of a Cylinder in a Viscous Liquid. 593 



From this table it appears that, except for dilute gases, R 

 has to be relatively small in order that the second part may 

 be neglected with respect to the first. For instance, for atmo- 



Hf(cR) 

 spheric air with R = Oo cm. &R = 1*4 and — m — = 080, 

 F Hf(cR)| 



so that the amplitude of the second term of the frictional 

 couple is still 56 per cent, of that of the first (see equation (1 1)), 

 in which everything is calculated for a time of oscillation of 

 2ir seconds. 



There is a further special limiting case of equation (13), 

 which is of some interest. Let R become infinite, and let a 

 at the same time disappear, in such a manner that Ra 

 converges to a finite limit b. We thus approach the one- 

 dimensional problem of the oscillation of an unlimited flat 

 plate in its own plane in an infinitely extended liquid. The 

 frictional force per unit of area is found from (13) to be 



a formula which is well known from hydrodynamics*. A 

 term analogous to — 47r / u,R 2 &) does not occur in the one- 

 dimensional problem, the reason evidently being that with a 

 uniform translation of the plate a condition of equilibrium 

 does not arise, until the whole liquid away to infinity proceeds 

 with the velocity of the plate. 



Finally it is of importance to ascertain for what frequency 

 the amplitude of the forced vibration becomes a maximum, 

 in other words, to what frequency the system cylinder-liquid 

 resounds, if the cylinder is urged back to the position of 

 equilibrium by a quasi-elastic force. 



The differential equation for the forced oscillation in 

 complex notation is as follows : 



Here in our case L is a complex quantity L=L'-H'L", 

 where 



L' =(47r/xR 2 + v /27r^R 3 ) 



L" = ^/27Tfim\ 

 If we only concern ourselves with the particular solution 

 * Cf. Lamb, ' Hydrodynamics/ 3rd edition, 1905, p. 559. 



