Fig. 21Fig. 21.
Fig. 22Fig. 22.
If an impulse be given to a top or gyrostat in the direction of the precession, it will rise in opposition to the force of gravity, and should at any instant the precessional velocity be greater than what it ought to be for the balance of the force of gravity, the top or gyrostat will rise, its precessional velocity diminishing. If the precessional velocity is too small, the top will fall, and as it falls the precessional velocity increases.
Now I say that all these facts, which are mere facts of observation, agree with our rule. I wish I dare ask you to remember them all. You will observe that in this wall sheet I have made a list of them. I speak of gravity as causing the precession, but the forces may be any others than such as are due to gravity.
I.Rule.When forces act upon a spinning body, tending to cause rotation about any other axis than the spinning axis, the spinning axis sets itself in better agreement with the new axis of rotation. Perfect agreement would mean perfect parallelism, the directions of rotation being the same.
II. Hurry on the precession, and the body rises in opposition to gravity.
III. Delay the precession and the body falls, as gravity would make it do if it were not spinning.
IV. A common top precesses in the same direction as that in which it spins.
V. A top supported above its centre of gravity, or a body which would be in stable equilibrium if not spinning, precesses in the opposite direction to that of its spinning.
VI. The last two statements come to this:—When the forces acting on a spinning body tend to make theangleof precession greater, the precession is in the same direction as the spinning, andvice versâ.
Having by observation obtained a rule, every natural philosopher tries to make his rule a rational one; tries to explain it. I hope you know what we mean when we say that we explain a phenomenon; we really mean that we show the phenomenon to be consistent with other better known phenomena. Thus when you unmask a spiritualist and show that the phenomena exhibited by him are due to mere sleight-of-hand and trickery, you explain the phenomena. When you show that they are all consistent with well-observed and established mesmeric influences, you are also said to explain the phenomena. When you show that they can be effected by means of telegraphic messages, or by reflection of light from mirrors, you explain thephenomena, although in all these cases you do not really know the nature of mesmerism, electricity, light, or moral obliquity.
The meanest kind of criticism is that of the man who cheapens a scientific explanation by saying that the very simplest facts of nature are unexplainable. Such a man prefers the chaotic and indiscriminate wonder of the savage to the reverence of a Sir Isaac Newton.
Fig. 23Fig. 23.
The explanation of our rule is easy. Here is a gyrostat (Fig. 23) something like the earth in shape, and it is at rest. I am sorry to say that I am compelled to support this globe in a very visible manner by gymbal rings. If this globe were just floating in the air, if it had no tendency to fall, my explanation would be easier to understand, and I could illustrate it better experimentally. Observe the point P. If I move the globe slightly about the axis A, the point P moves to Q. But suppose instead of this that the globe and inner gymbalring had been moved about the axis B; the point P would have moved to R. Well, suppose both those rotations took place simultaneously. You all know that the point P would move neither to Q nor to R, but it would move to S; P S being the diagonal of the little parallelogram. The resultant motion then is neither about the axis O A in space, nor about the axis O B, but it is about some such axis as O C.
To this globe I have given two rotations simultaneously. Suppose a little being to exist on this globe which could not see the gymbals, but was able to observe other objects in the room. It would say that the direction of rotation is neither about O A nor about O B, but that the real axis of its earth is some line intermediate, O C in fact.
If then a ball is suddenly struck in two different directions at the same instant, to understand how it will spin we must first find how much spin each blow would produce if it acted alone, and about what axis. A spin of three turns per second about the axis O A (Fig. 24), and a spin of two turns per second about the axis O B, really mean that the ball will spin about the axis O C with a spin of three and a half turns per second. To arrive at this result, I made O A, 3 feet long (any other scale of representation would have been right)and O B, 2 feet long, and I found the diagonal O C of the parallelogram shown on the figure to be 3½ feet long.
