244 VORTEX MOTION. [CHAP. VII 



where f denotes the value of f at the point (#', y'), and r now 

 stands for 



The ' complementary function ' fa may be any solution of 



V^o-O ........................... (6); 



it enables us to satisfy the boundary-conditions. 



In the case of an unlimited mass of liquid, at rest at infinity, 

 we have fa = const. The formulae (3) and (5) then give 



(7). 





Hence a vortex-filament whose coordinates are #', t y' and whose 

 strength is ra' contributes to the motion at (oc, y) a velocity whose 

 components are 



m/ y~y' , m' x x' 



-- . - /- , and . . 



TT r 2 TT r 2 



This velocity is perpendicular to the line joining the points (#, y), 

 (a?', y'\ and its amount is m'/rrr. 



Let us calculate the integrals jju^dxdy, and ffvdxdy, where 

 the integrations include all portions of the plane xy for which f 

 does not vanish. We have 



jfedxdy = - IfjjfeljfMb dx'dy', 



where each double integration includes the sections of all the 

 vortices. Now, corresponding to any term 



^y^/' dxdydx'dy' 



of this result, we have another term 



and these two terms neutralize one another. Hence 



........................... (8), 



