surface
,the order and
rank of the curve which is the section of the surface by anyquantic of the first degree. Thus
,consider the case of five
independent variables,then a system of three equations con
stitutes a surface,and if their orders be 7x, n, v, the order of the
surface w ill be M W, and its rank M W (7t + p + v); these beingthe order and rank of the curve got by uniting with the givenequations an additional one of the first d egree.Now, first let k— l quantics have a surface in common
,
whose order and rank are a, B ; they will also in general
have common besides a complementary curve whose order isreadily found. For j oin with the given quantics another of
the first degree, and we then have a system of k* quantics,hav ing a curve common
,and therefore by Art. 258 intersecting
in lmnr a (1+m n+ r) B points besides. But these are thepoints in which the quantic of the first degree meets the
complementary curve, and therefore this is the order of thatcurve.
To fix the ideas we take k 5 in the next article.
ORDER OF RESTRICTED SYSTEMS OF EQUATIONS. 231
265 . Next let us investigate the number of points in which .
the surface and complementary curve intersect each other.Let U
,V, W”, Y bs respectively of the forms
Au Cw o,
A'
u 2.B'
v +aa.) = o,
B'
s + 0"
w o,
A n B'
a m 0.
Then the points common to U,V,W
,Y which do not make
u,v,w 0
,will satisfy the system of determinants
A,A
’
,A A
B,B
'
,B B
0,C
"
,O
"
,0 0.
But Since A is of the order l 7x,B of the order I n, A
' of theorder m A
,&c.
,it follows (Art. 249)that the order of the set
of determinants is Q where P and Q are the sum
and sum of products in pa irs Of l,m,n,r,wherep = 7t + p. V
,
and p, = 7t2
p.”
V2
M1. nV VA. If now we combine thissystem Of determinants (equ ivalent to two conditions), withthe 7c— 2 conditions which constitute the surface
,we determine
the points common to the surface and complementary curve.And their number is the order of the system Of determinants
,
multiplied by M W. Writing then. a and B for the order and
rank of the surface M W, M W (A+ 11. v), and denoting by ry the
new characteristic N W(A2
V“=M1. [.W which we
may call the class of the surface,we find
f= aQ— BF+ 7.
266. If then we have an addit ional quantic Z also con
taining the given surface,and if it be required to find how
manv points not on the surface are common to all Inquantics,
these will be evidently the points of intersection of the com
plementary curve with Z,less the number of points of intersec
tion Of the complementary curve with the surface. If thenI,m,n,r,s be the orders of the quantics, the number sought
will be got by subtracting from
s {lmnr
232 ORDER OF RESTRICTED SYSTEMS or EQUATIONS.
the number
a lr + mr + nr)— B (l+ m+ + 7
And the difference islmnrs a Q BP 7,
which is the formula required.
267. Next let us consider the case where a system of
quantics have common not only a surface,whose characteristics
are a, B, y, but also a curve
,whose characteristics are a
'
, B'
,
intersecting the surface in 75 points. A s before,consider first
k 1 of the quantics. Their intersection we have seen con
sists Of the surface and Of a complementary curve,whose order is
Zmnr
And if the complementary curve be itself complex,consisting
in part of the curve a ',and also of another curve
,whose order
is a"
,we have ev idently
a"= lmnr
The points therefore which we desire to determine are got bysubtracting from the a
"
s points Of intersection of the curve a"
with the remaining one of the given system Of la quantics,8+ 8
’
where 8 Is the number of po ints in which the curve a"
meets the surface (a , B, y), and 8’
i s the number of points whereit meets the curve. But we know 8
,since we know
,by
Art. 265,the number of po ints where the surface is met by
the entire curve complementary to it ; and therefore have
and we know knowing,by Art. 257, the number of points
in which the curve a ’
is met by the entire curve complementaryto it
,and therefore have
—B'
.
Substituting the values thence derived for 8 and 8 In a 8— 8
we getZmnrs a Q+ BP 7
3 a'
P +B'
In other words, the diminution from the number hnnrs produced by curve and surface together is equal to the sum of
their separate diminutions lessened by three times the numberOf their common points.
234 ORDER or RESTRICTED SYSTEMS or EQUATIONS.
Adding which we havelmnr (Apv+ { (l m+ n+ r)(s
'
s lm
a'
In other words,the combined effect of the two surfaces is
equal to the sum of the effects of the surfaces separately con
sidered,diminished by six times the number of their common
points. When there are only four variables,two surfaces
always must have common points of intersection.
270. Lastly,let the two surfaces have a common curve
whose order and rank are a"
, B”. Proceeding
,as in the
last article,we find that the system UVWYZ ’
have commonindeed the surface Nev ; and the curve Nn
’
V'
s. But sincethis curve is a complex one
,consisting in part of the curve
a"
, B" which lies on 7t
,u.V, we are only to take into account
the complementary curve which,by Art. 259 , has for its order
N'
p'
v’
s a"
,while its rank is
B"
2a (N s’
v'
and the complementary curve intersects a"B"
in
a"
(a'
pl v’
s'
) 6" points.
The number of intersections is thereforelmnrs
'
s'
+ - ' ry~
a"
)(l m n r s'
)
,8"
2a (A'
+ n’
+ 17
+ s'
) 3a (N+ 11 + v'
+ s'
) 33
Similarly the intersections for UVWYZ are
lmnrs"
lm+&c.} 'y'
(Rays a"
)(1+m n r s"
)— 2a
"
) )+ 3a — 3B
Adding, we havehnnr (s
'
s"
) (MW {(l m n r)(s'
s by: &c.}
fy— 4B
In other words, the two surfaces make up a complex surface,
whose order is the sum of their orders,whose rank is the s um
of their ranks increased by twice the order Of the common
ORDER or RESTRICTED SYSTEMS or EQUATIONS. 235
curve,and whose class is the sum of their classes increased by
four times the rank Of the common curve and diminished by
We must leave untouched some other cases which ought tobe discussed in order to complete the subject ; in particularthe case where the surfaces touch in points or along a curve.The cases examined sufii ce to enable us to ascertain the orderOf a set Of determinants in whose matrix the number Of columnsexceeds that of the rows by three. Taking four determinantsOf the system we find that they have a surface common ; andby determining the three characteristics of that surface and
applying the formula of Art. 265 , the order Of the system can
be obtained.
271. We come now to the problem Of finding the order Ofthe system Of conditions that three ternary quantics shouldhave two common points. The method followed is the sameas that given by Mr. Cayley for . eliminating between threehomogeneous equations in three variables
,and which we have
expla ined (Art. Let the three equations be Of the degreesZ,m,n. Multiply the first by all the terms yac
mm ’ 4&c.
of an equation of the degree m + n— 3,the second in like
manner by all the terms of an equation of the degree n+ l— 3 ,and the third by all the terms of an equation of the degreel+ m 3 . We have thus in all
7} (m+n
equations of the degree l+ m+ n — 3 ; from which we are toeliminate the ’
g (l m n 1)(l m n 2)terms &c.
But as it has been shewn,in the place referred to , the equations
we use are not independent but are connected by
s(Z -1)(l — 1)(m 202 — 2)
relations. Subtracting then the number of relations, the numberof independent equations is found to be one less than the numberOf quantities to be eliminated ; and we have a matrix in whichthe number Of columns is one more than the number Of rows,the case considered in Art. 249. But
,as was shewn, Art. 89 ,
when we are given a number of equations connected by rela
tions,the determinants formed by taking a sufficient number of
236 ORDER OF RESTRICTED SYSTEMS OF EQUATIONS.
the equations,require to be reduced by dividing out extraneous
factors,these factors being determinants formed with the co
efficients of the equations of relation. If then, in the present
case,we took a sufficient number of the equations and deter
mined the order by the rule of Art. 249 , our result would requireto be reduced by a number which we proceed to determine.
272. Let us commence with the simplest case where we haveInequations in 71: variables, the equations being connected by a
single relation. To fix the ideas we write down the systemwith three rows
0 b 0 7A.
(1 b c'
N
a b c )t"
where we mean to imply that the quantities involved are con
nected by the relations
M N'
a"
0,Nb Nb
'
N7)"
o,M A c N
'
s"
0.
We also suppose that h a,Na
’
,N
’
a”are of the same order
,so
that the orders A. a 7t’
a.’
A."
a"
. Now let us,in the first
place,suppose that the two first equations are in the simplest
form,and that A"
1. The true order then is that determinedfrom the first two equations ; that is to say, if we indicate theorders, as in Art. 249 , Now supposethat we had omitted the first row
,the order deduced from the
second and third,would be Q + P (B+ ry) B + 7
2
+ Bry ; whichwe see
,in order to give the true order
,requires to be reduced
by (fy— a) in other words,by the order of 7x
multiplied by the order of the determinant obtained from the
three equations. And the general rule to which we are thusled is : Leave out one of the rows and determine the order of
the remaining system by the rule of Art . 249 from the numberso found
,subtract the order of the determinant formed from all
the equations,multiplied by the order of the term in the relation
column belonging to the omitted row. It is easy to verify,that
we are thus led to the same result whatever be the omittedrow. Thus
- 7t
238 ORDER or RESTRICTED SYSTEMS or EQUATIONS.
are of the orders A+ a,7m a
'
,and so on
,the orders of the
coefficients increasing by a for every power of g, and by a'
for every power of z . Then the terms in the first columnconsist first of 1} (m n 1)(m n 2)terms whose orders are
7x; k— a
, A— a
’
; 7t— 2a
,A— a — a
'
,A— 2a
’
,&c. ; secondly, of
5 (h l 1)(n l 2)terms whose orders are p , p. a, ,u.— a
'
,
&c.,and thirdly of 11, (l+m l)(l +m 2)similar terms in v.
These may be taken for the numbers a, ,8 , «y, &c. of Art. 249 .
The numbers a,b,c,&c. of that article are 0 , a , a
'
2a,a + a
'
,
2a'
,&c.
,there being in all 1} (l+m n 1)(1+m n 2)such
terms. Lastly,the numbers A
,B,C,&c. of the last article are
found to consist of Q (l— l)(l— 2)terms, p + V, ,u.+v— a , ,
u,+v—1a'
;together w ith 1} (m 1)(m 2)and Hn 1)(n 2)corresponding terms in v and 7t + p h In calculating I have found itconvenient to throw the formula of the last article into the shape
1} {(PH ? P72
(3 2 153where 3
2denotes
'
the sum of the squares of the terms a,b,c,&c.
Also if (b (a) Al4Bl
8Cl
2DZ E
,it is convenient to take
notice that
96 1)12Ahmn (l m n) 6B lmn E.
I have thus arrived at the result, that the order of the system is7377272 07111 1)N
2
{ ml (ul 1),u.2
glm (lm 1)v”
{ (nl 1)(kn- 1) t (l 1) [ w
— l)(mn 1) (m 1)(m — 2)VA
1)(nm 1) t (n 1)<n A)»
+mn7t {lmn 1+ 1
+ ulft {lmn — m+ 1 — 5
+ v {lmn— u + 1 — %I
glmn (lmn— Z— m—n+ 2)(a2
+ oc &lmn(2Imn— l—m—n+ 1)aa'
.
If the order of all the terms in the first equation be 7t,in
the second a , in the third v, we have only to make a and a’
in the preceding formula. In this case,supposing k : —
,u. y= 1
0
,
and Z : m :— n,the order becomes (n 1)(12
2+ 92
( 239 )
LESSON XIX.
APPLICATIONS OF SYMBOLICAL METHODS.
275 . IN this Lesson,which is supplementary to Lesson X1V.
