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
(y − ab) (y − ac) (y − ad) (y − bc) (y − bd) (y − cd)
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.