8G6 
MR. G. H. DARWIN ON THE SECULAR CHANGES IN 
revolves with a uniform precessional motion, and (so long as the earth is rigid) the 
inclinations of the orbit and equator to the invariable plane remain constant. 
Here also inequalities, which depend on the longitude of the common node, are 
harmonically periodic in time, and can lead to no cumulative effects. 
But if the lunar proper plane he not inclined at a small angle to the ecliptic, the 
nodes of the orbit may either revolve with much irregularity, or may oscillate about a 
mean position* on the ecliptic. In this case the inclinations of the orbit and equator 
to the ecliptic may oscillate considerably. 
Here then inequalities, which depend on the longitudes of the node and of the 
equinoctial line, are not simply periodic in time, and may and will lead to cumulative 
effects. 
This explains what was stated above, namely, that we cannot entirely ignore the 
motion of the two nodes. 
Our problem is thus divisible into three cases :— 
(i.) Where the nodes revolve uniformly on the ecliptic, and where there is a second 
disturbing satellite, viz. : the sun. 
(ii.) Where the earth and moon are the only two bodies in existence. 
(iii.) Where the nodes either oscillate, or do not revolve uniformly. 
The cases (i.) and (ii.) are distinguished by our being able to ignore the nodes. 
They afford the subject matter for the whole of Part II. 
It is proved in § 5 that the tides raised by any one satellite can produce directly 
no secular change in the mean distance of any other satellite. This is true for all 
three of the above cases. 
It is also shown that, in cases (i.) and (ii.), the tides raised by any one satellite can 
produce directly no secular change in the inclination of the orbit of any other satellite 
to the plane of reference. This is not true for case (iii.). 
The change of inclination of the moon’s orbit in case (i.) is considered in § 6. The 
equation expressive of the rate of change of inclination is given in (61) and (62). In 
§ 7 this is applied in the case where the earth is viscous. Fig. 4 illustrates the 
physical meaning of the equation, and the reader is referred to § 7 for an explanation 
of the figure. From this figure we learn that the effect of the frictional tides is in 
general to diminish the inclination of the lunar orbit to the ecliptic, unless the obli¬ 
quity of the ecliptic be large, when the inclination will increase. The curves also 
show that for moderate viscosities the rate of decrease of inclination is most rapid 
when the obliquity of the ecliptic is zero, but for larger viscosities the rate of decrease 
has a maximum value, when the obliquity is between 30° and 40°. 
If the viscosity be small the equation for the rate of decrease of inclination is 
reducible to a very simple form ; this is given in (64) § 7. 
In §§ 8, 9, is found the law of increase of the square root of the moon’s distance 
from the earth under the influence of tidal reaction. The law differs but little from 
* It is true that this mean position will itself have a slow precessional motion. 
