1899-im] Dr Muir on the Theory of Alternants. 
93 
The Theory of Alternants in the Historical Order of its 
Development up to 1841. By Thomas Muir, LL.D. 
(Read March 19, 1900.) 
The first traces of the special functions now known as alter- 
nating functions are said by Cauchy to be discernible in certain 
work of Vandermonde’s; and if we view the functions as origina- 
ting in the study of the number of values which a function can 
assume through permutation of its variables,* such an early date 
may in a certain sense be justifiable. To all intents and purposes, 
however, the theory is a creation of Cauchy’s, and it is almost 
absolutely certain that its connection with determinants was never 
thought of until his time. 
PROXY (1795). 
[Leyons d’analyse. Considerations sur les principes de la methode 
inverse des differences. Journ. de VEc. Polyt ., i. (pp. 211- 
273) pp. 264, 265.] 
In the course of his investigations Prony comes upon a set of 
equations 
Pl + 
P'2 + • 
. . . + 
Pl/*1 + 
/ ) 2/ / '2 4" • 
. . . + 
P rP n 
2 
Pllh + 
P 2 P' 2 + • 
. . . + 
PA 
-1 
71-1 
n-l 
Pi ! + p 2 p 2 + • • • • +P n = * n _i.J 
where the coefficients of each unknown are the 0 th , 1 st , 2 nd , &c., 
powers of the same quantity, and where, therefore, the determinant 
of the set is that special form long afterwards known as the 
* The history of this subject is referred to in Serret, M. J.-A.: “Sur le 
nombre de valeurs qui peut prendre une fonction quand on y permute les 
lettres qu’elle renferme,” Liouville’s Journ. de Math ., xv. pp. 1-70 (1849). 
