199 



kR := J.4 and 



0.80, so that the amplitude of the 2'"^ term 



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

 be relatively small in order that the 2'"' part may be neglected with 

 respect to the first. For instance for atmospheric air with /^=: 0.5 cm. 



of the frictional couple is still 56 7o o^ ^^^^^ o^ tl^® ^ï^'st (see 

 equation (11)), every thing calculated for a time of oscillation of 

 2 n: seconds. 



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

 some interest. Let R become intinite, 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 uidimited tlat plate in its own plane in an infinitely extended 

 liquid. The frictional force per unit of surface is found from (13) to be 



d 



F = 



dt 



h cos I pt \- 



n 



(U) 



a formnla which is well-known from hydrodynamics *). A term 

 analogous to — 4 Jiii R^ o) 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. 



^) At these pressures n has not become much smaller. Gomp. Kundt u. War- 

 burg. Pogg. Ann. 1875 Band GLV. 



') Comp. Lamb. Hydrodynamics, S'^* edition 1905, p. 559. 



