702 
DR. J. CASEY ON CYCLIDES AND SPHERO-QDARTICS. 
quartic, their points of contact with the quartic lie on the polar circle of the given pair 
of inverse points. 
311. The discriminant of the equation (182) is 
Q'Q"=P 2 . 
Hence the equation of the pair of generating circles of the sphero-quartic which pass 
through (a, ft, y) is 
q«'={(4+p’!+4)q} 2 ( is4 ) 
This equation can also, as is evident, be written in the form 
a, Ji , 
h, b, 
I / 
fty — fty , y 'a — y a, aft — aft 0, 
In like manner the equation of the four points in which the circle ^a-b^/3-f-^y cuts the 
quartic is 
2' 2"— p 2 = 0 (see art. 308), (180) 
g, fty— (By', 
f , y'a-ya', 
C, a! ft — a' ft 
= 0. . . . (185) 
which may also be written in the form 
A, 
H, 
G, 
yv 
— Pi, 
H, 
B, 
F, 
— ^1, 
G, 
F, 
c, 
A,y,' 
— 
[j^v — yv,, 
V- 
■ — A/Aj 
0. 
(187) 
CHAPTER XY. 
Invariants and Covariants of Cyclides. 
312. It is always possible in an infinity of ways to choose four spheres a, ft, y, b so 
that the equations of two cyclides having a common sphere of inversion can be thrown 
into the forms 
W (a , b , c , d , l , m , n , p , g, r fa, ft, y, b) 2 = 0, 
W '=(«', V, o', d', V, in', n',f, q', r'fa, ft, y, bf= 0. 
For each of these equations contains explicitly nine constants, and each of the spheres 
a, ft, y, b contains implicitly four constants, so that we have thirty-four constants at our 
disposal, and we require but twenty-two. For the two cyclides are determined when the 
common sphere of inversion and the two focal quadrics are given ; hence the number of 
constants required is 4-}-9x2 = 22. 
