504 
MR. R. LACHLAN ON SYSTEMS OP CIRCLES AND SPHERES. 
whence 
-1, 
COS <f), 
cos <f>, 
COS (fj, 
COS (fj 
COS (f), 
-1, 
COS 
COS COj^ 2 ? 
COS CJ h4 , 
cos <j>, 
cos w 2il , 
-i, 
COS C0 2j gj 
cos co 2j 4 , 
COS (fj , 
COS CO^ ^ ? 
COS <Wg i2 , 
-1, 
COS CO 3 4 
cos <f>, 
COS 6 ) 4}1? 
COS OJ^ 2 , 
cos 
-1 
See., are 
the angles 
of intersection of the 
system 
(1 
= 0; 
We have then to determine <f>, the equation 
— sec 3 <£, 
1, 
1, 
1 , 
1, 
1, 
-1, 
COS OJ 2 5 0} 
COS Cl) ]_ } 3 9 
COS COq^ 
1, 
cos w 2)1 , 
-1, 
COS Gig, 35 
COS CO 2^ ^ 
1, 
COS CO^ 
COS COg^ 2? 
- 1 , 
COS CO3 ^ 
1, 
COS 00 4> 
cos co 4j2 , 
COS W43, 
-1 
= 0 
(51) 
Also since 
/S, 1, 2, 3, 4\ . 
n \0,l,2,3,4 ’ 
we obtain at once, if p be the radius of the circle, 
sec <p 
1 
1 
1 
1 
P 
? 
T2 
V 
A 
1, 
— T 
COS coq ? 2? 
COS CO-j^ 35 
COS oj 
1, 
cos co 2j ], 5 
-1, 
COS Wo, 3, 
cos oj 
1. 
COS COg q ? 
COS W 3j2 , 
1} 
COS OJ 
1 1, 
COS <^>4, x? 
COS 
COS C0 4) gj 
— 1 
1.4 
2.4 
= 0 
• (52) 
We thus see that only one circle can be drawn ; (51) determines the angle at which 
this circle cuts the given system. 
For instance, if the four given circles cut orthogonally, we have 
sec <£=2, 
2 1,1,11 
— + d-h • 
P O J 2 »8 T i 
This circle will be imaginary, since one of the four (1, 2, 3, 4) is so. 
