128 Mr. W. Williams on the 



which becomes Fourier's series when n = co . Grouping to- 

 gether corresponding terms in nx, and summing the series 

 so formed, he gets 



1 fir \ n Cn 



— - 1 F(v) ~d v + — S I F (v) (cos nx cos nv + sin nx sin nv) d v, 



1 f tt In r v 



=-f- F(v)3w+-S F(v) cos?i(v-.i')^v, 



2^J_ W ^ 1 J-* 



= ~r F(«)[i+'SooBn(i»-*)]a», 



'V-zr " 1 



=ij> 



sin (2/z+ l)^(v — .r) 

 sin J(u— #) 





where s ^ i^ + l)l(v-x) . § of ft + £ cos »(*-*)] , 



by a well-known summation in ordinary trigonometry. 



This final expression may be called the integral sum of the 

 series. It involves two variables, or rather it involves the 

 same variable twice over, namely, once in determining the 

 coefficients of the series, and then in assigning to the series 

 its different values. This double use of the same variable is 

 denoted by the different symbols employed in the two cases, 

 namely, v in the one case, and x in the other. We may, 

 therefore, call v the variable of integration, and x the variable 

 of summation. Denoting the expression by S a , Dirichlet's 

 problem is to determine the limiting value of S» when 7i = oo 

 for all values of x between + 7T. This limiting value we may 

 conveniently denote by S<» . 



9. As a result of his investigation, Dirichlet proved that if 

 the function F is finite, and single-valued between +7T, and 

 has only a finite number of discontinuities and maxima and 

 minima between those limits, then Fourier's series is con- 

 vergent, and tends to the value ¥(x) for all values of x 

 except those wdiich correspond to the discontinuities and the 

 limits + it ; the value of the series at a point of discontinuity 

 being the mean of the values of the parent function on either 

 side of the discontinuity and infinitely close to it, and its 

 value at either limit the mean of the values of the parent 

 function at the two limits. This result has been made the 

 subject of further inquiry by later mathematicians, notably 

 Riemann, Heine, Cantor, and P. Du Bois-Eeymond , the 

 inquiry relating to the necessity for the conditions laid down 

 by Dirichlet. For an account of these investigations, and of 



