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 dimmish their velocity perpen- 

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

 AA' 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 



m x m 



* -' *- 



where r 2 = 3?+y 2 . By division we obtain the differential equation of the 

 path, viz. 



dx 



whence a?(x 2 +y 2 ) 



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



Also since xi/-yx=ml < 2,Tr i 



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 



* See Greenhill, " 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. Rep. 1882, pp. 41...; Love, "On 

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

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



