p1 = Σa, p2 = Σab, &c.
and
xn − p1xn−1 + ... = x − a·x − b. &c.
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
nan−1 − (n − 1) p1an−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 = 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, |