841 



ttr=i ae ■' cos 2jr — , . . . . . . . (2) 



where 7' is the new time of vibration and ó the logarithmic decre- 

 ment of the elongations for one vibration.') The problem before us 

 is, how fS and T depend upon the specific properties of the litpiid, 

 in particular on tlie viscosity »/, and how ?/ may be calculated from 

 observations on tlie two quantities in question. 



2. We shall couline our investigation to the two cases in which 

 the liquid is either externall}- unlimited (i.e. practically speaking, 

 fills a space the dimensions of which are very large compared with 

 the radius of the sphere] or is limited by a stationary spherical 

 surface which is concentric with the oscillating sphere ; in these 

 cases we may naturally assume, tlial the motion in the liquid is 

 such, that it divides itself into spherical, concentric layers, which 

 each separately oscillates as a solid shell about the same axis as the 

 sphere, with the same periodic lime and the same logarithmic decre- 

 ment ; it will be shown further down that this assumed state of 

 motion is actually a possible one, at least when the motion is very 

 slow'). In that case it is only the anqilitude and the phase of the 

 motion which differ from one shell to another, and foi- a shell of 

 radius v we may therefore put : 



' t 



T ''') (=^) 



where rf, and 'ƒ, are functions of r. If we further assume that the 

 liquid layer which is contiguous to the sphere, adheres to it, as is 

 well known to be generally the case, expression (3) must become 

 identical with (2) for r= R, thus a/^ ^ a and '//? = 0. 



3. In order lo find the functions a/t and </ ^ we proceed l(» estab- 

 lish the equation of motion for a spherical liquid shell. For this 

 pur|)Ose we shall consider the ring whose section is ^4 /iC/J ^ r.c/f.r/c 

 (comp. adjoining figure) and whose radius is Q:=rcoiie. On its side- 

 faces AB and CD this ring according to our assumption does not 

 experience any friclion ; on the innei' surface AB, owing to friction 

 against a shell closer to the centre, it ex|)eriences a tangential force 

 F per unit area in the direction of its motion, and on the outer 



surface /iC' similarly a force — f F -\- ~ dr j ; writing down the 



') If the motion of the sphere without friction were a compound harmonic motion, 

 as would be the case, if the sphere were coupled to otlier oscillating systems, the 

 motion with friction would be compounded of damped liarmonic vibrations. 



-) For the necessary condition of slowness of the motion see note in Comm N". \iSd. 



54* 



