SB 



494 Prof. Helmholtz on Integrals expressing Vortex -motion. 



gular velocity, in a portion of a vortex -filament containing the same 

 element of fluid, remains constant during the motion of that element. 

 From equations (2) it follows directly that 



^ + ^ + ^ = 0. 



dx dy dz 



And, further, from this, that 



the integration being extended over any given portion S of the 

 fluid mass. Integrating partially we have 



jj i dy dz -f f fi? due dz -j- J f %d® dy = 0, 



where the integration extends to the whole surface of S. Calling 

 dco an element of this surface, and a, /3, <y the three angles made 

 with the axes by the normal to dco drawn outwards, w r e have 



dy dz = dco cos a, dx dz=dco cos {3, dx dy = dco cos 7. 

 Hence 



\ J (f cos « -f 7] cos j3 +?cos7) Bco = 0' } 



or if we call q the resultant angular velocity, and 6 the angle be- 

 tween its axis and the normal, 



\ f q cos d.dco = 0, 



the integration extending to the whole surface of S. 



Now let S be a portion of a vortex-filament bounded by two in- 

 definitely small planes co { and co u perpendicular to the axis of the 

 filament ; cos 6 is equal to 1 at one of these, and — 1 at the other, 

 and equal to for the rest of the surface of S ; hence if q i and 

 q u be the angular velocities in co t and co jj} the last equation re- 

 duces itself to 



whence it follows that the product of the velocity of rotation and 

 the cross section is constant throughout the whole length of any 

 one vortex-filament. That it does not alter by the motion of 

 the filament itself has been already proved. 



It also follows from this that a vortex -filament can never end 

 ivithin a fluid, but must either return ring-shaped into itself 

 within the fluid, or reach to the boundaries of the fluid, since, if 

 a vortex-filament ended anywhere within a fluid, a closed surface 

 could be constructed for which f q cos 6 dco would not vanish. 



§3. 



If the motion of the vortexr filaments in a fluid can be deter- 



