TRANSACTIONS OF SECTION A. 563 



or. taking the notation of Thomson and Tait, in which (1 +/-)(! -e=) = 1, the 

 critical value of f is nearly equal to 3-14. If / be greater than this value, the 

 motion is unstable. Also M. Poincare ' bas proved the statement of Thomson and 

 Tait that the motion is secularly unstable for any degree of viscosity, however 

 small, when /exceeds 1-39467, and he has sho-mi that it is thoroughly stable for 

 all displacements when f is less than this value. 



In this paper Riernann's equations of motion are applied to find the small 

 oscillations of a perfectly inviscid spheroidal mass of liquid, rotating as if rigid 

 about its polar axis, whose direction is fixed in space, the displacement contem- 

 plated being of such a kind that the bounding surface remains ellipsoidal. It appears 

 that there are two periods of oscillation, viz.: if n\'2iv be the frequency, we have either 



n' = 167ryp/(3 +/=) - 2a)-(3 + 8/= +/^)/(3/^ +/*), 

 or TT = 24^vp(l +/=)/(3 +f) - coHl +/=)(27 + 18/^ -/*)/(3 ^ff, 



•where p is the density of the spheroid, a the angular velocity, and y the constant 

 of gravitation. In these qj and/ are connected by the condition of steady motion, 



viz. : — 



(3 +P)tan-'f=M + io"-r-l2^yp). 



These values of 7i vary very little for different small values of a, so that for a 

 spheroid rotating in any period longer than about three hours the period varies 

 very nearly inversely as the square root of the density. 



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 Yiscous Spheroid and the Remote History of the Earth ' ^ saw 

 reason to reject Laplace's hypothesis that the moon separated from the earth as a 

 ring because the angular velocity was too great for stability. In the light of 

 Riemann's and Poincar^'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 rotating 

 mass may be estimated at something between two and six hours (more probably 

 between two and four hours),^ so that, even allowing for continued contraction of 

 the two cooling bodies, and remembering that the present mean density of the 

 moon is about 37, it seems highly improbable that La:place's hypothesis as to 

 instability can be correct. Professor Darwin suggested as one alternative that 

 possibly the spheroid might have a period of free oscillation not far removed from 

 the semi-diurnal tidal period, in which case the solar tides would be of enormous 

 height, and we should not then have to make additional demands on the lapse of 

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



Now it is shown in this paper that with a density of 5-67 the longer period is 

 very nearly equal to 1^ hours, while with a density of 3 this period is veiy 

 nearly equal to two hours, whatever the rate of rotation may be, provided that 

 it is not faster than once in three hours ; so that, if the period of rotation ever 

 came to be between three and four hours, the density may easily have been such 

 as to make the period of free oscillation very nearly identical with that of the 

 semi-diurnal solar tides. Thus the possibility of Professor Darwin's guess is 

 confirmed. 



5. Waves in a Viscous Liquid* By A. B. Basset, M.A. 



The propagation of waves in a viscous liquid was first considered by Professor 

 Stokes,^ who showed that when the viscosity is small, and the depth of the liquid 



' Acta 3fathemat!ca, vii. 1S8.^. = Phil. Trans. 1879. 



s ' On the Secular Changes in the Elements of the Orbit of a Satellite,' § 22, 

 PJdl. Trans. 1880. 



■• The original paper is published in the author's treatise on Hydro-dyrMmics, vol. 

 ii. §§ 510-523. 



* Trans. Camh. Phil. Soc. vol. ix. 



o 2 



