340 The Hon. J. W. Strutt on BesseFs Functions. 



In the theory of the vibrations of a gas, or the flow of heat 

 within a sphere, the notation of BesseFs functions may advanta- 

 geously be introduced. The expansion in Spherical Harmonics 

 replaces the series of Fourier, and instead of (13) the equation 

 satisfied by the coefficient of S„ becomes 



$ + 'f-^^ + «^<^=0, . . (36) 

 which may be also written 



^+(«^-"-^)'-* = 0. . . . (37) 

 Equation (13), however, may be put into the form 



from which we see that, since the solution of (38) (subject to 

 the condition atr=0) is 



= J^(/cr), 

 or 



r40 = ?4J^(A;r), 



the solution of (37) under a like condition must be 



r^ = r*J„+|(A:r) 

 or 



<^ = r-*J,^.(/cr).S«, (39) 



if the angular factor be restored. J^+i is, as we have seen, ex- 

 pressible in finite terms; but it is not easy thence to derive the 

 approximate form when r is small. As it is, the known theorems 

 about BesseFs functions allow us to expand (39) at once in an 

 ascending series. For the problem of vibrations in a rigid 



sphere, we require the roots of j^ =0 ; or if r = l, 



J,+^(/^)=2/cJUdW (40) 



An investigation of this problem wnll be found in the Mathema- 

 tical Society's ^Proceedings^ for 1872. 



For the conduction question, we have a linear relation between 



and ~- when r=l, say. 



dr 



M'6 + N'7=0. 

 dr 



Hence from (39), when r=], 



(M'-iNOJn-M(/^)+N'/cJ',^-^(/c), .... (41) 

 which comes under the form (18). 



