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

ELIMINATION

95

aO 0 Co 0
aC ao c1 Co
a2 aC C2 C1
0 a2 0 c2
This vanishing determinant is the resultant.
[To recall the reason for this, observe that if each member of
the four equations I is divided by A2, we have really only three
unknown quantities, viz. A, C2 - If their values, which
A2' A2' A2
may be obtained from the first three equations, are substituted
in the fourth equation, then we obtain a relation between the
coefficients of the two given equations which is the same as that
expressed by the above determinant.]
75. Sylvester's Dialytic Method of Elimination. To eliminate
x between f(x) = and F(x) = 0, equations of the degrees n
and m, defined as in ~ 73, multiply the first successively by
x~, x, x2,..., xm-l, and the second successively by x~, x', x2,, x'-1.
We obtain thus the m + n equations
f ()= 0, )xf(x) = 0, 2f(x) = 0O., "-lf() = 0,
F(x) = 0, zF(x) = 0, x2F(x) - 0,..., -F(x) = 0.
The highest power of x is m + nt - 1. If f(x) = and
F(x) = 0 have a coimmon root, it will satisfy all the m + n
equations. If the different powers of x, viz. x, x2, x',..,... n-l,
be taken as m + n —1 unknown quantities satisfying m +n
linear equations, it is evident that a relation must exist between
the coefficients of the equations. This condition of consistency
is the vanishing of the resultant.+
* The above proof of Sylvester's method is the one usually given.
Attention should be called to the fact that it is not shown there that the
different powers of x have values that are consistent.