MR. R, LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
585 
becomes 
fel,2,3,4,5\ 
W, 1,2, 3,4, 5/ 
which mav be written 
7T, 
77 ■>',!/> 
77 'x, 15 
77 j\ 2 > 
TTr, 5 
77 y, n 
^l.n 
0, 
o, 
o 5 
0 
7Ti J,2> 
o, 
7To j oj 
o, 
o, 
0 
n !/,3’ 
o. 
0, 
77 3,3’ 
0, 
0 
77 y, -0 
0, 
0, 
0, 
0, 
7D.5 
77 y,^’ 
0, 
0, 
0, 
7 r 4, 5> 
0 
• 7r '/.l 1 
TT^.Tr^ 
1 77x ,3-' T7 !/,3 | 7r - t 
,4^,5 4" TTar.g 
IT 
~r 
i,i 
7To j 2 
1 
71- 3 
3 
’D.i 
(231) 
Denoting the radii of the spheres by r l5 r. 2 , r 3 , and the distance between their common 
points by e, we have— 
if x denote any sphere, radius r x , 
ct _ 7r -'.l i 77 x , 2 I Tx.S ct 7r - r . l + 7r - r ,5 j 
"— „ 2 a - „ 2 ~r „ 2 — z ° ! 
? 1 7 o 1 q 1 
'1 '2 3 
o_ 7 Tj- 3 " 77 x.i' 77 x , 5 
4;v~ = 
?v r., 3 
J 
if a? denote any plane, 
TT.r i ( TT.^o j TTj- g „ A 71” .n, - 
r l 
?v 
._ I 1 tt^s" _ / rr - i \ i - 7r * 
o I o I o 0 
O V 7 V e 
Y- • 
(232) 
(233) 
Also, taking x and y as coincident with the plane at infinity, (231) becomes 
1,1,1 4 
t>- 0+ o-f 0- o- 
r \ T 2 *3 C 
(234) 
Spheres touching one another. —§§ 215, 216. 
215. If the system of spheres (l, 2, 3, 4, 5) touch each other externally, the 
equation 
% 1, 2, 3, 4, 5 N 
\h, 1, 2, 3, 4, 5/ 
4 F 
n 
= o, 
MltCCCLXXXYI. 
