334 The Hon. J. W. Strutt on Bessel's Functions, 



a simpler form. If /c be a root of J„(^) =0, we have 



2rVrfr[J„(«r)]^=[J'„W]^=[J„+,(«)]^ . . (20) 



in virtue of (9). 



If /c be a root of J'„ {z) = 0, 



2J V«V[J„(«r)]^=(l- 5)[J„W]^= -J„(«) J"„W. (21) 



In particular, if ?i = 0, 



Jo 

 More generally, if a; be a root of (18), 



i^\dr[J„{'cr)Y= {l + ^ (S -»')} [J»(«)]' 



= {l + («^-«^)|^'}[J'»W]' (23) 



Results corresponding to equations (17) and (22) are given 

 by Fourier [The'orie de la Chaleur, Chap. VI.) for the special 

 case of w = 0. The extension to the general value of n (I am 

 not sure that he contemplates fractional values) is due to Lom- 

 mel; but he drops generality in another direction by confining 

 himself to the case where k satisfies J„(«:) = 0. His results are 

 accordingly equivalent to (17) and (20), So far as I am aware, 

 the general equations (17), (18), (22) are new. 



It should be noticed that the same method is applicable to the 

 hollow cylinder bounded by r = rj, r=-rc^, provided that instead 

 of J„ the complete integral of (1) is used. In general the form 

 will be AJ„ + BJ_„, where A and B are arbitrary constants; 

 but if n be integral, J„ must be replaced by a more complicated 

 function denoted by Y„ (see Lommel) . The complete primitive 

 may be used, because the space through which Green's theorem 

 is to be applied does not include the axis r = 0; it must be used 

 in order to get a general solution, because there will now be two 

 boundary conditions to be satisfied. If fn{f<^r) be the complete 

 integral, and k, k' are subject to the conditions 

 M,/./„'(«n)+N,/„(«r,) = 0,-l 

 =0,/ 



with similar equations for /c', the equivalent of (17) stands 

 y„{Kr)UK'r)rdr=0, 



always provided that k and k are different. The arbitrary con- 

 stants contained in the expression of /„ allow of an indefinite 



f 



