486 
ME. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
This is easily proved by multiplying together the two matrices 
i, 2 g 5 
2/l 
c l 
C 6’ 
O 
CJi 
1 
/ 6 > 
1 
2/*, 
c 7 , 
~9i’ 
~fl’ 
1 
2/v 
C 3 
C 8’ 
00 
1 
1 
2/4, 
-99’ 
1 
2/ 5> 
c 5 
C 10> 
1 
O 
' 103 
1 
equation 
^l.G’ 
77 hi’ 
^l, 9? 
ff l,10 
= 0, . 
• 
77 2, G’ 
77 2,1’ 
7r 2,8» 
77 M’ 
77 2 ,10 
*8,83 
77 'i, 9’ 
77 z, 10 
77 hG’ 
77 hi’ 
7r U, 8 ’ 
77 h 9J 
nao 
77 5, G’ 
77 5,75 
77 5,8’ 
77 o,9’ 
77 0 ,10 
( 6 ) 
which may be conveniently written 
'i, 2, 3, 4, 5 
6, 7, 8, 9, 10,' 
( 7 ) 
7. An important particular case is when 6 is a member of both systems : then we 
have 
(8) 
n U 5, 6, 7, 8, 1 °’ 
or 
0, 
1, 
1, 
1, 
1 
1, 
^1,53 
7r l,7» 
CO 
1, 
77 2, 0 ’ 
7r 2, O’ 
7Tq 7 , 
•O') / 7 
’hj.s 
l. 
^3,53 
^G’ 
77 Z,1’ 
3 
CO 
OO 
1, 
77 hh’ 
77 h 03 
77 hi’ 
77 h 8 
= 0. 
Whence, denoting the angle of intersection of the two circles ( x , y) by o)^ y , we 
have, provided none of the circles reduce to points, 
0 , 
1 
1 
1 
1 
5 
5 
J 
) 
r. 
’’6 
r l 
1 
V 
cos 
COS Ct)^ gj 
cos w li7 , 
cos o) ^ g 
1 
r 2 
COS 5 
COS Ct>2, G? 
cos an 7 , 
COS 6)^ g 
1 
V 
COS C0 3i5 , 
cos &J 3iG , 
cos oi 3)7 , 
COS Ct)g^ g 
1 
COS 0)^5, 
COS C0 4j gj 
COS Cd 4i7 , 
cos C0 4i8 
= 0 . 
( 9 ) 
