502 
MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
Hence the two circles S, S' are real, coincident, or imaginary, according as C ° S 0 W is 
positive, zero, or negative. 
But by § 18 the sign of r z is the same as the sign of 
n 
or cos 2 a + cos 2 /3 + cos 2 y + 2 cos a cos /3 cos y — 1. 
Hence, by (45), S, S' are real, coincident, or imaginary, according as 
-1, 
cos 6, 
COS <j), 
COS i f) 
cos 6, 
-1, 
cos y, 
COS /3 
COS (f), 
cos y, 
-L 
COS a 
cos xjj, 
cos /3, 
cos a. 
— 1 
is negative, zero, or positive, i.e., according as 
n 
/S, 1, 2, 3\ 
\S, 1,2,3 
is negative, zero, or positive. 
36. It is clear that four pairs of circles can be drawn to cut the given system 
(1, 2, 3) at angles equal to 6, <f), xf/, or their supplements. The radii of these eight 
circles are connected by a remarkable relation. Let them be denoted by p, p'; p L , p\ ; 
p. 2 , P % P?j’ p's,- By equation (46) we have 
1 . 1 
1 
1 
1 
+ n 
— 
P P 
r l 
^ 2 
cos 9, 
-1, 
cos y, 
cos /3 
COS (f>, 
cos y, 
-4, 
cos a 
COS xjj, 
cos ft, 
COS a, 
— 1 
= 0 , 
or 
similarly, we shall have, 
Hence 
- + * ,=F cos #+G cos <£+H cos i//; 
P P 
L +i=-F cos 0+G cos c?>d-H cos xb. 
Pi P i 
—+- 7 -= F cos 6— G cos d>-|-H cos \b, 
P -2 P 2 
1 + w = F cos 0+G cos </> — H cos \b. 
Ps P 3 
1 , 1 1 , 1 , 1 , 1,11 
(48) 
