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Full text of "Scientific Papers - Vi"

J.C/JLOJ             <Ji\    XJaJtli   i5LiJL'J."lJlittli>Hjr   UJt)    JjJUjJtl'JL'   JtS X    SJfJUJUJK.J.UAL   SJtUlJ.Lj.bS                     Oil
is nearly in the direction of the primary ray continued (/3 = 7 = 0).    In this case m is very small,
sin m     cos m     1
~~m?         ^~ = 3'
and | P   is independent of k, and is proportional to R3.    The intensity is then
The haze immediately surrounding a small source of light seen through a foggy medium is of relatively great intensity. And the cause is simply that the contributions from the various parts of a small obstacle agree in phase.
But in general when R is great, so also is m, and | P \ varies rapidly and periodically with k along the spectrum. We might then be concerned mainly with the mean value of P2 . Now
P2  = (K - I)2 . 4V3 Rs (sin m - m cos m)2 m~G, of which the mean value is
(K - I)2 . 87r2^6 (1 + m2) m~\ or approximately, since m is great,
(K - I)2 . 87r2J?fl m~\ When we introduce the value of m from (6), this becomes
,,   ,   „,,     -           -
Mean  P2  =~r-± — ~, - = on  .     , ,     ............. (10)
1              *4               24-                  V      !
The occurrence of X4 shows that this is in general very small in comparison with (9).
If, instead of a sphere of uniform quality, we have to deal with one where (K - 1) is variable, we must employ (7). The case of greatest interest is when (K — 1), besides a constant, includes also a periodic part. For the constant part the integration proceeds as' before, and for the periodic part, where (K— 1) varies as a circular function of R, it presents no difficulty. It may suffice to consider the particular case where (K— 1) is proportional to sinm, m as before being given by (6); for this supposition evidently leads to a large augmentation of P, analogous to what occurs in crystals of chlorate of potash, to which a plane periodic structure is attributed*. It will be observed that the wave-length of the structure now supposed varies with ^, as well as with k or X. Thus, if K — 1 = /3 sin m,
P = _ VBJ <-«-*) ^ m2 - m sin 2m + HI -cos 2m)    ......
m3                          ......
Phil. Mag. Vol. xxvi. p. 256 (1888) ; Scientific Papers, Vol. in. p. 204. consider the phases represented by the factor eik (x~^'] in P. For the point 0, x = 0, r = p, and the exponential factor is e~ikt>. As in the ordinary theory of specular reflection, the same is true for every point in the plane AOA and therefore for the element of surface at A A whose volume is 27rRdRd£. For points in a plane