90 
Proceedings of Royal Society of Edinburgh. [sess. 
In Catalan’s notation it is 
det. (A : + A 2 + . . . + A* , A x - A 2 , A 2 - A 3 , . . . , A w _ x - A n ) 
= (-ly-'n • det. (A 1} A 2 , ... , A n ) , 
although, strange to say, it is never so formulated by him. 
A generalisation of it is next given by saying : — 
“Si la premiere ligne du systeme (B) avait renferme 
seulement p des quantites A 1 , A 2 , . . . , A n , nous aurions 
trouve, pour la determinant de ce systeme, 
A' = (- A,” 
and then there follow a number of applications to the evaluation 
of certain special determinants. 
Thus, to take the simplest example, having 
. 1 . . 
. 1 . 
. . . 1 
the theorem gives 
1 1 1 1 =(-l) 3 4A = -4. 
1-1 . . 
1-1 
1 -1 I 
The other illustrations all concern determinants of the special 
form afterwards known as “circulants ” • for example, C ( - 1 , 1 , 
1, 1), C(-l,-l, 1, 1 1), etc, C(l, 1, 1, 0), 
0(1, 1 , . . . , 1,0,0), etc. They therefore fall to be dealt 
with in a different place. 
Sarrus, P. E. (1846). 
[Finck, P. J. E. Elements d’Algebre. Seconde edition. 
iv + 544 pages. Strasbourg.] 
In the course of his discussion of the solution of a set of linear 
equations with three unknowns, the author interjects the following 
paragraph (No. 52, p. 95) : — 