,
we wish to Show how the symbolical notation there explainedaffords a calculus by means of which invariants and covariantscan be transformed
,and the identity of different expressions
ascertained. The basis of the whole,as applied to binary
quantics,is the following identical equation : Let D
,denote
(Dd—(
i; +g (é and as before let 12dia
ldig all d
o
;is easy to verify that
,identically
D123 D
23 1 0
312 0
, (A),for on expansion the coefficients of w and g separately vanish.To illustrate the use to be made of this equation, we write outthe first application we make of it in greater fulness of detailthan we Shall think it necessary to do afterwards. Squaring
equation A ,which may be written D1
23 = D213 “ D
312:we get
2D2D312 .13 D313
2 D3
212
2D323 (B).
In this form the equat ion is true , even if the functions U1) U2 , Q ,
supposed to be operated on,are all different in form. But the
case which we Shall exclusively consider in this Lesson is whenwe are forming derivatives of a single function U
,with which
U] ,U;, Us all become identical when their suffixes are sup
pressed. And in this case we have seen (Art. 149) that
D1
223
2
,D
231
2
,D
8
212
2are all only different expressions for the
same thing. Hence,equation B becomes
D312”
2D2D
812 . 13 .
Now, by the theorem of homogeneous functions, the operationDperformed on any function only affects it with a numerical multiplier. On the left-hand side of the equation, since D3
2 onlyaffects U; which is not there elsewhere differentiated, that multiplier is n (n on the right-hand side D
2,D
8 ,each . affect
functions which have been once differentiated besides ; and
then it
240 APPLICATIONS OF SYMBOLICAL METHODS.
therefore,introduces the numerical multiplier n 1 conse
quently,if we expand the equation drop
suffixes,and div ide both Sides by 2 (n we get
fl yd2U dU dU d
’U
(dU
(n " 1) dx”dy dandy dx dy W 75:
d2U d
2U
(d
”U
dx"dy
’dxdy
276. It will be Observed, that whenever by transformationthe number offigures in the symbol is diminished
,it shows that
the quantic U is itself a factor in the derivative. Thus,in the
example chosen,we proved that differs only by a numeri
cal factor from But as what we operate on is the productUIUQUQ,and as the symbol 12 does not affect U
8at all ; when
we drop the suffixes, U remains as a factor.
277. And in general any symbol may be so transformed thatthe highest power of any factor 12 may be even. For thesignification of the symbol is not altered if we interchange thefigures 1 and 2 ; therefore,
and by the help of equation A the quantity 961 432 can be so
transformed as to be div isible by 12. Thus,for example
,the
derivative 12m .13,where m is odd
,differs only by a numerical
multiplier from For
13 {B 213 D
123} D
812
“,
or' if n be the degree of U the function Operated on,
2 (n m) 13 n
278. In addition to the identical equation already mentioned,
we employ another which can be easily verified, viz . :
By the help of these equations we can reduce all symbols tocertain standard forms, which we denote by special letters
,as
follows : For two factors we take as our standard form the
242 APPLICATIONS OF SYMBOLICAL METHODS.
and assembling the identical terms, we have4 (n Q, D,
‘ 4D,2D,,
2
3a (a 1)(a 2)(n 3)3 2 17+ 4a 2 (a
It remains to bring to the stand forms. Ji aise to the fourth powerD l 23 D
2 13 D, 12, when collecting identical terms, we get
8D,3D2 12
3.13 D3
4
But, as inArt. 277, 8D33D2 12
3.13 4D3
4 12‘. Hence
2 (n 2)(n 3) n (n 1)SH.
Otherwise thus :We know by elementary algebra that the equation a b c 0,
implies a 4 b‘ c‘ 2a”bz 26202 0. Hence, from aluation A, we have
D,‘23‘ D2
431‘ D,‘12‘ 2D1
2D22 2D2
2D,,2 312.122 0
and, as before, D,“124.
We have then
4 (n (n 2)(rt — 3)Q, — 3n (n 1)(n 2)2 (n UH 2 + 2n3 (n UBE.
It is eas ily seen in general that when in a symbol &c. we substitute for
each pair of factors by equation (B), the efiect is to diminish the number of
figures which enter into the symbol, and therefore U is a factor. The values of
Q, and Q, are calculated (Cambridge a/nd DublinMathematica l Journa l, vol. Ix. p.Ex. 3. To reduce Multiply equation (D)by 122, when we have
2D,t D,
‘12“ 2D 12D2
2 122.232.312 4.1)22D ,
2
2 (n 2)(n 3)(n— 4)(n 5) n(n 1)(n 2)(n 3)AU
—2 (fl)— 4)2 (n T.
Ex.4. To reduce Multiply the three equations
2D,D, 13.14 D3132 D ,2 142 D ,
2 342,
2D,D214.12 D3142 D3 122 B 12242,
when, after assembling the identical terms, and reducing, we have6D2
2D32D ,
2 4D,“ 6D8
2D,‘
where, as the value Of has been already calculated, the value of
is determined.Ex. 5. To reduce We may either by multiplying equation (D)by
342 obtain an expression for the derivative in question in terms of 12t .132, and
which have been already calculated, or else proceed directly as follows.Multiply by 142 the product of2D2D, D3 122 D,
2 132 B 12232 2D2D, D3 242 D ,
2342,z 232,
and we have
4D221J,
2 121 3243 4 142 D ,2D,
2 ma y.)
But from equation (0142 132.24z
APPLICATIONS OF em somcm. METHODS. 243
4mm; 2mm;and hence reducing, we have
612221231224 423 42 21 2
1 1222 424 12,
6 (u 2)(n 3 (a 2)(n 3)53 + 215 (n 1)TU.
280. We wi sh next to show how to form the symbol expressing any derivative Of a. derivative. In the first place it isobvious that if we have any function of an as“, a s,
&c.,we shall
get the same result whether we suppress all the suffixes and
then differentiate with regard to x,or whether we take the sum
of the differentials with regard to w‘, 903,&c., and then sup
press the sufiixes. The differential,then,with regard to w
of any derivative symbol containing the figures 1, 2 , 3 , &c.
(such as 12”
is got by operating on this symbol with
a
d
} ; £1 £4 &c. The manner in which other derivatives
are expressed will sufficiently appear if we take a particular example
,and show how to express symbolically the Hessian Of the
Hessian. It will be remembered that the Hessian itself is expressed by forming the product of U
1by the same function
written with different letters U2,
and then operating with
(fins where'
E,‘0) denote differentials. To form the
Hessian of 12“we form the product and Operate on it
with fem)“ where E, d
ig?!£2; E, = d
im. £2
And thus it is easy to see that the Hessian of the Hessian is
(13 14 23 which expanded is equivalent to
4 (122. 4 8
Or, to take a more complicated example, let us endeavour toexpress symbolically the result Of performing the Operation
on three different functions, the first being the
Hessian the second being T, or and the thirdthe function U itself. Then we must use different figures for
the two derivatives, and we therefore form the product of 12’by
and operate on it with
£37192
$1773)a
,
244 APPLICATIONS OF SYMBOLICAL METHODS.
d d d d” div
,
+dx dx
+6133 :
x4 6
and for E9 , d‘i and the symbol required is found by expanding6
where we are to put for g
(13 14+ 15+23 24+ 25)(36 46 (16 122.
Ex . To reduce the Hessian of the Hessian to the form (2 83 BTU (see p.We have already seen, Ex . 5 , p. 242, that can be expressed in this form.
It therefore only remains to prove the same thing for the otherterms in the expressionfor the Hessian of the Hessian given in this article. Multiply by 122 the pro ductof the equations
2D,,Dl D,2 342 D,
2 132 D,2 142 ;
2D2D, D3 342 242 D3 29 ;whenwe have
4D,D2D,D, D17D2
2 12’.34‘ 2D,
4
Which, putting in for its value already found, gives12 (n 3)a 6 (n (n 3)SH 4n (n 1)(n 2)TU.
Again, to calculate we have only to substitute for from equation (B), when we have
2D3D, 22 2 1223 421 32 B ,2
and substituting as before, we get6 (n a)2 2n (n 1)TU,
by the help of whi ch values the Hessian of the Hessian is expressed in the desiredform.
281. It would evidently be convenient if a general symbolical expression could be given for the result of eliminationbetween two equations. When one equation is simple it iseasily seen (as in Art. 279)that the result Of elimination betweenit and an equation of the nth degree is &c.
,where
the symbol (1)relates to the equation of the nth degree
,and the
remaining symbols to the simple equation. Let us examinewhether any similar general formula can be found if one equation is a quadratic.* The result is to be of the second degreein the coefficients of the general equation
,and of the n
min
those of the quadratic ; there must be therefore in its symbolical
The general formula for the resultant of a quadratic and an equation of the
nth degree was given by me in 1853 (Cambridge and Dublin Mathematic a l Journal,vol. 111 . p. The theorem was re-discovered by Clebsch in 1860, and extended byhim to t he case of a. system of any number of equations, one of the second, one of
the um, and the rest of the first degree (Crelle, vol. In this article the formulais inMr. Cayley’
s notation as originally given ; inArt. 285 I give Clebsch ’s extension.
246 APPLICATIONS or SYMBOLICAL METHODS.
The eliminant then which in this notation is [n, is
(1A)(2A)&c.— 920A)(21
V)&0 '
n (u 3)(1A)”
(1B)°
(2MV(2N)”&c.
16
the coefficients being those which occur in the expansion of thesum of the inverse n‘" powers of the roots of a quadratic
,the
next coefficients for instance being
n (u — 4)(n— 5) n (u — 5)(n— 6)(n— 7)
282. The preceding series may be transformed so as to
proceed by powers of the discriminant of the quadratic. For,
since (1A)(2B) (13 )(2A) (12)(AB), we have
2 (1A)(2A)(1B)(2B) (1A)2
(2B)2
(1B)2
(2A)”
(AB)’
2 (IA)2
(2B)2
(AB)2
by the help of which substitution which is the discriminant of the quadratic, is introduced. Thus
,in the case of two
quadratics,the formula given by the last article is2 (14 )(2A)(13 )(23 )
which,by the substitution
,becomes
which is,in other words
,Mr. Boole ’s formula (Art. In
the case of the quadratic and cubic, the formula of the lastarticle is
4 (l a) (2b)(10)(2c) 3 (1a)”
(2b)"
(l o)which
,in like manner becomes
(l a)8
(2b)(l o)(20) (ab)2
(l o)
283. In like manner, it may be inv estigated whether a
general formula can be given for the discriminant of an equation. Formula for invariants of the same degree can easilybe written down. The discriminant is to be Of the degree2 (n Take
,then, two sets of (n 1)symbols, 1, 3, 5 , &c.,
2,4,6, &c., and form a product of n 1 factors
,each factor
containing one of each set. Thus,when n 4
,we form
(12)(34)(56)=A. Then by cyclically permuting one of the
APPLICATIONS or SYMBOLICAL METHODS. 247
sets of symbols we form (n 1)products in all. Thus,for
n= 4, we have (14)(36)(52)= B , (16)(32)(54) 0. Then anyfunction of the n degree of these quantities A
,B,&c.
,will
denote an invariant of the same degree as the discriminant.The two difficulties in the general theory are
,first
,to ascertain
which of these functions is the discriminant,and secondly
,to
discover the relations which ex ist between the different invariants which may hé so formed, and to find the simplest formsto wh ich they may be reduced. Thus
,for instance
,when n is
even,it is easy to see that k+ 1 being the
number of factors in the product A .
In the case then of an equation of the fourth degree,the
different invariants that can be formed are A4
,A
2B
2
,A
2B 0
,
omitting ABB which we know to be only iA
“. The first problem
is to form a combination of these invariants,which will vanish
identically if the given function be of the form nfio,where u
is of the first degree. To do this,suppose the function to be
of this form ; substitute for 1, a a (a referring to v and a to
for 2,b
,3 , &c., and examine the effect of this substitution on
each of the invariants in question. It is obv ious that the termsin which either a or a enter above the second degree vanish ;and denoting by M the condition (aa)
2
,that u should be a
factor in v,we can find without much trouble that
,on this sub
stitution, we should have
A4
2 161116
,A 23 2
66M "
,A 2B 0 = 42111 6 ;
and hence that the discriminant may be expressed by any of
the three equationsA
4A
zB
"A
2B C'
36 4 7
284. The methods here explained apply equally to quanticsin any number of variables. The identical equations principallyused in the case of ternary quantics are
D,123 D
,234 D
2134 (E),
o (F),
For the reduction of these to the standard fo'
rm, and the deduction thence of
the ordinary form of the discriminant of a quartic, see Cambridge and DublinMathematical Journal, vol.Ix.p.33.
