763 
1 888-89.] Dr T. Muir on the Theory of Determinants. 
which is easily expanded, though from the mere number of 
terms the process is somewhat long.” 
Than this no better example could have been chosen to illustrate 
what has just been said above regarding the great advantages of 
Cayley’s notation. As is well known, the result arrived at had 
been given in forms, lengthy and forbidding, many years before by 
Lagrange and Carnot. What Cayley did was to rob it of all 
disguise, by expressing it as the vanishing of an elegantly formed 
determinant ; and secondly, to show that the said determinant 
vanished because it was eight times the square* of another deter- 
minant whose zero character could not be overlooked. As has been 
implied, the result is purely algebraical, its geometrical character only 
appearing when x , y, z are taken to denote the coordinates of a point. 
The corresponding identities for the cases of four points in a 
plane and three points in a straight line are given ; and the latter 
of the two is most interestingly shown to be deducible also from 
the general theory of elimination. This is done as follows : — 
“ Let 
x u - x tit = a, x tn - x t = /3, X,-X n = y; 
then 
12 2 = y 2 , 23 2 = a 2 , 
^2 
31 =/3 2 , and a + fi + y = 0 ; 
from which a, (3, y are to be eliminated. Multiplying the last 
equation by (3y, ya, a/3, and reducing by the three first, 
O.a -f 12 2 ./3 + 
f2 2 .a +• 0./3 + 
-f 
a + (3 
31 2 .y + 
31 2 .a 
23 + 
23 .y + 
0 .y + 
y + 
from which, eliminating a, j3, y, aj3y by the general theory of 
simple equations 
a/?y = 0, 
a/?y = 0, 
a/3y = 0, 
O.a (3y = 0; 
0, 
si*. 
31 2 , 
1 , 
12 2 . 
0. 
32 2 , 
1 , 
13 2 , 
23 2 , 
0, 
1 . 
0 .” 
The first factor being 16 times the second, and the w’s unnecessary. 
