THE ELEMENTS OF THE ORBIT OF A SATELLITE. 
Fourth period of integration. 
863 
The procedure is now changed in the same way, and for the same reason, as in the 
fourth period of § 17 of “Precession.” 
71/ 
Let N=~ (as in that paper). Then the equation of tidal friction is 
dN i • T ” /1 w 
-^=ism4f—(1-X) 
and the equation for the change in g may be written approximately 
njc 11 —18\ 
£ 1-X 
Since X or ft/n is no longer small, this expression will be integrated by quadratures. 
Using the numerical values given in § 17 of “Precession,” I find the following 
corresponding values. 
N= l'OOO 1-107 1-214 1-321 
kn n 11 —18A, 
T ———=15-469 17-665 19-465 1P994 
g J- — A 
Then integrating by quadratures with the common difference dN equal to '107, we 
find the integral equal to 5 "5715. 
Whence 77 = 44-273 X 10~ 10 , and e= '00009411. 
The results of the whole integration are given in the following table, of which the 
first two columns are taken from the paper on “ Precession.” 
Table XVI. 
Day in m. s. hours 
and minutes. 
Moon’s 
sidereal period in 
m. s. days. 
Eccentricity of 
lunar orbit. 
h. 
m. 
Days. 
23 
56 
27-32 
•054908 
15 
28 
18-62 
•027894 
9 
55 
8-17 
•006446 
7 
49 
3-59 
•001526 
5 
55 
12 hours 
•000094 
Beyond this the eccentricity would decrease very little more, because this inte¬ 
gration stops where X is about and the eccentricity ceases to diminish when X is y-§-. 
The final eccentricity in the above table is only g-g ^th of the initial eccentricity, 
and the orbit is very nearly circular. 
5 s 2 
