PAPER BY PROF. HELMHOLTZ. 51 



and so on through all the other pairs of sums. Now U is the velocity 

 in the direction of x, of the center of gravity of the masses m h m 2 , etc., 

 multiplied by the sum of these masses; similarly V is the velocity 

 taken in the directiou of y. Both velocities are therefore zero, unless 

 the sum of the masses is zero, in which case there is no center of grav- 

 ity at all. Therefore the center of gravity of the vortex filaments 

 remains unchanged during their motion, aud since this proposition 

 holds good for every distribution of the vortex filaments, therefore we 

 may also apply it to the individual filaments of infinitely small cross 

 section. 

 Hence result the following consequences : 



(1) If we have but one individual rectilinear vortex filament of infi- 

 nitely small cross-section within a liquid mass of infinite extent in all 

 directions perpendicular to the filament, then the movement of the par- 

 ticles of water at a finite distance from the filament depends only on the 

 product £ da db = m, or the velocity of rotation multiplied by the area 

 of the cross-section, and not on the form of the cross-section. The 

 liquid particles rotate about the filament with the tangential velocity 



7)1 



— where r denotes the distance from the center of gravity of the vor- 



tex filament. The location of the center of gravity, the velocity of 

 rotation, the area of the cross section, aud therefore also the quantity 

 m remains unchanged although the form of the infinitely small cross- 

 section may change. 



(2) If we have two rectilinear vortex filaments of infinitely small cross- 

 sections and an indefinitely large liquid mass, each will drive the other 

 in a direction that is perpendicular to the line joining them together. 

 The length of this connecting line will not be changed thereby; there- 

 fore both will revolve about their common center of gravity, remain- 

 ing at equal constant distances therefrom. If the rotatory velocity is in 

 the same direction in the two filaments and therefore has the same 

 sign, then their center of gravity must lie between them. If the rota- 

 tions are mutually opposed to each other and therefore of opposite signs, 

 then their center of gravity lies in the prolongation of the line connect- 

 ing the filaments. If the products of the rotatory velocity by the cross 

 section are numerically equal for the two but of opposite signs, thereby 

 causing the center of gravity to be at an infinite distance, then both 

 filaments advance with equal velocity and in the same direction per- 

 pendicular to their connecting line. 



The case where a vortex filament of infinitely small section lies close 

 to an infinitely extended plane surface parallel to it can be reduced to 

 this last case. The boundary condition for the movement of the liquid 

 along a plane (i. e., that the motion must be parallel to this plane) is 

 satisfied when we imagine a second vortex filament, which is as the re- 

 flected image of the first, introduced on the other side of the plane. 

 Hence it follows that the vortex filament within the liquid mass ad- 



