220 WISCONSIN ACADEMY SCIENCES, ARTS, AND LETTERS. 



axis of magnetization and the radius vector; and tlie direction of 

 the vector-potential is perpendicular to the plane of the axis of 

 magnetization and the radius-vector, and is such that to an eye 

 looking in the positive direction along the axis of magnetization, 

 the vector-potential will be drawn m the direction of rotation of the 

 hands of a icatchy The results just leferred to above, show 

 that the distance-action of vortex-tilaments is similar to the electro- 

 magnetic action of current-conducting wires; for they prove that 

 'Wc/i rotating element of fiuid (a) implies in each other element (b) 

 of the same fluid mass, a velocitij ivhose direction is perpendicular to 

 the plane through (b) and the axis of rotation (a). The magnitude 

 of this velocitij is directly proportional to the volume of (a), its angu- 

 lar velocitij, and the sine of the angle between the line (a) (b) and that 

 axis of rotation, and inversely proportional to the square of the dis- 

 tance between (a) and {h)." 



Thus the vector-potential of the electric-current, in free space 

 surrounding the wire, has for its analogue the velocity of the fluid 

 element, due to a vortex-filament supposed to occupy the place of 

 the current. 



Many other curious analogies between vortex-motions in fluids 

 and the action of magnets and electric currents have been pointed 

 out by Sir Wm. Thompson.* 



Of course it^is possible that these analogies may be merely for- 

 mal, and that they arise from the fulfillment of similar mathematical 

 conditions by both the electric current and vortex-motion in fluids. 



But whether the relationship shall or shall not ultimately be 

 found to consist in a closer connection than mere formal analogy, 

 one thing is certain. The discovery of the laws governing vortex- 

 motion in fluids constitutes an era in physical science. The differen- 

 tial equations of the motions of fluids although'handled by such 

 masters as LaGrange, LaPlace, Euler, and Green, had only been in- 

 tegrated on the special assumption of a velocity-potential; which 

 condition we have seen to hold only in the space outside of those 

 portions ot the fluid which are in rotation. It remained for Helmholtz 

 to make the next great step by integrating these equations under the 

 supposition that no velocity-potential exists; and to show that while 

 the establishment of vertex-motion in fluids, is, on the one hand, 

 a consequence of fluid friction, on the other, that when vortex-fila- 



* Sir Wm. Thompsou's. "Papers on Electrostatics and Magnetism." 



