582 
MR. R, LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
gives us 
12 3 4 
3, ±5, 
7T. rl .n(7 7 :)—7r r o.n( 7 7 7 :)+7j> i3 .n 
1, 2, 3, 4 
1, 3, 4, 5, 
w h2, 3, 4\ 
1, 2, 3, 4 
1, 2, 4, r v 
1, 2, 3, 4 
applying the theorem of § 210, we have 
7r, -. 
n 
1, 2, 3, 4 
1,2, 3,4 
= ^.i- n 2 3 
2, 3, 4, 5 
l, 3, 4, 5 
+77,,,. n 
3, 4, 5, 1\U 
3, 4, 5, ljj 
But if r denote the radius of the common orthogonal sphere, we have by 
equation (218) 
1, 2, 3, 4 
2, 3, 4, 5 
3, 4, 5, 1 
2 _ " [1, 2, 3, 4/ IT U 3, 4, 5/ 11 \3, 4, 5, l) _ 11 \4, 5, 1, 2)_ 11 U b 2, 3 ) . 
Y(l, 2, 3, 4) Y(2, 3, 4, 5) V(3, 4, 5, 1) Y(4, 5, 1, 2) V(5, 1, 2, 3) ’ 
4, 5, 1, 2 
5, 1, 2, 3 
and thus our equation becomes 
7r,, 5 .V(l, 2, 3, 4) = 7r* fl .V(2, 3, 4, 5) + 7 t, i2 .Y(3, 4, 5, 1) 
+ 77,; 3 .V(4, 5, 1, 2)+7r, i4 .Y(5, 1, 2, 3). 
Thus, if any five spheres have a common orthogonal sphere, and the tetrahedral 
coordinates of the centre of one of them, (5) say, referred to the tetrahedron formed 
by the centres of the other four (1, 2, 3, 4), be a, (3, y, 8, then the powers of any other 
sphere are connected by the relation 
TT.V, 5 = a .77.,, ] -f- # .77V , 2 + y . 7T. r> 3 + § • 7T.,, 4 . (222) 
As a particular case, if A, B, C, D be the centres of (I, 2, 3, 4), P any point on 
the sphere cutting these orthogonally, and O be any other point, 
OF= a .(OA 3 -r 1 3 )+/3.(OB 3 -r 3 3 )+ r .(Oa--7v) + 8.(OD 2 -r 4 2 ), 
a, f3, y, § being the tetrahedral coordinates of P referred to ABCD. 
Orthogonal Systems of Spheres. —§§ 212-214. 
212. Five spheres may be said to form an orthogonal system if they cut one 
another orthogonally. It is clear that the centres of any four must form a tetra¬ 
hedron, such that the perpendiculars from the angular points on the opposite faces 
meet in a point, viz., the centre of the fifth. 
