872 
MR. G. H. DARWIN ON THE SECULAR CHANGES IN 
That which we here call a small viscosity is, when estimated by terrestrial standards, 
very great (see the summary of “ Precession ”). 
To return, however, to the case in hand :—We begin with the present configuration 
of the three bodies, when the moon’s proper plane is almost identical with the ecliptic, 
and when the inclination of the equator to its proper plane is very small. This is the 
case (i.) of the first problem :— 
It appears that the solution of “ Precession” is sufficiently accurate for this stage of 
the solution, and accordingly the parallel values of the day, month, and obliquity of 
the earth’s proper plane (or mean equator) are taken from § 17 of that paper; but the 
change in the new element, the inclination of the lunar orbit, has to be computed. 
The results of the solution are given in Table I., § 18, to which the reader is referred. 
This method of solution is not applicable unless the lunar proper plane is inclined at 
a small angle to the ecliptic, and unless the equator is inclined at a small angle to its 
proper plane. How at the beginning of the integration, that is to say with a homo¬ 
genous earth, and with the moon and sun in their present configuration, the moon’s 
proper plane is inclined to the ecliptic at 13", and the equator is inclined to the earth’s 
proper plane at 12" (for the heterogeneous earth these angles are about 8"'3 and 9"‘0); 
and at the end of this integration, when the day is 9 hrs. 55 m. and the month 
8'17 rn. s. days, the former angle has increased to 57' 31 // , and the latter to 22' 42”. 
These last results show that the nutations of the system have already become con¬ 
siderable, and although subsequent considerations show that this method of solution 
has not been overstrained, yet it here becomes advisable to carry out the solution into 
the more remote past by the methods of Part III. 
It was desirable to postpone the transition as long as possible, because the method 
used up to this point does not postulate the smallness of the inclinations, whereas the 
subsequent procedure does make that supposition. 
In § 19 the solution is continued by the new method, the viscosity of the earth still 
being supposed to be small. After laborious computations results are obtained, the 
physical meaning of which is embodied in Table VIII. The last two columns give the 
periods of the two processional motions by which the system is affected. The preces¬ 
sion of the pair of proper planes is, as it were, the ancestor of the actual luni-solar 
precession, and the revolution of the two nodes on their proper planes is the ancestor 
of the present revolution of the lunar nodes on the ecliptic, and of the 19-yearly 
nutation of the earth’s axis. 
This table exhibits a continued approach of the two proper planes to one another, so 
that at the point where the integration is stopped they are only separated by 1° 18'; 
at the present time they are of course separated by 23° 28'. 
The most remarkable feature in this table is that (speaking retrospectively) the 
inclination of the lunar orbit to its proper plane first increases, then diminishes, and 
then increases again. 
If it were desired to carry the solution still further back, we might without much 
