588 
MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
From the equation 
we may deduce 
n 
X, 1, 
4,5 
x, 1, 2, 3, 4, 5 
= 0 *, 
-1, 
cos 6 , 
cos <f>, 
COS l//, 
cos X, 
COS CO 
cos 6, 
— b 
COS (0 ^ 
cos <u li3 , 
COS oq )4 , 
0 
COS cf), 
COS an l5 
— b 
COS to.-,, g, 
COS too 4, 
0 
COS l j), 
COS to 31 , 
COS ( 0 ^ 2? 
-b 
COS to 3t4 , 
0 
cos X, 
cos to 4)1 , 
cos co 4?2 ) 
cos oq 3, 
-b 
0 
COS to, 
0, 
0, 
o, 
0, 
— 1 
whence we have 
-b 
COS Cl) 2j 
COS CO 4} 3j 
COS to 14 
COS Ct) 2} 1 5 
-b 
COS to 2i3 , 
COS too A 
A) 
COS (1)3^ 
COS Cl)^ 2? 
-b 
COS to 3i4 
COS to^ 4 , cos to 4j3 , cos to 4( 3 , 1 
= 0 ; 
0, 
cos 6, 
COS cf), 
COS V f), 
cos x 
cos 6 , 
-b 
COS Ct)^^ 2 j 
COS ^2,35 
COS to L4 
COS cf), 
COS to 3t 
-b 
COS to 3i 3, 
COS to 3i4 
COS if), 
COS to 3j 4 , 
COS 
-b 
COS 4 
cos X, 
COS to 44 , 
COS Ct) 4> 2 5 
COS to 4i3 , 
— 1 
(238) 
Let p denote the radius of the sphere ( x), then by the equation 
we have 
uh 1} 2 ' 3 ’ 4) ;, )=o 
\0, 1, 2, 3, 4, 5 u ’ 
1 
1 
1 
1 
1 
1 
5 
5 
^ 9 
9 
' 
p 
r l 
? 2 
? 3 
n 
cos 0, 
-b 
COS Ct) 4 ^ o> 
COS Cc), 35 
cos oq 4 , 
0 
COS cf), 
COS Ct> 2 , i5 
-b 
COS 3 ? 
COS <^45 
0 
COS xfj, 
COS to 3il , 
COS Ci)^ oj 
-b 
COS ^ 3 , 4 ,? 
0 
cos x, 
COS Oq l5 
COS Cl) 4j 2 5 
COS oq 3 , 
-b 
0 
COS to, 
0 , 
0 , 
0 , 
0 , 
— 
= 0 ; 
