An introduction to the modern theory of equations, by Florian Cajori.

THEORY OF EQUATIONS

The terms on the right-hand side constitute three sets, represented in our notation, respectively, by lam+qyP, Scan/fp+q,
acmpyP. Hence
sntpnL = +,m/+q p + ramf3p+ + cmppy.
Transposing and substituting for the symmetric functions whose
terms involve only two roots their values as determined by 1I
we obtain
amCCmpyq = SSpsq - Sq+pSq - S+qSp- SSp+q J- 2 Sm+p+q. II
This supposes that m, p, q are unequal. If m =p, we have
2: (ap)"y = q sJS - - S2., ~- 2 sm+qSn + 2 Sm+q.
If m =p = q, we obtain for S(ejpPyq the value 2 * 3 (c/y)" ) aaid
6 mlpqpyQ -Sm3 - 3 S2-m3Sn + 2 53m,.
Thus, tmfiPy7 may always be expressed rationally in terms of
the coefficients of the given equation.
This method may be continued to any extent, and the proof
may be given for any function mf"pPyq'....
In every symmetric function thus far considered all the terms
were of the same degree; the function was homogeneous. If
any rational symmetric integral function is not homogeneous,
then it is the sum of two or more homogeneous symmetric
integral functions, such as a +,3 + y + - a 7 + - py +y. Hence
it is evident that a rational symmetric integral function can be
expressed rationally in terms of the coefficients, whether the
function be homogeneous or not.
Finally, we' observe that no firactional function can be symmetric unless it can be so reduced that its numerator and
denominator are each integral symmetric functions. Hence,
also, a fractional rational symmetric function can be expressed
rationally in terms of the coefficients, and our theorem is
established.