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



In our further investigations, we should remember that if 

 e — when £ = 0, it can be proved that* 



= k + /c' r alt 



2tt J c 2 +(a- + O -'- O - -o- / )/7r' ' " ~ { J 



where c* , a , aj are the values of c, a, a' respectively 

 when £ = 0. 



The equations (27) and (28) show that there are two 

 distinct types of vibrations of u 2 ' and fij of gradually varying 

 periods given by 



U3-*"> (? ' o) 



17 Jo ( '0 



*. o 



dt 



= 2/i7T, . (31) 



where n is any integer. Evidently the period given by (30) 

 is shorter than the period given by (31). 



Suppose « 2 ' = /3 2 ' = ; since 2e — — 1 jj is always 

 negative, we get "J 



ai = _ C03 (£ j *)£ _|_ [, sin (g_ j *•_ 2.) 



-K£j3)ri[-(£j?--) 



so that vibrations are becoming more and more prominent 

 as the vortices are approaching one another ; since 



is always real and bjc is also small, the value of a 2 ' cannot 

 increase indefinitely so that the motion is stable f . 



We shall now proceed to consider some interesting 

 particular cases. 



* Chree, I c. p. 109. 



t This point will be clearer from equations (25) which are of the type 



dc 2 _ d ,„ ,„, „^, % , , „ . ,„ lb 



Since 



,r=2— 6 2 (Chree, p. 108), the function F ( — 2 ) continually 

 decreases as t increases so that in the limit the integral must be finite. 