Observe that if the rotation about the axis O A iswiththe hands of a watch looking from O to A, the rotation about the axis O B looking from O to B, must also be with the hands of a watch, and the resultant rotation about the axis O C is also in a direction with the hands of a watch looking from O to C. Fig. 25 shows in two diagrams how necessary it is that on looking from O along either O A or O B, the rotation should be in the same direction as regards the hands of a watch. These constructions are well known to all who have studied elementary mechanical principles. Obviously if the rotation about O A is very much greater than the rotation about O B, then the position of the new axis O C must be much nearer O A than O B.
Fig. 24Fig. 24.
Fig. 25Fig. 25.
We see then that if a body is spinning about an axis O A, and we apply forces to it whichwould, if it were at rest, turn it about the axis O B; the effect is to cause the spinning axis to be altered to O C; that is, the spinning axis sets itself in better agreement with the new axis of rotation. This is the first statement on our wall sheet, the rule from which all our other statements are derived, assuming that they were not really derived from observation. Now I do not say that I have here given a complete proof for all cases, for the fly-wheels in these gyrostats are running in bearings, and the bearings constrain the axes to take the new positions, whereas there is no suchconstraint in this top; but in the limited time of a popular lecture like this it is not possible, even if it were desirable, to give an exhaustive proof of such a universal rule as ours is. That I have not exhausted all that might be said on this subject will be evident from what follows.
If we have a spinning ball and we give to it a new kind of rotation, what will happen? Suppose, for example, that the earth were a homogeneous sphere, and that there were suddenly impressed upon it a new rotatory motion tending to send Africa southwards; the axis of this new spin would have its pole at Java, and this spin combined with the old one would cause the earth to have its true pole somewhere between the present pole and Java. It would no longer rotate about its present axis. In fact the axis of rotation would be altered, and there would be no tendency for anything further to occur, because a homogeneous sphere will as readily rotate about one axis as another. But if such a thing were to happen to this earth of ours, which is not a sphere but a flattened spheroid like an orange, its polar diameter being the one-third of one per cent. shorter than the equatorial diameter; then as soon as the new axis was established, the axis of symmetry would resent the change and would try to become again the axis of rotation, and a great wobbling motion would ensue.I put the matter in popular language when I speak of the resentment of an axis; perhaps it is better to explain more exactly what I mean. I am going to use the expression Centrifugal Force. Now there are captious critics who object to this term, but all engineers use it, and I like to use it, and our captious critics submit to all sorts of ignominious involution of language in evading the use of it. It means the force with which any body acts upon its constraints when it is constrained to move in a curved path. The force is always directed away from the centre of the curve. When a ball is whirled round in a curve at the end of a string its centrifugal force tends to break the string. When any body keyed to a shaft is revolving with the shaft, it may be that the centrifugal forces of all the parts just balance one another; but sometimes they do not, and then we say that the shaft is out of balance. Here, for example, is a disc of wood rotating. It is in balance. But I stop its motion and fix this piece of lead, A, to it, and you observe when it rotates that it is so much out of balance that the bearings of the shaft and the frame that holds them, and even the lecture-table, are shaking. Now I will put things in balance again by placing another piece of lead, B, on the side of the spindle remote from A, and when I again rotate the disc (Fig. 26) thereis no longer any shaking of the framework. When the crank-shaft of a locomotive has not been put in balance by means of weights suitably placed on the driving-wheels, there is nobody in the train who does not feel the effects. Yes, and the coal-bill shows the effects, for an unbalanced engine tugs the train spasmodically instead of exerting an efficient steady pull. My friend Professor Milne, of Japan, places earthquake measuring instruments on engines and in trains for measuring this and other wants of balance, and he has shown unmistakably that two engines of nearly the same general design, one balanced properly and the other not, consume very different amounts of coal in making the same journey at the same speed.
Fig. 26Fig. 26.
If a rotating body is in balance, not only does the axis of rotation pass through the centre of gravity (or rather centre of mass) of the body, butthe axis of rotation must be one of the three principal axes through the centre of mass of the body. Here, for example, is an ellipsoid of wood; A A, B B, and C C (Fig. 27) are its three principal axes, and it would be in balance if it rotated about any one of these three axes, and it would not be in balance if it rotated about any other axis, unless, indeed, it were like a homogeneous sphere, every diameter of which is a principal axis.
