768 
MR. G. H. DARWIN ON THE SECULAR CHANGES IN 
Therefore 
And by (127) 
b b_I^ I 
n 
approximately 
a 
Key "j” CC 
a ■ =m- approximately 
t + t' 
Ko= —-1, 
n 
K-, — /c, = n approximately 
t It 
j =m-l 
‘ n 
Now we have shown above that — k . 2 is the common angular velocity of the pair of 
proper planes, and the above results show that it is in fact the luni-solar precession. 
/Co — /Cj is the angular velocity of the two nodes on their proper planes, and it is 
nearly equal to It. 
The ratio of the amplitude of the 19-yearly nutation to the inclination of the lunar 
orbit is l/ll. 
The ratio of the inclination of the lunar proper plane to the obliquity of the ecliptic 
is ml/tt. 
In this case, therefore, the lunar proper plane is inclined at a small angle to the 
ecliptic, and if the earth were spherical would be identical with the ecliptic. 
Secondly, suppose that U is small compared with l. 
. tT . 
Then d fortiori - is small compared with l. Hence we may put /3=b. 
HI) 
Therefore 
Therefore 
a—b 
k 2 —k x — \/\a — /3) 2 + 4ab = a+b-}- pyyn, nearly 
/ , , m—1 
= in+i H—ntt 
v ' tn + l 
/Co-j-zq^ — (m+ 1)1 — 
n - , m 
K, - — , K X — —(111 —j— 1) l— | T IT 
1 m+l 1 v ’ m+l 
/Cq T - f3 1 n 
+ a_ 1/ 1 n' 
m + l T’ a — ml m +1 I 
1 n N 
1 n'' 
