24 INTEGKATION OF THE EQUATIONS IN SPECIAL CASES. [CHAP. IT. 



or, using G in the same sense as before, 



+ lg 2 -f V=C ...................... (10), 



which is what the equation (9) 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 



I pd (-} , or + I , per unit mass. The addition of these 



J \P P J P 



terms converts the equation (10) into the equation (9). 



In most cases of motion of gases, the relation (4) of Art. 7 

 holds, and (9) then becomes 



30. Equations (10) and (11) shew that, in steady motion 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, is obvious if we reflect that a 

 particle passing from a place of higher to one of lower pressure 

 must have its motion accelerated, and vice versa. Some interesting 

 practical illustrations and applications of the principle are given by 

 Mr Froude in Nature, Vol. xm. 1875. 



It follows that in any case to which the aforesaid equations 

 apply there is a limit which the velocity cannot exceed if the 

 motion be continuous. For instance, let us suppose that we have 

 a liquid flowing from a reservoir where the motion is sensibly 

 zero, and the pressure equal to P, and that we may neglect the 



P 



external impressed forces. We have then in (10) C = , and 



P 

 therefore 



2P 



Hence if (f exceed , p becomes negative, whereas we know that 



actual fluids are unable to support more than a very slight, if any, 



* Tait s Thermodynamics, Art. 174 (first edition). 



t This restriction is, by (9), unnecessary when a velocity-potential exists. 



