J (^2) ^s^s=-J (S2) 9s^s 



since interchange of the limits reverses the direction of motion along the path and hence the 

 positive direction for q^ . Hence 



) qds= I (e,) « ^, 



^1 ^1 



•2' 'is^'s 



It follows that, if qS is defined as 



n 



-0 



J 



Is <is t6a] 



along any path joining any point P to a fixed point P^, ^ will be a single-valued function of the 

 coordinates x, y, z of P. 

 Furthermore, 



2 

 dx" ' dy' dz' ^ \dx/ \dy> 



—• — — :-^=Q^Q^o '--^-' 



where u, v, w are the components of the particle velocity at P. For, let the path be drawn so 



as to run from P straight to a neighboring point P', which is displaced a distance 8x from P 



toward + x without change of y or 2, as in Figure 6b. Then, if 84> is the difference in the 



values of (A at P ' and at P, 



p, . 



Is ^« 



'P 



since the path from P' differs from that drawn from P only by the omission of the additional 



stretch PP\ But along this stretch q^ = u, and is constant in the limit as P ' approaches P. 



Hence 



P/ 



ds = - u8x 



P 



Equation [6b] follows; and Equations [6c, d] can be similarly obtained. 



The function thus introduced is called the velocity potential. If it is known at all 

 points, the particle velocity can be found from it by differentiation. The sign has been chosen 

 in such a way that the potential decreases in the direction of the particle velocity. The 

 relation between the velocity and the velocity potential is the same as the relation between 

 the electric intensity and the electrostatic potential. 



Since the position of P^ is arbitrary, the velocity potential, like the electrostatic 

 potential, contains an arbitrary additive constant. A surface over which ^ has a constant 

 value is called an equipotential surface. As the derivative of (/> with respect to any element 



-I 



