500 
MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
whence as a particular case 
1 +r+r+rYr+-+-+- 
r, r 0 
rj \r- r, r n 
: 2 ( a +; l,l + a 
r t\r, 
r„r 
2 6 
• (43) 
Chapter IV.— Circles Connected with a Triangle. 
The properties of several of these circles have been discussed in detail by the author 
in a paper published in the ‘ Quarterly Journal of Mathematics’ (vol. 21, 1885). It is 
only proposed to discuss here some of the more general cases, which can be deduced at 
once from the general equation in § 6, and which by this method admit of immediate 
extension. The first case considered, viz., the circles which cut three given circles at 
given angles, is discussed by Darboux (‘ Annales de l’ficole Normale,’ vol. 1, 1872). 
By triangle is meant the general case of a triangle formed by three circles. 
Circles cutting Three Given Circles at Given Angles. —§§ 35-38. 
35. Let (1, 2, 3) denote any given system of three circles, which cut at angles 
a, /3, y. If then S be any circle which cuts them at angles 9, <j>, \}j, we have at once 
by § 7—denoting the radius by p — 
o, 
1 
1 
1 
1 
9 
9 
9 
— 
P 
r \ 
r 3 
1 
9 
P 
-L 
cos 9, 
COS <f>, 
COS xp 
1 
*i’ 
cos 9, 
— 1, 
cos y, 
cos /3 
1 
COS <f), 
r 2 ’ 
cos y, 
— l, 
COS a 
1 
V 
COS ifj, 
cos ft, 
COS a, 
— 1 
quadratic for p ; 
and two 
circles 
caai in 
= 0 . 
• • (44) 
the given circles at the given angles. 
Now let either of these circles cut the orthogonal circle of the system (1, 2, 3), at 
the angle w—denoting the orthogonal circle by the symbol 4, and its radius by r; the 
equation 
/S, 4, 1, 2, 3\ 
