MR. H. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
509 
K cos 3 co =2(1 + COS a)( 1+ cos ft(l + cos r)'] 
K cos 3 co 1 = 2(1+ cos a)( 1 — cos j8)( 1 — cos y) 
K cos 3 <y 3 =2(l — cos a)( 1 + cos /3)(l — cos y) 
K cos 2 w 3 =2(1 — cos a)(l — cos /?)(! + cos y) 
1 . 
(65) 
where 
K= cos 3 a + cos 3 /3 + cos 3 y + 2 cos a cos ft cos y — 1. 
44. The radii of the circles are given at once by the formula 
/SU.U.MN 
U 1. 2, 3, 4/ 
Thus, if p be the radius of circle touching the system (1, 2, 3) externally, we have 
1 = 0 , 
r 
0 
0 
0 
1 
1 
1 
1 
P 
L’ 
+ 
V 
1, 
— 1, 
cos 
V> 
cos 
ft 
h 
COS y, 
— ] 
cos 
a, 
1, 
cos (3, 
cos 
a, 
— 
L, 
cos 
oj, 0, 
0 , 
0 , 
or, 
1 
cos co 1 
1 
1 
p 
h >• ’ V 
V 
V 
1, 
L 
cos y, 
cos (3 
1, cosy, 
— 
1, 
COS a 
1, cos j3, 
COS a, 
— 1 
If we 
denote the 
radii of the pairs of tangent 
circles 
(P'3’ P 3 )? 
we have, by 
(49) 
1 
1 1 
1 1 
1 
1 1 
+ , — ~ + 
7 
+ 
+ / 
p 
P Pi 
Pi P 
2 
P 2 
Pi P: 
a theorem first given 
by Mr. 
Cox.—(‘ Quart. 
Journ, 
Math. 
= 0. 
• ( 66 ) 
45. Let (5, 6, 7, 8) denote a system of circles formed by taking one of each pair of 
tangent circles of the system (1, 2, 3). This can be done in sixteen ways :—We may 
show that eight of these sixteen groups are touched respectively by eight other 
circles. 
Let z he the circle which touches the group (5, 6, 7, 8): let z touch 5 internally and 
(6, 7, 8) externally ; then, since 
n(+++=o, 
\x, 5, 6, 7, 8 J 
