Convergence of Fourier's Series. 127 



this investigation, there are two of particular importance on 

 account of the results to which they have led, and the fact 

 that they are still the methods most generally employed in 

 mathematical text-books. These are the methods of Poisson 

 and Dirichlet. 



6. Poisson proceeds * by forming, from the given Fourier 

 series, another derived from it by multiplying each term of 

 the latter in succession by ascending powers of a quantity g 

 less than unity, and then finding to what limit this derived 

 series tends when g tends to the value 1. This method has 

 given rise to numerous and interesting investigations. In 

 particular, the method in the hands of Stokes in England led 

 to the discovery of the infinitely slow convergence of a 

 periodic series in the neighbourhood of a discontinuity. 

 Stokes showed that when a periodic series represents a dis- 

 continuous function, the rate of convergence of the series 

 increases indefinitely at the point of discontinuity, or that, 

 if a certain number of terms is required to represent the 

 continuous portion of the function to a given degree of 

 approximation, the number required to represent the function 

 to the same degree of approximation becomes greater and 

 greater as we approach a discontinuity. This important 

 discovery was published, in Dec. 1847, in a paper " On the 

 critical values of the Sums of Periodic Series " (Cambridge 

 Philosophical Society) . The subject was independently in- 

 vestigated, and the same result discovered by Seidel, and pub- 

 lished in 1848 (Journal of the Bavarian Academy, 1847-49), 

 another remarkable instance of two investigators proceeding 

 independently along the same line of inquiry. 



7. Dirichlet's method of proceeding is to form an expres- 

 sion for the sum of the first n terms of the series taken in 

 order, and to find the limit to which this tends when n is in- 

 creased indefinitely. This method was given by Dirichlet in 

 1829 {Journal de Crelle, vol. iv. p. 157), in a paper which 

 contains the first rigorous investigation into the convergency 

 of Fourier's series. The method is more direct than Poisson's, 

 it enables us to investigate the limitations more simply and 

 effectively, and it has formed the basis for most of the 

 researches that have been subsequently made into the subject. 



8. Dirichlet starts with the finite series 



1 Tt l n rir 



-pr— I F(v)B^ + — 2cosn<r| F(v) cosnv'dv 



+ -2sin?i*i'| F(r) sin nc~dv, 



1 J —TV 



* Memoires de V Academic des Sciences, 1823, p. o74. 



L2 



