192, 193, 194, 195] VELOCITY DUE TO VORTEX. 515 



194. Kinetic Energy of Vortex. The kinetic energy of the 

 incompressible liquid moving vertically is 



80) T - <>(u* + v* + w^ dr 



dlJ cW 



V\ 

 - - Ti) 



= 2 / / K* 7 w - w F ) cos ( Wfl O + (wUuW) cos (ny) 



-f 



If the integral be taken over all space, since the motion is supposed 

 to vanish at infinity the surface integrals vanish, and 



81) T= <>fff[ Vt + Vrt + Wf\ dr, 



or inserting the values of U, V, W from 70) 



and the integration may now be restricted to the vortices. 

 If again we integrate by filaments, we find 



where the integration is expressed as over the length of each of the 

 double infinity of vortex -filaments constituting the vortices. This is 

 the form obtained for the energy of two electric currents by Franz 

 Neumann. 



195. Straight parallel Vortices. Let us now consider the 

 case in which the vorticity is everywhere parallel to a single direc- 

 tion, that of the axis of g. Let the motion be uniplanar, that is 

 parallel to a single plane, the X3^-plane, and the same in all planes 

 parallel to it. All quantities are therefore independent of g. The 

 vortices are columnar and either of infinite length or end at the free 

 surface of the liquid. Such vortices may be produced standing 

 vertically in a tank with a horizontal bottom. Under the conditions 

 imposed we have 



QA\ du dv CM c. rr rr 



84) = w = TT- = Q- = ^ = $ = 7? = c7=F, 



dz dz dz 



and 



QKX dW dW 



85) u = -5> v = -- ;> 



cy Ox 



33* 



