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XII. On the Stability of two Rectilinear Vortices of Com- 

 pressible Fluid moving in an Incompressible Liquid. By 

 Bibhutibhusan Datta, M.Sc. y Lecturer in Applied Mathe- 

 matics, University of Calcutta, India*. 



THE stability of the circular form in two rectilinear 

 vortices has been discussed by Sir J. J. Thomson f for 

 incompressible fluid. 



Dr. Chree % attempted to extend his treatment to a 

 compressible fluid but did not succeed except in some special 

 cases. The object of the present paper is to complete the 

 work begun by Dr. Chree. 



When the vortex-lines are straight lines parallel to the 

 axis of 0, the velocity components u, v at any point ($, y) in 

 the fluid are given by § 



_ d0_dV r _ 30 .c^. 

 ~d* ~dy ' "dy d^ ' 



• (1) 



Vi 2 tf>=-0, Vi 2 ^ = ?, • • • 



• (2) 



„ du dv 2 B 2 3 2 

 ■di\dy' Vl ~"ba? W 





(3) 



hence 

 where 

 Then by the theory of attraction, 



^=- 2 MiVlogr^'^'+0o, 



'f = ^ 1 P '* 0g r dx ' dy ' + ^ 0; 



6', f being respectively the value of 6 and f at the point 

 {js' f y ' ) and 



r={( x -xy + (y- y <y\\ Vi 2 (/>o = 0, Vi 2 ^o = 0. 



if — denote partial differentiation, and -n- differentiation 

 dt r Dt 



* Communicated by the Author. 



t 'Motion of Vortex Rings,' p. 71. 



X Messenger of Mathematics, vol. xvii. p. 113 (1888). 



§ Lamb, ' Hydrodynamics/ 4th ed. p. :213 (1916). 



