628 Prof. T. EL Havelock on the Decay of Oscillation 

 Comparing with (32) it follows that 



A,= 2itvV/fl 2 {^ + *(*+ 4 )}' ■ • • ( 35 ) 

 With this expression for the solving function, (28) gives 



dt~I a 2 L J A. 2 + k{kj- 4) dT ' ' • W 



By expanding r 2 by (33) and putting r = l, it can be 



shown that 



i-%KSS>- -.- ■• <"> 



Carrying out the integration in (36) and using (37), we 

 find 



<fc ~I + iir/wi 4 + I *** + *(* +4)" ' ' . V ; 



The angular acceleration has a finite limiting value in this 

 case, the same as if the cylinder and enclosed liquid were 

 rotating like a rigid body. We notice that in this case zero 

 is excluded from the roots of the equation (29) for the 

 exponents cf the nucleus. 



Integrating (38) we obtain the angular velocity of the 

 cylinder at any time ; then, using the differential equation 

 (26), we may complete the solution by writing down the 

 angular velocity of the liquid. It is found to be given by 



N f r 2 k + 6 



N _ 1 r* _ k + 6 ~\ 

 I + i7rpa i V + Sv 12(*+4)J 



2£N a*J 1 (\r/a)e-*W'* 



^\ 2 {X, 2 + A</f-h4)}J 1 (X)' 



LXXIV. On the Decay of Oscillation of a Solid Body in a 

 Viscous Fluid. By T. H. Havelock, F.R.S.* 



I. TT^HE decay of rotational oscillation of a cylinder 



I or a sphere in a viscous liquid is a well-known 



problem in Hydrodynamics ; among more recent researches, 



reference may be made to the work of Verschaffelt +, 



Coster J, and others. In those papers it is remarked that 



* Communicated by the Author. 



t Gr. E. Verschaffelt, Amsterdam Proc. xviii. p. 840 (1916) ; also 

 Comm. Leiden, cli. (1917). 



X D. Coster, Phil. Mag. xxxvii. p. 587 (1919). 



