516 XI. HYDRODYNAMICS. 



86 ) 2 ? = ^-|' 



and by equations 57) 



so that is independent of the time for any given vortex -filament. 

 The function W is not a velocity potential, but is said to be 

 conjugate to a velocity potential (p for which 



87 ) fe If- 



The function TF has a simple physical meaning. If we find the 

 amount of liquid which flows across a cylindrical surface with 

 generators parallel to the #-axis of height unity in unit of time, 

 we have B B 



. 88) iff = I q cos (qn) ds = I [u cos (nx) + v cos (ny)\ ds, 



A A 



the line integral being taken around any orthogonal section of the 

 cylinder. Now we have 



ds cos (nx) = dy, 



ds cos (ny) = dx, 

 so that 



89) * -{ ~ vdx) -dx + dy -W,- W A . 



A function, the difference of whose values at two points A and J5 

 gives the quantity flowing in unit time across a cylinder of unit 

 height drawn on any curve with ends at A and jE?, is called a flux 

 or current function. The quantity crossing is independent of the 

 curve because the fluid is incompressible. In the present case the 

 vector potential "FT is a current function. The stream lines being 

 lines across which no current flows are given by the equation ^ = const. 

 Substituting the values of u and v from 85) in 86), we have 



But this is the equation for a logarithmic potential with density 

 138, 61), so that we have as the integral 



91) W=- 



as may also be found from equation 70) by integrating over the 

 infinite cylinder as in 135, subject to the difficulty mentioned on 

 p. 385. The value of W given in equation 91) satisfies the equation 



