Contents -BIBLIOGRAPHIC RECORD

Scientific Papers.
The Harvard Classics. 1909–14.

Sir William Thomson (Lord Kelvin)

The Tides

Appendix A

[Extracts from a Lecture on “The Tides,” given to the Glasgow Science Lectures Association, not hitherto published, and now included as explaining in greater detail certain paragraphs of the preceding Lecture.]

(1) Gravitation.—The great theory of gravitation put before us by Newton asserts that every portion of matter in the universe attracts every other portion; and that the force depends on the masses of the two portions considered, and on the distance between them. Now, the first great point of Newton’s theory is, that bodies which have equal masses are equally attracted by any other body, a body of double mass experiencing double force. This may seem only what is to be expected. It would take more time than we have to spare were I to point out all that is included in this statement; but let me first explain to you how the motions of different kinds of matter depend on a property called inertia. I might show you a mass of iron as here. Consider that if I apply force to it, it gets into a state of motion; greater force applied to it, during the same time, gives it increased velocity, and so on. Now, instead of a mass of iron, I might hang up a mass of lead, or a mass of wood, to test the equality of the mass by the equality of the motion which is produced in the same time by the action of the same force, or in equal times by the action of equal forces. Thus, quite irrespectively of the kind of matter concerned, we have a test of the quantity of matter. You might weigh a pound of tea against a pound of brass without ever putting them into the balance at all. You might hang up one body by a proper suspension, and you might, by a spring, measure the force applied, first to the one body, and then to the other. If the one body is found to acquire equal velocity under the influence of equal force for equal times as compared with the other body, then the mass of the one is said to be equal to the mass of the other.

I have spoken of mutual forces between any two masses. Let us consider the weight or heaviness of a body on the earth’s surface. Newton explained that the attraction of the whole earth upon a body—for example, this 56 pounds mass of iron—causes its heaviness or weight. Well, now, take 56 pounds of iron here, and take a mass of lead, which, when put in the balance, is found to be of equal weight. You see we have quite a new idea here. You weigh this mass of iron against a mass of lead, or to weigh out a commodity for sale; as, for instance, to weigh out pounds of tea, to weigh them with brass weights is to compare their gravitations towards the earth—to compare the heaviness of the different bodies. But the first subject that I asked you to think of had nothing to do with heaviness. The first subject was the mass of the different bodies as tested by their resistance to force tending to set them in motion. I may just say that the property of resistance against being set into motion, and again against resistance to being stopped when in motion, is the property of matter called inertia.

The first great point in Newton’s discovery shows, then, that if the property of inertia is possessed to an equal degree by two different substances, they have equal heaviness. One of his proofs was founded on the celebrated guinea and feather experiment, showing that the guinea and feather fall at the same rate when the resistance of the air is removed. Another was founded upon making pendulums of different substances—lead, iron, and wood—to vibrate, and observing their times of vibration. Newton thus discovered that bodies which have equal heaviness have equal inertia.

The other point of the law of gravitation is, that the force between any two bodies diminishes as the distance increases, according to the law of the inverse square of the distance. That law expresses that, with double distance, the force is reduced one quarter, at treble distance the force is reduced to one-ninth part. Suppose we compare forces at the distance of one million miles, then again at the distance of two and a half million miles, we have to square the one number, then square the other, and find the proportion of the square of the one number to the square of the other. The forces are inversely as the squares of the distance, that is the most commonly quoted part of the law of gravitation; but the law is incomplete without the first part, which establishes the relation between two apparently different properties of matter. Newton founded this law upon a great variety of different natural phenomena. The motion of the planets round the sun, and the moon round the earth, proved that for each planet the force varies inversely as the square of its distance from the sun; and that from planet to planet the forces on equal portions of their masses are inversely as the squares of their distances. The last link in the great chain of this theory is the tides.

(2) Tide-Generating Force.—And now we are nearly ready to complete the theory of tide-generating force. The first rough view of the case, which is not always incorrect, is that the moon attracts the waters of the earth towards herself and heaps them up, therefore, on one side of the earth. It is not so. It would be so if the earth and moon were at rest and prevented from falling together by a rigid bar or column. If the earth and moon were stuck on the two ends of a strong bar, and put at rest in space, then the attraction of the moon would draw the waters of the earth to the side of the earth next to the moon. But in reality things are very different from that supposition. There is no rigid bar connecting the moon and the earth. Why then does not the moon fall towards the earth? According to Newton’s theory, the moon is always falling towards the earth. Newton compared the fall of the moon, in his celebrated statement, with the fall of a stone at the earth’s surface, as he recounted, after the fall of an apple from the tree, which he perceived when sitting in his garden musing on his great theory. The moon is falling towards the earth, and falls in an hour as far as a stone falls in a second. It chances that the number 60 is nearly enough, as I have said before, a numerical expression for the distance of the moon from the earth in terms of the earth’s radius. It is only by that chance that the comparison between the second and hour can be here introduced. Since there are 60 times 60 seconds in an hour, and about 60 radii of the earth in the distance from the moon, we are led to the comparison now indicated, but I am inverting the direction of Newton’s comparison. He found by observation that the moon falls as far in an hour as a stone falls in a second, and hence inferred that the force on the moon is a 60th of the 60th of the force per equal mass on the earth’s surface. Then he learned from accurate observations, and from the earth’s dimensions, what I have mentioned as the moon’s distance, and perceived the law of variation between the weight of a body at the earth’s surface and the force that keeps the moon in her orbit. The moon in Newton’s theory was always falling towards the earth. Why does it not come down? Can it be always falling and never come down? That seems impossible. It is always falling, but it has also a motion perpendicular to the direction in which it is falling, and the result of that continual falling is simply a change of direction of this motion.

