189, 190] 



CIRCULATION. 



507 



constant % = aS is called the strength of the filament. The strength 

 of any vortex is the sum or integral of the strengths of all its 

 filaments. If co is finite , S cannot be zero, so that a vortex -filament 

 cannot end except at the free surface of the fluid. We see a case 

 of this in the vortices produced by an oar or paddle in rowing and 

 by a spoon drawn across the surface of a cup of coffee. We see by 

 equation 46) that the strength of a vortex is equal to one -half the 

 circulation around any closed path drawn embracing the vortex on 

 its surface , which is independent of the path. In particular, in any 

 non- vortical region the circulation around any closed path is zero, 

 or the circulation along an open path <p AB is independent of the 

 path, depending only on A and J5, or the velocity is a lamellar 



vector. We then have 



4r .^ dm dw dcp 



48) u = ^-> v = -^-9 w = --> 



ox y dz 



and qp is called the velocity potential, a term introduced by Lagrange. 

 (When there is vorticity there is no velocity potential.) 



Before 1858 only cases of motion had been treated in which a 

 velocity potential existed. In that year appeared the remarkable paper 

 by Helmholtz 1 ) on Yortex Motion. 



Let us now find the change of circulation along a path moving 

 with the fluid, that is, composed 

 of the same particles, the forces 

 being conservative. 



Our equations of motion 3) may 

 be written, putting U' = (F-f P) 



49) 



du 

 ~dt 

 dv 



dw 

 ~dt 



dU' 



vdz 



dU' 



Fig. 161. 



The change of circulation along the path AB is 



50) 



dt 



d 

 dt 



A 



B 



I (udx -f vdy + wdz), 



in which dx 9 dy, dz vary with the time, being the projections of an 

 arc ds composed of parts which move about. If after a time dt the 

 arc ds assumes a length ds' whose components are dx', dy', dz* we 

 have (Fig. 161) 



1) Vber Integrate der hydrodynamischen Gleichungen, welche den Wirbel- 

 bewegungen entsprechen. Wissenschaftliche Abhandkmgen I, p. 101. 



