552 
MR. R. LACHLAN ON SYSTEMS OE CIRCLES AND SPHERES. 
139. The system of circles (1, 2, 3, 4) may be called a “ semi-orthogonal” system, if 
(1, 2) cut orthogonally in the points (3, 4). Then x, y denoting any circles we have 
by 
(x, 1, 2, 3, 4N 
the equation 
nr’:’:’;/: =o, 
\l/> L 4:/ 
_1 I ' 7r -i',2- 77 ~'/.2 L 7r - r . 3 ,7r y, i ~t 77 ~.f, 4- 7r '/, 3 
TTj\ v — "T” 
7T 
1,1 
TT o o 
' 3,4 
■ (152) 
If 2e denote the arc between (3, 4) we have as a particular case, 
— 1 = cot 2 7^4- cot 2 r 2 — cosec 2 e. . . . 
(153) 
Circles touching one another. —§§ 140-144. 
140. If the four circles (1, 2, 3, 4) touch one another externally, we shall have from 
the equation 
/», 1, 2, 3, 4\ 
\», 1, 2, 3, i) U ’ 
4+ cot 2 iq-f- cot 2 r 2 -f- cot 2 r 3 + cot 2 r 4 
= 2{cot cot r 2 + cot cot ?’ 3 + cot ?q cot r 4 
+ cot r 2 cot r 3 + cot r 0 cot ?q-f- cot r 3 cot ? 
43 j 
whence 
cot r 4 ,= cot 7q-(- cot r«- (- cot r 3 dz2{cot r 2 cot r 3 + cot r 3 cot 74 + cot cot r 2 — 1 }*. (154) 
141. We may also easily extend the formulae (39) and (40) in § 31. Thus, let two 
circles (1, 2) be described, with angular radii a, y, and let another circle radius r be 
described touching these internally, and having its pole on their common diameter. 
Let S 4 be a circle touching this circle internally, and (1, 2) externally; and let a 
series of circles S 2 , S 3 , S 4 , &c. be described touching externally (1, 2) and the preceding 
one in the series; and let the radii of these circles be ?q, r. z , r 3 , &c. 
We shall have, since S„_ 1 and S, ;+1 touch S,„ 
cot r H+1 — 2 cot r n -\- cot r„_ 1 = 2(cot a+ cot y); 
whence, exactly as in § 31, 
cot r n = n 2 (cot a -f cot y) — cot r 
n 2 sin r 
sin a sin 7 
cot r. 
