DE. J. CASEY ON CYCLIDES AND SPHEEO-QUABTICS. 661 
forming the discriminant of this and returning to x, y, z coordinates, we get the equation 
of the tangent cone to be 
I 3 — 27J 2 =0, 133) 
where 
I = AE(H ++ + s 2 ) 2 + 4BD(H +?/ 2 + s 2 ) + 3C 2 , (134) 
J=ACE(a- 2 +/H-s 2 ) 2 + 2BCD(a’ 2 +r+2 2 )-AD 2 (a’ 2 H- < ?/ 2 +^) 2 -EB 3 (a' 2 +/4-^) 2 -C 3 . (135) 
194. From the form of the equation of the tangent cone, I 3 — 27J 2 =0, it has twenty-four 
cuspidal edges ; but from the forms of I and J we see that they have respectively, with the 
imaginary circle at infinity, contacts of the first and second order at each of the points 
where the cone C meets that circle. Hence the cuspidal edges coincide six by six with 
the four lines from the origin to these imaginary points ; and it hence follows that, when 
we omit the factor (x 2 -\- y 2 z 2 ) 2 in the equation (133), the remaining part, which represents 
a cone of the eighth degree, has no cuspidal edge. This equation is 
A a EY-12A 2 BDE 2 ? 6 
- (6 AB 2 D 2 E + 1 8A 2 C 2 E 2 — 5 4A 2 CD 2 E + 2 7 A 2 D 1 - 54 AB 2 CE 2 +27 B 4 E' 2 ) ? 4 
- ( 1 80 AB 2 C 2 DE — 1 0 8 ABCD 3 + 6 4B 3 D 3 )g 2 ^ ^ 136 ^ 
- (54 ACT) 2 - 81 AC 4 E + 5 4B 2 C 3 E + 3 6B 2 C 2 D 2 ) = 0 • 
In this equation for shortness we have written f for x 2 -\-y 2 -\-z 2 . 
Cor. If the origin be on the cyclide E=0, and the tangent cone reduces to the square 
of the tangent plane to the cyclide at the origin and a cone of the sixth degree, 
27A 2 DY + 4BD(16B 2 — 27AC)§ 2 — 18C 2 (2B 2 + 3AC)=0. . . . (137) 
195. The cone J is such that every edge of it is cut harmonically by the cyclide; and 
therefore, if any edge of it meet the cyclide in two coincident points, there must be a 
third point coincident ; therefore, since the imaginary circle at infinity is a double line on 
the surface, the points where J meets it are such that every edge which passes through 
it is an inflectional tangent. Hence from any point can be drawn to a cyclide twelve 
lines, which are inflectional tangents to it at the imaginary circle at infinity ; and these 
lines are distributed into four sets of three lines each, each triad consisting of three con- 
secutive lines. 
196. If the cyclide W has a double point, its class is diminished by two; if it has a 
biplanar node, its class will be diminished by three ; if it has two nodes, its class will 
be diminished by four. The following Table contains the singularities of the tangent 
cones for each of these cases : — 
No node. 
Conic node. 
Biplanar node. 
Two nodes. 
m = 8, 
m = 8, 
m = 8, 
m= 8, 
n =16, 
n =14, 
n =13, 
n =12, 
* = o, 
K = 6, 
* = 9, 
* = 8, 
l =20, 
l =12, 
l = 8, 
a =10, 
i =24, 
* =24, 
/ =24, 
/ =20, 
o' 
CO 
II 
u 
r =51, 
r =38, 
r =32. 
4x2 
