A Method for Determining the Number of Impossible Roots in Adfected AEquations. By Mr. George Campbell

A Method for Determining the Number of Impossible Roots in Adfected AEquations. By Mr.George CampbellAuthor(s): George CampbellSource: Philosophical Transactions (1683-1775), Vol. 35 (1727 - 1728), pp. 515-531Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/103709 .

Accessed: 25/06/2014 06:08

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

The Royal Society is collaborating with JSTOR to digitize, preserve and extend access to PhilosophicalTransactions (1683-1775).

http://www.jstor.org

This content downloaded from 185.44.78.105 on Wed, 25 Jun 2014 06:08:32 AMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=rsl

http://www.jstor.org/stable/103709?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp


( 515 )

II. ,# Method for deter?stisatrtg the lVtsxber of iano poWble (toots in adfieAed 42atiotzs b Mdr Gcorge Campbell.

LE M M A I TN every adfF&ed quadratick:quatiOn vxBs 1 X-os whofe RootsR are reals a fourt1ll Pare of the Square of the CQeflicient of tlle fecond Term is greater tllan the Re&angle under the Coefficient of the firft lNerm and tile abColute Nutnber or 4 B2 > aX Js 3nA vice verfa if 4 B 2 2 a X 2<X tlle Roo.s of the Aquatioa vx:_Bx+X=s,willbereal. Butif+B38ax tne Roots will be itupoEbles This is evldetzt frQ n the

E><oots Qf tho 1Equation I7cing ' + i 4 *uX s

_

2, B-io4 B2_axS a

LE M M A ISS \N7lz;atever bP tlle Nutnber of ilnpoElble ROQtS in tiln

Equation xt_ B , n I -}_ C%' n-a _ s ,tr-* _0_ gr +dX3+CX2ibn¢+X-oXtllereate jufRastrarlyin

tl1e Aquation -Jx "-bx-x + c X n- 2-d"- 3 _ J, +sX3 + CX2 +BS T = 0* For tXne RQDt5 of the laA if4quation are the Reciprocals of thofe of the fir{t as is evident froln cotnlnon Algebra. LeF tIze Roo.s of the biquadratick X.quation ;¢ 4-B 3: ! X @5s: q) ,% +S w a be a, b, , d, mrhnreof 1CF cs d lb- inpofliblea then tlle Roots of tlle ql rxttDn

Zzz z SwA



( 5t6 ) q) X 3 _ C ,,a _ B , + I-O Will be

, -,-, -, and therefore tvo of thetn to wit , t a b a d s d impoIllble.

L E ̂ EzI M A lIIe In every Gquation A¢n Bs + Czs2

n-3 _X_ E X n-4 fg. +e X 4 Jr d XZ ur a nc a+

bs A o, all w-hofe Roots are realX if each Tertn be multiply'd by tlle Index of Z in that Term- and each Produd be divided by x, the refulting iSquation 5 xt.-1 _ ff - - - x B n¢ " 2 + n z C X n-3

si_ 3 q)Xt-4+n,-"-4.ESnS WC. +4encg + 3 d nc44r EL {X+b = o ihall have all its Roots real. Thus if all tlle Roots of tlleGquation- 4 9 f B X3 + CXX_S X +XW | O be realX tllen all the Roots of the -Equation 4 X 3-3 B wt + z C tD - o will alfo be real. rRllis Lelutna doth not 11old converlly for there are an Infinity of Cafes where all tlle Roots oftheLEquation X w" t-X _ X B t-2 +

^ C ?s-s _ ff _ 3 q) ,% n-4 + &Sa + 3 d X 2+

oncAb_o are real) at the fame Time fotne or

perhaps all tlle Roots of the Gquation tn 1;9 Sn § 1

C,%B2_D Z > + Wr.Ads3+cz2+bx+X_o are itnpoilble: Bllt wrllatever be the Nutnber of impo

fible Roors in the l!Equation X X 7} - I g _ I B X 73 2

p

n X Cs" ^-bc. + o a X + b _ o, there are at

leaR as many in tlle jEquation ,% n fl B ,% I _{_

CS7t- W. + CS2 + b x AS= o. Thus a11 the RQ0tS of the Aquatoorl 4 X 3 _ 3 B 6t 2 + ^ C st

55) _ a may lJe realX and ye, turo or perhaps a11 tEle



( 517 )

four Roots of the Aquation s BS 3 +C ,t4'-

q), +C=o tnay be impoElble) but lf tsvo vS tlne Roots of the iEqufation 4 X 3 X-3 B nc 2 + t C i) = o be impolllble) there IllUR be at lealt t.vo in

pofllble Roots in the Aquation c 4 B X 3 + CS 2 - -Z-

q>X+-g=o. All this hath beendeluonttratedby AlgebraicalWritersX particularly by Mr Reyneag in his Snv0Jen DeenontrS arld is eafily made cvident by the Met-llod of the Masima and Miaima.

