MR, R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
583 
If the system be denoted by (1, 2, 3, 4, 5). then (x, y) denoting any other spheres, 
we have, since 
the formula 
, x , 1, 2, 3, 4, 5 
n( -i ’ ’ , _ =o, 
yj, 1, 2, 3, 4, 5/ 
_ 7r -c,l- 7r .'/,l I I 7r %3 7r 7/,3 1 7r -<\4' 7r '/,l I 7r %3 7r '/,5 
77- ,. ^—-1-1-- -p- —p- ; 
77*1,7 77*0 2 77% 3 77% , 77% 
3,3 
4,1 
or if the radii of the spheres be r l5 r. : , r. 6 , r±, r 5 , we have, 
_ — 
— •• X h - 
Ni;, l '77*y, i 
+ 
7T, o.7T 
y,2 
77% o.7T 
?/,3 
77% i7T 
7V* 
r 
r-~ 
From this we can deduce at once the formulae— 
(223) 
—2=" d- 77 '? + 
4 r*= 
?v 
. 7Tj-,i" 
Q 
TF.r, 2 
o 
r x 
where x denotes any sphere. 
Also 
77% •> , 77% 
0= 7r ^d- 7 % J + "T + "V + "A 
d 
4 = 7 Y + "ft d- "72 +" ft +I 
r i r 2 r S 'V r 5 J 
?V 
77*,' 0 "* . 7T 
^ , 
(224) 
(225) 
where x denotes any plane. 
Again, taking x and y as coinciding with the plane at infinity, we have, since 
7r e,e = ^, 
(226) 
whence we see that one of the five spheres is imaginary. 
213. If any system of spheres, say (1, 2, 3, 4, 5), be given, then the five spheres, 
each of which is orthogonal to four of the given system, form a system, say 
(6, 7, 8, 9, 10), which may be called the “orthogonal” system of the former. 
If (x, y) denote any other spheres, the equation 
