634 Decay of Oscillation of a Solid Body in Viscous Fluid. 



displacement of the sphere, a its radius and I its moment 

 of inertia ; and let h be the radius of: the fixed outer sphere. 

 Then the equation of motion of the rotating sphere is 



' *§ -:M£). + *™=°. • • • (26) 



with = <9 O and cW/dt = Q for t = 0. 

 In the fluid we have 



with co — Q r = b, and w — d6\dt for r = a. 

 Using the method of § 3, we write 



(o= —.{ue^dcc: 6=^[ V e at du. . . (28) 



Then equation (27) gives the solution 



_ a» dO Vb-v) cos h {k(b -r)\ + (k*br-l) sinh {ft(fr-r )} 

 M ~ r 3 di A- (6 -a) cosh {k(b-a) } + ( **6a - 1) sinh {£(£— a) } ' 



• ' (29) 



where k = a?/vK 



Modifying (26) so as to take account of the initial con- 

 ditions, we have for ?; the equation 



,/(a) = F(«), (30) 



where 



/(a) = la 2 -f 8/*7ra 3 « + l/> 2 + f.Trpa»a? 



bkcosh. k(b — a) — s'mh k(b—a) 

 X k{b-a) cosh A'(6-a)+ (£ 2 aft-l) sinh k{b-a) ' 

 F(a) = la -f 8ya7ra 3 -f- f 7rpa 5 a 



bk cosh k(b — a) —sinh k(b — a) 

 X A- (ft -a) cosh £(6 -a) + (//aft-1) sinh £(6 -a) * 

 The angular displacement of the sphere is then 



. ;#•""»•#!<" • • • m 



where the summation extends over the roots of/(a) = 0, and 

 it is assumed that these are all simple roots. 



In practice we may usually separate the roots into two 

 classes : first a pair of roots which may be either complex or 

 real and negative, then a series of real negative roots in the 

 neighbourhood of — irv'^b — a) 2 , — 4:7r 2 v/(b — a) 2 and so on. 

 In deducing the form of (31) for a sphere in an infinite 

 liquid the sum of the terms from the latter series of roots 

 must be replaced by a corresponding infinite integral. 



2m ! . 



