2 
ON A THEOREM IN THE GEOMETRY OF POSITION. 
[1 
(This theorem admits of a generalisation which we shall not have occasion to 
make use of, and which therefore we may notice at another opportunity.) 
To find the relation that exists between the distances of five points in space. 
We have, in general, whatever x 1} y lt z 1} w lt &c. denote 
#i 2 + 2A 2 + ¿i 2 + Wi 2 , — 2«!, -2y x , -2z lt -2w x , 1 
x 2 + y% + z 2 + vj 2 — 2x 2) —2y 2 > 2^2} 2 w 2} 1 
+ y 5 2 + zi + wf, — 2x s , — 2y 5 
1 
2 z s 
2 «/, 
multiplied into 
1, 
1, 
1, 
0, 
x 1} 
x 2 , 
> 
0 
, 0 
o 
o 
2/i> 
*i, 
Wi, 
%1 + Vi + Z 1 + W 1 
V 2 , 
¿2, 
w 2 , 
x* + V? + + w 2 2 
2/s, 
z 5 , 
W 5) 
X? + y 5 + Z i + W 5 2 
0, 
o, 
o 
1 
1 
æ x - x x + y 1 -y 1 +z 1 - z 1 + w 1 -w 1 , 
2 2 
X\ X 2 “J" • •., X x X 3 4~ ... , X x X x “p ... , Xj Xij • • • , 1 
X 2 — X \ + • • * 
x 2 — X 2 +..., #2— «3+..., #2 — #4 4“ • • • > X 2 — X 5 + ..., 1 
«5-^1+ ••• 
2 2 2 
X 5 — X 2 +..., x 5 -x 3 + ..., x 5 — x i + 
1,1,1 
, x 5 -x 5 +..., 1 
, 1 , o 
Putting the w’s equal to 0, each factor of the first side of the equation vanishes, 
and therefore in this case the second side of the equation becomes equal to zero. 
Hence x 1} y 1} z 1} x 2 , y 2 , z 2 , &c. being the coordinates of the points 1, 2, &c. situated 
arbitrarily in space, and 12, 13, &c. denoting the squares of the distances between 
these points, we have immediately the required relation 
0, 
2 
12, 
2 
13, 
2 
14, 
2 
15, 
1 
2 
21, 
0, 
2 
23, 
2 
24, 
2 
25, 
1 
2 
31, 
2 
32, 
0, 
2 
34, 
2 
35, 
1 
2 
41, 
2 
42, 
2 
43, 
o, 
2 
45, 
1 
2 
51, 
2 
52, 
2 
53, 
2 
54, 
0, 
1 
1, 
1, 
1, 
1, 
1, 
0