504 Prof. Hclmholtz on Integrals expressing Vortex -motion, 



x m V\> ®z> Vii & c - m a fl u ^ mass indefinite in the directions of 

 w and y, and denote the product of the section and the angular 

 velocity in them by m v m 2 , &c, then, forming the sums 



V==m l u 1 + m 2 u 2 -f W3W3 &c, 

 V= m l v 1 + m 2 v<2 -f ni 3 v 3 &c, 



these will be each equal to 0, since the portion of the sum V 

 which arises from the effect of the second vortex-filament on the 

 first, is destroyed by the effect of the first on the second. They 

 are respectively 



711/0 X I "— Xn -. in 1 Xn """" X I 



w»i . — • o and m«a . -— • > 



7r r 4 - ir r z 



and so for other pairs in each sum. But U is the velocity of 

 the centre of gravity of the masses m v m^, ... in the direction 

 of x multiplied by the sum of these masses ; so of V in the di- 

 rection of y. Both velocities are thus zero, unless the sum of 

 the masses be zero, in which case there is no centre of gravity. 

 The centre of gravity of the vortex-filaments remains, therefore, 

 stationary during their motions about one another; and since 

 this is true for any distribution of vortex- filaments, it will also be 

 true of isolated ones of indefinitely small section. 

 From this we derive the following consequences : — 



1 . If there be a single rectilinear vortex-filament of indefinitely 



small section in a fluid infinite in all directions perpendicular to 



it, the motion of an element of the fluid at finite distance from 



it depends only on the product (t;dadb = m) of the velocity of 



rotation and the section, not on the form of that section. The 



elements of the fluid revolve about it with tangential velocity 



in 

 = — > where r is the distance from the centre of gravity of the 



filament. The position of the centre of gravity, the angular velo- 

 city, the area of the section, and therefore, of course, the magni- 

 tude m remain unaltered, even if the form of the indefinitely 

 small section may alter. 



2. If there be two rectilinear vortex-filaments of indefinitely 

 small section in an unlimited fluid, each will cause the other to 

 move in a direction perpendicular to the line joining them. 

 Thus the length of this joining line will not be altered. They 

 will thus turn about their common centre of gravity at constant 

 distances from it. If the rotation be in the same direction for 

 both (that is, of the same sign) their centre of gravity lies be- 

 tween them. If in opposite directions (that is, of different signs), 

 their centre of gravity lies in the line joining them produced. 

 And if, in addition, the product of the velocity and the section 

 be the same for both, so that the centre of gravity is at an infi- 



