22 INTEGRATION OF THE EQUATIONS IN SPECIAL CASES. [CHAP. II. 



Taking ds in the direction of the normal to the surface &amp;lt; = const, 

 we see that if a series of such surfaces be drawn so that the differ 

 ence between the values of (/&amp;gt; for two consecutive surfaces is con 

 stant and infinitely small, the velocity at any point will be inversely 

 proportional to the distance between two consecutive surfaces in 

 the neighbourhood of that point. 



Hence, if any equipotential surface intersect itself, the velocity 

 is zero at every point of the intersection. 



The intersection of two distinct equipotential surfaces would 

 imply an infinite velocity at all points of the intersection. 



Steady Motion. 



28. When at every point the velocity at that point is constant 

 in magnitude and direction, i. e. when 



du A dv dw /H7N 



dT dt = dt= (7) 



everywhere, the motion is said to be steady. 



In steady motion the lines of motion coincide with the paths 

 of the particles and are in this case called stream-lines. For let 

 P, Q be two consecutive points on a line of motion. A particle 

 which is at any instant at P is moving in the direction of the 

 tangent at P, and will, therefore, after an infinitely short time 

 arrive at Q. The motion being steady, the lines of motion remain 

 the same. Hence the direction of motion at Q is along the tangent 

 to the same line of motion, i.e. the particle continues to describe 

 the line of motion. 



In steady motion the equation (3) becomes 



^ = - V- i? 2 + constant (8). 



P 



The equations of motion may however in this case be inte 

 grated to a certain extent without assuming the existence of a 

 velocity-potential. For if ds denote an element of a stream-line, 



we have u = q -y- , &c. Substituting in the equations of motion 



we have, remembering (7), 



du ^ 1 dp 

 0--J-X -f , 



* ds p ax 



