847 



in such a manner that, independently of the algebraic dependence 

 on r, the amplitude is reduced in the ratio A:l over a distance 1, 

 where A = ö~*'. 



With a small value of ö according to (20) the damping increases 

 as T becomes smaller and with a sufticiently small value of T it 

 may happen, that even a comparatively narrowly bounded liquid is 

 practically unbounded, because the motion which starts from the 

 sphere is practically completely damped, before it reaches the external 

 boundary; to this point we shall return later on (§ 12). 



8. We can now proceed to calculate the time of swing and the 

 logarithmic decrement of the damped oscillations of the sphere from 

 the specific constants of the liquid (viz. the viscosity i] and the 

 density (x). The equation of motion of the oscillating sphere is 



iT^-'- C-f i¥« = U,') (23) 



where C, the moment of the frictional forces, is given by (comp. § 3) 



C = - Cf . 2jtR' cos' 6 rff = i :^R'ii ( -^ j . . (-23') 



According to (10) and (12) we may write 



do) /te*' 



Or R" 



and therefore 



— ]==-— lP{bUV^UR + 'ó) - Q{b-'R:^~--dbR+:\)\ = 



=z [P {b'R' + UR+d) - Q{PR' -UR+3)]'-^ , 



aR^ at 



so that for the case of a damped harmonic motion we may write 



d^a da 

 K \-L [-M« = 0, n (24) 



dt' ^ dt ^ ' ^ ' 



where 



1) The equation once more expresses the fact that the sphere oscillates freely. 



2) In the case of a not purely harmonic damped motion the proportionality of 



G with — no longer exists. As far as I can see, it is in that case impossible to 



say, how in general C depends on the motion, so that it will then probably be 

 impossible to establish a general differential equation for y. 



