MR. R, LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
540 
where R is the radius of the sphere, and V(l, 2, 3) denotes the volume of the tetra¬ 
hedron formed by the centre of the sphere and the poles of the circles (1,2, 3). 
Hence the radius of the orthogonal circle of the system (1, 2, 3) is given by 
tan 5 " r 
sin" r Y sin" r 2 sin" r s 
36 
E 6 
{V(l, 2, 3)} 
oX 
-1, 
COS 04 , )2 , 
COS CO^ 0 
COS CO.2, 1, 
-1, 
cos co 2i g 
COS 03,1, 
COS W3 2, 
— 1 
(142) 
134. If the three circles meet in a point, r must be zero, hence 
0 9 3 > 
X, o 
n 
1 , 2 , 
= 0. 
Four Circles having a Common or Orthogonal Circle. —§§ 135, 136. 
135. Let ( x ) denote the common orthogonal circle of the system (1, 2, 3, 4), then, 
since 
it follows that we must have 
1, 2, 3, 4\ 
W, 1, 2, 3, 4/ ' 
nd 3 ’ 3 ' 4 W 
,1.2,3, ij 
(143) 
This is clearly the necessary and sufficient condition that the system (1, 2, 3, 4) 
may have a common orthogonal circle. 
136. If (5, 6, 7, 8) be any other system of circles, we must also have 
As a particular case, we have 
where (x) denotes any other circle. 
We deduce that 
n 
n 
1, 2, 3, 4 
5, 6, 7, 8 
1, 2, 3, 4 
■L 1, 2, 3 
= 0 . 
(144) 
= 0, 
„ /2, 3, 4\ _ /l, 4, 3\ /l, 2, 4\ A, 2, 3 
^ l ' n U 2,s) +lr ^- U \l, 2 , d +,r '>»- n U 2, 3 j-^*- n b, 2, 3 
But from (144) we can deduce, as in § 24, that, 
(145) 
