Regular Reflexion of Light by Gas Molecules, 95 



emitted are plane-polarized, with electric vector parallel 

 to z ; all else belongs to the irregular disturbance. If a now 

 represents the amplitude at distance / due to a single secon- 

 dary radiator, when the axis of the radiator is parallel to the 

 ,2-axis, and the distance / perpendicular thereto, it is evident 

 that the resultant amplitude at any point of the diffraction 

 pattern contains as a new factor the average value of cos 2 (£, 

 where <j> may be regarded as the co-latitude of any point on 

 the surface of a sphere. The average of cos 2 $ is -J, so that 

 at each point of the diffraction pattern the amplitude of 

 disturbance is J as great and the energy-flux -^ as great as if 

 the secondary radiators had been of type I. At the same 

 time the expression for the scattered energy contains a new 

 factor av.cos 2 <£ or ^; and accordingly the regularly re- 

 flected energy bears to the scattered energy a ratio one-third 

 as great as that given by (50), namely, 



[Type II.] (7X' 2 /8tt or 7ra/2v 2 (51) 



43. Now let the whole plane of yz be scattered over with 

 secondary radiators, whether of type I. or of type II. ; the 

 distribution (of g radiators per unit of area) being completely 

 irregular, while the average vector-potential due to a single 

 radiator is given, as regards direction and magnitude, by 



a»'=0, 0, 0r» 1 expt(p*-vr n -7). . . (52) 



Then the vector-potential in the plane-waves propagated in 

 the direction of x increasing will be 



a" = 0, 0, XCi'n 1 exp i(pt — vr n —y). 



As in the diagram in Part I., let be the origin, P a 

 point (x, 0, 0) and p 2 — ij 2 + z 2 . If Q is any point in the 

 ?/r-plane, distant p from the origin and 5 from the point P, 

 the angle e made by PQ with the axis of z is given by 



cos e = sin 6 cos<£ ; 



where 6 is the angle OPQ, and c/> the angle made by OQ with 

 the axis of c. For the moment we are concerned only with 

 the regular waves, which depend on average values, and not 

 with the diffuse radiation, which depends on deviations from 

 the average : thus from symmetry the resultant vector 

 potential at any point is always parallel to the c-axis. The 

 contribution of an elementary area in the neighbourhood 



of Q has to be twice resolved through an angle k — €, so 



that the corresponding element of the vector-potential at P 



