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



da, ' 

 so that to our degree of approximation —^-=0; similarly 



dS ' 

 we p'et — =-^-=0. Hence, so far as these terms are con- 

 ° dt 



cerned, compressibility of fluid makes no change in the 



motion or shape of the vortices. 



Equating the coefficients of cos 26' and sin 26', we get to 



the same degree of approximation 



d*2 , k' a i d> • o b da . . 



-df + 2^i^ = i^ sm2e -2^-dt COs2e ' • (21 > 



dBJ k' , Kb _ b da . n /n -». 



-dT-^V a ^-Y^ cos2e -¥^W %m2e - (22) 



About these equations Dr. Chree remarks : " It is scarcely 

 likely that these equations admit of a complete solution." * 

 But the solution of them is of essential importance, for on 

 it depends the determination of the exact change in the 

 form of the cross section. I shall presently show that they 



a 3 

 can be solved. The terms in a 3 , /3 3 will involve — d , and thus 



c 



will be relatively unimportant, so we shall not determine 



them here. 



where i= y-l.J.. (24) 



V = a 2 —?p 2 , ) 



Multiply the equation (22) by i : the two equations (21), 

 (22) become, by addition and subtraction, 



du . k! . Kb „. ■ b da 



dt 2irb 2 2irc 2 2m c 2 dt 



(24) 

 dv . k .Kb „._ b da . 



-T- 4 i cr~T^ v==l n — ^<? — 7T- —r e~ 2ie . 

 dt 2irb- 2ttc 2 2?rc 2 dt 



These are linear differential equations. On integration 

 we set 



ve 



J/ rw + a^AJ *• 



1/ I'w-^^I^J 



(25) 



«-' 



where w and r are initial values of w and v. If o^', oA/ De 

 the values of « 2 ', # 2 ' respectively when t—0, we must have 



W(3 = a 2 / + ?'oA' 5 v = ct 2 / — i f3 2 '. . . . (26) 



* X. c pp. 115, 118. 



