First Appell functions of two variables were studied in detail by A. Erdélyi [Acta Math. 83, 131-164 (1950; Zbl 0041.39402)] who analysed its contour integral solutions and second Appell functions \(F\) were studied at length by P. O. M. Olsson [J. Math. Phys. 18, 1285-1294 (1977; Zbl 0348.33002)]. In this paper, the behaviour of the function \(F_4\) defined by \[ F_4(a, b; c, c'; x, y)= \sum^\infty_{m, n= 0} {(a, m+ n)(b, m+ n)\over (c, m)(c', n)m!n!} x^m y^n, |x|^{{1\over 2}}+ |y|^{{1\over 2}}< 1 \] has been investigated by studying the associated system of partial differential equations and obtaining their explicit solutions by replacing the Gauss hypergeometric functions in the expansion of the functions, in their terms, by their analytic continuations. Solutions near the point \((0, 1)\) and \((\infty, \infty)\) are obtained by using a continuation formula of \(_2F_1\) and convergence of the new series representation is investigated by Horn’s method. A list of analytic continuation formulae for the different \(F_4\) functions is given as an appendix.