600 
DE. J. CASEY ON CYCLLDES AND SPHEEO-QUAETICS. 
cyclide , the focal quadric is inscribed in the tetrahedron formed by the centres of the spheres 
of reference . 
32. Let " W=(a , b , c, d , l, m, n, p, q , rfa, (3, y, e>) 2 =0 be the equation of a cyclide, 
and we know that the square of the Jacobian of a, (3, y, 6 is given by the equation (29). 
Now, substituting A, p, v, § for a, [3, y, l in these equations, we have the tangential 
equations of the focal quadric F of the cyclide and the sphere U ; but if F and U be 
referred to their common self-conjugate tetrahedron, their equations will take the form 
«A 2 + bf -j- cv 2 + dg 2 = 0, A 2 +/P+P+£ 2 = 0. 
Hence we have W and TP given by the equations 
W = aa 2 + + cy 2 + dP = 0 , 
U 2 ^-(a 2 +/3 2 +y 2 +S 3 )=0 (see art. 28. Cor.). 
Now if, for the sake of uniformity, we represent U by s, we have the following equation 
identically true, 
« 2 +/3 2 +y 2 h^ 2 hr=0; (32) 
and since the addition of any multiple of an expression which vanishes identically does 
not alter a function, we see that the equation of a cyclide may be written in the fol- 
lowing form by adding e(a 2 + /3 2 + y 2 + P + a 2 ) to aa 2 + fy3 2 + cy 2 +£$ 2 =0, and afterwards 
putting a for (a-j-e), b for (b-\-e) &c., 
W ^cia 2 -\-b 2 (3-\-cy 2 -\-d'h 2 -\-es 2 =0, (33) 
in which each of the spheres of reference is cut orthogonally by all the others. 
We could show a priori that W can be expressed in either of the forms 
aur -f- b(3 2 + cy 2 -j- db 2 = 0, 
or 
aod + b(3 2 -f cy 2 + dd 2 -f- ez 2 = 0 ; 
for the first form contains explicitly three constants, and each sphere contains four 
constants, so that there are 3 + 4x4 = 19 constants at our disposal ; but each pair 
of spheres being mutually orthogonal is equivalent to six conditions. Hence we 
have thirteen constants, which is the number required. Similarly, in the second form 
we have twenty-four constants; but these are subject to the equation of condition 
a 2 +/3 2 + y‘ 2 + P+s 2 , which does not vanish identically except by the incorporation of 
certain constants, and the condition of orthogonality of each pair of spheres is equivalent 
to ten conditions. Hence, as before, we have thirteen conditions remaining. 
33. By means of the identical relation c4 2 +/3' 2 + y 2 +P + £ 2 =0, we can eliminate in 
succession each of the five spheres a, (3, y, S, s from the equation (33) of the cyclide; 
and we see that the same cyclide can be written in five different ways, the letter elimi- 
nated representing the sphere used in the corresponding equation of the surface. 
Hence we have 
