592 
ME. R, LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
we deduce then from (243), that 
5 - 77 - 5 , V(P, 2, 3, 4) +V(Q, 3,4,1) + V(E, 4,1, 2) + Y(S, 1, 2, 3) 
7T,,, ^ ' V(l,2,3,4) 
i.e., denoting the radius of the sphere (P, Q, It, S) by p, and its angle of intersection 
with (5) by cj, 
p cos w _ Y(l,2, 3,4) 
r 5 “ V(l, 2, 3,4) + V(P, 2, 3, 4) + Y(Q, 3, 4,1) + Y(E, 4,1, 2) + V(S, 1, 2, 3f * 
If P be without the tetrahedron formed by the centres of the spheres (1, 2, 3, 4) 
the sign of V(P, 2, 3, 4) must be changed, and thus the powers of the spheres with 
respect to the sphere orthogonal to (1, 2, 3, 4) can be written down. 
Also we can easily deduce from (244), the equation 
co j 
v 7 P i.n 
1 ^ 
Ci 
1 
Y 7 p3,3> 
V 7 Pi, 4 
A,i> 
Pl,U 
Pi, 2’ 
Pi, 35 
Pi,4 
V 7 P-0,0, 
p2,l> 
p 2, 2 j 
p2,3’ 
p2,4 
V 7 ,P 3 ,3) 
^3,1> 
^3,2’ 
p3,3’ 
p3,4 
V 7 p4, 4; 
P 4 , n 
Pr, 2? 
p4,3’ 
Pi, 4 
where p L1 , p 13 , &c. are the minors of 7r lil} 7 t 1i3 , &c. in the determinant n(^’ 
Thus p ltl = — 288{V(P, 2, 3, 4)} 3 , and thus the sign of is positive or negative 
according as P is within or without the tetrahedron formed by the centres of the 
spheres (1, 2, 3, 4). 
221. The system of spheres (1, 2, 3, 4, x) is orthogonal to the system (P, Q, P, S, 5), 
and we get some interesting theorems by aid of § 213. 
Thus, by equation (230), we have 
—■+—-+—= 0 . 
^P.l 7r R,3 7r *. 5 
(247) 
And if x denote the sphere passing through P', Q', IV, S', the inverse points of 
P. Q, P, S with respect to (5), we have 
—■+—■+ —+—+—=0. 
1 P',1 ^Q'.S 7r S', 4 ^'.5 
But since x, x' are inverse with respect to (5), 
