VII. Memoir on symmetric functions of the roots of systems of equations

The theory of the symmetrical functions of a single system of quantities has been investigated in a large number of memoirs, but so far, only a few attempts have been made to develop an analogous theory with regard to several systems of quantities. The chief authors are Schläfli and Cayley, both of whom have, however, restricted themselves to the outlines of the commencement of such a theory. In the theory of the single system it is found convenient to regard the quantities as the roots of an equation, since the coefficients of such an equation are themselves those particular symmetric functions of the quantities which have been variously termed fundamental, elementary, and unitary; they are fundamental because all other rational integral functions are expressible by their products of the same or lower degree; elementary because they are those which, first of all, naturally arise; unitary because their partitions are composed wholly of units.