Fig. 27Fig. 27.
Every body has three such principal axes through its centre of mass, and this body (Fig. 27) has them; but I have here constrained it to rotate about the axis D D, and you all observe the effect of the unbalanced centrifugal forces, which is nearly great enough to tear the framework in pieces. The higher the speed the more important this want of balance is. If the speed is doubled, the centrifugal forces become four times as great; and modern mechanical engineers with their quick speed engines, some of which revolve, like the fan-engines of torpedo-boats, at 1700 revolutions per minute, require to pay great attention to this subject, which the older engineers never troubled theirheads about. You must remember that even when want of balance does not actually fracture the framework of an engine, it will shake everything, so that nuts and keys and other fastenings are pretty sure to get loose.
I have seen, on a badly-balanced machine, a securely-fastened pair of nuts, one supposed to be locking the other, quietly revolving on their bolt at the same time, and gently lifting themselves at a regular but fairly rapid rate, until they both tumbled from the end of the bolt into my hand. If my hand had not been there, the bolts would have tumbled into a receptacle in which they would have produced interesting but most destructive phenomena. You would have somebody else lecturing to you to-night if that event had come off.
Suppose, then, that our earth were spinning about any other axis than its present axis, the axis of figure. If spun about any diameter of the equator for example, centrifugal forces would just keep things in a state of unstable equilibrium, and no great change might be produced until some accidental cause effected a slight alteration in the spinning axis, and after that the earth would wobble very greatly. How long and how violently it would wobble, would depend on a number of circumstances about which I will not now venture to guess. If youtell me that on the whole, in spite of the violence of the wobbling, it would not get shaken into a new form altogether, then I know that in consequence of tidal and other friction it would eventually come to a quiet state of spinning about its present axis.
You see, then, that although every body has three axes about which it will rotate in a balanced fashion without any tendency to wobble, this balance of the centrifugal forces is really an unstable balance in two out of the three cases, and there is only one axis about which a perfectly stable balanced kind of rotation will take place, and a spinning body generally comes to rotate about this axis in the long run if left to itself, and if there is friction to still the wobbling.
To illustrate this, I have here a method of spinning bodies which enables them to choose as their spinning axis that one principal axis about which their rotation is most stable. The various bodies can be hung at the end of this string, and I cause the pulley from which the string hangs to rotate. Observe that at first the disc (Fig. 28a) rotates soberly about the axis A A, but you note the small beginning of the wobble; now it gets quite violent, and now the disc is stably and smoothly rotating about the axis B B, which is the most important of its principal axes.
Fig. 28Fig. 28.
Again, this cone (Fig. 28b) rotates smoothly at first about the axis A A, but the wobble begins and gets very great, and eventually the cone rotates smoothly about the axis B B, which is the most important of its principal axes. Here again is a rod hung from one end (Fig. 28d).
See also this anchor ring. But you may be more interested in this limp ring of chain (Fig. 28c). See how at first it hangs from the cord vertically, and how the wobbles and vibrations end in its becoming a perfectly circular ring lying all in a horizontal plane. This experiment illustrates also the quasi-rigidity given to a flexible body by rapid motion.
To return to this balanced gyrostat of ours (Fig. 13). It is not precessing, so you know that the weight W just balances the gyrostat F. Now if I leave the instrument to itself after I give a downward impulse to F, not exerting merely a steady pressure, you will notice that F swings to the right for the reason already given; but it swings too fast and too far, just like any other swinging body, and it is easy from what I have already said, to see that this wobbling motion (Fig. 29) should be the result, and that it should continue until friction stills it, and F takes its permanent new position only after some time elapses.
You see that I can impose this wobble or noddingmotion upon the gyrostat whether it has a motion of precession or not. It is now nodding as it processes round and round—that is, it is rising and falling as it precesses.
Fig. 29Fig. 29.