248 APPLICATIONS or SYMBOLICAL msrnons.
to which may be added the corresponding equations for con
travariant symbols (Art. 156)
P 123 = Dla23 + .D
9a3 l + .D
8a12
a 12.a34 a3 1.a24
where P is our By ryz . For reasons already given (Art. 183)I touch but slightly
,in this volume, on those parts of the subject
which admit of geometrical explanations,and therefore only add
one or two examples ; referring the reader, for the present, to theCambridge and DublinMa thema tica l Journa l
,vol. IX. p. 27.
Ex. 1. It has been stated (p. 129)that a ternary cubic has an invariant of the
fourth order It i s now required to find the symbolical expression foranother got by operating on the Hessian with the evectant of thi s invariant. The
evectant (p. 129)is and the method of Art. 280 requires us now
to combine with this the ad ditional factor 4562 and substitute for a,4 5 6. And
as we are to omi t any term which contains any of the symbols 4, 5 , 6 more than three
times the required symbol reduces to123 124 .235 .316 .4562.
Ex. 2. In the theory of double tangents to plane curves, explained, Higher P laneCur ves, p. 81, it is necessary to calculate the result of substituting in the successive
emanants, v
y% fidz
U’ a % 7 22
7, fi—Z 11 3—
U, for a , y, e ; and to show that
each result is of the form PnU+ Q” (ax By We shall in thi s and in the
next two examples perform the calculation of Q2 , Q, , Q, . The symbolical expressionfor the result of substitution, it is easy to see, is (2 12 .a 13 .a 14 .a 15, &c.
Nowfirst to calculate a 12 a 13 , we have only to square equation (G), whenwe getP 2 1232 .1 122 6D2D3a 12
Or if we denote the Hessian 1232 byH, and the bordered Hessian by G, we have
6 (n a 12 .a 13 3n (n 1)GU P23 .
Ex. 8. To calculate a 12 .a 13 a 14. From equation (0)we have2DZD , a 12 a 13=P2 1232 2PD , 123 (1 23 B ,
241 23
2 D22a 312 D 3
2a 122.
Multiply by a 14,two of the terms vanish identically, and
2 (a a 12 .a 13 a 14 2a (a 1)021 122 .
To prove that 123 a23 .a 14 vanishes identically, we have only to multiply equation (17)by 123, when, since the terms in it difier only by a permutation of the
figures 1, 2, 3, each must separately 0. In like manner, it is proved that123 .a23 .a 14 .a 15 is identically 0.
285 . In the preceding articles I have used exclusively Mr.Cayley
’s notation. To illustrate now the method of working
with Mr. Aronhold’s notation
,explained (Art. I give
Clebsch’s investigation of the problem (_Art. 281)to eliminate.
250 APPLICATION or SYMBOLICAL METHODS.
If then we substitute the first set of values in we
get the determinanta a
s)as)
“4
p i 7 p s7 ps7 p e
an “a,
as)
“4
B] , 621 Ba) 3 4 7
which we may write (a I The result of elimination thenmay be written symbolically R (a 1 We use
in the second factor the symbol 6 instead of a,for the reason
explained (Art. in order to obtain powers of the coefficientsof d); but it is understood that the b symbols have exactly thesame meaning as the a
,since after expansion we equally replace
the products by the corresponding coefficientof 98, am”
, We may then write the result of elimination inthe more symmetrical form
213 (a.melt)”
ft)”
(hpaaafil
for this after expansion will be only double the former ex
pression.
Let us now write
(em it) A,
(611319 864)(bigaasB-i) B)
p 2a813 4)(5195 384) (a rgaaefla)(bipaasleai) 2 07
may be easily expressed in terms of A ,B, 0.
—V(02
01‘ 0
n
n ('
n 1)(77.— 2)(71.— 3)
286. It remains now to examine more closely the expressionsfor A ,
B,C,and to get rid of the quantities p and g which we
have introduced, so that the result may be expressed in termsof the coefficients of the given quadratic.
APPLICATION OF SYMBOLICAL METHODS. 251
.
Now A, which is the product of two determinants, may be
Wri tten as a single determinant,
p ig! 7 2“ (p 1Q2+p 291)7 2" (pigs-l-p ag1)7 “17 17
“1
psgz 7 2(pags+p sg2)7 2(27 da7 '8 27 “
2
2(29 2(paqs+psq2)7 p age 7 2(psgs+p 4qa)7 as7 18 87 as2 2" (pegs "2“ (173944194Qa)7 27494 7
“47B47 “
4
a a al 2 8
multiplied however by where m is the number of variables, that is to say, in the present case , 4.
For every constituent of this determinant must contain a
constituent from each of the last three rows and columns ; it istherefore of the first degree in the termspg l ,
11441919, +p 2q1), &c. ;
and if the coefficient of any of these terms be examined , it willbe found
,according to the number of variables
,to be either the
same,or the same with sign changed
,as in the product of the
two determinants. Now in this determinant we are to substitutefrom equation (A),
PI91 u all] Bi l‘u
(29192 +p 2gl) “12 (al z
aaxi) 2 (Bi lL2 &0 '
But when this change has been made,if we subtract from each
of the first four rows and columns the a row and column eachmultiplied by and the ,8 row and column each multipliedby Tha i , the additional terms disappear
,and the determinant
reduces to
um “121
“i s
“147
“n B“217
“227
”23
“247
“27 18 27 as
“817
“827
”997
“347
“37 BM “
3
“417
”427
“4117
“447
“41 13 47" a4
s.
“17
“2 ,
“3,
a4
0 0 0 0 0 0 0 0 0 0 0 0
B] , 6 Ba, B4 c c c c c c c d o o c u
as,
a4
o o o o o o o o o o o o
Clebsch denotes the above determinant, in which the matrix
252 APPLICATION OF SYMBOLICAL METHODS.‘
of the discriminant of a quadratic function is bordered by rowsand columns
,01, B, &c., by the abbrev iation, (see p. 15)
c1a01
, ,8 , a
the upper line denoting the columns, the second the rows bywh ich the matrix is bordered. Thus, then, we find for any
number of variables
A = ( 1) (11
,g,0 0 . aO . b
B = la, B,
In like manner l (a , ,8 , 5)And in the same way
a B ‘0 . al 7 70 (a , 3 , 0 0 0 b)
8
Now it was proved, Ex. 2,p. 26
,that Z) A (Z
,Z)12
where A is the discriminant of the quadratic function. And it
is proved in the same way, in general, that
A (01, B,
01, ,8,
01,
01, B,
If,then
,we call the last written function D, we have
0 ’ AB AD,
and the formula of Art. 285 becomes
The read erwill find some applications of this to geometrical problems in Clebsch’s
paper (Crelle, vol. In the case of binary quantics, to translate thi s result into
the notation we have used, ,we must write for the product of 91. pairs of factors
1A .2A, 1B .2B , &c., and for D, 122. I refer also to Clebsch ’
s paper (Crelle, vol.“Usher symbolische Darstellung algebraischer Formen,
” for a rule for obtaining a
general symbolic formula for the resultant of two binary quantics or for the discrimi
Bezout’s method of elimination, explained (Art . 83)to two quantics written sym
bolically (aim; (am but the resulting fule is, as may be ex pected,
254 APPENDIX.
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APPENDIX. 255
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256 APPENDIX.
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APPENDIX. 259
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260 APPENDIX.
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264 APPENDIX.
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NO TES.
HISTORY or DETERMINANTS. (PageTHE following historical notices are taken from Baltz er’
s T heory of Determinants ;and from the sketch prefixed to Spottiswoode’
s Elementary Theorems relating to Deter
minants. The first idea of determinants is due to Leibnitz , as Dirichlet has pointedout. In Leibnitz ’
s letter to L’
p ital, 28 April , 1693, (Leibnitz ’s Ma thematical Works,
published by Gerhardt, vol. IL, p. is to be found the first example of theformation of these functions , and of their application to the solution of linear equations ; the double sufix notation (p. 7)is employed, and he expresses his convictionof the fertility of his idea. But nowhere else in his writings is there to be foundany proof that he sought to draw any new fruits from his discovery ; and the methodwas lost until re-discovered by Cramer in 1750. Cramer in his Introduction a l'Ana lysedos l-ignes Cow bes (Appendix), has exhibited the determinants arising from linearequations in the case of two and three variables, .and has indicated the law accordingto which they would be formed in the case of a greater number. The rule of signsby the method of displacements (Note, p. 5) is given by Cramer. The equivalenceof the other method by permutation of suffixes was afterwards proved by Bez outand Laplace. In the Histoire de l’Aoadémie Royakf dea Sciences, Année 1764, (published in Bezout has investigated the degree of the equation resulting from theelimination of unknown quantities from a given system of equations, and has at thesame time noticed several cases of. determinants, without however entering upon thegeneral law of formation, or the properties of these functions. The Hiatoi/re dc
l’Acade’
mie, An. 1772, part II. (published in contains papers by Laplace and
Vandermonde relating to determinants of the second, third, fourth, ao. orders. The
former, in discussing a system of simultaneous differential equations,has given the
law of formation, and shown that when two rows or columns are interchanged , thesign of the determinant is changed , and that when two a re identical , the determinantvanishes. The latter employs a notation in substance identical with that which, afterMr. Sylvester, we have called the umbral notation, and explained p. 7. In hisMemoiron Pyramids (Memoires de l'Académie dc Ber lin , Lagrange made an extensiveuse of determinants of the third order, and demonstrated tha t the square of sucha determinant can itself be expressed as a determinant. The next impulse to thestudy was given by Gauss, Disquisitiones Arithmeticaz, 1801, who showed , in the caseof the second and third orders , that the product of two determinants is a determinant,and very completely discussed the case of determinants of the second order arisingfrom quadratic functions ,‘ t.e. of the form 62 ac. In 1812 Binet published a memoiron this subject (Journa l de l’Ecole Polytechnlque, tome Ix., cahier in which heestablishes the principal theorems for determinants of the second, third, and fourthorders, and applies them to geometrical problems. The next volume of the sameseries contains a paper, written at the same time, by Cauchy, on functions which onlychange sign when the variables which they contain are transposed. The second partof this paper refers immediately to determinants, and contains a large number of verygeneral theorems. Cauchy introduced the name “determinants,” alread y appli ed byGauss to the functions considered by him, and called by him “determinants of
quadratic forms.” In 1826 Jacobi took possession of the new calculus, and the
NOTES. 267
volumes of Crelle’s Journa l contain brilliant proofs of the power of the instrument
in the hand of such a master. By his memoirs in 1841, Deforma tions ccproprietatibus
determinantium, and De determinamibusfunctiom libus (Crelle, vol. determinantsfirst became easily accessible to all mathematicians. Of later papers on this subject
,
perhaps the most important are Cayley’s papers on Skew Determinants (Cretle,vols. xxxn. and Of elementary treatises on this subject, I have tomention Spottiswoode's E lementary 777m m relating to Determinants, LondonBrioschi, La teorica dei determinanti, Pavia, 1854 ; and Baltz er, Theorie and Amoco
dimg der Dcterminanten, Leipz ig, 1857 second' editi on, 1864. French translationsboth of Brioschi 's and Baltz er’s works have been published.
