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

152

THEORY OF EQUATIONS

Next, replace the roots a,, y, y, by any other combination
ac, /, yl, y... of the roots, one root being taken from each equation. We get = a + b,1 + cy+
m - &al + bfi+ + cy/ +..*.
Similarly we obtain p27 p3, *,. The total number of p's is
equal to the total number of possible combinations, which is
* n o..., where m, n, o are respectively the degrees of the
equations. By assigning appropriate values to a, b, c,.., all
the p's will be distinct from each other.
Now construct the function F(t), thus:
F(t) - (t - p) (t- p,) (t - p,) II
F(t) is not altered if a is replaced by ac, or /3 by fi,. Hence
the coefficients of II, obtained by performing the indicated
multiplications, are symmetric functions of the roots of each
one of the equations fi(x) = 0, f2(x)= 0,...; therefore, the
coefficients are numbers in 2, and F(t) is a function in 2.
Now, any number IV in is(,,,v..) is a rational function of
a, /, y, *... Let N go over into IV1, N2, *.. for the substitutions
which convert p into pi, P2, *'. With these construct the new
function G(t), defined as follows:
G(t)-F_ (t) -- + + -Nl III
t- p t - p t- p2 )
G(t) is symmetrical with respect to the roots of f(x) = 0,
f(x) = 0, *... Hence its coefficients lie in,. For t = p, F(t)
vanishes, as appears from II. But the denominator t -p vanishes also.
Hence for t p, we have by ~ 20
G(p) = XF(p) = AT(p),
p-P
where F'(t) is the first differential coefficient of F(t).
Hence, N= ()
Z,(p)