486 Prof. Helmholtz on Integrals expressing Vortex-motion. 



sisted in the fact that no idea had been formed of the species of 

 motion which friction produces in fluids. Hence it appeared to 

 me to be of importance to investigate the species of motion for 

 which there is no velocity-potential. 



The following investigation shows that when there is a velo- 

 city-potential the elements of the fluid have no rotation, but 

 that there is at least a portion of the fluid elements in rotation 

 when there is no velocity-potential. 



By vortex-lines (Wirbellinien) I denote lines drawn through 

 the fluid so as at every point to coincide with the instantaneous 

 axis of rotation of the corresponding fluid element. 



By vortex-filaments (Wirbelfaden) I denote portions of the fluid 

 bounded by vortex-lines drawn through every point of the boun- 

 dary of an infinitely small closed curve. 



The investigation shows that, if all the forces which act on the 

 fluid have a potential, — 



1. No element of the fluid which was not originally in rota- 

 tion is made to rotate. 



2. Tbe elements which at any time belong to one vortex-line, 

 however they may be translated, remain on one vortex-line. 



3. The product of the section and the angular velocity of an 

 infinitely thin vortex-filament is constant throughout its whole 

 length, and retains the same value during all displacements of the 

 filament. Hence vortex-filaments must either be closed curves, 

 or must have their ends in the bounding surface of the fluid. 



This last theorem enables us to determine the angular velocity 

 when the form of the vortex- filament at different times is given. 

 Besides, there is given a solution of the problem of finding 

 the velocities of the fluid elements at any instant, if at that in- 

 stant the angular velocities are given : an arbitrary function, 

 however, remains undetermined, and is to be applied to satisfy 

 the boundary conditions, 



This last example leads to a remarkable analogy between the 

 vortex-motion of fluids and the electro-magnetic action of elec- 

 tric currents. If, for instance, in a simply-connected * (einfach 

 zusammen hang end) space full of fluid there be a velocity-poten- 

 tial, 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 or electric currents on its surface. But, if vortex-fila- 



* I use this expression in the sense in which Riernann (Crelle, vol. liv. 

 p. 103) speaks of simply and complexly connected surfaces. An wly con- 

 nected space is thus one which can be cut through by n— 1, but no more, 

 surfaces, without being separated into two detached portions. In this 

 sense a ring is a doubly-connected space. The cutting surfaces must be 

 completely enclosed within the lines in which they cut the bounding surface 

 of the space considered. 



