1888 - 89 .] Dr T. Muir on Distances of Points in Space. 87 
In 1841 Cayley published a paper “ On a Theorem in the Geometry 
of Position,” Camb. Math. Jour., ii. pp. 267-271, in which he 
expressed the relation in question by equating to zero the deter- 
minant — 
c" c' a g 1 
c" . c a' g' 1 
c' c . a" g" 1 
a a' a" . f 1 
9 9' 9" f • 1 
11111 . |. 
These two equations, Lagrange’s and Cayley’s, ought of course to 
agree; and Cayley, in a note just published in the Messenger of 
Mathematics , shows that not only is this the case, but that if the 
term 4 A 2 / in Lagrange’s equation be taken to the other side, the 
expression put equal to zero in the one equation is really identical 
with the corresponding expression in the other. This conclusion 
is reached by examining only the coefficient of / 2 , and showing 
that in both cases it is 
= - (c 2 + c' 2 + c" 2 - 2 c'c" - 2c" c - 2 cc') . 
A still more interesting question, it seemed to me, was the 
possibility of direct transformation of the one into the other. On 
trial I was surprised to find that this could be settled in a few lines, 
and that considerable interest attached to the transformation, 
because it brought to light a third expression different from either 
Lagrange’s or Cayley’s, and useful as a link not only between these 
two, but between either of them and a well-known fourth. 
Starting with Cayley’s determinant, and subtracting the 4 th column 
from the 1 st , 2 nd , 3 rd , 5 th columns, and thereafter the 4 th row from 
the 1 st , 2 nd , 3 rd , 5 th rows, we obtain the form 
- 2a 
c" — a - a' 
c 
-a — a" 
a 
9 
-a -f 
c" - a - a 
-2 a' 
c 
- a - a" 
a' 
9' 
-a! -f 
e 
i 
i 
c - a" - a' 
-2 a" 
a!' 
9" 
a 
a' 
a" 
/ 
e 
i 
i 
g'-f -a' 
9" 
-f -a" 
f 
-2/ 
