195] COLUMNAR VORTICES. 517 



outside of the vortices and equation 90) at points within them, as 

 shown in 138. If we have a single vortex filament of cross-section 

 dS and strength n = 'dS, 



93) W = - ^-logrdS=- -logr, 



7T 7C 



and the lines of flow are circles, r = const. Then 



= = ^ y-y 



dy 'XT' r 

 cW x x-x 1 



V = 



ex Ttr r 



95) 



,2 



the velocity is perpendicular to the radius joining the point x, y 

 with the vortex and inversely proportional to its length. It is to be 

 observed that although the motion is whirling, every point describing 

 a circle about the center, the motion is irrotational except at the 

 center, where the vortex -filament is situated, each particle describing 

 its path without turning about itself, like a body of soldiers obliquing 

 or changing direction while each man faces in the same unchanging 

 direction. The motion in the vortex on the contrary is similar to 

 that of a body of soldiers wheeling or changing direction like a rigid 

 body rotating. 



If we have a number of vortices of strengths ^, 3 2 , . . . %, and 

 form the linear functions of the velocities of each, 



Z7 



r, 



where u, 9 v sy is the velocity at the vortex s both vanish. For any 

 pair of vortices r and s we have 



where u r is the part of the velocity at x r , y r due to the vortex of 

 strength x, situated at x s y s . Thus 



while similarly 



x r -x s 



X S U S = K s X r -^ > 



so that the terms of the sum destroy each other in pairs. 



1) U and V have nothing here to do with the components of the vector- 

 potential. 