Perhaps I had better put the matter a little more clearly. You see the same phenomenon in this top. If the top is precessing too fast for the force of gravity the top rises, and the precession diminishes in consequence; the precession being now too slow to balance gravity, the top falls a little and theprecession increases again, and this sort of vibration about a mean position goes on just as the vibration of a pendulum goes on till friction destroys it, and the top precesses more regularly in the mean position. This nodding is more evident in the nearly horizontal balanced gyrostat than in a top, because in a top the turning effect of gravity is less in the higher positions.
When scientific men try to popularize their discoveries, for the sake of making some fact very plain they will often tell slight untruths, making statements which become rather misleading when their students reach the higher levels. Thus astronomers tell the public that the earth goes round the sun in an elliptic path, whereas the attractions of the planets cause the path to be only approximately elliptic; and electricians tell the public that electric energy is conveyed through wires, whereas it is really conveyed by all other space than that occupied by the wires. In this lecture I have to some small extent taken advantage of you in this way; for example, at first you will remember, I neglected the nodding or wobbling produced when an impulse is given to a top or gyrostat, and, all through, I neglect the fact that the instantaneous axis of rotation is only nearly coincident with the axis of figure of a precessing gyrostat or top. And indeed you may generallytake it that if all one's statements were absolutely accurate, it would be necessary to use hundreds of technical terms and involved sentences with explanatory, police-like parentheses; and to listen to many such statements would be absolutely impossible, even for a scientific man. You would hardly expect, however, that so great a scientific man as the late Professor Rankine, when he was seized with the poetic fervour, would err even more than the popular lecturer in making his accuracy of statement subservient to the exigencies of the rhyme as well as to the necessity for simplicity of statement. He in his poem,The Mathematician in Love, has the following lines—
"The lady loved dancing;—he therefore appliedTo the polka and waltz, an equation;But when to rotate on his axis he tried,His centre of gravity swayed to one side,And he fell by the earth's gravitation."
"The lady loved dancing;—he therefore appliedTo the polka and waltz, an equation;But when to rotate on his axis he tried,His centre of gravity swayed to one side,And he fell by the earth's gravitation."
"The lady loved dancing;—he therefore applied
To the polka and waltz, an equation;
But when to rotate on his axis he tried,
His centre of gravity swayed to one side,
And he fell by the earth's gravitation."
Now I have no doubt that this is as good "dropping into poetry" as can be expected in a scientific man, and ——'s science is as good as can be expected in a man who calls himself a poet; but in both cases we have illustrations of the incompatibility of science and rhyming.
Fig. 17Fig. 17.
The motion of this gyrostat can be made even more complicated than it was when we hadnutation and precession, but there is really nothing in it which is not readily explainable by the simple principles I have put before you. Look, for example, at this well-balanced gyrostat (Fig. 17). When I strike this inner gymbal ring in any way you see that it wriggles quickly just as if it were a lump of jelly, its rapid vibrations dying away just like the rapid vibrations of any yielding elastic body. This strange elasticity is of very great interest when we consider it in relation to the molecular properties of matter. Here again (Fig. 30) we have an example which is even more interesting. I have supported the casedgyrostat of Figs. 5 and 6 upon a pair of stilts, and you will observe that it is moving about a perfectly stable position with a very curious staggering kind of vibratory motion; but there is nothing in these motions, however curious, that you cannot easily explain if you have followed me so far.
Fig. 30Fig. 30.
Some of you who are more observant than the others, will have remarked that all these precessing gyrostats gradually fall lower and lower, just as they would do, only more quickly, if they were not spinning. And if you cast your eye upon the third statement of our wall sheet (p.49) you will readily understand why it is so.
"Delay the precession and the body falls, as gravity would make it do if it were not spinning."Well, the precession of every one of these is resisted by friction, and so they fall lower and lower.
