197 



second term on the right of (Ji) disappears on two grounds: 



(1) becausec = (2) Lm; — ; = 0; only the first term then 



remains, which agrees with (2'). 

 Moreover 



I Am ~ = — I. 



It appears from the accompanying graphs ^) of the module and 

 argument of — - — -- that tiiis limiting value is practically reached at 



\cR\=k.R— 10 



2^1/2 ;..... (12) 



Ic| =k = ~~^-~ (cf. 8')| 



The condition \c R\^^0 means, that the radius of the cylinder 

 must be about equal to or larger than the wave-length. If R is 

 small compared with / the second part of the frictional couple is 

 negligible. For \c R\^\0 the 2"^* term on the right-hand side of 

 ''10) becomes 



. / . TT \ / in 



— a ci e^^' = — a k e \ ^ ' I since c = k e '^ 



Hence equation (11) now becomes: 



K= — 4 Tin R' oj — 2:^ IxkR^ -(acos (pt + jjj . (13) 



where 



d 

 a> = — (a cos pt). 

 dt 



The frictional couple thus divides into two parts, one which does 

 not contain the density of the liquid and another, in which it 

 occurs and which therefore refers to the emission of waves. In the 

 transition to the limit of uniform rotation the first part only remains. 



In the discussion of the 2"^' part of the frictional moment the 



quantity k = 1/ ^^ is an important factor. If we take a time of 



, I /^ 



oscillation of 2 .t seconds, so that /^=1, we have k=\X — . 



This gives the following values for k. 



1) Gomp. J. u. E. 1. c. 



2) Tables for Hoih and //o(2; will be found J. u. E. p. 139, 140. 



