855 

 If d is small at the same time, we have in first approximation 



U..LJ-=^^.R^l/^, (34) 



from wliifh, b}' (28') 



jr/J^ , T—T„ if 



<i = ^^\/^iinJ\ , — ^=.^. . . . (35) 



This extreme case is discussed by Kikchhoff in his Vorlesungen 



iiber mathematische Phvsik, N°. 26; it occurs when is a very 



fiR"- 



small number'). This case would be realized, if in a liquid 

 with small - (say a liquid gas) a large sphere was made to swing 



quickly; taking say =r 0,001, in order to have //7i=rlOOO with 

 ft 



R = 10, it wonld be necessary for T to be 0,3. Apart from the not 



very practical natnre of these conditions, it may be considered very 



doubtful, whether with the comparatively high velocities, involved in 



a rapid vibration of that kind the preceding theory would still hold. 



It seems to me, therefore, that the extreme case in question has no 



experimental physical importance. 



When b' R and h" R are only moderately large L' and L" may 



1 1 



be developed according tot ascending powers ot ^ and — ; if in 



b R h" R 



addition the series (20) and (28'), are introduced, and the development 



is stopped at a definite point, formulae such as those of Lampk'), 



Klkmencic '^), BoLTZMANN ') and KöNiü') are obtained. 



1) KiRCHHOFF assumes ^ to be very small, which must of course be taken to 

 mean : under otherwise normal circumstances, for, taken absolutely, it has no 

 sense to suppose a quantify which is hot dimensionless to be very small, seeing 

 that the value depends on the choice of units. For the rest, the liquid need not 

 necessarily have a very small viscosity in order to obtain the simple case in 

 question ; a small friction would even be a disadvantage, if combined with a small 



density, as in the case of gases. For air for mstance — is about 0,2, and thus 



much larger than for water, notwithstanding the much smaller value of » 

 (comp. 12 note). 



2) loc. cit. 



3) Vid. Lampe. Wien Ber. II. 93. 291, 1886. These formulae are as a rule not 

 very suitable for accurate calculations, because a sufficient accuracy cannot be 

 obtained with only a few terms; as an instance, König's experiments can be cal- 

 culated much more simply and accurately in the manner of section 15 of this 

 paper, than by Konio's own method. From one of KöNia's experiments (the last 



55 

 Proceedings Royal Acad. Amsterdam. Vol. XVIII. 



