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



or, using C in the same sense as before, 



(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 



-K) M?- 



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 



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



and the equation (3) then takes the form 



~i- p = - n -^+ G .................. () 



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

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

 ccvteris 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 versd&quot;^. 



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 

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

 (4), C = PQ/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, &quot; Ueber discontinuirliche Fliissigkeitsbewegungen,&quot; Berl. 

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

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



