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 using the following words: "Since the left-hand side is the sum of two functions of distinct and independent magnitudes, the form of the equation shews 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.