HELMHOLTZ IN BERLIN 



magnetism require rotating elements ; and vorticity of 

 some kind seems to be an essential feature of electro- 

 magnetic action. 



Wherever in a fluid vortex motion exists, there is 

 rotation of the smallest imaginable particle about some 

 axis. Starting from this conception, which Stokes was 

 the first to describe clearly, Helmholtz proceeds to 

 investigate the forms and behaviour of vortices. A 

 vortex line he defines as a line drawn through the 

 fluid in such a way that its direction at any point is 

 the axis of rotation of the element at that point. A 

 vortex filament is part of the fluid marked off from 

 the surrounding fluid by drawing the corresponding 

 vortex lines through all points of the circumference of 

 an infinitely small plain area. Thus each vortex 

 filament may be imagined to be shut off from the 

 surrounding fluid by a thin layer or mantle of vortex 

 lines. Vortex lines form closed curves in a finite 

 fluid ; and vortex filaments form closed filaments or 

 rings, simple or knotted. The simplest type of a 

 vortex ring is a circular anchor ring, every element 

 rotating round the circular axis. Such a vortex ring 

 advances through the fluid in the direction of motion 

 of the elements on the inside of the ring. An ordinary 

 smoke-ring shows the essential nature of the motion 

 very well. If two vortex rings have the same axis 

 and the same sense of rotation, and if they both 

 advance through the fluid in the same direction, 

 the first ring, a, will widen, and suffer retardation, 

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