518 XL HYDRODYNAMICS. 



Similarly for vortices continuously distributed, the strength of 

 any elementary filament being 



97) U 



which again vanishes, since every point is covered by both dS and dS'. 

 If we define the center of the vortex as x where 





then since g does not depend upon t, if we follow the particle 

 differentiating, 

 99) 



the integrals being taken over areas moving with the liquid. Therefore 



or the center of all columnar vortices present remains at rest. 



If we have a single vortex filament of infinitesimal cross- section S, 

 for which 

 101) 



the velocity depends on the current function W = - log r. In the 



vortex and close to it, if x is finite, g, W, u, v are infinite. But at 

 the center u = v = 0, the vortex stands still and the fluid moves about 



it in circles with velocity The angular velocity and the area of 



the cross -section remain constant, although the shape of the latter 



may vary. If we have two such vortex- 

 filaments each urges the other in a direc- 

 tion perpendicular to the line joining 

 them, they accordingly revolve about their 

 center, maintaining a constant distance 

 from each other. If they are whirling in 

 the same direction the center is between 

 them (Fig. 162), but if in opposite direc- 

 tions, it is outside, and if they are equal 

 it lies at infinity. Such a pair of vortices 

 may be called a vortex-couple or doublet, 

 jijg 162 and they advance at a constant velocity, 



keeping symmetrical with respect to the 



plane bisecting perpendicularly the line joining them. This plane is 

 a stream -plane and may accordingly be taken as a boundary of the 



