MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
555 
Hence we obtain 
' 7r 4 . 4 ~ 7r -i.^ _ V(P, 2 » 3) cos ?-! + ¥(?, 3, l)co3r g +V(P, 1, 2) cos ?- 3 _ 
■nq,* Y(b 2, 3) 
or if o) be the angle of intersection of the circle PQR with the orthogonal circle to 
(1, 2, 3), we may write 
tan r, 
V(l, 2, 3) 
tan r sec co Y(l, 2, 3) + Y(P, 2, 3)cosr 1 + V(Q, 3, l)cosr 2 + Y(R, 1, 2) cos? 1 
146. Again, if p L1 , p 1>3 , &c., denote the minors of 7r L1 , 7t 1i2 &c., in II 
shall have, as in § 39, 
: • (159) 
7T x 7T 4,4 7T ^ j 
4, x 
. n 
1, 2, 3\] 2 
1, 2,3 
o, vV&s, 
V 7 pi, l> P'1,1) Pi,2* 
'3,3 
Pi,3 
h 2, 3\ 
1, 2, 3> we 
(160) 
v p2,2) p2,u p2,2* p2,3 
\/ p3,3- P.3,1’ P.3,2’ p3,3 
But 
hence 
36 
R« 
P 1 , 1 = n (2’ 3) = iT5 se ° 3 r s see 3 ^3{V(P, 2, 3 )} 3 ; 
tan 3 <» sec 3 sec 2 r 3 sec 2 r 3 tan 4 r 2 . (V(l, 2, 3)} 4 
V(P, 2, 3) cos r l5 V(Q, 3, 1) cos r 3 , V(R, 1, 2) cos i\ 
P 1 , n Pi, 2 ? 
P'3,1* p 2 , 2 * 
p3,1* p2 3* 
whence may be deduced, 
cos 3 w= sec .s.cos (s—a).cos (s—/3).cos (,?—y), . 
0 , 
V(P, 2, 3) cos jy 
V(Q, 3, 1) cos r 3 , 
Y(R, 1, 2) cos r 3 , 
Pi, 3 
p2,3 
P.3,3 
where 
(161) 
2 s=a+,8+ y . 
If the three given circles are great circles, then the imaginary circle 6 will be their 
orthogonal circle, in this case equation (160) reduces to, 
— cotsec .s.cos (s — a).cos (.s' — /3).cos (s—y ); 
the ordinary formula for finding the radius of the circum-circle of a spherical triangle 
4 B 2 
