8 



Mr. G. H. Darwin. 



[Dec. 18, 



viscosity, the periods of rotation and revolution start from identity 

 and end with identity ; hence the eccentricity rises from zero to a 

 maximum, and ultimately decreases to zero again. 



It is also proved, that in the history of a single satellite revolving 

 about a planet of large viscosity, the eccentricity rises very rapidly to 

 a maximum, decreases slowly to a minimum, and then increases again ; 

 but the actual degree of viscosity has an important influence on the 

 results. 



The following considerations (in substitution for the analytical 

 treatment of the paper) throw some light on the physical causes of 

 these results. 



Consider a satellite revolving about a planet in an elliptic orbit, 

 with a periodic time which is long compared with the period of 

 rotation of the planet ; and suppose that frictional tides are raised in 

 the planet. 



The major axis of the tidal spheroid always points in advance of 

 the satellite, and exercises a force on the satellite which tends to 

 accelerate its linear velocity. 



When the satellite is in perigee the tides are higher, and this dis- 

 turbing force is greater than when the satellite is in apogee. 



The disturbing force may, therefore, be represented as a constant 

 force, always tending to accelerate the motion of the satellite, and a 

 periodic force which accelerates in perigee and retards in apogee. 

 The constant force causes a secular increase of the satellite's mean 

 distance and a retardation of its mean motion. 



The accelerating force in perigee causes the satellite to swing out 

 further than it would otherwise have done, so that when it comes 

 round to apogee it is more remote from the planet. The retarding 

 force in apogee acts exactly inversely, and diminishes *the perigeean 

 distance. Thus, the apogeean distance increases and the perigeean 

 distance diminishes, or in other words, the eccentricity of the orbit 

 increases. 



Now consider another case, and suppose the satellite's periodic 

 time to be identical with that of the planet's rotation. Then when 

 the satellite is in perigee it is moving faster than the planet rotates, 

 and when in apogee it is moving slower ; hence at apogee the tides 

 lag, and at perigee they are accelerated. ~Now the lagging apogeean 

 tides give rise to an accelerating force on the satellite, and increase 

 the perigeean distance, whilst the accelerated perigeean tides give rise 

 to a retarding force, and decrease the apogeean distance. Hence in 

 this case the eccentricity of the orbit will diminish. 



It follows from these two results that there must be some inter- 

 mediate periodic time of the satellite, for which the eccentricity does 

 not tend to vary.* 



# The substance of the preceding general explanation was suggested to me in con- 



