142] VORTEX-FILAMENTS. 223 



' vortex-tube.' The fluid contained within such a tube constitutes 

 what is called a ' vortex-filament/ or simply a ' vortex.' 



Let ABC, A'B'C' be any two circuits drawn on the surface of a 

 vortex-tube and embracing it, and let AA' be a connecting line 

 also drawn on the surface. Let us apply the theorem of Art. 33 to 

 the circuit A EGA A'C'B'AA and the part of the surface of the 



tube bounded by it. Since / 4- m?j + n is zero at every point of 

 this surface, the line-integral 



f(udx + vdy + wdz), 



taken round the circuit, must vanish; i.e. in the notation of 

 Art. 32 



which reduces to 



Hence the circulation is the same in all circuits embracing the 

 same vortex-tube. 



Again, it appears from Art. 32 that the circulation round the 

 boundary of any cross-section of the tube, made normal to its 

 length, is 2o)cr, where o>, = (f 2 + rf + 2 )*> is the angular velocity of 

 the fluid, and a the infinitely small area of the section. 



Combining these results we see that the product of the angular 

 velocity into the cross- section is the same at all points of a vortex. 

 This product is conveniently termed the 'strength ' of the vortex. 



The foregoing proof is due to Lord Kelvin ; the theorem itself was first 

 given by von Helmholtz, as a deduction from the relation 



