140 Mr. W. Williams on the 



diminish. Hence the integral 



'& j 



vanishes by the above when n = co. But this integral is 

 equal to 



*) «/ 



and therefore, since the first term and the difference of the 

 two are both infinitely small, the second term must also be 

 infinitely small. Thus in both cases the integral vanishes, 

 so that 800= ¥(x). It is interesting to note that the alter- 

 nating series which appears in Dirichlet's investigation appears 

 also here, but in a different manner. For whereas in the 

 former case it appears with terms of finite magnitude, here its 

 terms are infinitely small, because the two portions of the 

 integral S ra which lie outside the infinitely thin strip bounded 

 by +A have already been disposed of. There is therefore no 

 trouble in manipulating the series ; for all that we have to do 

 is to show that the terms decrease numerically, since the 

 series can then be neglected, the first term being infinitely 

 small. 



29. Functions having an infinite number of maxima and 

 minima are of two kinds, according as to whether the ampli- 

 tudes of the oscillations are finite or infinitely small. In the 

 former case the functions are discontinuous, for they violate 

 the definition in (24) ; in the latter case they are determinate 

 and continuous. Dirichlet maintained that all functions 

 which have only a finite number of indeterminate values, and 

 are elsewhere continuous, give rise to convergent Fourier 

 series*; but Du Bois-Reymond and Schwarz have given 

 examples of functions which are determinate and continuous, 

 but for which Fourier's series is divergent! . These functions 

 are of the class mentioned in (26) for which the integral 



JV(g+«)-r ( ,)] 8in <yW ag 



is infinite or indeterminate. 



30. The condition that F(#-f#) must not have an infinite 

 number of maxima and minima is not a necessary condition 

 in order that Fourier's series may tend to the value F(#). 

 For Lipschitz J has shown that the series may be still con- 



* Sachse's Essay, p. 19. t Ibid. p. 49. \ Ibid. p. 21. 



