PAPER BY PROF. HELMHOLTZ. 57 



We can now in general see how two circular vortex threads having 

 a common axis will behave with respect to each other, since each one 

 independent of its own translatory motion also follows the movement 

 of the liquid particles caused by the other filament. If they have the 

 same direction of rotation, then they both advance in the same direc- 

 tion, and at first the preceding one enlarges, then it advances more 

 slowly while the following one diminishes and advances more rapidly; 

 finally, if the progressive velocities are not too different, the second 

 catches up with the first and passes through it. Then the same perform- 

 ance is repeated by the one that is now in the rear so that the rings 

 alternately pass through each other. 



If the vortex filaments have the same radii, but equal and opposite 

 rotatory velocities, then they will approach each other and simultane- 

 ously enlarge, so that finally when they have come very close together 

 their movement towards each other grows continually feebler, while on 

 the other baud the enlargement goes on with increasing rapidity. If 

 the two vortex threads are perfectly symmetrical, then midway be- 

 tween them the velocity of the liquid particles in the direction parallel 

 to the axis is equal to z ero. Therefore one can imagine a rigid wall 

 located here without disturbing the motion and thus obtain the case of 

 a vortex ring that encounters a rigid wall. 



I remark further that we can easily study these movements of circular 

 vortex rings in nature if we draw a half-immersed circular disk or the 

 approximately semicircular end of a spoon rapidly for a short distance 

 along the surface of a liquid and then quickly draw it out. There then 

 remain in the liquid semi- vortex riugs whose axes lie in the free upper 

 surface of the liquid. The free upper surface thus forms, for the liquid 

 mass, a boundary plane that passes thr ough the axis whereby no im- 

 portant change is made in the motions. The vortex rings advance, 

 broaden when they encounter a screen, and are enlarged or diminished 

 by the action of other vortex rings precisely as we have deduced from 

 the theory. 



