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XI. 



EXTENSIONS OF FOUEIER'S AND THE BESSEL-EOURIEE 



THEOEEMS. 



By WILLIAM M'FADDEN OEE, M.A. 

 Pakt I. — Integrals. 



Read December 14, 1908. Ordered for Publication January 27. Published June 14, 1909. 



Introduction. 



The investigations in this paper were suggested by problems connected 

 with vibratory motion in the space outside a sphere or an infinitely long 

 cylinder. 



In the former case the equation V'^ = c~-d-<p/dt' is satisfied by 



where /S'„ is a solid harmonic of order n ; and, accordingly, if the disturbance 

 be supposed to involve a surface harmonic of assigned type, the solution is 

 readily obtainable by the aid of these general functions.! 



The problem might, however, be approached from another point of view. 

 If, for example, it is required that <p should vanish at the surface of the 

 bounding sphere, supposed of radius a, elementary type-solutions satisfying 

 this condition are 



"■^'"'s^T^^''^^'' 



/ d \" /cos Xci\ \.J '^ Y f^^^ ^^\\ ^^^ \ 



\adaj \ a J ' \adaj \ a J) cos 



Any function of r ought, therefore, for values greater than a, to be expressible 

 as an integral in A, whose element is the expression in large brackets multi- 

 plied by d\ ; and, if r - a be replaced by x, this becomes of the form 



{ C (A) cos {\x) + >S' (A) sin (A.r) j dX, 



where C, S are certain polynomials. This suggested the question discussed in 

 Art. 1. 



* Love, Phil. Trans., cxcvii., 1901 ; see also Lord Kelvin, "Baltimore Lectures," p. 193. 

 t For examples, see Love, " Some Illustrations of Modes of Decay of Vibratory Motions," 

 Proc. Lond. Math. Soc, ser. 2, vol. ii., part 2. 



K, I. A. PROC, VOL. XXVII., SECT. A. . [30] 



