452 
Proceedings of Royal Society of Edinburgh. [july is. 
17. The Theory of Determinants in the Historical Order of 
its Development. By Thomas Muir, M.A., LL.D. 
Part I. (continued). Determinants in General (1779-1812). 
Now it is at once manifest that the successive developments here 
obtained of the determinant \xyzt\ are letter by letter identical with 
the successive “ lignes ” obtained by Bdzout from the unreal product 
xyzt ; but that instead of having one arbitrary step succeeding 
another, as in the application of Bezout’s rule, there is here a fluent 
reasonableness characterising the whole process.* As for the 
peculiarities requiring elucidation in the series of special examples 
above referred to, they are seen, when looked at in this light, to be 
but matters of course. 
Not only so, but it will be found that the translation of xy into 
\xy\ &c., is an unfailing key to much that follows in Bdzout in con- 
nection with the subject. Por example, let us take the wide exten- 
sion of the rule which is expounded later on in the treatise, in a 
section headed 
* If the fact at the basis of the process were made use of nowadays, it would 
be advantageous, of course, in the first instance to simplify the determinant as 
much as possible. For example, the equations being (Bezout, p. 178) 
2x + 4?/ + 5s = 22 \ 
3a? + 5y + 2s=30 ( 
5x + 6y + 4s=43 ) , 
we might proceed as follows: — 
2 
4 
5 
-22 
0 
2 
n 
-6 
3 
5 
2 
-30 
= 
1 
1 
— 3 
-8 
5 
6 
4 
-43 
0 
-3 
-3 
9 
X 
y 
z 
t 
X 
y 
z 
t 
0 
0 
9 
0 
0 
0 
1 
0 
1 
0 
-4 
-5 
= 27 
1 
0 
0 
-5 
0 
-1 
-1 
3 
0 
-1 
0 
3 
X 
V 
z 
t 
X 
y 
z 
t 
27 { t + Oz 3 y — 5 ^| ; 
whence x = 5, y=3, z= 0 . 
