MR, R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
553 
So that 
sin « sin 7 sin r , 
tan r u = - 5 —~ ; -:-:—. (1 
sin 2 r— cos r sin a sm 7 v 
Similarly if S' 1} S' 2 , S' 3 , &c. be a series of circles touching (1, 2) and one another 
externally, and S' x touching the common diameter of (I, 2), we shall find 
142. Since 
tan r n — — 
4 sin « sin 7 sin r 
(2n — l ) 2 sin 2 r— 4 cos r sin « sin 7 ’ 
TTx,y-\- \/ TTx,z-TTy,y = t an r x tail r y (cOS to^fi-l), 
(156) 
we infer from § 32 that if the four circles (1, 2, 3, 4) are all touched by another circle 
externally, then we must have 
cos -Jjcoj^.cos -gW 3 , 4 rt;oos i 5 <y 1)3 .cos -g&^itcos ^a> lj4 ,.cos 0, 
co l 2 being the angle of intersection of the circles ( 1 , 2 ). 
This formula must be slightly modified if the tangent circle does not touch them all 
externally: if, for instance, the circles ( 1 , 2 ) have contact with the tangent circle of 
opposite nature, then cos|<y li2 must be replaced by sin 
143. If this condition be satisfied the radius of the circle touching the circles 
( 1 , 2 , 3, 4) may be easily found by means of § 138. Thus, let the orthogonal system 
of (1, 2, 3, 4) be (5, 6 , 7, 8 ) ; and let the contact be external in each case. 
Then since 
we shall have 
7T x. 2 , 7T, 
7T, 
+ —■+—+ — = 1 , 
^2,6 ^3,7 7r 4,8 
cot 
tan r, , tan r 9 , tan r, , tan r, 
r x — +- - H-H- 
%5 77 2,6 77 3,7 7r 4,8 
(157) 
144. If the system of circles ( 1 , 2 , 3, 4) be such that four other circles (5, 6 , 7, 8 ) 
can be drawn to touch them all, symmetrically, say let each of the latter touch one of 
the former internally and the others externally; e.g., let (5) touch (2, 3, 4) externally: 
then since 
n 
(X, 1, 2, 3, 4\ 
\y, 5, 6 , 7, 8 / 
= 0 , 
where ( x , y ) denote any other circles, we have 
— 4 7 r*, y = 22 % v n Xi 5 cot r l cot r 5 — ( 2 %^.cot r 1 )( 2 % 5 .cot r 5 ) ; 
MDCCCLXXXVI. 4 B 
