500 Prof. Helmholtz on Integrals expressing Vortex-motion. 



blem belongs to those which can be solved by the assumption of 

 a velocity-potential. 



In the hydrodynamic integrals of the first class the velocities 

 of the fluid elements are in the direction of, and proportional to, 

 the forces which a determinate magnetic distribution outside the 

 fluid would exert on a magnetic particle in the places of the 

 elements. 



In the hydrodynamic integrals of the second class the veloci- 

 ties of the fluid elements are in the direction of, and proportional 

 to, the forces which would act on a particle of magnetism if 

 closed electric currents passed through the vortex-filaments with a 

 density proportional to the angular velocity in these filaments, 

 combined with magnetic masses outside the fluid. The electric 

 currents inside the fluid must move with their respective vortex- 

 filaments and have constant intensity. The assumed distribution 

 of magnetic matter outside or at the surface of the fluid must be 

 taken so as to satisfy the conditions at the surface. Each mag- 

 netic mass can also, as we know, be replaced by electric currents. 

 Thus, instead of using for the values of u, v, w the potential- 

 function P of an external mass k, we get quite as general a solu- 

 tion if we give £, iq, and f outside of, or at the bounding surface 

 of, the fluid any values such that only closed current-filaments 

 exist; and then the integration in (5 a) must be extended to all 

 space in which £, tj, and J are different from zero. 



§4, 



In hydrodynamic integrals of the first class it is sufficient, as 

 I have shown above, to know the motion of the surface. By this 

 the whole motion in the interior is determined. In integrals of 

 the second class, on the other hand, the motion of the vortex- 

 filaments in the interior of the fluid must be found with reference 

 to their mutual action, and with attention to the conditions at the 

 surface, by which the problem becomes much more complicated. 

 Even this problem can be solved in certain simple cases — namely, 

 when rotation of the fluid elements takes place only in known 

 surfaces or lines, and the form of these surfaces or lines remains 

 unchanged during the motion. 



The properties of surfaces bounded by an indefinitely thin 

 sheet of rotating elements can be easily deduced from (5 a). If 

 f, 7), f differ from zero only in an indefinitely thin sheet, their 

 potential functions L, M, N will, by known theorems, have equal 

 values on both sides of the sheet; but their differential coeffi- 

 cients, taken in the direction of the normal to the sheet, will be 

 different. Suppose the coordinate axes so placed that at the 

 point of the vortex-sheet we are considering the axis of z is the 

 normal to the sheet, and that of x the axis of rotation of the 