C o R o L A R Y. Let all tlle Roots of the Aquation d¢"-BXnt + C,"-2,,_gi)4%4?s3+ExF-4-

Fxt-5+ eF. +fxS+ex4+ jX34 X2+ Xf

S_ o be realX and by this Lelnma a11 the Roots of t11e Equation nw"-t-n_xBac"-2 n^Cs"- 3_ n, 3rDw" 4+n-4EX"s_n_5FUN64r ei. +yfs++4ex3+3dx2+Xa,Ab o witl be real7 and therefore (by the fame Leln;ana) all the Roots of tlle Aquation ff X n _ I ACt 2 _ 3W-I X

n _ ^ B "- 3 + g-2 X n-3 C Sn 4 n-3 X

a 7X_ 459X?1b 5-|-8 4xB-iEff-6 8-SXg-6 Fx"7+ eSr. i 20fXx3+ x zext+6dwizc-a

or (dividing all by t) of n X - x n- 2X_n _ I X

" SL n- 3 - B t-3+8 --%X CXn4- SCx +

_

Tofx3+6ex2-F 3 dr+a_o will be real. After the fanac Manner all tlle Roots of tlle GEquation

. a_ X _t X 2 - 3

XX X Xn3 r"- l-n_ X X f X 3 t- 3

Bsa-++



( 5 t 8 )

Bx"4 + n-X x 3 x 4c,tn_-S_ g. +

sofx2+4exAd=o will be realj and thus we uay defcend until we arrive at tlle quadratick ASquav

tion n x S I St g _ I Bx + C_ o. The fatne

iEquations do aScend tllus a x x 2 _ ss_ I Bs +

X-X n_ 2 tg-z C_O,GX X S3-n-IX Bx2+

3

B T #S ff-3 X-tCS-s-°,GX X X S4

2v 3 4

X t g 3 ib ff 3 22_TX X Bx3 +X_ 2v x x

tWI n X C ncz-X-3 i) X + E-0 ff x x x

3

_

S-3 '--r4 ff-^ ff 3 n -4 X w-n-IX ^ X X

4 S ^ 3 4 ff-3 B -- 4 ^ 4

Bxr +ffx x Csa-n-3x 2@ 3

iO x2+ tw4 Enc_F= oX and fo an. Let Mre preSent any-of tlle Coeffictents of the A!quation

,xn_B,% n-X Cxn-2_9Xa-3 +E"-+ a

rt X = o, and let L 1f be the ad jacent Coefficicntss let M l)e tlle Exponent of tIze Coefficient M: By the li;xponent of a Coefficiens I lnean tlle Number w7hich

expreINetll



( sl9 ) expreSetlo the Place mthich it hath among tlle Coefli cientss thus if M repreSent tlae Coefficient E (and tllerefore L i) and Z\l= F) then m-- 4. It will be eafy to feeX thatX aluongA tlle foregoIng aCcending Equations, that wl:lich hatll its abSolute

NutnlDer X will be gw x x X x Cc.

3

B _

X-m | n X "- m

Xm t _ _ I X - X &;C - ---- -: B wb +

m+ I ^- tS X - iM

X-S^X Wir. Cs -s _ F!c. i n-tn xX m-x

"- L X 2 i: n-m Mx + 1\1-os all xvhofe Roots are real when all the Roots of the iE:quation rn_B,%"I + CNC-2-<Frt AS_ o are real. Let i\1- F and tllerefore M= E, L _ i) and ?n-4, tllen tllat of tlle aCcending Aquations whoSe

abfivlute Nutnl)er is BX will be X x x x _ 5 3 n-3 n-4 ^^ X-3 n 4 - X ws_n-TX X X

4 S _ X 3 4

z 3 3 X

9X+n-4ExF_ o,

p px o



P R o P O S t t l o N I Let t8 _ B xrt + C Sn-2 _ 9 S83 +

ESn-4r - &§¢e i+ f VC4 + d x s a S2 + b x + X_ 0

.

