530 MR. G. H. DARWIN ON THE PRECESSION OE A VISCOUS SPHEROID 
3 hours, and therefore this must be regarded as a very moderate degree of visco¬ 
sity. It was chosen because initially it makes the rate of change of obliquity a 
maximum, and although it is not that degree of viscosity which will make all the 
changes proceed with the greatest possible rapidity, yet it is sufficiently near that 
value to enable us to estimate very well the smallest time which can possibly have 
elapsed in the history of the earth, if changes of the kind found really have taken 
place. This estimate of time is confirmed by a second method, which v 7 ill be referred 
to later. 
The changes were tracked backwards in time from the present epoch, and for con¬ 
venience of diction I shall also reverse the form of speech— e.g., a true loss of energy 
as the time increases will be spoken of as a gain of energy as we look backwards. 
I shall not enter at all into the mathematical difficulties of the problem, but shall 
proceed at once to comment on the series of tables at the end of Section 15, which 
give the results of the solution. 
The whole process, as traced backwards, exhibits a gain of kinetic energy to the 
system (of which more presently), accompanied by a transference of moment of 
momentum from that of orbital motion of the moon and earth to that of rotation of 
the earth. The last column but one of Table IV. exhibits the fall of the ratio of the 
two moments of momentum from 4 - 01 down to '44. The whole moment of momentum 
of the moon-earth system rises slightly, because of solar tidal friction. The change is 
investigated in Section 19. 
Looked at in detail, we see the day, month, and obliquity all diminishing, and the 
changes proceeding at a rapidly increasing rate, so that an amount of change which at 
the beginning required many millions of years, at the end only requires as many thou¬ 
sands. The reason of this is that the moon's distance diminishes with great rapidity; 
and as the effects vary as the square of the tide-generating force, they vary as the 
inverse sixth power of the moon’s distance, or, in physical language, the height of the 
tides increases with great rapidity, and so also does the moon’s attraction. But there 
is a counteracting principle, which to some extent makes the changes proceed slower. 
It is obvious that a disturbing body will not have time to raise such high tides in a 
rapidly rotating spheroid as in one which rotates slowly. As the earth’s rotation 
increases, the lagging of the tides increases. The first column of Table I. shows the 
angle by which the crest of the lunar semi-diurnal tide precedes the moon ; we see 
that the angle is almost doubled at the end of the series of changes, as traced back¬ 
wards. It is not quite so easy to give a physical meaning to the other columns, 
although it might be done. In fact, as the rotation increases, the effect of each tide 
rises to a maximum, and then dies away; the tides of longer period reach their maxi¬ 
mum effect much more slowly than the ones of short period. At the point where I 
have found it convenient to stop the solution (see Table IV.), the semi-diurnal effect has 
passed its maximum, the diurnal tide has just come to give its maximum effect, whilst 
the fortnightly tide has not nearly risen to that point. 
