318 Dr. C. Burton on Plane and Spherical 



opposite to that with which the surface of discontinuity at 

 first moves, to be impressed upon the whole mass of fluid. In 

 order to find the relations which must subsist between the 

 velocity and density on the one side (u^ p Y ) and the velocity 

 and density on the other side (w 2 , p 2 ), we notice in the first 

 place that by the principle of the conservation of matter 

 p 2 u 2 =p 1 u 1 . Again, if we consider the momentum of a slice 

 bounded by parallel planes and including the surface of dis- 

 continuity, we see that the momentum leaving the slice in the 

 unit of time is for each unit of area (p2 u 2 — Pi u i) u 2i while the 

 momentum entering it is piU\. The difference of momentum 

 must be balanced by the pressures acting at the boundaries of 

 the slice, so that 



whence 



pxUifa — u x ) =p x —p 2 = a 2 (p x - p 2 ), 



The motion thus determined is, however, not possible ; it 

 satisfies indeed the conditions of mass and momentum, but it 

 violates the condition of energy expressed by the equation 



i u 2 2 — i V = a 2 log p x — a 2 log p 2 ." 



2. The assumed motion here criticised is one in which 

 density and velocity are constant for all points on the same 

 side of the surface of discontinuity, while this surface itself is 

 propagated through the fluid with constant velocity. It is 

 easily shown, however, that the same objection applies when, 

 on either side of the surface, velocity and density vary con- 

 tinuously in the direction of propagation, while the velocity 

 of propagation of the surface is also allowed to vary. For 

 let IS (fig. 1) be a surface of discontinuity which 

 is being propagated through the fluid, while the Fig. 1. 

 planes A, B, parallel to S and lying on either 

 side of it, are fixed in the fluid. At a given 

 instant let 



distance of S from A = m, 



„ B „ S = n; 



density and velocity of fluid just to the left 



of S=/0i, u u 



density and velocity of fluid, just to the right 



ofS=p 2 ,« 2 ; AS B 



velocity with which 8 is travelling = V. 



