94 Lord .Rayleigh on the Propagation of 



When X is sufficiently great we may ignore altogether 

 the space between x x and x 2 , that is we may suppose that the 

 pressures are the same at these two places and that the total 

 flow is also the same, as if the fluid were incompressible. 

 As there is now no need to distinguish between x 2 and # 2 , 

 we may as well suppose both to be zero. The condition 

 <£i = <£ 2 g^es 



A + B = C, (25) 



and the condition cr x d§ildx — cr 2 d§ 2 \dx gives 



oi(-A + B) = -<r 2 C (26) 



Thus B ai _ a a 2 



A <T\ + CT2 A Vl-tO: 



(27) 



These are Poisson's formulae *. If a i and <x 2 are equal, 

 we have of course B = 0, C = A. Our task is now to 

 proceed to a closer approximation, still supposing that the 

 region of irregularity is small. 



For this purpose both of the conditions just now employed 

 need correction. Since the volume V of the irregular region 

 is to be regarded as sensible and the fluid is really susceptible 

 of condensation (.<?), we have 



y ds = d&_ to, 



dt dx x dx 2 



and since in general 6-= —a 2 d^>/dt, we may take 



dt" a dt* ^ ~~ a ~dt^ 



the distinction being negligible in this approximation in 

 virtue of the smallness of V. Thus 



<7 # 1 _ <r2 ^ == _V^ = A2 y^. . . (28 ) 



dx x dx 2 a ,J dt 2 



In like manner, assimilating the flow to that of an 

 incompressible fluid, we have for the second condition 



fc-fc = Roig, (29) 



where R may be defined in electrical language as the 

 resistance between X\ and x 2 , when the material supposed to 

 be bounded by non-conducting walls coincident with the 

 walls of the tube is of unit specific resistance. 



* Compare « Theory of Sound,' § 264. 



