On Non-Euclidean Properties of Conics

On Non-Euclidean Properties of ConicsAuthor(s): William E. StorySource: American Journal of Mathematics, Vol. 5, No. 1 (1882), pp. 358-381Published by: The Johns Hopkins University PressStable URL: http://www.jstor.org/stable/2369551 .

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On Non-Eucltidean Properties of Conies.

BY WILLIAM E. STORY.

In this paper I apply Professor Cayley's projective measurement,* general- ized by Professor Kleint and still farther extended by me,t to a conic; the complete theory of this application is, of course, the complete theory of the projective properties of a conic in its relation to an arbitrary fixed conic, here called the absolute.

Corresponding to the ordinaryr classification of conics as ellipses, hyperbolas, parabolas and circles, we have here a classification of real conics according to their relation to the absolute as

Ellipses cutting the absolute in four imyaginary points, Hyperbolas cutting the absolute in four real points, Semi-Hyperbolas cutting the absolute in two real and two imaginary points, Elliptic Parabolas meeting the absolute in two coincident points and cutting

it in two other imaginary points, Hyperbolic Parabolas mneeting the absolute in two coincident points and

cutting it in two other real points, Semi- CGircatlar Parabolas meeting the absolute in three coincident points and

cutting it in one other real point, Circular Parabolas meeting the absolute in four coincident points, Circles having double contact with the absolute. Either the outside of the absolute is to be regarded-as citra-infinite and the

inside ultra-infinite or vice versa2. Of ellipses there are then two varieties, one citra-infinite and one ultra-infinite, each having, a single closed branch; of hyperbolas two varieties each having two citra-infinite and two ultra-infinite branches alternating, one being cut by every entirely finite real straight line in

* Sixth Memoir upon Quantics. Phil. Trans., 1859. t Ueber die sogenannte Nicht-Euklidische Geometrie. Math. Ann., Vol. IV. I This volume, pp. 180-211.



STORY: On Non-Euclidlean Properties of Conics. 359

real points, the other cut in real points only by some entirely finite real straight lines; of semi-hyperbolas one variety having one citra-infinite and one ultra- infinite branch; of elliptic parabolas, two varieties, one citra-infinite and one ultra-infinite, each having on e closed branch; of hyperbolic parabolas two varieties, one having the points adjacent to the contact with the absolute ultra- infinite and one having the adjacent points citra-infinite, each having one citra- infinite and one ultra-infinite branch (the relations of these two varieties to the citra-infinite portion of the plane are quite different, one goes to infinity in two and the other in three different directions, some entirely finite real straight lines will cut the one in two imaginary points and others will cut it in two real points, whereas, every entirely finite real straight line cuts the second in two real points); of semi-circular parabolas one variety having one citra- infinite and one ultra-infinite branch; of circular parabolas two varieties, one citra-infinite and one ultra-infinite, each having a single closed branch; of circles four varieties, one citra-infinite with real contacts, one ultra-infinite with real contacts, one citra-infinite with inmaginary contacts, and one ultra-infinite xvith imaginary contacts, each having a single closed branch. -Circular parabolas may be regarded as special cases of hyperbolic parabolas or of circles.

This classification is useful only when the absolute is real. If the absolute is imaginary, every conic is ail ellipse or circle with respect to it, and there is no real ultra-infinite portion of the plane.

Every conic has four intersections with the absolute, say the absolu'te points of the conic, and four tangents in common wl'h the absolute, say the absolute

tangents of the conic; the four absolute points lie by twos on six straight lines, say the focal lines of the conic; the four absolute tangents pass by twos through six points, say the foci of the conic; the tangents to the -conic at its absolute points are its four asymj)totes or asgmiptotic tangents; the contacts of the absolute tangents with the conic are its four asymptotic points; the six intersections of the asymptotic tangents, also poles of the focal lilies with respect to the conic, are the director points or simply di'rettors of the conic; the six junctions of the asymptotic points, also polars of the foci with respect to the conic, are the directrices of the conic; there is one self-conjugate triangle common to the conic arid the absolute, whose sides are the three axes, and whose vertices are the three centres of the conic. The six focal lines, six foci, six directorsy and six directrices may be conveniently grouped in pairs, namely each focal line is the junction of two absoltite points, and the other focal line



360 STORY: On Non-Euclidean Pr operties of Conics.

of the same pair is the junction of the other two absolute points; each focus is the intersection of two absolute tangents, and the other focus of the saine pair is the intersection of the other two absolute tangents; each director is the intersection of two asymptotes, and the other director of the same pair is the intersection of the other two asymptotes; each directrix is the junction of two asymptotic points, and the other directrix of the same pair is the juniction of the other two asymptotic points. Moreover, to each focal line corresponds a definite director which is its pole with respect to the conic, and to each focus corresponds a definite directrix which is its polar with respect to the conic. Finally, the poles and polars of these characteristic lines and points with respect to the absolute may be considered as themselves characteristic of the conic, but their introduction here seems unnecessary.

It is evident that the focal lines pass by twos through the centres, the foci lie by twos on the axes, the asymptotes intersect by twos oii the axes, the asymptotic points lie by twos on lines through the centres, the directors lie by twos on the axes, anid the directrices pass by twos through the centre. Each axis cuts the conic in tw'o points, its vertices, and through each centre pass two tangents, its vertical tangents, whose points of contact are the vertices on the

opposite axis. Just as we have employed the terms citra-infinite and ultra-infinite to

denote on this side and on the other side of the absolute, so it will be convenient to use intra-absolute anid extra-absolute to denote inside and outside the absolute. The former distinction is conventional, the latter actual. We may call a real straight line semi-in)finite or finite according as it does or does not cut the absolute in real points. Then, in the Euclidean geometry, every real straight line is semi-infinite; in the non-Euclidean geometry with an imaginary absolute every real straight line is finite ; and in the non-Euclidean geometry withi a real absolute (true conic) some straight lines are finite and some semi-infinite, the limit between the two is a tangent to the absolute, which might properly be called an infinite straight line, since it makes an infinite angle with any other straight line ; then, in the Euclidean geometry, every straight line is infinite.

