1918] ON THE DISPERSAL OF LIGHT BY A DIELECTRIC CYLINDER
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60
In Fig. 2 are plotted curves showing the variation with z at given angles of 6 = 0°, 60°, and 90°. At 0° the polarization is all in one direction over the whole range from 0 to 2'4. At 60° there are reversals of polarization at ^ = 1'5 and z = 2'05. At 90° these reversals occur when z — l-7 and z = 2-3.
The curves stop at z = 2'4. It would have been of interest to carry them further, but the calculations would soon become very laborious. As it is, they apply only to visible light dispersed by the very finest fibres, inasmuch as z is the ratio of the circumference of the cylinder to the wave-length of the light.
When z, or kc, is greater than 2'4, we may get an idea of the course of events by falling back upon the case where the refractivity (/j, — 1) is very small, treated in my 1881 paper. In our present notation the light dispersed in direction 6 depends upon
J^Zz cos $0).........................(29)
x 2 ' ^ }
z cos u When 0 = 180n, i.e. in the direction of primary propagation,
Ji (2# cos \ 0) = z cos 10,
and (29) reduces to ire2. In this direction every element of the obstacle acts alike, and the dispersed light is a maximum*. In leaving this direction the dispersed light first vanishes when
cos £0 = 3-8317/2*, and afterwards when
2^ cos|0 = 7-0156, 10-173, 13-324, etc.
The factor (29) is applicable, whether the primary vibrations be parallel or perpendicular to the axis of the cylinder. The remaining factors may be deduced by comparison with the case of an infinitely small cylinder. Thus for vibrations parallel to the axis, we obtain from (6)
__i a\
J> .........(30)
\'2ikrJ cos|-0
applicable however large c may be, provided (k' — k} be small enough.
In like manner for vibrations perpendicular to the axis we get from (9) i
(kc — k'c) cos 0.«/! (2/70 cos •
gi Cnt—kr) ^
:, ......(31)
vanishing when 6 — 90°, whatever may be the value of Tec. It will be seen that (30) and (31) differ only by the factor — cos 0, and tha't this is unity in the direction of the primary light.
[* The successive maxima occur at the roots of J2 (2z cos £0) = 0, viz. %z cos £0=0, 5 '135, 8-417, 11-620, etc. W. P. S.]
36-2P. S.] J Reports for 1913, p. 115; 1914, p. 75. 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