96 Dr. C. V. Burton on the Scattering and 



will contain a factor sin 2 e = 1 — sin 2 6 cos 2 <f> ; the average 

 value of which, for any given value of 0, is 



l-i«n»*=i(l + $). 



Thus to the ^-component of the vector-potential at P the 

 annulus 2irpdp (or 2irsds) contributes 



Os _1 expz (pt — vs — y) .o-.2irsds .^(l + - 2 h 



the integral of which is 



TnrCJ exyi(pt-vs-y)(l + ~ \ds. . . (53) 



As in § 11, the limits of integration are s = x and s = R, 

 where R is very great compared with x. 



44. The first term in (53) is (within a constant term) 



— iircrCv' 1 exp i {pt — vx—y) 



= — TraCu' 1 exipi (pt — vx—y + i7r). 



The second term in (53) depends on the integration of 

 s~ 2 exp(— ivs)ds between the limits s = x and s = R. Its 

 value is readily found by successive integration by parts, if 

 we remember that vx is large and may be chosen as large as 

 we please, provided only that R : x is very large. The value 

 finally obtained for (53) is 



— ZttgCv' 1 exp i (pt — vx — y +^7r), 



and the waves propagated in either sense from the plane 

 of yz are given by 



a" ; a'" = 0, 0, — 27raCv^ 1 exp i (pt + vx-y + ^ir). (54) 



45. Suppose now that the secondary radiators, irregularly 

 distributed in the plane of yz, are sending forth disturbances 

 (represented on an average by (52)) owing to the incidence 

 of primary waves 



a = 0, 0, Aexpz {pt — vx) ; .... (55) 



and let <rX 2 be so small that each radiator is sensibly 

 uninfluenced by the disturbances reaching it from its 

 neighbours. If the radiators are of type I. we can apply 

 (49) ; y being the phase-lag in the case of an isolated 

 secondary radiator. The secondary plane waves are thus 



[Type I.] 



a"; a"' = 0, 0, — 37rcrAu" 2 sin 7 exp z (pt + vx — y + ^7r). (56) 