It may be assumned for coinvenience that, if either the conic under consideration or the absolute is imaginary, the coefficients of its equation referred to any real system of coordinates are nevertheless real. Then the three axes and the three centres of any conic other than a semi-hyperbola are real. It will be



STORY: On Nons-Euclidean Properties of Conies. 361

readily seen in which of the results obtained in this paper this assumption may be dropped.

We may call a real axis transverse or conJugate according as the vertices on it are real or imaginary, and a real centre exterior or interior according as the vertical tangents through it are real or imaginary. Then an interior centre is always opposite a conjugate axis, and an exterior centre is opposite a transverse axis. It is evident that any irnaginary conic has three real interior centres and three real conjugate axes, and a real conic other than a semi-hyperbola has one real interior and two real exterior centres and one real conjugate and two real transverse axes. Also, if the absolute is imaginary, all three centres of any conic are intra-absolute and all three axes finite; but if the absolute is real and a true conic, one centre of any conic other than a semi-hyperbola is intra- absolute, two centres extra-absolute, one axis finite and two axes semi-infinite. In the Euclidean geometry, ain ellipse has one finite interior centre, two infinite exteriors centres, two transverse infinite axes, and one conjugate axis situated altogether at infinity; while an hyperbola has one finite exterior centre, one infinite exterior centre, onie infinite interior centre, one infinite conjugate axis, one infinite transverse axis, and one transverse axis situated altogether at infinity. In the non-Euclidean geometry with a real absolute, every conic other than a semi-hyperbola has either one intra-absolute interior centre, two extra-absolute exterior centres, one finite conjugate axis, and two semi-infinite transverse axes; or one extra-absolute interior centre, one extra-absolute exterior centre, one intra-absolute exterior centre, onle semi-infinite conjugate axis, onle semi-infinite transverse axis, and one finite transverse axis. This mlight be made the basis of a subdivision of the classification given above.

Most of the results contained in this paper miay be obtained readily by purely geometrical mnetlhods based on the definitions of distances involving anharmonic ratios, but I have preferred to treat the subject analytically, in order to illustrate better the nature of the non-Euclidean geometry as a metrical geometry.

I emnploy the same notation, with some slight modifications, as in the paper above cited, namely

Ql is the absolute, QO-=0 is the equation of the absolute,

1 = 0 is the conidition that the point 1 lies on Q, VOL. V.



362 STORY: On Non-Etclidean Properties of CJonics.

flo= 0 is the equatioll of the polar of the point 1 with respect to ?2,

?212= 0 is the condition that the point 2 lies on the polar of the point l

with respect to ?2,

12 is the projective (non-Euclidean) distance between the points 1 and 2, or what is the same thing, between the polars of 1 and 2 with respect to ?2,

12 is the projective distance between the point 1 and the polar of the point 2 with respect to ?2; further, I take for convenience, the constants involved in the projective measures so that

2ik= 2ik'= 2ik"'= . . . 1, which will cause no amnbiguity, as we may always pass back to the general case by dividing each distance by the constant 2ik belonging to that species of measurement. Then

(1) 12=cos'l S212

(2) 12 = sin-( j ' )

Let also S be the conic under consideration, to which Soo, S,1, Slo, S12, bear the same relation as noo, ?11, Q1o, ?12, to Q.

If the points 1 and 2 are the poles, qua2 ?, of the two focal lines of S of the same pair, we may write (3) Soo= /?2 ?oo2- 2?210o?20,

where the signification of the parameter X is determined by the condition that for every point 0 of S ___ A2

&2T =20 = 2 V21 222 !20 _0

Q12 Q200 fg VS20 Y21A V2S2 !240

2 cos 10. cos 20 cos 12

so that l l

(4) cos 10 cos 20 = sin 10.sin 20 -1 X cos 12 = const., i. e. the product of the sines of the distances of any point of a conic from its focal lines of either pair is constant. It is to be noticed that, with the projective measurement, every theorem, descriptive or metfical, has a perfect reciprocal, hence the product of the sines of the distances of any tangent to a conic from its foci of either pair is constant.

If 1 is a focus of S and 2 the pole, qua ?2, of the corresponding directrix, we may write (5) 00:?- ?22 (?ll2 E - ?2IO) - yl201



STORY: On Non-Euclidean Properties of Conires. 363

hence for every point 0 of S ,11 Qoo- JQ _Q20

A1 IOO I22 IOO

i. e. sin210 ,sin220,

whence

(6) sin 10

sin 20 i. e. the ratio of the sines of the distances of anty point of a conic from either focuts and the corresponding directrix is constant, and reciprocally the ratio of the sines of the distances of any tangent of a conic from either focal line an(d the corresponding director is constant. The values of these -constants, which correspond to the eccentricity in the ordinary theory, depend on thie pairs to which the focus and focal line belong.

If 1 and 2 are the foci of S of either pair, the directrix corresponding to 1 will pass through the intersection of f1lO = 0 and i20 = 0, so that its equation will be of the form nloN/`Q22 f20/Q11= 0,

and we may write S- X 22 (Qlll ?0O - nQ10) (V nlONV222 - n2O VQ1)2,

but the tangents to Ql from 2, whose equation is

f22 lOO - n20 0

are also the tangents to S, hence the result of substituting the value of 2oo from this equation in the left-hand member of the preceding must be a perfect square, i. e. x-fll 2o Q22 O - y(rV (lo -20V/ 210)2

=(2- ~t)ll n20 + 2yiV V n!222io 0n2- (2 + Y.v2) n22 ni2

must be a perfect square, and hence 52

r2+ ( -)( + yi,2) 0,

i. e.x 2,

and SOO becomes

(7) AS'O- (1 -2) QlllQ22?l00- (f?22fn2O + lllQ22O) + 22 Q11 n22 10n2O, which mnay also be written

SOO-Qll 222 OO (1 -72) (cos2 IP + cos2 0) + 2v COS 10 COS 201 - Qnl 22 lo[cos (1o + 20) - i'] [cos (To- -) --]

i. e. for any point of 8, (8) 10+20=i4cos'lv or 10-20=i= cos-',

hence either the sum or difference of the di&tances of any point of a conic from the two foci of either pair is constant, and reciprocally either the sum or difference



364 STORY: On Non-Eactidean Properties of Conies.

Of the distances (angles) of any tangent of a conic fronm the two focal lines of either pair is constant.

