566 VISCOSITY. [CHAP, xi 



whence, substituting from (v) and (ix), 



~=-(n-l}(2n+l}^A ........................ (xi). 



This shews that A&amp;lt;z e~ tlr , where 



1 a 2 



The most remarkable feature of this result is the excessively minute 

 extent to which the oscillations of a globe of moderate dimensions are affected 

 by such a degree of viscosity as is ordinarily met with in nature. For a globe 

 of the size of the earth, and of the same kinematic viscosity as water, we 

 have, on the c.G. s. system, a = 6*37 x 10 8 , i/= 0178, and the value of r for the 

 gravitational oscillation of longest period (n = 2) is 



T = 1*44 x 10 11 years. 



Even with the value found by Darwin f for the viscosity of pitch near the 

 freezing temperature, viz. /i = l 3 x 10 8 x#, we find, taking # = 980, the value 



r = 180 hours 



for the modulus of decay of the slowest oscillation of a globe of the size of the 

 earth, having the density of water and the viscosity of pitch. Since this is 

 still large compared with the period of 1 h. 34 m. found in Art. 241 , it appears 

 that such a globe would oscillate almost like a perfect fluid. 



The investigation by which (xii) was obtained does not involve any special 

 assumption as to the nature of the forces which produce the tendency to the 

 spherical form. The result applies, therefore, equally well to the vibrations 

 of a liquid globule under the surface-tension of the bounding film. The 

 modulus of decay of the slowest oscillation of a globule of water is, in seconds, 



r=ll 2a 2 , 

 where the unit of a is the centimetre. 



The formula (xii) includes of course the case of waves on a plane surface. 

 When n is very great we find, putting X = 2n-a/?i, 



r = A 2 /87T 2 v .. ............................... (xiii), 



in agreement with Art. 301. 



The same method, applied to the case of a spherical bubble, gives 





where v is the viscosity of the surrounding liquid. If this be water we have, 

 for n = 2, r = 2-8a 2 . 



The above results all postulate that 2yrr is a considerable multiple of the 

 period. The opposite extreme, where the viscosity is so great that the motion 



* Proc. Lond. Math. Soc., t. xiii., pp. 61, 65 (1881). 



t &quot;On the Bodily Tides of Viscous and Semi-Elastic Spheroids,...,&quot; Phil. 

 Trans., 1879. 



