832 Dr. J. W. Nicholson on the 



The equation may be shown to have a root zero, but no 

 others except complex values whose real part is positive. 

 Thus the vibratory terms ultimately decay. 



We proceed to the case of a sphere without Newtonian 

 mass. Taking the mass at first as very small, the non- 



1 2 e 



vibratory part of ~ Ft 2 — - — x becomes, on reduction, 



_ o (IC 



m¥ 



im c 



fc 2 t 2 + 2act + 2a 2 -~}, 



and by (7) this is the non-vibratory part of ra£. The 

 vibratory portion is of the form 



m2D*- Xc ^sin(^+e), .... (13) 



where the root of the period equation is now written X ;£*/*, 

 and D and € depend on X and //,. 



In order that f may satisfy the appropriate conditions at 

 / = 0, it is necessary that 



^ T . . m a 2 F k 

 ZD sin e= 7 . - --„- . ^, 



a 2 F 

 SODcose— XDsine) = —& • • • ( 14 ) 



m c 



the summation being for all roots \±i/jl of the period 

 equation 



tanh kKx 



= 1+—!^ (15) 



K'X K — 1 — KX 



The acceleration is always finite, whatever the distribution 

 of the vibrations among the possible periods. The deter- 

 mination of this distribution is difficult, but is not necessary 

 for the present purpose. In addition to the decrease in 

 amplitude which may be expected as the vibrations recede 

 from the fundamental, there will be increased damping^ in 

 the higher modes. When /c is not too great, the damping 

 will not be slight even for the fundamental, which will then 

 be the only vibration needing attention. If this is so, and 

 if the amplitude of this vibration is sufficiently preponderant, 

 we may write 



f-B?(^ , +*-+S9+ D ^ > **.('F + ') 



