598 
MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
which may be written 
and 
which may be written 
or 
7 . , 7 I z. I L f • 
A ' 1 3\ + 4/3 0/z, + ^ dv + ^ dp + 0<r ~ ’ 
( 256 ) 
/L 1,2,3,4,5\ , 
n ( L, 1, 2, 3, 4, 5 / * ’ 
^|+^?+^=+ 2 . 
[X, v, p, cr)—l. 
(257) 
The Sphere .—§§ 228-231. 
228. Let P be any point on the sphere S whose coordinates are (£p[,co—), then since 
77p s =0, and 
/P, 1, 2, 3, 4, 5\ 
U, 1,2, 3, 4, 5/ ’ 
we see that the equation of the sphere is 
0£ X + 0t; y + 0r " + 0a> + 0^ ° ' 
and hence the general equation of the first degree, 
ax+by+ cz-\-dw-{-ev= 0, 
will, in general, represent a sphere, whose coordinates are given by 
_9 
0\|r 07^ d\jr d\ft' d'yp 
0£ 07; 0f 007 0y 
(t. b c (l c cilc^ -|- hie.*, -f- cA’g "F 
by equation (255). 
229. Given any two spheres (^r^wTrr) (iz'yftjbizs'), their power it is, since 
(258) 
given by 
■ < 2 n={ ^+ V w +co g-+sr ^ 
ct; 
(259) 
