MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
497 
system discussed in § 25 ; which is of some importance. Thus, let (1, 2) be any two 
circles cutting orthogonally, and let (3, 4) be their two points of intersection ; the 
equation 
becomes 
7 T 
x ,y 
uh I- 
w i. 
2, 3, 4\ 
2, 3, 4 :) 
= 0 
yy 
77 x, 1) 
77 x, 2> 
7D, 3> 
'H’x, 4 
= 0 
77 1,V 
0, 
0, 
0 
77 y, 2, 
o, 
0, 
0 
77 y, 3’ 
0, 
o, 
0, 
77 y, 4.5 
0. 
0, 
77 4, 3? 
0 
put 
o 
II 
Tp 
CO 
2 
5 
77 y, l j 77 ■ 
i-.o-TT,/ n 
77x ,3' 77 !/, 
4."t TV Xt 4 . 77^3 
. 0 . 5 * 
TV 
1,1 
7T O O 
a particular case of which is 
0 ——-4- 1 _4 
^ W 2 „2 • 
/ 1 / o V 
(36) 
Circles touching one another .—§§ 30-34. 
30. Two circles may be said to touch externally, or internally, according as their 
angle of intersection =0 or n, i.e., if we denote the circles by x, y, then they 
touch externally if ir. K!J = + (77V, *. Tr, h y )% and internally if tv X!J — — {Tv.,^ x .Tv^.^f. 
If the four circles (1, 2, 3, 4) touch externally, we have at once, from the equation 
n (*• 2 ’ 3 > 4 Wo- 
V 1 , 2, 3, 4/ ’ 
0, 
1 
1 
i 
1 
V 
V 
r 4 
1 
-1, 
h 
I, 
1 
? 1 
1 
rC 
1, 
-i, 
1, 
1 
1 
V 
1, 
i, 
-1, 
1 
1 
1, 
i, 
1, 
— 1 
= 0 . 
3 s 
MDCCCLXXXVI. 
