240-241] OSCILLATIONS OF A LIQUID GLOBE. 437 



to be satisfied when r = a, gives 



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



where 7 is the gravitation-constant. Putting 



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

 we find 



Q = **. + 02" -^fc, ................. (6). 



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



(7), 



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



d 



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



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



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-j = 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. 



" 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, 



