438 Lord Rayleigh on the Light emitted from a 



As regards the former part, we observe that since -1 sin 6 

 can never exceed unity, 



dx' sin/c(V — w) x . . 



T' k(x'-x) K V ' ' ' * ( ' 



j: 



in which again ,#<X. The result of the second integration 

 leaves us with a quantity less than X 2 /^ 2 . The anomalous 

 part, both ends included, is less than 



nr(?+i). < 25 > 



which is small in comparison with the principal part, of the 

 order ir\kl and itself negligible. We conclude that here 

 again the mean intensity in a great number of trials is 2 

 simply. It may be remarked that this would not apply to 

 the mean intensity in a specified direction, as we may see 

 from the case where the initial phases are the same. In a 

 direction perpendicular to the line on which the sources lie, 

 the phases on arrival are always in agreement, and the 

 intensity is 4, wherever upon the line the sources may be 

 situated. The conclusion involves the mean in all directions, 

 as well as the mean of a large number of trials. 



Under a certain restriction this argument may be extended 

 to a large number n of unit sources, since it applies to every 

 term under the summation in (19). But inasmuch as the 

 evanescence is but approximate, we have to consider what 

 may happen when n is exceedingly great. The number 

 of terms is of order n 2 , so that the question arises whether 

 n 2 7r/kl can be neglected in comparison with n. The ratio is 

 of the order n\/l, and it cannot be neglected unless the 

 mean distance of consecutive sources is much greater than X. 

 It is only under this restriction that we can assert the 

 reduction of the mean intensity to the value n when the 

 initial phases are not at random. 



The next problem proposed is the application of (19) when 

 the n sources are distributed at random over the volume of 

 a sphere of radius R. In this case the distinction between 

 the mean in one direction and in the mean of all directions 

 disappears. If for the moment we limit our attention to a 

 single pair of sources, the chance of the first source lying in 

 the element of volume dY is dY/Y, and similarly of the 

 second source lying in dY' is dY'/Y. As the individual 

 sources may be interchanged, the chance of the pair 



