Random Distribution of Luminous Sources. 447 



intensity which is proved to be n. In any particular arrange- 

 ment of particles the intensity may be anything from to 

 n 2 . But in the application to a gas dispersing light, the 

 motion of the particles ensures that a random redistribution 

 of phases takes place any number of times during an interval 

 of time less than any which the eye could appreciate, so that 

 in ordinary observation we are concerned only with what is 

 called the expectation. 



It is hoped that the explanations and calculations here given 

 may help to remove the difficulties which have been felt in 

 connexion with this subject. The main point would seem to 

 be the interpretation of the n 2 term as representing the sur- 

 face reflexion when a cloud is supposed to be abruptly 

 terminated. For myself, I have always regarded the light 

 internally dispersed as proportional to n, even when n is 

 very great, though it may have been rather by instinct than 

 on sufficiently reasoned grounds. Any other view would 

 appear to be inconsistent with the results of my son's 

 recent laboratory experiments on dust-free air. 



The reader interested in optics may be reminded of the 

 application of similar ideas to a grating on which fall plane 

 waves of homogeneous light. If the spacing be quite uni- 

 form, the light behind is limited to special directions. Seen 

 from other directions the interior of the grating appears 

 dark. But if the ruling be irregular, light is emitted in all 

 directions and the interior of the grating, previously dark, 

 becomes luminous. 



In the problems considered above the space occupied 

 by a source, whether primary or secondary, has been sup- 

 posed infinitely small. Probably it would be premature to 

 try to include sources of finite extension, but merely as an 

 illustration of what is to be expected we may take the ques- 

 tion of n phases distributed at random over a complete period 

 (27r), but under the limitation that the distance between 

 neighbours is never to be less than a fixed quantity o\ All 

 other situations along the range are to be regarded as equally 

 probable. • • * • 



As we have seen, the expectation of intensity may be 

 equated to 



7i + 2 JJJ ... . 2 cos {6n-e T )de l d0 2 .... dd n 



-Jjj....^....^, . (40) 



and the question turns upon the limits of the integrals. 

 The case where there are onlv two phases (n=2) is simple 



2H2 



i 



