859 



manner (30) will then give : 



/ b'R—l 



L'=z^ jrR'ii { b'R— 2 + 



(b'R—iy + i>"'R' 



L" — f jtR'L"',! (1 1 .... (52) 



•* ' \^ {b'R—iy + b"'R'J ^ ' 



The calculations are to be carried out as in § 15. 



When h' R and b" R are very large, the same formulae (33) are 

 arrived at as before, which means that, when the motion is com- 

 pletely extinguished at a very short distance from the oscillatjng 

 sphere, it makes no difference whether the friction is internal or 

 external ; this might of course have been foreseen. ^) 



23. When bR, and therefore also b?\ are very small, that is: 

 when the wavelengtli is very large compared to the radius of the 

 sphere, as would probably be the case with very viscous liquids 

 (comp. § 18), it follows from (49) that // = a, i. e. the sphere swings 

 as a completely solid mass, as might have been expected a priori. 

 There will thus be no damping and the time of swing must be 

 that of a system the moment of inertia of which is equal to A' with 

 the addition of the moment of inertia of the liquid. 



This actually follows from the above formulae, for (50) then 

 reduces to 



L = y**-g- jr b^R'^ti =z -J^ jt^iR^k, 



and introducing this into (26'), we find that 



1 j2 2 

 k' = and — =: = - <K+ A"). 



where K' =^ ^ jtfiR^, the moment of inertia of the liquid.^) 



1) In PiOTROWSKi's experiments the aforesaid condition was not fuliilled, no 

 more than in König's experiments; R was = 12,5, T'=30, and hence b'R = 7,ü 

 about. Still this value is sufficiently large to make the application of (51) allow- 

 able, and as in König's experiments, this leads without difficulty to the value 

 of V). Similarly in Zemplén's experiments with air equntion (51) is applicable to 

 the inside-friction on the oscillating sphere, for with ;x = 0,0012, ■, = 0,0002, 



2'= 30 and i? = 5 one finds b' = I / -"^^^ = 0,8, hence e--''''l{ = e- 8 = 



y 111 2000 



about. 



-) This result may be expressed as follows ; L is imaginary in this case and 



L' = and L" = -^ jtfiR^k", 



showing that the addition to the moment of inertia (comp § 10), is here equal 



to the actual moment of inertia of the liquid, and the equation of motion of the 



sphere becomes (29'): 



d'a' 

 {K+K') +■ J\Ia' — 0. 



