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' (/ d 1 , , d 1 , ,\ , . 



=cr \\T-'--dy - j ~,-.dz ) ............ (1). 



27rJ \dz r d r J 



j ~, 

 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, 



m, (dR dQ\ (dP dR\ /dQ dP\] 



= -U (x~/-T^ ) + m (j->- j->) + ( T?- j->) 



J J { \dy dz / \dz dx) \dx' dy )} 

 If we put 



we find 



dR _ dQ _ _ / d 2 ^\ 1 _ d?_ 1 

 dy' dz' ~ \dy f * + dz 2 ) r ~ dx'* r' ' 



dP_dR_ _!? ! 



dz' dx' dx'dy' r' ' 



_^ _ _ 



dx dy' ~ dx'dz r ' 



so that (1) may be written 



d d d \ d 1 



-- - - 



, j, , t . 



x dy dz J dx r 



Hence, and by similar reasoning, we have, since 

 dldx' . r- 1 = - d/dx . r~\ 



dd> dd> dd> /ox 



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



dx dy dz 



where 



m' /Y/ 7 d d d\ 1 , , /QA 



(/, = -- u 1 - 7 + m -j-f -I- ?i -y-, ) - a^S ............ (3). 



27rjj\ ote c^2/ ^/ ^ 



Here , m, n denote the direction-cosines of the normal to the 

 element SS' of any surface bounded by the vortex-filament. 



