256 VORTEX MOTION. [CHAP. VII 



To determine ^jr in terms of the (arbitrary) distribution of 

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

 which give 



F=0, 



i t [(*&amp;lt;** 



H = ^rjlj-^ 



ft) COS ^ / 7 / 7 / 7 f 



TS d^ dx d ur 



(4), 



where r = {(as # ) 2 + cr 2 + &quot; /2 2wj cos (^ & ))*. 



Since 27r^ denotes (Art. 93) the flux, in the direction of 

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



+4 II W If 



J J V dy o 



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

 Stokes Theorem, this gives 



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

 our present coordinates, 





provided 



f i ^ ^ i = r - _jBos^d0_ , , 



7 V, sr } Jo {(a? - # ) 2 + ^ 2 + ^ /2 ~ 2 ^ OT/ cos ^1- 

 where ^ has been written for $ & . 



It is plain that the function here denned is symmetrical with 

 respect to the two sets of variables a?, -BJ and x , *& . 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 xw. If we denote it by &amp;lt;S\ we have F= 0, G = - S sin ^, H = S cos ^, 

 so that (7) is equivalent to 



\f= -TffS. 



