4 
ON A THEOREM IN THE GEOMETRY OE POSITION. 
[1 
from which, eliminating a, /3, 7, ci/3y by the general theory of simple equations, 
0, 
—2 
12, 
—2 
13, 
2 
21, 
0, 
—2 
23, 
—2 
31, 
—2 
32, 
0, 
1, 
1, 
1, 
1 =0. 
1 
1 
0 
The (additional) equation that exists between the distances of five points on a 
sphere or four points in a circle, has such a remarkable analogy with the preceding, 
that they almost require to be noticed at the same time. 
If a, /3, 7, r be the coordinates of the centre, and the radius of the sphere, and 
8 = a 2 + (3 2 + 7 2 — r 2 , we have immediately 
x i + Vi + ¿i 2 — 2a«! — 2/3y 1 — 2y Zl + 8 = 0, 
a? + yi + zi - 2 ax 5 - 2/% 5 - 27^ +8 = 6; 
whence eliminating a, ¡3, 7, 8, 
whence, multiplying by 
we have immediately 
+ y/ + z 2 , 
+ y?+z 2 , - 
- 2«i, - 
-2# 5 , - 
2yi, - 
2y B , - 
2*j, 1 
2^5, 1 
= 0; 
Xi, 
Vi, 
Zu 
^1 2 + y? + z 2 
*®5, 
2/5 > 
Z 5i 
X? + y 5 + z 2 
0, 
—2 
12, 
2 
13, 
2 
14, 
—2 
15 
= 0. 
—2 
21, 
0, 
2 
23, 
—2 
24, 
—2 
25 
—2 
31, 
—2 
32, 
0, 
—2 
34, 
—2 
35 
—2 
41, 
—2 
42, 
—2 
43, 
0, 
—2 
45 
—2 
51, 
2 
52," 
—2 
53, 
—2 
54, 
0 
equation 
for four points in 
a circle, 
—2 
2 . 12 
—2 
34 
2 2 
13 24- 
-2 . 14 
—2 —2 —2 
23 13 24-2 
readily deduce 
which is the rational, and therefore analytically thé most simple form of 
12 34+14 23=13 24. 
Euclid, B. vi., last proposition. 
(It may be remarked that the two factors we have employed in the preceding 
eliminations, only differ by a numerical factor.)