754 Proceedings of Royal Society of Edinburgh. [sess. 
being, as in the memoir of 1812, to include all kinds of the latter 
as special cases of the former. The two pages which Cauchy 
devotes to the subject are curious to read, and deserve a little 
attention. He says (p. 161) : — 
“ Concevons maintenant que la fonction 
K x >y& • • • ) 
se reduise au produit de divers facteurs dont chacun renferme 
une suite des variables 
x , &«,•••■ 
en sorte que l’on ait, par exemple, 
i{x,y,z, . . . ) = ^{x) x {y)^(z) .... 
alors, pour obtenir la somme alternee 
*= s[±#%G#(z) • . • ] 
il sufhra ...” 
and having shown the mode of formation, and given the examples 
s — < t > ( x )x(y) ~ 4 > {y)x( x )’ 
s=4,{x)x(y)<!'(z)-<t>{x)x(z)'l'{y) + • • • 
he adds 
“Les sommes de cette espece sont celles que M. Laplace a 
designees sous le nom de resultantes.” 
In regard to this the first comment clearly must be that it is not a 
little misleading. The sums referred to are only a very special class 
of those functions which Laplace called resultants ; they belong, 
in fact, to that peculiar type for which in later times the name 
alternant was coined. In the second place, Cauchy’s virtual renun- 
ciation of his own word “ determinant ” must be noted, — a renuncia- 
tion all the more curious when we consider that the word had now 
been adopted by Jacobi, and had thereby become the recognised 
term in Germany. It may be that Laplace’s word “ resultant ” had 
proved more acceptable in France, and that Cauchy merely bowed 
to the fact; but there is little or no evidence to support this.* 
* Liouville, in a paper published in the same year as Cauchy’s memoirs, 
uses resultant, but adds in a footnote, ‘ ‘ Au lieu du mot rlsultante, les geometrgs 
emploient souvent le mot determinant ” {Liouville' s Journ., vi. p. 348). 
