x^n - p1x^(n-1) + ... [+-]p_n = 0,
and we here _assume_ the existence of roots, viz. we assume that there
are n quantities a, b, c ... (in general all of them different, but
which in particular cases may become equal in sets in any manner), such
that
x^n - p1x^(n - 1) + ... [+-] p_n = 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 = [Sigma]a, p2 = [Sigma]ab, ... p_n = abc ...;
i.e. regarding the coefficients p1, p2 ... p_n as given, then we assume
the existence of roots a, b, c, ... such that p1 = [Sigma]a, &c.; or,
regarding the roots as given, then we write p1, p2, &c., to denote the
functions [Sigma]a, [Sigma]ab, &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 x^2 - px + q = 0 has a root x = 1/2{p + [root](p^2 - 4q)}, we
mean that writing this value for x the equation becomes an identity,
[1/2{p + [root](p^2 - 4q)}]^2 - p[1/2{p + [root](p^2 - 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([alpha], [beta]) as a representation of [f](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.