590 
MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
Hence, by equation (238), the two spheres which cut (1, 2, 3, 4) at the angles 
(0, <f), xp, y) will be real, coincident, or imaginary, according as the expression 
-1, 
cos 0, 
cos <p, 
cos Xp, 
cos 
X 
cos 0, 
-1, 
COS 
COS CtJj g, 
cos 
w l,4 
cos <p. 
COS (0. 2i J, 
-1, 
COS O).^ 05 
cos 
<42,4 
COS Xp, 
COS 2 5 
COS 
-1, 
cos 
w 3,4 
cosy, 
COS 0)^ 2 ? 
COS 0)^ 2 j 
COS ca 43 , 
— 1 
negative: 
i i.e., according as 
x, 1 , 2 , 3 , 4 
U \x, 1 , 2 , 3 , 4 j 
is positive, zero, or negative. 
218. It is evident that eight pairs of spheres can be drawn to cut the four given 
spheres at angles whose cosines are i/q, zb *23 dr* 3 , fit/q. If we denote the radii of 
these pairs by p l5 p\ ; p. 2 , p 2 ; &c., we have by equation (240) 
“ d~ h = zh H k 1 dz G- x-2 dz H. k 3 dz Iv. /q. 
Hence we have the relation 
-d - 4 -, 
pi pi 
1 + T, 
Pi Pi 
- L d- T, 
P i P :? 
1+1; 
Pi Pi 
1 , 1 
d~ / j 
Pi Pi 
1 1 
— + / 3 
Pg P G 
1+1 
/ 5 
Pi Pi 
A + ~ 
Ps P 8 
i, 
-1, 
1, 
1, 
1, 
-1, 
1, 
1 
i, 
1, 
-1, 
1, 
1, 
1, 
-1, 
1 
i, 
1, 
1, 
-1, 
1, 
1, 
1, 
-1 
i, 
1, 
1, 
1, 
-1, 
-1, 
-1, 
— 1 
= 0. 
(241) 
219. In the case of a tetrahedron formed by four planes, the orthogonal sphere 
becomes the plane at infinity; and so the pair of spheres which cut the faces at angles 
(i 0<f) xpx ) coincide. Thus eight spheres can be drawn cutting the faces of a tetrahedron 
at angles equal or supplemental to ( 0 , <p, xp, y); and their radii are connected by the set 
of equations 
= 0. . . (242) 
1 
1 
1 
1 
1 
1 
1 
1 
Pi 
3 
Pi 
Pi 
Pi 
Pi 
Pg 
3 
P7 
Ps 
1, 
-1, 
1, 
1, 
1, 
-1, 
1, 
1 
1, 
1, 
-1, 
1, 
1, 
1, 
“I, 
1 
1, 
1, 
1, 
-1, 
1, 
1, 
1, 
— 1 
1, 
1, 
1. 
1, 
"I, 
-1, 
“I, 
-1 
