Prof. Helmholtz on Integrals expressing Vortex -motion. 499 



<f) which satisfies the above equation can have more than one 

 value; and it must be so if it represent currents reentering, 

 since the velocities of the fluid outside the vortex-filaments are 

 proportional to the differential coefficients of <f), and therefore the 

 motion of the fluid must correspond to ever increasing values 

 of (j>. If the current returns into itself, we come again to a 

 point where it formerly was, and find there a second greater 

 value of <p. Since this may occur indefinitely, there must be for 

 every point of such a complexly-connected space an infinite num- 

 ber of distinct values of <f> differing by equal quantities like those 



of tan -1 -; which is such a many- valued function and satisfies 



the differential equation. 



Such also is the case with the electromagnetic effects of a 

 closed electric current. This acts at a distance just as a deter- 

 minate arrangement of magnetic matter on a surface bounded by 

 the conductor. Exterior to the current, therefore, the forces ex- 

 erted on a particle of magnetism may be considered as the dif- 

 ferential coefficients of a function V which satisfies the equation 



dx 2 dif- dz 2 



But in this case also the space in which this equation holds is 

 complexly connected, and V has more than one value. 



Thus in the vortex-motion of fluids, as in electromagnetic 

 effects, the velocities or forces external to the vortex-filaments (or 

 electric-current-penetrated space) depend upon potential functions 

 with mGre values than one, which satisfy the general differential 

 equation of magnetic potential functions; while within the vor- 

 tex-filaments or the space traversed by electric currents, velocities 

 and electromagnetic forces can be expressed (both in an analo- 

 gous manner) by those functions which appear in the equations 

 (4), (5), and (5 a). On the other hand, in simply streaming 

 fluid-motion and magnetic forces we have to do with potential 

 functions with only one value, just as in the cases of gravitation, 

 electric attractions, and constant currents of heat and electricitjr. 



The latter integrals of the hydrodynamical equations, in which 

 a single-valued velocity-potential exists, we may call integrals of 

 the first class ; those, on the other hand, where there is rotation 

 of some of the elements of the fluid, and in consequence a velo- 

 city-potential with more than one value in the non-rotating ele- 

 ments, integrals of the second class. It may occur that in the 

 latter case only such portions of space are to be treated in the 

 example as contain no rotating elements, — for instance, the mo- 

 tion of fluids in ring-shaped vessels, where a vortex-filament may 

 be supposed to lie along the axis of the vessel, and where the pro- 



