﻿Regular Reflexion of Light by Gas Molecules. 631 



of the transmitted waves yjr + yjr^ and of the reflected waves 

 ty'", gives for the energy diffusely scattered 



The energy of the reflected waves, referred to the same 

 standard, is w 2 /(l + w>) 2 ; and the ratio of regularly reflected 

 to scattered energy is w/2 = 7t<t/u 2 , in agreement with (5). 



14. The more general case where the vibrators are not 

 tuned as resonators (y^^ir) need not be discussed at length, 

 since its solution can be derived (§ 23) from that of a still 

 more general problem. But it will be useful to write down 

 the expressions for the secondary plane waves when 27ro-/u 2 

 is negligible in comparison with unity. From (2), (3), and 

 (7) these are readily seen to be 



• rt i m 2tt<tK sin <y / .-— , i \ 



•sfr f/ , yjr //f =— L cos (pt + vx — y + \ir) 



corresponding to the primary waves (1). 



15. It will now be convenient to introduce complex 

 quantities. When, in place of (1), we write for the primary 

 waves 



i/r = Aexpi (pt — vx); .... (10) 



the plane waves emitted by <t secondary vibrators per unit of 

 area in the plane of yz are 



, „ , ... 27rcrA sin 7 . r — . •> 



Y > Y — 2 Qwp i {pt + vx — y + %7r} 



= —kAexj)i(pt + vx), (11) 



where k= - 2 — -exp i(iir — y). . . . (12) 



16. Suppose, now, that in the space between the planes 

 ^• = and x=L there is a statistically homogeneous swarm 

 of secondary vibrators, the average number of vibrators per 

 unit of volume being v. For the most part, no restriction is 

 imposed on the value of v, but when v is small enough for 

 the aggregate bulk of the vibrators in any considerable 

 volume to be but an insignificant fraction of that volume — 

 the vibrators being then distributed like the molecules of a 

 gas — some of the results already obtained become applicable. 

 Consider the lamina, bounded by the planes .<•', x' + dx': da' 

 being in any case very small compared with the wave-length 

 of the primary disturbance (10) ; for the moment let it also 

 be chosen so small that 'IttvcIx' Jv~ is a negligible fraction. 



