The Solar System—Orbits nearly Plane—Satellites, Saturn’s Ring, Spiral NebulÆ—An Explanation of this Tendency of a System towards Flatness—The Energy of a System—Loss of Energy by Collision and Tidal Action—A System within a System—Movements of Translation and Movements of Rotation—The General Law of Conservation of Moment of Momentum—Illustrations of the Principle—The Conception of the Principal Plane—The Utility of this principle arises from its independence of Collisions or Friction—Nature does not do Things infinitely Improbable—The Decline of Energy and the Preservation of Moment of Momentum—Explanation of the Motions in one Plane and in the same Direction—The Satellites of Uranus—The Rotation of Uranus—Why the Orbits are not exactly in the same Plane—The Evolution of a Nebula—The Inevitable Tendency towards the Spiral—The Explanation of the Spiral.

WE have to consider in this chapter the light which the laws of mathematics throw upon certain features which are possessed by a very large number of celestial objects. Let us first describe, as clearly as the circumstances will permit, the nature of these common features to which we now refer, and of which mathematics will suggest the explanation.

We shall begin with our solar system, in which the earth describes an orbit around the sun. That orbit is contained within a plane, which plane passes through the centre of the sun. We may neglect for the present the earth’s occasional slight deviations from this plane which are caused by the attractions of the other planets. If we consider the other bodies of our system, such, for instance, as Venus or Jupiter, we find that the orbit of Venus also lies in a plane, and that plane also passes through the centre of the sun. The orbit of Jupiter is found to be contained within a plane, and it, too, passes through the sun’s centre. Each of the remaining planets in like manner is found to revolve in an orbit which is contained in a plane, and all these planes have one common point, that point being the centre of the sun.

It is a remarkable fact that the mutual inclinations are very small, so that the several planes are nearly coincident. If we take the plane of our earth’s orbit, which we call the ecliptic, as the standard, then the greatest inclination of the orbit of any other important planet is seven degrees, which is found in the case of Mercury. The inclinations to the ecliptic of the planes of the orbits of a few of the asteroids are much more considerable; to take an extreme case, the orbit of Pallas is inclined at an angle of no less than thirty-four degrees. It must, however, be remembered that the asteroids are very small objects, as the collective masses of the five hundred which are at present known would amount to no more than an unimportant fraction of the mass of one of the great planets of our system. Three-fourths of the asteroids have inclinations under ten degrees. We may, therefore, leave these bodies out of consideration for the present, though we may find occasion to refer to them again later on. Still less need we pay attention at present to the comets, for though these bodies belong to our system, and though they move in plane orbits, which like the orbits of the planets pass through the centre of the sun, yet their orbits are inclined at angles of very varying magnitudes. Indeed, we cannot detect any tendency in the orbits of comets to approximate to the plane of the ecliptic. The masses of comets are, however, inconsiderable in comparison with the robust globes which form the planets, while the origin of comets has been apparently so different from that of the planets, that we may leave them out of consideration in our present argument. There is nothing in the motion of either asteroids or comets to invalidate the general proposition which affirms, that the planes of the orbits of the heaviest and most important bodies in the solar system are very nearly coincident.

Many of the planets are accompanied by satellites, and these satellites revolve round the planets, just as the planet accompanied by its satellites revolves round the sun. The orbit of each satellite is contained within a plane, and that plane passes through the centre of the planet to which it is appended. We thus have a system of planes appropriate to the satellites, just as there is a system of planes appropriate to the planets. The orbits of the satellites of each planet are very nearly in the same plane, with notable exceptions in the cases of Uranus and Neptune, which it will be necessary to consider at full length later on. This plane is very nearly coincident with the planes in which the planets themselves move. Omitting the exceptions, which are unimportant as to magnitude, though otherwise extremely interesting and instructive, the fundamental characteristic of the movements of the principal bodies in our system is that their orbits are nearly parallel to the same plane. We draw an average plane through these closely adjacent planes and we term it the principal plane of our system. It is not, indeed, coincident with the plane of the orbit of any one planet, yet the actual plane of the orbit of every important planet, and of the important satellites, lies exceedingly close to this principal plane. This is a noteworthy circumstance in the arrangement of the planetary system, and we expect that it must admit of some physical explanation.

When we look into the details of the planetary groups composing the solar system, we find striking indications of the tendency of the orbits of the bodies in each subordinate system to become adjusted to a plane. The most striking instance is that exhibited by the Rings of Saturn. It has been demonstrated that these wonderful rings are composed of myriads of separate particles. Each of these particles follows an independent orbit round Saturn. Each such orbit is contained in a plane, and all these planes appear, so far as our observations go, to be absolutely coincident. It is further to be noted that the plane, thus remarkably related to the system of rings revolving around Saturn, is substantially identical with the plane in which the satellites of Saturn themselves revolve, and this plane again is inclined at an angle no greater than twenty-eight degrees to the plane of the ecliptic, and close to that in which Saturn itself revolves around the sun.

Overlooking, as we may for the present, the varieties in detail which such natural phenomena present, we may say that the most noticeable characteristic of the revolutions in the solar system is expressed by the statement that they lie approximately in the same plane.

