Mr. G. G. Stokes on a difficulty in the Theory of Sound. 353 



which (A.) first becomes infinite, cannot influence a particle at 

 a finite distance from the former until after the expiration of 

 a finite time. Consequently, even after the change in the 

 nature of the motion, our original expressions are applicable, 

 at least for a certain time, to a certain portion of the fluid. It 

 was for this reason that I inserted the words " without limita- 

 tion," in saying that we are not at liberty to use our original 

 results without limitation beyond a certain value of t. The 

 full discussion of the motion which would take place after the 

 change above alluded to, if possible at all, would probably 

 require more pains than the result would be worth. 



I proceed now to consider the possibility of the existence 

 of a surface of discontinuity, and the conditions which must 

 be satisfied at such a surface. Although I was led to the 

 subject by considering the interpretation of the integral (1.), 

 the consideration of a discontinuous motion is not here intro- 

 duced in connexion with that interpretation, but simply for its 

 own sake ; and I wish the two subject? to be considered as 

 quite distinct. 



Suppose that in passing across a jfoint Q in the axis of # 

 the velocity and density change suddenly from w, p to «/, p', 

 and let a be the velocity of propagation of the surface of dis- 

 continuity. Let us first investigate the equation which ex- 

 presses that there is no generation or destruction of mass at 

 the surface of discontinuity, the equation in fact which takes 

 the place of the equation of continuit •, which has to be satisfied 

 elsewhere. 



Take two points A, B in the axis of z, the first on the ne- 

 gative and the second on the positive side of Q, and let QA = /z, 

 QB=//. Take also QQ'= tsdt, so that Q' is the point where 

 the surface of discontinuity cuts the axis of z at the time t + dt. 

 The quantities h, h' are supposed to be very small, and will 

 be made to vanish after QQ'. Consider the portion of space 

 comprised within a cylinder whose ends consist of two planes, 

 of area unity, drawn through the points A, B perpendicular 

 to the axis. In the time dt, the mass of fluid which flows in 

 at the plane A is ultimately pwdt, and that which flows out at 

 the plane B is ultimately p'ro'dt. Hence the gain of mass 

 within the cylinder is ultimately (pio—p'w^dt. Now the mass 

 at the time t is ultimately ph + p'h', and that at the time t + dt is 



(p + ?ldt)(h + »dt) + (pi+^di)(h>-»dt); 



and therefore the gain of mass is a(p—p')dt, h and h' being 

 omitted, since they are to be made to vanish in the end. Equa- 

 ting the two expressions for the gain of mass, we get 



pw—p'w'=(p—p')n (2.) 



