1879.] On Secular Changes in the Orbit of a Satellite. 7 



the lunar orbit to the proper plane, for during one part of the moon's 

 history, the inclination to the proper plane would have increased, even 

 if the viscosity of the earth had been small. 



It does, however, follow, from the analytical investigation, that if 

 the lunar orbit was primitively coincident with the earth's equator, 

 then the present amount of inclination of the lunar orbit to the 

 ecliptic (viz., 5° 9') is not explicable on the hypothesis of small 

 viscosity of the earth, but is explicable if we suppose that the viscosity 

 of the earth has always been large, as it certainly is at present. The 

 theory which gives the most perfect account of the present amount of 

 inclination of the lunar orbit is, that the more recent changes in the 

 system have been principally due to oceanic tidal friction, and the more 

 ancient principally to bodily tidal friction, with a large degree of 

 viscosity of the earth's mass. 



The presence of the sun rendered it expedient to divide the problem 

 of the inclination into three cases : — 



1st. When the solar influence is large, as it is at present. 



2nd. When the solar influence is small, or nil. This is the case to 

 which the above general considerations apply. 



And the third case is intermediate between the first and second 

 cases. 



For the third case the theory of the proper planes of the moon and 

 earth had to be investigated, and the problem resolved itself into the 

 determination of the secular changes of the positions of the two 

 proper planes, and of the inclinations of the plaues of motion of the 

 two parts of the system to their respective proper planes. 



The questions involved in these three cases are, however, so com- 

 plex that it does not seem advisable to enter on them in this abstract. 



The second of the two problems, that of the eccentricity of the 

 orbit, is also treated by the method of the disturbing function. 



The result, for a viscous planet, shows that in general the eccen- 

 tricity of orbit will increase; but if the obliquity of the planet's 

 equator be nearly 90°, or if the viscosity be so great as to approach 

 perfect rigidity, or if the periodic time of the satellite (measured 

 in rotations of the planet) be short, the eccentricity will slowly 

 diminish. 



When the viscosity is small the law of variation of eccentricity is 

 very simple, and it appears that if eleven periods of the satellite 

 occupy a longer time than eighteen rotations of the planet, the eccen- 

 tricity increases, and vice versa. Hence, in the case of small viscosity, 

 a circular orbit is only dynamically stable if the eleven periods are 

 shorter than the eighteen rotations. 



In the history of a single satellite revolving about a planet of small 