If 1, 2, 3 are the centres of S as above defined,

and the equations of the axes are

f'10=?0 f'20--7 f' 30= 0

and we may write (provided S has not contact with Q)

(9) SOO xn20o + -I--2 + r Q30;

if then 4, 5 are the intersections of S with any chord through 1, we may write X5X1+ pX4, Y5=Y1+ y47 Z5=Zl+ PZ4

f55 = lll + 2pQ14 + P2 n44,

f15 =111 + P1147 25 p=24 f35 =P34

and, since 4, 5 are points of S, 214 + 6tf'224 + VQ34 = 0

(lll + p9?14)2 + Jtp2 f224 + VP2 234 =0

and, by subtraction of the first from the second, and division by XQll,

Q11 + ,?D14 &211?21 ?ill + 2pflM =O, i. e. fi - 211

then cos 15 +Qll+ 0 14 4) 2, VQ S11 (Q211 + 2po?24 + t[I ?24) 2,o, rv'Q1 Q244

S14 - ___ = T cos 14,

VYfl21 S244 i. e., neglecting multiples of the length of the whole line and noticing that in

general 4 and 5 are different points, (10) 15 -14,

hence either centre of a conic bisects any chord througlh it, and reciprocally either

axis of a conic bisects the di,itance (azgle) between the tangents from any point of it.

This justifies the naine centre. In general, if 6 is the bisector of the line joining 4 and 5, we may write

X6 =X4 + pX5, 6 =Y4 + PY5 6 =Z4 + PZ5 ,

f66 = f44 + 2p?i45 + p2f155

f46 = D144 + Pf'45 7 f56 =f45 + PfA55 ,

and the condition for p is cos 56 = :1: cos 46,

_ _6 _246

5. e. V!4566 V/ S4!266



STORY: 02n Non-Ectlidean Properties of Conies. 365

or Q44Q56 Q55Q46 f144( 45 + p5)2 - 55 (44 + pf45)2

=Q44 45)(P2f55 Q44) =0,

e. e.p

and there are two bisectors

(X4VNI255 + X5V1244, y/4VQ55 + YN5V244, Z4VS255 + Z5VQ44)

() (X4VQ55- X5VQ44, 4V7 VS5 5 - V 44, Z4VQ 55 Z5VQ44)X

namely, these bisect respectively the two segments of the line betweein 4 and 5, and are distant from each other by half the length of the whole line, i. e. each lies on the polar of the other, qua Ql; in other words, the bisectors of 45 are the foci of the involution of which 4, 5 and the intersections of the line 45 with fl are pairs of conjugates. Hence if a point bisects every chord of S through it, its polar with respect to S is at the same time its polar with respect to Ql, i. e. the vertices of the self-conjugate triangle common to S and Ql are the only centres of S.

There exists an identical relation between the distances of any point 0 fromn three arbitrary points not in one straight line, say 1, 2, 3, or from three arbitrary straight lines not passing through one point, say the polars of 1, 2, 3. Namely, if 1, 2, 3 are not collinear, we may write

Xo XX1 + YX2 + VX3, Yo =l/i + 6kZ/2 + 'V3X Zo= ?%Z1 + YZ2 + VZ3,

then 0- X2 flll + Y2 f22 + V2 133 + 2yf'{23 + 2V%ti31 + 2X2fI12,

lO - lll + 4112 + Pl13,

f20 %21 + Yn22 + VPf237

3O Z-%31 + Y?32 + Vfl33,

so that A, ~t, v can be expressed as homogeneous linear functions of lo, f120, f130

and these values give by substitution loo as a homogeneous quadratic function of n1o) Q2O0 n30, i. e. there exists a (non-homogeneous) quadratic relation

between cos 10, cos 20, cos 30, i. e. also between sin 10, sin 20, sin 30. These relations are easily seen to be

fl ll n12 n13 nlO

(12) n21 n22 n23 n20 0 f131 f132 f33 f30



366 STORY: On Non-Euclidean Properties of Conies.

1 cosl2 cosl3 coslO = 1 sin 12 sin 13 sin 10 =0.

(13) cos 21 1 cos 23 cos 20 sin 21 1 sin 23 sin 20

cos 31 cos 32 1 cos 30 sin 31 sin 32 1 sin 30 ___ I ~ ~ ~ ~~~~~~~ 1 I

cos 10 cos 20 cos 30 1 sin 10 sin 20 sin 30 1 If 1, 2, 3 are vertices of any self-conjugate triangle, qua Q, f23 - 31

f12 = 0, and the relations are

(14) '11 f22 f33 fOO f22 f33 1O 4 '33 lll 220 + 11 22 f230

r c _5_ I I I | COS2 10 + cos2 20 + COS2 30 1 sin2 20 + sin2 30-1 (15) - or

L sin2 10 + sin2 20 + sin2 30-cos210 + cos2 20 + cos2 30 2. By virtue of these relations equation (9) gives, for any point on S,

xfll cos2 10 + yL22 COS2 20 + VPn33 Cos 30

r (%l1- 133) COS2 P1 + (yf'22 - VQf33) cs2 20 + VQ33

6 I = (6Q2XIi - ?i33) sin2 110 + (Yn22 - vf'33)sin2 20 + VK33 = 0, or

L(2lll - vf33) sin216 + (afi22 - Xf33) sin220 - (ll + YQ22 -V33)- 0 i. e. there exists a linear relation between the squares of the sines or cosines of the distances of any puint of a conic from any two centres or axes, and reciprocally a similar relation exists between the squares of the sines or cosines of the distances

of any tangent of a conic from any two axes or centres. Let 1, 2 be two directors of S of one pair, then, just as S0, was written in

the form (3), so ?0J can iiow be written in the form

(17) QOO- /%S12 SOO 2S1 S20, where So = 0 and S20 = 0 are the focal lines of S corresponding to 1 and 2

respectively. Take any point 3 on 8, the tangent at this point meets SO and

S20 in 4 and 5 respectively, say. Then

S33-0, S34=0, S35= 0, S14= 0 S25=0, an1d by (17)

