"(?^t) 



Vj^ = U COS + 2 l~:^ + -] cos Q (44) 



Vq = - U sin e + f ^ - I ^ sin e (45) 



13 3 



A = ^ Ua , B = - f- Ua 

 4 4 



Here the first and second terms represent components due to the uniform 

 stream and a doublet at the center of the sphere, and the last terms are 

 due to vorticity. We shall now show that the velocity at points of the 

 X-axis, (G = 0, R > a), can be obtained from the vorticity outside the 

 sphere, together with the uniform stream. 



The vorticity, determined from the last terms of (44) and (45) , is 

 given by 



2B 



w_ = (jOq = 0, oj, = -^ sin (46) 



K u *'' R 



At constant R and 0, a vortex ring of unit strength induces a velocity (by 

 the Biot-Savart Law), at a point x of the x-axis, given by 



^2 . 2 

 R sxn 



2 2 3/2 



2(x -2xR cos 0+R ) ' 



The velocity due to the vorticity in the space exterior to the sphere is 

 then given by 



°° IT 



p I I R sin d0 dR ,,^. 



u = B — ^ ;-^-pr (47) 



/I 



2 2 3/2 



^ «„ (x -2xR cos 0+R ) ' 

 a 



17 



