J\fass carried forward by a Vortex. 



601 



P, P' any symmetric points on opposite sides of the axis Oy. 

 Then the stream function ^ at P = flow through PN due to 



A, A' = flow through PP' due to A alone = flow across P'Q 

 whereAQ = AP. Hence 



AP 



a I dr fju . AP' 



AQ 



Also velocity of: translation is velocity at A' due to A, or 



u=, /t -. 



Hence when brought to rest the stream function is 



X 



- \JX + 7— log — 



2 1 „,2 



or 



tt x , (^4-a) 2 + y 



The velocity at the centre relative to the surrounding liquid 

 is /ju/Tra or 4U. Consequently in the case of parallel straight 

 vortices the cyclic boundary {2, 3) is singly-connected. The 

 stream function along it has the same value as for the axis 

 and is zero. The equation to the boundary is therefore, 

 taking the unit of length = a, 



. (x+iy+f 



*(x-l) 2 +y 2 



x 



or 



y 2 = 2x co t\\ 2 



1. 



