of distance along the surface is zero, there can be no component of the particle velocity 

 tangent to an equipotential surface; compare Equation [6f] below. Thus the direction of the 

 velocity is everywhere normal to the equipotential surfaces; and, since the streamlines have 

 everywhere the direction of the velocity, the streamlines cut the equipotential surfaces per- 

 pendicularly. 



Certain other relations between the velocity and the potential may be noted. The 

 component of the velocity in any given direction can be written 



1. =- 



dcf> 



[6f] 



here dcfj/ds is the space derivative of in the given direction or 



(9^ 



lim^ 



As-»o As 



where As is a displacement in that direction and A0 is the corresponding change in <^. The 

 proof is similar to that of Equations [6b, c, d]. By integrating Equation [6f] it is seen that the 

 change in from one end to the other along any path is 



A0 = -/^^c?s [6g] 



where the integral is taken along the path. 



It is often convenient to use spherical polar coordinates r, 0, co. Here r is the distance 

 from a fixed origin 0, is the angle between the line O;* and a fixed line or axis through 0, and 

 CO is the angle between the plane containing d and a fixed plane drawn through the fixed axis. 

 The definition is illustrated in Figure 7. A set of Cartesian coordinates is also shown having 



(r.e.cj) 



Figure 7 - Illustrating polar coordinates. 



10 



