856 



18. The opposite extreme case is that, in which h R a,nd h( R' — R) 

 are very small numbers; in that ease R' cannot of course be 

 intinite, i. e. the liquid must be bounded. With normal dimensions 

 of the spheres and usual times of swing this case might be realized 

 with liquids of very high viscosity; for ordinary liquids the time 

 of swing would have to be mu(;h greater than practice allows. 



In that case (24') leads to : 



■ ' (36) 



(37) 



, - (38) 



/ beitig the wave-length in the li(|uid, the physical meaning of the 



given simplifying condition is thus, that the radii R and R' are 



small as conipared to the wavelength. In that case all the spherical 



shells i« the liquid swing practically in the same phase ') (</> and y 



are nearly zero, so that u becomes real ; in that case u = x (sect. 4) 



and equation (1J) reduces to the first equation (7)); at the same 



time approximately e—^''('^'—^)=^e^''(^'—^)^i, i.e. the waves are 



propagated without being appreciably damped, as they move forward. 



The resulting equation is this time -. 



R' R''-r' ') 

 u = « (39) 



with sphere 3) I find for water of 15° •/ = 0,01103, whereas KoxiG himself found 

 0,01140. 



1) This is the simplifying condition used by Zemplén (Ann. d. Phys. (4) 19, 783, 1906) 

 as the basis in the deduction of the formulae which served for the calculation of 

 the results of his experiments ; thereby he overlooked the fact, that in that case 

 his coefficient m (our factor b') is very small, so that cosm {R~r') and sin m 

 (B — r') ought to have been developed according to powers of m (R — r') ; 

 carrying out this development, his equation (14) leads to our equation (39) (it 

 may be noted here, thot a small error has crept into his equation (14) ; the terms 

 m-Rr" and m~RrJ should be m'-Rr and m-Rr\ respectively). As a matter of fact 

 in Zemplén's experiin.ents the assumed approximation is not applicable, for in his 

 case A — 9, and thus not large as compared to the radii of the spheres( ü = 5, 

 2J' = 6); his result is, therefore, very doubtful. Later on (Ann. d. Physik. 29, 899, 

 1909) he discovered this himsolf and gave a more accurate treatment of the 

 problem; but owing to the very comphcated nature of the correct formulae he 

 did not submit his experiments to a new calculation. 



2) This distribution of velocities is the same as the one found for uniform 

 rotation ^comp. for instance Bkillouin I.e. p. 89); this explains itself by theconsi- 



