MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
491 
the equation 
jt ( X > Z > 1 > 2 ’ 3 \_A 
n U, 2, 1, 2, 3 ) — 
gives us 
whence we have 
ITX ? 
TTx y z) 
o, 
o, 
0 
^ Z, X) 
IT*, *9 
{b-cf, 
, (c—«) 2 , 
(a 
o, 
(h~cf, 
0, 
c 2 
4’ 
6 2 
4 
o, 
(c-af, 
o 
cr 
4* 
o, 
a 2 
4 
0, 
(a— by, 
P 
4’ 
« 2 
4’ 
0 
7T r .r z 
o 
4=77% 
= 0; 
or the circle which passes through the mid-points of the sides of a triangle touches 
the inscribed circle. Similarly it can be proved to touch the escribed circles. 
Circle Cutting three Circles Orthogonally. —§§ 18-20. 
18. Let the circle cutting the system (1,2, 3) orthogonally be denoted by (a 1 ) : 
then since 
we have 
But 
te.x, i,2,,v . 
U r, 1. 2, 3,, 1 ’ 
ie.1. 2,3\_ n, 2,3 
* r ’*- U \6,l,2, s) n (l, 2, 3 
n 
6, 1, 2, 3 
\0, 1, 2, 3 
0, 
1, 
1, 
77,1, 77 
1,1’ “1,25 “1,3 
1) ^2,1’ 7r 2,2’ 77 -2 ,3 
1) 7r 3 ,l> 77 ’?„ 2’ 7r 3,3 
and if the equations of the circles be 
•? 3 + 3/ a + 2g&+%-y+c, —0, 
3 R 2 
where r=(l, 2, 3) 
(14) 
