Stahiliti/ of Rectilinear Vortices of Compressible Fluid. 1 39 

 following the fluid, the equation of continuity is 



dp dpu dpv _ 



dt + dx "*" dy 



••• *-sj}?¥ io « r ^ , * ,+ ^ • • • (4) 



If p be the mean value of the density of the fluid at the 

 cross section a of a circular vortex, defined by the relation 



]]p'd X 'dy' = p,a, (5) 



we get <£= J _£° log »• + <£„. 



Again, the equation of continuity is 

 D .p a 



Dt 



= 0, 



1 D Po , 1 Da n 

 or fw +~ -ft: =0. 



Hence 0= - ^r^ lo g ?, + 0o ( 6 ) 



Since a always refers to a definite column of fluid, there 



is no difference between =r- and — . 



Vt at 



Let us consider a truly circular cross section of radius a ; 

 if the vorticity be the same at any instant at all points of 

 the section, we have, when r>a, 



1 da 



logr+U,| 



(7) 

 logr + D: I 



'LIT J 



where k is the strength of the vortex, C and D being 





