24 INTEGRATION OF THE EQUATIONS IN SPECIAL CASES. [CHAP. II 



or, using C in the same sense as before, 



= -n-Jg + C ........................ (4), 



which is what the equation (3) becomes when p is constant. 



To prove the corresponding formula for compressible fluids, we 

 remark that the fluid entering at A now brings with it, in addition 

 to its energies of motion and position, the intrinsic energy 



per unit mass. The addition of these terms to (4) gives the 

 equation (3). 



The motion of a gas is as a rule subject to the adiabatic law 



P/Po = (p/Po) y ........................... (5), 



and the equation (3) then takes the form 





(6). 



24. The preceding equations shew that, in steady motion, 

 and for points along any one stream-line*, the pressure is, 

 cceteris paribus, greatest where the velocity is least, and vice versa. 

 This statement, though opposed to popular notions, becomes 

 evident when we reflect that a particle passing from a place of 

 higher to one of lower pressure must have its motion accelerated, 

 and vice 



It follows that in any case to which the equations of the last 

 Art. apply there is a limit which the velocity cannot exceed J. For 

 instance, let us suppose that we have a liquid flowing from a 

 reservoir where the motion may be neglected, and the pressure is 

 p , and that we may neglect extraneous forces. We have then, in 

 (4), C = p /p, and therefore 



Now although it is found that a liquid from which all traces 



* This restriction is unnecessary when a velocity-potential exists. 



t Some interesting practical illustrations of this principle are given by Froude, 

 Nature, t. xiii., 1875. 



J Cf. von Helmholtz, " Ueber discontinuirliche Fliissigkeitsbewegungen," Berl. 

 Monatsber., April, 1868; Phil. Mag., Nov. 1868; Gesammelte Abhandlungen, Leipzig, 

 1882-3, t. i., p. 146. 



