234 Proceedings of the Royal Irish Academy, 



Contents. 



1. A Generalized Trigonometrical Expansion. 



2. Example of Bessel Expansion : ^(r), a <r < h, expressed as a Series. 

 each term of which is a Multiple of 



A being so chosen that each term vanishes at r = 5. 



3. Generalization of preceding result. 



4. Case in which a = 0. 



5. Nature of the Convergence. Order of Magnitude of the Terms. Term 

 by Term Differentiation. 



6. A^alidity of discussion of Vibratory Motion in Space between r = a, 

 r = h, by Bessel-Fourier Analysis. 



Aet. 1. A Generalized Trigonometrical Expansion. 



The type of expansions in trigonometrical series which is usually required 

 in Physics is that in which it is required to express (f> (x), an arbitrary function 

 of X, between the limits a, h, in a series of the form 



S (C cos \x + S sin Xx) , (1 ) 



where C, S, X are determined so as to satisfy the equations 



{d/da + hi) {C cos Xa + S sin Xa) = 0, (2) 



(d/dh + h) (C'cos Xh+S sin A&) = 0, (3) 



hi, ho being given constants, (including, as such, zero and infinity). 



Eeplacing the trigonometrical functions by exponential, the expansion 

 above is seen to be a particular case of one which I proceed to consider, in 

 which each term is of the form 



Ae^'' + Be-'^'^, (4) 



where A, B, fx have to satisfy the equations 



Ae>'''Fi{fx,a) + Be->'^F,{- ix,a) = 0, (5) 



Ae'^^F.ifx, h) + Be-y-^F,{- n>h) = 0; (6) 



the F's being given polynomials,* which I suppose unconnected with one 



* That is, in M ) «, ^> and the minus sign are introduced into the notation in the hope of making 

 it more suggestive. 