We shall also find that this tendency of the movements in a system to range themselves in orbits which lie in the same plane, is exhibited in other parts of the universe. Let us consider from this point of view the spiral nebulÆ, those remarkable objects which, in the last chapter, we have seen to be so numerous and so characteristic. It is obvious that a spiral nebula must be a flat object. Its thickness is small in comparison with its diameter. When a spiral nebula is looked at edgewise (Fig. 45), then it seems long and thin, so much so that it presents the appearance of a ray such as we have shown in Fig. 33, which represents a type of object very familiar to those astronomers who are acquainted with nebulÆ. The characteristics of these objects seem consistent only with the supposition that there is a tendency in the materials which enter into a spiral nebula to adapt their movements to a particular plane, just as there is a tendency for the objects in Saturn’s ring to remain in a particular plane, and just as there has been a tendency among the bodies belonging to the solar system themselves to revolve in a particular plane. Remembering also that there seems excellent reason to believe that spiral nebulÆ exhibiting this characteristic are to be reckoned in scores of thousands, it is evident that the fundamental feature in which they all agree must be one of very great importance in the universe.

We may mention yet one more illustration of the remarkable tendency, so frequently exhibited by an organised system in space, to place its parts ultimately in or near the same plane, or at all events, to assume a shape of which one dimension is small in comparison with the two others. We have, in the last chapter, referred to the Milky Way, and we have alluded to the significance of the obvious fact that, however the mass of stars which form the Milky Way may be arranged, they are so disposed that the thickness of the mass is certainly much less than its two other dimensions. Herschel’s famous illustration of a grindstone to represent the shape of the Milky Way will serve to illustrate the form we are now considering.

When we meet with a characteristic form so widely diffused through the universe, exhibited not only in the systems attending on the single planets, not only in the systems of planets which revolve round a single sun, but also in that marvellous aggregation of innumerable suns which we find in the Milky Way, and in scores of thousands of nebulÆ in all directions, at all distances, and apparently of every grade of importance, we are tempted to ask whether there may not be some physical explanation of a characteristic so universal and so remarkable.

Let us see whether mathematics can provide any suggestion as to the cause of this tendency towards flatness which seems to affect those systems in the universe which are sufficiently isolated to escape from any large disturbance of their parts by outside interference. We must begin by putting, as it were, the problem into shape, and by enumerating certain conditions which, though they may not be absolutely fulfilled in nature, are often so very nearly fulfilled that we make no appreciable error by supposing them to be so.

Let us suppose that a myriad bodies of various sizes, shapes, materials and masses, are launched in space in any order whatever, at any distances from each other, and that they are started with very different movements. Some may be going very fast, some going slowly, or not at all; some may be moving up or down or to the right or to the left—there may be, in fact, every variety in their distances and their velocities, and in the directions in which they are started.

We assume that each pair of masses attract each other by the well-known law of gravitation, which expresses that the force between any two bodies is proportional directly to the product of their masses and inversely to the square of their distance. We have one further supposition to make, and it is an important one. We shall assume that though each one of the bodies which we are considering is affecting all the others, and is in turn affected by them, yet that they are subjected to no appreciable disturbing influence from other bodies not included in the system to which they belong. This may seem at first to make the problem we are about to consider a purely imaginary one, such as could only be applicable to systems different from those which are actually presented to us in nature. It must be admitted that the condition we have inferred can only be approximately fulfilled. But a little consideration will show that the supposition is not an unreasonable one. Take, for instance, the solar system, consisting of the sun, the planets, and their satellites. Every one of these bodies attracts every other body, and the movement of each of the bodies is produced by the joint effects of the forces exerted upon it by all the others. Assuredly this gives a problem quite difficult enough for all the resources that are at our command. But in such investigations we omit altogether the influence of the stars. Sirius, for example, does exercise some attraction on the bodies of our system, but owing to its enormous distance, in comparison with the distances in our solar system, the effect of the disturbance of Sirius on the relative movements of the planets is wholly inappreciable. Indeed, we may add that the disturbances in the solar system produced by all the stars, even including the myriads of the Milky Way, are absolutely negligible. The movements in our solar system, so far as our observations reveal them, are performed precisely as if all bodies of the universe foreign to the solar system were non-existent. This consideration shows that in the problem we are now to consider, we are introducing no unreasonable element when we premise that the system whose movements we are to investigate is to be regarded as free from appreciable disturbance by any foreign influence.

To follow the fortunes of a system of bodies, large or small, starting under any arbitrary conditions at the commencement, and then abandoned to their mutual attractions, is a problem for the mathematician. It certainly presents to him questions of very great difficulty, and many of these he has to confess are insoluble; there are, however, certain important laws which must be obeyed in all the vicissitudes of the motion. There are certain theorems known to the mathematician which apply to such a system, and it is these theorems which afford us most interesting and instructive information. I am well aware that the subject upon which I am about to enter is not a very easy one, but its importance is such that I must make the effort to explain it.

Let me commence by describing what is meant when we speak of the energy of a system. Take, first, the case of merely two bodies, and let us suppose that they were initially at rest. The energy of a system of this very simple type is represented by the quantity of work which could be done by allowing these two bodies to come together. If, instead of being in the beginning simply at rest, the bodies had each been in motion, the energy of the system would be correspondingly greater. The energy of a moving body, or its capacity of doing work in virtue of its movement, is proportional jointly to its mass and to the square of its velocity. The energy of the two moving bodies will therefore be represented by three parts; first, there will be that due to their distance apart; secondly, there will be that due to the velocity of one of them; and, thirdly, there is that due to the velocity of the other. In the case of a number of bodies, the energy will consist in the first place of a part which is due to the separation of the bodies, and measured by the quantity of work that would be produced if, in obedience to their mutual attraction, all the bodies were allowed to come together into one mass. In the second place, the bodies are to be supposed to have been originally started with certain velocities, and the energy of each of the bodies, in virtue of its motion, is to be measured by the product of one-half its mass into the square of its velocity. The total energy of the system consists, therefore, of the sum of the parts due to the velocities of the bodies, and that which is due to their mutual separation.

