498 
MR, R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
Whence 
o+ o L —21 -j-h -j- 
L A” r 3 2 r 4 2 \r x r 2 LA LL r : 
— + T“T + 
or if r x , r 2 , r 3 be known, 
2L3 AL AL 
1 V 
• • • (37) 
— + -f- i 2 ( - b -f —).(38) 
A A A A \AA AA A A/ 
31. This formula is given by Steiner (‘ Crelle, Jonrn. Math.,’ vol. 1, 1826), in a 
paper, in which several interesting cases of series of circles touching one another are 
discussed : two cases may be noticed here. 
Let two circles (1, 2), radii a, c, be described touching each other externally and 
touching another circle, radius r, internally, whose centre lies on the common diameter 
of the other two. Now let a circle S x be described touching the two former externally 
and the latter internally, and let a series of circles S a , S 3 , &c., be described, ail touch¬ 
ing (1, 2) externally, as well as the preceding one in the series. Let their radii be 
r l5 r 2 , &c. 
Clearly r n _ x , r n+l are the roots of the equation 
therefore 
1 1 1 l 2/1 1 1\ 2 2 2 
p 2 W 2+ C 2 p\a~^~ c r„)ac~^~r n ar n c 
1 ~4+±=&+$=* r 
V n— 
And we easily find that 
n—\ 'n 'n+\ 
ac 
acr 
* n o o 
n~r~ — ac 
■ (39) 
If, however, S\ be drawn touching (1,2) externally, and also the line joining their 
centres, we shall have, if be the n th circle of this second series 
4a cr 
(2n — l) 2 r 3 — 4ac 
(40) 
32. If the system of circles (1, 2, 3, 4) have a common tangent circle x, say, the 
equation 
may be written 
uh 2 ' 3 ’ 4 Wo 
U l 2,3, 4 / 
L 
77 L,l> 
V TT^ 
^ 77 3, 3j 
V 77 
7r l, 1> 
77 1,2’ 
7r l, 3? 
77 XA 
V' / 7T 2i 0 , 
77 % 1’ 
77 2,3> 
77 2A 
C 7T 3,3> 
’A.n 
^3,25 
3» 
—t 1 
CO 
C 7T4.4, 
h,i> 
77 4, 
77 \, 3> 
At, 4 
=0 
(41) 