bn an RqSuatiorl of any Diluentions all sxthoSe Roo*s are realX let R1 be any Coedicient of t11ls Equatonn IJ 1\! the adlacent CoefficientsX and tn the E;xponent of M rtllen the Square of any Coefficierlt X moulttm

pld by the Enra*ion Mxn-m - -- will alw fn+IXg-m+x

urays csceed the Re&angle under tlle adjacent C./oefli cients L x lEle trhUS iN tlle ARquation wr _ Bs > .ir C, Z - - q) X + X = 0 wllere s& - -: 4, tnaking 1k1 = C:^ and tllerefore L = B, N_ vD) and m = ^, then

z X 4-^ X (72 or 4 C2 will esceed SxS

CL I X 4 X + I 9

prosriding all the Roots-of the Equation be real Becaufe (by Leln. 3.) the Roots of tlze gtladratick

X I

ffiquation X x * X 2 _ n - I B X + G a 0 are .

realX therefbre (by Letn T*) A8llixB2 lnull: be n-x

greater rllan nx xC and (dividitag borll by

8 X X I ) n I x B 2 greater tllan I X C. Therefore sn

B tXlle ARquation w2'-B Sw ' ' X + (;fXa ; q) S +

W. + S= o ofT tlle X DegreeX all whoCe Roots are realX t}le Square of B the CCoeScient of the fecond

r n . . errnX

¢ sao )



)

Term, multiply'd by the FraAion isgreaterthan x xC tlle Re&angle under tlle adjacent Ctefficient¢ But (lJy Letn. ̂.) all the Roots of rlle ARquation ,2ffX§1W_;ZZ-t _1_r#-2_ , iCt + Bx +

I _ O or (dividing by S) of z_ Xq x* l +

-X82_!a +-,yaq_-X+-- & are real, X -X X - S therefore (frotn what hatll been juIi now faid) n-I bt c

> x , muR be greater than I X X ar.d conGew

#I

quently_^ bs greater tilan sxS. Therefore in an Etquation wZ-Bs0-X+Cz-*-Yz + -¢ X D T 6 X + X-o, of the X Degree, all wllofe Roots are realX the Square of tlle Coefficiont of X

multiply'd by the Frad[ion _ is greater than the

Reflcangle under tlle Coefficient of t2 and the -abSolllte Nulnber. But by Cor. Lem . all tlle Roots of the

n-x n-* X-m

ADqluat1on 7W X X X Wc. X --- X S + X ^ 3 m T

-n^ X S "_TX X6¢.x - E3¢Jrn-X x W{ 2 m

* Cxt 'W¢.An--m+lx _xL} T n-X z



( 522 )

ff _ Es X X + hl-o are reaI) tllerefore (feeing tIlis fliquation is of t;ze m + E Degree) the Square of

, . m + I I

s_w x M lllUltiply d by tne Fracilon z x w t

will be greater than tlle ReAangb under X-m I X . . .

ff wmS X L and 1\1; tllat is xx-ml 2 X

5 ' tSS+t

M wili be greater tilan m + I X - X L x 1f

X

and therefore (dividing both by oD-m + I X )

_

raxn-m X g2 greater than L x 1\1:

m + l x X _ m + T

C O R O L A R Y Make a Series of FraAions

B # I ff ^ ff--3 T

, - , - , WrX unto - * who2d Deu x X 3 4 ff

notninatXs are Nutnbers going on in the Progreflion Is t 3 + eSc. unto the Nutnber X whicll is tlle Diw

menr1ons of the Equation xn-B Xn-I _ CXn-i_

Sc. + S_ oX and wllofie Nlltnerators are the farne Progrefiion inverted. D5ride tlle fbcond of theX Frac- tiorls by the fi-rkX the third by tlle fecon-d} the fourth by the third) and fo onX and place the FraAions which reEult frotn this Divi¢on above the middle Terms of

"_t 2X n-X

.. . . _ . . .

kn 329X r

tlle iEquationX thus xtx-Bs"-X Cxta_ * - sUi-}.+



t5t3 )

