﻿Regular Reflexion of Light by Gas Molecules, 635 



lamina. The last written result can be readily put in 

 the form 



ijr", y=. — Aw(l + 2w sin Y + w; 2 )~*exp i(pt + x + ±Tr — 7 + e) 



sin w cos % l + io sin <y f (24) 



where e= —7^ — -^ = ^. 



cos x /(l + 2iv$my + w 2 ) 



24. If the vibrators are tuned as resonators, r y = ^7r and 

 (24) becomes identical with (9), (9 a). Now in the deduction 

 of (24) no limit has been imposed on the closeness of packing 

 of the vibrators, except the condition that their aggregate 

 bulk is but a small fraction of the space through which 

 they are distributed. If we can conceive of the vibrators as 

 indefinitely small, and as retaining always their property of 

 scattering without absorbing wave-energy, the number a 

 per unit area of the lamina may be as great as we please 

 without invalidating (24) or its particular form (9), (9a). 

 The restriction provisionally imposed in § 10 is thus found 

 to be unnecessary. 



25. From (9) together with (5) a good idea is gained of 

 the change of behaviour of a sheet (or thin lamina) of 

 resonators as the number a per unit of area is gradually 

 increased. The proportion of the incident energy contained 

 in the regularly reflected beam is 



4tt 2 <7 2 /u 4 



(1 + 27T(7/U 2 ) 2 ' 



which gradually approximates to unity as a is increased. 

 At the same time the proportion of the incident energy which 

 becomes diffusely scattered is v^jira times this expression, 

 that is 



4 77 <t/i/ 2 

 (1 + 27TO-/1; 2 ) 2 ' 



which becomes insignificant both for very small and for very 

 large values of <t ; attaining its maximum when 2 r TTajv 2 = l', 

 that is when this scattered energy is half the energy of the 

 regularly reflected train. 



26. A further point should now be remarked : if the 

 secondary vibrators become so closely crowded together that 

 the freedom of position for any given one is sensibly restricted 

 by the presence of the others, the irregularity of the distri- 

 bution will no longer be complete, and the proportion of 

 energy scattered will be less. As the swarm of vibrators 

 becomes more and more compressed, though retaining as 

 complete an irregularity as still remains possible, the distri- 

 bution will resemble that of the molecules of a liquid rather 



