604 
DR. J. CASEY ON CYCLIDES AND SPHERO-QTTARTICS. 
Fig. 1. 
Demonstration . Let F be the focal sphero-conic, a the centre of a, one of the circles 
of reference ; then if P, P' be limiting points of the system, composed of J and any 
tangent TT' to F ; P, P' are points on the sphero-quartic. Let K be the pole of the great 
circle TT'. Now if at be the tangent from a to J, by art. 36, the result of substituting 
the coordinates of P in the small circle a=cos at — cos«P, this may be written 
but 
and 
a = cosatf— cos«P; 
, cos aO sin A sin OT + cos A cos OT cos OK« 
cos at — 7TT = rT. 1 
cos(J£ cos (Jt 
cos aP = sin A sin PT J r cos X cos PT cos OKa 
cos A cos OT cos QKa 
cos 
= sin X sin PT - 
Hence, by substitution, we get 
. fsinOT . -pnp) 
K = sm q3soi _sm]PT i ; 
and putting for cos Ot its value cosOT 4 - cos PT (see art. 24), we get 
sin A sin OP 
“ = cosOT 
Hence the results of substituting the coordinates of any point P of the sphero-quartic 
in the equation of the small circles a, (3, y are proportional to the sines of the arcs from 
the centres of a, j3, y to a great circle tangential to the sphero-conic F, and hence the 
proposition is proved. 
41. If the cyclide W be expressed in terms of four spheres a, j3, y, § which are mutually 
orthogonal, then the sphero-quartic WU will be expressed in terms of four circles which 
