156 VORTEX MOTION. [CHAP. VI. 



a way that the product (0 say) of 6 into the thickness e is finite. 

 The normal velocities at adjacent points on opposite sides of the 

 film will then differ by d . 



Velocity-Potential due to a Vortex. 



133. At points external to the vortices there exists of course 

 a velocity-potential, whose value may be found by integration 

 of (15), as follows. Taking, for shortness, the case of a single 

 closed vortex, we write dx dy dz = ads, where ds is an element 

 of the length of the filament, a its section. Also we may write 



o, _ &dx f , _ tody ., to dz 

 ~~ &quot;~ ~~ ~- 



so that u=-~ &quot; - dy - J, - dz } (21), 



2-7T J \dz r dy r J 



w r here the product wcr , the strength of the vortex, being constant, 

 is placed^ outside the sign of integration, which is taken right 

 round the filament. Now the analytical theorem (7) of Art. 40 

 enables us to replace a line-integral taken round a closed curve 

 by a surface-integral taken over any surface bounded by that 

 curve. To apply this to our case, we write, in the formula cited, 



d 1 d I 



u = } v = - w = -j-,-, 

 dz r dy r 



which, give 



dy~dz f== 



du dw _ d* 1 



dz dx dx dy r 



dv du _ d* 1 



dx dy dx dz r 



Hence (21) becomes 



d d d \ d 1 , 



or since - - = - - 



dx r dx r 



cfy 



