812 
MR. G. H. DARWIN ON THE SECULAR CHANGES IN 
=1 + ,' 
n n A it, 
-AS 
1 —Z+Tsi-J (1 — P)-fr 4 -sm i tan i (1 — £ 7 ) 
\ T o/ T o 
+ 1 • o, •‘^o 1 
i sm ^ 
?i 0 ftrc 0 -t-l\f 
1 “h 3 
i-l)(7+'eT?) + i 
£ +1 
./2 n 
1 
n 0 (kn 0 + !) 
Q lO 
3 ■ Lv '&e 
7vi 0 +1 — ^ 
kn 0 % 
( 258 ) 
In this equation we attribute to i, as it occurs on the right-hand side, an average 
value. 
By means of this equation, I had already computed a series of values of n corres¬ 
ponding to equidistant values of £ 
On dividing the first of (256) by the second of (257) we get an expression which 
differs from the d log tan 3 ^ i/d£ of (84) of “ Precession” by the presence of a common 
factor cos 3 j, and by sec j occurring in some of the small terms. Hence we may, 
without much error, accept the results of the integration for i in § 17 of “ Precession.” 
Lastly, dividing the first of (257) by the second, we have 
-1 
fl b : 
-sec i sec/ 
n J / 
This equation has now to be integrated by quadratures. 
All the numerical values were already computed for § 17 of “Precession,” and only 
required to be combined. 
The present mean inclination of the lunar orbit is 5° 9', so that j 0 =5° 9'. I then 
conjecture 5° 12' as a proper mean value to be assigned to j, as it occurs on the right- 
hand side of (259) for the first period of integration, which extends from ^=1 to '88. 
-logsin,=- 
f 1 
(259) 
First period of integration. 
From £= 1 to *88, four equidistant values were computed. 
From the computation for § 17 of “ Precession ” I extract the following. 
I 
n 
•96 
•92 
•88 
log-sec 7+10 = 8-59979 8*57309 8-56411 8’56746 
Then introducing j= 5° 12', I find 
n 
28 1 — - sec i sec j 
= 1 
= •5208 
•96 
•5412 
•92 
*5643 
•88 
•5901 
