THE ELEMENTS OF THE ORBIT OF A SATELLITE. 
881 
At some early stage in the history of the system, the moon has conglomerated into 
a spheroidal form, and has acquired a rotation about an axis nearly parallel with that 
of the earth. We will now follow the moon itself for a time. 
The axial rotation of the moon is retarded by the attraction of the earth on the tides 
raised in the moon, and this retardation takes place at a far greater rate than the 
similar retardation of the earth’s rotation.* * * § As soon as the moon rotates round her 
axis with twice the angular velocity with which she revolves in her orbit, the position 
of her axis of rotation (parallel with the earth’s axis) becomes dynamically unstable.! 
The obliquity of the lunar equator to the plane of the orbit increases, attains a 
maximum, and then diminishes. Meanwhile the lunar axial rotation is being reduced 
towards identity with the orbital motion. 
Finally her equator is neaidy coincident with the plane of her orbit, and the attrac¬ 
tion of the earth on a tide, which degenerates into a permanent ellipticity of the lunar 
equator, causes her always to show the same face to the earth.| Laplace has shown 
that this is a necessary consequence of the elliptic form of the lunar equator. 
All this must have taken place early in the history of the earth, to which I now 
return. 
As the month increases in length the lunar orbit becomes eccentric, and the eccen¬ 
tricity reaches a maximum when the month occupies about a rotation and a half of the 
earth. The maximum of eccentricity is probably not large. After this the eccentricity 
diminishes. § 
The plane of the lunar orbit is at first practically identical with the earth’s equator, 
but as the moon recedes from the earth the sun’s attraction begins to make itself felt. 
Here then we must introduce the conception of the two ideal planes (here called the 
proper planes), to which the motion of the earth and moon must be referred.|| The 
lunar proper plane is at first inclined at a very small angle to the earth’s proper plane, 
and the orbit and equator coincide with their respective proper planes. 
As soon as the earth rotates with twice the angular velocity with which the moon 
revolves in her orbit, a new instability sets in. The month is then about 12 of our 
present hours, and the day is about 6 of our present hours in length. 
The inclinations of the lunar orbit and of the equator to their respective proper planes 
* “ Precession,” § 23. 
f “ Precession,” § 17. It is of course possible that the lunar rotation was very rapidly reduced by the 
earth’s attraction on the lagging tides, and was never permitted to be more than twice the orbital motion. 
In this case the lunar equator has never deviated much from the plane of the orbit. 
X Heliuholtz, I believe, first suggested the reduction of the moon’s axial rotation by means of tidal 
friction. 
§ Parts V. and YI. The exact history of the eccentricity is somewhat uncertain, because of the 
uncertainty as to the degree of viscosity of the earth. 
|| See Parts III. and IY. (and the summaries thereof in Part VII.) for this and what follows about 
proper planes. 
