222 ON THE DIFFRACTION OF LIGHT BY [381
These are the formulae previously given. I had not then noticed that the integral in (4) can be expressed in terms of circular functions. By a general theorem due to Hobson *
i77 - . ..... / ( TT \ , , , sin m cos m
f*77 r- , ^ ..7. // ^ \r / \ smw cosm f .
j /, (m cos ^) cos2 <f>d<j> = yi (^J /| (TO) = —- - - -^-,......(6)
sothat p,.(^i).4,2?.^^fei-^f............(7)
in agreement with (5). The secondary disturbance vanishes with P, viz., when tan in = m, or m = 2&E cos & = 7r (1-4303, 2-4590, 3-4709, 4-4774, 5-4818, etc.)f. ...(8)
The smallest value of kR for which P vanishes occurs when % = 0, i.e. in the direction backwards along the primary ray. In terms of X the diameter is
2# = 0'715X..................................(9)
In directions nearly along the primary ray forwards, cos |^ is small, arid evanescence of P requires much larger ratios of M to X. As was formerly fully discussed, the secondary disturbance vanishes, independently of P, in the direction of primary vibration (a = 0, /? = 0).
In general, the intensity of the secondary disturbance is given by
in which P0 denotes P with the factor ei (nt~kri omitted, and is a function of ^ the angle between the secondary ray and the axis of x. If we take polar coordinates (%, <£) round the axis of x,
(11)
and the intensity at distance r and direction (%, <£) may be expressed in terms of these quantities. In order to find the effect upon the transmitted light, we have to integrate (10) over the whole surface of the sphere r. Thus
//,2P\2
-^-0 (1 + cos*%)
0
7./rr _., „. /""• . , Xl , , (sin TW — m cos m)2 = 7r^^-l)2EBJo sm^^a+cos2^)^---------—--------i
} ....... (12)
* Lond. Math. Soc. Proc. Vol. xxv. p. 71 (1893). f See Theory of Sound, Vol. n. § 207.hich m is written for 27dS cos fa. It is to be observed that in this solution there is no limitation upon the value of R if (K — I)2 is neglected absolutely. In practice it will suffice that (K— 1) R/\ be small, X (equal to 2?r/&) being the wave-length.