32 THE MECHANICS OF THE EARTH's ATMOSPHERE. 



mostly iu the fact that we had no idea of the forms of motion that fric- 

 tion produces in the fluid. Therefore in this respect an investigation 

 of those forms of motion in which no velocity potential exists seems to 

 me to be of importance. 



The following investigation will now show that in those cases in 

 which a velocity potential does exist the smallest particles of liquid 

 have no motion of rotation, but that when no velocity potential exists 

 then a part at least of the liquid particles are in the act of rotation. 



By vortex lines (Wirbellinien) I designate lines that are so drawn 

 through the mass of liquid that their directions everywhere coincide 

 with the direction of the instantaneous axis of rotation of the liquid 

 particles at that point of the line. 



By vortex filament (Wirbelfaden) I designate the portion of the mass 

 of liquid that is cut out when we construct the corresponding vortex 

 lines passing through every point of the circumference of an infinitely 

 small element of the surface. 



The following investigation shows that when a force potential exists 

 for all the forces that act upon the fluid then: 



(1) No x)article of liquid acquires rotation that was not in rotation 

 from the beginning. 



{'2) The jjarticles of liquid that at any moment belong to the same 

 v^ortex line remain belonging to the same vortex line, even although 

 they have a motion of translation. 



(3) The product of the sectional area by the velocity of rotation of 

 an infinitely slender vortex filament is constant along the whole length 

 of the filament and also retains the same value during the translatory 

 motion of the filament. Therefore the vortex filaments must return into 

 themselves within the liquid or can o\i\y have their ends at the bounda- 

 ries of the fluid. 



This last proposition makes it possible to determine the velocities of 

 rotation when the form of a particular vortex filament is given at dif- 

 ferent moments of time. Further we solve the problem to determine 

 the velocity of the particles of liquid for a given moment of time when 

 the velocities of rotation are given for this moment, but in the solution 

 there remains undetermined one arbitrary function that must be util- 

 ized to satisfy the boundary conditions. 



This last problem leads to a remarkable analogy between the vortex 

 motions of liquids and the electro-magnetic actions of electric currents. 



When in a simf»ly connected space* filled with moving liquid a ve- 

 locity potential exists, the velocities of the liquid particles are equal to 

 and in the same direction as the forces that a certain distribution of 



* I use this expression (eiufach znsammeuliiingenden Raume) in the same sense in 

 ■which Riemann (t7o«j'>m//Mr die reitie «wd angeivandte Mathematik, 1857, Liv, p. 108) 

 speaks of simple and multiple-connected surfaces. A space that is n-times connected 

 is therefore one such that n — 1 but not more intersecting surfaces can pass through, 

 it without cutting the space into two completely separate portions. A ring is there- 

 fore in this sense a doubly-connected space. The intersecting surfaces must be com- 

 pletely surrounded by the lines in which they cut the surface of the space. 



