201 



But tlie amplitude of this system is iiiliiiite at infinity, so that our 

 first boundary-condition would not be satisfied. 



We may also choose our boundary -conditions diiTerently. We may 

 for instance imagine the liquid limited on the outside by a second 

 cylinder co-axial with the first and at rest. It is then advisable to 

 write the general solution of equation (^4) in the following form 



ii=z—{C //j(2) (cr) + D H,-^ {cr) \ 



(19) 



At a sufficient distance from the axis of the cylinders two systems 

 of waves then arise, one of which is propagated outwards and the 

 other inwards. At the surface of the exterior cylinder we obtain 

 refiection with reversal of phase, so that the liquid thei'e is at rest. 

 For the determination of the integration-constants C and D we 

 obtain comparatively complicated relations which may be omitted 

 here as they do not yield anything of fui-ther interest. 



The problem of the free oscillation does not now give any further 

 special difficulties. 



We must now seek a solution of equation (1) of the form 



ar = f[r) ^' - ^''' COS {k"t — (f {>')), 



which for r =z R becomes a/^ = a e~^'^ cos k"t. Again we may write 

 a = u e"^ , where n = — k' -\- Ik". 



The same method of solution may now be followed. Instead or 

 (7) we obtain : 



i7j(2) (c'R) r 



ant 



(20) 



7HJ 



wliere c = ' " 



part is chosen. Hence 

 da,. 



, if for c' the root with the negative imaginary 



+ ac' 



//.(■2' (c'R) 



Lim 





e"f. 



(21) 



'/?l=<« H,i^)(c'R) 



Therefore 



da,- 

 ~d7 



2a 

 R 



1,11 1 



(22) 



if for 



IIQ 



ft' 



we take the root with the positive real term. 



The frictional moment now becomes 



