224 VORTEX MOTION. [CHAP. VII 



which follows at once from the values of , 77, f given by (1). In fact, writing 

 in Art. 42 (1), & 77, ( for U, V, W, respectively, we find 



SS(l + mi + nQdS=0 (ii), 



where the integration extends over any closed surface lying wholly in the 

 fluid. Applying this to the closed surface formed by two cross-sections of a 

 vortex-tube and the portion of the tube intercepted between them, we find 

 ( 1 o- 1 =o) 2 o- 2 , where o lt o> 2 denote the angular velocities at the sections cr lt o- 2 , 

 respectively. 



Lord Kelvin's proof shews that the theorem is true even when , 77, are 

 discontinuous (in which case there may be an abrupt bend at some point of a 

 vortex), provided only that u, v, w are continuous. 



An important consequence of the above theorem is that a 

 vortex-line cannot begin or end at any point in the interior of 

 the fluid. Any vortex-lines which exist must either form closed 

 curves, or else traverse the fluid, beginning and ending on its 

 boundaries. Compare Art. 37. 



The theorem of Art. 33 (4) may now be enunciated as follows : 

 The circulation in any circuit is equal to twice the sum of the 

 strengths of all the vortices which it embraces. 



143. It was proved in Art. 34 that, in a perfect fluid whose 

 density is either uniform or a function of the pressure only, and 

 which is subject to extraneous forces having a single-valued 

 potential, the circulation in any circuit moving with the fluid is 

 constant. 



Applying this theorem to a circuit embracing a vortex-tube we 

 find that the strength of any vortex is constant. 



If we take at any instant a surface composed wholly of vortex- 

 lines, the circulation in any circuit drawn on it is zero, by Art. 33, 

 for we have 2-f-mty+ji=0 at every point of the surface. The 

 preceding article shews that if the surface be now supposed to 

 move with the fluid, the circulation will always be zero in any 

 circuit drawn on it, and therefore the surface will always consist 

 of vortex-lines. Again, considering two such surfaces, it is plain 

 that their intersection must always be a vortex-line, whence 

 we derive the theorem that the vortex-lines move with the 

 fluid. 



. This remarkable theorem was first given by von Helmholtz for 

 the case of liquids ; the preceding proof, by Lord Kelvin, shews 

 that it holds for all fluids subject to the conditions above stated. 



