MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
589 
or 
COS CO 
r. 
-1, 
COS 
cos 
COS CO J 
COS CO^ -j^j 
-1, 
cos 
COS CO iA 
COS Ct) 3i ] , 
COS 
-1, 
COS OJ 3i4 
COS OJ 4)1 , 
cos co^, 
COS W 4i3 , 
— ] 
1 
] 
1 
1 
0, 
3 
3 
3 
3 
r \ 
r 2 
G 
r i 
cos 0, 
COS 0 )^ 23 
COS CO j ? gj 
COS (O lA 
cos <f>. 
COS Cx)^ i? 
-i, 
COS CO^ 35 
COS C 0 2A 
COS xjf, 
cos w 3; 13 
COS 0 ) 3 i 2 , 
-i, 
COS 6j 3i 4 
cos y, 
COS 0 )^ 23 
COS 2, 
cos &> 4)3 , 
-1 
= 0. . . (239) 
We infer, then, that two spheres can be drawn to cut the given spheres at the 
angles (0, <j>, \p, y), or else at angles supplementary to them. If the two spheres cut 
(l, 2, 3, 4) at the same angles, they cut the orthogonal sphere at supplementary 
angles, and vice versd; and evidently one is the inverse of the other with respect 
to the orthogonal sphere. 
Denoting their radii by p, p we have 
1 1 2 cos u> _. 
/“t" M ^3 
P P r -o 
and 
-+-,=F.cos 0+Chcos <£+H .cos x}j-\- K.cos y, 
(240) 
where F, G, H, K are independent of (0, <f>, y). 
We see at once, then, that the two spheres will be real, coincident, or imaginary, 
according as 
cos^ co 
is positive, zero, or negative. 
But by equation (218) we see that the sign of r-~ is opposite to that of 
i.e., opposite to the sign of 
5 
-1, 
COS Cl>2, 23 
COS Op, 3, 
COS W2,4 
COS COc 2 5 
- 1 , 
COS Ci)^3? 
COS 4, 
COS CO^ 2 3 
COS 
- 1 , 
COS W 3) 4 
cos w 41 , 
COS 0 . 
cos a> 4i3 , 
— 1 
