140 Mr. B. Datta on Stability of two Rectilinear Vortices of 

 constants. Also when r<a, 



*=-iS" 2+c ''j (8) 



Let the equation of the boundary of the cross section at 

 any instant be given by 



r = a + 2(a»cosw0 + #»sinw0), . . • (9) 



where a H and /3 n are functions of t, independent of 6, and 

 very small compared with a. 

 Outside the vortex, r>a, let 



4> = C - ~ ^log r+ 1 (E. cos nO + F» sin n6) ("J, 



r) ' 

 inside the vortex, r < a, 



^ = D' + ^2 r 2 + 2(A„' cos nO + B n ' sin n0) Q". J 



Since $ and ty must be continuous when crossing the 

 boundary, we at once get, neglecting products of small 

 quantities, 



A„' = A W , B,/ = B n , E n ' = E n , F n ' = F tt . . (12) 



The radial and tangential velocities for points just inside 

 and just outside the vortex must be the same. Therefore 

 when 



r = a-\- 2(« n cos n6 -r (3 n sin nO) 

 r "dd ~dr r ~d0 ~&r 



and — -^ r + -~ = _ "5T" + ~^a • 



Or rod or rov 



