503 
in 
AND ON THE REMOTE HISTORY OF THE EARTH. 
First period. From £= 1 to '88 ; p= 4'0074; {=20° 28'; IV=1'5478. 
— in the expression for JV added '0012 to the value of iV. 
The term 
Second period. From £=1 to ‘76; /x = 2'2784; 7=17° 4'; iV=l'5590. The term 
in — added '0011 to the value of N. 
n 0 
Third period. From ^=1 to '76; p.= l'1107; 7=15° 22'; iV=:l'2677. 
in — added '0007 to the value of N. 
The term 
It may be observed that during the first period of integration — diminishes, and 
reaches its minimum about the end of the period. During the rest of the integration 
it increases. If we neglect the solar action and the obliquity, it is easy to find the 
, n n 0 1 , , . , dN sn 
minimum value ot —. Jbor —=——— and reaches its minimum when —= — ; but 
n n n 0 Ap at, £ 
dN 
— —p. Therefore N= 
%. Now A 7 =l+p,(] —£), and hence £=f 
If p,= 4, 
(1 O ... yti 
9375. This value of £ is passed through at near the end of the first period 
of integration. At this period there are 19"2 mean solar hours in the day; 22mean 
solar days in the sidereal month ; and 281- rotations of the earth in the sidereal 
month. This result of 28y is, of course, only approximate, the true result being 
about 29.'" 
The physical meaning of these results is given in a table below. 
At the end of the third period of integration the solar terms (those in —) have 
become small in all the equations, and as they are rapidly diminishing they may be 
safely neglected. To continue the integration from this point a slight variation of 
method will be convenient. 
Our equations may now be written approximately 
n= i+MW) 
2/2 
, . , 1 —— sec i 
d , , U 1 n 
— — log tarn- = - 
dN 6 ^ a. 
N 
1 
n 
n 
In order to find how large a diminution of obliquity is possible if the integration 
be continued, we require to stop at the point where n cos 7—2/2. 
Now the equation K= 1 +/x(l — f) may be written 
n 
n 0 
+d l ~ '\f n)- 
* The subject is referred to from a more general point of view in a paper on the “ Secular Effects of 
Tidal Friction,” see ‘ Proc. Roy. Soc.,’ No. 197, 1879. 
MDCCCLXXIX. 3 T 
