RECENT PROGRESS IN THEORETICAL PHYSICS. 213 



For these cases there is no velocity potentiah " It is only Avhen 

 there is no velocity potential that some fluid elements can rotate 

 and that others can move round aloua: a closed curve in a simply- 

 connected space." 



Helmholtz calls the motions that have no velocity potential, 

 generally, vortex motions. 



He shows that in a frictionless fluid, these vortices when once 

 instituted in the fluid, have a wonderful tenacity of existence; 

 that they may go on widening, changing their form under 

 the influence of other vortices, moving about, attracting and re- 

 pelling each other in consequence of combining their motions; and 

 that they may play amongst themselves all sorts of fantastic games, 

 yet preserve unchanged their identity and living force (i. e. their 

 kinetic energ}') so as to be the very types of the unchanging atoms 

 of matter, which are never destroyed. 



One simple instance of Helmholtz' results I will state, to make 

 the matter plain. If there be, for example, a single circular vortex 

 ring set up in an indefinitely extended fluid, the center of gravity of 

 the section of the ring (section supposed small) will have from the 

 commencement an approximately constant and very great velocity 

 parallel to the axis of the ring, and this will be directed toward 

 the side to lohich the fluid flows through the ring. 



Two ring-formed vortex-filaments having the same axis would 

 rautuall}' afiect each other, since each, in addition to its own proper 

 motion has that of its elements of fluid as produced by the other. 

 If they have the same direction of rotation, they travel in the same 

 direction; the foremost ring widens and travels more slowly, the 

 pursuer shrinks and travels faster, till finally, if their velocities are 

 not too different, it overtakes the first and penetrates it. Then the 

 same game goes on in the opposite order, so that the rings pass 

 through each other alternately. If they have equal radii and equal 

 and opposite angular velocities, they will approach each other and 

 widen one another. So also one will widen on coming to a fixed 

 wall. '* The motions of circular vortex rings can be studied by 

 drawing rapidly for a short space along the surface of a fluid a 

 half immersed circular disk, or the nearly semi-circular point of a 

 spoon, and quickly withdrawing it. There remain in the fluid 

 half-vortex rings whose axis is in the free surface. These vortex 

 rings travel and widen when they come to a Avall, and are widened 



