Prof. Helmholtz on Integrals expressing Vortex-motion. 487 



merits exist in such a space, the velocities of the fluid elements are 

 represented by the forces exerted on a magnetic particle by closed 

 electric currents which flow partly through the vortex-filaments 

 in the interior of the fluid mass, partly on its surface, their in- 

 tensity being proportional to the product of the section of the 

 vortex-filament and the angular velocity. 



I shall therefore frequently avail myself of the analogy of 

 magnetic masses or electric currents, simply to give a briefer and 

 more vivid representation of quantities which are just such func- 

 tions of the coordinates as the attractive forces exerted by these 

 masses or currents on a magnetic particle, or the corresponding 

 potential functions. 



By means of these theorems the various species of fluid- 

 motion which are concealed in the yet unstudied integrals of 

 the hydrodynamical equations can at least be represented, even 

 although the complete integration is possible only in a few of 

 the simplest cases — as when we have one or two straight, or cir- 

 cular vortex-filaments in a mass of fluid which is either infinite in 

 all directions or bounded in one direction by an infinite plane. 



It can be shown that straight parallel vortex-filaments, in a fluid 

 mass which is limited only by planes perpendicular to the fila- 

 ments, revolve about their common centre of gravity, if we deter- 

 mine this point by employing the angular velocity as we would the 

 density of a mass. The position of the centre of gravity remains 

 unaltered. On the other hand, in the case of circular vortex-fila-. 

 ments which are all perpendicular to a common axis, the centre 

 of gravity of their section moves on parallel to the axis. 



§1- 



At a point % } y, z in a liquid let/? be the pressure, u, v, w the 

 rectangular components of the velocity, X, Y, Z the components 

 of external forces acting on unit of mass, and h the density 

 (whose variations will be supposed indefinitely small), all at 

 time t. Then we have the following known equations of motion 

 for the interior particles of the fluid, : — 



v 1 dp du du du du 



■A-— T ' i = ~T7 -\~ u 1 rV -. \~VJ -f-J 



h dx dt dx dy dz 



, r 1 dp do dv dv dv 



I i dy dt ax ay dz 



rj 1 dp dw dw div dw 



/ _ ~.-j- = — -f uj- -\-v -j- + iv^-j 



h dz dt dx dy dz 



n — ^ u _i_ ^ v d w 



dx- ' dy dz 



2K2 



(1) 



