228 VISCOSITY. [CHAP. ix. 



To solve this equation we assume 



f(r) = 2Ar n , 

 which gives (n + 1) (n - 2) (n 1) (n 4) = 0, 



so that / (r) = ^ + Br + Cr* + Dr\ 



The conditions (35) are then satisfied by 



D = 0, 2(7 = -F. ...................... (3G). 



We have still to introduce the conditions to be satisfied at the 

 surface of the sphere. On the hypothesis of no slipping, these are 



5 = 0, = ............... ..... ....(37), 



when r = a (the radius) , or 



/(o) = 0, /=&amp;lt;&amp;gt; .................. (38). 



Combined with (36), these give 



A = -\VOL\ B = lVa .................. (39), 



but we retain for the present the symbols A, B. The resulting 

 value of -^r is simplified if we restore the problem to its original 

 form by removing the impressed velocity V in the direction of x, 

 i. e. if we add to the term 



Thus ,lr = ~ + ]fr8m*0 .................. (40), 



whence 



The resistance experienced by the sphere is most readily calculated 

 by means of the dissipation-function F of Art. 179. Let us take 

 at any point a subsidiary system of rectangular axes, in the direc 

 tions of E, , and of a normal to the plane of 6, respectively ; and 

 let [a], [b], [c], [/], [g\ y \li] have the same meanings with respect 

 to these axes that a, b, c,f, g, h have with respect to a?, y, z. We 

 find by simple calculations 



_ dU d R R 



dR 



