MR, R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
577 
whence we deduce at once, denoting the spheres by S x , S 2 , and the inverse spheres by 
S'], S'* 
S. 2 __ TTS',, S' a 
V TTo.Si’^O.Sg 's/ 7r O,S' 1 -77 ’0,S , 2 ’ 
(0) denoting the origin. 
And generally, if x, y denote either spheres, points, or planes, we infer that the 
expression 
TTx, U 
is unaltered by inverting on the point (0). 
General Theorems .—§§ 202-205. 
202 . If we have a system of six spheres, say ( 1 , 2 , 3, 4, 5, 6 ), their powers with 
respect to any other system of six spheres, say (7, 8 , 9, 10, 11, 12), are connected 
by the relation 
n 
(1, 2, 3, 4, 5, 6 \ 
\7, 8, 9, 10, 11, 12/ 
= 0 . 
For if we multiply together the matrices 
1 , 
2/n 
2 9d 
2 K, 
C 1 
c 7 , 
-1/75 
-4, 
1 
1 , 
24 
2 cj 2 , 
2 h 2 , 
Co 
C 8 i 
—.4 
— f/85 
-4. 
] 
1 , 
-A, 
2* s , 
C 3 
C 9) 
J 95 
-t/05 
-K 
1 
h 
% 
2 9i, 
2/1,, 
<h 
C 10’ 
Ao> 
”0105 
— ^ 10’ 
1 
1 , 
2 / 5 . 
%9o, 
2*5, 
G 
c m 
/in 
0115 
4l’ 
1 
1 , 
2/e, 
~9<>, 
2 K 
G 
C 13> 
t 125 
“hl25 
^12> 
1 
we have at once the equation, 
7r l,7’ 
^1,85 
7T 1,95 
77 1,105 
hm 
77 1 13 
’L’, 7’ 
85 
7r 2, 9> 
^2,105 
^san 
^*2j 12 
77 3,7’ 
7r 3,85 
95 
^ 105 
^san 
^aa 
77 i, 7’ 
7r r,85 
77 ’r, 95 
105 
^ai’ 
^,13 
^ 0,75 
77 0 , 8 ’ 
7r 5, 95 
^5, 105 
^oai’ 
^saa 
77 6,7) 
77 6,8> 
7r 6,95 
7r 0,105 
^can 
^ is 
<: 
2, 3, 4, 
8, 9, 10, 
5, 
11, 12; 
)— 0 ) 
4 E 
MDCCCLXXXVI. 
