36 



Now let us choose for u an axisymmetric potential function and let 

 ^(x, y) be the corresponding stream function. Then 



y dn ds 



and 



f>t 



ds = ^r. f%4l ds 



In Jo QS 



Also let U be the total velocity along the body when the flow is made steady 

 by superposing a stream of unit velocity in the positive x-direction. Then 



U = -t? + cos y 

 ds 



Furthermore, we have dx = ds cos y, dy = ds sin y. Then [89] may be written 



\ p r p , r p 



cj>ip - \l> (cos y - U)ds = - yw dy 

 'o Jo Jo 



U^ds = I {^dx - ywdy) 



■'0 Jn 



M [90; 



But, since u> and ip are corresponding axisymmetric potential and stream func- 

 tions, we have 



dw _ d± 

 * dx dy 



Hence ^dx - ywdy is an exact differential defining a function Q(x, y) such 

 that 



dQ , dQ 



- *• w = -y w [91 ] 



But since also 



we obtain from [91 ] 



6>x ^' ay 



v |w = d± 



J dy dx 



d 2 Q 6fQ = J_dQ 



ex 2 d Y 2 y a y 



