302 Lord Rayleigh on the Production 



The mean value of (18) over angular space is thus 

 TA^ - OK' -^1-^^ <^Qg {€i — €o) sin ^D 



where D denotes the distance between the specimen pair 

 of sources. If all the sources are in the same phase 

 cos (ei — €9) = 1. It' the distance between everv pair of sources 

 is a multiple of ^X, sin /jD = and (21) reduces* to its first term. 

 We fall back upon a former particular case if we suppose 

 that there are only two sources, that these are units_, and are 

 in the same phase. (21) then becomes 



^ sin kD 



agreeing with (11) ^ which represents the work done b^' 

 each source. 



If the question of the phases of the two unit sotu*ces be 

 left open (21) gives 



9 + 9cos(6i-e,)!i?^. . . . (-22) 



It D bi- small, this reduces to 



2 + 2 cos (ei — e^), 



which is zero if the sources be in opposite phases, and is 



equal to 1 if the phases be the same. 



If. however, sin J:D he equal to —1, the case is altered. 

 Thus when D=4\. we get 



2— ;t- cos (61—69), 



and this is a minimum (and not a maximum) when the 

 phases are the same. 



In {22) if the phases are 90° apart, the cosine vanishes. 

 The work done is then simply the double of what would l>e 

 done by each source acting alone, and this whatever the dis- 

 tance I) may be. 



Contiii nous Distributions. 



If the distribution of a source be continuous, the sum in 

 (21) is to be replaced by a double integral. As an example, 

 consider the case of a source all in one phase and imiformly 

 distributed over a complete circular arc of radius c. If I) 

 be the distance of two elements d6. d6'. we have to consider 

 the integral 



i!\^dOd0: (23) 



where 



I) = -2csmi{e-e") . . . (24) 



