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PROCEEDINGS OF THE AMERICAN ACADEMY 



Fig. 4. 

 The value at the point (x, y) of the flow function is 

 x^/ z= m,\ tan~ ^ — ^— ^ + tan~ ^ — ^ — - — tan~^ — ^ — ^ — tan~^ ^ 



X — 8 





= m\ tan-^ _— — J-— + tan"! 



a: + 8 



7M * 



8 



8M^' + y') 



If, now, 8 be made to approach zero as a limit, and m to increase in 

 such a way that 2 m 8 is equal to the constant ^, 



^o^Limit^^-^( -^+/;-/^ ); (6) 



an expression which is equal to zero at every point of the circumfer- 

 ence, x^ -\- ■if- ^ r^. 



The velocity components Mq, Vq, obtained by differentiating (6), are 

 identical with those given by equations (4) and (5). 



Fig. 5. 



5. In Figure SJet OA = OA^ ^a, 0B= OB' = b, and OA OB 

 = OA' OB' = OC'^ = r\ and let there be at A and B sources of 

 strength m, and at A' and B' sinks of equal strength m ; then one 

 of the lines of flow due to this combination of sources and sinks in an 



