254 VORTEX MOTION. [CHAP. VII 



of either vortex moves perpendicular to AB with a velocity 

 m/Tra.cot^AB. The two vortices therefore describe parallel 

 and equal small circles, remaining at a constant distance from 

 each other. 



Circular Vortices. 



160. Let us next take the case where all the vortices present 

 in the liquid (supposed unlimited as before) are circular, having 

 the axis of a? as a common axis. Let CT denote the distance of any 

 point P from this axis, ^ the angle which -cr makes with the plane 

 xy, v the velocity in the direction of -cr, and co the angular 

 velocity of the fluid at P. It is evident that u, v, co are functions 

 of x, & only, and that the axis of the rotation is perpendicular 

 to X-GT. We have then 



y = in cos ^, = -or sin ^, } 



v= ucos^, w=usin^, > ............... (1). 



f = 0, 7j = co sin ^, f = w cos ^ J 



The impulse of the vortex-system now reduces to a force along 

 Ox. Substituting from (1) in the first formula of Art. 150 (12) 

 we find 



(2), 



where the integration is to extend over the sections of all the 

 vortices. If we denote by m the strength a>&p8tz7 of an elementary 

 vortex-filament whose coordinates are x, OT, this may be written 



= 2-7T/3 . 2m . CTo 2 ..................... (3), 



The quantity r c , thus defined, may be called the ' mean-radius ' 

 of the whole system of circular vortices. Since m is constant for 

 each vortex, the constancy of the impulse requires that the mean- 

 radius shall be constant with respect to the time. 



The formula for the kinetic energy (Art. 153 (4)) becomes, in 

 the present case, 



T = 4oFpff{*FU xv) vr(i)dxd'& = 4nrp%m (VFU xv) TV ...... (5). 



