OP ARTS AND SCIENCES. 



223 



indefinitely extended plate will consist in part of the circumference of 

 which 00 is a radius. This shall be called the circumference 0. 



The value, at P, the co-ordinates of which are x and y, of the flow 

 function is 



^p= —m\ta.n~^^ + tan ^-l -^ — tan~^ ^ 



( a; — a cos ^ x — Oisosd x — a 



^-m ^tan-^ ^^ ~ ^^ ^^ - a sin 6) —y (x ~ a cos 0) 

 I {x — a) [x — a cos t>) + y {y — a sin 0) 



— tan 



-1 y > 



x — b) 



j^^- 1 {x - h) (y -bsine)-y(x-b cos ^) ) 



■ sin ej\' 



(x — b) {x — b cos 0) -\- y (y — b sin 



If now we let 6 approach zero as a limit and m increase in such a 

 way that the product of m and the straight line AA' shall always be 

 equal to the constant fi, we have 



.2 



ij/Q = Limit iff =: -{- fi- 



X — a 



+ 



X — — 

 a 



= ^{x-ar + f ' a^ L-'lX + f 



the real part of the function 



,(7) 



w 



1 



z — a a' 



At every point of 0, \f/o has the value 



2 — — 



a 



t. 

 a 



The velocity components at P corresponding to i/^o are 

 M = — 2/xy 



X — a 



[_{x - af + y'Y "^ a 



r 



X 



a 



v = -\- fl- 



(x — ay — y^ 



[{X - af + ff 



+ -, 



(^-^T-^1 

 (^-^T-' 



a' 



(^-9'-^T 



(8) 



(9) 



If a is made to approach zero as a limit, these expressions approach 

 respectively the limits 



Wo = 



_ — 2 [xxy 



PTTP 



/ a;2 - ^2 1 X 



(10) 



(11) 



