In conclusion, let us revert again to the two critical conditions of the earth-moon system. As to what happened before the first critical period, the tides tell us nothing, and every other line of reasoning very little; we can to some extent foresee what may happen after the second critical epoch is reached, at a time so remote that I do not venture even to express the number of ciphers which ought to follow the significant digit in the expression for the number of years. I mentioned, however, that at this time the sun tides will produce the effect of applying a still further brake to the rotation of the earth, so that ultimately the month will have become a shorter period than the day. It is therefore interesting for us to trace out the tidal history of a system in which the satellite revolves around the primary in less time than the primary takes to go round on its own axis--such a system, in fact, as Mars would present at this moment were the outer satellite to be abstracted. The effect of the tides on the planet raised by its satellite would then be to accelerate its rotation; for as the planet, so to speak, lags behind the tides, friction would now manifest itself by the continuous endeavour to drag the primary round faster. The gain of speed, however, thus attained would involve the primary in performing more than its original share of the moment of momentum; less moment of momentum would therefore remain to be done by the satellite, and the only way to accomplish this would be for the satellite to come inwards and revolve in a smaller orbit. We might indeed have inferred this from the considerations of energy alone, for whatever happens in the deformation of the orbit, heat is produced by the friction, and this heat is lost, and the total energy of the system must consequently decline. Now if it be a consequence of the tides that the velocity of the primary is accelerated, the energy corresponding to that velocity is also increased. Hence the primary has more energy than it had before; this energy must have been obtained at the expense of the satellite; the satellite must therefore draw inwards until it has yielded up enough of energy not alone to account for the increased energy of the primary, but also for the absolute loss of energy by which the whole operation is characterized. It therefore appears that in the excessively remote future the retreat of the moon will not only be checked, but that the moon may actually return to a point to be determined by the changes in the earth's rotation. It is, however, extremely difficult to follow up the study of a case where the problem of three bodies has become even more complicated than usual.