MR. R, LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES 
493 
we have 
whence 
77 II. 1 5 
77 j/ t O) 
77 y, 35 
7r //, 4. 
^ 1 , 1 * 
7T 1,2 J 
7r l,3j 
*1,4 
77 % 1 > 
0 5 
77 2, V 
*2,4 
77 Z, 1 > 
A;, 2’ 
77 % 3> 
3 
CO 
77 y, n 
77 4, 2; 
7r 4,3» 
*4,4 
n ( 
W 2, 3, 
Vb 2, 3, 
o' 
'-H 
= 0 ; 
(18) 
which is the condition that the system has a common orthogonal circle. 
22. It is evident also that (5, 6, 7, 8) being any other system of circles, we must 
have, since 
(x, 5, 6, 7, 8\_ 
b 2, 3, 4/ 
/b 2, 3, 4\ 
\5, 6, 7, 
(19) 
It follows by symmetry that this result must be true if either of the two systems 
(1, 2, 3, 4) or (5, 6, 7, 8) have a common orthogonal circle. 
Hence we must have 
l 1 ’ 2 ’ 3 ’ 4 '\ 
\5, 6, 7, 8/ 
2 
=n 
(1, 2, 3, 4\ 
\b 2, 3, ±) 
xn 
/5, 6, 7, 8\ 
\5, 6, 7, 8 )• 
23. We may notice that any three circles, whose centres lie on a straight line, may 
be considered as having, with the line at infinity, a common orthogonal circle. 
24. The system (1, 2, 3, 4) having a common orthogonal circle, and (5, 6, 7, 8) 
being any other system of circles, we have 
n 
1, 2, 3, 4 
5, 6, 7, 8 
= 0 
and, as in § 8, we may prove that 
As a particular case, we have 
n 
x , 1, 2, 3 
1, 2, 3, 4 
= 0 
(20) 
1. 2. 3 
"•>- n U i4)+ w '-* T \l , X» + w «* n 
1 . 2 , 
—7 T. r> 4 .n 
whence 
