DR. J. CASEY ON CYCLIDES AND SPHEEO-QUAETICS. 
713 
to W will be a generating sphere of W -j-77W. We have then in <r 4 - ler -f- 7tV 4 - 7rV 
to substitute for X, g, v, g : the result is expressible in terras of the 
dot dp dy do ‘ b 
covariants ; it is 
AW -f 7i’( 0W — AW') + T^ffiW-T') + Ic\ 0' W - T) = 0 (220) 
In like manner the locus of inverse pairs of points whose polar spheres with respect to 
W' are generating spheres of W-j-TW' is 
0W-T' + Jc{ $ W' — T) + 7r 2 ( 0 W' — A W) + k 3 A' W'= 0 (221 ) 
340. To find the locus of a pair of inverse points whose polar spheres with respect to 
W, W' form a conjugate system with respect to W 4-TWb Let 
W = au 2 + W + cy 2 + do 2 , W' = a' a 2 + 6'3 2 + c'y 2 + d'Z 2 , 
and the locus will be 
a-\-Jca\ 
0, 
0, 
0, 
au, 
0, 
6 +& 6 ', 
0, 
0, 
63 , 
o, 
0, 
c 4~ Jed , 
0, 
cy. 
0, 
0, 
o, 
d ”f” Jbd! ,) 
d\ 
ciu, 
63, 
w 
db, 
o, 
This can be expressed in terms of the covariants, and is the cyclide 
AW , +£T+7r 2 T'+/LA'W=0 (compare 212) (223) 
(341). To find the condition that a given circle should have a given pair of inverse 
points common to the curve of intersection of two given cyclides W, W'. Suppose we 
have formed the condition (see art. 327) E=0, that the given circle should have double 
contact with W, and that we substitute in it for each coefficient a , a-\-ica', See., the con- 
dition becomes 
E+7’x+7rE'=0 ; (224) 
and if the given circle has any arbitrary position, we can, by solving this quadratic for 
Jc, determine two cyclides through the intersection of W and W', each having double 
contact with the given circle ; but if the given circle has a pair of inverse points in 
common with the curve (WW) , the cyclides having double contact which can be drawn 
through (WW') become coincident, and the equation (224) becomes a perfect square. 
Hence the required condition is the discriminant 
4EE' — t 2 =0 (225) 
Cor. 7r = 0 is the condition that the pair of segments which W intercepts on the given 
circle should be harmonic conjugates to those which W' intercepts on it. 
342. If W=au 2J r b[3 2 + cy 2J r dh 2 ==;0, ~ s N'=a!u?-\- b'p 2 -}-c'y 2 -\-d l h 2 =0, and if the circle 
be the intersection of the spheres 
Xu -j- [Jj \ 3 -j- vy -f- ^ = 0, x!u-\- (Jj'i 3 -}- v'y -j- = 0, 