COMMUTANTS. (Page
commutants, which are but an extension of the same idea. If we wri te for brevi tyE, n, for ii
i-
y, it is easy to see what, according to the rule of the umbral notation,
is meaut byE, ”9 £2: "
’y
E: ’h E? ) 511» "a.
We compound the partial constituents in each column in order to find the factors
products obtained by permuting the terms in the lower row. Thus the first exampledenotes which is the Hessian ; and the second denotes
E‘-E’n’-n‘ 410»
which is the Ordinary cubinvariant of a quartic.Again, since multiplication is performed by addition of indices, it will be readily
combined by addition instead of by multiplication. Thus, considering the quantics“I; “OI”: “3: “a, an 001 2’
s
the invariants in the last two examplesmay be written1, 0, 2, 1, 0.
1, 0, 2, 1, 0,
Which expand-ed are “gag d ial “‘w e (z‘a-la , &cc
All these commutants with only two rows may be Written as determinants, butit is a natural extension of the above notation to form commutante with more thantwo rows, such as
E, m 1, 0, E”, En, n”
E! ’l! 1! 0! £2, E", "2
E, 'b I: 0, £2! E", 712.
E; m 19 0, £1: E": 11’
These all denote the sum of a number of products, each product consisting of as manyfactors as there are columns in the commutant, and each factor being formed bycompounding the constituents of the same column ; and where we permute in everypossible way the constituents in each row after the first
:Thus the first and second
examples denote the same thing, namely, the quadrinvanant of a quartic expressed
268 Norms;
ineitherof the -
forms 4 4 3& R5‘nfi or c‘a , 3d2a2,
“ while “
the third example -E‘fl‘-fl° &c. denotes the cubinvariant of an octavic given at
length, p. 124.We have seen that the two invariants of a binary quartic can be expressed as
commutants, but it will be found impossible to express in the same way the dis
criminant of a cubic. Thus, the lead ing term in it being af ar,2 or £8£3n3n3, we arenaturally led to expect that it might be the commutant
Er 'b E: 7b
.5’ 7h E, 7 bE) 7b E; 771
but this commutant, instead of giving the discriminant, will be found to vanishidentically. It may, however, be mad e to yield the discriminant by placing certainrestrictions on the permutations which are allowable. For further details I refer tothe papers of Messrs. Cayley and Sylvester in the Cambr idge and Dublin Mathematica l Journa l,
HESSIANS. (Page
The name was given by Sylvester after Professor Otto Hesse of Heidelberg, whohas made much use of the f unctions in question, which he called functional determinants. They are a particular case of those studi ed under the same name byJacobi ,(Crelle, vol. the constituents of which are the differentials of a series of ashomogeneous functions in n variables. It is so convenient to have short distinctivenames for the functions of whi ch we have repeatedly occasion to speak, that I havefollowed Sylvester in calling the formerHessians, the latter Jacobians, see p. 69.
SYMMETRIC FUNCTIONS. (Page
The rules for the weight and order of symmetric functions I believe to be Mr.Cayley’
s, though I cannot give the reference. The formula , Art. 55, I have takenfrom Serret
’s Lessons on Higher Algebra. The differential equation, Art. 56, is
an application of the differenti al equation for invariants, of which I speak afterwards.Brioschi ’s expression, Art. 61, I know from the use mad e of it by Mr.M. Roberts,Qua rterly Journa l, vol. IV., p. 168.
ELIMINATION. (Page
The name eliminant’ was introduced I think by ProfessorDeMorgan : I believeI have done wrong in using a second appellation when a name to which there wasno objection was already in use. The older name resultant’ was employed by Bez out,Histoire de l’Académie de Par is, 1764. The method of elimination by symmetricfunctions is due to Euler (Ber linMemoir s, The reduction of the resultant toa linear system was mad e simultaneously by Eul er (BerlinMemoirs, 1764)and Bez out(Par is Memoir s, The theorem as to the degree of the resultant is Bezout’s.
270 ,NOTES.“
of the coefficients of an equation posses s this property of invari ance. He found thatit was not peculiar to discriminants
,and he discovered other functions of the cc
efficients of an equation, at first called by him ‘hyper-determinants, ’ possessing the
same property. Mr. Cayley’s first results were published in 1845 (Cambr idgeMathe
mat ica l Journa l, vol. IV .,
p. From this discovery of Cayley’s, the modern algebra
which forms the subject of the bulk of this volume may be said to take its rise.
Among the first invariants distinct from discriminants,which were thus brought to
light, were the quad rinvariants of binary quantics, and in particular the invariant Sof a quartic. Mr.Boole next discovered the other invariant T of a quartic, and theexpression of the discriminant in terms of S and T (CambridgeMathema tic a l Journal,vol. IV., p. It is worthy of notice that both the functions 8 and T had been
used by Eisenstein (Crelle, 1844, xxvn.,p. 81)in hi s expression for the general solution
of a quartic, but their property of invariance was unknown to him, as well as the
expression for the discriminant in terms of them. Mr. Cayley next (1846)publi shedthe symbolical method of finding invariants, explained in Lesson X IV. (Cambridge andDublin Mathemat ica l Journal, vol. L, p. 104, C'relle, vol. The next importantpaper was by Aronhold, 1849, (Crelle, vol. xxxrx.,
p. in which the existence of
the invariants S and T of a ternary cubic was demonstrated. Early in 1851Mr. Boolereproduced, with additions, his paper on Linear Transformations (Camb ridge and
Dublin Mathemat ica l Journa l, vol.VL, p. and Mr. Sylvester began his series of
papers in the same Journal on the Calculus of Forms, after which discoveries followedin rapid succession. I can scarcely pretend to be able to assign to their properauthors the merits of the several steps ; and, as between Messrs. Cayley and
Sylvester, perhaps these gentlemen themselves, who were in constant communicationwith each other at the time, would now find it hard to say how much properlybelongs to each. To Mr. Boole is, I believe, due the principle that in a binary
a
d
; a?)may be substituted for a: and y (Cambr idge
and DublinMathematica l Journa l, vol. VL, p. 95, January, The principle wasextended to quantics in general byMr. Sylvester, .to whom is to be ascribed the generalstatement of the theory of contravariante, Cambr idge andDublinMathematica lJourna l,
vol. VL, p. 291 ; al though particular appli cations of contravariants had pre
viously beenmad e in Geometry in the theory of Polar Reciprocals , and in the theoryof ternary quad ratic forms by Gauss (Disguisit iones Ar ithmeticw, Art. who
gives the reciprocal under the name of the ad junctive form, and establishes itsinvariance under what he calls the “transformed substitution.” Mr. Sylvester alsoremarked that wemight not only replace contravariant by operative symbols, but alsoby the actual differentials 35 ,
(Art. 121)that invariants of emanante are covari ants of the quantic 1842, Cambr idgeMathematic a l Journa l, vol. p. 110, though Boole’
s methods were generalized byMr. Sylvester, Cambridge and Dub lin Mathematica l Journa l
,vol. VL, p. 190. Some
of the first steps in the general theory of covariants may thus be ascribed to Boole,though a remarkable use of such a function had been made by Hesse in determiningthe points of infiexion of plane curves. I had myself been led to study the same
functions both for curves and surfaces, in ignorance of what Hesse had done
(Cambr idge and Dublin Mathematica l Journa l, vol. IL, p. The discovery of
evectants (Art. 130)is Hermi te’s, Cambr idge andDublin Mathematica l Journa l, vol .VL,
p. 292. InMr. Cayley’s first paper he gave a system of partial difierential equations
satisfied by invariants of functions linear in any number of sets of variables. The
partial differential equations (p. 113)satisfied by the invariants and covariants of
b inary quantics were, as far as I know,first given in print byMr.Sylvester (Cambridge
and DublinMathematical Journa l, vol. e , p. Mr.Sylvester there acknowledges
quantic the operative symbols
&c. To Boole I would ascribe the principle
NOTES. 271
himself to have been indebted to an idea communicated to him in conversation byMr. Cayley ; and he also Speaks of having heard it sai d that Arch bold was also in possessionof a system of diiferential equations. These are not made use of in Aronhold
's
paper (Crelle, vol. xxxxx .) already referred to, but he refers, Crelle, vol. a .,
to a communicationmad e by him in 1851 to the Philosophical Faculty at K‘
dnigsberg,which, if it ever appeared in print, I have not seen. Very probably there may be otherparts of the theoryto which Aronhold may justly lay claim. After the publication inCrelle, vol.xxx , ofMr. Cayley
’
s paper, in which the symbolical method of forming ih
variants was fully explained, Aronhold worked at the theory inGermany simul taneouslywith the labours of Cayley and Sylvester inEngland ; and the mas tery of the subjectexhibited by his papers lead s me to suppose that of some of the principles he mustbe able to claim independent if not prior discovery. The method in which the subjectis
,
introduced (Art. 117)is taken from hi s paper (Crelle, vol. wi l l). I refer in a
subsequent note to the valuable paper byHermite (Cambr idge and DublinMathematica l
Journa l, vol. p. 172)in which the theorem of reciprocity was established, whichhad at first suggested itself to Sylvester, but was has tily rejected by him ; and inwhich the whole theory of quintics received important additions. Mixed con
comitants are Mr. Sylvester’s (Cambr idge and Dublin Mathematica l Journa l
,vol. VIL,
p. The theorem,Art. 131, is Cayley
’s and Sylvester’
s. The application of sym
metric functions to the invariants bf binary quartics was , I believe, first made in theAppendix to my Higher P lane Curves The method (Art. 134)of thencefinding
conditions for systems of equalities between the roots is Mr. Cayley’s (Phi losop hica l
Transactions, 1857, p. With regard to the subject generally, reference mustbe made to the important series of papers by Mr. Sylvester, beginning in the six thvblume of the Cambr idge and Dublin Mathematica l Journa l ; to a series of paperson Quantics published by Mr. Cayley in the PhilosoPhica l Transactions ; and to
Aronhold’s Memoir on Invariants (C relle, vol. mi n). The name ‘invariant
,
’as well
as much of the rest of the nomenclature, is Mr. Sylvester’s.
ON THE NUMBER or INVARIANTS on dovs amn'rs or A BINARY
QUANTIC. (Page
The following is an abridged sketch of the method pursued in Mr. Cayley's investigation (Phi losop hica l Transactions, 1855, p. The following illustration willshew the kind of formulae obtained and the interpretation to be put on them. A cubic
we have seen has three distinct covariants, viz . U, H, J, whose orders in the coefficientsare 1, 2, 3, and degrees in the variables 3 , 2, 3. To these we may add the discriminantA which is of the order 4 and of the degree 0 in the variables. These covariants are
not independent ; but H 3, J
2,and AU2 are connected by a linear relation (Art.
A ssuming then that these are the only distinct covariants, any other covariant mustbe of the form either UPH GA” or JUPE 9M . The number of the covariants of the
order 0 of the first form is equal to the number of ways in which 6 can be expressedin the formp 2g 4r ; that is to say, it is the coefficient of a“in the expansion of
1“
(f w“
) we)(1 r)‘
form is the coefficient of m0 in as (1 x)(1 a ?)(1 The total number then
is the coefficient of no in the expansion of 1 as (1 a)(i x2)(1 z
‘) or, whatcomes, to the same thing, in
In like manner, the number of covariants of the second
l — me
272 norms.
