708 DE. J. CASEY ON CYCLIDES AND SPHEEO-QUAETICS. 
but this is also the condition that the squares should be connected by the linear 
relation. 
325. Propositions similar to those of the three last articles hold for sphero-quartics 
and also for bicirculars. Thus the analogue of art. 322 is, being given five sphero-quartics 
S 15 S 2 , $ 3 , S 4 , S 5 , and if multiples l x , l 2 . . l 5 be determined so that + ^ + 
may be the square of a circle v, then the envelope of v is a sphero-quartic. The analogue 
of art. 324 is, if six circles be generators of the same sphero-quartic, their equations are 
connected by a linear relation. 
326. To find the equation of the binodal cyclide formed by the generating spheres 
which touch W along the sphero-quartic in which W is intersected by the sphere 
Xa-f-^j3-|-j'y-}-gS, where X, p, v, g are multiples. 
The equation of any cyclide touching W along this curve will be of the form 
kW T (Xa-j -[Jjfi -\-vy-\- ft) 2 : 
and it is required to determine Jc so that this cyclide will be binodal. We find in this 
case 0', A' all =0 ; the invariant equation has therefore three roots =0 ; and if we 
denote by <r the tangential expression (A, B, C, D, L, M, N, P, Q, ITj(X, fa b ft 2 , the 
equation of the required binodal will be 
ffW = A(\a -\- (6$ i"y -)r ft) 2 ( 201 ) 
Cor. 1. When Xa+^+vy+g^ is a generating sphere of W, the binodal (201) reduces 
to <7=0. 
Cor. 2. If a!, ft, ft, h' be the sphero-coordinates of the points polar to Xa+^ + v yTg§ 
with respect to W, and if W' be the result of substituting a!, ft, ft, V in W, then we have 
< 7 = AW' (202) 
For the binodal circumscribed to W , whose nodes are a, ft, ft, d, is 
W W ' = (Xa -f- |U/j3 -f- Vy -f- gc)) 2 
(see art. 280), and eliminating Xa+^|3-f-j'y-{-g& between this and ( 201 ), we get ( 202 ). 
327. To find the condition that the circle of intersection of two spheres shall have 
double contact with W. Let W be given by the general equation, and let the spheres 
be Xa+jM/ 3+vy-f-gei, X'a+^'0+v'y -J-g'^, then the required condition is the determinant 
a. 
n, 
m. 
1C 
A, 
N, 
n, 
b, 
l , 
(C 
ft, 
m, 
l , 
c , 
r, 
v , 
r , 
d, 
?» 
* , 
v , 
b', 
ft> 
v', 
ft 
o 
II 
(see 
art. 
326), 
, that the 
= 0 . 
(203) 
generator of W, is a contravariant of the third order in the coefficients of W. Hence, if 
