Let a 1, a 2 . . a n be the reciprocals of the roots of the expression 1 − s 1 x+s 2 x 2 − . . ±s n−1 x n−1∓s n x n , which call ϕx. We have then ϕx = (1 −a 1 x) (1 −a 2 x) . . . (1 −a n x), 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 a 1, a 2 . . a n.

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.