134, 135.] MOTION OF VORTEX-LIXES. 159 



portion of a fluid mass in motion the conditions = 0, 77 = 0, = 

 obtain at any one instant, the same is true for the same portion 

 of the mass at every other instant, which is the theorem of 

 Art. 23. 



It follows that rotational motion cannot be produced in any 

 part of a fluid mass by the action of forces which have a single- 

 valued potential, and that such a motion, if already existent, 

 cannot be destroyed by the action of such forces. 



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. 40, 

 for we have ?f + mrj + n= 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 Helmholtz* for 

 the case of liquids ; the preceding proof, by Thomson, shews it 

 to be applicable to all fluids satisfying the conditions stated at 

 the beginning of this article. 



Kinetic Energy. 

 135. The formula for the kinetic energy, viz. 



2T=fffp(u* + v* + w 2 )dxdydz (25), 



may be put into several remarkable and useful forms. We confine 

 ourselves, for simplicity, to the case where the fluid (supposed in 

 compressible) extends to infinity and is at rest there, and where 

 further all the vortices present are within a finite distance of the 

 origin. 



We have in this case, 6 = 0, P = 0, p = const., so that (25) 

 becomes on substitution from (6), 



ALV dM\ fdL dN\ (dM dL\\ , , 



ill -j f-)+tM-j- ~ -j- }+w(-j -j- }\dxdydz. 



\dy dz ) \dz dxj \dx dy/} 



* See note (D). 



