147-148] MOTION DUE TO AN ISOLATED VORTEX. 233 



Velocity-Potential due to a Vortex. 



148. At points external to the vortices there exists of course a 

 velocity-potential, whose value may be obtained as follows. Taking 

 for shortness the case of a single re-entrant vortex, it was found 

 in the preceding Art. that, in the case of an incompressible fluid, 



m [i rf 1 7 , d 1 , A 



= o~ I \ j- - dy 7 -,-.dz } (1). 



27rJ \dz r J dy r v 



dy 



By Stokes Theorem (Art. 33 (5)) we can replace a line-integral ex 

 tending round a closed curve by a surface-integral taken over any 

 surface bounded by that curve ; viz. we have, with a slight change 

 of notation, 



fdR dQ\ fdP dR\ fdQ dP\] , 



T-/-T^ + m j-/- j^l + ( T%-^r-,)td&. 



\dy dz) \dz doc) \dx dy )} 



If we put 



we find 



^ 1 p d I 

 - tf- 



dz r r dy r* 



dR_dQ = _/d* _^1\1_^11 

 dy dz ~ (dy* dz&quot;&amp;gt;) r ~ d^ r f 



dP _dR_ d* !_ 



dz dx ~ dx dy r 



dQ_dP = _fc _! 



dx dy dx dz r&quot; 



so that (1) may be written 



in f [[/, d d d \ d 1 70( 



u = x 1 1 U -r, + m -J-, +n -j- f - r -&amp;gt; - ao . 

 27rJJ\ d c^/ dz J dx r 



Hence, and by similar reasoning, we have, since 



f . r~ l = - dldx . r~\ 



dd&amp;gt; dd&amp;gt; dd&amp;gt; /ON 



u = - -f, -y = - y^, w = --^ , ............... (2), 



dx dy dz 



where 



m f/Yj rf rf rf\ 1 7ry- /ox 



= er U TT + ^ J&quot;7+ * :r&amp;gt; J - &amp;lt;* ............ w&amp;gt; 



27rJJV rf^ rf/ dW r* 



Here ^, m, ?i denote the direction-cosines of the normal to the 

 element 8$ of any surface bounded by the vortex-filament. 



