130 
Proceedings of Royal Society of Edinburgh. [sess. 
des variables x, y, z, . . . est evidemment elle-meme une 
fonction alternee de ces variables. Reciproquement, si une 
fonction alternee de x } y, z, . . . se trouve representee par 
une fraction rationnelle dont le denominateur se reduise a 
une fonction symetrique, le numerateur de la meme fraction 
rationnelle sera necessairement une autre fonction alternee 
dex,y,z, . . ” 
This prepares us for the consideration of the alternating aggre- 
gate 
S[±f(x,y,z, . . .)] 
where / is fractional and rational, and where, although Cauchy 
does not explicitly say so, the numerator and denominator are 
integral. In regard to this he asserts that the various fractions 
which compose the aggregate may be combined into one fraction 
U/V, where V is an integral symmetric function divisible by all 
the denominators, and where, therefore, U will necessarily be an 
integral alternating function and, as such, be divisible by the 
difference-product of its variables. We are thus led to the propo- 
sition that the given alternating function of x, y, z, . . . can be 
resolved into two factors, one of which is the difference-product (P) 
of x, y, z, . . ., and the other of the form W/V, where W and Y 
are integral symmetric functions of the same variables. 
As an illustration of this, full consideration is given to the case 
where 
f( x ,y& •-••)“ (a? - a)(y - b)(z - c) . . . .’ 
the number of variables being n. The appropriate symmetric 
function Y, which is divisible by all the denominators of the 
aggregate ±f(x,y,z, . . .)] is evidently in this case 
(x - a)(x - b)(x -c)....(y- a)(y — b)(y -c)....(z- a)(z - b)(z - c ) . . . . 
or, say, 
F(x).-F(y).F(z) ; 
and the corresponding numerator U, always divisible by the differ- 
ence-product of x, y, z, . . . is in this case, because of the peculiar 
form* of the denominator of the function/, also divisible by the 
* The form is such that the result of any interchange among x, y, z, . . _ 
is attainable by a corresponding interchange among a, b, c, . . . . 
