CHAPTER IV. 



MOTION OF A LIQUID IN TWO DIMENSIONS. 



59. IF the velocities u, v be functions of x, y only, whilst w 

 is zero, the motion takes place in a series of planes parallel to xy, 

 and is the same in each of these planes. The investigation of the 

 motion of a liquid under these circumstances is characterized by 

 certain analytical peculiarities; and the solutions of several pro- 

 blems of great interest are readily obtained. 



Since the whole motion is known when we know that in the 

 plane 2 = 0, we may confine our attention to that plane. When 

 we speak of points and lines drawn in it, we shall understand 

 them to represent respectively the straight lines parallel to the 

 axis of z, and the cylindrical surfaces having their generating 

 lines parallel to the axis of z, of which they are the traces. 



By the flux across any curve we shall understand the volume 

 of fluid which in unit time crosses that portion of the cylindrical 

 surface, having the curve as base, which is included between the 

 planes z = 0, z 1. 



Let A, P be any two points in the plane xy. The flux across 

 any two lines joining AP is the same, provided they can be 

 reconciled without passing out of the region occupied by the 

 moving liquid ; for otherwise the space included between these 

 two lines would be gaining or losing matter. Hence if A be 

 fixed, and P variable, the flux across any line AP is a function 

 of the position of P. Let -x/r be this function ; more precisely, let 

 ^r denote the flux across AP from right to left, as regards an 

 observer placed on the curve, and looking along it from A in the 

 direction of P. Analytically, if I, m be the direction-cosines of the 