And conversely,if this expression for the number of distinct covariants were
established independently, it would indicate on inspection that there were four irreducible covariants of the orders 1, 2, 3, 4 respectively, and connected by an equationof the order 6.Now it was proved (Art. 144)that the number of covariants of the order 6 and
weight g is equal to the difierence of the number of ways in which 9 and q 1 can
be expressed as the sum of 6 numbers from O to n inclusive. Now the number ofways in whi ch q may be so ei pressed may eas ily be seen to be the coefficient ofafl z e in the development of
1
(1 z)(1 (1 x“z)’
where the expansion is to be effected in ascending powers of 2 . This equal to1_
— at"" l (1 — W 1)(1 2
l — m (1 — azz)
z ” w"
1 _ um 1the general term being(1 $ 9
or, what is the same thing,
(1
It remains then to find the coefficient of $ 7 in the part multiplying 2 9. To transformthis expression, the equation is used
2x9 (1
(l -i-z z) -
1 _ w (1 — w)(1the general term beingz2 &c.,
and the series is a. finite one, the last term being that corresponding to s n ; viz .
si t-mm ». Writing x9 for z , and substituting the resulting value of
(1 (1
in the preceding formula, the number we are investigating is found to beM m“)
(1 z )(1 y )(1 w)(1 art-t)
wh ere the sum extends from s 0 to s u, but it is of course unnecessary to includeany value of s which makes the index of s in the numerator greater than g. If wewri te 9 1 (n6 a), the formul a last writtenmay be transformed into
2 , coeflicient of x9 in
where the sum extends when n is even from a : 0 to s gm, I, and whenn is odd
Thus, suppose it were required to find the number of terms in an invariant of a
cubic of the order 6, we have to calculate the number of ways in which the weight36 can be made up as the sum of 6 numbers from 0 to 3 inclusive. We have thena 0, and the sum consists of two terms
, viz .
the coeff.of $ 124
The following will illustrate the process by which these are transformed. Writing
274 NOTES.
was the resultant of a, v multiplied by a power of (Xp’ Mu). Mr. Sylvester's
results, Arts. 182, 186, 189, are given in the Cmnp tes Rendus, vol. LVIII., p. 1071. The
reference to Lesson xvrn.,p. 158, was made before I had decided on omi tting the
Lesson on the Applications of the Theory to Ternary Quanti cs.
APPLICATIONS TO BINARY QUANTICS. (Page
The discussion in this Lesson of the quadratic, cubic and quartic, is mainlyMr. Cayley’s. See hi s Memoirs on Quantics in the Pkib sopklca l Transactions, 1854.The second form of the resultant of two quadratics, p. 160, is as elsewhere stated,Dr. Boole’s. The discussion of the systems of quadratic and cubic, two cubics, andtwo quarti cs, is I believe for the most part new. The form for the resultant of twocubics, p. 165, has been published by Clebsch (Crelle, vol. LX IV., p. and was
obtained by him by a different method, but had been previously in my possessionby the method here given. The proof (Art. 202)that every invariant of a quarticis a rational function of S and T is slightly modified from Mr. Sylvester’s (Phi losop hica l Magaz ine, April , The theorem, p. 176, that the quartic may bereduced to its canonical form by real substitutions, is Legendre’s (Tra itédes FonctionsEllip tiques, chap. IL). The canonical form of the quintic at “ by
5 ozs, which so
much facilitates its discussion, was given by Mr. Sylvester in his Essay on CanonicalForms, 1851. The invariants J and K were calculated by Mr. Cayley. The val ue ofthe discriminant and its resolution into the sum of products (p. was given by mein 1850 (Cambr idge and Dublin Mathematical Jour na l
, vol. v. p. Some mostimportant steps in the theory of the quintic were made in Hermite’s paper in theCambridge and DublinMathematic a l Journa l, 1854, vol. IX. p. 172, where the numberof independent invariants was established, the invariant I [in which it may be statedthe highest power of a , a ’
, has for multipli er f (df e2)5] was discovered ; attentionwas called to the linear covariants, and the possibility demonstrated of expressing byinvariants the conditions of the reality of the roots of all equations of odd degrees.The theory of the
' quintic was further advanced by Mr. Sylvester's Trilogy,”(Philosopkia z l Transactions, 1864, p. 57 and in Hermite’
s series of papers in the
Comp tes Rendus for the present year (1866)already referred to. The values of theinvariants A , B , C of the sextic were given byMr. Cayley in his papers on Quantics,and the existence of the invariant E pointed out. The rest of what is stated in thetext about the sextic is new.
THE QUINTIC. (Page
With respect to the special form a: (m2 a !)(x2 62)used, pp. 198, 201, I havenoticed since that its characteristic is that Hermi te ’
s invariant I vanishes. This formmay therefore be safely used in calculating any invariant“ fimctions whose order isdivisible by 4 and is below 36, since such forms cannot contain I. For the calculationtherefore at p. 201, it was sufficient to use this special form. More generally, if thealternate terms be wanting in any equation, every skew invariant vanishes. For
the weight of a skew invariant is an odd number ; and if the degree of the equation
NOTES. 275
be odd, the order of every invariant is even. Now an odd number can neither bemade up as the sum of an even number of odd numbers, nor of any number ofeven numbers. In the special form just referred to, a: and y are the linear covariants.What has been just stated leads to a. simple proof for the expression of I in
terms of the roots. It has in fact appeared that when I vanishes, one of the rootsis one of the foci of the involution determined by the other two pair. I is thereforethe product of the fifteen determinants of the form 2a 6 7 , a (3 '
y) 237l2a — d e
,a (6 — 2de
since if any of them vanish, the equation is reducible to the special form in question.And the vanishing of any of these determinants may be expressed as at p. 202,
(a - fi)(a — 6)(v —V)(a — e)(fl
Mr. Sylvester had also communicated a simple proof of the same thing depending onthe fact (p. 194)that a quintic for which I vanishes, is linearly transformable into arecurring equation. In like manner, what is stated (p. 210)gives at once the expression for the skew invariant of the sextic in terms of the roots : viz . that it isthe product of the fifteen determinants of the form
1 1
3, 7 4‘ s,
76 ,54)
Until my attention was called to it by Mr. Sylvester, I had omitted to notice(Art. 224)the use made by M.Hermite of the fact, that the quintic as well as everyequation of odd degree is reducible to a forme-typ e, in which the a: and y are thelinear covariants and the coefii cients are invariants. It follows immediately, thatby applying Sturm’
s theorem to the forme-type, the condi tions for reality of rootsmay be expressed by invariants. Hermite extends hi s theorem to equations of evendegree above the fourth, by the method explained at the end of the next note.I think it therefore worth while now to give the coefficients of the f orme-type of thequintic. They were given by Hermite (Cambr idge and Dublin Mathematica l Jowrna l,
vol. III. p. and re-calculated by me before I found out the key for the translationof Hermite’s notation into Mr. Sylvester’
s, which is A J, J2 K,J3 JK 9L.
I write now J 2 3K M,JK 9L N 5 and Q a numerical multiple of Hermite’s I,
Such thatQ¢=JK 2M 2 2MNK (J 2 + 12K ) JN 2 (J2 + 72K )— 48N 3
,
then the coefficients of thefarme-tgp e areA QMB JEM 2 MN (J 2 ISK ) 30JN 2
,
C : Q (JM — 12N ),
D J'ZKM 2 JMN (J 2 30K ) N 2 (42J 2 144K ),
E Q (J ZM 24JN),
F : J 3K ]l{ 2 J 2MN (J ? 42K ) N 2J (54J 2 288K ) 1152N 3.
I thus find the first Sturmian constant 3 2 A0 to be36N 2 { (MX ti JN)2 16MN 2}.
The Sturmian constants being essentially unsymmetrical , there seems no reason toexpect that the discussion of these forms would lead to any results of practical interest.The coeflicients of thefam e-type, as M.Hermi te remarked, satisfy the relations
AJ 2 BJ2 — 2DJ + F — 1152N 3,
.4E — 1‘
24N 5, AF BF
276 norms.
Thus then the quadratic covari ant is N 5 (a2 Jy’) and operating wi th this on the
quintic, we get the canoniz ant in the formN 5 (AJ 0, BJ D, CJ E, DJ PI”,
the coeficients inside the parentheses being all further divisible by N. Hence, we haveACE 2B CD AD12 EB 2 0 8 4 , 12‘N 6Q,
and the second Sturmian constant is got immedi ately by substituting the values justfound for B2 A C, AE
ILBD A CE 2B C’D &c., in the formula of Art225. I have not thought it worth while to calculate the third constant.
THE TSCHIRNHAUSEN TRAN SFORMATION. (Page
The Tschirnhausen transformation consists in taking a new variable
y a Ba: yaz hmm l
;
then there are n values of y corresponding to the n values of x , and the coefiicientsof the new equa tion in 3; are readi ly found in terms of those of the given equation bythe method of symmetric functions, the firs t for example being (1 80 B3 , 7 32 &c.
The coeffi cient of g’H is evidently a linear homogeneous function of a , B, &c., thatof y" ? a quadratic, of y” a cubic function, and so on. In the case of the quintic,the transformation is y a. Ba: 6a“
, and we have four constants a, B, 7 , 8
at our disposal . Mr. Jat tard pointed out that the coeficient of ya being a quadraticfunction of a , B, «
y , 6 was (Art. 162)capable of being written as the algebraic sum
of four squares, say t2 a2 02 10 2. It can therefore be mad e to vanish, by
a ssuming two linear relations between a , B, y , 6 , t a 0,v w 0. If we combine
with these two that linear rela tion which makes the coefficient of g“vanish, we havethre e rela tions enabling us to express three of the constants a
, B, '
y , 6 linearly in termsof the fourth. We can then by solving a cubic only make the coeflicient of 312also vanish, or else by solving a biquad ratic make the coefficient of g vanish. In thiswayMr. Jerrard showed, that by the solution of equations of inferior orders, a quinticmay be reduced to either of the trinomial forms y5 by c, or g
5 by2
c. The
actual performance of the transformations would be a work of great labour, butM. Hermite showed how by somewhat alteri ng the form of substitution, we canavai l ourselves of the help of the cal culus of invari ants.
If we have to transform the equation aa"! be?” can-2 &c Hermite's form
is to takey z aX-l-(aaz -l
then in the first place the transformed equa tion will be divisible by a ; and secondly,if the givenequation be linearly transformed, and if the corresponding substitutionfor the transformed equa tion be
Y=AN + (AX + B)a'
+ (AXH—BX -i 0)
then he has shewn that the expressions for a'
, B’
, &c. in terms of a , B, &c. involveonly the coefficients of linear transformation, and not those of the given equati on.It is not so with respect to the first coefficient A, whi ch we have therefore da ignatedby a special letter. But the theory of linear substitutions will be directly applicableto all functions of the coefficients of the transformed equation which do not contain A.
278 NOTES.
In like manner, for the second operation, we have, by operating on the originalequation,
a :(a, b, 0, dis , E
a: (b, c, d, 8133, 0.
But the original equationmay be writtenas (a , b, a.di m. (b, c, orel se.as o.
dHence z
”' The part of(T:due to the variation of a: is therefore
aw’a (2am:a 4bw’
)B (3az ‘ 'y.
The remaining part is(46a: 30)a (4barz 1202: 6d)B (4bx3 12da' 36)7 .
Adding, the coefficient of '
y vanishes in virtue of the original equation,remaining part is found to be
dVa
a?2B
which completes the proof of the theorem.
When thi s transformation is applied to a cubic, if we consider a , B as variables,the coefficients of the transformed equation in 3; will be covariants of the given equation. The transformed in fac t has been calculated by Mr. Cayley, and found to be313 3Hy J, where H is the Hessian (ac b?)a2 + &c., and J is the covariant(Art. (c2d 2abc 263)a
3 &c.
Mr. Cayley has also calculated the result of transformation as applied to a quartic.Take the two quantics
(a : by c, d: 61 37, (a , firVIII, it )?
Let A denote the invariant got by squaring the second equation, introducingdifferential symbols and Operating on the first, viz .
a a 2 4baB c (2ay 4B?) 4dB-y my”;
and let B denote the invariant got by operating similarly on the Hessian of thefirst, viz .
(ac— b?)a 2+2 (cd—be)aB+ (cc— 2bd -l-cf)ay+4 (bd— cz)B2+2 (be— cd)B7 + (ce—d2)72;
let 0 denote the result of Operating-with the cube of the quadratic on the covariantJ (p. 173)of the quartic, viz .