I wonder if any of you have followed me so well as to know already why a spinning top rises. Perhaps you have not yet had time to think it out, but I have accentuated several times the particular fact which explains this phenomenon. Friction makes the gyrostats fall, what is it that causes a top to rise? Rapid rising to the upright position is the invariable sign of rapid rotation in a top, and I recollect that when quite vertical we used to say, "She sleeps!" Such was the endearing way in which the youthful experimenter thought of the beautiful object of his tender regard.
All so well known as this rising tendency of a top has been ever since tops were first spun, I question if any person in this hall knows the explanation, and I question its being known to more than a few persons anywhere. Any great mathematician will tell you that the explanation is surely to be found published inRouth, or that at all events he knows men at Cambridge who surely know it, and he thinks that he himself must have known it, although he has now forgotten those elaborate mathematical demonstrations which he once exercised his mind upon. I believe that all such statements are made in error, but I cannotbe sure.[6]A partial theory of the phenomenon was given by Mr. Archibald Smith in theCambridge Mathematical Journalmany years ago, but the problem was solved by Sir William Thomson and Professor Blackburn when they stayed together one year at the seaside, reading for the great Cambridge mathematical examination. It must have alarmed a person interested in Thomson's success to notice that the seaside holiday was really spent by him and his friend in spinning all sorts of rounded stones which they picked up on the beach.
And I will now show you the curious phenomenon that puzzled him that year. This ellipsoid (Fig. 31) will represent a waterworn stone. It is lying in its most stable state on the table, and I give it a spin. You see that for a second or two it was inclined to go on spinning about the axis A A, but it began to wobble violently, and after a while, when these wobbles stilled, you saw that it was spinning nicely with its axis B B vertical; but then a new series of wobblings began and became more violent, and when they ceased you saw that the object had at length reached a settled state ofspinning, standing upright upon its longest axis. This is an extraordinary phenomenon to any person who knows about the great inclination of this body to spin in the very way in which I first started it spinning. You will find that nearly any rounded stone when spun will get up in this way upon its longest axis, if the spin is only vigorous enough, and in the very same way this spinning top tends to get more and more upright.
Fig. 31Fig. 31.
I believe that there are very few mathematical explanations of phenomena which may not be given in quite ordinary language to people who have an ordinary amount of experience. In most cases the symbolical algebraic explanation must be given first by somebody, and then comes the time for its translation into ordinary language. This is the foundation of the new thing called Technical Education, which assumes that aworkman may be taught the principles underlying the operations which go on in his trade, if we base our explanations on the experience which the man has acquired already, without tiring him with a four years' course of study in elementary things such as is most suitable for inexperienced children and youths at public schools and the universities.
Fig. 32Fig. 32.
Fig. 33Fig. 33.
With your present experience the explanation of the rising of the top becomes ridiculously simple. If you look at statementtwoon this wall sheet (p. 48) and reflect a little, some of you will be able, without any elaborate mathematics, to give the simple reason for this that Thomson gave me sixteen years ago. "Hurry on the precession, and the body rises in opposition to gravity." Well, as I am not touching the top, and as the body does rise, we look at once for something that is hurrying on the precession, and we naturally look to the way in which its peg is rubbing on the table, for, with the exception of the atmosphere this top is touching nothing else than the table. Observe carefully how any of these objects precesses. Fig. 32 shows the way in which a top spins. Looked at from above, if the top is spinning in the direction of the hands of a watch, we know from the fourth statement of our wall sheet, or by mere observation, that it also precesses in the direction of the handsof a watch; that is, its precession is such as to make the peg roll at B into the paper. For you will observe that the peg is rolling round a circular path on the table, G being nearly motionless, and the axis A G A describing nearly a cone in space whose vertex is G, above the table. Fig. 33shows the peg enlarged, and it is evident that the point B touching the table is really like the bottom of a wheel B B', and as this wheel is rotating, the rotation causes it to rollintothe paper, away from us. But observe that its mere precession is making it rollintothe paper, and that the spin if great enough wants to roll the top faster than the precession lets it roll, so that it hurries on the precession, and therefore the top rises. That is the simple explanation; the spin, so long as it isgreat enough, is always hurrying on the precession, and if you will cast your recollection back to the days of your youth, when a top was supported on your hand as this is now on mine (Fig. 34), and the spin had grown to be quite small, and was unable to keep the top upright, you will remember that you dexterously helped the precession by giving your hand a circling motion so as to get from your top the advantages as to uprightness of a slightly longer spin.
