814 
MR. G. H. DARWIN ON THE SECULAR CHANGES IN 
The results of the present integration are embodied in the following table, of which 
the first three columns are taken from the table in § 17 of “ Precession.” 
Table I. 
Sidereal day in m.s. 
hours and minutes. 
Moon’s sidereal 
period in m.s. 
days. 
Inclination of 
mean equator to 
ecliptic. 
Inclination of 
lunar orbit to lunar 
proper plane. 
h. 
m. 
Days. 
o / 
O 
/ 
Initial 
23 
56 
27-32 
23 28 
5 
9 
15 
28 
18-62 
20 28 
5 
30 
Final 
9 
55 
8-17 
17 4 
6 
21 
We will now consider what amount of oscillation the equator and the plane of the 
lunar orbit undergo, as the nodes revolve, in the initial and final conditions represented 
in the above table. 
It appears from (119) that sin 2j oscillates between sin 2/ 0= La sin 2i Q /( y K 2 J r a), and 
that sin 2 i oscillates between sin 2?’ o: fi(/<-, fi-a) sin 2/ 0 /a, where i Q and j Q are the mean 
values of i and j. 
With the numerical values corresponding to the initial condition (that is to say in 
the present configurations of earth, moon, and sun), it will be found on substituting in 
/ /2 / \ ^ / 12 \ . T 
(115) and (112), with a 2 =§ ( —) 1 — f — )/2 instead of simply , that 
n 
o' 
n 
sr 
cl= ’341251, ^=-000318, a=-000059, b = -000150, 
when the present tropical year is the unit of time. 
Since 4ab is very small compared with (a—/3)' 2 , it follows that we have to a close 
degree of approximation 
k 1 = — a, k. 2 = — /3 
Then since (/c 1 -f-a)/a=b/(K: 1 +/3), it follows that sin 2 j oscillates between 
sin 2/ o: Pa sin 2/ 0 /(a — fi), and sin 2 i between sin 2?’ u =: f = b sin 2j 0 /(a—/3). 
Let Sj and Si be the oscillations of j and i on each side of the mean, then 
S sin 2/=a sin 2i/(a.—/3) and 8 sin 2i=h sin 2// (a — /3). 
Hence in seconds of arc 
^. 648000 1 
„ 2 
sin 
i 2 i ^ 
a.—(3 COS 2j 
0 . 648000 . b 
bi = - A.- 
sin 2 j 
a — f3 cos 2 i 
( 260 ) 
