Compressible Fluid moving in Incompressible Liquid. 147 



Case 1. — Two columns of fluids of varying density 

 possessing no vorticity. 



In this case k = k' = e = 0, 



so that the equations (27) and (2S) become 



, i" /> da 

 ** = °**-\wdt dt ' 



which agree with the result obtained by Dr. Chree. Since 



--=27ra^-, « 2 ' would increase or decrease according as a 

 tit at 



was decreasing or increasing. " Thus, i£ both columns were 



diminishing in cross section, and so approaching, there would 



be a decided tendency in both cross sections to assume an 



elliptical sort of outline, the major axes coinciding with the 



line joining their centres/' 



Case II. — Two rectilinear vortices in an 

 incompressible fluid. 



As previously mentioned, this case has been studied by 

 Sir J. J. Thomson. AVe shall simply show that his results 

 can be deduced from our more general results. Let 

 a 2 / = /3 2 ' = ; also if n be the angular velocity of the central 



line, e = nt. Then putting -j-=0 in (27) we get, since 



a: = cos (£t\ r |™ sin f'2nt- 1 1) dt 



+ sin(|^)£|Jcos(2n-| / ^^ 



or a 2 '= -^ (cos2nt— cos(^) \; 



2c^-2n) 1 V2/) 



similarly from (28), 



ft'*, Vf* { s i D2w ^sin(gA}. 

 2c2/| -2n\ L K " y J 



These will be found identical with the results obtained by 

 Sir J. J. Thomson* when we remember that f=2w, and 



?' = 2a>'. 



* ' Motion of Vortex Rings," p. 77. 



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