(azd 3abc 2b”)a." (a ’e 2abd 9coz 6620)a
fiB (abe 3acd 2b2d)(1 27
(4abe 12acd s d)aB" 6 (ad z bfle)aB'y 4 (adz b’e)B3
(ads 3bce 26d?)(2 72 (4ade 12bce 8bd2)Bz 'y
(ae’ 2bde 9628 60d?)B72 (be2 3cde 2d”)7 3
let S and T denote the two invariants of the quartic, and A the discriminanta
'
y B2 of the quadratic, then the transformed equation in y is(GB 28A)y? 403, SAZ 33 2 182A2 12TAA 23 3 A.
Mr. Cayley has also calculated the S and T of the transformed equation. In
making the calculation, it is useful to observe that since the square of J, from which 0was derived, can be expressed in terms of the other invariants, so also may the squareof C the actual expression found by him being
0 2 TA8 SAzB 4B“ (182A2 12TAB 453 2)A BISITAA2 16T2A3.
The result then is that the new 8 is33
211 2 12TAA ,
and the new T is TAa fiSzA’A 4:8TAA2 A3 (16T2
NOTES. 279
Finally, he has observed that these are the S and T of AU+ 4AH, as may beverified by the formulas of Art. 211. It follows
,then
,that the effect of the Tschirn
hausen transformation ‘is always to change a quartic into an equation having the same
invariants as one of the form,U+ RH,
and therefore reducible by linear transformationto the latter form. Mr. Cayley has not
, as yet at least, made the correspondingcalculations for the quintic. The following is the form in which M. Hermite hasappli ed his methods to the quintic.Let a be a quantic (as, u , , a , its differentials wi th regard to ac and y let be
a covariant, which we take of the degree n 2 in order that the equation we are aboutto use may be homogeneous
'
ia a: and 31 ; then the coefficients of the transformedequation, obtained by putting z are all invariants of a. The equa tion “
in z is“I
got by eliminating z and 3] between zu, y¢=0, and or, what comes to thesame thing, z a , an}; 0, which follows from the other two. If we linearly transform a and y, the new equation in z is got, in like manner, by eliminating betweenz fl l Y0 0, But, if z =XX + pY, A=Nu
' —Np ,we have s '
w—py, AY=Xy— h
’w, and, Art. 126, U1=Xu1 + NumU
and, since is a covariant, we have <I> Making these substitutions, the equation ih 3 , corresponding to the transformed equa tion, is got by eliminating betweenzow, Na.) AH ¢ow r s) o, zow. flu.) are as as» 0.
Multiply the first by y', the second by N , and subtract, and we have Asa, A‘y¢ 0.
In likemanner, multiplying the first by p , the second by h , and subtracting, we getAw, A‘mcp 0. In other words, we have the two ori ginal equations, except that zis replaced byfi Consequently, the equations in z corresponding to the originalequation, and to the same linearly transformed, only differ in having the powers of zmultiplied by different powers of the modulus of transformation A, and therefore theseveral coeficients of the powers of z are invariants.The actual form of the equation in z will be
A B —3z " &c. 0.
It is easy to see that the discriminant will appear in the denominator ; and the coeficient of z“ will vanish, since, if be any funcfion of the order a —2, the sum of
the results of substituting all the roots of U in5; vanishes. In fact,l
of this sum are brought to a common denominator, the numerator is the sum of 4mmultiplied by the differences of all the roots except a , and this is a function of theorder n—2 in a , which vanishes
’
for n— l values of a , a =B, a : and must
”Ur “(P1 34’s 74’s “My
where are four covariant cubics of the orders 8, 5, 7, 9 respectively inthe coeflicients ; is the canoni z ant ; (p, is the covariant cubic of the fifth order,noticed p. 191 ; and for the general equation, its leading term or source (p. whenceall the other terms can be deri ved, isa’cef 8a2d2f + 2afi’de2 ab’qf -l 14abcdf l l abce2 abd’e 9aa°f + 14ac2de
671 c Sbsdf + 9ble2 66207: Iwede 862023 3bc'e 2bc’d’ .
On inspecting this, we see that it vanishes if both a and b vanish, consequently, ifthe given quintic has two equal roots, their common value satisfies this covariant.We can forma covariant cubic of the seventh order from in the sameway that
280 NOTES.
was formed from and by ad ding multiplied by J and a numerical coeflicient,can obtain such that its source vanishes when a and b vanish ; and, in likemanner,«p, can be mad e to possess the same property. When this substitution is made, thecoeficient of z ’ is a quad ratic function of a , fi, 7 , 8. Hermite finds for its actualvalue (a result which may be verified by working with the special form, p. 200)
{Fa ’ GKDa '
y D (F IOJK )72} 1) {Km was (91m loAF)
where F : Q (I6L- JK ), and vanishes when the quintic has two distinct pairs of
equal roots. By breaking up into factors each of the parts into which this coeficient
has been divided, the two linear relations between a , 7 fi, 8, which will make itto vanish, can readily be obtained ; as also by another process which I shall not delayto explain. The discussion of this coefficient is also the basis of Hermite’s laterinvestigations as to the cri teria for reality of the roots. He avai ls himself of a
principle of Jacobi ’s (Chalk, vol. that if a , B, 'y , &c. be the roots of a given equa
(t as a‘lv cu
r (t flu Bar &c.,
be brought by real substitution to a sum of squares, the number of negativesquares will be equal to the number of pairs of imaginary roots in the equation.Hermite shows, by an easy extension of this principle, that the number of
pairs of imaginary roots of the quintic is found by ascertaining the number of
negative squares, when the coeflicient of 2 3 just written is resolved into a sum of
squares. And since the same process is applicable to every equation whose degreeis above the fourth, he concludes that the conditions for reality of roots in everyequation above the fourth can be expressed by invariants.
NOTE ON THE ORDER OF SYSTEMS OF EQUATIONS. (Page
Mr.Cayley, in the Cambridge and Dublin Mathematical Journa l, vol. IV.,p. 134,
determined the order of a. matrix with k rows and 70 + 1 columns, in the particular
lished, Quar terly Journa l, vol. L, p. 246, and in the Appendix to my Geometry ofThreeDunensiom. These are, as far as I know, the only papers published on the
subject of this Lesson. Since the Lesson was printed, Mr. Samuel Roberts has communicated to me some extensions of the theory th ere developed. His method is tosuppose each quantic resolved into factors, and to deal with the combinations of the
ro binary quantics which can always be resolved into factors, and in the oase of ternaryand higher quantics, it would seem that the questi on whether or not they can be soresolved does not affect the problems here discussed , and that the orders determinedin the case of quantics which are the products of factors must be generally true. Thus,to determine the order of the resultant of two binary quantics of the degrees m, a ;if the order of the terms in the first be A, X+ c , X+ 2a , &c., it may be resolvedinto the product of m factors cm+ by, the orders of u. and 6 being 3, $4 a re
spectively ; similarly, for the second quantic ; and the resultant is the product of mufactors, the order of each being3+23: a ; and therefore mn times this numberwillbe the order of the resultant. Now he argues that we may deal in the same manner
282 NOTES.
In this way the order of conditions that a curv e should have two double points isfound to be
1)(n (u + 1)
1)(n 2){15X2 + lOn (11 + 6)a a
'
+ 272 (2711— 3)
Mr. Roberts investigates other problems by the same method ; as, for instance, theorder of conditions that four curves may have two points common ; or that a surface
mayhave a b i-planar double point. For these I must refer to his paper which I hope
will be soon publi shed . I only give the following result : The order of conditions thatthree b inary quantics should have two roots common is
"F fl v 'é E r r)rzmnu— l)<m — l)<n + 1
+ Eglmn (m
+ Eg-lmn (n -\
l-+ a)
With regard to the other subject discussed in this Lesson, I find (see Faa de Bruno
On E limination, p. 94)that Bez out gave a formul a for the degree of the re sultant oftwo equations from which some terms are wanting, viz . that if m,
n be the degrees
of two ternary quantics , and if the highest powers of the variables a: and y, whichoccur in the quantics, be only a , 13 ; a
'
, fi’respectively ; then the number of their
common values is reduced from mn to ma (m a)(n a'
) (m B)(n And
the same case of quantics from whi ch certain terms are wanting has been investigatedby Minding (Crelle, vol. xx .) But I noticed these references too late to be able to
study the papers in question, and to compare them with the theory I have given of
the cas es where the order of resultants falls below the ordinary number. The theoryof elimination cannot be said to be perfect until rules have been given for determiningin every case the exact order of the resultant.
BEZ OUTIANTS. (Page
It has been shown (Art. 81)that the resul tant of two equations of the nth degree isexpressed by Bez out ’
s method as a symmetrical determinant. This may be considered(Art . 114)as the discriminant of a quadratic function which Mr. Sylvester has called
the Bez outiant of the system. When the quantics are the two differential s of the
same quantic, then if we resolve the Bez outiant into a sum of squares (Art. the
number of negative squares in this sum will indicate the number of pairs of imaginaryroots in the quantic. The number of negative squares is found by adding (as inArt. 44))x to each of the terms in the leading diagonal of the matrix of the Bez outiant,
and then determining by Des Cartes’rule the number of negative roots in the equation
for X. The result of this method is to substitute for the leading terms in Sturm’s
functions , terms which are symmetrical with respect to both ends of the quantic ;that is to say, which do not al ter when for ac we substitute i (see Mr. Sylvester’
s
Memoir, Phi losophica l Transactions, 1853, p.
( 283 )
TABLES.
IN the preceding pages, the equations are usuallywritten with binomia lcoefficients, but as in practice it is often necessary to apply formulae to
equations not so written, we give for convenience some of the principal
results of elimination as applied to equations written without binomialcoefficients.
(1) The resultant of the two quadratics
(A , B , 01x, (a , b, ejie , y)” is (A c Cu)”(A b .Ba)(B a Cb),
a’ C ”
abB C + ac (.B2 2A 0) b2A 0 bcAB 0
2A”.
2. The resultant of the quadratic (A , B , 01 33, y)2 and the cubit
(a , b, 0, di sc , is
a‘C’“
abB C "ac (7(B
3 2A O’
) aml (B8 3A B C)
b’A O’2 beAB C +MA (192 2A 0 ) c
’A ’O c A "
622A 3
3 . The resultant of quadratic and quartic isa
’ C‘abB O 3
acC ’
(Ba 2A 0 ) ad0 (B
sSAB C)
as (B‘ 4B 2A 0 2A ’ O’ 2
) b’A C ’ bcA B C " bdA C (B 2 2A 0 )
bad (B3 3AB C ) c
‘A ’ C ” c ’B C caA 2
(B9 2A 0 )
d i‘A i‘O' deA ’B e’A‘
.
4. The resultant of quadratic and quintic isabB C "
(B’ 2A 0 ) adC’ 2
( Bs 3AB C’
)
a (B‘ 4AB ’O’
+ af (35 53 3A 0 + 5A
’B C”)
b’A‘
O " bcAB C 8 bdA C" (B3 2A 0) beA C (B
8 3AB C’
)
bjA (B‘ 4AB‘C + 2A
30 ”) c
iA ’C s c 2190 8ceA ’ O (B
2 2A 0)
q’
(B3 3AB C ) d ’A ’ C” deA ’B C’ J
r d"
(B2 2A 0)
e'A‘C efBA
‘+f
’A 5
.
5. Discriminant of cubic is
27A‘D‘ 4A 0 3 4DB 8 B 2
0 ”
6. Resultant of two cubics (A , B , O', Di x, (a , b, 0, di s ,
The value expressed in terms of the determinants of the form A b B a.
is given in p. 63 and p. 165 . Expanded it is
(,s a’bCD“
a’cD ( 0
2 2BD) a’d ( 0
3 33 CD 3AD’
)
ab’BD‘abcD (B 0 SAD) abd (.BO
'2 2B ’D A CD)
ac’D (B
‘ 2A 0) acci (2A C”3 +ABD B i c) ad’
(B3 3AB C + 3A
2D]
bi’AD2 630A CD b‘dA ( C2 2BD) bc BD bc (B O 3AD)
MM (38 2A 0) c
’A’D c
fldA flo i cd=A=B (PA S.
