88 Proceedings of Royal Society of Edinburgh. [sess. 
The simultaneous use of binome alterne and resultante is far 
from happy.* 
Catalan (1846). 
[Recherches sur les determinants. Bull, de V Acad. roy. ... de 
Belgique, xiii. pp. 534-555.] 
As is known, Catalan had already dealt with determinants in 
the year 1839 in a memoir regarding the change of variables in a 
multiple integral. In the paper which we have now come to he 
leads up to examples of the same kind of transformation ; but the 
greater part of it — seventeen out of the total twenty-two pages — 
is occupied with determinants pure and simple. Half of this 
amount consists of an elementary exposition of known properties, 
and calls for no remark save that what Cauchy called “produit 
principal ” or “terme indicatif ” is here called “ terme carac- 
teristique,” and that he makes constant use of the symbolism 
det. (A, B, C, . . . ) 
to stand for the determinant whose first row consists of a’s, second 
row of V s, and so on : for example, 
det.(B, A, C, . . . ) = - det. (A, B, C, . . . ) , 
det. (A, A, C, . . . ) = 0 , 
det. (A + M, B) = det. (A, B) + det.(M, B) , 
When we come to § 13, however, we find fresh ground struck. 
The exact words are : — 
“Supposons maintenant qu’etant donne le systeme — 
(A) 
A„ ) 
* Two years later we find him, in referring to a paper of Cayley’s where the 
determinant l L T S M 
i T M R 7? 
S R N C 
£ v C 
occurs, calling it a “ fonction cramerienne, ” and writing it 
r L T S | 'j 