If the bodies could really be perfectly rigid, unyielding masses, so that they have no movements analogous to tides, and if their movements be such that collisions will not take place among them, then the laws of mechanics tell us that the quantity of energy in that system will remain for ever unaltered. The velocities of the particles may vary, and the mutual distances of the particles may vary, but those variations will be always conducted, subject to the fundamental condition that if we multiply the square of the velocity of each body by one-half its mass, and add all those quantities together, and if we increase the sum thus obtained by the quantity of energy equivalent to the separation of the particles, the total amount thus obtained is constant. This is the fundamental law of mechanics known as the conservation of energy.

For such material systems as the universe presents to us, the conservation of energy, in the sense in which I have here expressed it, will not be maintained; for the necessary conditions cannot be fulfilled. Let us suppose that the incessant movements of the bodies in the system, rushing about under the influence of their mutual attractions, has at last been productive of a collision between two of the bodies. We have already explained in Chapter VI. how in the collision of two masses the energy which they possess in virtue of their movements may be to a large extent transformed into heat; there is consequently an immediate increase in the temperature of the bodies concerned, and then follows the operation of that fundamental law of heat, by which the excess of heat so arising will be radiated away. Some of it will, no doubt, be intercepted by falling on other bodies in the system, and the amount that might be thus possibly retained would, of course, not be lost to the system. The bodies of the solar system at least are so widely scattered, that the greater part of the heat would certainly escape into space, and the corresponding quantity of energy would be totally lost to the system. We may generally assume that a collision among the bodies would be most certainly productive of a loss of energy from the system.

No doubt collisions can hardly be expected to occur in a system consisting of large, isolated bodies like the planets. Even in any system of solid bodies collisions may be presumed to be infrequent in comparison with the numbers of the bodies. But if, instead of a system of few bodies of large mass, we have a gas or nebula composed of innumerable atoms or molecules, the collisions would be by no means infrequent, and every collision, in so far as it led to the production of heat, would be productive of loss of energy by radiation from the system.

It should also be added that, even independently of actual collisions, there is, and must be, loss of energy in the system from other causes. There are no absolutely rigid bodies known in nature, for the hardest mineral or the toughest steel must yield to some extent when large forces are applied to it, and as the bodies in the system are not mere points or particles of inconsiderable dimensions, they will experience stresses something like those to which our earth is subjected in that action of the moon and sun which produces the tides. In consequence of the influences of each body on the rest, there will be certain relative changes in the parts of each body; there will be, as it were, tidal movements in their liquid parts and even in their solid substance. These tides will produce friction, and this will produce heat. This heat will be radiated from the system, but the heat radiated corresponds to a certain amount of energy; the energy is therefore lost to the system, so that even without actual collisions we still find that energy must be gradually lost to the system.

Thus we have been conducted to an important conclusion, which may be stated in the following way. Let there be any system of bodies, subject to their mutual attractions, and sufficiently isolated from the disturbing influence of all bodies which do not belong to the system, then the original energy with which that system is started must be undergoing a continual decline. It must at least decline until such a condition of the system has been reached that collisions are no longer possible and that tidal influences have ceased. These conditions might be fulfilled if all the bodies of the system coalesced into a single mass.

As illustrations of the systems we are now considering, we may take the sun and planets as a whole. A spiral nebula is a system in the present sense, while perhaps the grandest illustration of all is provided by the Milky Way.

It will be noted that we may have a system which is isolated so far as our present argument is concerned, even while it forms a part of another system of a higher order of magnitude. For instance, Saturn with his rings and satellites is sufficiently isolated from the rest of the solar system and the rest of the universe, to enable us to trace the consequences of the gradual decline of energy in his attendant system. The solar system in which Saturn appears merely as a unit, is itself sufficiently isolated from the stars in the Milky Way to permit us to study the decline of energy in the solar system, without considering the action of those stars.

This general law of the decline of energy in an isolated system, is supplemented by another law often known as the conservation of moment of momentum. It may at first seem difficult to grasp the notion which this law involves. The effort is, however, worth making, for the law in question is of fundamental importance in the study of the mechanics of the universe. In the Appendix will be found an investigation by elementary geometry of the important mechanical principles which are involved in this subject.

Whatever may have been the origin of the primÆval nebula, and whatever may have been the forces concerned in its production we may feel confident that it was not originally at rest. We do not indeed know any object which is at rest. Not one of the heavenly bodies is at rest, nothing on earth is at rest, for even the molecules of rigid matter are in rapid motion. Rest seems unknown in the universe. It would be, therefore, infinitely improbable that a primÆval nebula, whatever may have been the agency by which it was started on that career which we are considering, was initially in a condition of absolute rest. We assume without hesitation that the nebula was to some extent in motion, and we may feel assured that the motions were of a highly complicated description. It is fortunate for us that our argument does not require us to know the precise character of the movements, as such knowledge would obviously be quite unattainable. We can, however, invoke the laws of mechanics as an unerring guide. They will tell us not indeed everything about those motions, but they will set forth certain characteristics which the movements must have had, and these characteristics suffice for our argument.