Q34 - 13 824, Q'44 as12 S44, Q35 S23 S15, Q1255 = X812 S55

Q133 2S13 S23; and by (12) applied to 1, 2, 3, 4 and 8,

0 Sl S12 S13 0 _ 811 S12 813 844+S123S24,

S21 S22 S23 S24 S21 S22 S23

S31 832 o o S31 S32 0 0 S42 0 S44



STORY: On Non-Eutclidean Properties of Conics. 367

i. e. 8l 812 813 S44_ S238S%24,

821 822 823

831 832 0

and simnilarly 811 812 813

821 822 S23 255 - 23 215;

831 832 0

from these equations follow

cos2 4 - ____ __ Sl

812 813 Cos 234 Q34 =_ S13S224 81 22 823 3

3244 2)812 S23 S44 22812 813 S23 831 323 0

cos2K~ ____ S,S2 1 S 1 SI 8 13

COS2-5 = Q35 =_ S23S125 - 1S21 S22 823

233 55- 2281 2S55 2)S12 S813 S23 1 8 o

hence cos2 35= cos2 34, and evidently

(18) 35 34 i. e. any point of a conic is equally distant from the intersections of the focal lines of either )pair withi thte tangent at the point, and reciprocally any tangent to a conic

makes equal angles with the junctions of the foci of either pair to its point of contact.

Still using equation (17), let any transversal cut S in 3 and 4, SEo in 5,

820 in 6, then

833 = , 844 =07 15=0 8 26 =0

X5 pX3+ GX4, 7/5 pY3 + aY4, Z5 pZ3 + rZ4,

S15 =pS3 + crS14 =? 7 *.. p S14 S13

X5 =SI X3 - S13 X4, Y5 =S4 Y3 S13 Y4 Z5 S14 Z3 S13 Z4,

similarly X6 = S24X3 - S23X4, Y6 = S24Y3 - S234 Z6 = S24z3 - 23Z4

and hence S25 S14 S23 S13 S24, S35 = - S13 S34, S55 - 21 3814 S34,

S16 = S13 S24 S14 S23, S46 S24 S34, S66 : - 2S23 S24 S34,

?233 = - 2S13 S23 7 ?55= 2%S12 S55 = - 2aS12 S13 S14 S34 ,

?235 = XS12 S35 - S13 S25 - S13 (XS12 S34 + A14 S23 - S13 S24)

f244 = 2S14 S24 ?266= XS12 S66 = - 2XS12 S23 S24 S34,

n46 S12 S46 - S16 S24 S24 (XS12 S34 - S13 S24 + S14S23),

Cos 235 _Q35 O ('S122S84+S14S23-S13S24) cos235 23 255 4)S12 S14 S23 34



368 STORY: On Nor-Euclideant Propverties of Conics.

Cos246 -46 (2S12S34- S13S24+S14S28)2 cos2 35, Q44 66 4S12 S14 S23 S34

46- i 35,

where the sign can be determined by the case of a tangent, for which evidently the positive sign cannot be taken, hence in general

(19) 46 - 35,

i. e. on any transversal to a conic, the intercepts between the curve and the two focal lines of either pair are of equal lengthis and opposite signs, and reciprocally the two tangents from any point to a conic make equal (opposite) angles with the junctions of the point to the (different) foci of either pair. A more accurate statement of the theorem is this: any straight line intersects a conic in two points and its focal lines of either pair in two points, and either point of the conic is just as far in one direction from the point of either focal line a.s the other point of the conic is in the opposite direction from the point of the oth.er focal linze; and reciprocally through any point pass twvo tangents to a conic and two junctions of the point to its foci of either pair, and either tangent to the conic makes the same angle in one direction withi the junction to either focus as the other tangent to the conic makes in the opposite direction with the junction to the other focus.

Again use equation (17) and let 3, 4 be any fixed points of S, and 5 a variable point of S, and let 6, 7 be the points in which the lines 35 and 45 respectively nmeet So = 0; then, as in the proof of the preceding theorem, siince

S16 = 0 and % 7=- 0,

'6 Sl3 X5 S15 X3, Y6 S13Y5 S- 15Y3 6 S135 -S1537

X7 14 X5 S15 X4, f7 = S145 - S15 Y4 7 7=S14 Z5-S15 Z4 X

S33= 0, S44= 0, S55= 0,

S66 - 2S13 S15 S35, S77 - 2S14 S15 S45,

S67 = S15 (AS13 S45 + S14 S35 -S15 S34)

'66 = S12 S66I f'77 = aS12 S77, 67= XS12 S67X

COS267 = 27 -6 - (S18 S45 + S14 S35 - S15 34)2

!27 6 277 S66 S77 4S1s S14 535SO

but, by (12) applied to 1, 3, 4, 5, and S,

0 = Sll S13 S14 815 = (SlS 45 + S14 S35-S,5 S34)2-2S35 S45(2S13 S4-811 S34), S31 0 S34 S35

S41 S43 0 S45

S51 S53 S54 0



STORY: On Non-Euclidean Properties of Conics. 369

hence cos267 = 2S18S14 -SllSU 281s514

(20) sin2 67 S21534 2SjsS14

which is independent of 5, i. e. the segments of any focal line cut otut by the

junctions of a variable point of the conic with two fixed points of the same is constant,

and reciprocally the angle subtended at either focus by the segment cut out of a

variable tangent of the conic by two fixed tangents of the same is constant. The theorems already proved show the inaccuracy of the assumption

usually made, explicitly or implicitly, that the principal of duality or reciprocity

is inapplicable to all cases in which metrical relations are involved. It is, for instance, noticeable that in the chapter on "reciprocal polars" in Salmon's Conic

Sections no example is given of reciprocation between a theorem involving distances between points and one similarly involving angles between liines. The

fact is the method of reciprocal polars can be applied to all theoremrs involving distances between points and angles between lines in which all the points and lines can

be constructed by purely descriptive processes. It will be observed that the properties of the focal lines of a conic above proved are known properties of the asymptotes of a conic in the Euclidean geometry, . e. the asymptotes have these properties in the ordinary theory by virtue of their coinci(ence with focal lines,

and not as tangents at infinity. Let us consider more carefully the equation (9) of the conic S referred to

its axes. This form of equation can be used, of course, only when S is an

ellipse or hyperbola. In either centre, say 3, meet two axes = 0 Q20 ?0; Q10= 0 meets S in two vertices, of which 4 may be one; similarly Q20 = 0 meets

S in two vertices of which 5 may be one; 34 and 35 are then the lengths of two

semi-axes meeting in the centre 3; and we have

14--o, Q25: = 0 tt24+ ?34-0, = JfQ.15+ V?35 0

and by (14) _Q10 LO20+ Q20

511 Q222 IQ3 hence

%Q44 - + 34 [ + (A]34' IQ22 A83 '- 2+ 4

9234 lQ22

c3 S= 9 44 - P}22 2 9 VOL. V.



