direction, namely, the line parallel to the axis of rotation. 



9. THE PRESSURE EQUATION FOR IRROTATIONAL FLOW 



Let the external forces per unit mass be given by 



X = -^, y = -iil,Z = -^ [9a, b, c] 



d X d y d z 



in terms of a potential function Q, as is true for forces due to gravity; and let the fluid have 

 the property that its pressure is a definite function of its density. Then, if the motion is 

 irrotational, the equations of motion for a frictionless fluid can be integrated. 



On these assumptions, Equation [3c] can be written, after replacing m by - dcji/dx in the 

 first term, 



dt dx dx dy dz dx p dx 



or, by means of Equations [8b] and [8c], after changing signs, 



^x dt dx dx dx dx p dx 



Equations [3d] and [3e] become similarly 



dy dt dy dy dy dy p dy 



d ^'^.- u ^ - v^ - w^"^ - ^^^ - ^- 



and 



dz dt dz dz dz dz P dz 



Let these last three equations be multiplied through by dx, dy, dz and added; and let the time 

 t be held constant. Then the first terms give 



dx A ^+dy A ii +d2 ± -= dl— ] 

 dx dt dy dt dz dt \dt I 



In eeneral, d — would contain also a term dt ^\ — |; but here it is assumed that dt = 0. The 



^ dt dtXdt) 



next terms in the equations give in the sum 



ui^dx + u^dy + u^dz = ±il uAdx -dUJ] 



dx dy dz dx\2 / \^ I 



The remaining terms give similar results. Thus the final result is 



df 

 P 

 This can be integrated to give 



d{^\-d(^.-^.y^\-d^- 



\dtl \2 2 2 I 



J P dt 2^ 



15 



