66] SPECIAL CASES. 83 



of a liquid flowing from one side to the other of a thin plane 

 partition, through an aperture of breadth 2c ; the velocity at the 

 edges is however infinite. 



If the ellipses be taken as the stream-lines we get the case of 

 a liquid circulating round an elliptic cylinder, or, as an extreme 

 case, round a rigid lamina whose section is the line joining the foci 

 (c,0). 



At an infinite distance from the origin < is infinite, of the 

 order log r, where r is the radius vector ; and the velocity is 

 infinitely small of the order l/r. 



2. Let z = w + e w < (5), 



or x (f> -f e* cos ty, y = ty + e* sin ty (6). 



The stream-line ^ = coincides with the axis of x. Again the 

 portion of the line y= f rr between x = oo and x = 1, considered 



as a line bent back on itself, forms the stream-line ty = TT ; viz. as 

 <[> decreases from + oo through to oo , x increases from x to 

 1 and then decreases to oo again. Similarly for the stream- 

 line yjr = 7T. 



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