To incorporate in this work the discoveries of the last twenty years would necessitate the rewriting of at least Chapters XII-XIX; my interest in Bessel functions, however, has waned since 1922, and I am consequently not prepared to undertake such a task to the detriment of tny other activities. In the preparation of this new edition 1 have therefore limited myself to the correction of minor errors and misprints and to the emendation of a few assertions (such as those about the unproven character of Bourget's hypo- hypothesis) which, though they may have been true in 1922, would have been definitely false had they been, made in 1941. My thanks are due to many friends for their kindness in informing me of errors which they had noticed; in.particular, I cannot miss this opportunity of expressing my gratitude to Professor J. R. Wilton for the vigilance which he must have exercised in the compilation of his list of corrigenda.