Observe that if, for instance, a = b, then the equations a^n - p1a^(n - 1) + ... = 0, b^n - p1b^(n - 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 x^n - p1x^(n - 1) +, &c; but by considering b as the limit of a + h, h indefinitely small, we obtain a second equation na^(n - 1) - (n - 1)p1a^(n - 2) + ... = 0, 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 = [Sigma]a, &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 [Sigma]a^2, [Sigma]a^3, &c., are rational and integral functions of the coefficients. The process originally employed for the expression of other functions [Sigma]a^[alpha] b^[beta], &c., in terms of the coefficients is to make them depend upon the sums of powers: for instance, [Sigma]a^[alpha] b^[beta] = [Sigma]a^[alpha] [Sigma]a^[beta] - [Sigma]a^([alpha] + [beta]); but this is very objectionable; the true theory consists in showing that we have systems of equations p1 = [Sigma]a, p2 = [Sigma]ab, p1^2 = [Sigma]a^2 + 2[Sigma]ab, p3 = [Sigma]abc, p1p2 = [Sigma]a^2 b + 3[Sigma]abc, p1^3 = [Sigma]a^3 + 3[Sigma]a^2 b + 6[Sigma]abc, 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 [Sigma]abc, [Sigma]a^2 b, [Sigma]a^3. Hence in each system the root-functions can be determined linearly in terms of the powers and products of the coefficients: [Sigma]ab = p2, [Sigma]a^2 = p1^2 - 2p2, [Sigma]abc = p3, [Sigma]a^2 b = p1p2 - 3p3, [Sigma]a^3 = p1^3 - 3p1p2 + 3p3, and so on. The other process, if applied consistently, would derive the originally assumed value [Sigma]ab = p2, from the two equations [Sigma]a = p, [Sigma]a^2 = p1^2 - 2p2; i.e. we have 2[Sigma]ab = [Sigma]a.[Sigma]a - [Sigma]a^2,= p1^2 - (p1^2 - 2p2), = 2p2.