370 STORY: On Non-Etelidean Properties of Conies.

or, putting tan 34 = b, and similarly tan 35 a,

b2_ _ P283, a2 9533, A2 /222 il11

1 1 _1 t. e.

A:,u:v-~~~~~~~a2.Qll b2 222 238

so that Soo can be written

(21) coa2 + b2"222 s

where a, b are the tangents of the lengths of the semi-axes meeting in 3. Equation (21) may be made the basis of an investigation of the properties

of conjugate diamneters through the centre 3. Let 4 be the extremity of any semi-diameter 34 whose equation,is

pV/fi22 nio - VN21 n20 = 0;

now - 0 + 20 +30

1? 21 + 222 + 2833

and hence for any point 0 of S,

Q??-( a2 + 1) 5 +

(b2 + ) 520

hence

f124 = 222f11

A,~~~Q4 f144 = 2+ I + p( 2+ 1) 14,

-2 a QP2 b Q24 Q2

- 2 _ _ __ p2a2 + b 2

cos s44 2a2 (1 + b2) + b2(1 + a2)'

or, putting tan 34= p,

(22) = ((p2 + 1)a b2 pP(2 a2+ +

If 34 and 35 are two conjugate semi-dianmeters of S whose equations are respectively

p ^Vi22 o - N/ll Q20 = 0 and a - i212 10 - I1 Q20 , =

i. e. if the pole 6 of 34, qu , lies on 35, PI 16 +210 +226 Q20 _236 Q30

S60 a 2 11+ b2A2 2

Q3




must be proportional to p -/fV22 lo- / n2o i. e.

K36 = , b2 V/722/ f16 + pa2 V/lll 26 =0, and

C VfI22 f16 - fll Q26 = 0, whence

(23) paa2+ b2= 0, or - = a2' '

which is then the condition that p Vn?22 ? lo- V?/ n2O and a -N/V?lo V (f20

are conjugate diameters. If we put tan 3 = q we obtain, as in (22),

2 (a2 + l)a2b2 - p2a4 + b4. q 2da.+b2 p2a2+ b2

hence (24) p2+q2 =2+b27 i. e. the sum of the squares of the tangents of the lengths of any two conjugate semi-

diameters through either centre is constant, and reciprocally the sum of the squares

of the tangents of the angles which two tangents, one from each of two conjugate

points on either axis, make with that axis is constant.

Let now 5 be the angle between the two conjugate semi-diameters 34 and 3,

then cos 2 (pOa+ 1)2 2 (a2-b2)2

Cos p2 +l)(a21+lj) (p02+1J)(p2a4+b4)

5121 5 (p 2+

-)2 (p2 a2+ b2)2 a2b2 2 __

-+1)(, + (p2+1)(o2a4+b4) p2qg2

hence (25) pq sin = ab,

i. e. the product of the tangents of the lengths of two conjugate semi-diameter.s

through either centre into the sine of the angle between them is constant, and recip-

rocally the product of the tanagents of the angles which two tangents, one from each

of two conjugate points on either axis, make with that axis into the sine of the

distance between the points is constant. The equation of the tangent to S at 4 is 84o = 0, where

Q14 Q10 +224 220 234 Q30 40-a2 l b2 &22 53B

- 14 ( o + 20 Vp2a2A+ b2O> v- l-i Ka2VS +P b2V2 abV3 2,

and the angle qp which this tangent makes with the diameter 35 conjugate to 34



372 STORY: On Non-Euclidean Properties of Conics.

is given by

2 C a2 bz J (pa2-,b2)2 COS1~ ~ + =(I +1-~ + p2 a2 + 62 ) (2 + 1) [po2a4(1 + b2) + b4(1 + a2)]

/2 a b4 ab2

a2+ 1

aM(l + aM) + (1 + b2) hence tan2 (p _

a2 q.2 ] 1

or (26) q tan (p = ab, t. e. the tangent of the length of a semi-diameter through either centre into the tangent of the angle whqich it makes with thte tangent at the extremity of the conjugate semi- diameter through the same centre is constant, and reciprocally the tangent of the angle which a tangent from a point on either axis makes with th?at axis into the tangent of the distance of that point from the point of contact of either tangent from the conjugate point on the same axis is constant. The corresponding theorem in the Euclidean geometry will be found by introducing the constants k and 1X' with

the values co and respectively; then (26) becomes q ~a b

2ik tan (p __ 2ik

.ab i. e. tanp 2ik(p

hence (p = 0, i. e. any diameter is parallel to the tangents at the extremities of the conjugate diameter. In the general case this is not so, it is not even true that the tangents at the extremities of a diameter are parallel, in fact these tangents meet in the point where the conjugate diameter through the same centre inltersects the axis opposite that centre; the angle between either tangent and the conjugate diameter is the angle (p in (26), and the angle between the two tangents is 2(p. In the Euclidean case the reciprocal of the theorem above stated has a real interpretation only for conjugate points on the conjugate axis of an hyperbola. Then any point on the conjugate axis has the same ordinate as the point of contact of a tangent fron the conjugate point on the same axis, and the theorem may be expressed thus: the intetcept on the conjugate axis of an hyperbola between any point of that axis and the perpendicular let fall from a point of the hyperbola having the same ordinate as the given point upon the tangent from the given point is constant.




