On the Oscillations of a Rotating Liquid Spheroid. 255 

 by the condition of steady motion. 



(3 +f) tan- V =/(3 + <oYI^'^lP)- 

 These values of n vary very little for different small values of 

 &), so that for a spheroid rotating in any period longer than 

 about three hours the period of oscillation varies inversely 

 as the square root of the density approximately. 



The determination of these periods of oscillation has an 

 important bearing on the question of the origin of the Moon. 

 Professor Darwin, in his paper " On the Precession of a 

 Viscous Spheroid and the Remote History of the Earth/' * 

 saw reason to reject Laplace's hypothesis, that the moon sepa- 

 rated from the earth as a ring because the angular velocity 

 was too great for stability. In the light of Hiemann and 

 Poincare's researches above referred to it is clear that, when 

 the density is not less than 3 and the period of rotation longer 

 than three hours, the motion is certainly stable. According 

 to Professor Darwin, the period of rotation of the earth-moon 

 system when the two bodies formed a single mass may be 

 estimated at something between two and six hours, more 

 probably between two and fom- hoursf ; and if we take account 

 of the continued contraction of the two cooling bodies since 

 the date of the disruption and remember that the present 

 mean density of the moon is about 3*7, it seems highly im- 

 probable that Laplace's hypothesis as to instability can be 

 correct. As an alternative Professor Darwin has suggested 

 that possibly the spheroid might have a period of free oscilla- 

 tion not far removed from the semidiurnal tidal period, in 

 which case the solar tides would be of enormous height. This 

 is a new cause of instability in the otherwise stable dynamical 

 system, and its consequence would be a division of the mass at 

 a moment of greatest elongation. Thus on this hypothesis 

 the moon broke off from the earth as a single mass and not 

 as a ring, and we have not in our account of the history of 

 the system to make additional demands upon the lapse of time 

 with a view to the consolidation of the ring-moon into one 

 body. The hypothesis gains greatly in credibility Avhen it is 

 shown that the spheroid really has a period of oscillation of 

 the requisite length. This is done in the present paper. It 

 is proved that for a liquid spheroid of the same mean density 

 as the earth the longest period is always very nearly equal to 

 1^ hours, while for a spheroid whose density is 3 this period 

 is very nearly equal to 2 hours, whatever the rate of rotation 



* Phil. Trans. (1879). 



t " On the Secular Changes in the Elements of the Orbit of a Satellite," 

 § 22, Phil. Trans. 1880. 



