JJ 



440 Lord Rayleigh on the Light emitted from a 



This potential is next to be multiplied by 4-7rR 2 iR and 

 integrated from to R. We find 



S -^dYdV' = ^(smm-m C osm)\ . (27) 



We have now to divide by V 2 , or 167r 2 R 6 /9 ; and finally 

 we get 



(19) = ti+ 9 ^R6 1) (sin *R-£Rcos&R) 2 , . (28)* 



where &R will now be regarded as very large. When n is 

 moderate, or at any rate does not exceed PR 3 , the second 

 term is relatively negligible, that is reduction occurs to n 

 simply, provided n be not higher than of order R 3 /\ 3 , 

 corresponding to one source for each cubic wave-length f. 

 But evidently n may be so great that this reduction fails, 

 unless otherwise justified by a random distribution of initial 

 phases. 



At the other extreme of an altogether preponderant n, the 

 second term in (19) dominates the first, and we get in the 

 case of constant initial phases and a very large &R, 



(19)= 9nWm (29) 



Under the suppositions hitherto made of a random spatial 

 distribution within the sphere (R), and of uniformity of 

 initial phases, there is no escape from the conclusion that 

 the reduction to the simple value n fails when n is great 

 enough. Nevertheless, there is a sense in which the reduction 

 may take place, and the point is of importance, especially in 

 the application to the dispersal of primary waves by a cloud 

 of small obstacles. In order better to understand the 

 significance of the term in n 2 , let us calculate the intensity 

 due to an absolutely uniform distribution of source of total 

 amount n over the spherical volume. Since there is complete 

 symmetry, it suffices to consider a a single specified direction 

 which we take as axis of z. As in (15), we have 



-47rR <£=^-^ \}\e ikx dxdydz, . . (30) 



as the symbolical expression for the velocity-potential, from 



* We may confirm (28) by supposing JcR very small, when the right- 

 hand member reduces to n 2 . 



t The number of molecules per cubic wave-length in a gas under 

 standard conditions is of the order of a million. 



