414] 
ON POLYZOMAL CURVES. 
569 
other two given circles; and S, S', i(z — S") for the required circle; the equations of 
the given circles will be 
(x — a) 2 + (y — a'y + {z — a") 2 = 0, 
(x - b) 2 + (y - b' ) 2 + (z- b"f = 0, 
(x - c ) 2 + (y - c' ) 2 + (z- c" ) 2 = 0, 
and that of the required circle will be 
(x - S) 2 + (y- S') 2 + (z- S") 2 = 0. 
In order that this may touch the given circles, the distances of its centre from the 
centres of the given circles must be i (S" — a"), i (S" — b"), i (S" — c") respectively; the 
conditions of contact then are 
(S - a) 2 + (S' - a') 2 + (S" - a") 2 = 0, 
(S - b) 2 + (S' — b') 2 + (S" - b") 2 = 0, 
(-Si - c) 2 + (S' — c'y + (S" - c" ) 2 = 0, 
or we have from these equations to determine S, S', S". But taking (a, a’, a"), 
(b, b', b"), (c, c', c") for the coordinates of three given points in space, and (S, S', S") 
for the coordinates of the centre of the cone-sphere through these points, we have the 
very same equations for the determination of (S, S', S"), and the identity of the two 
problems thus appears. 
I will presently give the direct analytical solution of this system of equations. 
But to obtain a solution in the form required, I remark that the equation of the 
cone-sphere in question is nothing else than the relation that exists between the 
coordinates of any four points on a cone-sphere; to find this, consider any five points in 
space, 1, 2, 3, 4, 5; and let 12, &c. denote the distances between the points 1 and 2, &c.; 
then we have between the distances of the five points the relation 
0, 
1, 
1, 
1, 
1, 
1 
1, 
0, 
12, 
9 
13', 
2 
14, 
2 
15 
1, 
2l\ 
o, 
23, 
24, 
2 
25' 
1, 
Sl\ 
32, 
o, 
34, 
35 
1, 
41 2 , 
2 
42, 
2 
43, 
0, 45 
1, 
51 2 , 
52, 
53, 
54, 0 
whence taking 5 to be the centre of the cone-sphere through the points 1, 2, 3, 4, 
we have 15 = 25 = 35 = 45 = 0; and the equation becomes 
o, 
12, 
13, 
14 
= 0 
2?, 
o, 
2 
23, 
2 
24 
2 
31, 
2 
32', 
0, 
2 
34 
2 
41, 
2 
42, 
2 
43, 
0 
C. VI. 
7 2