248 VORTEX MOTION. [CHAP. VII 



the effect of the presence of the pair A, A on B, B is to separate 

 them, and at the same time to diminish their velocity perpen 

 dicular to the line joining them. The planes which bisect AB, 

 A A at right angles may (either or both) be taken as fixed rigid 

 boundaries. We thus get the case where a pair of vortices, of 

 equal and opposite strengths, move towards (or from) a plane 

 wall, or where a single vortex moves in the angle between two 

 perpendicular walls. 



If x, y be the coordinates of the vortex B relative to the planes of 

 symmetry, we readily find 



where r 2 = j?+y 2 . By division we obtain the differential equation of the 

 path, viz. 



dx dy 



-^ + -4=0, 

 * 



whence a 2 (x 2 + y 2 } 



a being an arbitrary constant, or, transforming to polar coordinates, 



r= a/sin 20 .................................... (ii). 



Also since 4$f-y m/2r, 



the vortex moves as if under a centre of force at the origin. This force is 

 repulsive, and its law is that of the inverse cube*. 



156. When, as in the case of a vortex-pair, or a system of 

 vortex-pairs, the algebraic sum of the strengths of all the vortices 

 is zero, we may work out a theory of the impulse/ in two di 

 mensions, analogous to that given in Arts. 116, 149 for the 

 case of a finite vortex-system. The detailed examination of this 

 must be left to the reader. If P, Q denote the components of the 

 impulse parallel to x and y, and N its moment about Oz, all 

 reckoned per unit depth of the fluid parallel to z t it will be found 

 that 



Q=- pSJx^dxdy, ) 



+ y*)Sdxdy 1 ........ 



* See Greenhill, &quot; On plane vortex-motion, Quart. Journ. Math., t. xv. (1877), 

 where some other interesting cases of motion of rectilinear vortex-filaments are 

 discussed. 



The literature of special problems in this part of the subject is somewhat 

 extensive; for references see Hicks, Brit. Ass. Ecp. 1882, pp. 41...; Love, &quot;On 

 Kecent English Eesearches in Vortex Motion,&quot; Math. Ann., t. xxx., p. 326 (1887) ; 

 Winkelmann, Handbuch der Physik, t. i., pp. 446-451. 



