514 XL HYDRODYNAMICS. 



But since a is solenoidal the volume integral vanishes, and, since at 

 the surface of the vortices, which are composed of vortex -lines, co is 

 tangential, the surface integral vanishes and div. Q = 0. 



193. Velocity due to Vortex. Let us now consider in an 

 incompressible fluid the velocity at any point due to vortical motion. 

 We have 



Thus the portions of the velocity contributed by an element dt' of 

 the vortex are: 



or the velocity at the point x, y, 2 due to the element dr' is - - s 



multiplied by the vector -product of the vorticity and the vector r 

 drawn from the element dr' of the vortex to the point #, i/, #. If 

 dq be the magnitude of the resultant of du, dv, dw we thus have 



78) dq = 



Let us take for the element dr f a length ds of a vortex filament of 

 cross-section S. Then dr' = Sds and since So , the strength of 

 the filament, 



79) dq= * d frfo r ). 



The velocity is connected with the vorticity in the same 4nay that 

 the magnetic field is connected with the electric current density 

 producing it, and equation 79) gives us the magnetic field produced 



by a linear current element of length ds and strength ~-- r ) 



1) See the author's treatise on The Theory of Electricity and Magnetism, 

 222 226. 



