MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
505 
The Circles which pass through three of the points of intersection of three Circles. 
§§ 39-41. 
39. Let the given systems of circles be denoted by (1, 2, 3), and their orthogonal 
circle by the symbol (4). Let P, Q, R, P', Q', TP be the six points of intersection of 
the three circles, the points P, Q, It being situated within the triangle formed by the 
centres of the circles. Let S be the circle which passes through the points P, Q, P. 
We have, then, 
nP 2 ' 3 )=uh 3 ' 1 ')=n(' s ’ 
VS, 2, 3/ U VS, 3, l) "VS, 1, 2/ U ' 
Hence we have by a theorem of determinants 
and 
But since 
we 
or 
have 
o \ /S, 1, 2, 3 
xn ’ 
2,3y VS, 1, 2, 3/ J 
nf s ’ 2 ’ 3 ’ Vo 
VS, 1, 2, 3,4 1 u ’ 
TT/!. 2 , 3, «\_ n /8, 1, 2, 3\_ 
7rs,4 ' n \l, 2, 3, s) n (s, 1, 2, 3/ °* 
/S, 1, 2, 3\_ r /l, 2, 3\ _/S, 2,3\ /S, 3, 1\ 
n VS, 1,2,3/ 7rs ’ 4 j r \b2,3j n (l, 2, 3 ) (l,2,3) 1 
s 1 2 
*“5 ± 5 
12 3 
and since 
n /S,l,2,3,4\ 
VS, 1, 2, 3, 4 ’ 
(53) 
(54) 
But by § 18, 
/S, 1, 2, 3\_ 3 /1, 2,3\ 
hi'Hly 1 0 -J — 77 s ’ 4,11 \i, 2, 3/ 
n 
.s, 1, 2, 3 
b 2,3 
1 2 
= -16tt, > 4 . A(l, 2, 3) , 
(55) 
hence by means of (53), (54) becomes 
7r 4,4~' 7r 4,S 
7r 4, S 
,4A(i, 2 , 3) = {n( 2 ’j)} ! +{n(^)}A^ 
But 
? 
n (, D=i6{a(P, 2 , 3 )}=. 
3 T 
1,2 
1,2 
MDCCCLXXXVI. 
