ME. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
579 
205. Again 
we 
have 
(0, b 2, 3, 4\ 
- 
o, 
0, 
o, 
0 , 
1 
X 
1, 
0, 
0 , 
0 , 
0 
\0, h 2, 3, 4/ 
h 
2/i, 
2 dn 
2/H, 
Cl 
c i> 
~/n 
“Sh 
1 
VI 
2^2, 
2k 2 , 
c. 2 
Co, 
"/s’ 
“1/27 
^'2> 
1 
1, 
2 / 3 , 
C 3 
c 3 , 
-/s» 
.*73’ 
^3-’ 
1 
1, 
2/u 
2^7-1, 
c 4 
~/n 
—9* 
-K 
1 
= 8 
-/n 
-K 
1 
2 
-K 
1 
~A> 
b,3) 
1 
-A, 
— 
1 
= 288.{Y(l f 2, 3, 4)} 2 ; .(216) 
where V(t, 2, 3, 4) denotes the volume of the tetrahedron, whose vertices are the 
centres of the spheres (1, 2, 3, 4). 
Again, let P be the common point of the spheres (1, 2, 3), then if P be denoted by 
the symbol (4) we have 
288. { V(] , 2, 3, P)} 2 =n 
0. 1, 2, 3, 4 
0, b 2, 3, 4 
0 , 
h 
1, 
1, 
1 
1 , 
Yu 
7r l,2’ 
^l.S. 
0 
h 
71 2, i> 
77 2 , 0 
7^2,3, 
0 
b 
YU 2 
hs, 3) 
*8,4’ 
0 
h 
0, 
0, 
0, 
0 
= —n 
1, 2, 3 
d, 2, 3/ 
Thus, if P be a common point of the system (1, 2, 3), 
n| 'l; 2p) = -288.{V(l,2, 3, P)} 2 . 
(217) 
4 E 2 
