576 
ME, R, LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
where 7 r L3 denotes their power, r l5 r 3 their radii, cj L2 their angle of intersection, d l2 
the distance between their centres. 
The definition is due to Darboux (‘Annales d.e l’Ecole Normale Superieure,’ vol. 1, 
1872); it is also given in a paper to be found in Clifford’s ‘Mathematical Papers,’ 
p. 332 ; the date of which paper is assigned by the editor as 1868 (see note on p. 332). 
200 . If the equations of two spheres be 
,x 3 + y~ -j- z~2fx +2 gy +2 hz -f e =0, 
x ' + V ~b z 2 + 2 \f\x + 2 g'y -fi 2 h z fi- c'= 0 ; 
we have at once for their power 
tt =c+ c — 2 ff — 2 gg' — 2 hii, 
( 211 ) 
Extending the definition given in § 4, the power of a sphere and a plane may be 
defined as twice the perpendicular distance of the centre from the plane ; thus the 
power of the sphere 
* 9 +/+a 8 +^e+2^ + 2^+c=0, 
and the plane 
x cos oi.-\-y cos /3-{-z cosy— p=0, 
will be 
7 T—- 2 y> — 2 /’cos a — 2 g cos / 3 — 2 h cos y 
( 212 ) 
And similarly the power of two planes may be defined as twice the cosine of the 
angle between them. 
Also if 0 denote the plane at infinity, S any sphere, or point (considered as a sphere 
of indefinitely small radius), we shall have 
77 e, s— 1; 
and if L be any plane, v fl)L —0 ; and also TT e e —Q. 
201 . If we take the inverse spheres, with respect to a sphere whose centre is the 
origin and radius II, of the spheres 
« 3 +r + 2 2 +2/x+2f/y q _ 2 hz +c =0, 
+ y ' + z 2 + %f ' x +2 g'y +2 h 2+c =0 ; 
we see at once, that the power 7/ of the inverse spheres is connected with the 
power of the original spheres by the formula 