These results, so far as they refer to centres and axes, only apply in general to ellipses and hyperbolas. The equation of a conic of any other species cannot be put into the form (21). It can, however, be put into an almost equally simple form. The forms which I give below in equations (27)-(30) are substantially those given by Clebsch ("Vorlesungen iiber Gfeometrie," Iter Band, 2t- Abtheilung, VI). Clebsch has given also in the same place a method by which the species to which a conic belongs can be determined. If

Qoo -ax2 + f3y2 + )z2 + 2pyz + 2Xzx + 2?zy, S ax2 + by2 + cz2 + 2fyz + 2gzx + 2hxy,

the species to which S belongs depends upon the cubic equation Xx-a - a , A;~- h, A% -g - 0;

A;-h, 24-b-, A+p-f

namely, if the roots of A (%) = 0 are all different, S meets Q) in four distinct p6ints; if two roots are equal, S has contact with fn; if two roots are equal and for them all the first minors of A (x%) vanish, S has double contact with Ql; if all the roots are equal, S meets Ql in three consecutive points; and if all the roots are equal and for them all the first minors of A (X) vanish, S meets Q in four consecutive points. Let A (x) be developed and the terms arranged according to powers of A, say

X A X A (x)- F p (X3 _s12I + s2-s3),

then the discriminant of A (X) is D: 4 (3S2 - A2)(3s153- s )-(9S3 s1s2)2;

it is assuined that the system of coordinates is real, and that the coefficients in the equations of SI and S are all real. I. n2 imaginary.

a) If D > 0, S is an ellipse; b) if D = 0, S is a circle.

II. Ql real, S real. a) If D >0, S is an ellipse or an hyperbola; b) if D < 0, S is a semi-hyperbola; c) if D = 0, S is an elliptic or hyperbolic parabola; d) if D = 0 and all the first minors of A(X) vanish for the double root of

A (X)=0, S is a circle;



374 STORY: On Non-Euclidean Properties of Conics.

e) if - - = - (and therefore D = 0), S is a semi-circular parabola;

__ 2 383 f) if - and all the first minors of As (%) vanish for the triple root

of A (x) =0, S is a circular parabola. III. Ql real, S imaginary.

a) If D > 0, S is an imaginary ellipse; b) if D =0, S is an imaginary circle. It is hardly necessary to state the extra and intra-absolute positions of the

centres and foci, and the finite and semi-infinite positions of the axes and focal lines, as they are geometrically evident. It inay, however, be worth while to remark that a semi-hyperbola has only one real centre, one real axis, one pair of real foci and one pair of real focal lines; the elliptic or hyperbolic parabola has a douible centre and a double axis coincident with the point of contact with Ql and the tangent at that point respectively, two double foci and two double focal lines; a circle has one isolated centre (axial centre), every line through which is an axis, and one isolated axis (central axis), every point of which is a centre, the axial centre is a quadruple focus, and the central axis is a quadruple focal line; a seini-circular parabola has an infinite triple centre, a triple axis tangent to Q, two triple foci of which one is the triple centre, and two triple focal lines of which one is the triple axis; a circular parabola has an infinite double axial centre and a double central axis tangent to Ql, a sextuple focus coincident with the axial centre and a sextuple focal line coincident with the central axis.

The equations above given for Q and S may be regarded as equations in trilinear coordinates x, y, z, where these are constant multiples of joI, Q120t IQ30.

The forms of equation given by Clebsch, for what we have called parabolas of various species and circles are substantially then the following: for an hyperbolic or elliptic parabola, let 1 be the single centre, 2 the double centre, and 3 the pole (qua Q1) of the single focal line other than the double axis, then 212=?,

(227) -i =O13 =

02_

00 p20 &233D20 Q235 Q0 Q2 Q3 Q2+

(27) - 2 - + 2 ],D Soo ^if 222

for a circle, let 1 be the axial centre, 2 and 3 the single foci, then

f22= 0, ?330= 12 = ? =13 ?0

', 2 f20Q30 (28) floo= '-

+ 2 5220Q30 - Soo

Q20 3.

2 + 23




for a semi-circular parabola, let 1 be the triple centre, 3 the pole (qua Q2) of the triple focal line other than the axis, and 2 the pole (qua ?2) of the anharmonic conjugate of ?230= 0 with respect to the tangents to Ql and S at their single intersection, then 2I1=0, n21 = 0, ?22?233 - Lu23 0,

Q20 Q23 O2O A20 A?80 A2.-20 -2 (29) 2oo= - -2 QQ +2 _Q_

'Qs Soo 2 -Q 2- -Q8-0

A?22 2P22 A2222z zS2 A?

for a circular parabola, let 1 be the axial centre, 2 any point on the central axis, and 3 any point not on the central axis, then ll? = 0, ?212 = =0,

fIO 2 - Q28 -Q)lo X0 23 2lo'Q20 +2 f2lo'Q30

| ?? =- 5~222 f2123 + Q22 S222 S213 5413 (30) f A2A8A~)?Q+A20 2 A2 ?0A2 L ) I - (522.2 QA-?23) f210

+`20 -2 923 S10 S20 + 2 0lo 20 Soo _

- - +% Sl2 -2-5 +2- L 1222 is S2~~~~A22 'S222 AS S210

There is, however, a simple form of equation which applies to all conics exceptinig semi-circular parabolas, which I give below as (31). Every ellipse,

hyperbola, semi-hyperbola, elliptic, hyperbolic or circular parabola, or circle, has at least one real finite (i. e. not situated on ?2) centre, such that the opposite axis is real and transverse (both to ?2 and S), not a chord of double contact of S and ?2. In the case of the circle the centre in question is any point of the central axis, anid the axis in question passes through the axial centre. That

such a real centre exists for all the conics named excepting the semi-hyperbola

is evident, and for the semi-hyperbola it is only nlecessary to prove that the real

axis is transverse. To prove this let z_ 0 be the chord joining the real inter-

sections of S and SI (both real conics), y = 0 the real axis, x 0 the polar,

qua ?2, of (1, 0, 0); then QO, and Soo can evidently be written

oo= X2 i4 2 %2, SOO=X2 ++ -+xy y

The intersections of a and S lie on the pair of lines

(xT 2 1)y2+ltxy 0,

?I e, the imaginary intersections lie on

zFp1)y + tX = 0,

and for these intersections then

{( 1 )2 i :L y2-jtz2z 0,

hence (F T 1)2 4i t2 < 0, and therefore the lower sign must be taken, and

OO2x_X2- y22, S00=Xr+ Vy2 Z2+yXY,

where (2 + 1)2 - ~2 < 0; the intersections of S with the real axis ares given by

Z=0, Xa+2.y2+yxy=0,



376 STORY: On Non-Euolidean Proper-ties of Conies.

and by virtue of the condition for X and It the factors of the last equation are real; hence the axis is transverse to S, and it is evidently transverse to Q.

