MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
503 
37. We also obtain some interesting results by considering another group of circles 
allied to the pair considered in § 35. Thus let 26=d+</>+i//; let p L , p\ be the radii 
of the pair of circles cutting the system (1, 2, 3) at angles 2s, if/, </>; let p. z , p\ 2 be the 
radii of the pair cutting the system at angles i/j, 2s, 6 ; and p 3 , p' ?j the radii of the 
pair cutting the system at angles </>, 0, 2s. 
As in (48), we have 
1 + t = F cos e +g cos <A+H COS ib, 
P P 
-f y =F COS 26' +G COS ip +H cos 6, 
Pi p 1 
1 + F cos if/ +G cos 26+Id cos 6, 
Pi Pi 
+ \ — F cos (b +G cos 9 +H cos 25. 
Pi P 3 
Hence, by addition, 
1 
P 
+ i ,+ I +T+ 1 + !+ T + ~ 
P Pi Pi Pi Pi Pi Pi 
= (F+G+H)(cos d+ cos COS l//+ COS 25) 
=(+b) 4 cos Ui>+i’)- ms ; 
(50) 
where It, It' are the radii of the circles which touch the given circles externally. 
Similarly 
11,1.1 1 1 1 1 
I / I - ~T" / / " ” / 
P P Pi Pi Pi Pi Pi Pi 
= (F —G—H)(cos d+ cos 25— cos <£— cos if/) 
cos i//).sin ^(r//+d).sin !(#+<£); 
where It 1; B,\ are the radii of the circles which touch the circle (1) internally and the 
ch’cles (2, 3) externally. 
38. The problem of drawing a circle to cut four given circles at equal angles has 
been discussed by Darboux, in his paper cited above, who makes the solution depend 
on that of drawing a circle to cut three given circles at given angles. Given four 
circles, say (1, 2, 3, 4), we can easily find the angle at which a circle can cut them, 
and the radius of the circle. For let this angle be + then, if we denote the circle by 
S, we have 
n 
S, 1, 2, 3, 4' 
S, 1, 2, 3, 4 
— 0 , 
