1878.] the Precession of a Viscous Spheroid, §c. 185 



and then a formula expressive of the shape of the distorted spheroid 

 may be at once written down. 



The spheroid or earth is found to be distorted by tides of seven 

 different periods ; three nearly semi-diurnal, three nearly diurnal, and 

 one fortnightly tide. 



Each such tide has a height which is a different fraction of the 

 corresponding equilibrium tide of a perfectly fluid spheroid, and is 

 differently altered in phase. Throughout nearly the whole investiga- 

 tion it is, however, sufficiently accurate, if the three semi-diurnal tides 

 are grouped together, and so also with the three diurnal tides ; by this 

 approximation the earth is regarded as distorted by only three tides. 



The next process is the formation of the couples acting on the earth, 

 which are caused by the attraction of the moon on the several tidal 

 protuberances. In the development of these couples only those terms 

 are retained which can give rise to secular alterations in the precession, 

 the obliquity to the ecliptic, and the length of day. These expressions 

 are then substituted in the differential equations of motion, and the 

 equations are integrated ; whence follow the correction to the uniform 

 precession of the earth considered as a rigid body, and differential 

 equations expressive of the rate of change of obliquity, and the rate of 

 retardation of the earth's diurnal rotation. 



It appears that, if the tides do not lag (as with a perfectly fluid or 

 perfectly elastic spheroid), the obliquity and rotation are unaffected, 

 and, whether they lag or not, the correction to the precession is but a 

 small fraction of the whole precession. 



Henceforth it is only the changes of obliquity and rotation which 

 are of interest. 



A second disturbing body~-the sun — is now introduced. A new set 

 of bodily tides are of course raised, and the expressions for the couples 

 are augmented by the addition of solar terms, and also by terms de- 

 pending on the attraction of the sun on the lunar tides and the moon 

 on the solar tides. It seems paradoxical that there should be these 

 combined effects, for the sun's and moon's periods have no common 

 multiple. But, as far as concerns their interaction, the sun and moon 

 may be conceived to be replaced by two annular satellites of masses 

 equal to those of the two bodies. The combined effects vanish with 

 the obliquity, and depend solely on those tides which run through 

 their periods in a sidereal day, and in twelve sidereal hours. Up to 

 this point all the analysis is conducted so that the solutions may be 

 applied either to a viscous, elastic, or imperfectly elastic spheroid. 



In the case where the earth is purely viscous a graphical examina- 

 tion of the equation, giving the rate of change of obliquity, shows that 

 the obliquity sometimes tends to increase and sometimes to diminish, 

 according as the obliquity and viscosity vary. There are also a number 

 of positions of dynamical equilibrium, some stable and some unstable ; 