To illustrate the important principle on which we are now entering I must mention the famous problem of three bodies which has engaged the attention of the greatest mathematicians. Let there be a body A, and another B, and another C. We shall suppose that these bodies are so small that they may be regarded merely as points in comparison with the distances by which they are separated. We shall suppose that they are all moving in the same plane, and we shall suppose that each of them attracts the others, but that except these attractions there are no other forces in the system. To discover all about the motions of these bodies is so difficult a problem that mathematicians have never been able to solve it. But though we are not able to solve the problem completely, we can learn something with regard to it.

We represent by arrows in Fig. 36 the directions in which A, B, and C are moving at the moment. We choose any point O in the plane, and for simplicity we have so drawn the figure that A, B, and C are forces tending to turn round O in the same direction. The velocity of a body multiplied into its mass is termed the momentum of the body. Draw the perpendicular from O to the direction in which the body A is moving, then the product of this perpendicular and the momentum of A is called the moment of momentum of A around O. In like manner we form the moment of momentum of B and C, and if we add them together we obtain the total moment of momentum of the system.

We can now give expression to a great discovery which mathematicians have made. No matter how complicated may be the movements of A, B, and C; no matter to what extent these particles approximate or how widely they separate; no matter what changes may occur in their velocities, or even what actual collision may take place, the sum of the moments of momentum must remain for ever unaltered. This most important principle in dynamics is known as the conservation of moment of momentum.

Though I have only mentioned three particles, yet the same principle will be true for any number. If it should happen that any of them are turning round O in the opposite direction, then their moments of momentum are to be taken as negative. In this case we add the moments tending in one direction together; and then subtract all the opposite moments. The remainder is the quantity which remains constant.

We may state this principle in a somewhat different manner as follows: Let us consider a multitude of particles in a plane; let them be severally started in any directions in the plane, and then be abandoned to their mutual attractions, it being understood that there are no forces produced by bodies external to the system; if we then choose any point in the plane, and measure the areas described round that point by the several moving bodies in one second, and if we multiply each of, those areas by the mass of the corresponding body, then, if all the bodies are moving in the same direction round the point, the sum of the quantities so obtained is constant. It will be the same a hundred or a thousand years hence as it is at the present moment, or as it was a hundred or a thousand years ago. If any of the particles had been turning round the point in the opposite direction, then the products belonging to such particles are to be subtracted from the others instead of added.

We have now to express in a still more general manner the important principle that is here involved. Let us consider any system of attracting particles, no matter what their masses or whether their movements be restricted to a plane or not. Let us start them into motion in any directions and with any initial velocities, and then abandon them to the influence of their mutual attractions, withholding at the same time the interference of any forces from bodies exterior to the system. Draw any plane whatever, and let fall perpendiculars upon this plane from the different particles of the system. It will be obvious that as the particles move the feet of the perpendiculars must move in correspondence with the particles from which the perpendiculars were let fall. We may regard the foot of every perpendicular as the actual position of a moving point, and it can be proved that if the mass of each particle be multiplied into the area which the foot of its perpendicular describes in a second round any point in the plane, and then be added to the similar products from all the other particles, only observing the proper precautions as to sign, the sum will remain constant, i.e., in any other second the total quantity arrived at will be exactly the same. This is a general law of dynamics. It is not a law of merely approximate truth, it is a law true with absolute accuracy during unlimited periods of time.

The actual value of the constant will depend both on the system and on the plane. For a given system the constant will differ for the different planes which may be drawn, and there will be some planes in which that sum will be zero. In other words, in those planes the areas described by the feet of the perpendiculars, multiplied by the masses of the particles which are moving in one way, will be precisely equal to the similar sum obtained from the particles moving in the opposite direction.

But among all possible planes there is one of special significance in its relation to the system. It is called the “principal plane,” and it is characterised by the fact that the sum (with due attention to sign) of the areas described each second by the feet of the perpendiculars, multiplied into the masses of the corresponding particle, is greater than the like magnitude for any other plane, and is thus a maximum. For all planes parallel to this principal plane, the result will be, of course, the same; it is the direction of the plane and not its absolute situation that is material. We thus see that while this remarkable quantity is constant in any plane, for all time, yet the actual value of that constant depends upon the aspect of the plane; for some planes it is zero, for others the constant has intermediate values, and there is one plane for which the constant is a maximum. This is the principal plane, and a knowledge of it is of vital importance in endeavouring to understand the nebular theory. Nor are the principles under consideration limited only to a system consisting of sun and planets; they apply, with suitable modifications, to many other celestial systems as well.

The instructive character of this dynamical principle will be seen when we deduce its consequences. The term “moment of momentum” of a particle, with reference to a certain point in a plane, expresses double the product of the rate at which the area is described by the foot of the perpendicular to this plane, multiplied by the mass of the particle. The moment of momentum of the system, with reference to the principal plane, is a maximum in comparison with all other planes; that moment of momentum retains precisely the same value throughout all time, from the first instant the system was started onwards. And it retains this value, no matter what changes or disturbances may happen in the system, provided only that the influence of external forces is withheld. Subject to this condition, the transformations of the system may be any whatever. The several bodies may be forced into wide changes of their orbits, so that there may even be collisions among them; yet, notwithstanding those collisions, and notwithstanding the violent alterations which may be thus produced in the movements of the bodies, the moment of momentum will not alter. No matter what tides may be produced, even if those tides be so great as to produce disruption in the masses and force the orbits to change their character radically, yet the moment of momentum will be conserved without alteration.

