Time and Tide by Sir Robert Stawell Ball (read with me txt) 📖

of the moon--in the immediate vicinity of the earth. There, however, dynamics leaves us, and unfortunately withholds its accurate illumination from the events which immediately preceded that state of things.

The theory of tidal evolution which I am describing in these lectures is mainly the work of Professor George H. Darwin of Cambridge. Much of the original parts of the theory of the tides was due to Sir William Thomson, and I have also mentioned how Professor Purser contributed an important element to the dynamical theory. It is, however, Darwin who has persistently deduced from the theory all the various consequences which can be legitimately drawn from it. Darwin, for instance, pointed out that as the moon is receding from us, it must, if we only look far enough back, have been once in practical contact with the earth. It is to Darwin also that we owe many of the other parts of a fascinating theory, either in its mathematical or astronomical aspect; but I must take this opportunity of saying, that I do not propose to make Professor Darwin or any of the other mathematicians I have named responsible for all that I shall say in these lectures. I must be myself accountable for the way in which the subject is being treated, as well as for many of the illustrations used, and some of the deductions I have drawn from the subject.

It is almost unavoidable for us to make a surmise as to the cause by which the moon had come into this remarkable position close to the earth at the most critical epoch of earth-moon history.

With reference to this Professor Darwin has offered an explanation, which seems so exceedingly plausible that it is impossible to resist the notion that it must be correct. I will ask you to think of the earth not as a solid body covered largely with ocean, but as a glowing globe of molten material. In a globe of this kind it is possible for great undulations to be set up. Here is a large vase of water, and by displacing it I can cause the water to undulate with a period which depends on the size of the vessel; undulations can be set up in a bucket of water, the period of these undulations being dependent upon the dimensions of the bucket. Similarly in a vast globe of molten material certain undulations could be set up, and those undulations would have a period depending upon the dimensions of this vibrating mass. We may conjecture a mode in which such vibrations could be originated. Imagine a thin shell of rigid material which just encases the globe; suppose this be divided into four quarters, like the four quarters of an orange, and that two of these opposite quarters be rejected, leaving two quarters on the liquid. Now suppose that these two quarters be suddenly pressed in, and then be as suddenly removed--they will produce depressions, of course, on the two opposite quarters, while the uncompressed quarters will become protuberant. In virtue of the mutual attractions between the different particles of the mass, an effort will be made to restore the globular form, but this will of course rather overshoot the mark; and therefore a series of undulations will be originated by which two opposite quarters of the sphere will alternately shrink in and become protuberant. There will be a particular period to this oscillation. For our globe it would appear to be somewhere about an hour and a half or two hours; but there is necessarily a good deal of uncertainty about this point.

We have seen how in those primitive days the earth was spinning around very rapidly; and I have also stated that the earth might at this very critical epoch of its history be compared with a grindstone which is being driven so rapidly that it is on the very brink of rupture. It is remarkable to note, that a cause tending to precipitate a rupture of the earth was at hand. The sun then raised tides in the earth as it does at present. When the earth revolved in a period of some four hours or thereabouts, the high tides caused by the sun succeeded each other at intervals of about two hours. When I speak of tides in this respect, of course I am not alluding to oceanic tides; these were the days long before ocean existed, at least in the liquid form. The tides I am speaking about were raised in the fluids and materials which then constituted the whole of the glowing earth; those tides rose and fell under the throb produced by the sun, just as truly as tides produced in an ordinary ocean. But now note the significant coincidence between the period of the throb produced by the sun-raised tides, and that natural period of vibration which belonged to our earth as a mass of molten material. It therefore follows, that the impulse given to the earth by the sun harmonized in time with that period in which the earth itself was disposed to oscillate. A well-known dynamical principle here comes into play. You see a heavy weight hanging by a string, and in my hand I hold a little slip of wood no heavier than a common pencil; ordinarily speaking, I might strike that heavy weight with this slip of wood, and no effect is produced; but if I take care to time the little blows that I give so that they shall harmonize with the vibrations which the weight is naturally disposed to make, then the effect of many small blows will be cumulative, so much so, that after a short time the weight begins to respond to my efforts, and now you see it has acquired a swing of very considerable amplitude. In Professor Fitzgerald's address to the British Association at Bath last autumn, he gives an account of those astounding experiments of Hertz, in which well-timed electrical impulses broke down an air resistance, and revealed to us ethereal vibrations which could never have been made manifest except by the principle we are here discussing. The ingenious conjecture has been made, that when the earth was thrown into tidal vibrations in those primeval days, these slight vibrations, harmonizing as they did with the natural period of the earth, gradually acquired amplitude; the result being that 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 alteration of the energy of the system may be expressed by multiplying the relative rotation by the change in the earth's angular velocity._ This result will explain many points to us in the theory, but just at present I am only going to make a single inference from it.

I must advert for a moment to the familiar conception of a maximum or a minimum. If a magnitude be increasing, that is, gradually growing greater and greater, it has obviously not attained a maximum so long as the growth is in progress. Nor if the object be actually decreasing can it be said to be at a maximum either; for then it was greater a second ago than it is now, and therefore it cannot be at a maximum at present. We may illustrate this by the familiar example of a stone thrown up into the air; at first it gradually rises, being higher at each instant than it was previously, until a culminating point is reached, when just for a moment the stone is poised at the summit of its path ere it commences its return to earth again. In this case the maximum point is obtained when the stone, having ceased to ascend, and not having yet commenced to descend, is momentarily at rest.

The same principles apply to the determination of a minimum. As long as the magnitude is declining the minimum has not been reached; it is only when the decline has ceased, and an increase is on the point of setting in, that the minimum can be said to be touched.

The earth-moon system contains at any moment a certain store of energy, and to every conceivable condition of the earth-moon system a certain quantity of energy is appropriate. It is instructive for us to study the different positions in which the earth and the moon might lie, and to examine the different quantities of energy which the system will contain in each of those varied positions. It is however to be understood that the different cases all presuppose the same total moment of momentum.

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