It would occupy too much of our time to go into this theory. It is simply the dynamical theory of centrifugal force. There is a continual falling away from the line of motion, as illustrated in a stone thrown from the hand describing an ordinary curve. You know that if a stone is thrown horizontally it describes a parabola—the stone falling away from the line in which it was thrown. The moon is continually falling away from the line in which it moves at any instant, falling away towards the point of the earth’s centre, and falling away towards that point in the varying direction from itself. You can see it may be always falling, now from the present direction, now from the altered direction, now from the farther altered direction in a further altered line; and so it may be always falling and never coming down. The parts of the moon nearest to the earth tend to fall most rapidly, the parts furthest from the earth, least rapidly; in its own circle, each is falling away and the result is as if we had the moon falling directly.

But while the moon is always falling towards the earth, the earth is always falling towards the moon; and each preserves a constant distance, or very nearly a constant distance from the common centre of gravity of the two. The parts of the earth nearest to the moon are drawn towards the moon with more force than an equal mass at the average distance; the most distant parts are drawn towards the moon with less force than corresponds to the average distance. The solid mass of the earth, as a whole, experiences, according to its mass, a force depending on the average distance; while each portion of the water on the surface of the earth experiences an attractive force due to its own distance from the moon. The result clearly is, then, a tendency to protuberance towards the moon and from the moon; and thus, in a necessarily most imperfect manner, I have explained to you how it is that the waters are not heaped up on the side next the moon, but are drawn up towards the moon and left away from the moon so as to tend to form an oval figure. The diagram (FIG. 125, p. 284) shows the protuberance of water towards and from the moon. It shows also the sun on the far side, I need scarcely say, with an enormous distortion of proportions, because without that it would be impossible in a diagram to show the three bodies. This illustrates the tendency of the tide-generating forces.

(3) Elastic Tides.—But another question arises. This great force of gravity operating in different directions, pulling at one place, pressing in at another, will it not squeeze the earth out of shape? I perceive signs of incredulity; you think it impossible it can produce any sensible effect. Well, I will just tell you that instead of being impossible, instead of it not producing any such effect, we have to suppose the earth to be of exceedingly rigid material, in order that the effect of these distorting influences on it may not mask the phenomenon of the tides altogether.

There is a very favourite geological hypothesis which I have no doubt many here present have heard, which perhaps till this moment many here present have believed, but which I hope no one will go out of this room believing, and that is that the earth is a mere crust, a solid shell thirty, or forty, or fifty miles thick at the most, and that it is filled with molten liquid lava. This is not a supposition to be dismissed as absurd, as ludicrous, as absolutely unfounded and unreasonable. It is a theory based on hypothesis which requires most careful weighing. But it has been carefully weighed and found wanting in conformity to the truth. On a great many different essential points it has been found at variance with the truth. One of these points is, that unless the material of this supposed shell were preternaturally rigid, were scores of times more rigid than steel, the shell would yield so freely to the tide-generating forces that it would take the figure of equilibrium, and there would be no rise and fall of the water, relatively to the solid land, left to show us the phenomena of the tides.

Imagine that this (FIG. 134, p. 299) represents a solid shell with water outside, you can understand if the solid shell yields with sufficiently great freedom, there will be exceedingly little tidal yielding left for the water to show. It may seem strange when I say that hard steel would yield so freely. But consider the great hardness of steel and the smaller hardness of india-rubber. Consider the greatness of the earth, and think of a little hollow india-rubber ball, how freely it yields to the pressure of the hand, or even to its own weight when laid on a table. Now, take a great body like the earth: the greater the mass the more it is disposed to yield to the attraction of distorting forces when these forces increase with the whole mass. I cannot just now fully demonstrate to you this conclusion; but I say that a careful calculation of the forces shows that in virtue of the greatness of the mass it would require an enormously increased rigidity in order to keep in shape. So that if we take the actual dimensions of the earth at forty-two million feet diameter, and the crust at fifty miles thick, or two hundred and fifty thousand feet, and with these proportions make the calculation, we find that something scores of times more rigid than steel would be required to keep the shape so well as to leave any appreciable degree of difference from the shape of hydrostatic equilibrium, and allow the water to indicate, by relative displacement, its tendency to take the figure of equilibrium; that is to say, to give us the phenomena of tides. The geological inference from this conclusion is, that not only must we deny the fluidity of the earth and the assertion that it is encased by a thin shell, but we must say that the earth has, on the whole, a rigidity greater than that of a solid globe of glass of the same dimensions; and perhaps greater than that of a globe of steel of the same dimensions. But that it cannot be less rigid than a globe of glass, we are assured. It is not to be denied that there may be a very large space occupied by liquid. We know there are large spaces occupied by lava; but we do not know how large they may be, although we can certainly say that there are no such spaces, as can in volume be compared with the supposed hollow shell, occupied by liquid constituting the interior of the earth. The earth as a whole must be rigid, and perhaps exceedingly rigid, probably rendered more rigid than it is at the surface strata by the greater pressure in the greater depths.

The phenomena of underground temperature, which led geologists to that supposition, are explained otherwise than by their assumption of a thin shell full of liquid; and further, every view we can take of underground temperature, in the past history of the earth, confirms the statement that we have no right to assume interior fluidity.