A SPHERE IN A VISCOUS LIQUID. 57 



Putting ? a -f a = 2u v /('f) this becomes 



7-'=' J = + (i_=)exp. (-'" Vl "}\t-*du. (45) 



/ *ff I ** Iff I * A> * ' 



If r > a it follows that v f = wheu t = 0. When r = a and t = the lower 

 limit of the definite integral (45) becomes indeterminate ; but since, in this case, we 

 are to have v' = aw sin 6, it follows that if we put k = r a the quantities k and t 

 must vanish in such a manner that when k = and t = 0, k/2^/(vt) = 0. 



When t = oo we obtain 



, a*u> sin 

 v = (40) 



This equation gives the value of v' after a sufficient time has elapsed for the motion 

 to have become steady, and agrees with Professor STOKES'S result. 



11. Since the tangential stress per unit of area which opposes the motion of the 

 sphere is 



T =-^ 



dr 

 the opposing couple is 



= Zirvpa* j ( - 1 

 * dr \r/. 



If, therefore, the sphere be acted upon by a couple, N', it equation of motion will 

 be 



A** 5 * + G = N', 

 or 



<rw d fv\ XT , ... 



-- v . (- = N, ........ (4/) 



>p dr \r/ 



where 



N = 3pN'/8a 4 . 



When the motion of the sphere commences from rest the value of v or v' cosec 6 will 

 be obtained from (45) by changing / into T, <a into F' (t r)dr, and integrating the 

 result with respect to T from t to 0, where F (t) is the variable angular velocity of the 

 sphere. 



MPCCCLXXXVin. A. I 



