(1) As is well known, the simplest form of alternant is

and the problem of multiplying it by any symmetric function of a, b, c, d, … has been in a manner fully solved.

(2) When the symmetric function is linear in each of the variables—that is to say, when it takes any of the forms Σa, Σab, Σabc, ….—the result is an alternant got from the multiplicand by increasing the last index, the last two indices, the last three indices, …. respectively by 1. Thus, writing for shortness' sake five variables only, we have

This was first established in 1825 by Schweins in his Theorie der Differenzen und Differentiale, p. 378; but it is also barely possible that it was known to Prony in 1795 (see Journ. de l'Ec. Polyt., i. pp. 264, 265), and Cauchy in 1812 (see Journ. de l'Ec. Polyt., x. pp. 49, 50).

(3) When the symmetric function is non-linear, the result takes the form not of one alternant, but of an aggregate of alternants. These cannot be so readily specified, but the mode of obtaining them can be made clear without any difficulty. Let us take the case of the function Σa3b the multiplicand being ∣a0b1c2d3∣.