508 
MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
Again, from equation (31), we shall have 
1 + ^+^+^=l, . 
"P, 1 7r Q,2 7r E,3 7r S,4 
oi, if S cut the system (1, 2, 3) at angles <f> l} <f>. 2 , <f> 3 , this may be written 
(62) 
1_ 2 r x cos <p x 2r 2 cos <£ 2 2r g cos <£ 3 1 
7r fit 3 r cos &)' 
p, i 
Q,2 
(63) 
This with equation (60) is sufficient to determine w and p. Also by drawing a 
figure we shall see at once that 
4*2 + ^3 + a — ^3+ ^1 + = </>l + <^2+ y = 7T. 
So that, if 2.v=:a-j-/3+y, 
<£i = 27r—(s—a); ( s ~P)'> ^ > 3=i 77 ' — (^ — r)- 
The Circles which touch three given Circles .—§§ 42-46. 
42. Let (1, 2, 3) be the given system, and let (4) denote the orthogonal circle of 
the system : then if S be the circle which touches all the circles externally, and a> its 
angle of intersection with (4), we shall have, since, 
and 
7r“s,i — 7r ],n 7r s,s 5 & c *5 
^4,4- n 
s, 1, 2, 
S, 1, 2, 
b 2, 3\ 
1, 2, S) ’ 
b 2, 3\ 
b 2, ?>) 
0, 
V 7 TTl.l, 
\ZtT% g, 
I CO 
CO 
! 
v 7 Ai. 
77"l, 1, 
77" 1,2, 
77-1,3 
V 7 TTo, 2 ’ 
TT’a, i> 
Oj 
77-2,3 
V 7 ^3,3> 
77-3,1, 
77-3,2, 
CO 
CO 
(64) 
By giving the expressions \/ tt ]A , &c., different signs, we obtain the values of cos o> 
for the other pairs of tangent circles ; and it is clear that there are four pairs of such 
circles. 
43. If a, /3, y be the angles of intersection of the system (1, 2, 3), and (o, <u ]} co s 
the angles of intersection of the four pairs of tangent circles with the orthogonal circle 
of the system, we can easily deduce from (64) the formulae 
