8G8 
MR. G. H. DARWIN ON THE SECULAR CHANGES IN 
to introduce a new conception, viz. : that of a second proper plane to which the 
motion of the earth is referred. It is proved that the motion of the system may then 
be defined as follows :— 
The two proper planes intersect one another on the ecliptic, and their common 
node regredes on the ecliptic with a slow precessional motion. The lunar orbit and 
the equator are respectively inclined at constant angles to their proper planes, and 
their nodes on their respective planes also regrede uniformly and at the same speed. 
The motions are timed in such a way that when the inclination of the orbit to the 
ecliptic is at the maximum, the obliquity of the equator to the ecliptic is at the 
minimum, and vice versa. 
Now let us call the angular velocity with which the nodes of the orbit would 
regrede on the ecliptic, if the earth were spherical, the nodal velocity. 
And let us call the angular velocity with which the common node of the orbit and 
equator would regrede on the invariable plane of the system, if the sun did not exist, 
the 2 orecessional velocity. 
If the various obliquities and inclinations be not large, the precessional velocity is 
in fact the purely lunar precession. 
Then if the nodal velocity be large compared with the precessional velocity, the 
lunar proper plane is inclined at a small angle to the ecliptic, and the equator is 
inclined at a small angle to the earth’s proper plane. 
This is the case with the earth, moon, and sun at present, because the nodal period 
is about 18 -et years, and the purely lunar precession would have a period of between 
20,000 and 30,000 years. It is not usual to speak of a proper plane of the earth, 
because it is more simple to conceive a mean equator, about which the true equator 
nutates with a period of-about 18 Jr years. 
Here the precessional motion of the two proper planes is the whole limi-solar 
precession, and the regression of the nodes on the proper planes is practically the 
same as the regression of the lunar nodes on the ecliptic. 
A comparison of my result with the formula ordinarily given will be found at the 
end of § 13, and in a note to § 18. 
Secondly, if the nodal velocity be small compared with the precessional velocity, 
the lunar proper plane is inclined at a small angle to the earth’s proper plane. 
Also the inclination of the equator to the earth’s proper plane bears .very nearly 
the same ratio to the inclination of the orbit to the moon’s proper plane as the orbital 
moment of momentum of the two bodies hears to that of the rotation of the earth. 
In the planets of the solar system, on account of the immense mass of the sun, the 
nodal velocity is never small compared with the precessional velocity, unless the 
satellite moves with a very short periodic time round its planet, or unless the satellite 
be very small ; and if either of these be the case the ratio of the two moments of 
momentum is small. 
Hence it follows that in our system, if the nodal velocity be small compared with 
