4 Lord Rayleigh on the Correction to the 



a and b, so that <j>(hr) vanishes for r = a, r = b. In the usual 

 notation we may write 



A (ua - J o( Ar ) Y o( Jir ) fqx 



* (/ir) -J^)-Yo(Aay' * * ' * (9 > 



with the further condition 



Y (ha)J o (hb)-J {ha)Y (hb)=0, . . . (10) 



determining the values of h. The function <f> satisfies the 

 same differential equation as do J and Y . 



Considering for the present only one term of the series 

 (8), we have to find for the positive side a function which 

 shall satisfy the other necessary conditions and when z = Q< 

 make V = from to a, and V' = H$(7ir) from a to b* 

 As before, such a function may be expressed by 



V'=B!J 8 (V) e-^+B 2 J (k 2 r)e-^+..., (11) 



and the only remaining question is to find the coefficients B„ 

 For this purpose we require to evaluate 



j <f)(hr) J (Jcr) rdr. 



v a 



From the differential equation satisfied by J and </> we get 

 V [' J,(fa)+(*r)r dr = - [r .$ . §>f + £ J Jr *, 



and 



hj\(kr)4>(hr)rdr= - [r . J . g]* + ^ d £rdr 



so that 



(P-A 2 ) f J (kr)cl>(hr)r dr= [rJ ~ -^°*Y 



= —JiaJ (ka)(f)'(ha) } . (12) 



since here ${ha) = <f>(lib) = 0, and also J (&£) = 0. Thus in 

 (11), corresponding to a single term of (8), 



(A 2 -F)6^J 1 2 (^) ^ 



The exact solution demands the inclusion in (8) of all the 



