and integrate once by parts. Then 



c^ = A\—+— ^ -- I -4- [134b] 



1 a 1 1 r ^^2 



where 



This is the potential due to a point source of strength A located on the a;-axis at a; = c^, 

 together with that due to another source of strength aA/c^ at a? = c^ and a line of sinks of 

 uniform strength -A/ a per unit length extending from the origin to the point x - c . The 

 second source and the line of sinks may be regarded as the image of the first source in the 

 rigid sphere of radius a. It must be assumed that c > a. 



Evaluating the last integral, and treating xfj similarly and dropping a useless constant 

 term. 



1 1 T+x \ 1 a 1 1 r+r^ + c^ 

 In ^ A — + - — In 



r + r„ - c„ 



'2 " '2 2/ \'l ""1 '2 "■ ''22/ 



[134c] 



, r = y/x^ +1:;^ . [134d,e] 



These formulas represent the flow in fluid that is at rest at infinity, caused by a 

 source outside a fixed sphere of radius a. With the origin at the center of the sphere, the 

 source is on the a;-axis at a; = Cj', sr= vy^ + 2^ and denotes distance from the a;-axis. If 

 ^ < 0, the source becomes a sink and all velocities are reversed. 



On the sphere itself r - a, r^/r = a/c„ = c ./a by similar triangles, x - a cos 6 and 



r^= {a'^ + c^ -200^003 d)^^^ 



in terms of the polar angle 6 measured from a radius drawn toward the source. Hence, on the 

 sphere, ip = - A, which shows that the sphere is a stream surface, and 



A /2 c a + r + c \ 



<^ = — ^ -In — - — -]; nun 



337 