Fig. 34Fig. 34.
I must ask you now by observation, and the application of exactly the same argument, to explain the struggle for uprightness on its longer axis of any rounded stone when it spins on a table. I may tell you that some of these large rounded-looking objects which I now spin before you in illustration, are made hollow, and they are either of wood or zinc, because I have not the skill necessary to spin large solid objects, and yet I wanted to have objects which you would be able to see. This small one (Fig. 31) is the largest solid one to which my fingers are able to give sufficient spin. Here is a very interesting object (Fig. 35), sphericalin shape, but its centre of gravity is not exactly at its centre of figure, so when I lay it on the table it always gets to its position of stable equilibrium, the white spot touching the table as at A. Some of you know that if this sphere is thrown into the air it seems to have very curious motions, because one is so apt to forget that it is the motion of its centre of gravity which follows a simple path, and the boundary is eccentric to the centre of gravity. Its motions when set to roll upon a carpet are also extremely curious.
Fig. 35Fig. 35.
Now for the very reasons that I have already given, when this sphere is made to spin on the table, it always endeavours to get its white spot uppermost, as in C, Fig. 35; to get into the position in which when not spinning it would be unstable.
Fig. 36Fig. 36.
The precession of a top or gyrostat leads us at once to think of the precession of the great spinning body on which we live. You know that the earthspins on its axis a little more than once every twenty-four hours, as this orange is revolving, and that it goes round the sun once in a year, as this orange is now going round a model sun, or as is shown in the diagram (Fig. 36). Its spinning axis points in the direction shown, very nearly to the star which is called the pole star, almost infinitely far away. In the figure and model I have greatly exaggerated the elliptic nature of the earth's path, as is quite usual, although it may be a little misleading, because the earth's path is much more nearly circular than many people imagine. As a matter of fact the earth is about three million miles nearer the sun in winter than it is in summer. This seems at first paradoxical, but we get to understand it when we reflect that, because of the slope of the earth's axis to the ecliptic, we people who live in the northern hemisphere have the sun less vertically above us, and have a shorter day in the winter, and hence each square foot of our part of the earth's surface receives much less heat every day, and so we feel colder. Now in about 13,000 years the earth will have precessed just half a revolution (seeFig. 38); the axis will then be sloped towards the sun when it is nearest, instead of away from it as it is now; consequently we shall be much warmer in summer and colder in winter than we are now. Indeed we shall then be much worse off than the southernhemisphere people are now, for they have plenty of oceanic water to temper their climate. It is easy to see the nature of the change from figures 36, 37, and 38, or from the model as I carry the orange and its symbolic knitting-needle round the model sun. Let us imagine an observer placed above this model, far above the north pole of the earth. He sees the earth rotating against the direction of the hands of a watch, and he finds that it precesses with the hands of a watch, so that spin and precession are in opposite directions. Indeed it is because of this that we have the word "precession," which we now apply to the motion of a top, although the precession of a top is in the same direction as that of the spin.
Fig. 37Fig. 37.
Fig. 38Fig. 38.
The practical astronomer, in explaining theluni-solar precession of the equinoxesto you, will not probably refer to tops or gyrostats. He will tell you that thelongitudeandright ascensionof a star seem to alter; in fact that the point on the ecliptic from which he makes his measurements, namely, the spring equinox, is slowly travelling round the ecliptic in a direction opposite to that of the earth in its orbit, or to the apparent path of the sun. The spring equinox is to him for heavenly measurements what the longitude of Greenwich is to the navigator. He will tell you that aberration of light, and parallax of the stars,but more than both, this precession of the equinoxes, are the three most important things which prevent us from seeing in an observatory by transit observations of the stars, that the earth is revolving with perfect uniformity. But his way of describing the precession must not disguise for you the physical fact that his phenomenon and ours are identical, and that to us who are acquainted with spinning tops, the slow conical motion of a spinning axis is more readily understood than those details of his measurements in which an astronomer's mind is bound up, and which so often condemn a man of great intellectual power to the life of drudgery which we generally associate with the idea of the pound-a-week cheap clerk.
