of a Solid Body in a Viscous Fluid. 634 
displacement of the sphere, a its radius and I its moment 
of inertia ; and let 6 be the radius of the fixed outer sphere. 
Then the equation of motion of the rotating sphere is 
reas ree (2) +pl@=0 (26) 
diz 3 dr}, ? Op eed int 
with 0=6, and d@/dt=0 for t=0. 
In the fluid we have 
Oe =1(S8 4 2) 
Oi NO TP OPY 
with a=0 r=b, and w=d6/dt for r=a. 
Using the method of § 3, we write 
1 1 ans 
o= sefuer da ; 0=5- netda. . . (28) 
p 
(27) 
Then equation (27) gives the solution 
ak a? dO k’b—r) cosh {k(b—r)} + (br —1) sinh {k(b—7)} 
~ 73 dt k(b—a) cosh {k(b —a)} + (ba —1) sinh {k(b—a)}’ 
where k=a3/v'. 
Modifying (26) so as to take account of the mitial con- 
ditions, we have for 7 the equation 
a mia=N@,  s 6 5 0 o @ 
f(a) =a? + 8urra*a + Ip? + $rrpara? 
bk cosh k(b—a)—sinh k(b—a) 
‘ k(b—a) cosh k(b—a)+ (Pab—1) sinh k(b—a)’ 
F@)=Ie+ Sutra? + Sarpara 
bk cosh k(b—a) —sinh k(b—a) 
X k(b—a) cosh k(b—a) + (Kab —1) sinh (6 =a)’ 
The angular displacement of the sphere is then 
— 9 (EC) pat ag — 6,3 ED pat 
9 ori) Flay eo Fa) 
where the summation extends over the roots of f(4)=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 —7v/(b—a)?, —47°v/(6—a)? 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. 
(31) 
191 