It is essential to notice the fundamental difference between the principle which has been called the conservation of energy in the system, and the conservation of moment of momentum. We have pointed out that when collisions take place, part of the energy due to motion is transformed into heat, and energy in that form admits of radiation through space, and thus becomes lost to the system, with the result that the total energy declines. Even without actual collision, we have shown how certain effects of tides, or other consequences of friction, necessarily involve the squandering of energy with which the system was originally endowed. A system started with a certain endowment of energy may conserve that energy indefinitely, if all such actions as collisions or frictions are absent. If collisions or frictions are present the system will gradually dissipate energy. Our interpretation of the future of such a system must always take account of this fundamental fact.

It is, of course, conceivable that the moment of momentum with which a system was originally endowed might have happened to be zero. A system of particles could be so constructed and so started on their movements that their moment of momentum with regard to a certain plane should be zero. It might happen that the moment of momentum of the system with regard to a second plane, perpendicular to the former one, should be also zero; and, finally, that the moment of momentum of the system with regard to a third plane perpendicular to each of the other two, should be also zero. If these three conditions were found to prevail at the commencement, they would prevail throughout the movement, and, more generally still, we may state that in such circumstances the moment of momentum of the system would be zero about any plane whatever. There would be no principal plane in such a system. We thus note that though it is inconceivable that a group of mutually attracting bodies should be started into movement without a suitable endowment of energy, it is yet quite conceivable that a system could be started without having any moment of momentum. And if at the beginning the system had no moment of momentum, then no matter what may be the future vicissitudes of its motion, no moment of momentum can ever be acquired by it to all eternity, so long as the interference of external forces is excluded.

But having said this much as to the conceivability of the initiation of a system with no moment of momentum, we now hasten to add that, so far as Nature is actually concerned, this bare possibility may be set aside as one which is infinitely improbable. Nature does not do things which are infinitely improbable, and, therefore, we may affirm that all material systems, with which we shall have to deal, do possess moment of momentum. However the system may have originated, whatever may have been the actions of forces by which it was brought into being, we may feel assured that the system received at its initiation some endowment of moment of momentum, as well as of energy. Hence we may conclude that every such system as is presented to us in the infinite variety of Nature, must stand in intimate relation to some particular plane, being that which is known as the principal plane of moment of momentum. In our effort to interpret Nature, the physical importance of this fact can hardly be over-estimated.

In a future chapter we shall make some attempt to sketch the natural operations by which individual systems have been started on their careers. Postponing, then, such questions, we propose to deal now with the phenomena which the principles of dynamics declare must accompany the evolution of a system under the action of the exclusive attraction of the various parts of that system for each other. The system commences its career with a certain endowment of energy, with a certain endowment of moment of momentum, and with a certain principal plane to which that moment of momentum is specially related. In the course of the evolution through which, in myriads of ages, the system is destined to pass, the energy that it contains will undergo vast loss by dissipation. On the other hand, the moment of momentum will never vary, and the position of the principal plane will remain the same for all time. We have to consider what features, connected with the evolution, may be attributed to the operation of these dynamical laws. We have, in fact, to deduce the consequences which seem to follow from the fact that, in consequence of collisions, and in consequence of friction, an isolated system in space must gradually part with its initial store of energy, but that, notwithstanding any collisions and any friction, the total moment of momentum of the system suffers no abatement.

As the system advances in development, we have to deal with a gradual decline in the ratio of the original store of energy to the original store of moment of momentum. And hence we must expect that a system will ultimately tend towards a form in which, while preserving its moment of momentum, it shall do so with such a distribution of the bodies of which it consists as shall be compatible with a diminishing quantity of energy. It is not hard to see that in the course of ages this tends, as one consequence, to make the movements of each of the bodies in the system ultimately approximate to movements in a plane.

Let us, for simplicity, begin with the case of three attracting particles, A, B and C. Let B be started in any direction in the plane L, and let A be started in an orbit round it, and in the same plane L. Now let C be started into motion, in any direction, from some point also in L. It is certain that the sum of the areas projected parallel to any plane, which are described in a second by these three bodies, must be constant, each of the areas being, as usual, multiplied by the mass of the corresponding body. Let us specially consider the plane L in which the motions of A and B already lie. It is on this plane that the area described by C has to be projected. The essential point now to remember is that the projected area is less than the actual area. It is plain that if C has to describe a certain projected area in a certain time, the velocity with which C has to move must be greater when C starts off at an inclination to the plane than would have been necessary if C had started in the plane, other things being the same. Thus we see that, if the three bodies were all moving in the same plane, they could, speaking generally, maintain more easily the requisite description of areas, that is, the requisite moment of momentum with smaller velocities than if they were moving in directions which were not so regulated; that is to say, the moment of momentum can be kept up with less energy when the particles move in the same plane.

In a more general manner we see that any system in which the bodies are moving in the same plane will, for equal moment of momentum, require less energy than it would have done had the bodies been moving in directions which were not limited to a plane. Thus we are led to the conclusion that the ultimate result of the collisions and the friction and the tides, which are caused by the action of one particle on another, is to make the movements tend towards the same plane.

In this dynamical principle we have in all probability a physical explanation of that remarkable characteristic of celestial movements to which we have referred. The solar system possesses less energy in proportion to its moment of momentum than it would require to have if the orbits of the important planets, instead of lying practically in the same plane, were inclined at various angles. Whatever may have been the original disposition of the materials forming the solar system, they must once have contained much more energy than they have at present. The moment of momentum in the principal plane, at the beginning, was not, however, different from the moment of momentum that the system now possesses. As the energy of the system gradually declined, the system has gradually been compelled to adjust itself in such a manner that, with the reduced quantity of energy, the requisite moment of momentum shall still be preserved. This is the reason why, in the course of the myriads of ages during which the solar system has been acquiring its present form, the movements have gradually become nearly conformed to a plane.

