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for large x from that of co(n) for large n. References to most of the literature before 1936 are given in Ford's book [2](1) which contains a systematic account of the theory with improvements and simplifications. The most important early articles are due to Barnes [1] and Watson [7]. Barnes' methods have been developed and extended by Ford [2], Hughes [31 and Newsom [5, 6], while my own work [11, 14] is perhaps more closely connected with Watson's. In the present article I carry the theory a stage further by relaxing the conditions on cO(fn) so that the class of functionsf(x) studied is considerably wider than before. We are led very naturally to this generalisation when we consider a new but closely associated problem, namely, that of deducing the asymptotic expansion for large n of c,(n), where X is any number and

The asymptotic expansion of integral functions and of the coefficients in their Taylor series