TEANSACTIONS OF SECTION A. 455 



d'-u o A- d~u /, V 



.^-'^" = .?^ ...... (1) 



and it is deduced from this result that the ordinary differential equation 



^_a'«=KZ±A)« (2) 



Las for its complete integral, when ^ is a positive integer, u = AP+BQ, where A 

 and B are constants, and P and Q denote the coefficients of /t'+* in the respective 

 expansions of e''^'-^"+-^'" and e~''^''^''+-^''' in ascending powers of h. 



The present note relates to the partial differential equation (1), which is satis- 

 fied by the three following values of u ; 



r r ^w ^ 



M=| COS«^(^( -5— "2 l«?i (1.) 



"■rsf^H'-c-D}*- • • • • w 



where <^ denotes an arbitrary function. 



To verify that (i.) satisfies the differential equation (1), we find by differentia- 

 tion that 



flnd, by a double integration of parts, 



find the differential equation is therefore satisfied, subject only to the conditions 

 that the integral (i.) is finite, and that the expression 



/ hx \ , 2hxt ,,C hx \ 



«sm«^<^(^p^J-cos«^^^^,^j^<^ (.^^T^'j 



vanishes when taken between the limits and oo , 



Similarly it can be shown that (ii.) satisfies the differential equation if the 

 integi-al is finite and if 



fa.rsinrt^ 2.rfcosa<T , f,/ t"\^ 2htcosat . , f , f t'^\'\ 



vanishes when taken between the limits and 00 of t. 

 With regard to the integral (iii.), putting 



we find 



d-u r -'''""f fa^-x"' a\^ fhx\ 2ahx.,fhx\ h\„/hx\l 



and ^,=J^ e ^ -^ (^-jc?^; 



whence 



Integrating by parts, we have 