The operation of the principle, now before us, may be seen in a striking manner in Saturn’s ring. (Fig. 37.) The particles constituting this exquisite object, so far as observations have revealed them, seem to present to us an almost absolutely plane movement. The fact that the movements of the constituents of Saturn’s ring lie in a plane is doubtless to be accounted for by the operation of the fundamental dynamical principle to which we have referred. Saturn, in its great motion round the luminary, is, of course, controlled by the sun, yet the system attached to Saturn is so close to that globe as to be attracted by the sun in a manner which need not here be distinguished from the solar attraction on Saturn itself. It follows that the differential action, so to speak, of the sun on Saturn, and on the myriad objects which constitute its ring, may be disregarded. We are therefore entitled, as already mentioned, to view Saturn and its system as an isolated group, not acted upon by any forces exterior to the system. It is therefore subject to the laws which declare that, though the energy declines, the moment of momentum is to remain unaltered. This it is which has apparently caused the extreme flatness of Saturn’s ring. The energy of the rotation of that system has been expended until it might seem that no more energy has been left than just suffices to preserve the unalterable moment of momentum, under the most economical conditions, so far as energy is concerned.

Let us suppose that one of the innumerable myriads of particles which constitute the ring of Saturn were to forsake the plane in which it now revolves, and move in an orbit inclined to the present plane. We shall suppose that the original track of the orbit was a circle, and we shall assume that in the new plane to which the motion is transferred the motion is also circular. That particle will have still to do its share of preserving the requisite total moment of momentum, for we are to suppose that each of the other particles remains unaltered in its pace and in the other circumstances of its motion. The aberrant particle will describe, in a second, an area which, for the purpose of the present calculation, must be projected upon the plane containing the other particles. The area, when projected, must still be as large as the area that the particle would have described if it had remained in the plane. It is therefore necessary that the area swept over by the particle in the inclined plane, in one second, shall be greater than the area which sufficed in the original plane. This requires the circle in which the particle revolves to be enlarged, and this necessitates that its energy should be increased. In other words, while the moment of momentum was no greater than before, the energy of the system would have to be greater. We thus see that inasmuch as the particles forming the rings of Saturn move in circles in the same plane, they require a smaller amount of energy in the system to preserve the requisite moment of momentum than would be required if they moved in circular orbits which were not in the same plane. In such a system as Saturn’s ring, in which the particles are excessively numerous and excessively close together, it may be presumed that there may once have been sufficient collisions and frictions among the particles to cause the exhaustion of energy to the lowest point at which the moment of momentum would be sustained. In the course of ages this has been accomplished by the remarkable adjustment of the movements to that plane in which we now find them.

The importance of this subject is so great that we shall present the matter in a somewhat different manner as follows: We shall simplify the matter by regarding the orbits of the planets or other bodies as circles The fact that these orbits are ellipses, which are, however, very nearly circles, will not appreciably affect the argument.

Let us, then, suppose a single planet revolving round a fixed sun, in the centre. The energy of this system has two parts. There is first the energy due to the velocity of the planet, and this is found by taking half the product of the mass of the planet and the square of its velocity. The second part of the energy depends, as we have already explained, on the distance of the planet from the sun. The planet possesses energy on account of its situation, for the attraction of the sun on the planet is capable of doing work. The further the planet is from the sun the larger is the quantity of energy that it possesses from this cause. On the other hand, the further the planet is from the sun the smaller is its velocity, and the less is the quantity of energy that it possesses of the first kind. We unite the two parts, and we find that the net result may be expressed in the following manner: If a planet be revolving in a circular path round the sun, then the total energy of that system (apart from any rotation of the sun and planet on their axes), when added to the reciprocal of the distance between the two bodies, measured with a proper unit of length, is the same for all distances of the same two bodies. This shows the connection between the energy and the distance of the planet from the sun.

Thus we see that if the circle is enlarged the energy of the system increases. The moment of momentum of the system is proportional to the square root of the distance of the two bodies. If, therefore, the distance of the two bodies is increased, the moment of momentum increases also.

It will illustrate the application of the argument to take a particular case in which a system of particles is revolving round a central sun in circular orbits, all of which lie in the same plane. Let us suppose that, while the moment of momentum of the system of particles is to remain unaltered, one of the particles is to be shifted into a plane which is inclined at an angle of 60° to the plane of the other orbits; it can easily be seen that an area in the new plane, when projected down into the original plane, will be reduced to half its amount. Hence, as the moment of momentum of the whole system is to be kept up, it will be necessary for the particle to have a moment of momentum in the circle which it describes in the new plane which is double that which it had in the original plane. It follows that the radius of the circle in the new plane must be four times the radius of the circle which defined the orbit of the particle in the old plane. The energy of the particle in this orbit is therefore correspondingly greater, and thus the energy of the whole system is increased. This illustrates how a system, in which the circular orbits are in different planes, requires more energy for a given moment of momentum than would suffice if the circular orbits had all been in the same plane. So long as the orbits are in different planes there will still remain a reserve of energy for possible dissipation. But the dissipation is always in progress, and hence there is an incessant tendency towards a flattening of the system by the mutual actions of its parts.

