556 ON THE DISPERSAL OF LIGHT BY A DIELECTRIC CYLINDER [434
with the understanding that the 2 is to be omitted when m = 0. Corresponding to the primary wave ei(nt+kx), we find as the [approximate] expression of the secondary wave at a great distance from the cylinder,
K'-k*
The term in cos 6 is now the leading term ; so that the secondary disturbance approximately vanishes in the direction of the primary electrical displacements,1 agreeably with what has been proved before. It should be stated here that (9) is not complete to the order 7^c4 in the terms containing cos 6. The calculation of the part omitted is somewhat tedious in general ; but if we introduce the supposition that the difference between k'2 and k- is small, its effect is to bring in the factor (1 - %k2c2).
" Extracting the factor (k'2 - k'), we may conveniently write (9)
in which
, 7cV 4 k"-cn- Fc2 0/) - 7cV-7c2c2 7c2c2 0/) /11X
cos 6 -- ^ ---- ^- cos 20 = cos 0 -- r^ --- j- cos2(9. . . .(11) ID o ID {±
" In the directions cos 6 — 0, the secondary light is thus not only of high order in kc, but is also of the second order in (k' — k). For the direction in which the secondary light vanishes to the next approximation, we have
z.2/,2 ir> _ ir
o T> »\ h v •"• •"• /tn\
--fc"C-)=-- ~ ................ (I2)
This...is true if kc, k'c be small enough, whatever may be the relation of k' and k. For the cylinder, as for the sphere, the direction is such that the primary light would be bent through an angle greater than a right angle...."
"If we suppose the cylinder to be extremely small, we may confine ourselves to the leading terms in (6) and (9). Let us compare the intensities of the secondary lights emitted in the two cases along 0 = 0, i.e. directly backwards. From (6)
ty cc ^ (k'~c~ — 7t;2c2), while from (9)
\fr oc — k~cr (k'" — k2}/(k'* + k").
The opposition of sign is apparent only, and relates to the different methods of measurement adopted in the two cases. In (6) the primary and secondary disturbances are represented by hjK, but in (9) by the magnetic function c...."
It may be remarked that Ignatowski's equation agrees with (5) for this case,'and that his corresponding equation (11) for tlie second case also agreesion.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