137, 138.] RECTILINEAR VORTICES. 163 



since we are concerned only with the differential coefficients of N, 

 we may write 



N=--Jj?logrdx dy .. .. . ............ (33). 



The formulae (6) then give 



dN 



dN 1 



We see that N is identical with the function -fy of Chapter IV. 



A vortex-filament whose co-ordinates are # , y and strength 

 m contributes to the motion at (a?, y) a velocity whose components 

 are 



V ~ y . m x x r 

 5 , anQ . 



2 



5 



7T ? 2 7T 



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





(x, y ), and its amount is - - . 





Let us calculate the integrals jju^dxdy, and }Jvdxdy, where 

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

 does not vanish. We have 



where each double integration includes the sections of all the 

 vortices. Now, corresponding to any term 



of this integral, we have another term 



and these terms neutralize one another. Hence 



JfuSdxdy = ........................... (34); 



and, by the same reasoning, 



........................... (35). 



112 



