558 ON THE DISPERSAL OF LIGHT BY A DIELECTRIC CYLINDER [434
where Nm = zGm'(z) Jm(*'} -z G.M(z} Jmf(z)
= sQn^l(s')Jm(zf)-e'Gm(e)Jmr.1(z').............(20)
Thus
_ 2t'm m==JV ID 4- iiV...........................^)
•Liml-um i 2 47r
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
1___ , ;2(-i;
h^7T+f iVm/£
Here JVm, Dm are given by (19), (20), and are real. In like manner (8) may be put into the form
9/i'm
R ' -______________ /SON
^W Nm'IDm' + tiir'..........................M}
' = *Jm 00 0«r. (*) - «/«' (/) &,» (*)
= z'Jm (z'} (?m_x (0) - eJ^ (Z'} Grm (z)
•-~] Jm(z'} GM,........................(24)
z J
» Jm-i (z) - z'Jm (z') Jm-i (z)
- ^VroOO/mC*').........................(25)
z /
And, as in (22), the expression for the diverging wave at a distance is
__ I 2 (- !)"»
^ + TSv5V
When we fix the refractive index at 1'5, the value of z'lz — zjz1 in (24), (25) is 5/6.
The values of Nlt NI, A. A' inay be deduced from the corresponding quantities with m — 0 by means of the relations
N^W, ^/-JV.-KiCO^^),..................(27)
A = A', A^A-H/zOO/zOO...................(28)
For numerical calculation we have also to specify the values of z, or ko. For this purpose'we take ^ = '4, '8, T2,1'6, 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 + $i7r}~1 and of (N'jD' + ^irr)"1 may then be tabulated.
The next step is the calculation of the series included in the square brackets of (22) and (26) for various values of d from 0 = 0 in the direction backwards along the primary ray to 6 —180° in the direction of the primaryht-hand side of (5)