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



pressures, p', p" . Of these two sets of values only one can 

 represent the actual motion of the fluid ; let this be q\ p',p'. 

 Necessarily q', p' are everywhere positive, according to the 

 first paragraph, since the motion is actual ; consequently q" , p" 

 are everywhere positive, according to the preceding paragraph. 

 Hence the motion represented by q'\ p n ,p", which may be 

 designated the conjugate motion, is physically real, and is con- 

 nected with the actual motion by the relation, contained in 

 (1), (2), that in each case the same quantity of energy, CM, 

 and the same quantity of matter, M, pass any section in unit 

 time. 



Pass now to a consecutive tube. The parameters and vari- 

 ables of the actual motion, as contained in equations (1), (2), 

 (3), suffer an infinitesimal variation in this passage, one factor 

 of the variation being the distance between the tubes. As 

 this distance is brought to its limiting value, zero, the varia- 

 tion in p' becomes zero, so that the normal pressures on oppo 

 site sides of a stream-line at every point are equal. That this 

 condition of equilibrium exists is necessary to the hypothesis 

 that there is an actual flow along these stream-lines. But the 

 conjugate flow has the same parameters as the actual flow, and, 

 therefore, in the passage from tube to tube, also suffers an 

 infinitesimal variation that vanishes at a stream-line. Conse- 

 quently, the conjugate flow is in equilibrium along the same 

 stream-lines. It thus appears that any actual case of steady 

 motion of a perfect compressible fluid whose compressibility 

 is defined in expression (3), is one of two physically real solu- 

 tions of the equations ; and that the motions represented by 

 these solutions differ in velocity at every point, but have the 

 same stream-lines and boundaries, and discharge in unit time 

 across any fixed mathematical surface the same quantity of 

 matter and the same quantity of energy. 



Differentiate (4) with respect to s, for the moment designat- 

 ing the quantity after <f> by r : 



dp dr _ ( fdp ) 1 dp ,r\nr> ^ \ \ ^Q dq 



dr ds 



4>' 



/"flirt =*'{<->- f!i-f-«ll-<'») 



Divide by the differential equation of (1) with respect to s ; 

 there results 



*{/ 



JSTow differentiate (5), using the several values of p, dp-i-ds, 

 and <j>' in (4), (10), and (11) as convenience directs : 



p dT +q l& \-~ds-- q ds ] = -Vdi> (12 > 



