where 



•S-=(2/2 + z^)\ r^ = {{x- b^f +7^'^i\ T^ = [(a;- h^)"^ +7Z^'\\ [131b, c,d] 



The stream surface for i/i = is given by 



T^/Tl^-i,^/y.^,oxT^/rl=k 



where 



^ = (-/i2/Mi)'^'>0, 

 or also, after replacing r and t by their equivalents, 



2,-2 ,. r/^ ;, \2 . ■~~2^ 



{x- b^Y +~'- ^k[{x- b^ +~^] 



(1 -k){x^ +—2) + 2 {kb^ - ^2^ ^ = '^^1^ " ^2- 



This is the equation of a sphere. 



Let the origin be transferred to its center. Then the term in x disappears from its 

 equation; hence the new values of b^ and b^ are Siich that kb^ = b^, and the radius a of the 

 sphere is given by 



kb^-b^ 

 a^ = = kb^ = b, 6,. [131e] 



Thus the dipoles are located at inverse points with respect to the sphere; see Figure 217, 

 in which two alternative cases are illustrated. Either b^ or b^ must exceed a. 



The formulas may represent either, if b^ > a, the flow around a sphere of radius a 

 caused by a dipole of moment //j placed at a distance b from the center of the sphere and 

 with its axis directed radially, or, if b^ < a, the flow inside a spherical shell of radius a 

 caused by a dipole similarly placed inside it. In either case the a^axis is to be drawn from 

 the center through the location of the dipole; and the second dipole, at a distance 

 ^2 = d^/b^ from the center, becomes a fictitious one that can be regarded as the image of 

 the first. If /i, > 0, the axis of the dipole is directed outward from the center. The potential 

 and stream function are 



327 



