432 Lord Rayleigh on the Light emitted from a 



If in (6) K = n absolutely, the second factor is unity, and 

 since the first factor increases indefinitely with m, there is 

 a concentration of probability upon the value n, as compared 

 with what obtains for a single combination. 



In general we have to consider what becomes of 



y/m . {xe l ~ x } 



(7) 



when w = co, and x, written for K/rc, is positive. Here 

 xe i ~ x vanishes when x = and when ^ = qo, and it has but 

 one maximum when x=l, xe 1 ~ x =l. We conclude that 

 xe ] ~ x is a positive quantity, in general less than unity. The 

 ratio of consecutive values when m in (7) increases to m 4-1 

 is x ^"^^/(l + l/m), and thus when m = cc, (7) diminishes 

 without limit, unless x=l absolutely. Ultimately there is 

 no probability of any mean value K which is not infinitely 

 near the value n. 



Fig. 1 gives a plot of R in (5) as a function of x, or K/w, 

 for m = 2, 4, 6. It will be observed that for m > 2, d/Rjdx—O 

 when#=0, but that for ra = 2, dR/dx = 4:. 



Fig.l. 

















' <v ^<^ 







// 



V 









J 



\ 









The corresponding question for J may be worth a moment's 

 notice. We have 



K' = (3):(l) 



. m I 



(8) 



so that R/ goes to zero as m increases, if J be comparable 

 with n, as might have been expected 



It must not be overlooked that when the random distri- 

 bution of phases is due to a random spatial distribution of 



