In this way a dipole located near a sphere, as in Section 131, can be replaced by a 

 point source. In the formulas of Section 131 let b be replaced by a;/ and fi. by - Adx' where 

 ^ is a constant. Then the flow is represented due to a dipole of moment -Adx', located on 

 the element dx^' of the a; '-axis at the position x^. Let such dipoles be located on all elements 

 from ajj' = c^ to + ~; and integrate to obtain the flow due to all of them. The potential and 

 stream function thus obtained are, from Equations [131f, g]. 



- A 



I 



dx^, 



where 



<A = ^ 



i, ^ <' \<l 



dx', 



1/2 



2' = a /ajj', r^' = [(x - x^'y + oT^] , r^ = [{x - x^f + £^ 1 



.1/2 



Here x and co are the coordinates of a fixed point in space and are constant in the integration; 

 see Figure 222. 



Figure 222 - See Section 134. 



The first term of the integral for can be evaluated at once. In the second term take 

 x^ as the variable of integration with limits and C2 where 



c^-^a /Cj, 



[134a] 



336 