Fig. 22Fig. 22.
The precession of the earth is then of the same nature as that of a gyrostat suspended above its centre of gravity, of a body which would be stable and not top-heavy if it were not spinning. In fact the precession of the earth is of the same nature as that of this large gyrostat (Fig. 22), which is suspended in gymbals, so that it has a vibration like a pendulum when not spinning. I will now spin it, so that looked at from above it goes against the hands of a watch, and you observe that it precesses with the hands of a watch. Here again is a hemispherical wooden ship, in which there is a gyrostat with its axis vertical. It is in stableequilibrium. When the gyrostat is not spinning, the ship vibrates slowly when put out of equilibrium; when the gyrostat is spinning the ship gets a motion of precession which is opposite in direction to that of the spinning. Astronomers, beginning with Hipparchus, have made observations of the earth's motion for us, and we have observed the motions of gyrostats, and we naturally seek for an explanation of the precessional motion of the earth. The equator of the earth makes an angle of 23½° with the ecliptic, which is the plane of the earth's orbit. Or the spinning axis of the earth is always at angle of 23½° with a perpendicular to the ecliptic, and makes a complete revolution in 26,000 years. The surface of the water on which this wooden ship is floating represents the ecliptic. The axisof spinning of the gyrostat is about 23½° to the vertical; the precession is in two minutes instead of 26,000 years; and only that this ship does not revolve in a great circular path, we should have in its precession a pretty exact illustration of the earth's precession.
The precessional motion of the ship, or of the gyrostat (Fig. 22), is explainable, and in the same way the earth's precession is at once explained if we find that there are forces from external bodies tending to put its spinning axis at right angles to the ecliptic. The earth is a nearly spherical body. If it were exactly spherical and homogeneous, the resultant force of attraction upon it, of a distant body, would be in a line through its centre. And again, if it were spherical and non-homogeneous, but if its mass were arranged in uniformly dense, spherical layers, like the coats of an onion. But the earth is not spherical, and to find what is the nature of the attraction of a distant body, it has been necessary to make pendulum observations all over the earth. You know that if a pendulum does not alter in length as we take it about to various places, its time of vibration at each place enables the force of gravity at each place to be determined; and Mr. Green proved that if we know the force of gravity at all places on the surface of the earth, although we may know nothing about thestate of the inside of the earth, we can calculate with absolute accuracy the force exerted by the earth on matter placed anywhere outside the earth; for instance, at any part of the moon's orbit, or at the sun. And hence we know the equal and opposite force with which such matter will act on the earth. Now pendulum observations have been made at a great many places on the earth, and we know, although of course not with absolute accuracy, the attraction on the earth, of matter outside the earth. For instance, we know that the resultant attraction of the sun on the earth is a force which does not pass through the centre of the earth's mass. You may comprehend the result better if I refer to this diagram of the earth at midwinter (Fig. 39), and use a popular method of description. A and B may roughly be called the protuberant parts of the earth—that protuberant belt of matter which makes theearth orange-shaped instead of spherical. On the spherical portion inside, assumed roughly to be homogeneous, the resultant attraction is a force through the centre.
Fig. 39Fig. 39.
