336 The Hon. J. W. Strutt on BessePs Functions. 



whence 



A, = 2J'V<;ry{r)J>r)^[J'„W]^ + (l-^^)[J„(«)]^ (24) 



In particular, if /c be a root of J„(/c) =0, 



Ap=2jVrfr/W J„H^ [J'„W]^ : 



or if /c be a root of J'w(/f) =0, the same integral must be divided 



(l-^)[J„(«)]^or -J„(«).J'„(«). 



An application of these formulae will be found below. 



If y\r) does not fulfil the above specified condition, but remains 

 finite when r=0, we may, following Lommel, write 



whence 



2 r V»+ y(r) J„ {Kr) dr = A \'2rdr [J„(/«r)] 2, 

 Jo Jo 



as before. 



The application to the problem of vibration in two dimensions 

 is very easy. The particular solution of (12) is 



(^ = cos [Kat + e) . J^ {Kr) . cos (n6 + a), . . (25) 



K being determined from (18) or one of its special forms accord- 

 ing to the circumstances of the case. The most interesting in 

 the application to acoustics is when the cylinder r = I is rigid, 

 so that J'n{K) =0. The lower values of k (calculated from Han- 

 sen's Tables by means of the relations allowing J„ to be expressed 

 in terms of Jq and J,) are given in the following Table : — 



Order of Harmonic. 



1 i 





0. 



1. 



2. 



3. 







3-832 



1-841 



3054 



4-201 



^2 



1 



7-015 



5-332 



6-705 



8-015 



-1 



2 



10-174 



8-536 



9-965 



11-344 



il 



3 



13-324 



11-706 







4 



16-471 



14-864 







^ 



' 



19-616 



18016 







When K is very great the roots are given by (6), the series 

 being the same for the alternate functions. The trouble of the 

 calculation of the earlier roots increases rapidly with n. When 

 n is great, it would appear probable from physical considerations 

 that the first root varies as n ; but I have not succeeded in put- 



