CHAPTER XII.. 
ALTERNANTS FROM THE YEAR 1771 TO 1841. 
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 
originating 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. 
PEONY (1795). 
[Leçons d’analyse. Considerations sur les principes de la 
méthode inverse des différences. Journ. de VEc. Polyt, i. 
(pp. 211-273) pp. 264, 265.] 
In the course of his investigations Prony comes upon a set of 
equations 
Ml + M‘2 + • • • • + Un = z o 
PiMi + 
p 2 Ma + • • 
• • “b pnUn Z \ 
mX + 
plu 2 + • • 
. • + = «2 
riTVi + 
jn—1 ., \ 
P 2 M 2 + • • 
• • + pîTX = z n-i j 
* 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,” Journ. {de Liouville) de Math., xv. pp. 1-70 (1849).