THE ELEMENTS OE THE ORBIT OF A SATELLITE. 
873 
error here make the transition to the method of case (ii.) of the first problem, and neg¬ 
lecting the solar influence entirely, refer the motion to the invariable plane of the 
moon-earth system. This invariable plane would have to be taken as somewhere 
between the two proper planes, and therefore inclined to the ecliptic at about 11 45'; 
the invariable plane would then really continue to have a precessional motion due to 
the solar influence on the system formed by the earth and moon together, but this 
would not much affect the treatment of the plane as though it were fixed in space. 
We should then have to take the obliquity of the equator to the invariable plane as 
about 3°, and the inclination of the lunar orbit to the same plane as about 5° 30'. 
In the more remote past the obliquity of the equator to the invariable plane would 
go on diminishing, but at a slower and slower rate, until the moon’s period is 12 hours 
and the day is 6 hours, when it would no longer diminish; and the inclination of the 
orbit to the invariable plane would go on increasing, until the day and month come to 
an identity, and at an ever increasing rate. 
It follows from this, that if we continued to trace the changes backwards, until the 
day and month are identical, we should find the lunar orbit inclined at a consider¬ 
able angle to the equator. If this were necessarily the case, it would be difficult to 
believe that the moon is a portion of the primeval planet detached by rapid rotation, 
or by other causes. But the previous results are based on the hypothesis that the 
viscosity of the earth is small, and it therefore now became important to consider 
how a different hypothesis concerning the constitution of the earth might modify the 
results. 
In § 20 the solution of the problem is resumed, at the point where the methods of 
Part III. were first applied, but with the hypothesis that the viscosity of the earth is 
very large, instead of very small. The results for any intermediate degree of viscosity 
must certainly he between those found before and those to be found now. 
Then having retraversed the same ground, but with the new hypothesis, I found 
the results given in Table XY. 
The inclinations of the two proper planes to the ecliptic are found to be very nearly 
the same as in the case of small viscosity. But the inclination of the lunar orbit to 
its proper plane increases at first and then continues diminishing, without the subse¬ 
quent reversal of motion found in the previous solution. 
If the solution were carried back into the more remote past, the motion being 
referred to the invariable plane, we should find both the obliquity of the equator and 
the inclination of the orbit diminishing at a rate which tends to become infinite, if the 
viscosity is infinitely great. Infinite viscosity is of course the same as perfect rigidity, 
and if the earth were perfectly rigid the system would not change at all. The true 
interpretation to put on this result is that the rate of change of inclination becomes 
large, if the viscosity be large. This diminution would continue until the day was 
6 hours and the month 12 hours. For an analysis of the state of things further back 
than this, the reader is referred to § 20. 
