received Theory of Sound. 55 



Now the function y( .2) is merely defined as an integral of 

 f\z)dzi and we may suppose the integral so chosen as to 

 vanish when z=.b, and therefore when z has any smaller value. 

 Consequently we get from (4.), for every point within the 

 sphere which forms the inner boundary of the wave of con- 

 densation, 



^=-^ (5.) 



Again, if we put /(c) = A, so thaty(2r) = A when 2>-c, we 

 have for any point outside the wave of condensation, 



A + D .^, 



v= ^:^-. ..... (6.) 



The velocities expressed by (5.) and (6.) are evidently such 

 as could take place in an incompressible fluid. Now Professor 

 Challis's reasoning requires that the fluid be at rest beyond 

 the limits of the wave of condensation, since otherwise the 

 conclusion cannot be drawn that the matter increases with the 

 time. Consequently we must have D = 0, A = ; butifA = 

 the reasoning at p. ^es evidently falls to the ground. 



2. We may if we please consider an outward travelling 

 wave which arose from a disturbance originally confined to a 

 sphere of radius 5. At p. 463 Professor Challis has referred 

 to Poisson's expressions relating to this case. It should be 

 observed that Poisson's expressions at page 706 of the Traits 

 de Mecanique (second edition) do not apply to the whole wave 

 from r = ai — s to r=ai + e, but only to the portion from 

 r=:ai — s to r=:ai; the expressions which apply to the re- 

 mainder are those given near the bottom of page 705. We 

 may of course represent the condensation s by a single function 



— x(f—(^i)i where 



X{-Z)=f{z\ %(^) = F(.), 

 z being positive ; and we shall have 



K=f\{z)dz=f{s) -/(O) + F(s)- F(0). 



Now Poisson has proved, and moreover expressly stated at 

 page 706, that the functions F,/ vanish at the limits of the 

 wave; so that/(s) = 0, F(6)=:0. Also Poisson's equations (6.) 

 give in the limiting case for which 3 = 0, /(0)+F(0) = 0, so 

 that A = as before. 



3. We may evidently without absurdity conceive the velo- 

 city and condensation to be both given arbitrarily for the in- 

 stant at which we begin to consider the motion ; but then we 



