J. W. Davis — Motion of Compressible Fluids. 113 



the critical section the density in one of the flows is every- 

 where considerably greater than in the other, and, therefore, 

 the mass of fluid is considerably different in the two flows. 

 Now suppose the quantity, M, of fluid passing any section in 

 unit time increases from zero to the capacity of the critical 

 section. Through all this range the velocity of the fluid 

 everywhere in the tube remains on one side of the velocity of 

 sound until at the limit it may still remain everywhere on that 

 side, or may pass from one side to the other at the critical 

 section. Hence one solution at the limit is continuous with 

 all the previous solutions, and the other is isolated. The iso- 

 lated case cannot originate from a flow that increases to the 

 capacity of the tube, for this would involve at the final instant 

 in the total mass of the fluid an instantaneous change of con- 

 siderable magnitude. Neither can it exist in a tube that 

 returns into itself and has only one section of least area or of 

 greatest potential. 



The behavior of the conjugate flows in any space may be 

 summed up as follows : If in either flow the velocity of the 

 fluid at any point is less than the velocity of sound at that 

 point, then, in general, the velocity of the fluid is everywhere 

 less than the velocity of sound, or may reach the latter as a 

 limit. If the velocity is greater than that of sound at one 

 place, it is greater at all places, or may diminish to the velocity 

 of sound as a limit. One conjugate velocity is everywhere 

 greater than the other conjugate velocity, or may diminish to 

 equivalence with the latter as a limit where each becomes 

 equal to the velocity of sound. The places of greatest 

 velocity and least density in one case are the places of least 

 velocity and greatest density in the other. In the approach to 

 a gorge or place of higher potential the more rapid flow is 

 retarded and the other is accelerated ; the recession from a 

 gorge or place of higher potential produces inverse effects. 

 Under very special conditions the velocity of the fluid is not 

 limited by the velocity of sound. 



There is an inferior limit, p , to the density of a compres- 

 sible liquid, whereas the density in the pressure-density formula, 

 (3), of that liquid, and, therefore, in equation (4), is continu- 

 ous from zero to infinity. Hence the foregoing conclusions 

 have a limited application to liquids. Suppose the line p — p 

 to be drawn in figure 1. If now the actual motion is such 

 that both intersections of (4) and (2) always occur on the posi- 

 tive side of the line p — p for every elemental tube of flow, 

 there is also a conjugate motion ; otherwise, there is no conju- 

 gate motion, and the conclusions of the preceding paragraph 

 are only partially applicable. 



