1878.] 



the Precession of a Viscous Spheroid, fyc. 



187 



If the substance of the earth were a perfect fluid or perfectly elastic, 

 the apices of the tidal spheroid would be at M and M'. If, however, 

 there is internal friction due to any sort of viscosity, the tides will lag, 

 and we may suppose the tidal apices to be at T and T". 



Now, suppose the tidal protuberances to be replaced by two equal 

 heavy particles at T and T, which are instantaneously rigidly con- 

 nected with the earth. Then the attraction of the moon on T is 

 greater than on T ; and of the anti-moon on T' is greater than on T. 

 The resultant of these forces is clearly a pair of forces acting on the 

 earth in the direction of TM, T'M'. 



The effect on the obliquity will be considered first. 



These forces TM, T'M', clearly cause a couple about the axis in the 

 equator, which lies in the same meridian as the moon and anti-moon. 

 The direction of the couple is shown by the curved arrows at L, L'. 



Now, if the effects of this couple be compounded with the existing 

 rotation of the earth, according to the principle of the gyroscope, it 

 will be seen that the South Pole S tends to approach M, and the 

 North Pole to approach M'. Hence supposing the moon to move in 

 the ecliptic, the inclination of the earth's axis to the ecliptic dimi- 

 nishes, or the obliquity increases. 



Next, the forces TM, T'M', clearly produce a couple about the 

 earth's polar axis, which tends to retard the diurnal rotation. 



Lastly, since action and reaction are equal and opposite, and since the 

 moon and anti-moon cause the forces TM, T'M', on the earth, therefore 

 the earth must cause forces on those two bodies (or on their equiva- 

 lent single moon) in the directions MT and M'T'. These forces are 

 in the direction of the moon's orbitual motion, and therefore her 

 linear velocity is augmented. Since the centrifugal force of her or- 

 bitual motion must remain constant, her distance increases, and with 



