494 MR. G. H. DARWIN ON THE PRECESSION OF A VISCOUS SPHEROID 
Table IY. —Showing the physical meaning of the results of the integration. 
Time 
(-<)• 
Sidereal 
day in m. s. 
hours. 
Moon’s side¬ 
real period 
in m. s. days. 
Obliquity of 
ecliptic 
(0- 
Reciprocal 
of elliptieity 
of figure. 
Moon’s 
distance in 
earth’s mean 
radii. 
Ratio of m. 
of m. of orbi¬ 
tal motion to 
m. of m. of 
earth’s 
rotation. 
Hrat gene¬ 
rated (see 
Section 16). 
Initial 
state. 
Tears. 
0 
li. 
23 
m. 
56 
d. 
2732 
23° 28' 
232 
60-4 
4-01 
Degrees Eahr. 
0° 
I. 
46,300,000 
15 
30 
18-62 
20° 40' 
96 
46-8 
2-28 
225° 
II. 
56,600,000 
9 
55 
8-17 
17° 20' 
40 
27-0 
1-11 
760° 
III. 
56,800,000 
7 
50 
3-59 
15° 30'* 
25 
15‘6 
•67 
1300° 
IV. 
56,810,000 
6 
45 
1-58 
14° 25'* 
18 
9-0 
•44 
1760° 
The whole of these results are based on the supposition that the plane of the lunar 
orbit will remain very nearly coincident with the ecliptic throughout these changes. 
I now (July, 1879), however, see reason to believe that the secular changes in the 
plane of the lunar orbit will have an important influence on the obliquity of the 
ecliptic. Up to the end of the second period the change of obliquity as given in 
Table IY. will be approximately correct, but I tind that during the third and fourth 
periods of integration there will be a phase of considerable nutation. The results in 
the column of obliquity marked (” r ) have not, therefore, very much value as far as 
regards the explanation of the obliquity of the ecliptic; they are, however, retained as 
being instructive from a dynamical point of view. 
§16. The loss of energy of th e system. 
It is obvious that as there is tidal friction the moon-earth system must be losing 
energy, and I shall now examine how much of this lost energy turns into heat in the 
interior of the earth. The expressions potential and kinetic energy will be abbreviated 
by writing them p.e. and k.e. 
The k.e. of the earth’s rotation is }Mcrnr. 
The k.e. of the earth’s and moon’s orbital motion round their common centre of 
inertia is 
m 
mx \ 3 
m 
+ M) 
fi' 2 -\-hn 
Mx 
m + M, 
n-: 
flf 
+ v 
But since the moon’s orbit is circular f2 2 r=g{ - 
whole k.e. of the moon-earth system is 
-1 + z' ,1 , 42 2 r- ga 
, so that —- = — 
v 1+v vx 
Hence the 
