MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
591 
A particular case is that of the eight tangent spheres, i.e., one inscribed, four 
escribed to one face, three escribed, to two faces, of a tetrahedron. 
The Spheres Circumscribing the Tetrahedron Formed by Four Spheres .—§§ 220, 221. 
220. Let the four given spheres be denoted by (1, 2, 3, 4), their orthogonal sphere 
by the symbol (5). It is evident that eight pairs of spheres can he drawn. Let 
P, Q, R, S be the four points in which they intersect, which lie within the tetrahedron 
formed by the centre of the spheres. Let x denote the sphere circumscribing PQRS. 
We have by equation (220), 
(x, 2, 3, 4\ fx, 3, 4, 1\ (x, 4, 1, 2\ (x, 1, 2, 3 
U 2, 3, 4/ U 3, 4, 1) \x, 4, 1, 2/ \x, 1, 2, 3 
Hence, by a theorem of determinants, 
= 0 . 
n 
n 
' 9 % 1 , 2 \ wTT /®, 1 , 2 , 3 , 4 \^ 
,1, 2)* n [x,l,2,3,4:] j 
2, 3, 4 
x, 1, 2, 3, 4\ | 
But 5 
since 
we have 
'x, 1, 2, 3 ,, „ 
W 1.2, i)\ ~ 
uh 1 ’ 2 ’ 3> 4> d )= o 
U L 2, 3, 4, 6 ’ 
or 
-"liiiMi: 
2 : 3 .’.')}•• (“*) 
Also 5 
since 
we have 
But by equation (218), 
and by equation (217), 
(x, 1, 2, 3, 4, o\_ 0 
U 1,2, 3,4,5/ ’ 
/a, 1, 2, 3, 4\ 8 /l, 2, 3, 4' 
,wn Ul,2 1 3,4 "'’‘11,2, 3, 4 
n lJ’ 2 *;J)=288ir M .{V(l, 2, 3, 4)}*; 
(244) 
n(J; 2 ’ i) = “288.{V(X, 2, 3, S)} 3 ; 
