<;i = aln - = aln [123d] 



T,+ X- T + Tl - 0+ a 



■A = a(r^-ri^) [123e] 



where r^ and r^ denote distances from the ends of the line of sources, or 



r^ = [(x-af + Z^f\ r^ = [(x- bf ^-To^f 



The identity of the two forms given for <•> is easily verified by eliminating x. 

 The components of velocity are, from [118h,i], 



since "w^ = r^^ - (x-a)-^ = Tj^ - (x- b)."^ 



The equipotential or <f> surfaces are ellipsoids, the stream surfaces for i/f = constant 

 are hyperboloids; all have common foci at (a, 0,0), (6,0,0). 



If a < 0, there is a line of sinks instead of actual sources. (See Reference 2, Section 

 15.24; Reference 7, p.60.) 



124. LINE OF POINT SOURCES IN A STREAM 



Suppose that a uniform distribution of point sources exists along the stretch of the 

 a;-axis from ;» = - a to a; = 0, and that the fluid at infinity is also streaming at velocity V 

 toward negative x; see Figure 205. From [119a, b] and [123d, e], in which now a -* - a, 6 -» 0, 



r^ + x+a r+r. + a 



</) = f a; + aln = Ux + aln [124:a] 



r+x T+r.-a 



(A = - f/S^ +a (r.-r) [124b] 



2 



where 



r=(a;^ + w^) , Tj = [(a;+ a)-^ + 3^] 



and &) denotes distance from the a;-axis. The volume of fluid emitted per second from unit 

 length of the line of sources is 4n- c. Let a and V have the same sign. 



312 



