434 Lord Rayleigh on the Light emitted from a 



At the first source where r = 



4:7rr 2 d(j)/dr = cos kat—kr sin kat, 

 dd> , d-dr ha sin kat , ka . -, , . -^\ 



di + It " 1ST" + SD »in K*-D). 



The work done by the source at r = is accordingly pro- 

 portional to 



i+^*5, • (12) 



and an equal amount of work is done by the source at r' = Q. 

 If D be infinitely great, the sources act independently, and 

 thus the scale of measurement in (12) is such that unity 

 represents the work done by each source when isolated. If 

 D = 0, the work* done by each source is doubled, and the 

 sources become equivalent to one of doubled magnitude. 



If D be equal to ^ A, or to any multiple thereof, sin &D = 0, 

 and we see from (12) that the work done by each source is 

 unaffected by the presence of the other. This conclusion 

 may he generalized. If any number (n) of equal sources in 

 the same phase be arranged in (say a vertical) line so that the 

 distance between immediate neighbours is -J A., the work done 

 by each is the same as if the others did not exist. The whole 

 work accordingly is n, whereas the work to be done by a 

 single> source of magnitude n would be n 2 . Thus if sound be 

 wanted only in the horizontal plane where there is agree- 

 ment of phase, the distribution into n parts effects an 

 economy in the proportion of n : 1. 



A similar calculation would apply when the initial phases 

 differ, but we will now take up the problem in a more general 

 form where there are any number (n) of unit sources, and by 

 another method *. The various centres are situated at points 

 finitely distant from the origin 0. The velocity-potential 

 of one of these at (#, y, z), estimated at any point Q, is 



cos (pt + e-kU) 



where R is the distance between Q and (x, y, z). At a 

 great distance from the origin we may identify R in the 

 denominator with OQ, or R ; while under the cosine we 

 write 



R=R — (lx + my + nz), . . . . (14) 



* " On the Production and Distribution of Sound," Phil. Mag. vol. vi. 

 p. 289 (1903) j Scientific Papers, vol. v. p. 136. 



