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Full text of "Scientific Papers - Vi"

1918]      ON THE DISPERSAL  OF  LIGHT  BY  A  DIELECTRIC  CYLINDER          555
analytically identical with that treated in my book on Sound*,  343, to which I must refer for more detailed explanations. The incident plane waves are represented by
aint aikx  pint aikr cos 6 &      \j      ~~~ \j     \j
= eint (Jo (kr} + 2t/! (kr} cos (9 + ...
+ 2imJm(kr} cos mO + ...};............(3)
and we have to find for each value of m an internal motion finite at the centre, and an external motion representing a divergent wave, which shall in conjunction with (3) satisfy at the surface of the cylinder (r  c} the condition that the function [b/K] and its differential coefficient with respect to r shall be continuous. The divergent wave is expressed by
BQ^ + B^COS 6 +-82^2cos %0+ ...,..................(4?)
where ->Jr0, i^, etc. are the functions of kr defined in  341. The coefficients B are determined in accordance with
^ Jm (k'c} - k'c^m -^rr Jm (k'c}l
w/. /CC                                                 CL  nt C                         \
= 2im {k'cJm(kc}Jm' (k'c) - kcJm(k'c)J,n (kc)},.........(5)
except in the case of m  0, when 2im on the right-hand side is to be replaced by im-\. In working out the result we suppose kc and k'c to be small; and we find approximately for the secondary disturbance corresponding to (3)
-._ ( * N* i<M-to [l^tf-tfc*    /2c2(/</2c2-/<;2c2)
,
r     \ftkrj  "         L       2
showing, as was to be expected, that the leading term is independent of 9.
"For case 2, which is of greater interest, we have [from the general equations]
This is of "the same form as (2) within a uniform medium, but gives a different boundary condition at a surface of transition. In both cases the function itself is to be continuous ; but in that with which we are now concerned the second condition requires the continuity of the differential coefficient after division by k-. The equation for B,m [or J5m' as we may write it for dis-tinctiveness] is therefore
d.kc
d.k'c c) Jm'(k'c) - k'cJm(k'c)Jmf(kc)}, ...... (8)
* Theory of Sound, Vol. n.    MacmUlan, 1st ed. 1878, 2nd ed. 1896. f Here k' relates to the cylindrical obstacle and k to the external medium. J In (7) c is the magnetic component, and not the radius of the cylinder.  So many letters are employed in the electromagnetic theory, that it is difficult to hit upon a satisfactory notation.again, as in (1), so that the vibration is A sin2 6 + C cos2 6, reducing to G simply, if A = G. This is the result for a single particle whose axis is at W. What we are aiming