It may help to elucidate this subject to state the matter as follows: The more the system contracts, the faster it must generally revolve; this is the universal law when disturbing influences are excluded. Take, for instance, the sun, which is at this moment contracting on account of its loss of heat. In consequence of that contraction it is essential that the sun shall gradually turn faster round on its axis. At present the sun requires twenty-five days, four hours and twenty-nine minutes for each rotation. That period must certainly be diminishing, although no doubt the rate of diminution is very slow. Indeed, it is too slow for us to observe; nevertheless, some diminution must be in progress. Applying the same principle to the primitive nebula, we see, that as the contraction of the original volume proceeds, the speed with which the several parts will rotate must increase.

The periodic times of the planets are here instructive. The materials now forming Jupiter were situated towards the exterior of the nebula, so that, as the nebula contracted, it tended to leave Jupiter behind. The period in which Jupiter now revolves round the sun may give some notion of the period of the rotation of the nebula at the time that it extended so far as Jupiter. Subsequently to the formation, and the detachment of Jupiter, a body which was henceforth no longer in contact with the nebula, the latter proceeded further in its contraction. Passing over the intermediate stages, we find the nebula contracting until it extended no further than the line now marked by the earth’s orbit; the speed with which the nebula was rotating must have been increasing all the time, so that though the nebula required several years to go round when it extended as far as Jupiter, only a fraction of that period was necessary when it had reached the position indicated by the earth’s track at the present time. Leaving the earth behind it, just as it had previously left Jupiter, the nebula started on a still further condensation. It drew in, until at last it reached a further stage by contraction into the sun, which rotates in less than a month. Thus the period of Jupiter namely, twelve years, the period of the earth, namely, one year, and the period of the sun, namely, twenty-five days, illustrate the successive accelerations of the rotation of the nebula in the process of contraction. No doubt these statements must be received with much qualification, but they will illustrate the nature of the argument.

We may also here mention the satellites of Uranus, all the more so because it has been frequently urged as an objection to the nebular theory that the orbits of the satellites of Uranus lie in a plane which is inclined at a very large angle; no less than 82° to the general plane of the solar system. I shall refer in a later chapter to this subject, and consider what explanation can be offered with regard to the great inclination of this plane, which is one of the anomalies of our system. For the present I merely draw attention to the fact that the movements of all four satellites of Uranus do actually lie in the same plane, though, as already indicated, it stands nearly at right angles to the ecliptic.

Professor Newcomb has shown that the four satellites of Uranus revolve in orbits which are almost exactly circular, and which, so far as observation shows, are absolutely in the same plane. From our present point of view this is a matter of much interest. Whatever may have been the influence by which this plane departs so widely from the plane of the ecliptic, it seems certain that it must be regarded as having acted at a very early period in the evolution of the Uranian system; and when this system had once started on its course of evolution, the operation of that dynamical principle to which we have so often referred was gradually brought to bear on the orbits of the satellites. We have here another isolated case resembling that of Saturn and its rings. The fundamental law ordained that the moment of momentum of Uranus and its moons must remain constant, though the total quantity of energy in that system should decline. In the course of ages this has led to the adjustment of the orbits of the four satellites into the same plane.

I ought here to mention that the rotation of Uranus on its axis presents a problem which has not yet been solved by telescopic observation. It is extremely interesting to note that, as a rule, the axes on which the important planets rotate are inclined at no great angles to the principal plane of the solar system. The great distance of Uranus has, however, prevented astronomers from studying the rotation of that planet in the ordinary manner, by observation of the displacement of marks on its surface. So far as telescopic observations are concerned, we are therefore in ignorance as to the axis about which Uranus revolves. If, following the analogy of Jupiter, or Saturn, or Mars, or the earth, the rotation of Uranus was conducted about an axis, not greatly inclined from the perpendicular to the ecliptic, then the rotation of Uranus would be about an axis very far from perpendicular to the plane in which its satellites revolve. The analogy of the other planets seems to suggest that the rotation of a planet should be nearly perpendicular to the plane in which its satellites revolve. As the question is one which does not admit of being decided by observation, we may venture to remark that the necessity for a declining ratio of energy to moment of momentum in the Uranian system provides a suggestion. The moment of momentum of a system, such as that of Uranus and its satellites, is derived partly from the movements of the satellites and partly from the rotation of the planet itself. From the illustrations we have already given, it is plain that the requisite moment of momentum is compatible with a comparatively small energy only when the system is so adjusted that the axis of rotation of the planet is perpendicular to the plane in which the satellites revolve, or in other words when the satellites revolve in the plane of the equator of the planet. We do not expect that this condition will be complied with to the fullest extent in any members of the solar system. There is indeed an obvious exception; for the moon, in its revolution about the earth, does not revolve exactly in the earth’s equator. We might, however, expect that the tendency would be for the movements to adjust themselves in this manner. It seems therefore likely that the direction of the axis of Uranus is perpendicular, or nearly so, to the plane of the movements of its satellites.

At this point we take occasion to answer an objection which may perhaps be urged against the doctrine of moment of momentum as here applied. I have shown that the tendency of this dynamical principle is to reduce the movements towards one plane. It may be objected that if there is this tendency, why is it that the movements have not all been brought into the same plane exactly? This has been accomplished in the case of the bodies forming Saturn’s ring, and perhaps in the satellites of Uranus. But why is it that all the great planets of our solar system have not been brought to revolve absolutely in the same plane?

