148 VORTEX MOTION. [CHAP. VI. 



connecting line also drawn on the surface. Let us apply the 

 theorem of Art. 40 to the circuit ABGAA C B A A and the part 



of the surface of the tube bounded by it. Since 1% + mrj + nf is 

 zero at every point of this surface, the line-integral 



J(udx + vdy + wdz\ 



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

 Art. 39 



I (ABC A) + 1 (AA) + I(AC B A) + 1 (A A) = 0, 

 which reduces to 



I(ABCA)=I(A ffC A ). 



Hence the circulation is the same in all circuits embracing the 

 same vortex-tube. 



Again, it appears from Art. 39 that the circulation round the 

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

 length, is 2&&amp;gt;er, where co = (%* + rf + f 2 )* is the angular velocity of 

 the fluid at the section, and cr 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 Thomson ; the theorem itself 

 was first given by Helmholtz, who deduced it from the relation 



? + + -o CD, 



dx dy dz 



which follows at once from the values of ( , rj, % given in Art. 38. 

 In fact, writing in Art. 64, Cor. 1, f, 77, f for u, v, w, respectively, 

 we find 



8~Q (2), 



