762 
Proceedings of Royal Society of Edinburgh. [sess. 
x i + Vi + z \ + w i 2 > ~ 2x v ~ 2 Sv ~ 2z j 
xf + yt + zf + wf, -2* 2 , -Zy* - 2s, 
+ + + -2* 6 , -2 y w 
1 , 0 , 0 , 
multiplied into 
' 2 %. 
0 , 
- 2 w v 1 
-2w 2 , 1 
- 2 W, 
0 , 0 
1 , X V Vx> z i> w i> x \ + V\ + z i + w \ 
1, X 2 , ?/ 2 , 2 2 , W 2 , £ 2 2 + 2/ 2 2 + *2 2 + W 2 2 
1, 
2/5> 
V W 5 > * 5 2 + ?/ 5 2 + 2 5 2 + “ , 5 2 
o, o , o , o , o , 
#1-*1 +01-^1 +Zl-*l +M>1 -Ml , ^1-^2 +—, ^1-^3 +•••> *l—#4 +•••» ^1-^5+—. 1 
^2—^i + . . 
, X2~X 2 +..., X 2 -X 3 +..., X 2 — X± + ..., x 2 -x 5 z +. 
^5-^1 + • 
, x 5 -x 2 +..., X 5 -X 3 +..., x 5 -x 5 +..., 1 
, 1 1 1 1,0 
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 v y v z v 
x 2 ,y 2i z 2 , &c., being the coordinates of the points 1, 2, &c., 
2 - 2 
situated arbitrarily in space, and 12 , 13 , &c., denoting the 
squares of the distances between these points, we have imme- 
diately the required relation 
0, 
l2 2 , 
I3 2 , 
1 42, 
l5 2 , 
1 
2T 2 , 
0, 
23 2 , 
24 2 , 
25 2 , 
1 
3l 2 , 
32 2 , 
0, 
3i 2 , 
35 2 , 
1 
£I 2 , 
42 2 , 
43 2 , 
0, 
45 2 , 
1 
5l 2 , 
52 2 , 
53 2 , 
54 2 , 
0, 
1 
1 , 
1, 
1, 
1 , 
1 , 
0 
0 , 
