240-241] OSCILLATIONS OF A LIQUID GLOBE. 437 



to be satisfied when r a, gives 



d n _ n ... 



~S- a &quot;- 



The gravitation-potential at the free surface is, by Art. 192, 



47T 7 pa 3 * 4,Trypa ( 



W Z i 27i+l&amp;gt;&quot; 



where 7 is the gravitation-constant. Putting 



g=%7rypa, r = a + 2? w , 

 we find 



n = const. +^ 2 2 ( ^r^ ................. ( 6 &amp;gt;- 



Substituting from (2) and (6) in the pressure equation 



(7), 



- 

 p at 



we find, since p must be constant over the surface, 



Eliminating S n between (4) and (8), we obtain 



This shews that f w oc cos (cr n t + e), where 



2n(n-l)jr 

 &quot; n 2 + l B&quot; 



For the same density of liquid, g oc a, and the frequency is 

 therefore independent of the dimensions of the globe. 



The formula makes o^ = 0, as we should expect, since in the 

 deformation expressed by a surface-harmonic of the first order the 

 surface remains spherical, and the period is therefore infinitely 

 long. 



&quot; For the case n = 2, or an ellipsoidal deformation, the length of 

 the isochronous simple pendulum becomes }a, or one and a quarter 

 times the earth s radius, for a homogeneous liquid globe of the 

 same mass and diameter as the earth ; and therefore for this case, 



