Let a1, a2 . . an be the reciprocals of the roots of the expression 1 − s1x+s2x2 − . . ±sn−1xn−1∓snxn, which call ϕx. We have then ϕx = (1 −a1x) (1 −a2x) . . . (1 −anx), from which the following series of values may be readily deduced, the Σ implying the sum of all the instances of the form placed under it, so that each expression is a symmetrical function of a1, a2 . . an.

The sign is positive when p is even, and negative when p is odd. A single term of the pth expression contains p factors of the form a, and n–p factors of the form 1 – a; and the expression itself is the sum of every term which can be so constructed.