844 



tt = a e''-' , (8') 



and u is a fiinelioii oF /• only, wliicli for r = li obtains the value 

 n. Putting 



k = k' + k"i (;=i/— 1), (9) 



it follows b}' equating (ö) to the real part of (8) that 



(S 2.T 

 k'^-j, and ^" = Y (9) 



The real angular velocity' <o is the real part of the complex 

 quantity 



ui^zkuel'' (10) 



the function u satisfying the equation 



cPu 4 du u 



^ H = •- ku, (11) 



di'' r dr t] 



which is obtained by substituting (10) in (5). ') 



5. The general solution of (11) is well known to be 

 u— ^ [A^f"- (èr + 1) + B,' '"■ {br - 1)J , 



u = - [Pe-'"-R (br + 1} + Qe'"--li(hr — 1)| , 

 where 



1) Equation (10) is a particular solution of equation (5). The mode of motion 

 which it represents is, therefor.', a possible one but not necessarily the actually 

 existing one. The reason why we only consider this solution is that we suppose 

 the sphere not to perform forced vibrations. In the case of a compound harmonic 

 motion w would consist of a number of terms, each with its own A:, the iCs of 

 which would satisfy as many equations (11). 



It is also obvious, that the condition of motion considered cannot exist from 

 the beginning, but can only be reached after a theoretically infinite period, so that 

 the motion of the sphere cannot correspond either to equation (2) from the moment 

 at which the motion begins. The experiments show, however, that the final 

 condition is practically reached after a comparatively short time (a few minutes), 

 i. e. very soon T and .- have become constant ; this may be expressed mathema- 

 tically by saying, that the assumed condition of motion is the limiting condition 

 to which the real motion approaches asymptotically and this approach is in general 

 so rapid, that even after a comparatively short time the deviations of the actual 

 motion from the final limit are within the limits of the errors of observation. The 

 question as to the real motion during the said period of approach is one which 

 would have to be settled by a separate theoretical and experimental investigation, 

 but is of no importance for our present purpose. 



