The Hon. J. W. Strutt on Bessel's Functions. 333 



Therefore, by Green's theorem, 



{k' — k) j rdrJn{Kr)3^ (/c'r) 

 Jo 



+ ^^i/c'J,(/cr) J'„(/c'r) -/.J'„(A:r)J„(/cV)| =0. . (15) 



Thus, if K and /c' are different, and such that 



//J„(«:)J'„(a(/)— /cJ'„(«:)J„(«0 = O, . . . (16) 

 we have 



Crdr^M)h['<^r) = Q (17) 



Jo 



The equation (16) may be satisfied in several ways. First, 

 we may have J„(«:), Jn(/c') equal to zero, so that /c, t^ are roots of 

 J„(^)=0; or /c, /c' may be roots of Vn{z) = 0; or lastly, they 

 may be roots of the more general equation 



M^J'„(^) + NJ„(^)=0, .... (18) 



which has an application in the theory of heat. In any of these 

 cases (17) holds good. 



If /t' = «r, (15), as it stands, becomes identical. We must take 

 K^ = K-\-hK, and seek the limiting form as hic tends to vanish. 

 Thus 



2 



Jo 

 _ ,. . r|A:J'„(«:r)J„(/cH-S/<:)r— (/c4-5/c)J„(Acr)J'n(/c^-8«;)rj■ 

 = ^j«r[J'„(«r)]^- J„(«r) [J'„(/o-) + «»-J"„(«;-)] ( . 

 Accordingly 



But by (1), 



J"„W + Jj'»W = -(i-5)j»M. 



so that 



2fV</r[J„H]^=[J'„(«)]^+(l-J)[J„(«)]^. . (19) 



(19) holds good for any value of k ; but in special cases it assumes 



