Now by the theory of symmetric functions, any symmetric functions of the mn values which satisfy the two **equations**, can be expressed in terms of the coefficient of those **equations**.

Hence, finally, the resultant is expressed in terms of the coefficients of the three **equations**, and since it is at once seen to be of degree mn in the coefficient of the third equation, by symmetry it must be of degrees np and pm in the coefficients of the first and second **equations** respectively.

The general theory of the resultant of k homogeneous **equations** in k variables presents no further difficulties when viewed in this manner.

Cayley, however, has shown that, whatever be the degrees of the three **equations**, it is possible to represent the resultant as the quotient of two determinants (Salmon, l.c. p. 89).

By solving k linear **equations** we similarly express the latter functions as linear functions of the former, and this table will also be symmetrical.