256 VORTEX MOTION. [CHAP. VII 



To determine ifr in terms of the (arbitrary) distribution of 

 angular velocity (&>), we may make use of the formulae of Art. 145, 

 which give 



F=0, 



1 ff/Vcos^' ,_,, ,, , 

 H= fiJJJr- "<**"*- I 



-W, 



where r = {(x - #') 2 + ^ 2 + ^' 2 - 2rr' cos (S - &')}*. 



Since 2?njr denotes (Art. 93) the flux, in the direction of 

 ^-negative, through the circle (a?, w), we have 



where the integration extends over the area of this circle. By 

 Stokes' Theorem, this gives 



2ir^ = -f(Gdy + Hdz) ..................... (6), 



the integral being taken round the circumference, or, in terms of 

 our present coordinates, 



provided 



cos OdO 



, <a> ~ o {(x - xj + vr* + vr'* - 2w' cos 6} 

 where has been written for ^. 



It is plain that the function here denned is symmetrical with 

 respect to the two sets of variables x, and x' t '. It can be 

 expressed in terms of elliptic integrals, as follows. If we put 



_ ( . 



- '- 



* The vector whose components are F, G, H is now perpendicular to the 

 meridian plane xiz. If we denote it by S, we have ^=0, G = - S sin ^, H = S cos ^, 

 so that (7) is equivalent to 