__

n-+

sn-2 S Xff-3

i) ,%sn 3 + E ,tn-4 gC, i J = 0. Then of a11 the Roots of t1we ffiqllation are reaX, 4*;tt,wt Square of any CotfficierIt -multlply'd ly tile Fradon szkica Itands above-utill be greater than the Redangle under the adjacent C;oefficients. Tllis Corolary dotll not llold conxternys for tllere are an Infinity of JEqua- tions in wttllch the Square of eacll Coeffisient mutti- ply'd by tlle Fradion above itX tnay be greater tllan tlle Re&angl-e under tlle adjacent CocfficientsX and n:>tvititllEandinD fotne or perhaps all of tlle Roots tnay I)e ilnpofllble. Tlleref&re xvhen tlle Square of a CoefE- cient mt}ltip-lysd by the Fra&ion altoveX is greater tllan the Reangle undr the ad jacent CoefficientsX frotrl this Circutnfl;ance notlling can be determined as to tlle PoIllbility or ItnpoElbility of tlle Roots of tlle ffiqua tion: But svhen the Square of a Coef icient multiply9d by the FracGtion aboxre itX is lefs than tl;ze ReAangle un sder tlle adjacent Coefficients it is a cerEain Indication of zwo itnpolfilDle Roots Froln svllat hath lzeen faids is itnLnediately deduced t1ze Demon0ration of that Rule which tlle luoft illuRrious Alewto} gtves for de- tertnining the Nutuber of itnpoflible R()ots itl any gi-

_ *

vel] iequatlon.

S C H O L I U M.

Let the Roots of the Equation s"+1-l Bs"-lt+ CXn-2_ i) fg-3 + E ,tn-4 _ ,t S -f_ (§f* +

X -- o (witll their Signs) be repreSented by the Let-^ ters a, b, , d, e, f; g, (gr tben (as is com;nonly knoutn) B will be the Sstn of all tlle Roots or-a + b + a + d + e + f + Sc. C the Sulm of the Produds

Aaaa X of



( 52-4 2 of all tlle Pairs of Roots or-4 4 + a ¢ + 4 d - f + ag + eir. S) the SUtM of the Predu&s of a11 tlle t;8ern.Xrstc1s of Roots or _ 4 b c + a b dw + a be + abf abg + Sc. E_abcd+abce+aArf7ur aheg+ Wr. F_4bade+bvXfRabGdg-+- bcdef + Sc. and fo on. Ler (as in this PropoZ- tion) M reprelent any of tnere Coefficients) L5 1\E the ad jacent Coefficientq, and a; the E;xp<nent of iTS; let Z rnprefent the Sutn of tlle Squares of all th.> prffible ]:)iSrences between tlle Tertn-s Qf the CoeEcivnt Ms et r be the Sum of all th()Sefl of tIle forel:;iid Squares vlzofe Tertns differ by onn LJetter) 3 tlle Suln of all tllofe Squares wllofe Terfxls cliffer by tWQ lLetters) s the Sutn- of tIzoX Squares thtllote Tertns differ by tllree LettersX a the Su-m of thofev Squares vlloSe Tertns differ bv four Letters and fo on. l'hus if 15I _F- abcd e +abadfRabadg+ WG

G s - -. nr .

tllen Z= abcde-a6cdf l 2 + a6rde-a6cdgl 2 47

46sde-abcfgl2+Acief-abfGgE2+ ga, 06 = a6aSeabadfl2+abade-ab c dgl2+ vvbcde-abcdDl2+badef-6rdegl2+ 23a ,$.= a.Acde,atbafgl2+abcde-abrfEl2+

Zdfhl2+ &S- 7==;;2+

abcdf a6eghl2 + Sc. a-abade-afgAkl2 +

a cdfg - a 6 e-h kl2 + Sc. Tllis being laid down I fay tlllat tlle Square of any Cvefficient M rnultiplyyd

lty. tlleZ Fradions exceeds rlle Refl>-

+ I X n-m + x

angte under thc adjacent CoeXhcient L x 1f- by - n IXZ



( 5:5 ) X + I XZ I t x x

S ...,. ffi - e - --

+ IXan+t < 3 4 5

k-&Sc. lUlle Series-_ ^ _ I o

z 3 4 Sc tnutt confiS cf S ]9tutnlDer of TerxnQ

tet tXle «q7latioN be xs _ B x + C t)- D w + E nc --- X = o, wlloSe ]?oots 1nt be a, b, c; 4 e in svlwich C:aSe X _ 5. Let M_ B_ a + h + a + d + e) rllen L _ x X-G w =

, . Z _ a _ b + {s (}2 + 4 _ dl2 + a-- e12 S

a - r12 + Wr = , therefore I x y-T

I + I X S I + T-

B or -B2 excceds x x C by 5 + t x Z s I + I X i =- I + l

- X -Z - X - ( 5ec3ure 2 = a ) Zt=-a _ b|2 + _ )<4 [|} + 4 isl2 _+

&3c xv1ate11 is alura5ts a poSitiste Nutn5er when. tTle RofDts ar) bX cX dX e ate reAl potitive or ngative

