MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
499 
where the positive sign is to be taken with the radicals for external contact, and the 
negative for internal contact. 
Now if idenote the direct common tangent of the circles (r, s) we have 
t — TTr,s I V^^V ,s > 
and if t' rs denote the transverse common tangent, then 
t' 3 
v r, s 
— 77V,s 
— \/ 77,. 
r^s, s* 
We can then deduce at once from 
(41) 
o, 
t 3 
f l,3 5 
t 3 
h, ib 
o, 
* 2|3 3 , 
t 3 
c 2,4 
f 2 
fc 3,l 5 
6 3,2 > 
0 , 
/ 3 
£ '3,4 
t 2 
1 4,1 > 
f 2 
0 
or 
h ,2 • Ha i h ,3 ■ 
^,2 dt t 
1,4 • ^2,3 
which is Dr. Casey’s well-known formula.* 
33. When the condition in the last article is satisfied, we can find the radius of the 
tangent circle by equation (31). 
Thus, let the system (5, 6, 7, 8) be the system orthogonal to the system (1, 2, 3, 4), 
then we have 
7 r 
*,i 
1 1,5 
' 2,6 
77" 
3,7 
7r *,4 _ | 
Thus if ( x) touch each of the circles externally, we have 
1 _ 2r i | 2r 3 _| 2?- 3 2r 4 
T x 7r 15 7 To iG 7r 3)7 7T 4 g 
(42) 
34. If the system (1, 2, 3, 4) be such that four other circles (5, 6, 7, 8) can be 
drawn to touch them all. symmetrically : say each of the latter touches three of the 
former externally and one internally : then the equation 
n 
x, 1, 2, 3, 4 
y, 5, 6, 7, 8 
= 0 
becomes 
5 7T r 
* r j V " 
:2^- 
■)\r- 
r. 
[* If the circle (r) touches the circles (1, 2) in opposite senses, then q 2 must be replaced by i' L2 in 
this formula.—October, 1886.] 
3 S 2 
