PROFESSOR AT BONN 169 



of velocities connotes the important law that each rotating 

 element of fluid implies in every other element of the same 

 fluid mass a velocity whose direction is perpendicular to the 

 plane through the second and the rotation axis of the first 

 element. The magnitude of this velocity is directly propor- 

 tional to the volume of the first particle, its angular velocity, 

 and the sine of the angle between the line that unites the two 

 elements and that axis of rotation, and inversely proportional 

 to the square of the distance between the two elements. But 

 since the same law holds for the force exerted by an electrical 

 current in the first element, parallel with its axis of rotation, 

 upon a magnetic particle in the second element, Helmholtz, 

 by means of the definition of w-dimensional space (i.e. one 

 which can be traversed by n i, but not more, surfaces, 

 without being separated into two detached portions), formu- 

 lates a law that has acquired great importance in electrical 

 theory. When, that is, a velocity-potential exists in a simply 

 connected space full of moving fluid, the velocities of the 

 fluid elements are equal to, and in the same direction as, 

 the forces exerted on a magnetic particle in the interior of 

 the space by a certain distribution of magnetic masses at its 

 surface. But if vortex-filaments exist in such a space, the 

 velocities of the fluid elements are represented by the forces 

 exerted on a magnetic particle by closed electrical currents, 

 which flow partly through the vortex-filaments in the interior 

 of the fluid mass, partly on its surface, their intensity being 

 proportional to the product of the cross-section of the vortex- 

 filament and the angular velocity. Since the first motion 

 implies a velocity-potential with only one value, while the 

 second in the non-rotating particles of water implies a velocity- 

 potential with more values than one, it is sufficient in hydro- 

 dynamic integrals of the first class to know the motion of the 

 surface ; in those of the second class, we must further deter- 

 mine the motion of the vortex-filaments in the interior of 

 the fluid, with reference to their mutual action, and having 

 regard to the limiting conditions. Helmholtz succeeded in 

 doing this for certain simple cases, in which the rotation of 

 the elements occurs only in given lines or surfaces, and the 

 form of these lines or areas remains unaltered during motion, 

 e.g. in straight, parallel, or circular vortex-filaments which 



