a 2 '= cos 



— sin 



148 Stability of Rectilinear Vortices of Compressible Fluid. 



Case III. — A Vortex-pair. 

 A combination of: two equal and opposite vortices is called 

 a "vortex-pair." Put k——k, a = b; then e = : we have 

 from (32) 



da' I k' ftf*\-| - 



da 1 . /k' fdt\-\ ,, 



so that in this case, the vortices have a single type of 

 disturbance of period given by 



Cdt 



k i dt _ 



2irn. 



As a vortex-pair is the two-dimensional analogue of a 

 circular vortex-ring some of the properties of the latter may 

 be deduced from it. 



Case IV. — A single vortex parallel to an infinite wall. 



This is really a subcase of Case III. For the velocity of 

 the fluid at all points of the plane of symmetry is wholly 

 tangential, so that this plane may be supposed to form a 

 rigid boundary of the fluid on either side of it ; thus we 

 obtain the case of a single rectilinear vortex in the neigh- 

 bourhood of a fixed plane wall to which it is parallel. If h 

 be the distance of the vortex from the wall we get 



< = c °* feJSf i ["' sin (If} v) 



da-' /«' Cdt\-\ , 



-w co \**)m dt 



da' . / k' Cdt\-\ , 



+ 



Hence the diameter parallel to the wall is greater if b is 

 increasing, i. e. the vortex is receding from the wall and 

 vice versa. 



University of Calcutta, 



1st "September, 1919. 



