MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
501 
becomes 
— L 
COS OJ, 
cos 9, 
cos <f>, 
COS iff 
COS OJ, 
-1, 
o, 
0, 
0 
cos 9, 
0, 
-1, 
cos y, 
cos /3 
cos </>, 
o, 
COS y, 
-1, 
cos a 
COS xfj, 
o, 
cos /3, 
COS a, 
— 1 
(45) 
— L 
cos y, 
COS ft 
— 
o, 
cos 9, 
cos <f>, 
cos xjj 
cos y, 
-1, 
COS a 
cos 9, 
— 1, 
cos y, 
cos /3 
cos /3, 
COS a, 
— 1 
COS <f), 
cos y, 
-L 
cos a 
cos iJj, 
cos j8, 
cos a, 
-1 
It thus appears that each of the circles, which can be drawn to cut the system 
(1, 2, 3) at angles 9, <f>, iff, cuts the orthogonal circle at one of the angles oj or tt — oj. 
It is otherwise evident that these two circles are such that one is the inverse of the 
other with respect to the circle which cuts the system (1, 2, 3) orthogonally. 
Let us denote the two circles by S, S', and their radii by p, p . We have by 
the equation 
n ( 
J 4, 1, 2, 3 N 
A 4, 1, 2 , 3y 
)=°> 
1 
1 
1 
1 
1 
5 
P 
r’ 
r i 
V 
’ T 3 ’ 
COS OJ, 
— 1 
, o, 
0, 
0 
cos 9, 
o, 
-1, 
cos y, 
cos j8 
COS (jj, 
o, 
COS y, 
-1, 
COS a 
COS ifj, 
0, 
cos j8, 
COS a, 
— 1 
(46) 
and we also get a similar equation for p. 
It appears, then, that the circle S', which is the inverse of S, cuts the orthogonal 
circle at the angle tt — oj. It may happen, however, that the roots of equation (44) 
are of opposite sign ; in this case, the circle S' evidently cuts the given system (1,2. 3) 
at angles tt — 9, tt — <f), tt — ifj, and the circle orthogonal to these at the angle oj. 
p and p being the roots of (44), we have at once by (46) 
1 1.2 cos oj 
— 0 . 
P P 
i.e., 
I _j_ JL_._I_.J_ 
^S ,4 ?’ 3 ^4,4 
r 
(47) 
