THE ELEMENTS OF THE ORBIT OF A SATELLITE. 
823 
Combining these terms according to the formulas (250), we have 
Table VII. 
£ 
1- 
•96 
•92 
•88 
d log tan -gJ /d£= 
+ '1496 
-•0754 
-•3298 
— •5625 
d log tan ^I/d£= 
+ 1-0601 
+ •8607 
+ •6354 
+ •4370 
By interpolation it appears that c/J/c/f vanishes when ^='9679. This value of £ 
corresponds with 7 lirs. 47 m. for the period of the earth’s rotation, and 3‘25 m. s. days 
for the period of the moon’s revolution. 
By the rules of the calculus of finite differences, integrating from £=1 to '88, 
log,, tan -gJ = log,, tan ^J 0 + '0244 
log,, tan -§1= log,, tan igl 0 — '0898 
Then with J 0 =6° 9', I 0 =13° 25' from the previous integration, we have J = 6° 18', 
1=12° 16'. 
When ^='88, the length of the day is 7 hrs. 15 m., and the moon’s sidereal period 
is 2'45 m. s. days. Also ^ = 3° 3', ^=10° 58'. 
Thus we have traced the changes back until the inclination of the proper planes to 
one another is only 12° 16'—10° 58' or l c 18'. 
In the same way as before it may be shown that, when ^=’88, the period of 
the precession of the proper planes is 609 years, and the period of the revolution of the 
two nodes on their moving proper planes is 22 years. The former of the two preces¬ 
sions is therefore at this stage getting slower, whilst the latter goes on increasing in 
speed. 
The physical results of the whole integration of the present section is embodied 
in the following table. 
O 
Table VIII.—Besults of integration in the case of small viscosity. 
Day in 
m. s. hours 
and min. 
Moon’s 
sidereal 
period in 
m. s. days. 
Inclination 
of earth’s 
proper 
plane to 
ecliptic. 
Inclination 
of equator 
to earth’s 
proper 
plane. 
Inclination 
of moon’s 
proper 
plane to 
ecliptic. 
Inclination 
of lunar orbit 
to moon’s 
proper 
plane. 
Processional 
period of 
the proper 
planes. 
Period of 
revolution of 
the two nodes 
on their moving 
proper planes. 
h. 
m. 
Days. 
O 
i 
/ 
/ 
Tears. 
Years. 
9 
55 
8T7 
17 
0 
6 22 
6 
57 
6 
0 
988 
60 
8 
45 
5-57 
15 
43 
1 16 
3 
39 
6 
17 
509 
77 
7 
49 
3-59 
13 
25 
2 43 
8 
54 
6 
9 
434 
51 
7 
15 
2-45 
12 
16 
3 3 
10 
58 
6 
18 
609 
22 
5 N 2 
