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



The kinetic energy of the fluid is given by 



2T= p Ud^ = pC*e-** 1^ cos-rjdrj 



J Jo 



= 7r / o& 2 U 2 ........................ (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 



whence 



These formulae 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, 



\lf= -(- J-) e~* (ubsinrj-vacosr)) ..................... (i). 



\a-oj 



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

 uy - v#= c (u sinh sin r) - v cosh cos ?;) ............... (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 



(iii), 



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. 



