Summable Double Series

We have to justify the term by term differentiation of u(x, y) with respect to x and y. It has been already supposed that the u's are such that u (x, y) is convergent in Pringsheim's sense for all values of x and y; and u(x, y) is a double power series in x and y, the associated radii of convergence of which are as large as possible. Therefore the double power series converges uniformly and can be differentiated any number of times with respect to either x or y or both.