478 WAVES OF EXPANSION. [CHAP. X 



In the notation of Arts. 254, 255 we may write 



[dp [teds 



= I - - = &s, 

 lp ! po 



7 



whence c 2 s = -i? (1). 



dit 



To form the equation of continuity we remark that, owing to 

 the difference of flux across the inner and outer surfaces, the 

 space included between the spheres r and r + Br is gaining mass 

 at the rate 



drV r dr, 

 Since the same rate is also expressed by dpjdt . 4?rr 2 Br we have 



This might also have been arrived at by direct transformation of 

 the general equation of continuity, Art. 8 (4). In the case of 

 infinitely small motions, (2) gives 



ds_l d d<t> 



whence, substituting from (1), 



_ 

 dt* ~ r 2 dr dr 



This may be put into the more convenient form 



*.*!>_ ffi.'r* ( } 



~W~ ~~d^~' 



so that the solution is 



) .................. (6). 



Hence the motion is made up of two systems of spherical waves, 

 travelling, one outwards, the other inwards, with velocity c. 

 Considering for a moment the first system alone, we have 



which shews that a condensation is propagated outwards with 

 velocity c, but diminishes as it proceeds, its amount varying 



