843 



Öü) 



F = — Ij ?■ co*' f — — , (4) 



dr 



when a>:=— ^ represents the angular velocity of the shell under con- 



ot 



sideration and ?j the viscosity of the liquid. The equation of motion 

 of the sphericall shell may now be written .in the form 



d'-'to 4 d(o ft du> 



1 r=z (5) 



dr' r dr tj dt 



4. This equation determines how co depends on r ; as it does 

 not contain the angle s, it is in accordance with our assumption, 

 that the individual shells oscillate to and fro as solid bodies '). As 

 regards the law of dependence of to on t, which we have already 

 presupposed in equation (3j, it appears that it also is compatible 

 with (5) ; substituting (3) in (5) and expressing the condition, that 

 equation (5) must be fulfilled at all times (by putting the coefficients 

 of cos and sin equal to zero), two differential equations are obtained, 

 which do not contain the time and which determine the functions 

 a,- and r/,-. 



This method is, however, very cumbrous. It is much simpler first 

 to reduce (3) to the form 



OS 2rr lj, + y sin 2jt^/j (6) 



where .c and >/ are new functions which for r = R become equal 

 to a and respectively and are determined by the two differential 

 equations : 



d'\v 4 d,v u 



dr r dr vJ 



' ] (7) 



d^y 4 dii a 



dr r dr t]l 



The simplest method of all is to consider (6) as ihe real part of 



an exponential function 



«, = we*' , (8) 



where u and k are in general complex quantities ; in that case (2) 



is the real part of 



') It should not be overlooked that in tliis manner the possibility of the afore- 

 said assumption has been proved, not its necessity (for this proof, see Lamb, 

 loc. cit.). It is moreover easily seen, that with a different law of friction, e g. 

 in which v- would also depend on the velocity itself, the assumption would become 

 unallowable. 



