370 Lord Rayleigh on the Dispersal of 



where N„ = z GJ [z) J m {z') -z'G m {z) J J (z') 



= zG m -i(z)J m (z , )-z'GJz).J m _ 1 (z r ). . (20) 



Thus 



2i m 



where, however, the 2 is to be omitted when m = 0. Thus 

 by (4) and (16) the divergent wave at a great distance r is 

 expressed by 



; \2ikr) LN0/D0+ i«r + <f N M /D W + *«r J * ^ 



Here N m , D ?n are given by (19), (20), and are real. 

 In like manner (8) may be put into the form 



2i m 

 B "'' = N„//I>l'+i*V' • • • • ( 23 ) 

 where 



K m ' = z'J m (z')G m '(z)-zJJ(z')Q m (z) 



=z , J m (z')G m - 1 {z)~z3 m - 1 <iz')GrJiz) 



-m(^-^)j m (z')G m (z), (24) 



D m = zo m (z) J m -i(z ) — z J m (z )J m . i(z) 



+ ™(^- Z j y m (z)J m (z') (2d) 



And, as in (22), the expression for the diverging wave at a 

 distance is 



e _ / jl.y<*-^) r 1 + f 2( - i )'" cos "^ i (26) 



\2ikr) e LN '/D ' + im + f N.'/D.' + iwJ* lZb> 



When we fix the refractive index at 1*5, the value of 

 z'/z-z/z' in (24), (25) is 5/6. 



The values of N l5 N^, I) 1? D/ may be deduced from the 

 corresponding quantities with m = by means of the 

 relations 



N,=N ' 5 N^No-fJ^G^), . . (27) 



D 1 = D ', D^Do + lJ^^JiW • • (28) 



For numerical calculation we have also to specify the 

 values of z, or he. For this purpose we take 2=*4, # 8, 1*2, 

 1*6, 2*0, 2*4, where z denotes the ratio of the circumference 

 of the cylinder to the wave-length in air ; the corresponding- 

 values of (N/D + izV)- 1 and of (N'/D' + ^tt)- 1 may then be 

 tabulated. 



