1. My only justification for presenting this paper to the Society lies in the fact that, so far as I am aware, the uniform convergence of the Fourier Series is nowhere alluded to, and far less discussed, in any English textbook; while the precautions that are necessary in differentiating the series are hardly ever mentioned even in treatises which give a very thorough treatment of its convergence. I have confined myself almost exclusively to what may be called ordinary functions, as a complete discussion of what has been done in recent years for functions that lie outside the category of “ordinary” would make the paper much too long. For information as to the original authorities, I would refer to the paper which I communicated to the Society last session On the History of the Fourier Series. It is sufficient to say here that the proof I now give is simply an adaptation of that of Heine (Kugelfunctionen, Bd. I. 57–64, Bd. II. 346–353) and of that of Neumann (Über die nach Kreis … Functionen fortsch. Entwickelungen, 26–52).