DR. J. CASEA ON CTCLIDES AND SPHERO-QUARTICS. 
681 
(157) 
Pig. 7. 
Hence we have the system of equations : 
tan 2 a =4 (sin 2 r —sin 2 r' )(sin 2 r — sin 2 r") : sin 2 2r 
tan 2 g' =4(snfV — sin 2 r")(siir r' — sin 2 r ) : sin 2 2r' , 
tan 2 = 4(sin 2 r" — sin 2 r )(sin 2 r" — sin 2 r' ) : sin 2 2r". 
247. Let us denote the radii of the three circles of inversion by J, J', J", the radii of 
the focal circles by r, r', r", and the distances by ci, l', l", 
as in recent articles. Now denoting the perpendicular 
from the centre of J to any tangent to E by yi, then taking 
OP=£ such that 
cosy> : cos(y) — ^) = cos J=4 suppose ; 
/C 
.•. (cos§ — k) cosy; 4 - sin § sinyi = 0 ; 
but from the spherical triangles O H M and O P H we get 
cosy) sin c) cos 3 + cos ^ siny)=sin r, 
cos g cos S -J- sin g sin c$ cos cos R. 
Hence, eliminating y> and Q, we get 
(1+& 2 — 2k cos £>)^sin r=k cos ci— cos R, 
with two similar expressions involving k', k" ; h', l". Hence we have the determinant 
(1+& 2 — 2k cosg)*sinr, k cos£, 1, 
—2k' cos %' y sin r ' , k' cosl' , 1, 
(1 -f k" 2 — 2k" cos sin r", k" cos h", 1 , 
= 0 ; 
and restoring the value of k, k', k", we get 
(1-fcos 2 d 
— 2 cos g cos d )* sin r , 
cos o , 
cosd , 
(1 + COS 2 d' 
— 2 cos g' cos d' )* sin r ' , 
COS C>' , 
cos d' , 
= 0. . 
. (158) 
(1 + cos 2 J" 
— 2 cos £>"cos d")^ sin r", 
O 
O 
Ul 
cy/ 
cosd", 
248. It is evident that 1 -j-7r 2 — 2&cos§ is equal to the square of the distance of the 
point P of the sphero-Cartesian from the pole of the plane of-. J with respect to the 
sphere U. Hence, if this distance be denoted by I), the first determinant of the last 
article may be written 
D sin r , cosci seed , 1, 
D' sin r ' , cos h' sec J ' , 1 , 
D"sinr", cos 'sec d", 1, 
= 0 
(159) 
