19-21] VELOCITY-POTENTIAL. 21 



equidistant values of <f>, the common difference being infinitely 

 small, the velocity at any point will be inversely proportional to the 

 distance between two consecutive surfaces in the neighbourhood 

 of the point. 



Hence, if any equipotential surface intersect itself, the velocity is zero at 

 the intersection. The intersection of two distinct equipotential surfaces would 

 imply an infinite velocity. 



21. Under the circumstances stated in Art. 18, the equations 

 of motion are at once integrable throughout that portion of the 

 fluid mass for which a velocity-potential exists. For in virtue 

 of the relations 



dv _dw dw _ du du _ dv 



dz ~ dy ' dx dz ' dy ~~ dx ' 



which are implied in (1), the equations of Art. 6 may be written 



d 2 (f> du dv dw _ cZfl __ 1 dp 

 dxdt dx dx dx dx p dx' 



These have the integral 



where q denotes the resultant velocity (u? + v' 2 + w 2 )*, and F (t) is 

 an arbitrary function of t. It is often convenient to suppose this 

 arbitrary function to be incorporated in the value of d<p/dt ; this 

 is permissible since, by (1), the values of u, v, w are not thereby 

 affected. 



Our equations take a specially simple form in the case of an 

 incompressible fluid ; viz. we then have 



with the equation of continuity 



,d*4>_ 

 ~ 



which is the equivalent of Art. 9 (1). When, as in many cases 

 which we shall have to consider, the boundary conditions are 

 purely kinematical, the process of solution consists in finding a 

 function which shall satisfy (5) and the prescribed surface-con- 

 ditions. The pressure p is then given by (4), and is thus far 



