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 ABCAA C B A A and the part of the surface of the 



tube bounded by it. Since 1% + mi] + n% is zero at every point of 

 this surface, the line-integral 



j(ndx + vdy -f wdz), 



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

 Art. 32 



/ (ABC A) + I(AA) + I(A C BA) + I(A A} = Q, 



which reduces to 



= I(AB C A). 



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 2&&amp;gt;cr, where co, = (f 2 + rf + f 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 



