Orr — Extensions of Fourier'' s and the Bessel- Fourier Theorems. 231 



ultimately of order A"^ at highest; while (70), (71), with ^ replaced by tP'\ 

 show that the only terms of this order are of the form 



S-4A"^ sin A (a? ± Xi), 



where x^ is a discontinuity in i//''. Thus, the coefficients in the integral for t//', 

 i.e. not its "proper" expansion, but that obtained by differentiating i//, or 

 integrating -^'^ under the sign of integration, are ultimately of order A"^ and 

 those in the integral for ^, of order A~^ These integrals for ip and t//' are 

 thus uniformly and absolutely convergent. 



The part of ^ which is due to i// is obtained by multiplying each element 

 of the integral by cos \ct. The integral thus remains uniformly and absolutely 

 convergent ; and also when each term is differentiated once with respect to x 

 or t. On differentiating a second time with respect to x or with respect to t, 

 the only terms which do not converge uniformly everywhere are those of the 



^A X'^ sinX(^' ± Xi) cosXddX ; 



and in these uniformity fails only near 



This is seen by making use of the identity 



2 sin \(x ± Xi) cos \ct = sm\(x ±Xi + d) + sin \{x ±Xi- d). (75) 



It may be thought that the identity (75) might quite as well have been 

 invoked throughout the solution ; but this would throw up the points under 

 discussion ; it would at the same time adopt a method inapplicable to Bessel 

 functions, whereas the above identity may be called in to settle questions 

 of convergence in the case of these functions, owing to the nature of their 

 asymptotic values. 



Thus, when each element of the integral for \p is multiplied by cos \d, a 

 new integral is obtained whose first and second differential coefficients with 

 respect to x and t can, (save for a finite number of isolated values of the 

 second differential coefficients), be obtained by differentiation under the 

 integral sign ; consequently, this new integral satisfies the differential equa- 

 tion, and also the boundary condition ; evidently, also, it is a continuous 

 function of f, since uniformly convergent, and therefore has the proper 

 initial value. 



Considering, next, the term arising from x. it is 



^ , , A cos Aa? + K sin Aic , 



= 2" I vT^? '^^ 



[cX)-\\ sin Aw - K cos Aw) sin \d . x [u] clu, (76) 



this being derived by integrating with respect to t, under the sign of integra- 

 tion, the corresponding expression for \. Here, again, the coefficient of d\ in ^ 

 contains only a finite number of terms of order as high as A'^ these arising 



[33*] 



