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.