If the dipole is at the center of a spherical shelly however, a limiting form of is 

 needed; for, as 6. ^0, 6 -> ~, Expanding by the binomial theorem in descending powers 

 of 6„, 



-J r) o Q— 3/2 Q— 3/2 o ,^~5/2 



hence 



{-2b^x+ x'^ +^^) = 62"^ + 3 62 



b-^ {x - b^) r-^ = - a-^ b^-2 a'^ x 



since 6 6 = a . The term -a ^ br. contributes a constant term in and may be omitted. 

 Then, as b^ -> 0, Equations [131f, g] become 



^1^4 + -J cos ^,0 = -^^-2/1 __J ^3ii^.] 



with use of polar coordinates at the center of the sphere such that r = (a;^ + a> ) , x = r cos 6. 

 All other terms in the series vanish as b -* ~. The components of velocity are 



d(f, /ll\ 1 <?<?!. /12\ 



o = = 2 u I — - — I cos 6, On = = M, I — + — sin 6. 



[131k, 1] 



Thus the presence of the shell superposes a uniform backward flow, with components of 

 velocity -2 fi^ cos d/a^ and 2 n^ sin d/a^, upon the flow due to the dipole alone. 

 The force F on the sphere or shell is of magnitude 



I 24:npa^ b^H^ 

 \F\ = — pfq^ cos e (277 a^ sin OdO) = . [131m] 



(if _ a^)' 



(The integration is long but easy.) The force tends to draw the nearest part of the sphere or 

 spherical shell toward the dipole. 



Streamlines for equally spaced values of 0, on a typical half-plane through the axis of 

 symmetry or a;-axis, are shown for an exterior dipole as solid lines in Figure 218. 



(See Reference 1, Article 96; Reference 2, Section 15.43.) 



329 