The other invariants of a system of two cubics are given (p.
284 TABLES.
7. The resultant of cubic
(A , B , C, Di x, y)3 and quartic (a , b, c, d, ”I f : is
a’D‘
aibCD
‘3u
’eD
’
( C‘ 2BD) a
’dD ( O’ s 3B CD 3AD ’
)
c’c ( C
‘ 4B 0 ’D 2B 2D ’ 4A CD”) ab’BD’
abcD'
(B 0 SAD)
abdD (B C,Yz 2B ’D A OB) abe (B 0
3 3B ' CD A C’QD 5ABD ’
)
ac’D 2
(B’2 2A 0 ) acdD (2A C
" ABD B ' O)
ace (B’ C ’ 2A 0 ° 2DB ’ 4A B CD 3A
’D ’
)
ctn (B3 3AB O + SA
'D) ade (B’C 3A B C ” AB ’D 5A ’ OB)
ac”(B
‘ 4AB ’ 0 + 2A’O ’ 4A ’BD) bSAD ’ b’
cA CD“
b’dAD ( 03 2BD) b’
eA ( O’s 3B C’D 3AD’
) bc’ABD’
bm p (B C 3AD) bceA (21321) A01) 190
2
) bd’AD (B2 2A 0 )
bdeA (B'C 2A 0 ” ABD) bc’A (B
s 3AB O + 3A2D)
csA zD’
c’dA‘CD c
’eA g
( C’ 2BD) crl gBD cdeA ’
(B 0 3AD)
ce”A a
(B2 2A dsA sD d’
eA 30 de’A ’B esA‘
.
8. The discriminant of a quartic written with b inomial coefficients,expanded is
use” 1211’1u1e2 18a”
c’e2 54azcd’
e 27a 2d" 54ab’ce’ Gabed'e 180abczde
108abcd’ Bl ac‘e Mercia?2 2711‘s2 l OSb’cde 641)"d8 54b’
cse 36b’cgd”
.
9. The discriminant of a quarticwrittenwithout binomial coefficients,4 (12ae 36d o
"
)s
(721106 c d 27ada 27eb’
or expanding and dividing by 27,256a3es 1920
260182 1286120282 1446190632 27a’d‘ 144abgce 6ab’d’
e
80abczde 18abcd’ 16ac‘e 4ac
"d2 27b‘e’ l i’cde
4bf’d3 4690949
10. The resultant of the two quartics
(A , B, C, D, E l m, (a , b, c, (I , el m,expanded is (see also p.
a4E‘
ai’bDE 3
a”cE 2
(D2 2CE) a
i’dE (D3 3 CDE
’
3BE 2
)
a’e (D
‘ 4 CD2E 4BDE 2 4AE ”) o
’bi c’E ’
a’bcE 2
( C’D 3BE) tr
id ( CD2 2 C ’E BDE 4A B 2
)
a2be ( CD
3 3 0 2DE BD ’E ADE”) 2BD)
cx’c ( C
’D 2BD2 B CE 5ADE)
a’ce ( O
’ZD2 2BD3 20313 4B ODE 2AD9E 3B 2E 2 2A CE ’
)a
’d ’E (O3 SB CD SAD2 SB ZE 3A CE )
aide ( C
’D 3B CD" 3AD’ B C ’E + 6B
’DE 2A CDE 5ABE ’
)
TABLES. 287
VIII. 2 u“ 6° 206‘3 if 16623
” 23‘ Sbbd 3269331 24bc’d
12b’d’ 8331a 83‘s 166313 43 2 8671663f Sdf 8629 839 866
6643q 23‘ 65d 11633 36 1763 936 56236”
Bed” 6‘ 10633 3 83 3 96d3 43” bff + 963f
8df + 839 66
23 4 26531 86’ 3d 96'd‘ 23 312
4323 166d3 43 ’ 2617+ 463f 8df
439 866
6’ 3 23‘ 3693 31 66326 362d’ 73da 3643 9693 3 8323
6333 43“ bcf 7df + 839 866
3‘ 463 ’d 26’d’ 4c 2 4393 86d3 63
”46?
864 4df + 469
9 439 466
bad 56’3d 563236 56W” 5cd” 4693 3 2323 96d3
43’ 6’f 363f + 8df 62
9 239 66
b’cd 363 ’d 62d“ 53dz 106313
83' Sbcf +°
3lf 33 9 439 9671
63931 262362 cdg 6933 106d3 83’ 4bi
’
f 1063f
dj 9629 1639 966
2 4327“ bgd‘ 2cd” 2623 3 43 ’ 26?f 463f Sdf
439 866
2 33W cal' 6363 432 5bcf 7e
°
239 866
2 1151376 6‘ 2323 4bde 43 ’ b’f 363f 34f
239 66
z a‘p
’
ya b’c bde 43 2 4bif l lbcf 9df +
1066
26313 23’ bcf 3df + 939 566
z usfii
y’a bale 43 ’ 3637
” 6df + 1766
2 a’
/32728” 3
’ 23” 239 266
2 a‘/3763 bff 363f + 3c 699 239 66 Si.
bcf 3df avg 839 1166
z a’fl
’
ygds clf 439 966 165.
Z asflydt z 209 610 i 85.
2 23 38734 39 666
66. 85.
Eaflydagnfl
288
IX. 2 ag
Eats/337
z afifity!
2 14flay!
TABLES;
6° 9670 3063 960‘ 4.456‘cd 546
’0
’d
93 3 13333 2 2760d° 338 36633 3 2763
’s
273 33 180d3 933 937 + 27b’cf 9c
2
f 181/elf
93f + 963
9 18609 9 319 906 963
670 14630”
763 4 6°d l 36‘cal 30620906
9esd 66312
1960d2 3368 653 126333
“bide 180d3 563” 6‘f 1 162
3f + 932
f + 1063lf
9ef 699 10609 9369 906 63°
6502 5633 3 560‘ l Ob‘cd 5623206 50836
l lbsde
1360d2 336
” 2653 86303 4 206’d3
40de 963 2 26‘f 6929 5 393+ 186df sef
4609 9319 2626 5 36 963°
Fe3 363‘ 36‘0d 96’0’d 3ci
‘d 369362 186cd’ 6d”
3653 960 23 962363 963 2 36ff 96eqf
9eff 3639 9319 9626 906 963°
63‘ 4623 ’d 3
3d 26W2 76031
2 Sd’ 46333 36023
136’al3 20d3 1163 2 46‘f s cf 0
2
f 2bdfl 13f 9639 18609
—°
9d9 9626 906 963°
b‘cl 66‘0d 9620236 2066 663332 l 2bcd2 3313 653
5633 3 56093 1162363 l l cde 563 2 bff + 46
20f
20ff l Obdf + 93f + 63
9 3609 9d9 236
62°
b‘cd 4620231 23336 63362 760362 336“ 3653
126023 1362363 40d3 36‘f + l lbi
cf 4off°
l Obdf + 183f + 363
9 8609 3626 436 1063
b’ogd 2o3d 26333
2élbcol2 3368 6
303 26323 562313
20113 663 2 467 + 15 620f 80’
f 56df 23f4639 10609 10671 1836 1063
°
(936 360362 3663 6023 5163 3 23d3 563 2 b’cf
07 + 66df ”ef + 563
9 14609 90l9 5626
906 563°
63362 360d’3369 26303 66023 63de 63 ’ 267 +66
'
qf86df ef + 26
3
9 4609 9019 566 962°
60d” Sol8 26093 bide Scale 263 2 56’0f 637°
26df 2qf + 563
9 4639 406 + 186i
bl3 Sede 363 2 307 + 36df + 3ef + 3609 6d93626 306 363
°
653 56303 56’de 50d3 5632 bff 4620f + 2e’
ff46df 93f 639 3609 3d9 626 236 bi 4
290 TABLES.
9693‘ 23 5 26731
c'zdg 106313 26°e
6333 246’de 2863 313 lOd‘e 116"3 ’z
265f + 2637 226’df 4ec1f + 206ef
5f’ 206319 1039 2636
4636 10316 2693°
63 3°
1069'
106.
235 3653 31 12693 231 263331 364312
632312 116313 186 3 3 103 33 1263313
363 313 113123 156232 367 + 126
’3f 963?
96’3_1f cdf + 2063f 5f’ 369 96239 6319
1039 3636 636 11316 106" 1033- 1069'
+ 106.
6” 23 5 4633231 863331 93
2312 2bds
1262023 103 33 869313 1263313 23193 96” 2 1433 12
467 + 16630f 18637 66’31f -1 203 31f 463f 6f
”
23 29 46319 1439 10636 20636
1033 10323 1033 1039'
1013.
3‘ 560931 5623 31” 5631” 562023 53
33 569313
563 313 5 3123 56232 53 32 5633f 1063ff + 106
2df
15cdf 153ef + l Of’ 153239 5 3
2
9 10n5 39 5696 10606 5316 5623 5 3 3
°
569°
573.
6731 765031 14693 331 763331 76‘d" 2 1623 31” 73 231”
7631" 2393 1369313 2663313
103123 66232 633” bff 5693f 5637 + 126
231f
‘
123 31f u bef + 5f"
43239 2329 116d9 1039
636 3636 10316 623°
23 3°
69°
106.
653 31 5633231 563 331 9623312 73612 46313
23693 23 4333 1663313 2160313 3123
1233 2 + 3bif 14630f + 12632f + l 36
’df 33df116239 43 29 106319 2039 3696
8636 316 3623°
43 3°
1169°
2073.
363331 26‘31g 6623 31” 332312 76313
563313 1563 313 133123 3623” 43 32 4617 19633f
186cff + 156231f 19331f 763f + 469
8329 46319 439 4636 10636 d6 203 3°
1139 203 .
60331 369031” 56313 563313 83123
43 3 ' b’cf bcff 126’df + 100df + 2363f 15f’I
83 29 76319 439
20dh 11b’3‘
2033 1167“
2073.
534139327:
2 9739“
E 371378
2 3 935918
Ea°fi’78
2 a‘fl‘78
2 415135
33
2 3439793
2 3 3 3 8
2 a 4fl9726’
TABLES. 291
232312 4631" 863 313
103 3 ’ 26ff 86’3f 4boff + 106’31f
4c31f‘863f + 5f
’86319 239
4636 10316 2623°
633°
106i 1013 .
623 31” 6313 262323 4093 63313 563 313 623 ’
1233 ’ 5633f 1360’
f 46'31f + 173 31f + 106qf 15f'
156 39 196319 2039 4: 5636 3636
313 123 3 433 z obj 2073.
2339 2383 213333 3383 2 233“ my26’31f + 12et1f + 463f + 5f
‘ 66239 103 99 46319
1439 6636 12636 10316 6623+ 233 106i 106.
631’ 363 313 d'e 36232 233“ 3boff 36’
31f + cq’f q f
59" 136319 239 3636 636 11316
103? 1033 10127 103 .
6° 2383 669313 1263 313 331“
6332bff -i 66
“0f 5boff 56’df + 5 3df + 1163f 59
”
649 232
9 46319 1039 636 3636 3316
201 19°
106.
2333 63313 763313 6632 633s 46‘f
196’3f 1760ff 196231f + 153 31f + 186ef — 15f’
633
9 156319 639 4636 11636 9316
4393 633°
1219 + 3073.
62393 263313 463 313 2623” 233” 6’qf2637 + 36
’df 4cdf 12bqf IQf"
156319 639 13636 9 316
2233 1267°
3013 .
