50G 
MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
Thus we may write this equation 
or 
tt,, 4 -tt 4 ,s A( P, 2, 3) + A( Q, 3, 1) + A(R, 1, 2) 
n,s ” A(l, 2, 3) 
p cos u> A(l, 2, 3) 
V = A(1, 2, 3) + A(P, 2, 3) +A(Q, 3, 1) + A(E, 1, 2)’ 
■ • ( 56 ) 
where a> is the angle of intersection of S and the orthogonal circle of the system 
(1, 2, 3), and p, r are the radii of the circles (S, 4). 
This formula is easily adapted for the circles (PQR), &c., by taking the area of the 
triangle (,P', 2, 3) as of opposite sign to A(P, 2, 3). 
( g 1 9 o\ 
g’ ' j; and if p hl , 
/I 2 3\ 
/r 1 o, &c. denote the minors of n 1 l5 n l 3 , &c. in H( ’ ’ ), we shall have from (55) 
\b z > 6 J 
[a® tt/ 1, 2 ’ 3 \ v Z 1 ’ 2 ’ 3 \_n/ /S ’ 1; 2 ’ v I 1 ’ 2 ’ 
^• n U 2, 3/ X ^\1, 2, 3/ n U b 2, 3/ X ^\l, 2, 
7T“ 
TT, 
TTS, S- K S, S> K S, 1? K S,%) k S,3 
K 1,S> p 1,1’ Pi,25 P 1,3 
K 2, S; p2,1? P' 2,2 J p 2,3 
/C 
3, Sj p3,n P3,2> p3,3 
But by (53) 
s,i 
= n 
S, 2, 3 
b 2,3 
= — n 
n 
S, 1, 2, 3 
S, 1, 2. 3 
jr( S, b 2, 3 
^ S, 1, 2,3 
Therefore 
7r s,s-'7D,4—'Trq.s w | tt/1’ 2, 3\] 3 = 
x n 
\b 2, 
0 , 
V 7 Pi, l> 
V^Ps,2’ 
VPs.3 
v 7 Pl,l, 
Pi, n 
Pi, 2> 
Pi,3 
V P2.2J 
p2,1’ 
Pa, 25 
p2,3 
1 <£> 
CO 
1 4. 
P 3,1’ 
P3.25 
p3,3 
But 
^S,S7T, M . 7I""4, S 
tan 3 
and 
77 4,S 
ftll =16{A(P, 2, 3)} 2 . 
• M 
