Dll. J. CASEY ON CYCLIDES AND SPHEllO-QUAKTICS. 
719 
F . . . F ; ", that is, the determinant 
dF 
d F 
dF 
dF 
d\ ’ 
dp 7 
dv 7 
dg ’ 
dF' 
dF' 
dF' 
dF' 
d\ ’ 
dp 7 
dv ’ 
df 
dF" 
dF" 
dF" 
dF" 
d\ ’ 
dp 7 
dv’ 
df 
dF'" 
dF"' 
dF'" 
dF'" 
, dk ’ 
dp ’ 
dv 5 
d? 5 
(235) 
354. If a cyclide of the systems &W -\-WW -{-WW have double contact with W"', 
the points of contact are evidently points on the Jacobian, and therefore lie somewhere on 
the curve of intersection of W" with the Jacobian. Again, if a cyclide of the system 
JcW -f fc'W' have double contact with the curve (W" W'"), that is to say, if at one of the 
pairs of inverse points where IcW -)- ’kW meets W" and W'" the generating spheres of 
(FW-f /FW'), W", W"' intersect in the same circle, the pair of points is evidently a pair 
of points on the Jacobian. It follows then that sixteen surfaces of the system FW -f/FW' 
can be described to have double contact with the curve (W" W'"), since the Jacobian is 
of the fourth degree in («, (3, y, ?$), and each of the cyclides W", W'" of the second degree, 
and each system of common values of a, /3, y, c) gives a pair of inverse points. 
355. Given three cyclides W, W', W", the locus of a pair of inverse points whose polar 
spheres with respect to W, W', W" have a common circle of intersection is the curve of 
the twelfth degree , which is common to the system of determinants 
dW 
dW 
dW 
dW 
da 5 
d/3’ 
dy 7 
dS 7 
dW' 
dW' 
dW 
dW' 
da 7 
d/3 ’ 
dy ’ 
“diT 7 
dW" 
dW" 
dW" 
dW 
da 
d/3 3 
dy 3 
dS 
356. To find the condition in the invariants that two cyclides W, W' shall be so 
related that four generating spheres a, (3 , y, e> of W' shall have the circles (aj 3), (ay), 
(h(3), (c)y) lying on W. 
The equation of W must be of the form L/3y-j-Pa&=0, and the coefficients a, h, c , d 
must be wanting in the general equation of W'. Hence we have 
A = L 2 P 2 , 0 = 2 LP(Lp + P l), 
d>=(Lp+PZ) 2 +2LP(Zp— mg— nr), 
&=2{lp—mg— nr){Lp + P /) . 
Hence the required condition is 
4A0ffi = 0 3 -l-SA 2 ©' 
( 237 ) 
