492 
MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
we have at once 
0, 
0, 
0 , 
1, 
29 n 
2/1, 
1, 
24 
1, 
2 <7 3 , 
2 / 3 , 
16{A(1, 2, 
3)} 2 
1 
1 , 
0 , 
0 , 
0 
X 
Cl, 
—9i> 
~fi> 
1 
% 
C-2, 
-fh, 
.4 
1 
C .3 
C3, 
~9 3, 
4 
1 
( 15 ) 
where A(l, 2, 3) denotes the area of the triangle formed by the centres of the circles 
(1, 2, 3). 
Hence by (14) we see that, if r denote the radius of the circle which cuts (l, 2, 3) 
orthogonally, 
or 
•v»2- 
n 
1, 2, 3\ 
\1, 2,3/ 
52{A(1, 2, 3)} 2 4{A(1, 2, 3)} 
X 
r — 
0 *i "To Tc 
[A(l, 2, 3)} 
3 cos s. cos (s— 
-1, 
COS O) j ? 2 ? 
cos oj 1-3 
5 
COS ]^j 
- 1 , 
COS g 
COS to li3 , 
COS (i)g^ gj 
-1 
COS (s — (o 
,l)-COS (s 
— “ 1 , 2)5 
• (16) 
w 2i3 , w 3il , co li3 being the angles of intersection of the circles (1, 2, 3), and 2s being 
equal to ci>o )3 d~ w 3) 
19. Since any point on a circle may be considered as a circle of infinitely small 
radius cutting the circle orthogonally, it follows that if three circles meet in a point, 
the radius of their orthogonal circle must vanish. Hence the condition that three 
circles meet in a point is 
ri 
= 0 
(17) 
20. The radius of the orthogonal circle of the system (1, 2, 3) is infinite when 
A(l, 2, 3) = 0, i.e., when the centres of the circles lie 011 a straight line; in which case 
the orthogonal circle degenerates into the straight line through their centres. 
Four circles having a Common Orthogonal Circle. —§§ 21-24. 
21. Suppose that the system (1, 2, 3, 4) has a common orthogonal circle, x say; 
then since 
fas, 1, 2, 3, 4 N \ 
\y, 1, 2, 3, 4/ 
n 
= 0 ; 
