1879.] On Secular Changes in the Orbit of a Satellite. 9 



But the preceding general explanations are in reality somewhat less 

 satisfactory than they seem, because they do not make clear the 

 existence of certain antagonistic influences. 



Imagine a satellite revolving about a planet, and subject to a con- 

 stant accelerating force, which we saw above would result from tidal 

 reaction. 



In a circular orbit a constant tangential force makes the satellite's 

 distance increase, but the larger the orbit the less does the given force 

 increase the mean distance. ISTow the satellite, moving in the eccentric 

 orbit, is in the apogeean part of its orbit like a satellite moving in a 

 circular orbit at a certain mean distance, but in the perigeean part of 

 the orbit it is like a satellite moving in a circular orbit but at a 

 smaller mean distance ; in both parts of the orbit it is subject to the 

 same tangential force. Then the distance at the perigeean part of the 

 orbit increases more rapidly than the distance at the apogeean part. 

 Hence the constant tangential force on the satellite in the eccentric 

 orbit will make the eccentricity diminish. It is not clear from the 

 preceding general explanation, when this cause for decreasing eccen- 

 tricity will be less important than the previous cause for increasing 

 eccentricity. 



The disturbing causes which tend to make the eccentricity diminish 

 are (i) the principal semi-diurnal tide, (ii) the "faster elliptic semi- 

 diurnal tide," (iii) the " elliptic monthly tide." The increase of eccen- 

 tricity depends entirely on (iv) the " slower elliptic semi-diurnal tide." 

 If the periodic time of the satellite be long, as measured in rotations 

 of the planet, the importance of the tides (i), (ii), and (iv) are as the 

 numbers 4, 1, and 49 respectively, and the importance of (iii) is very 

 small ; if the satellite were to move faster, the importance of (iv) 

 would decrease, and that of (i), (ii), and (iii) would increase. 



In the outline of results at the beginning of the abstract, it was 

 stated that the periodic times of revolution and rotation of the moon 

 and earth might be traced back to a common period of from 2 to 

 4 hours. In the paper on " Precession " the common period was 

 found to be a little over 5 hours in length ; but that result was 

 .avowedly based on a partial neglect of the sun's attraction. In this 

 memoir certain further considerations are adduced, which show that, 

 while the general principle remains intact, yet the common period of 

 revolution of the earth and moon must initially have been shorter than 

 •5 hours to an amount, which is uncertain but is probably large. 

 The period of from 2 to 4 hours is here assigned, because it is me- 

 chanically impossible for the moon to revolve about the earth in less 

 than 2 hours, and it is uncertain how the rupture of the primeval 

 planet took place. 



versation by Sir William Thomson, when I mentioned to him the results at which I 

 had arrived. 