Returning now to the general case for all the conics named, let y = 0 be the real axis, x = 0 the tangent at either vertex on y= 0 (other than the point of contact with QlI, if such exist), z = 0 the polar, qua Q 2, of (O, 0, 1), then ?iO and SOO may be written (31) .(o =_x2 y2 -Z2, Soo=ax2+ by2+cxz, where c2 + 4 (a + b) b = 0 (this condition is only necessary when c2 < a2). The conditions for the different species of conics are easily seen to be the following:

b 1 foranhyperbola, c2<a2, a <- 2-X andc2+4(a+b)b>0;

ellipse, e2<a2, b > 1 XC2 + 4(a+ b)b>O; a 2 a seni-hyperbola, C2 > a2;

i an hyperbolic parabola, C2 = a2 and - -_ -

a 2 6 1 "elliptic parabola, C2- a2 and -> - 2

a circular parabola, e2 = a2 and- - -

a 2'

for a circle having real contacts with lQ, c2 + 4 (a + b) b = 0 and > -2;

for a circle having imaginary contacts with SI, 2 + 4 (a + b) b =0 and a~ - 2.

If S is a semi-circular parabola we may write (32) Qoo-2xy- z2, S0o_ 2xy + 2byz -z2,

where y - 0 is the triple axis (and focal line) of S, z = 0 the other triple focal line, and x = 0 the tangent to Q at the single absolute point of S.

It may be remarked that (31) may be written

_ 10 D+20 + 1- 0

2- + b Q30

11 ! g22 33 1 OO=a22 / 1133

where 1 is intra-absolute, and 2 and 3 extra-absolute; and (32) may be written

(34) floo = 2 +Ql Q20 So0o 2 +Q1 20 0 +b 2 Q20 Q30

A2 A212 + 523 X S0 ~ 2Q12 +528 + /- 2 D X233

where 2 is the triple centre of S, 1 the single absolute point, and 3 the pole qud Qif of the junction of the triple and single absolute points.



STORY: On Non-Euclidecln Properties of Conics. 377

The circle is particularly interesting, on account of the simplicity of its metrical properties. Equation (3) becomes, for a circle, (35) So- %fll foo Q20,l where 1 is the axial centre, and ?llo= 0 the central axis. From this equation follows, for any point of S,

5Q11 O z. e.

(36) 10 Cos_l sL i. e. the distance of any point of a circle from its axial centre is constant, and reciprocally the angle which any tangent of a circle miakes with its central axis is constant, for the reciprocal, qua Ql, of a conic having double contact with Q1 is another conic having double contact with fl at the same points, i. e. a circle concentric with the former one, and every theorem concerning a circle has its exact reciprocal.

Instead of (35) we may evidently write

(37) loo PSI, Soo - 511

whence, if SOO O, S2OO_ o, f210= (2e.t - 1 Q - (7 -)S2

f2110

hence, by comparison with the previous values of the left-hand member, (38) X 1 -, 10 =sin1 /it. Let 2 be any point on the central axis of S, and 3 any point of S, the line 23 intersects S again in a point 4; then

x4= Px2 + ax3, Y4= pY2 + aY3, Z4=pZ2 + aZ3, S12=0, S33 0, S44- 0, but S44-p2 S22 + 2p(SA23'

and therefore p: a = 228: - 822, and say x4 -223x2 - 822 x3, y4 = 2823y2 - A22y3, z4= 2823Z2-

814 - 822 S13, 2S4 = S22 S23,

f23 = Ityll 823, 24 = S S24 =jSll S22 823,

a22- = p11 22 A 33 = - 123, Q44 - 14, Q28 _ 1SS3 - Q?24

i122 Q33 8 S22 18 222 944 . e. 24 23,

but 24 is not independent of the position of 2; hence every chord of a circle is bisected by the central axts, and reciprocally the angle between any two tangents of a

VOL. V.



378 STORY: On Non-Euclidean Properties of Conies.

circle is bisected by the junction of the axial centre with their intersection. In this sense, then, every point of the central axis is a centre, and every line through the axial centre is an axis.

Still using (37), let 2 be any point in the plane and 3 the point of contact of either tangent from 2 to S; then

S23- :0, S33 =, ?23 -S82 813, ?22 =(LSl S22 - S122 n33 -13,

AL _ S122 -122 212

Q222 P58 11522 - S1 (1 -i) Q11 Q222 -i&211 922 1 -

i.e. cos23 cosl12, or

(39) tan 23 - /Sll - ( S11S222 - Qd21 812 f1

which is independent of any choice between the two tangents from the point 2, i. e. the two tangents to a circle from any point are of equal length, and reciprocally any two tangents of a circle make equal angles with the chord joining their points of contact.

The tangent to S, given by (35), at any point 2 has the equation

S20-t211? 20-212? 1o = 0,

whose pole, qua ?, evidently lies on the radius 12, and hence any tangent of a circle is perpendicular to the radiuts to its point of contact from the axial centre, and reciprocally any point of a circle is perpendicular to the intersection of the tangent at that point with the central axis.

