752 Proceedings of Royal Society of Edinburgh. [sess. 
pour la seconde suite, par 
Voi Vu • • • • J Vn - 1 j 
pour la troisieme suite, par 
^0’ z i z n - 1 > 
etc., . . . . • et soit 
f • • • • J »«-!■ j 2/o> 2/lJ * • • • 1 Vn- 1 ; 2 0» 2 1» * * * * J ^»-l >•••*) 
une fonction donnee de ces divers termes. Si a cette fonction 
Ton ajoute toutes celles que l’on peut en deduire, a Taide d’un 
ou de plusieurs echanges operes entre les lettres 
y,z } . . . . 
prises deux h deux, chacune des nouvelles fonctious etant prise 
avec le signe + ou avec le signe - , suivant qu’elle se deduit 
de la premiere par un nombre pair, ou par un nombre impair 
d’echanges; le- resultat de cette addition sera une somme 
alternee par rapport aux suites dont il s’agit.” 
It is a little unfortunate that this definition proceeds on different 
lines from the others, being rather indeed a rule for the formation 
of an alternating function with respect to several sets of variables 
than a definition of such a function. It would have been much 
more appropriate and instructive to have said that a function was 
called alternating with respect to two or more sets of the same number 
of variables when the interchange of each member of a set with the 
corresponding member of another set altered the function in sign 
merely. Examples like the following could then have been given 
to make the two usages of the term perfectly clear, and to show the 
exact relation between them. To illustrate the first usage, the 
expressions 
ac-bc , 
(< a - b){c - d ) , 
(a - b)(a - c)(b - c ) , 
might be taken, where ac - be is an alternating function with respect 
to the variables a, b ; (a - b)(c - d) an alternating function with respect 
to «, b, and also with respect to c, d ; and (a - b)(a - c)(b - c) an 
alternating function with respect to «, b , with respect to a, c, and 
