earth's velocity of rotation at its lowest, we may suppose it
gradually and continually increased, and, as we have already seen, the
change in the energy of the system is to be expressed by multiplying
the relative rotation into the change of the earth's angular velocity.
It follows from the principles we have already explained, that the
maximum or minimum energy is attained at the moment when the
alteration is zero. It therefore follows, that the critical periods of
the system will arise when the relative rotation is zero, that is,
when the earth's rotation on its axis is performed with a velocity
equal to that with which the moon revolves around the earth. This is
truly a singular condition of the earth-moon system; the moon in such
a case would revolve around the earth as if the two bodies were bound
together by rigid bonds into what was practically a single solid body.
At the present moment no doubt to some extent this condition is
realized, because the moon always turns the same face to the earth (a
point on which we shall have something to say later on); but in the
original condition of the earth-moon system, the earth would also
constantly direct the same face to the moon, a condition of things
which is now very far from being realized.

It can be shown from the mathematical nature of the problem that there
are four states of the earth-moon system in which this condition may
be realized, and which are also compatible with the conservation of
the moment of momentum. We can express what this condition implies in
a somewhat more simple manner. Let us understand by the _day_ the
period of the earth's rotation on its axis, whatever that may be, and
let us understand by the _month_ the period of revolution of the moon
around the earth, whatever value it may have; then the condition of
maximum or minimum energy is attained when the day and the month have
become equal to each other. Of the four occasions mathematically
possible in which the day and the month can be equal, there are only
two which at present need engage our attention--one of these occurred
near the beginning of the earth and moon's history, the other remains
to be approached in the immeasurably remote future. The two remaining
solutions are futile, being what the mathematician would describe as
imaginary.

There is a fundamental difference between the dynamical conditions in
these critical epochs--in one of them the energy of the system has
attained a