Nulubers. L er M-C a h n a a + a d > a e + h c + f!a. then L = B X iD g = zX

Z _ 6z b-v s12 + a b-a ki2 + a b1. c -12 + b; -I j e12 + @¢* X-gb- a C:2 + 4 6 _ v dl2 +

s ss_v ot s t9 8 b a ef + W¢. 3-a b-c dlz + a a a el2 q_ .sb-d e;2+&Sr t;;Xerefo3re 5 1xC2

+XX,S zTI

ox



( 524 ) = ^

ot Gt furp.tX4SlCll B x s bY 5 Z ;t X 2

2 ' t+t M5+t

' M- 1 g = (beca^,re Z >, + n) _ t ^ n - r

N= 6Xab-rdl2ur 6 ab-cel2+ 6 X

a b-@e12 + Scv wrhicll is alwass a po(itixte Nutn- ber wwllen the Roots a, A, rX 4 e a;e real Wtumbers} pc)fotlve or negative. Ler 1ff= GD = a b c + a b d + a6e f acdEase+Sc. then L=C:; lXl _E, sw= 3 Z=-aba-abdl2 + abca6eli vtc_vdel2 +Sc. oc_ abs-abdf2 +elz + a6-vcdt2 + Wc. 6=abcadelt +a6c-cdei2 +

abc-bdela +Sc.>-o,therefore + i 3 x

92 or-S 2 exceeds Cx E by i + l x ^ 3 + I Xi-3 + I

I t I

Z -o.- . n = (becauSe -Z= X + N) = 6 x

ffi =,^ 6 xabr-add2 - 6 xabc-sde|i + 6

ab c b d el2 + br. whlcll is a poftive Number wllen tlzn Roo;s are real Numlrers. Let M = Ew =

vbsd abce Jr abd e bcd e + ScJ tLlen

L_S:D 21-A, m_4 Z=abcd_abve1t + v6cd _brde!2 + abad4rdet, + e§ -- : ocs

{5



t 527 )

6 = 8 e > _ & thexefore 4Xi-4+ 1< xE-vso 4 + I x5-4 + I

E2 exceeds s x X by - i + xZ - S 4+IxS4+t

-GG = 3 Z--X =-Z X dbc=46[et

2t S ^ IO I0 0

-x abcd-badeti + eSr. whicll is a pefitive

Nutnber svllen tlle Roots are real Nu£nber¢.

P R O P O S I t I O N IIv Let x-"Bxn-I Cza-2-Dx"3+Esa-<_

t. i a = O be an Aquation of any Degree) uthoi Roots witll tlleir Signs let be expretEed by tlle Letv ters a bv G d) eX 6 &fr. let M repreSent any Coeffi- cient of tllis A£quation, t lEl the oClgCients ad ja- cerlt to 1v; 1C,0 theCoeScientsadjacentto Lj19TX 1; tP tllolE adjacent to I; O; H, 2 tllOfe adjacent to 1; MPX and fo on. r Let m repreSent the Exponent of AI and lqt Z (as in the preceeding Propofition) epreSent the Suvm of the Squares of all tlle pO«l-

ble Diferences between the Terrns of the Cocfficit ent M. Therl the ProduE of the Square of arsy

Coefficient M multiply'd by the haAion - X

_ -<t '2 tsrtv- dOfu

X X -X-X br. X -

3

* alYo.v45Fs



( 5tg ) alrays exce>d L X AAx O + Ix Hx 2-u g{6

7

55-t - xv*.ich X X X X SC. X + 4

X 3 m is alwacts a poftive Nulnbera wllen tt^e Roots a bs , d.e.Sc. are real Nutlzbers pofiltive or rjegativer Ler tlle iEquation- be of the fexrentll Degree or x7Bncs q- css_SO..r4+Ex3_F.%Z__G,%_ X = o, wlaoSe Roots ler be a, b , dX e, fi g, in

w}lich CaCe _ 7. Let M=E_abcd+abce a 6 a f + 4 a r g + b r d-e + f!r tXllen m _

L _ _ 'D, 01 _ _ F, K = C, O -= G, I--BX 7?- X, Z=abad_abael2 + oubad-a6a f 1; +

a b c d-a b c sla + Xi. lherefore -x

. _

t xE;2or 7 E2exceedsOx 6 S 4 3S

7X - X X "- z 3 4

* tZ Z

F-_CxG+BxSbY 6 S 4. 7° 7x- x x --

-xa3car-abceli +-xabcd-abafl2 +

FrnX. this Propofitio¢) is dedllced the follomJito Rule for deterluining the NJutwber of itnpofi1ble Roots in any giNen Aquatior3. Froln each sf tlle IJnci< of tt<e luidd;le T-erins of that Power sof a Bino-Xr?*l)