We answer that the operations of the forces by which this adjustment is effected are necessarily extremely slow. The process is still going on, and it may ultimately reach completion. But it is to be particularly observed that the nearer the approach is made to the final adjustment, the slower must be the process of adjustment, and the less efficient are the forces tending to bring it about. For the purpose of illustrating this, we may estimate the efficiency of the forces in flattening down the system in the following manner. Suppose that there are two circular orbits at right angles to each other, and that we measure the efficiency of the action tending to bring the planes to coincide by 100. When the planes are at an angle of thirty degrees the efficiency is represented by 50, and when the inclination is only five degrees the efficiency is no more than 9, and the efficiency gradually lessens as the angle declines. As the angles of inclination of the planes in the solar system are so small, we see that the efficiency of the flattening operation in the solar system must have dwindled correspondingly. Hence we need not be surprised that the final reduction of the orbits into the same plane has not yet been absolutely completed.

Certainly the most numerous, and perhaps the grandest, illustrations of the operation of the great natural principles we have been considering are to be found in the case of the spiral nebulÆ. The characteristic appearance of these objects demands special explanation, and it is to dynamics we must look for that explanation.

As to the original cause of a nebula we shall have something to say in a future chapter. At present we are only considering how, when a nebula has come into existence, the action of known dynamical principles will mould that nebula into form. As an illustration of a nebula, in what we may describe as its comparatively primitive shape, we may take the Great Nebula in Orion. This stupendous mass of vaguely diffused vapour may probably be regarded as in an early stage when contrasted with the spirals. We have already shown how the spectroscopic evidence demonstrates that the famous nebula is actually a gaseous object. It stands thus in marked contrast with many other nebulÆ which, by not yielding a gaseous spectrum, seem to inform us that they are objects which have advanced to a further stage in their development than such masses of mere glowing gas as are found in the splendid object in Orion.

The development of a nebula must from dynamical principles proceed along the lines that we have already indicated. We shall assume that the nebula is sufficiently isolated from surrounding objects in space as to be practically free from disturbing influences produced by these objects. We shall therefore suppose that the evolution of the nebula proceeds solely in consequence of the mutual attractions of its various parts. In its original formation the nebula receives a certain endowment of energy and a certain endowment of moment of momentum; the mere fact that we see the nebula, the fact that it radiates light, shows that it must be expending energy, and the decline of the energy will proceed continuously from the formation of the object. The laws of dynamics assure us that no matter what may be the losses of energy which the nebula suffers through radiation or through the collisions of its particles, or through their tidal actions, or in any way whatever from their mutual actions, the moment of momentum must remain unchanged.

As the ages roll by, the nebula must gradually come to dispose itself, so that the moment of momentum shall be maintained, notwithstanding that the energy may have wasted away to no more than a fraction of its original amount. Originally there was, of course, one plane, in which the moment of momentum was a maximum. It is what we have called the principal plane of the system, and the evolution tends in the direction of making the nebula gradually settle down towards this plane. We have seen that the moment of momentum can be sustained with the utmost economy of energy by adjusting the movements of the particles so that they all take place in orbits parallel to this plane, and the mutual attractions of the several parts will gradually tend to bring the planes of the different orbits into coincidence. Every collision between two atoms, every ray of light sent forth, conduce to the final result. Hence it is that the nebula gradually tends to the form of a flat plane. This is the first point to be noticed in the formation of a spiral nebula.

But there is a further consideration. As the nebula radiates its light and its heat, and thus loses its energy, it must be undergoing continual contraction. Concurrently with its gradual assumption of a flat form, the nebula is also becoming smaller. Here again that fundamental conception of the conservation of moment of momentum will give us important information. If the nebula contracts, that is to say, if each of its particles draws in closer to the centre, the orbits of each of its particles will be reduced. But the quantity of areas to be described each second must be kept up. We have pointed out that it is infinitely improbable the system should have been started without any moment of momentum, and this condition of affairs being infinitely improbable, we dismiss any thought of its occurrence. As the particles settle towards the plane, the areas swept out by the movements to the right, and those areas swept out by the movements to the left, will not be identical; there will therefore be a balance on one side, and that balance must be maintained without the slightest alteration throughout all time. As the particles get closer together, and as their orbits lessen, it will necessarily happen that the velocities of the particles must increase, for not otherwise can the fundamental principle of the constant moment of momentum be maintained. And as the system gets smaller and smaller, by contraction from an original widely diffused nebulosity, like, perhaps, the nebula in Orion, down to a spiral nebula which may occupy not a thousandth or a millionth part of the original volume, the areas will be kept up by currents of particles moving in the two opposite ways around a central point. As the contraction proceeds, the opposing particles will occasionally collide, and consequently the tendency will be for the predominant side to assert itself more and more, until at last we may expect a condition to be reached in which all the movements will take place in one direction, and when the sum of the areas described in a second, by each of the particles, multiplied by their respective masses, will represent the original endowment of moment of momentum. Thus we find that the whole object becomes ultimately possessed of a movement of rotation.

The same argument will show that the inner parts of the nebula will revolve more rapidly than those in the exterior. Thus we find the whirlpool structure produced, and thus we obtain an explanation, not only of the flatness of the nebula, but also of the spiral form which it possesses. It is not too much to say that the operation of the causes we have specified, if external influence be withheld, tends ultimately to produce the spiral, whatever may have been the original form of the object. No longer, therefore, need we feel any hesitation in believing the assurance of Professor Keeler that out of the one hundred and twenty thousand nebulÆ, at least one-half must be spirals. We have found in dynamics an explanation of that remarkable type of object which we have now reason to think is one of the great fundamental forms of nature.