PAPER BY PROF. HELMHOLTZ. 45 



consists in the fact that in the case of liquid vortices there exists in 

 those parts of the liquid that have no rotation a velocity potential qj, 

 which satisfies the equation : 



tf<p , t?<p , A>_ n 



cic 2 " 1 " dy 2 d* 2 ' 



which equation fails to hold good only within the vortex filaments them- 

 selves. But when we imagine the vortex filaments as closed, either 

 within or without the mass of liquid, then the region in which the 

 above differencial equation for cp holds good is a manifold-connected 

 space, for it remains still connected when we imagine intersecting 

 planes passing through it, each of which is completely bounded by a 

 vortex filament. In such manifold-connected spaces a function cp that 

 satisfies the above differential equation becomes many-valued, and it 

 must be many- valued if it is to represent re-entering currents : for since 

 the velocities [u, v, w,] of the liquid particles outside of the vortex fila- 

 ments are proportional to the [partial] differential coefficients of cp [with 

 reference to x, y, z], therefore, following the liquid particle in its motion 

 one would find the values of cp steadily increasing. Therefore, if the 

 current returns into itself, and if one by following it comes finally back 

 to the place where he before was, he will find for this place a second 

 value of cp larger than before. Since we can repeat this process in- 

 definitely therefore for every point of such a manifold connected space, 

 there must be an infinite number of different values of cp, which differ 

 from each other by equal differences, like the different values of 



tang {y 



which is such a many-valued function as satisfies the above differ- 

 ential equation. 



The electro-magnetic effects of a closed electric current have relations 

 similar to the preceding. The current acts at a distance as would a 

 certain distribution of magnetic masses over a surface bounded by the 

 conductor. Therefore, outside of such a current the forces that it ex- 

 erts upon a magnetic particle can be considered as the differential 

 quotients of a potential function V which satisfies the equation 



-|2T7 V2T" ff „ 



}x* T dy 2 T dz 2 

 Here also the space that surrounds the closed conductor and through- 

 out which this equation holds good, is manifold-connected, and V is 

 many-valued. 



Therefore in the vortex motions of liquids, as in the electro-magnetic 

 actions, velocities or forces respectively external to the space occupied 

 by the vortex filaments or the electric currents depend upon many- 

 valued potential functions which moreover satisfy the general differ- 

 ential equations of the magnetic potential function, while on the other 

 hand within the space occupied by the vortex filaments or electric cur- 



