﻿212 Lord Rayleigh on the Mutual Influence of 



value of a from (10), we see that this is 



' <j> + ^ 1/a + l/r—ik ___ n ^ 



dtf + W/dr -1/r 2 ~ V ; * * W 



and this is the relation which must hold at a resonator so 

 tuned as to respond to the primary waves, when isolated 

 from all. other influences. 



The above calculation relates to the case of a single re- 

 sonator. For many purposes, especially in Optics, it would 

 be desirable to understand the operation of a company of 

 resonators. A strict investigation of this question requires 

 us to consider each resonator as under the influence, not 

 only of the primary waves, but also of the secondary waves 

 dispersed by its neighbours, and in this many difficulties are 

 encountered. If, however, the resonators are not too near 

 one another, or too numerous, they may be supposed to act 

 independently. From (11) it will be seen that the standard 

 of distance is the wave-length. 



The action of a number (?i) of similar and irregularly 

 situated centres of secondary disturbance has been con- 

 sidered in various papers on the light from the sky *. The 

 phase of the disturbance from a single centre as it reaches a 

 distant point, depends of course upon this distance and upon 

 the situation of the centre along the primary rays. If all 

 the circumstances are accurately prescribed, we can calcu- 

 late the aggregate effect at a distant point, and the resultant 

 intensity may be anything between and that corresponding^ 

 to complete agreement of phase among all the components. 

 But such a calculation would have little significance for our 

 present purpose. Owing to various departures from ideal 

 simplicity, e. g. want of homogeneity in the primary vibra- 

 tions, movement of the disturbing centres, the impossibility 

 of observing what takes place at a mathematical point, we 

 are in effect only concerned with the average, and the average 

 intensity is n times that due to a single centre. 



In the application to a cloud of acoustic resonators the 

 restriction was necessary that the resonators must not be 

 close compared with X ; otherwise they would react upon 

 one another too much. This restriction may appear to 

 exclude the case of the light from the sky, regarded as due 

 mainly to the molecules of air ; but these molecules are not 

 resonators — at any rate as regards visible radiations. We 

 can most easily argue about an otherwise uniform medium 



* Compare also "Wave Theory of Light," Enc. Brit. xxiv. (1888), §4; 

 Scientific Papers, vol. iii. pp. 53, 54, 



