(did Distribution of Sound. 301 



employed with a single source appears to be wasted, we are 

 left in doubt whether with the existino- arrangements economy 

 would be attained by breaking up the source. 



We will now investigate the expression for the energy 

 radiated from any number of sources of the same pitch 

 -situated at finitely distant points in the neighbourhood of the 

 origin 0. The velocity-potential <^ of the motion due to one 

 of the sources at (i*, j/, z) is at Q 



(/)=-^^cos(wi + e"^R), . . . (15) 



where R is the distance between Q and (.i*, ^, z). At a great 

 distance from the origin we may identify R in the denominator 

 with OQ, or p ; while under the cosine we write 



ll=p-{\x + fjLi/ + vz), .... (16) 



\, fi, V being the direction-cosines of OQ. On the whole 



— ^'Trp(^ = ^KQ,os{nt-\-e — kp-\-k{^Kx-\-iJLy-\'Vz)}, . (17) 



in w^hich p is a constant for all the sources, but A, e, <r, ?/, z 

 vary from one source to another. The intensity in the 

 direction \ yu,, v is thus represented by 



[tKco^{e^-ki\x + fiy^-vz)\Y+\tK^m{e + h[\x + fiy+vz)}']\ 

 or by 



SA2 + 22AiA2COs[€i-62 + A-{XG^,-^2) +)"• (3/1-3/2) + »^ (-^1-^2)}] , 



. . . (18) 



the second summation being for every pair of sources of 

 which Ai, A2 are specimens. We have now to integrate 

 (] 8) over angular space. 



It will suffice if we effect the integration for the specimen 

 term ; and this we shall do most easily if we take the line 

 through the points [xi^ i/i, z-^^ (^2> 2/2? ^2) ^s axis of reference, 

 the distance between them being denoted by D. If X, /jl, v 

 make an angle with D whose cosine is yu,, 



D/x = X-(.27i-.^'2)+/i(yi-^2) + »'(^i-'^2), . • (19) 

 and the mean value of the specimen term is 



Aj A 2 j cos { ei — €2 + /c D//, } dp.^ 



that is 



9A A 



^]^'sin/.Dcos(ei-62) (20) 



