Let \(f_ i(x)\) be non-constant rational functions over a field K \((i=1,2,...)\). A necessary and sufficient condition is given for reducibility over K of the numerator of the sum \(\sum^{n}_{i=1}f_ i(x_ i)\) in its reduced form provided \(n\geq 3\). If char \(K\neq 2\) the condition is simple and if char K\(=0\) it is never satisfied. This generalizes the results of A. Ehrenfeucht and A. Pełczyński and of H. Tverberg concerning polynomials [see Pr. Mat. 8, 117-118 (1964; Zbl 0125.284); Q. J. Math., Oxf. II. Ser. 17, 364-366 (1966; Zbl 0145.048)]. The method is similar to that used in Part I [Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 11, 633-638 (1963; Zbl 0122.019)]. A proof for the case char \(K\neq 2\) using a different method has since been found by M. Fried [Irreducibility results for separated variables equations, to appear in J. Pure Appl. Algebra].