354 Mr. G. G. Stokes on a difficulty in the Theory of Sound* 



It remains to form the equation of motion. Now we have 



moving force on plane A = a 2 p, ultimately; 



moving force on plane B = a 2 p', ultimately; 



.*. resultant moving force = a^(p—p'), 



.'.momentum generated in time dt=a\p—p')dt. 



Now the momentum of the mass contained within the cylinder 

 at the time / is ultimately pwh + p'w'h', and the momentum of 

 the same set of particles at the time t + dt is 



V W+ -^- dt ){h+^^wdt)-\- ypW + -^dt^{h' + wn~Zdt)[; 



and therefore the gain of momentum is ultimately 



{ (pw — p'lv 1 ) » — f>w 2 -f- pW 2 } dt ; 

 whence we have 



(pw— pW)e — (pw 2 — pV 2 )=a 2 (p— p'). . . . (3.) 

 By eliminating between (2.) and (3.), we get 



(uf-wYpp^a^p-p'f, .... (4.) 



an equation which we may if we please employ instead of (3.). 



The equations (2.), (3.) being satisfied, it appears that the 

 discontinuous motion is dynamically possible. This result, 

 however, is so strange, that it may be well to consider more 

 in detail the simplest possible case of such a motion. 



Conceive then an infinitely long straight tube filled with air, 

 of which the portion to the left of a certain section s is of a 

 uniform density p, and at rest, while the portion to the right 

 is of a uniform smaller density p', and is moving in the positive 

 direction with a uniform velocity w/, the surface of separation 

 5 at the same time travelling backwards into the first portion 

 with the uniform velocity ». The conception of such a motion 

 having been formed, consider next whether the motion is 

 possible or impossible; that is to say, not whether it is pos- 

 sible or impossible in the actual state of elastic fluids, but 

 whether it would or would not be consistent with dynamical 

 principles in the case of an ideal elastic fluid, in which the 

 pressure was equal in all directions about the same point, and 

 varied as the density. 



In the case under consideration, the fluid to the left of s is 

 in equilibrium in the simplest way. The fluid to the right is 

 of uniform pressure, and there is no generation or destruction 

 of velocity. The only question, then, can be as to the possi- 

 bility of the passage from the one state into the other taking 

 place in the way supposed. In the first place, it is evident 

 that, independently of any consideration of force, there must 

 be a relation between p } p' f w', and » ; for a length » t of con- 