The theory of the radical axes of two or three circles may be deduced geometrically (as in Salmon's Conic Sections, Art. 306), or we may proceed analytically. Let S', S", S"' be three circles, where

(40) SOO-Qlo-l,SOA/sQo Q0 l/A/f3Qo Q0 (40) 8~0~X'?211?200 - ~ ;V'fl22 2?20- flO 8O'O X" ?233 ?2OO - SI

Through the intersections of S' and S" pass three pairs of straight lines of which one pair is (41) xVI 222?_o- A1 2- 0,

and the peculiarity of this pair is that its lines intersect in the intersection of the central axes of S' and 8I". The common chords of two circles which pass through the intersection of the central axes of the circles may be called their radical axes. Every pair of circles has then two radical axes, and similarly every pair of circles has two radical centres, those intersections of their common



STORY: On Non-Eaclidean Properties of Conics. 379

tangenits which lie on the junction of their axial centres. From (41) follows directly, for any point of either radical axis,

(42) S 1Q - /'Q20

i. e., by comparison with (39), the foutr tangents to two circles from any point of either of their radical axes are of equal length, and reciprocally any line through either radical centre of two circles makes equal angles with7 the tangents to them at the four points where it intersects them. The three pairs of radical axes of the circles S', S", S"', taken two at a time, are

"'33 220 aX 22 f230, XIpQll 30 XI f33M2O 0, / n22(n2IO aXI nl 2o -?0

i. e. separately '%/%/If 33 S20 i%Xf 7 22 f30 - f A lllQf30 i4 '%//I f33 Qlo- X

V/2L' n22 Qo i vxAnA l S120 = 0,

from which it is evident that the six lines forming the three pairs of radical axes pass by threes through four points, the "orthogonal centres" of the three circles, namely through each orthogonal centre passes one radical axis of each pair; and reciprocally the six radical centres of thtree circles lie by threes on four straight lines, the " orthostatic axes " of the three circles, namely on each orthostatic axis lies one radical centre of each pair. The six tangents to the three circles from either orth&gonal centre are of equal length, and hence their points of contact lie on the same circle, whose centre is the orthogonal centre and which cuts the given circles perpendicularly in these points, i. e. it is orthtogonal to the given circles. Any three circles have then four orthogonal circles. Reciprocally either orthostatic axis makes equal angles with the tangents to the three circles at its six intersections with them, hence these tangents all touch the same circle, whose central axis is the orthostatic axis, and whose contacts with these tangents are perpendicular to the contacts of the given circles with the same, i. e. this circle is orthostatic to the given circles. Thus any three circles have four orthostatic circles. It may be remarked that two circles intersect in four points and have four common tangents, but two anid only two of the intersections can be orthogonal, and the contacts of two and only two of the tangents can be orthostatic. Let ll-- ia", t'Q22 = oP, XI//Q33 = 1/2; then (giving any possible combination of sigrns to a', a", a"') ob?I2o- aM30 = 0,

a'Q30 - aQ10 =0, a"Q)o - o20 = 0, are three radical axes meeting in the same orthogonal centre 4, for which then



380 STORY: On Nun-Euclidean Properties of Conies.

say (43) 14 -tal Q24- tE, Q34 tEtalX

and for the contacts of tangents from 4 to S' we have

J00?, AS0-?, i. e. a12??o - ?2 0, a'2?240- 214?2o10= a' (at '40- tS210) = 0,

from which follows (44) t2Qoo 4 - 0

which is symmetrical with respect to S', S" and S"', and hence is the equation of the orthogonal circle corresponding to the given combination of signs of at, a"' and a"'. The ratios

' 4:y:z:t are determined by (43) and these are to be substi- tuted in (44).

Similarly let (1 -') 2ll, fl2, (1- XI) Al22 = 9"' (1 -a"') n33 ,/ll2; then the radical centres of S' and S" are those points of the line 12 froin which tangents to S and S' will coincide, say one of these points has the coordinates

pxl +ax2 , pY1 +cy2, pzl + az2, and the equation of the pair of tangents from this point to 8' is found to be

[a2 (Qll, -22 12- ) -(p2 ?ll + 2p9?1j2 + a2 f222) f'2] ?o - (af2 f22 -p2 f2) 2o

-a2 (?2ll - f2) f24 + 2cr (arf12 - pff2) ?i210 = 0=0

i. e. a2 [(?lAl? - ?212) oo - 222f12?- SI +2?+2121o ?220]

- f 2 [( p2?ll + 2pa?12 + a2 f?22) ?200- p2 ?21-a2 ?20 + 2pa?21?o22o] = 0, and in order that these shall coincide with the tangents to S" it is necessary and sufficient that p2 a2 = /3112: /12, i. e. p: a = -": ? /' and the equation of the polar (qua Q2) of the radical centre is

(45) Mfl"?210 fl 'S20 0 ;

the combined equation of the polars, qua ?, of the two radical centres of S and S' is then (1 - ")222Al - (1;-')f11?220.

If 50- 0 is the equation of an orthostatic axis of 8', 8" and S"', then, if any combination of signs is given to /', /3" and 3"', by (45)

25: ?25 ?235 /3': f : f , i. e., say

(46) ?215 fl' ?25 = e'fl", ?35- et'fl

The locus of poles, qua' ?, of tangents to the orthostatic circle of S, S' and

8", say to u?55 o -Q50=?

is given by the equation

(47) (1 - P)Q5 oD 520;



STORY: On Non-Euclidean Properties of Conies. 381

and if 6 is either point of intersection of the orthostatic axis 150O 0 with S', we have (48) Q 0 ' /fi -=66- a 0,

and the tangent to S' at 6 is ;n1 n6O - n16 Q1o =0, 0 .- e. Q16 Q60 -Q66 Q10 = 0,

i. e. the pole of this tangent, qua' Qi, has the coordinates Q16 Z6 Qf6Bx1, Q16BY6 - 6Y11 16Z6 -Q66Z1

and this pole must then lie on the circle (47), whence follows, on reduction by (48), (1 -)(1-I)?n11Q55-?)5=O,

* Q~~~~~~~~~~~55 _ e2 . e. i t

and the equation of the orthostatic circle corresponding to the given combiNiation of signs of ', ", "' is (49) (QB5 -

0) OO Q 50=, and the locus of poles, qua Q2, of its tangents is given by the equation (50) t'2Q ?oo - ?)0 =0, where the ratios x5:y5:z5:te are determined by (46).

* BALTIMORE, JUne, 1883.



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On Non-Euclidean Properties of Conics