478 WAVES OF EXPANSION. [CHAP. X 



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



/&amp;gt; fKCfc 

 J P J Po 



whence tfs = -~ (1). 



(MI 



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 + $r is gaining mass 

 at the rate 



dH ^dr, 



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



dp d 



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 



_ _ 



dt~r*dr\ r dr 



whence, substituting from (1), 



^-^A( r 

 d? r* dr V dr 



This may be put into the more convenient form 



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 



= --F (r-ct), 



which shews that a condensation is propagated outwards with 

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



