Among the different cases that can be imagined, those will be of special interest in which the total quantity of energy in the system is a maximum or a minimum. We must for this purpose suppose the system gradually to run through all conceivable changes, with the earth and moon as near as possible, and as far as possible, and in all intermediate positions; we must also attribute to the earth every variety in the velocity of its rotation which is compatible with the preservation of the moment of momentum. Beginning then with the 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.