333 363313 33193 369 2 333“ 637 + 26
°31f + 331f 8639
“
5f’
32
9 36319 939 6636 17636 15316
6b" 1133'
619 + 1573.
360313 233' 960ff + 26
'31f
133 31f 63f + IQf‘ 569 176
239 186319
1839 5696 12636 9316 23 3°
2169°
306.
63 313 43 32 3baff°
+ 46’331f
°
5calf76
239 8399 156319 1239 126
36 21606 3316
266" 2833 429 6073.
23 3‘ cdf + 5bqf —.
5f’ 76319 239 4636
l l dh - 1033 76; + 1073.
6” 2 23 3 ’ 26’31f + 4colf 5f
’ 432
9 1039 2636
4636 10316 2623°
633°
1069°
106.
292
Eaafl
’y'd’
2 326
3763
2 31 532735
Ea‘p
’yde
2 a‘fl’
7’d£
1
2 3131137285
2 319.32? d
2 3125272623 2
Ea °fi76e§
Ea‘flz 'ydez
2 061437a
2 a’/327238 !
2 673734 3
2 313132754 3
2 31 292
72389,
2 a313756§119
Eazfiz'
yd &c.
Z a gfi'
y &c.
1 11 15 14 13 5 .
33' 2c3if 639
” 36319 939 7606 6316
76‘s“
3 3°
1519 + 1573.
bff 5bsgf 563’
ff 56‘31f 5 3 31f 5bef + 5f’ 69
2039 46319 439 636 3606 3316 691+203? 6j 1013.
b’cf 3bo‘ff 6’
31f + 5calf + bqf 5f’5 69
196319 1639 5636 14636 12316 833'
1331'
4073.
bcff 26’ 31f 031f 563/ Sf2 56319 839 6696
16636 12316 13693: 2431+ l 3bj 406.
6’31f 2031f bef 36319 1239 9696
23333 18316 9323 433 3319 6013.
031f 363f + 5j’ 2 43
9
9 66319 9636 24316 2 1621
2801 + 3339 6013.
bef 5f’ 36319 839 5606 2316 76" 8033+ 316i 4013.
=f2 239 2316 231+ 219
'
26.
46239 46319 439 636 3636 3316 6"z'
23 1 67°
+ 1013.
6239 6319 439 6636 17606 15316 6623:
1003: 146} 506.
26319 239 636 3316 7621 1301 767°
2573.
6319 439 5636 12316 123 3? 4619 + 10013.
39 4316 901 1619 2513.
bah 3336 3316 623°
23 3°
67‘
1073.
636 333 7323 123 3: 1519 6013.
316 631+ 2039“
5073.
693: 231 67’
+ 1073.
3 1 867'
3513.
bj 1073.
Mr. Cayley has noticed a certain symmetry in the coefficients of the
preceding formula; which may be more easily exhibited by using H irsch’s
notation. Let such a sum as 2 33313
27263! be denoted and let the
coefficients be a l , a” &c. so that (32919) will denote asa z’af
’
; then the
formulae for the sums of the fourth order may be written
v 1 00 N N
— 4 4 + 2 - 4 + 1
1 - 2 + 1
2 + 1
— 4 1
[ l‘]
2 9 4
INDEX.
Aronhold, on symbolical methods , 130.On the invariants of a ternary cubic,270.
On the diiferential equations of invariants, 271.
Bal tz er, on determinants, 266.Bez out, on elimination, 66, 266, 282.Binet, on determinants, 266.Bez outiants , 282.
Boole , on linear transformations, 94, 270.H is form for the resultant of twoquadratics
,148
,160.
Borchardt, proof that the equation of the
secular inequalities has all its rootsreal , 36.
Bord ered Hessians reduced , 15.
Symmetri cal determinants, value of,29
Bri oschi , expression for difierential equation of invariants in terms of roots
,
51.
On solution of the quintic, 201.On determinants, 266.
Burnside, investigation of radius of spherecircumscribing tetrahedron, 21.
Canonical forms, 133, 274.Canoniz ants, 138.
Catalecticants , 169, 203 .
Cauchy, on determinants, 266.Cayley, (see also p .
His expression for relation connecting mutual distances of five pointson a sphere, 21.of five points in space, 22.
Application of skew determinants tothe theory of ort hogonal substituti ons, 33 , 266.
Statement of Bez out’s method of
elimination, 68.General expression for resultants as
quotients of determinants, 71.Notation for quantics, 83 .
Discovery of invariants , 94On the number of invari ants of a
binary quartic, 116, 210.
Defini tion of covariants, 119.Symbolical method of expressing invariants atid covariants, 120Identifies two forms of canonizant ofequations of odd degree, 139.
Ondiscriminants of discriminants, 149.
Cayley, on tact-invari ants, 153 .Relation connecting covariants ofcubic, 162.
Solution of a cubic, 162.Solution of a quartic. 174.On covariants of system formed byquartic and its Hessian
,178.
On covariants of quintic, 190.Tab les of Sturmian functions
,192.
Tables of symmetri c functions, 285.On Tschirnhansen transformation,278.
Clebsch , on symbolical methods, 130.On
ganonical form of ternary quartics,
1 4.
His proof that every invariant may besymbolically expressed
,142.
H is form for resultant of two cubics,274.
Investigation of resultant of quadraticand
general equation, 248.
Genera expression for di scriminant,
252.
Cockle, on the solution of the quintic,201.
Combinants , 144 , 273 .
Invariant of invariant of U RV isa combinant, 167.
Of a system of two quartics,180.
Common roots determined,76.
Commutants, 267.
Concomitants, 104.Conditions that equations should have two
common factors, 63 , 82.For systems of equalities betweenroots, 109.
Th at U XV should have a cubicfactor, 147.
That two quartics should be differentials of same quintic, 184.
That quartic should have two squarefactors, 173.
That quintic shoul d have two square,or a.cub ic factor, 189.
That sextic should have two square,or
'
a cubic factor,205 .
That roots of sextic should be in involution, 210.See also Lesson XVIII.
Contragredience, 102.Contravariante
,101.
Covariants, 99.nr
éber of
,for a binary quantic,
INDEX .
Cramer, on determ inants, 266.Cri tical functions, 40.
Cubic di scussed , 160.Cubic quaternary, its canonical form,
144 .
Derivatives of derivatives expressed symboli cally, 243 .
Differential coefii cients of determinants,Of resultantswi th respect to quantitiesentering into all the quantics
,81.
Differential equation of functions of differences of roots, 47.
Of invariants,113 .
Differentiation mutual, of covariants and
contravariants, 110.
Discriminant of binary quantics expressedas determinant, 20.
Of products of two quantics, 86.Of discriminants, 148.
Enables us to distinguish whetherequation has even or odd numbersof pairs of imaginary roots, 191.
General symbolical expression for,246, 252.
Eisenstein , expression for general solutionof quartic, 270.
Emanants, 99.
Equalities between roots of an equation,conditions for, 109.
Euler, on the theory of orthogonal substitutions, 36.
On elimination, 64, 268.Evectants, 105 .
When discriminant vani shes, 107.Of di scriminant of cubic, 161.
Fad. de Bruno , calculates invariant of
quintic, 186.On elimination, 269, 282.
Forme-type of quintic, 278.Gauss, on linear transformations, 266
,
268.
H arley, on solution of a quintic, 201.Hermi te, law of reciprocity, 125 .Form for covarian ts of system form ed
by quartic and its Hessian, 178.
Discovery of skew invari ant of quintic,189.
Canonical form for quintic, 189.Forms-type of quintic, 91, 278.Expression by invariants of conditionsof reality of roots, 193 , 279.
Expression of invariants in terms of
roots, 202.
Solution of quintics by elliptic functions, 200.
On6
Tschirnhausen transformation,27
Hessians, 15 , 100, 268.Contain all square factors of originalquartic, 136.Of Hessians, 178.
Hirsch’s tables of symmetric functions,285.
Invari ants, 92.Absolute, 96.Skew,
115.
How many independent, 95 , 272.Relation connecting weight and orderof
,113 .
Involution, 145, 160.Condition roots of sextic shall be in,210.
Jacobi , on determinants and’
linear transformations, 266, 268.
Jacobian, of system of equations,69, 101,
272.
Its di scriminant discussed, 147.
Of quart ic and its Hessian , 177.
Jerrard, transformation of a. quintic, 188,2
Joachimsthal, expression for area of a.
triangle inscribed in an ellipse,21
Theorem on form of discriminant,
K ronecker, solution of quintic by ellipticfunctions
,201.
K ummer’s resolution into sum of squares’
of discriminant of cub ic whi ch determines axes of a quadric
,43 .
Minor determinants, 9,Of reciprocal system how related tothose of original , 25.
Multiplication of determinants, 17.Number of quadrics which can be de
scribed through five points to touchfour planes, 226.
Of
2
invariants of a binary quantic,71.
Order of determinants, 6.Of symmetric functions , 45.Of invariants, 114.
Of systems of equations, 213.Orthogonal transformations, 33.Osculants, 156.
Poisson’s method of forming symmetricfunctions of common roots of systems of equations, 268.
Lagrange, on the general solution of equations,200.
On condi tions that equation shouldhave two pai rs of equal roots,269.
On linear transformations,273.
Laplace, on determinants, 206.on equation of secular inequalities,36.
Leibnitz , hi s claim to invention of determinants, 266.
Linear covari ants of cubic and quadratic,163
Of quintic, 190.Of equations of odd degrees, 191,274.
296 INDEX.
Quadratic forms, reducible to sum ofsquares , 136Number of positive and negativesquares fixed
, 136.And equation of n
th degree, generalexpression for resultant
,244 .
Quadrinvariants of b inary quantics, 112.Quarti c, theory of
,169.
Reciprocal determinants, 24.
Reciprocity , Hermite’s law of
,125.
Reduciéig sextic for quintic, forms of,01.
Resul tant of two quadratics, 55.Two cubics , 63, 165 , 283.Two quartics, 180, 284.
Tables of, 283 .Roberts, Michael, on covariants, 117.On application of Sturm’
s theorem toquantics, 192, 268 .
Or;
equation of squares of differences,93
Roberts, Samuel, on orders of systems of
equations, 280.
Rodrig ues, on orthogonal transformations,36.
Skew symmetric determinants of evendegree are perf ect squares, 31.
Skew invariant of quintic, 189.Vanishes if quintic can be linearlytransformed to the recurring form,
194.
Or to one wanting alternate terms,274.
Of sextic, 210, 253.
Of all quantics vanish when quanticwants alternate terms
,274.
Source of covariants, 117.Sphere circumscrib ing tetrah edron, 21. Vandermonde, on determinants, 266.
Relations connectingmutual distancesof points on, 21, 22. Warren, on resultant of two cubics, 169.
THE END.
W . Metcalfe, Pri nter, Green Street, Cambridge.
Sturm’s functions, Sylvester’
s expressionsfor, 37.
In case of quartic, 175.of quintic
,192.
Extensions of, 269.Sylvester (see also p.
Umbral notation for determinants, 7.
Proof that equation of secular in
equalities has all real roots , 23 .Expression for Sturm '
s functions inte1ms of roots, 37.
Dialytic method of elimination, 65.Expression of resultant as determiminants, 70.
On nomenclature, 104.Canonical forms of odd and evendegrees, 137, 141.
Of quaternary cubic, 144 .
Expressions for discriminant with regard to variableswhich do not enterexplicitly, 181.
On osculants, 151.Investigation of expression by invari ants of conditions for reality of
roots of quintic, 193.On Bez outiants, 282 .
Symbolical expression for invariants, 121,239.
Symmetri c functions, 44 .
Their use in finding invariants, 108.Tables of
,285.
Tact-invari ants, 153 .Of complex curves , 155 .
Tetrahedron, radius of circumscribingsphere, 21.
Tschirnhausen, transformation of equations, 276.
