500 
Proceedings of Royal Society of Edinburgh, [july 18, 
nary symmetric functions by calling them “fonctions symetriques 
alternees,” and calling the other “ fonctions symetriques per- 
manentes.” Cauchy’s view of determinants may therefore now 
be described by saying that he considered them as a special class of 
alternating symmetric functions. 
To include them, however, either the adoption of a convention 
is necessary, or an extension of the definition must be made. For 
example, a 1 b 2 ~ °f\ is not an alternating function, unless the 
elements be so related that the interchange of a x and a 2 necessitates 
the interchange of b Y and b 2 at the same time; or unless the definition 
be so worded that interchange shall refer to suffixes, not to letters. 
Cauchy selects the former course, his words being (p. 30) 
“ concevons les di verses suites de quantites 
6^25 $2* • • • • j d'n 
b\> b 2 , . ... , b n 
Ci, c 2 , . ... , c n 
tellement liees entre elles, que la transposition de deux indices 
pris dans Tune des suites, necessite la merne transposition dans 
toutes les autres ; alors, les quantites 
bit ^i» • • • ) b 2 , c 2 , . . . , b g, c 3 , ... . 
pourront etre considerees comme des fonctions semblables de 
® • e e y 
et par suite, les fonctions de 
a v c i> 
, a 2 , b 2 , c 2 , ° . • , a n , b n , c n , .... 
qui ne changeront pas de valeur, mais tout au plus de signe, en 
vertu de transpositions operees entre les indices 1, 2, 3, .... n, 
devront etre rangees parmi les fonctions symetriques de 
«!, a 2 , . . . , a n , ou, ce que revient au meme, des indices 
1, 2, 3, . . . , n. Ainsi 
a \ + + 4 a x a 2 , 
+ a 2 b 2 + « 3 & 3 + 2C!C 2 C 3 , 
af 2 + a^b 3 + affi-^ 4- a 2 b^ •{■ a 3 b 2 + af 3 , 
cos ( a x - a 2 ) cos ( a x - a 3 ) cos ( a 2 - a 3 ) , 
