AND ON THE REMOTE HISTORY OF THE EARTH. 
509 
to the crude method of amending the solution, but also to the fact that they partly in¬ 
clude the obliquity in one way, whilst I partly include it in another way, and I include 
a large part of the solar tidal friction whilst they neglect it. It is interesting to note 
that the larger root, which gives the shorter length of day, is but little affected by the 
consideration of the earth’s heterogeneity. 
With respect to the second solution (56 days), it must be remarked that the sun’s 
tidal friction will go on lengthening the day even beyond this point, but then the 
lunar tides will again come into existence, and the lunar tidal friction will tend in part 
to counteract the solar. The tidal reaction will also be reversed, so that the moon 
will again approach the earth. Thus the effect of the sun is to make this a state of 
dynamical instability. 
The first solution, where both the day and month are 5 hours 36 minutes long, is 
the one which is of interest in the present inquiry, for this is the initial state towards 
which the integration has been running back. 
This state of things is one of dynamical instability, as may be shown as follows :— 
First consider the case where the sun does not exist. Suppose the earth to be 
rotating in about 5^ hours, and the moon moving orbitaliy around it in a little less 
than that time. Then the motion of the moon relatively to the earth is consentaneous 
with the earth’s rotation, and therefore the tidal friction, small though it be, tends to 
accelerate the earth’s rotation ; the tidal reaction is such as to tend to retard the moon’s 
linear velocity, and therefore increase her orbital angular velocity, and reduce her 
distance from the earth. The end will be that the moon falls into the earth. 
This subject is graphically illustrated in a paper on the “ Secular Effects of Tidal 
Friction,” read before the Royal Society on June 19, 1879. 
Secondly, take the case where the sun also exists, and suppose the system started 
in the same way as before Now the motion of the earth relatively to the sun is rapid, 
and such that the solar tidal friction retards the earth’s rotation ; whilst the lunar 
tidal friction is, as before, such as to accelerate the rotation. 
Hence if the viscosity be very large the earth’s rotation may be accelerated, but if it 
be not very large it will be retarded. The tidal reaction, which depends on the lunar 
tides alone, continues negative, and the moon approaches the earth as before. Thus 
after a short time the motion of the moon relatively to the earth is more rapid than 
in the previous case, whatever be the ratio between solar and lunar tidal friction. 
Hence in this case the moon will fall into the earth more rapidly than if the sun did 
not exist, and the dynamical instability is more marked. 
If, however, the day were shorter than the month, the moon must continually recede 
from the earth, until it reaches the outer limit of a day of 56 m. s. days. 
There is one circumstance which might perhaps decide that this should be the 
direction in which the equilibrium would break down ; for the earth was a cooling 
* From here to the end of the section a good many alterations have been made since the paper was 
presented.—July 5, 1879. 
