88 
Proceedings of Royal Society of Edinburgh. [sess. 
which, on account of the zero elements, manifestly reduces to 
- 2 a 
c" - d - a 
c' - a" - a 
9 -f ~ a 
c" — a — d 
- 2 ci 
c - a" - a 
g'-f 
c -a - a" 
c —a' - a' 
-2a" 
9" ~f -a" 
9 -a ~f 
9' ~ <*' ~f 
9" -a" -f 
- 2 / 
(A). 
Now this, strange to say, is the whole matter ; for if the form 
thus reached be expanded according to binary products of the last 
row and column, we obtain Lagrange’s expression 
- 4 A 2 / + a(a +f- g) 2 + 
The form (A) is seen to be axisymmetric like Cayley’s, but the 
full regularity of its presentment is not apparent until we do away 
with the letters and denote the squared distances by 12 2 , 13 2 , 
. . . . , 45 2 . Making these changes we have, as our expression 
of the relationship between the mutual distances of five points 
1, 2, 3, 4, 5 in space, the equation 
15 2 4 - 15 2 - ll 2 
152 + 252-122 
15 2 + 35 2 _13 2 
15 2 + 45 2 - 14 2 
25 2 + 15 2 - 21 2 
25 2 + 25 2 - 22 2 
25 2 + 35 2 - 23 2 
25 2 + 45 2 - 24 2 
35 2 + 15 2 - 31 2 
352 + 252-322 
35 2 + 35 2 - 33 2 
35 2 + 45 2 - 34 2 
452+ 152 _ 412 
45 2 + 25 2 - 42 2 
45 2 + 35 2 -43 2 
452 + 452-442 
from which the fourth identity above referred to is got by dividing 
in every case the s th row and & th column by s5. 
The Relation among Four Vectors. Note on Dr Muir’s 
Paper. By Prof. Tait. 
(Read March 4, 1889.) 
A system of five points is completely determined by the vectors 
joining one of them with the other four. If a, j3, y be three of 
these, the fourth is necessarily 8 = xa+y/3 + zy. Hence any property 
characteristic of a group of five points will remain when x , y , z are 
eliminated. But we have 
