DE. J. CASEY ON CYCLIDES AND SPHEEO-QUAETICS. 
707 
mine nine constants l x , l 2 , ... .I g so that /,W, + / 2 W 2 . . . . + / 9 W 9 may be a perfect square 
L 2 , or the product of two spheres, M and N; it is required to prove that the envelope 
of the sphere L is a cy elide, and that M and N are conjugate spheres with respect to it. 
Demonstration. We can determine a cyclide W («, b . . .\a, 3, y, bf so that the 
invariant 0 shall vanish for W and each of the nine cyclides, since we have nine equa- 
tions of the form 
Aff 1 + B5 1 +Cc 1 +D<Z 1 +2LZ 1 +2M?w I + 2Nra;+2P^ 1 +2Q2 I +2Kr 1 = 0, . . (198) 
A, B, C, &c. being the minors of the determinant A (see art. 313), and c 15 &c. the 
coefficients of Wj ; hence the mutual ratios of A, B, C are determined. Now if we 
have separately nine equations of the form (198), we have plainly also 
A(/i«i + / 2 « 2 • • • h a g) + & c - =0, 
that is, © vanishes for W and every cyclide of the system ^Wj liW 2 . . . / 9 W 9 . Hence 
the theorem is proved. 
Cor. If the sphere M be given, N passes through a given pair of inverse points, namely, 
the pole points of M with respect to W. 
323. If we are given only eight cyclides, W,, W 2 . . . W 8 , and seek to determine the 
cyclide W as in art. 322, so that the invariant © shall vanish for W and each of the 
eight cyclides, then, since we have only eight conditions, one of the tangential coefficients 
A, &c. remain undetermined ; but we can determine all the rest in terms of that one, so 
that the tangential equation of W is O-f KO'=0. Hence the focal quadric of W 
contains an indeterminate constant in the first degree, and therefore it passes through a 
given curve. 
324. If ten spheres, a„ a 2 . . . a 10 , be all generators of the same cyclide W, their equa- 
tions are connected by a linear relation, 
Z,a 2 -j- / 2 a 2 . . . — 9 (199) 
Demonstration. Let a 1 =:X 1 tt + i a 1 3 + f' 1 y + ^3= ; 0, &c. ; and writing down the conditions 
that a 15 a 2 , &c. are generating spheres of W, and eliminating linearly the ten quantities 
a 2 , 3 3 , y 2 , aft, ay, aS, 0y, y<$, 
we get the following determinant : — 
X 2 p* v\ o 2 \v l Ajg! 
>4 >A C X 2 /a 2 X 2 v . 2 k,0., (MJ>: v 2 0 2 
O 
O 
? 
( 200 ) 
MDCCCLXXI. 
