CHAPTER II. 



INTEGRATION OF THE EQUATIONS IN SPECIAL CASES. 



22. IN most cases of interest the external impressed forces 

 have a potential ; viz. we have 



dV dV dV 



~d^ y ~-fy ~Tz- 



In a large and important class of cases the component veloci 

 ties u, v, w can be similarly expressed as the partial differential 

 coefficients of a function &amp;lt;/&amp;gt;, so that 



d6 dd&amp;gt; d(j&amp;gt; , 



u=~, v = -j*-, w = -r- (2). 



dx dy dz 



Such a function is called a velocity-potential/ from its 

 analogy to the potential function which occurs in the theories of 

 Attractions, Electrostatics, &c. The general theory of the velocity- 

 potential is reserved for the next chapter ; but we give at once a 

 proof of the following important theorem : 



23. If a velocity-potential exist, at any one instant, for any 

 finite portion of a perfect fluid in motion under the action of 

 forces which have a potential, then, provided the density of the 

 fluid be either constant or a function of the pressure only, a 

 velocity-potential exists for the same portion of the fluid at all 

 subsequent instants. 



In the equations of Art. 18, let the instant at which the 

 velocity-potential &amp;lt; exists be taken as the origin of time; we 

 have then 



u da + v db + w Q dc d(f&amp;gt; Q , 



throughout the portion of the mass in question. Multiplying the 



