MR. R, LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
487 
This is true if any of the circles are replaced by straight lines. 
If we take the two systems (I, 2, 3, 4), (5, 6, 7, 8) as coincident, we have an 
equation which gives us the radii of the two circles which cut three given circles at 
given angles. 
Thus, if three given circles, radii r 2 , r 3 cut at angles a, /3, y, the radii of the 
circles which cut them at angles </> l5 <f> 2 , <f> 3 respectively, are the roots of the equation 
1 
1 
1 
1 
„ y 
> 
P 
r i 
^ 2 
?3 
— E 
cos y, 
cos /3, 
COS (f)y 
cos y, 
—i, 
COS a, 
COS (f>. 2 
cos /3, 
cos a, 
-B 
COS <j )3 
cos </> l5 
COS 
COS (f) 3, 
-1 
8. Another important theorem is easily deduced by the method employed in § 6, 
thus 
1, 
2 9i> 
2 /n c i 
X 
~9o, 
Jo> 
1 
E 
2 /s> 
C 6> 
9& 
J 6> 
1 
E 
2 9s> 
2f?ji c 3 
Cj, 
~9v 
-fl> 
1 
E 
2 9*, 
C 8’ 
~9s> 
1 
hence we have at once 
(l, 2, 3, 4\ 
\5, 6, 7, 8 ] 
n, 2, 3, 4\ 
VE 2, 3, 4/ 
xn 
/5, 6, 7, 8\ 
Vo, 6, 7, 8/ • 
( 11 ) 
Chapter II. —Extension oe Results of Chapter I. 
Preliminary Remarks .—§§ 9-12. 
9. Dr. Casey has shown in his memoir “ On Bicircular Quartics,’ that any two 
conics whose equations can be put in the form S — L 2 =0, and S — M~ —0, possess a 
pair of angles which he calls their anharmonic angles, and which he shows to be 
analogous to the angle of intersection of two circles. Thus, if S', S" denote the 
results of substituting the coordinates of the poles of L, M respectively in S, and R 
