Let $R(m,n)$ be defined on all the integers such that $R(m,0)=m, R(0,n)=n, R(m,n)=R(n,m)$ and $R(R(m,n),p)=R(m,R(n,p))$ for all integers $p$. Thus $R$ satisfies the *Abel associativity equation*. Let $f$ be defined for all integers, and $f(0)=0$. Then the Forsyth/Abel functional equation is

$$f(m)+f(n)=f(R(m,n))$$

The problem is to prove that $f(n)=n\theta$ where $\theta$ is a constant. It is the fundamental step in Forsyth's proof that a meromorphic function which satisfies an algebraic addition theorem and is not rational is necessarily periodic. (Theory of Functions, Chapter 18). A special case was treated in question 288554.