

f\ 



Random Distribution of Luminous Sources. 443 



With such a value of B/ — R the factor R'/R may be 

 disregarded *. 



It appears then that it is quite legitimate to regard the 

 intensity due to the simple sphere, expressed in (33), as a 

 surface effect; and this conclusion maybe extended to the 

 corresponding term involving n 2 in (28), relating to discrete 

 centres scattered at random. 



This extension being important, it may be well to illustrate 

 it further. Returning to the consideration of n sources in 

 the same initial phase distributed at random along a limited 

 straight line, let us inquire what is to be expected at a 

 distant point along the line produced. The first question 

 which suggests itself is — Are the phases on arrival distributed 

 at random ? Not in all cases, but only when the limited line 

 contains exactly an integral number of wave-lengths. Then 

 the phases on arrival are absolutely at random over the 

 whole period, and accordingly the expectation of intensity is 

 n precisely, if, however, there be a fractional part of a 

 wave-length outstanding, the arrival phases are no longer 

 absolutely at random, and the conclusion that the expecta- 

 tion cf intensity is n simply cannot be maintained. Suppose 

 further that n is so great that the average distance between 

 consecutive sources is a very small fraction of a wave-length. 

 The conclusion that when an exact number of wave-lengths 

 is included the expectation is n remains undisturbed, and 

 this although the effect due to any small part, supposed to 

 act alone, is proportional to n 2 . But the influence of any 

 outstanding fraction of a wave-length is now of increased 

 importance. If we do not look too minutely, the distribution 

 of sources is approximately uniform. If it were completely 

 so, the whole intensity would be attributable to the fractions 

 at the ends t, and would be proportional to ri 2 . In general 

 we may expect a part proportional to n 2 due to the ends and 

 another part proportional to n due to incomplete uniformity 

 of distribution over the whole length. When n is small the 

 latter part preponderates, but when n is great the situation is 

 reversed, unless the number of wave-lengths included be very 

 nearly integral. And it is apparent that the u 2 part has its 

 origin in the discontinuity involved in the sharp limitation of 

 the line, and may be got rid of by a tapering away of the 

 terminal distribution. 



Similar ideas are applicable to a random distribution in 

 three dimensions over a volume, such as a sphere, which may 

 be regarded as composed of chords parallel to the direction 



* The application to light is here especially in view. 



j It is indifferent how the fraction is divided between the two ends. 



* 



i 



