704 
DR. J. CASEY ON CYCLIDES AND SPHERO-QUARTICS. 
(1) 
CA. 
BD : 
BA. 
CD: 
= sin 2 S, 
(2) 
CB 
AD 
AB 
CD 
= cos 2 \ S, 
( 3 ) 
AC 
BD 
BC 
AD 
= — tan 2 \ S 
( 4 ) 
BA 
.CD 
CA 
BD 
= cosec 2 \ S, 
( 5 ) 
AB 
.CD 
CB 
.AD 
=sec 2 ^ S, 
(6) 
BC 
.AD 
AC 
.BD 
= — cot 2 ^ S 
317. Let there be given two conics referred to their common self-conjugate triangle 
S^x 2 -\-y 2 -\ -z 2 , =ax 2 -\-by 2 cz 2 , and let us denote by S', S", S’" the angles (see last article) 
of the anharmonic ratios of the three quartets of points in which the sides of the self- 
conjugate triangle is intersected by the two conics. Then for determining C we must 
find the anharmonic ratio in which the side x = 0 is intersected by the two conics; for 
that purpose we have the pencil formed by the two pairs of lines y 2 -\-z 2 =0 and by 2 -[-cz 2 =0, 
and we easily get 
21 ,_ (#+c») a 
COS 2 0 — • 
Hence 
sin 2 S' 
(b—c) 
4 be 
Now if we form the invariant equation in k for the two conics S, S', that is, if we 
form the discriminant of #S+S', and denote its roots by k', k", k" 1 , these roots are 
known to be — a, — b, — c. Hence we have the following system of equations: — 
sin 2 & = —(k" —k"’) 2 : ik" k l ", v 
sin 2 0" = — (k!"—k! ) 2 : A.k'"k' , l (192) 
sin 2 0"'=-(k' —k" f : 4k' k".J 
Hence the discriminant for the invariant equation of the two conics S, S' is 
— 64 A ' 2 . . _ _ — 64A' 2 
— (sin 2 S'. sin 2 S". sin 2 0'"), or omitting the multiplier — ^ — , which is numerical, the 
discriminant is sin 2 S'. sin 2 S". sin 2 0"' ; and as each sine squared is the product of two 
anharmonic ratios (see art. 316), we have the following theorem, which is the one 
referred to in art. 3] 5 : — 
The tact-invariant of two conics is the product of six anharmonic ratios, and the 
vanishing of any one of these six ratios is a condition of contact of the tivo conics. 
Cor. From the values given for the invariant angles S', S", S'" in this article, we get 
e 2S ' v “i —k" : k"', e 2 «" V=i —ft " : k' , e n - 6 '" V“i =# : k". 
Hence S' + 0" -j- (j" — 0, that is, the sum of the three invariant angles of two conics is equal 
to zero. 
