Page:A History of Mathematics (1893).djvu/357

 the $\scriptstyle{n}$th term, is made the basis of research. Kummer's is a criterion of the second kind. A criterion of the first kind, analogous to this, was invented by Pringsheim. From the general criteria established by Du Bois-Reymond and Pringsheim respectively, all the special criteria can be derived. The theory of Pringsheim is very complete, and offers, in addition to the criteria of the first kind and second kind, entirely new criteria of a third kind, and also generalised criteria of the second kind, which apply, however, only to series with never increasing terms. Those of the third kind rest mainly on the consideration of the limit of the difference either of consecutive terms or of their reciprocals. In the generalised criteria of the second kind he does not consider the ratio of two consecutive terms, but the ratio of any two terms however far apart, and deduces, among others, two criteria previously given by Kohn and Ermakoff respectively.

Difficult questions arose in the study of Fourier's series.[79] Cauchy was the first who felt the necessity of inquiring into its convergence. But his mode of proceeding was found by Dirichlet to be unsatisfactory. Dirichlet made the first thorough researches on this subject (Crelle, Vol. IV.). They culminate in the result that whenever the function does not become infinite, does not have an infinite number of discontinuities, and does not possess an infinite number of maxima and minima, then Fourier's series converges toward the value of that function at all places, except points of discontinuity, and there it converges toward the mean of the two boundary values. Schläfli of Bern and Du Bois-Reymond expressed doubts as to the correctness of the mean value, which were, however, not well founded. Dirichlet's conditions are sufficient, but not necessary. Lipschitz, of Bonn, proved that Fourier's series still represents the function when the number of discontinuities is infinite, and