THE ELEMENTS OF THE ORBIT OF A SATELLITE. 
831 
Therefore from X= 1 to ‘809 the inclination j decreases and the obliquity i 
increases. 
From \='809 to ’5 both inclination and obliquity decrease. 
From X=‘5 to ‘191 both inclination and obliquity increase. 
From X= i 9 L to 0 the inclination decreases and the obliquity increases. 
Now at the point where the above retrospective integration stopped, the moon’s 
period was 2‘45 days or 59 hours, and the day was 7"25 hours ; hence at this point 
X= '123, which falls between ’191 and '5. Hence both inclination and obliquity 
decrease retrospectively at a rate which tends to become infinite when we approach 
X='5, if the viscosity be infinitely great. For large, but not infinite, viscosity the 
rates become large and then rapidly decrease in the neighbourhood of X= ‘5. 
From this it follows that by supposing the viscosity large enough, the obliquity 
and inclination may be made as small as we please, when we arrive at the point 
where X='5. 
It was shown in § 17 of “Precession” that X='5 corresponds to a month of 
12 hours and a day of 6 hours. 
Between the values X=’5 and '809 the solutions for both the cases of small and of 
large viscosity concur in showing zero obliquity and inclination as dynamically stable. 
But between X=‘809 and 1 the obliquity is dynamically unstable for infinitely large, 
stable for infinitely small viscosity ; for these values of X zero inclination is dynamically 
stable both for large and small viscosity. 
From this it seems probable that for some large but finite viscosity, both zero 
inclination and zero obliquity would be dynamically stable for values of X between ‘809 
and unity. 
It appears to me therefore that we have only to accept the hypothesis that the 
viscosity of the earth has always been pretty large, as it certainly is at present, to 
obtain a satisfactory explanation of the obliquity of the ecliptic and of the inclination 
of the lunar orbit. This subject will be again discussed in the summary of Part VII. 
§ 21. Graphical illustration of the preceding integrations. 
A graphical illustration will much facilitate the comprehension of the numerical 
results of the last two sections. 
The integrations which have been carried out by quadratures are of course equivalent 
to finding the areas of certain curves, and these curves will afford a convenient illus¬ 
tration of the nature of those integrations. 
In §§ 19, 20 two separate points of departure were taken, the first proceeding from 
£=1 to •76, and the second from £=1 to "88. It is obvious that £ was referred to 
different initial values c 0 in the two integrations. 
5 o 2 
