CHAPTER II. 



INTEGRATION OF THE EQUATIONS IN SPECIAL CASES. 



18. IN a large and important class of cases the component 

 velocities u, v, w can be expressed in terms of a single function 

 &amp;lt;f&amp;gt;, as follows : 



d(f&amp;gt; dd&amp;gt; dd&amp;gt; ,, x 



u = f t v = -7 1 -, w = f (1). 



dv dy dz 



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

 with 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 : 



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 

 instants before or after*. 



In the equations of Art. 15, let the instant at which the 



* Lagrange, &quot; Me moire sur la Th6orie du Mouvement des Fluides,&quot; Nouv. 

 mim. de VAcad. de Berlin, 1781 ; Oeuvres, t. iv. p. 714. The argument is repro 

 duced in the Mecanique Anatytique. 



Lagrange s statement and proof were alike imperfect ; the first rigorous demon 

 stration is due to Cauchy, &quot; Memoire sur la Th6orie des Ondes,&quot; Mem. de VAcad. 

 roy. des Sciences, t. i. (1827) ; Oeuvres Completes, Paris, 1882..., l re Se&quot;rie, t. i. p. 38 ; 

 the date of the memoir is 1815. Another proof is given by Stokes, Camb. Trans, t. 

 viii. (1845) (see also Math, and Pliys. Papers, Cambridge, 1880..., t. i. pp. 106, 158, 

 and t. ii. p. 36), together with an excellent historical and critical account of the 

 whole matter, 



