158-159] ELLIPTIC VORTEX. 



where , e are arbitrary constants, so that the relative paths of the particles are 

 ellipses similar to the section of the vortex, described according to the harmonic 

 law. If of, y' be the coordinates relative to axes fixed in space, we find 



k k 



x' =37 cos nt -y sin nt = -(a + b) cos (2nt + e) + - (a- b) cos *, 



.(xii). 



k k 



y' = x sin nt +y cos nt = -(a + b) sin (2nt -f e) - ~ (a - b} sin e 

 _ 2i 



The absolute paths are therefore circles described with angular velocity 2n*. 



159. It was pointed out in Art. 81 that the motion of an 

 incompressible fluid in a curved stratum of small but uniform 

 thickness is completely defined by a stream -function ^, so that 

 any kinematical problem of this kind may be transformed by 

 projection into one relating to a plane stratum. If, further, the 

 projection be ' orthomorphic/ the kinetic energy of corresponding 

 portions of liquid, and the circulations in corresponding circuits, 

 are the same in the two motions. The latter statement shews that 

 vortices transform into vortices of equal strengths. It follows at 

 once from Art. 142 that in the case of a closed simply-connected 

 surface the algebraic sum of the strengths of all the vortices 

 present is zero. 



Let us apply this to motion in a spherical stratum. The 

 simplest case is that of a pair of isolated vortices situate at 

 antipodal points ; the stream-lines are then parallel small circles, 

 the velocity varying inversely as the radius of the circle. For 

 a vortex-pair situate at any two points A, B, the stream-lines are 

 coaxal circles as in Art. 81. It is easily found by the method of 

 stereographic projection that the velocity at any point P is the 

 resultant of two velocities w/7ra . cot %6^ and m/7ra . cot J# 2 > per- 

 pendicular respectively to the great-circle arcs AP, BP, where 

 1} #2 denote the lengths of these arcs, a the radius of the sphere, 

 and ra the strengths of the vortices. The centre f (see Art. 154) 



* For further researches in this connection see Hill, ' On the Motion of Fluid 

 part of which is moving rotationally and part irrotationally," Phil. Trans., 1884; 

 and Love, " On the Stability of certain Vortex Motions," Proc. Lond. Math. Soc., 

 t. xxv., p. 18 (1893). 



t To prevent possible misconception it may be remarked that the centres of 

 corresponding vortices are not necessarily corresponding points. The paths of these 

 centres are therefore not in general projective. 



The problem of transformation in piano has been treated by Kouth, "Some 

 Applications of Conjugate Functions," Proc. Lond. Math. Soc., t. xii., p. 73 (1881). 



