xn − p1xn−1 + ... ± pn = 0;
or looking at the question in a different point of view, and starting with the roots a, b, c ... as given, we express the product of the n factors x − a, x − b, ... in the foregoing form, and thus arrive at an equation of the order n having the n roots a, b, c.... In either case we have
p1 = Σa, p2 = Σab, ... pn = abc...;
i.e. regarding the coefficients p1, p2 ... pn as given, then we assume the existence of roots a, b, c, ... such that p1 = Σa, &c.; or, regarding the roots as given, then we write p1, p2, &c., to denote the functions Σa, Σab, &c.
As already explained, the epithet algebraical is not used in opposition to numerical; an algebraical equation is merely an equation wherein the coefficients are not restricted to denote, or are not explicitly considered as denoting, numbers. That the abstraction is legitimate, appears by the simplest example; in saying that the equation x² − px + q = 0 has a root x = ½ {p + √(p² − 4q) }, we mean that writing this value for x the equation becomes an identity, [½ {p + √(p² − 4q) }]² − p[½ {p + √(p² − 4q) }] + q = 0; and the verification of this identity in nowise depends upon p and q meaning numbers. But if it be asked what there is beyond numerical equations included in the term algebraical equation, or, again, what is the full extent of the meaning attributed to the term—the latter question at any rate it would be very difficult to answer; as to the former one, it may be said that the coefficients may, for instance, be symbols of operation. As regards such equations, there is certainly no proof that every equation has a root, or that an equation of the nth order has n roots; nor is it in any wise clear what the precise signification of the statement is. But it is found that the assumption of the existence of the n roots can be made without contradictory results; conclusions derived from it, if they involve the roots, rest on the same ground as the original assumption; but the conclusion may be independent of the roots altogether, and in this case it is undoubtedly valid; the reasoning, although actually conducted by aid of the assumption (and, it may be, most easily and elegantly in this manner), is really independent of the assumption. In illustration, we observe that it is allowable to express a function of p and q as follows,—that is, by means of a rational symmetrical function of a and b, this can, as a fact, be expressed as a rational function of a + b and ab; and if we prescribe that a + b and ab shall then be changed into p and q respectively, we have the required function of p, q. That is, we have F(α, β) as a representation of ƒ(p, q), obtained as if we had p = a + b, q = ab, but without in any wise assuming the existence of the a, b of these equations.
12. Starting from the equation
xn − p1xn−1 + ... = x − a·x − b. &c.
or the equivalent equations p1 = Σa, &c., we find
an − p1an−1 + ... = 0,
bn − p1bn−1 + ... = 0;
· · ·
· · ·
· · ·
(it is as satisfying these equations that a, b ... are said to be the roots of xn − p1xn−1 + ... = 0); and conversely from the last-mentioned equations, assuming that a, b ... are all different, we deduce