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
or the equivalent equations p1 = Σa, &c., we find
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
and
Observe that if, for instance, a = b, then the equations an − p1an−1 + ... = 0, bn − p1bn−1 + ... = 0 would reduce themselves to a single relation, which would not of itself express that a was a double root,—that is, that (x − a)2 was a factor of xn − p1xn−1 +, &c; but by considering b as the limit of a + h, h indefinitely small, we obtain a second equation
which, with the first, expresses that a is a double root; and then the whole system of equations leads as before to the equations p1 = Σa, &c. But the existence of a double root implies a certain relation between the coefficients; the general case is when the roots are all unequal.
We have then the theorem that every rational symmetrical function of the roots is a rational function of the coefficients. This is an easy consequence from the less general theorem, every rational and integral symmetrical function of the roots is a rational and integral function of the coefficients.
In particular, the sums of the powers Σa2, Σa3, &c., are rational and integral functions of the coefficients.
The process originally employed for the expression of other functions
Σaαbβ, &c., in terms of the coefficients is to make them depend upon the sums of powers: for instance,Σaαbβ = ΣaαΣaβ − Σaα+β; but this is very objectionable; the true theory consists in showing that we have systems of equations
| p1 | = Σa, | |
| p2 | =Σab, | |
| p12 | = Σa2 + 2Σab, | |
| p3 | = Σabc, | |
| p1p2 | = Σa2b + 3Σabc, | |
| p13 | = Σa3 + 3Σa2b + 6Σabc, |
where in each system there are precisely as many equations as there are root-functions on the right-hand side—e.g. 3 equations and 3 functions Σabc, Σa2b Σa3. Hence in each system the root-functions can be determined linearly in terms of the powers and products of the coefficients:
| Σab | = p2, | |
| Σa2 | = p12 − 2p2, | |
| Σabc | = p3, | |
| Σa2b | =p1p2 − 3p3, | |
| Σa3 | = p13 − 3p1p2 + 3p3, |
and so on. The other process, if applied consistently, would derive the originally assumed value Σab = p2, from the two equations Σa = p, Σa2 = p12 − 2p2; i.e. we have 2Σab = Σa·Σa − Σa2,= p12 − (p12 − 2p2), = 2p2.
13. It is convenient to mention here the theorem that, x being determined as above by an equation of the order n, any rational and integral function whatever of x, or more generally any rational function which does not become infinite in virtue of the equation itself, can be expressed as a rational and integral function of x, of the order n − 1, the coefficients being rational functions of the coefficients of the equation. Thus the equation gives xn a function of the form in question; multiplying each side by x, and on the right-hand side writing for xn its foregoing value, we have xn+1, a function of the form in question; and the like for any higher power of x, and therefore also for any rational and integral function of x. The proof in the case of a rational non-integral function is somewhat more complicated. The final result is of the form φ(x)/ψ(x) = I(x), or say φ(x) − ψ(x)I(x) = 0, where φ, ψ, I are rational and integral functions; in other words, this equation, being true if only ƒ(x) = 0, can only be so by reason that the left-hand side contains ƒ(x) as a factor, or we must have identically φ(x) − ψ(x)I(x) = M(x)ƒ(x). And it is, moreover, clear that the equation φ(x)/ψ(x) = I(x), being satisfied if only ƒ(x) = 0, must be satisfied by each root of the equation.
From the theorem that a rational symmetrical function of the roots is expressible in terms of the coefficients, it at once follows that it is possible to determine an equation (of an assignable order) having for its roots the several values of any given (unsymmetrical) function of the roots of the given equation. For example, in the case of a quartic equation, roots (a, b, c, d ), it is possible to find an equation having the roots ab, ac, ad, bc, bd, cd (being therefore a sextic equation): viz. in the product
the coefficients of the several powers of y will be symmetrical functions of a, b, c, d and therefore rational and integral functions of the coefficients of the quartic equation; hence, supposing the product so expressed, and equating it to zero, we have the required sextic equation. In the same manner can be found the sextic equation having the roots (a − b)2, (a − c)2, (a − d )2, (b − c)2, (b − d )2, (c − d )2, which is the equation of differences previously referred to; and similarly we obtain the equation of differences for a given equation of any order. Again, the equation sought for may be that having for its n roots the given rational functions φ(a), φ(b), ... of the several roots of the given equation. Any such rational function can (as was shown) be expressed as a rational and integral function of the order n − 1; and, retaining x in place of any one of the roots, the problem is to find y from the equations xn − p1xn−1 ... = 0, and y = M0xn−1 + M1xn−2 + ..., or, what is the same thing, from these two equations to eliminate x. This is in fact E. W. Tschirnhausen’s transformation (1683).
14. In connexion with what precedes, the question arises as to the number of values (obtained by permutations of the roots) of given unsymmetrical functions of the roots, or say of a given set of letters: for instance, with roots or letters (a, b, c, d ) as before, how many values are there of the function ab + cd, or better, how many functions are there of this form? The answer is 3, viz. ab + cd, ac + bd, ad + bc; or again we may ask whether, in the case of a given number of letters, there exist functions with a given number of values, 3-valued, 4-valued functions, &c.
It is at once seen that for any given number of letters there exist 2–valued functions; the product of the differences of the letters is such a function; however the letters are interchanged, it alters only its sign; or say the two values are Δ and −Δ. And if P, Q are symmetrical functions of the letters, then the general form of such a function is P + QΔ; this has only the two values P + QΔ, P − QΔ.
In the case of 4 letters there exist (as appears above) 3-valued functions: but in the case of 5 letters there does not exist any 3-valued or 4-valued function; and the only 5-valued functions are those which are symmetrical in regard to four of the letters, and can thus be expressed in terms of one letter and of symmetrical functions of all the letters. These last theorems present themselves in the demonstration of the non-existence of a solution of a quintic equation by radicals.
The theory is an extensive and important one, depending on the notions of substitutions and of groups (q.v.).
15. Returning to equations, we have the very important theorem that, given the value of any unsymmetrical function of the roots, e.g. in the case of a quartic equation, the function ab + cd, it is in general possible to determine rationally the value of any similar function, such as (a + b)3 + (c + d )3.
The a priori ground of this theorem may be illustrated by means of a numerical equation. Suppose that the roots of a quartic equation are 1, 2, 3, 4, then if it is given that ab + cd = 14, this in effect determines a, b to be 1, 2 and c, d to be 3, 4 (viz. a = 1, b = 2 or a = 2, b = 1,
