THE ELEMENTS OF THE ORBIT OP A SATELLITE. 
819 
The method pursued in the integration of the preceding section proceeds virtually 
on the assumption that the term y(K. 2 J ra)/hi[K l — k. 2 ) is the only important one in the 
expression for d log tan ^J/d£ and that the term 8(K. 2 -\-<x)/lcn(K. 2 — k x ) is the only 
important one in the expression for d log tan \l/d£. 
Now when £=1, at the beginning of the present integration, we see from Table III. 
that the said term in y is about 22 times as large as any other occurring in d log tan ^J, 
and that the said term in 8 is about 16 times as large as any other which occurs in 
d log tan 1,1. Hence the preceding integration must have given fairly satisfactory 
results. But after the first column these terms in y and 8 fail to maintain their rela¬ 
tive importance, so that when <f= 7 6, they have both become considerably less im¬ 
portant than other terms—notably b' a,/hi(K. 2 —/c 1 ) a and a,'b/kn(K 2 —/y) 2 . This is exactly 
what is to be expected, because the equations are tending towards the form which 
they would take if the solar influence were nil, and an inspection of (225) shows that 
these terms would then be prominent. 
Now if we combine these values of the several terms together according to (250), 
we obtain the seven equidistant values of d log tan -f J/d^ and d log tan bl/df exhibited 
in the following table:— 
Table IV. 
1- 
•96 
•92 
•88 
•84 
•80 
•76 
d log tan /cZ£= 
-•49386 
-•46660 
-•37218 
-15627 
+ -16138 
+ -35219 
+ -19330 
d log tan jdtj= 
+ -54460 
+ •58194 
+ -69284 
+ •93287 
+ 1-28273 
+ 1-51135 
+ 1-39323 
By interpolation it appears that dJ/d£ vanishes when ^=‘8603. This value of £ 
corresponds with 8 hrs. 36 m. for the period of the earth’s rotation, and 5 "20 m. s. days 
for the period of the moon’s revolution. 
Since d£ is negative in our integration, we see from these values that I, the inclina¬ 
tion of the earth’s proper plane to the ecliptic, will continue diminishing, and with 
increasing rapidity. On the other hand, the inclination J of the lunar orbit to its 
proper plane will increase at first, but at a diminishing rate, and will finally diminish. 
This is a point of the greatest importance in explaining the present inclination of the 
lunar orbit to the ecliptic, and we shall recur to it later on. 
Now combine the first four values by the rule of finite differences, viz.: 
K+%+3(wi+w 2 )]p, 
and all seven by W eddle’s rule, viz.: 
[>o ■-f %+ u 3 ++« 6 + 5 (u j+ u s +udjjrah 
where li is our d£, and the u’s are the several numbers given in the above Table IV.; 
then we have, on integration from 1 to ’88, 
