710 DE. J. CASEY ON CYCLIDES AND SPHERO-QUARTICS. 
being nodes for loth, if three harmonic spheres of one binodal he three generating spheres 
of the other. 
332. We can reciprocate the process of recent articles. Thus, let V, V' be the focal 
quadrics of two cyclides W, W' having a common inversion sphere, then V+#V' will 
be the focal quadric of a cyclide whose equation we can find as follows, viz. form the 
tangential equation of V + JcY 1 , and substitute a, 0, y, l for the variables, the required 
equation will be 
« 2 /3 2 t y 2 , S 2 A 
a - 1 + lea ' ~ 1 + b - 1 + kb'—' + c ~- 1 + kd~ 1 + 'dr - 1 + kd '~ 1 = 0 ’ * * * ( 20 0 
a~\ b~ x , See. are evidently the coefficients of V, since a, l , &c. are the coefficients of W. 
The discriminant of (207) with respect to Jc will be the envelope of all the cyclides 
which it can represent by varying Jc, that is, it will be the tore with which they all have 
double contact. The curve of taction of any of them with it will be of the eighth 
degree, being the intersection of tw 7 o cyclides of the fourth degree. 
The geometrical interpretation of the discriminant is, that it is the envelope of a variable 
sphere cutting U orthogonally, and whose centre moves along the twisted quartic (V V'). 
333. We can get the equation of the cuspidal edge as follows : differentiate (207) twice 
with respect to Jc, and we get a system reducible to the following equations : — 
[a~ l + ka!~f (A -1 + kb 1-1 ) 3 (c -1 + Arc ,_1 ) 3 (d ~ 1 + kd!~ 1 ) 3 * * • ( 20S ) 
(aa')- l cd (^V/3 2 (ccQ-y jdd 1 )-' 8 2 
(fl- 1 +A^- l ) 8 ." , "(i- 1 +*4 , - 1 ) 8 " t "(c- l + *c , " 1 )8T(rf- , + ^ 1 ) 8— * * * v^ uy ) 
{o!)--cd (£')~ 2 /3 2 (c')-y {d')-W 
(a- 1 + Ao , - 1 ) 8 + (6- 1 + *fi , - 1 ) a +(c- I + A:c , - 1 ) 8 +(rf- 1 + Arf , - I ) 8 -“ U * * * • ( 210 ) 
The result of eliminating Jc between these equations will be a pair of equations repre- 
senting two surfaces whose curve of intersection will be the cuspidal edge. 
Now solving the equations (208), (209), (210), we get 
I I i 
b 2 ’ c 2 ’ d 1 ’ 
( fl- 1 + &«'- 1 ) 3 — 
J_ 1 
bb h C(P 
Id" 
— A 2 suppose. 
1 1 2_ 
i/ 2 ’ d 2 ’ d' 25 
( c 2 \ 1 
jt V, with similar values for (b~ l -{-Jcb’- 1 ), See.; and substituting 
in the equation (207), we get 
(AV)H(B 2 0 4 )H(Cy)H(D^f=O (211) 
as another surface on which the cuspidal edge lies. But if we eliminate Jc between any 
three of the equations for a~ ] -\-Jca'~ l , b~ l -\-Jcb'~ l , See., we get four equations of binodal 
