234 DYNAMICAL THEORY OF SOUND 



where <I> represents a distribution of density (p = <l>/47r), or 

 of sources of heat, &c. The solution of (4) appropriate to 

 infinite space (when there are no sources at infinity) is known, 

 viz. it is 



where 4>' denotes the value of <3> at (of, y', z / ), r denotes distance 

 from this point to the point P, or (x, y, z), for which the value 

 of < is required, and the integration extends over all space for 

 which <E> differs from 0. For example, if we put <!>' = 4-Trp', 

 we get the ordinary expression for the gravitation potential of 

 a continuous distribution of matter. 

 The analogous solution of (3) is 



(6) 



This represents a distribution of simple sources through R, the 

 strength per unit volume being 4>, and it is therefore obvious 

 at once that the equation V 2 < + A^ = is satisfied at all points 

 P external to R. The only question of any difficulty arises 

 when P is inside R. We then divide R into two regions R l 

 and R 2) of which R 2 encloses P and is ultimately taken to be 

 infinitely small in all its dimensions. The parts of at P due to 

 the sources in R l and R% , respectively, may be denoted by fa and 

 < 2 - Since P is external to ^ we have V 2 ^ + A; 2 ^ = as before. 

 Within R 2 we may ultimately put e~ ikr l t and <f> 2 then 

 approximates to the gravitation potential of matter of density 

 4>/47r restricted to the space R 2 . We have then, ultimately, 

 on known principles, V 2 < 2 = <I> and < 2 = 0. Hence (1) is satisfied 

 by <f> = </>! + </> 2 . It is further evident that (6) is the solution 

 of (1) consistent with the condition that there are no sources of 

 sound except at the points to which <l> refers. 



When forces X, Y, Z per unit mass act at the various 

 points of a gaseous medium, the equations (4) of 68 are 

 replaced by 



9 <> +z , g = _ c ,^ + r , *?__<-* + *. ...(7) 



dt dx dt dt dt dz 



