or, from Equations [83f, g], [82i,j], [83d, e] and [82x], 



V 



r./ 



(a'a- b'b) cos (7/ -a) 

 a - b 2n 



[83k] 



U r a'+ b' 



(b'a-a'b) sin (17 -a) + In 



a — b 27T a + b 



[831] 



Thus i/( = on the ellipse ^ = ^^, whose equation from Equations [82g] and [83f, g], is 



This ellipse may represent the profile of a solid elliptic cylinder immersed in the fluid stream, 

 with its major axis parallel to the flow at infinity. In going once around the cylinder, rj in- 

 creases by 2 77 and <;6 decreases by F. Hence there is circulation F around the cylinder. 



If r = 0, the remainder of the streamline for = is defined by rj = ot on the forward 

 side or 7; = a + 77 on the rear side; it consists of hyperbolic arcs. 



The components of velocity at any point (x, y), respectively tangential and normal to 

 the ^-ellipse that passes through (x, y), or in the directions specified in Equations [82b', c'], 

 are, from Equations [82y, z] and [82i,j], [83f, g], 



^n- 1^- 



V {b' a - a' b) 



(a-b) (6 '2 + c^ sin^ ,,)'/' 



cos (77 - a), 



[83m] 



1 



?t=?. 



(b'^ + c'^sm^T])'^' 



a'a- b'b V 



V — sin (77-a) + — 



a- b zn 



[83n] 



On the ^-axis, where 77 = or tt and x = - a', respectively, 6'= yjx'^ - c^ , 



, U / b\x\ 



u = - g^ =- (a _ \ cos a , 



U I a\x\ 



V = - a, = — 



a - b\ f~2 ~2 



4x^ 



- b\ sin a - 



2ti^Ix'^ - 



[83o] 



L83p] 



On the y-axis, where 77 = 77/2 or 3;7/2, y = ± b', a' = \l b'^ + c^ = \]y^ + c^ , 



198 



