MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
547 
Q may be called the inverse point of P with respect to the circle whose radius is r, or 
simply the inverse of P with respect to O. 
It may be easily shown, if S, S' be any two small circles, by forming the equations 
of the inverse circles with respect to A (§ 125), that the expression 
7T S,S’ 
v 7 Ho, s-Wo. S' 
is invariable. 
General Theorems. —129-132. 
129. If (1, 2, 3, 4, 5), (6, 7, 8, 9, 10) denote any two systems of circles on the surface 
of a sphere, the powers of the former are connected with those of the latter by the 
identical relation 
n 
(1, 2, 3, 4, 5 \ 
\6, 7, 8, 9, 10/ 
= 0. 
This is at once proved by multiplying together the matrices, 
1, 
cq, 
K 
Cl 
> 
1, 
K 
C 6 
1, 
a 3 , 
K 
C 2 
1, 
— a 7 , 
&7, 
~ C 7 
1, 
«3> 
^3> 
C 3 
1, 
-cq, 
K 
c 8 
1, 
K 
1, 
—«9, 
K 
— c 9 
1, 
«5, 
K 
c 5 
1, 
a io- 
~ hr 
C 10 
Whence we get 
i.e. 
7r l, 65 
7r l, 7’ 
7T 1,8> 
7r l, 9> 
^1,10 
7r -2,e> 
77 2,7’ 
77 2, 8; 
77 2,9’ 
77 2, 10 
77 3,3’ 
77 3,7’ 
77 3 ,95 
^3,10 
77 4,3’ 
77 4,7’ 
77 4,8> 
7r 4,9’ 
77 4, 10 
77 5, GJ 
77 5,7 > 
77 5, 8? 
77 o,9’ 
77 o, 10 
n 
A, 2, 3, 4, 5 \ 
\6, 7, 8, 9, IQ/ 1 
= 0 
(138) 
130. It is evident that this result is true if one, or more, of the circles are great 
circles, provided that we interpreted the meaning of the symbol tt in accordance with 
the definition in § 127. Again, it is true, if the radius of any of the circles is zero. 
And we also see it is true if any circle of either system is such that the coordinates 
of its centre are zero; i.e., any circles of either system may be replaced by the 
4 a 2 
