{68 HERMANN VON HELMHOLTZ 



separate motions : a translation of the element of the fluid 

 through space, an expansion or contraction of the elements 

 in three principal directions of dilatation (in which any rect- 

 angular parallelepiped, whose edges are parallel with the 

 principal directions of dilatation, remains rectangular), and, 

 lastly, a revolution round some instantaneous axis of rotation 

 in any direction. In the first place he proves by rigid mathe- 

 matical deductions that the existence of a velocity-potential 

 is incompatible with the existence of a rotation of the fluid 

 elements, but when there is no velocity-potential some fluid 

 elements at least can rotate. On the assumption that all forces 

 acting on these fluids have a ' force-potential ', it follows neces- 

 sarily that such particles of water as have no initial rotary 

 motion cannot be thrown into rotation at a later period. If 

 lines drawn through the fluid so that their direction coincides 

 everywhere with the direction of the instantaneous axis of 

 rotation of the element of fluid lying there, are termed vortex- 

 lines, it follows again from the equations of hydrodynamics 

 that each vortex-line remains permanently composed of the 

 same elements of fluid, and swims along with them in the 

 fluid. Helmholtz terms a filament of the fluid with an 

 indefinitely small cross-section, a vortex-filament, when it is 

 produced by drawing vortex-lines through every point in the 

 circumference of any indefinitely small surface. Since, further, 

 the expressions for rotary velocity show that the magnitude 

 of the latter varies in any given element, in the same propor- 

 tion as the distance between this element and its neighbours 

 on the axis of rotation, it follows that the product of the angular 

 velocity and the cross-section in any portion of a vortex-filament 

 containing the same particles of fluid remains constant during 

 the motion of the filament, and further that this product does not 

 vary throughout the whole length of any given vortex-filament. 

 The vortex-filaments must accordingly return upon themselves 

 within the fluid, or end at its boundaries. But from this it 

 follows directly, that if the motion of the vortex-filaments in the 

 fluid can be determined, the velocity of rotation can be ascer- 

 tained, and that the velocities of the elements of the fluid are 

 determined for any given moment of time, when the angular 

 velocities are given, with the exception of an arbitrary function, 

 which covers the limiting conditions. This determination 



