668 
DR. J. CASEY ON CYCLIDES AND SPHERO-QUARTICS. 
we shall have 
W=A(tf 2 +3/ 2 +z 2 )+3B + 3C+D, (145) 
where 
A =lx -\-my+nz, B =(a, b, c,f g, hjx, y, z)\ 
C =px-\-qg -\-rz , B= constant; 
then, hy the method of art. 193, we find the tangent cone 
G 2 —4H 3 = discriminant xA 6 , (146) 
where 
G == A 2 D {x 1 +y* + zj + 2 B 3 + 3 ABC(a‘ 2 + f + z 2 ) = 0 , 
H~B 2 -AC(x 2 +y 2 +z 2 ). 
Hence the tangent cone is 
A 2 D 2 (a’ 2 +y~-\-z 2 ) 2 -ft (4 AC 3 — 6 ABCD)(a ,2 +^ 2 -f- z 2 ) + 4DB 3 — 3B 2 C 2 = 0. . (147) 
Cor. The plane A is parallel to the tangent planes to the cyclide at its centres of 
inversion. 
216. The cone G possesses the property that any edge of it meets the cyclide in three 
points, whose distances from the vertex are in arithmetical progression. Now, if we 
invert a cubic cyclide from the vertex of the cone, we get a quartic cyclide ; and since 
the cone G meets the cubic cyclide in points whose distances from the vertex are in 
arithmetical progression, it will meet the quartic cyclide in points whose distances are 
in harmonical progression. Hence the cone G is identical with the cone J of article 
193, when the vertex of J is on the surface. 
217. Since a cubic cyclide is determined when U and F are given, and U is deter- 
mined by four and F by eight conditions, we see that a cubic cyclide is determined by 
4-j-8 = 12 conditions. Hence it follows that every cubic cyclide can be written in the 
form 
A«=Bj3, , . (148) 
where A and B are planes, and a and ft spheres ; and in this form it is evident that the 
intersection of the radical plane of the spheres a, ft with the two planes A, B is a centre 
of inversion of the cyclide. From the equation (148), being the result of eliminating 
7c between the equations A— A:B = 0 and 7cu — ft = 0, we infer that if we have a system 
of planes passing through the same line , and a homographic system of spheres passing 
through the same circle , the locus of the circle of intersection of a sphere and its corre- 
sponding plane is a cubic cyclide. 
CHAPTER SI. 
Classification of Sphero-guartics. 
218. We have seen that if 'W^acP-ftbfth-ftcf-pdV', and U 2 =a 2 +|3 2 -{-y 2 -l-c5 2 , the 
sphero-quartic WU is also the curve of intersection of the quadric Y ~~ ax 2 -ft by~ + cz 2 -\- dnr 
and the sphere U=a ,2 +?/ 2 -l-;r-}-w 2 , the tetrahedron of reference being the one formed 
by the four planes of intersection of U with the four orthogonal spheres a, ft, y, 
