94 MOTION OF A LIQUID IN TWO DIMENSIONS. [CHAP. IV 



The kinetic energy of the fluid is given by 



2T= p Ud^ 



J 



o 

 ........................ (11), 



where b is the half-breadth of the cylinder perpendicular to the 

 direction of motion. 



If the units of length and time be properly chosen we may write 

 f% 



whence f*^(i+ 



These formula) are convenient for tracing the curves = const., ^ = const., 

 which are figured on the preceding page. 



By superposition of the results (8) and (9) we obtain, for the case of an 

 elliptic cylinder having a motion of translation whose components are u, V, 



^= - (^T V e~^ (u6 sin 77 - va cos 7?) ..................... (i). 



To find the motion relative to the cylinder we must add to this the expression 

 vy - vx c (u sinh sin rj - v cosh cos q) ............... (ii). 



For example, the stream-function for a current impinging at an angle of 45 

 on a plane lamina whose edges are at x=c is 



V^= - -TO 9o c sintl ( cos *) ~ sin ri ........................ (&quot;*) 



where q is the velocity at infinity. This immediately verifies, for it makes 

 ^ = for = 0, and gives 



for =oo. The stream-lines for this case are shewn in the annexed figure 



(turned through 45 for convenience). This will serve to illustrate some 

 results to be obtained later in Chap, vi, 



