What is the solution, $f(n)$, of the following functional equation:

$$mf(m)+nf(n)=(m+n+xmn)f(m+n+xmn) ,$$

where $f$ takes on integer values, $m$ and $n$ are integers, and $x$ is an indeterminate? It is a fundamental step in the proof of a famous theorem of Weierstrass that a non-rational meromorphic function which admits an algebraic addition theorem is necessarily periodic. The equation, due to A.R. Forsyth, is "solved" by him according to his following description: "Since the left-hand side is the sum of two functions of distinct and independent magnitudes, the form of the equation shows that it can be satisfied only if $x= 0$,so that..."

I am unable to follow this proof that necessarily $x=0$. If one can show it, then it is easy to show that the only solution of the functional equation is $f(n)= a$ constant.