the pulse of each successive vibration increased at
last to such an extent that the earth separated under the stress, and
threw off a portion of those semi-fluid materials of which it was
composed. In process of time these rejected portions contracted
together, and ultimately formed that moon we now see. Such is the
origin of the moon which the modern theory of tidal evolution has
presented to our notice.

There are two great epochs in the evolution of the earth-moon
system--two critical epochs which possess a unique dynamical
significance; one of these periods was early in the beginning, while
the other cannot arrive for countless ages yet to come. I am aware
that in discussing this matter I am entering somewhat largely into
mathematical principles; I must only endeavour to state the matter as
succinctly as the subject will admit.

In an earlier part of this lecture I have explained how, during all
the development of the earth-moon system, the quantity of moment of
momentum remains unaltered. The moment of momentum of the earth's
rotation added to the moment of momentum of the moon's revolution
remains constant; if one of these quantities increase the other must
decrease, and the progress of the evolution will have this result,
that energy shall be gradually lost in consequence of the friction
produced by the tides. The investigation is one appropriate for
mathematical formulae, such as those that can be found in Professor
Darwin's memoirs; but nature has in this instance dealt kindly with
us, for she has enabled an abstruse mathematical principle to be dealt
with in a singularly clear and concise manner. We want to obtain a
definite view of the alteration in the energy of the system which
shall correspond to a small change in the velocity of the earth's
rotation, the moon of course accommodating itself so that the moment
of momentum shall be preserved unaltered. We can use for this purpose
an angular velocity which represents the excess of the earth's
rotation over the angular revolution of the moon; it is, in fact, the
apparent angular velocity with which the moon appears to move round
the heavens. If we represent by N the angular velocity of the earth,
and by M the angular velocity of the moon in its orbit round the
earth, the quantity we desire to express is N--M; we shall call it the
relative rotation. The mathematical theorem which tells us what we
want can be enunciated in a concise manner as follows. _The al