MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
511 
T, T 1 , 
t 2> 
t 3 
T, Tj, 
T 2> 
t 3 
, 
/ / 
r, r l3 
r 3 , 
t 3 
T, T 1# 
t 2> 
t 3 
/ 
/ 
/ 
/ 
T, Tj, 
r 2> 
t 3 
L 
t 3 
, 
/ 
, 
T, Tj, 
T 2, 
t 3 
T, T x , 
r 3 
are touched by circles S, S l5 S 3 , S 3 ; S', S ' l5 S' 2 , S' 3 . Also each of these circles, S say, 
touches r internally and the others externally; and if S touch the group r, t 1? t 2 , t.., 
and S' the group t, t\, t' 2 , t' 3 , then S' is the inverse of S with respect to the 
orthogonal circle of the system (1, 2, 3). These circles are usually called Dr. Hart’s 
circles. 
46. We can easily deduce from formulae ( 66 ) that 
S, S' 
s» s\ 
So, S' s 
S 3 , S' 3 
cut (1, 2, 3) at angles /3—y, y — a, a—fi ~1 
„ (1, 2, 3) ,, ft— y, y+a, «ft 
» (P 2, 3) „ /H*y, y—«, «+/3 
„ (1, 2, 3) „ fi + y, y + a, a — ft j 
• (67) 
Also, if the angles of intersection with the orthogonal circle of ( 1 , 2 , 3) of the pairs 
(S, S'), &c., be 717, < 7 r tj 7 o, 73-3, we shall have 
cos 3 rs = 4 sec s . cos (s—a) 
cos 3 tt7 1 =4 cos s . sec (s—a) 
cos 3 73- 3 =4 cos 6 ' . cos (.s-— a) 
COS 3 73 - 3=4 cos s . cos (s — a) 
cos (.s- — /3) . cos (s—y) 1 
cos (s—(3) . cos (s—y) I 
sec (s—fi) . cos (s—y) | 
cos ( $—/3) . sec (s—y) J 
Referring to § 40 we see that if the given circles ( 1 , 2 , 3) intersect in the points 
P, Q, R, P', Q', R', and the circle (P, Q, R) cut the orthogonal circle to ( 1 , 2 , 3) at 
the angle to, then 
cos 73- =2 cos co. 
Hence, if p, p be the radii of the circles S, S', and R, R' the radii of the circles 
(P, Q, R), (P', Q', R'), then 
i_I =2 
P P’ »7 
Similarly each pair of Dr. Hart’s circles is connected with a corresponding pair of 
the circles which can be drawn through the points P, Q, R, P', Q', R' by a formula 
which is analogous to that which connects the radius of the nine-points circle of a 
plane triangle with the radius of the circum-circle. 
