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

 of the vortex, w« have from (15) 



3il — -- - 4 2 ( —t- cos ntf H r— sm n0 ) 



<i£ V «* »* / 



- 2ne(a,/ sin 710' —&' cos w0')- ■ • • ( 17 ) 



Since products and powers of a/, ft/ are to be neglected, 

 in (17) we may put 



(13) 



<H> 



2irl>' 



Again 



-_ 1*1_M:_ a±_a*_ Jf-faw-e) 



dt 



cos (0' -e), . . . (19) 



where r\ must be put equal to b 4 2(a»' cos nO' 4 ft/ sin n6 r ) 

 after differentiation, and e is the angle made by the line joining 

 the centres of the two vortices with its initial position. If 

 b/c be small, <f> and i|rcan be expressed in terms of r', #', 

 and e, such as 



, ,„ 1 dtrr, " (~iy- l /r'\ s //V 



+ 2 



2tt <ft 



1 r/cr a"- 1 



</£ 



-1 r- cc 



— (a rt cos 716 + ftj sill ?ie) 2 ( — l) s 

 L 5=0 



X — 



n(n-+l)...(n + s-l)(r'\* 



® 



cos s(0' — e) 



4 (ft, cos rie — a re sin >?e) 2 ( — l) 1 



*=i 



rcfra 4 1) ...(n + s — 1) //V . 



X l 



(^sin^'-e)], 



^^^h^^^C)-^'-^] 



_ V 



lira C 



X 



(a u cos ne 4 ft, sin we) 2 ( — 1) 



?l(« + !)...(*' 4- 5-1) /V 



S • 



^Vcoss((9'-e) 



4 (j8 n coS7ie — a„sinwe) 2 ( — I)*"' 1 



s=l 



w.(?i + l)..'.(n + * — l)/r'Y . 



5! 



gjsin<0'-e)]. 



