548 
MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
imaginary circle at infinity, which we will denote by 6; and we see that we must take 
77,, jj — 1, and 
7T s a =l if S be any small circle, 
if P be any point, 
TT St6 =0 if S be any great circle. 
131. If then co SiS , be the angle of intersection of the circles S, S' we may deduce at 
once from equation (138) :— 
1, 
cot r 5 , 
cot r 6 , 
cot r 7 , 
cot r 8 
cot r l5 
COS &) 1|5 , 
cos ct)^ g. 
cos <u li7 , 
COS Ci)^ g 
cot r 2 , 
COS 
cos 
cos on 7 , 
COS &) 3) g 
cot r 3 , 
COS (t) 3 5, 
COS co 3 0, 
COS 'j'j 
COS g 
cob r 4 , 
cos &) 4)5 , 
cos w 4 , 6 , 
cos CU 47 , 
COS 0) 4>8 
= 0 . 
132. Exactly as in § 8, we can prove that 
(139) 
(140) 
Chapter IT.—Special Systems of Circles. 
Circle Cutting Three Circles Orthogonally. —§§ 133, 134. 
133. Let the circle cutting the system (1, 2, 3) be denoted by ( x ), then since 
we 
have 
'9, x, 1, 2, 3\ 
Ll \0,x,l,2,3 
TTj-.X’ IT 
0, 1, 2, 3 
e, 1 , 2,3 
=n 
L 2, 3\ 
1 , 2 , 3 ’ 
and if the equations of the circles (1, 2, 3) be of the form 
a^ + hgj+c^l, 
we have at once 
/e, i , 2 ,3 
n l 
1, 2, 3 
1, 
0, 
o, 
0 
X 
1, 
o, 
0, 
0 
L 
—a i, 
-*i> 
“Cl 
L 
«i, 
K 
Cl 
i. 
a 2> 
^2’ 
^2 
L 
«8, 
K 
C 2 
L 
— «3, 
^3’ 
“~ C 3 
L 
«3> 
^3> 
C 3 
— — 
36 2 
- sec 3 
sec 3 To 
sec 3 r 3 . 
(V(L 2, 
3)] 2 
3 
. 
. 
(141) 
