578 
MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
a relation which is clearly true when any of the spheres are replaced by planes, or 
points, or the plane at infinity. 
203. An important case is when the plane at infinity is a member of both systems 
of spheres; thus taking the two systems as (9, 1, 2, 3, 4, 5), (9, 6, 7, 8, 9, 10), we 
have 
n 
0, 1, 2, 3, 4, 5 \_ 
0,6, 7, 8, 9, 10/ ^ ’ 
whence, if the radii of the spheres be all different from zero, and they cut at angles 
w li6 , &c., we have 
0, 
l 
1 
i 
1 
1 
5 
5 
— 
1 
r i 
To 
’’3 
4 
»*5 
4 
1 
COS 
COS Ct>2,0, 
cos c^g, 
COS 
cos w 5i6 
4 
1 
cos <w li7 , 
COS 
cos Ct) 3i7 , 
COS w ^ 7 , 
COS CO-, j 
T 8 
l 
COS 
COS g j 
COS Wg g, 
COS (t)^ g, 
cos w 5i8 
4 
l 
COS g 5 
COS (W^g, 
COS Wg g, 
COS to^g, 
COS <U 5 g 
V 
cos 
COS pQj 
COS O) 3il0 , 
COS (i)^ 7 q, 
cos O) 5tlo 
204. If we 
have two systems of five 
spheres 
each, say 
(1, 2, 3, 
4, 5), (0, 7, 8, 
9, 10), 
then we have 
n /l, 2, 3, 4, 5 N 
L, 
2/n 
2 l/u 
2A 15 
Cl 
X 
c 6 > 
6’ 
it;’ 4> 
1 
\ 6, 7, 8, 9, 10y 
1, 
2A 
2p 2 > 
2li z , 
Co 
C 7> 
“ .4 
“PA “4 
1 
1, 
2/s, 
2^35 
2h 3 , 
C 3 
c s> 
-4 
1 
1, 
2/u 
2^4, 
2h„ 
C 4 
C 9> 
— 4 
5^9’ 4, 
1 
1, 
2/», 
2 <7o> 
2A 5 , 
C 5 
C 10> 
'10> 
9io> 4o> 
1 
and hence we see that, 
n 
/1,2,3,4, 5 \ 
\6, 7, 8, 9, 10 ) 
(h 2, 3, 4, 5\ 
\b 2, 3, 4, b) 
xn 
/6, 7, 8, 9, 10\ 
\6, 7, 8, 9, 10/ 
(215) 