1 xvofe



t Ft9 ) wrllofe Index is the Dimen&oras of tlle propoCed ALqua; tlon, fabtradc Unity, th-en- dilride each Remainder by twice tlle Correfpondent i)nvia, and fet the FraEi ons whicll relult frotn this Divifion) abolre the Inid dle Tertlls of tlle gisren ASquation. And under arly of tlle middle Terms if its Square>tnultiplyed by tile Fra&ion Itandillg above it, be greater tllan the Re&* .jngle under the immediately adjacent -Terms, Sins he Redangle under the next adjacent Terms, P-s:Ww t:tue Reflcangle tlnder the Tertus tllell next adjacent > - Sc. place the Sign +, but if it be lefsX place t;:e Sign -* And under the firR and laR Terla place +* And there will lze at lea;It as tnany«ilnw

pofilble Rootss as tllere are Changes in the Series of the under-written Signs from + to-X or froter to +. Let it be required to detertnine the Num ber of ilnpoSlble Roots in the JEquation X 7 5X6 + I i SS _ X 3 S4 + I 8 3 + i 0 2

z 8 , + X 4 = o. Tlle Onaive of tIze middle Tertns of tlle 7th Pourer Qf a Sinolnial are 7, tI, 3S, 3S, o I, 7, frotn which fubtrading Unity) and dividing each of tlle Remainders by twice t]le correEpondent

sViv, the Quotients will be - - X , 3+, I4 4^ 7° 34 - to 6 3 to I7 I7 , 1- - or - | , 7o 4^ I4 7 - T 3s 3 S

X 3 which FraAions place al)ove tlle tnidd 7 3 rl n;

Terms of tl1e JEquationn has X* ". S r 6 _0_ I 5Xs S o + _ 1

B b b b

Y E

oof 2 ''t r



( 5v3° ) 1x At LQ s 3 $. 3 s 2 I 7

^3X4 + I8X-3 +tox:-t8X+t4_a. Tken _ + _ + +

becauSe tlle Square of-5-Xs tnultiplysd into the

Fra&ion over its Head 3 > to Wit 7S t13 S IeX

7

th3 t7 X I S X § or T ff Xt: I place the <ign- un der tlle Term- G *s BecauSe e Square Of X S ws

multiplysd by the Fradion over lts Head - * ̂ t

7°S - to wit - -*- - Xto is greatr tllan-Sx6 X-w-3 x+_

7

w7 X-I 8X3-9-7 wto I place the Sign + under

Ellerrerln I i,Vs, Secing - 993 xs (tlleSquare oftne

Tern t3Xa multiplysd nby the Fradrion QVef

its Head 7) ls 1efi than n S 3¢5 X I 8 t3- 3 5

sS6XIOX2 +X7X-^8w-tg%t8 T place the Sign - under the ertn z 3 Z+* 4Cuk

- 7 5Sts I g 3 | X-or S6 exceeds-^3X4XIOX '-

--- - . u w:p

tyws x-w8X-5Xs X^47Ow6 I*+ p1ace tXlbn Sign + unler the Tertn I 8 X 3 * Since I o X 2 1t X

lo or T:)C0Xr4 iS lefF3 than +I8X3X-L82- .zI %}

z 3 J4 X X 4--4 SXa I place e Sibn-un-der



( 53t )

the Term IOU'. BecauSe tSxlz x 3 Gr 3 36X2

is greater than X ow2 xx 4 = X 40X2 under X 8 X I place} + vllen under tl1pv firlE and 1a1E Terlus f place + ; and the 1is Changes of under-utritten Signs f}ews that there are fis itnpoflible RO03e

If rlle itupoSllble Roors were to be fiound by the Xewtondn Rule) tlle Operation would* £}and thus s

7 9 + 3 . 9

7-S X + I 5 Sv t 3 UX + T 8 z; + x O w

+ _ + + f ;+, 3

tSt + X 4= os by whtcll Rule tllere ar@ iund . .

Only two impoIElbIe Koots, whereas there are fix to warit x + /-3, t /-3, x + /=;

ff f''

ff- I + ff-tX t + v-I) tllC pes venth Root being-T

BbbS t lI1* X



Documents

A Method for Determining the Number of Impossible Roots in Adfected AEquations. By Mr. George Campbell