PAPER BY PROF. HELMHOLTZ. 33 



magnetic masses on the surface of the space would exert upon a mag- 

 netic particle in the interior. 



On the other hand, when vortex threads exist in any such space the 

 velocities of the liquid particles are equal to the forces exerted upon a 

 magnetic particle by a closed electric current that flows partly through 

 the vortex filaments in the interior of the mass and partly in the bound- 

 ary surface, and whose intensity is proportional to the product of the 

 sectional area of the vortex filament by its velocity of rotation. 



I shall therefore in the following lines often allow myself to hypoth- 

 ecate the presence of magnetic masses or of electric currents, simply 

 in order thereby to obtain shorter and more perspicuous expressions 

 for the nature of functions that are just the same functions of the co- 

 ordinates as the potential functions, or the attractive forces for a mag- 

 netic particle, are of the magnetic masses or electric currents. 



By these propositions the forms of motion concealed in that class of 

 integrals of the hydro-dynamic equations not hitherto treated of be- 

 come accessible at least to the imagination even although it be possible 

 to execute the complete integration only in a few of the simplest cases 

 where only oue or two rectilinear or circular vortex filaments are pres- 

 ent in masses of liquid that are either unlimited or partially bounded 

 by one infinite plane. 



It can be demonstrated that rectilinear parallel vortex filaments in a 

 mass of water that is bounded only by planes perpendicular to such 

 filaments, rotate about their common center of gravity, when in the 

 determination of this center we consider the velocity of rotation as 

 equivalent to the density of a mass. In this rotation the location of 

 the center of gravity remains unchanged. On the other hand, for cir- 

 cular vortex filaments, all stauding perpendicular to a common axis, 

 the center of gravity of their cross section advances parallel to the axis. 



I. DEFINITION OF ROTATION. 



At a point within a liquid whose position is defined by the rectangular 

 coordinates x, y, z, and at the time f, let the pressure be p, the three com- 

 ponents of the velocity u. v, w, the three components of the external 

 forces acting on the unit mass of the liquid X, Y, Z, and h be the den- 

 sity whose changes can be considered as negligible; the established 

 equations of motion for an interior point of the fluid are : 



_Upjm su du J* 1 

 A hdx Jt^ dx M dz 



Y JL®^ dv iv dv 

 x hdy dt dx W d z I 



z hdz tf & W & 





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