244 VORTEX MOTION. [CHAP. VII 



where f denotes the value of f at the point (x t y ), and r now 

 stands for 



The complementary function -&amp;lt;fr may be any solution of 



V, 2 V. = ........................... (6); 



it enables us to satisfy the boundary-conditions. 



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

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



(7). 



x x , , 



- 



Hence a vortex-filament whose coordinates are a? , y and whose 

 strength is ??^ contributes to the motion at (x, y} a velocity whose 

 components are 



m y y f , m x x 



-- . XffJ- , and . - . 



TT r 2 TT r 2 



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

 (# , y \ and its amount is m /irr. 



Let us calculate the integrals ffugdxdy, and ffvdacdy, where 

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

 does not vanish. We have 



gf. dxdy dx dy , 



where each double integration includes the sections of all the 

 vortices. Now, corresponding to any term 



K dxdydxdy 1 



of this result, we have another term 



and these two terms neutralize one another. Hence 



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



