based on the transformation theory of the elliptic modular functions. Later researches have tended in the direction of a still deeper study of particular problems, culminating in the exact formulae of Rademacher [6] and his followers, or in the direction of a broader and more elementary treatment giving less precise results than (1). But a specification of properties of the generating functions sufficient for the deduction of (1), but not highly extravagant, seems to be lacking. In this paper we show how to deduce (1) from an elementary knowledge of the asymptotic behaviour of the generating functions when z(= x + iy) approaches the principal singularity z = 1 in an arbitrarily wide 'Stolz angle' I y I < A (1 x), (O < A < oo). A general result to which the method naturally leads is set out as Theorem 2 in ?3, with indications of some of the more immediate applications. Theorem 2 is deduced from Theorem 1, a Tauberian theorem for the integral