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 x^n a function of the form in question; multiplying each side by x, and on the right-hand side writing for x^n its foregoing value, we have x^(n + 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 [phi](x)/[psi](x) = I(x), or say [phi](x) -[psi](x)I(x) = 0, where [phi], [psi], I are rational and integral functions; in other words, this equation, being true if only [f](x) = 0, can only be so by reason that the left-hand side contains [f](x) as a factor, or we must have identically [phi](x) - [psi](x)I(x) = M(x)[f](x). And it is, moreover, clear that the equation [phi](x)/[psi](x) = I(x), being satisfied if only [f](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)