857 



When bR and bR' are only moderately small numbers, L' and 

 L" can be de\ eloped according to powers of those quantities ; the 

 equations (36) are the first terms of the series which are obtained 

 in that manner. Probably j/ might be found by that method for 

 ordinary liquids at low temperature. 



19. The formulae become also very simple, when R' — R is 

 small with respect to R, a case which may possibly be of some 

 importance experimentally. In that case : 



R'—r ') 



"^'^^^ (^«) 



R 



L= jtR'v.— (41) 



' R'-R ^ 



20. Although probably not of any practical utility I will for 

 the sake of completeness discuss the case, in which the oscillating 

 sphere is hollow, contains the liquid and swings about a smaller 

 fixed sphere. Seeing that our general discussion of the state of 

 motion in the liquid is not altered thereby, the preceding treatment 

 retains in general its validity ; the boundary-conditions also remain 

 the same, so that equations (17) and (17') remain valid. Only owing 

 to the fact that R ^ i\> R', it is now more logical to write 



u =1 - [PV-'''(A' >-)(hr~\) + Qe'>{.^->-){hr + 1)] , . . (42) 



and the conditions at the boundaries now give 



F' .= ^ ^ - , Q'= ^ ^' (43) 



where 



D=(hR-\) (bR' + 1 ) .A/?-/? ] -(hR-t'l) (bR' — 1) <- -K'?-'^) , (44) 



As regards L, the expression given in ('24') still holds for it, 

 except that it has to be provided with the negative sign, because 

 now that the sphere undergoes friction on the inside, the tangential 

 force is not F but — F (comp. sect. 3 and 8); we thus have ^) : 



deration that, when the wave-length is large as compared to the radii of the 

 spheres, the condition may at any moment be considered as stationary. 



1) This distribution of velocities agrees with that between two parallel planes, 

 which move with respect to eacli otlier at constant speed ; this result could have 

 been expected. 



2) All the formulae for this case are obtained from the corresponding ones in 

 .") and 8 by giving B, R' and r everywhere the opposite sign ; this is quite 

 intelligible from a mathematical point of view. 



55* 



