DR. J. CASES' ON CYCLIDES AND SPHERO-QITARTICS. 655 
177. The sphero-quartic (WU) is the intersection of the two surfaces in tetrahedral 
coordinates, 
ax 2 J rby 2 -\-cz 2 -\- dvr = 0 , 
A’T//;-m 2 +w 2 3 
the first being the reciprocal of the focal quadric of W, and the second the sphere U. 
Now the osculating plane of WU at any point x', y', z 1 , w’ is (see Salmon’s ‘ Geometry 
of Three Dimensions,’ p. 291) 
(a—b)(a—c)(a — d)x' 3 x + (5 — d)(b — c)(b — d)y' 3 y 
+ (c — a) (c — b)(c — d)z ,3 z -f- (d — a)(d — b) (d — c)z' 3 w = 0. 
This may be written in a simpler manner : thus, if \}/(A) denotes a biquadratic whose 
roots are a , b, c, d, the coefficients of the above equation denote the results of substi- 
tuting the roots a, b , c, d respectively in \|/(A), so that the equation becomes 
■ty’(a)x ,3 x -fi -fy\b)y' 3 y + f ( c)z 3 z + 4/' ( d)w' 3 w =0 (1-1) 
Hence through any point can be drawn twelve planes to osculate a sphero-quartic. 
Cor. 1. Through any point on the sphere U can be described twelve osculating circles of 
WU. Hence Ciiasles’s characteristic g> for the osculating circles of a sphero-quartic is 
/a =12. 
Cor. 2. If the point be on the sphero-quartic itself , 9. 
Cor. 3. Every sphero-quartic is osculated by twelve great circles ; for twelve osculating 
planes can be drawn through the centre of U. 
Cor. 4. Let us consider any small circle Z on the surface of U ; then, since through the 
pole of the plane of Z can be drawn twelve planes osculating WU, we have the theorem 
that any circle on the surface of U is cut orthogonally by twelve osculating circles of WU. 
Cor. 5. By inversion we get the following theorem for bicirculars : — Any circle in the 
plane of a bicircular is cut orthogonally by twelve of its osculating circles. 
Cor. 6. The theorems that a bicircular quartic has twelve, and that a circular cubic 
has nine points of inflection, are the inversions of Cars. 1, 2. 
178. Since the cuspidal edge of 2 is the locus of the poles of the osculating planes of 
WU, it is plain that the cone whose vertex is any point of the cuspidal edge, and which 
circumscribes U, will touch U along an osculating circle of WU, and that it will be an 
osculating right cone of the cuspidal edge (see Salmon’s ‘ Geometry of Three Dimensions,’ 
art. 363). . Again, since twelve osculating planes of WU pass through any point, we see 
that the cuspidal edge is of the twelfth degree. This latter part corresponds to the 
theorem that the evolute of a bicircular quartic is of the twelfth degree. 
179. Since the cuspidal edge is of the twelfth degree, any quadric will cut it in 24 
points. Hence any cone will in general cut it in 24 points. If the cone circumscribe U, 
we have, by reciprocation, the theorem that any circle on the surface of U touches in 
general 24 osculating circles of WU. 
