580 
MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
Chapter II. —Special Systems of Spheres. 
Sphere Cutting Four given Spheres Orthogonally. —§§ 206-208. 
206. Let x denote the sphere which cuts the four given spheres ( 1 , 2 , 3, 4) 
orthogonally ; then, since 
we have 
/ft *1.2, 3, 4\ 
Vft * 1 , 2 , 3, 4/ 
fd. 1, 2,3, 4\ /l, 2,3, 4\ 
^• n U 1, 2, 3, 4) 11 11,2, 3,4/ ’ 
whence, by equation (216), 
n 
TT, 
1, 2, 3, 4 
1, 2, 3, 4 
288.{V(l. 2, 3, 4)} 2 * 
Hence the radius of the sphere is given by 
I-nd 2 ’ 3 ’ 4 Y ! 
1, 2. 3, 4 
24.V(1, 2, 3, 4) 
(218) 
207. If the radii of the spheres ( 1 , 2 , 3, 4) be all zero, and the sides of the tetra¬ 
hedron ( 1 , 2, 3, 4) be denoted by a, b, c, a, h', c, we have at once, 
—n 
2, 3, 4\ 
0 , 
a 8 ; 
c 3 
2, 3, 
a 3 , 
0 , 
'O 
C , 
?/ 3 
b\ 
c' 3 , 
0 , 
to 
a ~ 
' c 3 , 
a' 3 , 
0 
= 2 b°b 
/o 9 /o 
TO ~ 
+ 2 c 3 c ,: - 
0 
'era 
~ + 2 a 3 a 
'4 
where 
= 16 c r.(cr— aa){cr — bb')(cr — cc'); 
2cr—aa-\-bb'-\-cc'. 
Hence the radius of the sphere circumscribing a tetrahedron is equal to 
1 {cr(cr — cia')(o — bb')(<7 — cd)Y 
Y 
(219) 
where Y denotes the volume of the tetrahedron, which agrees with the known value 
(Todhunter, ‘ Spherical Trig.,’ § 163). 
