DK. J. CASEY ON CYCLIDES AND SPHEKO-QUAETICS. 
619 
coordinates of P in the equations of the spheres a, (3 respectively. Hence we have the 
following theorem : — 
The results of substituting the coordinates of any point P in the equations of two 
spheres , divided respectively by the diameters of those spheres, have a ratio which is unal- 
tered by inversion. 
70. The general equation of any cyclide, 
W- fa, b, c, d, l, m, n, p, q, r,fa, f3, y, S,) 2 =0, 
may be written in the form 
W=(a&?, bl:, ell, dll, Ilf, ml 3 l „ nof^plj, qKK rKKXf’ -p ^) 2 =0 ; 
CC /3 8 
and by the last article the six ratios f f — ' remain unaltered by inversion. 
z o 
Hence, denoting the inverses by the same letters accented, the cyclide W will be inverted 
into a cyclide W' given by the equation 
71. We have shown that the equation of any cyclide may be written in the form 
aot 2 -\-b(3 2 -\-cy 2J r dl 2 -\-ee 2 = 0 (see art. 32), 
where « 2 -|-j3 2 -|-y 2 -|-cr-f-s 2 = 0 is an identical relation. I shall call this form of the equa- 
tion of a cyclide the canonical form ; and we see by the last article that the equation of 
the inverse of a cyclide given by its canonical form is also in its canonical form. 
Cor. If the cyclide be of the form aoi 2 -\-bl3 2 -\-cy 2 =Q, the inverse cyclide will be of the 
same form, that is, the inverse of a binodal cyclide is a binodal cyclide. 
72. The five spheres a, (3, y, &, s of the canonical form are mutually orthogonal; and 
if we take the centre of a for a centre of inversion, and a for the sphere of inversion, 
each of the five spheres will be inverted into itself. Hence the centre of a is a centre 
of self-inversion of the cyclide. Similarly the centres of the spheres (3, y, l, e are centres 
of self-inversion. Hence we have the following theorem : — A cyclide is an anallagmatic 
surface, and the centres of the five spheres of the canonical form are its jive centres of 
inversion. 
73. We can confirm some of the foregoing results by Cartesian methods. Thus let 
the spheres of reference a, [3, y, £ be expressed in rectangular coordinates, and putting 
s for the sum of the coefficients a, b, c, & c., we get 
W:=4C+?/ 2 -f2 2 ) 2 +U 1 (^-f/-f5 2 ) + U 2 =0, (75) 
where Uj and U 2 are the general equations of the first and the second degree. Now 
transforming into polar coordinates by putting 
X=q COS 6, y = Q l COS <p, Z—q COS 'll, 
4 Q 
MDCCCLXXI. 
