244 Proceedings of the Royal Irish Academy. 



of riiTT : to both terms the arguments applied to the terms of a Fourier 

 series apply, and with similar results. i 



If ^ and its derivatives up to ^'-p) are finite and continuous and vanish 

 at the boundaries, and, if ^(''+1^ satisfies Dirichlet's conditions, the order 

 of the coefficient of the ^^^th term in 0(r) is ultimately less than that 

 of m~P~'^', and a discontinuity (or boundary value) at rx in ^''^'(r) gives 

 rise to a portion of the coefficient which is asymptotically a multiple of 



m-.-. sin t"'-;:);-^ ^'"{ ""7-1'"^ -^ (^ ' 1) i|('-./'-W*'"0-.)| ;. (55) 

 This is seen by replacing the asymptotic form of a term, viz. : — 



(P - «)"^ sm^ ^^ J^sm^ ^^ {ph')H{o)dg, (06) 



by its full expression and integrating by parts i? times in succession. 



The series can be differentiated term by term when ^ is finite and 

 continuous and vanishes at the boundaries, and 0' satisfies Dirichlet's 

 conditions ; and so on for successive differentiations. 



For the trigonometrical series this follows very easily by taking the 

 equation in the form (25), differentiating under the integral sign, and 

 comparing the result with that obtainable for the direct expansion of 

 <^\x). (Compare Part L, Art. 15.) 



For the Bessel series such an investigation is more difficult. The results 

 follow readily, however, from what precedes : when the series for is 

 differentiated term by term, integration by parts shows that the dominant 

 portion in the new term is, under the circumstances, of the form 



(5 -«)-^cos^ _^A_— J^^cos'^ ^^ (j)/r)if(p)f?f,. (57) 



On replacing the product of cosines by the sum of two the series is 

 exhibited as the sum of two : these are uniformly convergent everywhere 

 in the range save, in case of one, near infinities and discontinuities in ^'(p). 

 And, as the original series is uniformly convergent throughout the range, 

 term by term differentiation is thus legitimate. 



Aet. 6. Validitij of Discussion of Vibratory Motion in Si^ace hetiveen r = a, r = h, 



by Bessel-Fourier Analysis. 



I now proceed to justify to some extent the application of Bessel-Fourier 

 expansions of the type discussed to the following problem in the mathematics 

 of vibratory motion. A solution is wanted of 



d'(J)/dr~ + r'^dcp/dr + (1 - n-r'") (j> = f~d~<i)ldt~, (58) 



