252 VORTEX MOTION. [CHAP. VII 



is occupied by liquid having a uniform rotation , whilst the surround- 

 ing fluid is moving irrotationally. It will appear that the conditions 

 of the problem can all be satisfied if we imagine the elliptic boundary to 

 rotate without change of shape with a constant angular velocity (n, say), to 

 be determined. 



The formula for the external space can be at once written down from 

 Art. 72, 4 ; viz. we have 



,j, = %n(a + b) 2 e-^cos2T, + ab ..................... (ii), 



where , TJ now denote the elliptic coordinates of Art. 71, 3, and the cyclic 

 constant K has been put = 27r6, in conformity with Art. 142. 



The value of >// for the internal space has to satisfy 



with the boundary-condition 



ux vy x 'ii ,. ,. 



*+$'* &+**& ........................... (1V) - 



These conditions are both fulfilled by 



+=t(Ax* + By*) ................................. (v), 



provided A + B 



................................. (vi). 



= 1 , ) 



L 

 = (a?-V}) 



It remains to express that there is no tangential slipping at the boundary 

 of the vortex; i.e. that the values of d\jf/d obtained from (ii) and (v) 

 coincide. Putting x c cosh cos /, y = c sinh sin 17, where c = (a 2 - 6 2 )^, diffe- 

 rentiating, and equating coefficients of cos 2rj, we obtain the additional condition 



- \n (a + b) 2 e~^ = c 2 (A - B} cosh sinh , 

 which is equivalent to 



- 4 - B if-T (vii)> 



since, at points of the ellipse (i), cosh = a/c t sinh = b/c. 

 Combined with (vi) this gives 



Aa=Bb= -j (viii), 



and * = 7TTiA a f W- 



When a = 6, this agrees with our former approximate result. 



The component velocities x, y of a particle of the vortex relative to the 

 principal axes of the ellipse are given by 



3G 11 *U 1C 



whence we find -= ?&y, ^-n- .............................. (x). 



a b 1 b a 



Integrating, we find 



x = ka cos (nt + c ), y=ib8iu(nt+t) ..................... (xi), 