I will now consider the attraction on the protuberant equatorial belt indicated by A and B. The sun attracts a pound of matter at B more than it attracts a pound of matter at A, because B is nearer than A, and hence the total resultant force is in the direction M N rather than O O, through the centre of the earth's mass. But we know that a force in the direction M N is equivalent to a force O O parallel to M N, together with a tilting couple of forces tending to turn the equator edge on to the sun. You will get the true result as to the tilting tendency by imagining the earth to be motionless, and the sun's mass to be distributed as a circular ring of matter 184 millions of miles in diameter, inclined to the equator at 23½°. Under the influence of the attraction of this ring the earth would heave like a great ship on a calm sea, rolling very slowly; in fact, making one complete swing in about three years. But the earth is spinning, and the tilting couple or torque acts upon it just like the forces which are always tending to cause this ship-model to stand upright, and hence it has a precessional motion whose complete period is 26,000 years. When there is no spin in the ship, its complete oscillation takes place in three seconds, andwhen I spin the gyrostat on board the ship, the complete period of its precession is two minutes. In both cases the effect of the spin is to convert what would be an oscillation into a very much slower precession.
There is, however, a great difference between the earth and the gyrostat. The forces acting on the top are always the same, but the forces acting on the earth are continually altering. At midwinter and midsummer the tilting forces are greatest, and at the equinoxes in spring and autumn there are no such forces. So that the precessional motion changes its rate every quarter year from a maximum to nothing, or from nothing to a maximum. It is, however, always in the same direction—the direction opposed to the earth's spin. When we speak then of the precessional motion of the earth, we usually think of the mean or average motion, since the motion gets quicker and slower every quarter year.
Further, the moon is like the sun in its action. It tries to tilt the equatorial part of the earth into the plane of the moon's orbit. The plane of the moon's orbit is nearly the same as that of the ecliptic, and hence the average precession of the earth is of much the same kind as if only one of the two, the moon or the sun, alone acted. That is, the general phenomenon of precession of theearth's axis in a conical path in 26,000 years is the effect of the combined tilting actions of the sun and moon.
You will observe here an instance of the sort of untruth which it is almost imperative to tell in explaining natural phenomena. Hitherto I had spoken only of the sun as producing precession of the earth. This was convenient, because the plane of the ecliptic makes always almost exactly 23½° with the earth's equator, and although on the whole the moon's action is nearly identical with that of the sun, and about twice as great, yet it varies considerably. The superior tilting action of the moon, just like its tide-producing action, is due to its being so much nearer us than the sun, and exists in spite of the very small mass of the moon as compared with that of the sun.
As the ecliptic makes an angle of 23½° with the earth's equator, and the moon's orbit makes an angle 5½° with the ecliptic, we see that the moon's orbit sometimes makes an angle of 29° with the earth's equator, and sometimes only 18°, changing from 29° to 18°, and back to 29° again in about nineteen years. This causes what is called "Nutation," or the nodding of the earth, for the tilting action due to the sun is greatly helped and greatly modified by it. The result of the variable nature of the moon's action is then that the earth's axisrotates in an elliptic conical path round what might be called its mean position. We have also to remember that twice in every lunar month the moon's tilting action on the earth is greater, and twice it is zero, and that it continually varies in value.
On the whole, then, the moon and sun, and to a small extent the planets, produce the general effect of a precession, which repeats itself in a period of about 25,695 years. It is not perfectly uniform, being performed at a speed which is a maximum in summer and winter; that is, there is a change of speed whose period is half a year; and there is a change of speed whose period is half a lunar month, the precession being quicker to-night than it will be next Saturday, when it will increase for about another week, and diminish the next. Besides this, because of 5½° of angularity of the orbits, we have something like the nodding of our precessing gyrostat, and the inclination of the earth's axis to the ecliptic is not constant at 23½°, but is changing, its periodic time being nineteen years. Regarding the earth's centre as fixed at O we see then, as illustrated in this model and in Fig. 40, the axis of the earth describes almost a perfect circle on the celestial sphere once in 25,866 years, its speed fluctuating every half year and every half month. But it is not a perfect circle, it is really a wavyline, there being a complete wave every nineteen years, and there are smaller ripples in it, corresponding to the half-yearly and fortnightly periods. But the very cause of the nutation, the nineteen-yearly period of retrogression of the moon's nodes, as it is called, is itself really produced as the precession of a gyrostat is produced, that is, by tilting forces acting on a spinning body.