On some Dynamical Conditions applicable to Le Sage's Theory of Gravitation

I

1. THE tendency of modern science is undoubtedly to look to the existence of physical conditions or processes in those natural phenomena to which the theory of "action at a distance" has been applied. The gravitation theory of Le Sage has therefore of late naturally received a considerable share of attention. Le Sage finds it necessary, as a basis to his theory, to lay down certain conditions, some of which cannot but be regarded as arbitrary. Thus (as given in the paper by Sir William Thomson, ' Philosophical Magazine,' May 1873) Le Sage assumes among other conditions:—

1) That the direction of the streams of particles producing gravity is such that an equal number of particles are moving in all directions.

2) That the streams are all equally dense; or the total assemblage of matter forming the streams is of the same density in all parts.

3) That the mean velocity of the streams is everywhere the same.

2. These conditions cannot but be considered arbitrary. My object is to call attention to the fact (which, if it has been observed, would certainly appear to be deserving of more attention than it has received) that all these conditions which Le Sage, with the limited knowledge of his day, assumed to be arbitrary, are in reality inevitable deductions following [207] from the dynamical principles connected with the kinetic theory of gases, or that Le Sage unconsciously enunciated the inevitable principles of the kinetic theory — that, in short, all the conditions laid down in Le Sage's theory are perfectly satisfied by a gas whose particles are very minute, and consequently the mean length of path of whose particles is very great. In other words, it may be stated as a general proposition, that when two bodies are immersed in a gas at a less distance apart than the mean length of path of the particles of the gas, the two bodies will tend to be urged together. Thus all the arbitrary conditions of Le Sage's theory (and all the facts of gravity) would follow as inevitable deductions from the simple fundamental admission of the existence of matter in space, whose normal state is a state of motion.

3. The part of Le Sage's theory which most calls for explanation, and which he makes no attempt to explain, is (even if we allow as a purely arbitrary fact that the motion of his particles took place at one time uniformly or equally towards all directions) how this uniformity of motion of the particles could be kept up under the continual changes of direction resulting from the collisions of the particles against themselves and mundane matter. Now it has been proved mathematically by Professor Maxwell, in connexion with the kinetic theory of gases, that a self-acting adjustment goes on among a system of bodies or particles in free collision, such that the particles are caused to move equally towards all directions, this being the condition requisite to produce equilibrium of pressure. The method of calculating the rate of the above self-acting adjustment for any case is given in the Philosophical Transactions for 1866. This adjustment is of such a rigid character that, if by any artificial means the motions of the particles were interfered with and made to take place irregularly (i. e. unequally in different directions), the particles when left to themselves would in a very short time automatically return back to the above regular form of motion, i. e. so that an equal number of particles are moving in any two opposite directions. Thus it follows that when a system of particles are left in space with nothing to guide them, they will, by the rigid principles of dynamics adjust their motions in such a way as to be competent to produce the effects of gravity. In other words, the movement of streams of particles with perfect uniformity at all angles (which Le Sage assumed as a mere arbitrary postulate) is found to be the necessary consequence of dynamical principles; or the particles themselves adjust their motions so as to move in uniform streams in all directions ; and, further, when any disturbance of the uniformity [208] of the motion of the particles takes place due to their collisions with mundane matter, the particles themselves readjust the uniformity of motion.

4. Le Sage imagined that the collisions of the particles disturbed permanently the uniformity of their motions, and therefore supposed these collisions to take place only at intervals of time very remote from each other. Thus he assumes ".... that not more than one out of every hundred of the particles meets another during several thousands of years ; so that the uniformity of their motions is scarcely ever disturbed sensibly." We now know that, so far from the collisions of the particles among themselves disturbing the uniformity of their motions, this is the very cause which corrects and maintains the uniformity of motion, or preserves the uniformity of motion in opposition to external disturbing causes. The assumption, therefore, of the above enormous interval of time between the collisions of the particles, though admissible, is by no means necessary. The only necessary condition is that the path of the particle should be a certain length, not that a certain time should be occupied in traversing it. The time taken by the particle in traversing its path depends on its velocity; and this time might therefore be small, provided, under the conditions of the case, the velocity of the particle were high. Le Sage imagined that the collisions were detrimental, not only in destroying the uniformity of the motion of the particles, but also in destroying vis viva ; and he therefore supposed the collisions to take place as seldom as possible. This belief in the destruction of vis viva at collision was universal at the time of Le Sage ; and he therefore assumed that the gravific particles would finally come to rest, and gravity cease to exist. We now know that this is an error, and that motion is as naturally maintained among a system of particles as rest. Thus the one thing requiring to be admitted to account for all the effects of gravity is, that the universe is immersed in a gas the mean length of path of whose particles is great.

5. The other assumptions or postulates of Le Sage in connexion with his theory, viz. equal density in all parts of the streams of moving particles, equal mean velocity in all parts, follow no less as automatic consequences from the recent dynamical investigations connected with the kinetic theory of gases. Thus the conditions of Le Sage's theory become converted from a series of arbitrary assumptions or postulates, to a series of deductions following from the rigid principles of dynamics.

6. It forms a truly wonderful fact to consider, that a system of bodies or particles left to themselves, with nothing to guide them but their own collisions (which might well be regarded as fulfilling all the essentials of a chaos), produces and maintains the most rigid system of order, such that the number of particles contained in unit volume of the system (taken anywhere) is equal, the mean velocity equal in all parts, the mean distance of the particles the same in all parts, and the particles are moving uniformly towards all directions in all parts. 8uch is the result produced by pure dynamics. In fact it may be said that leaving the bodies to themselves constitutes the most perfect system of control, for any interference whatever would disturb the regularity of the motions. This regularity of movement is not only naturally continued, but forcibly and automatically maintained against any disturbance, — such that if it were imagined that a system of bodies were purposely put in motion in the most chaotic manner possible, the motion would of itself in a short time become regular, or the whole would become a system of order and uniformity.

7. Clausius, as is known, has investigated a relation between the mean length of path of the particles of a gas and the diameter of the particles. From this investigation it follows that the mean length of path of the particle of a gas (i. e. the average distance which the particle moves before encountering another particle) increases in proportion as the square of the diameter of the particle diminishes. Thus by making the particle small enough, its mean length of path may be increased to any extent. No objection, evidently, can be made to this, for à priori one size of particle is just as likely as another. This minute size would render it possible for the particle to possess a high velocity without producing thereby disturbance or displacement among the molecules of ordinary matter ; and this high velocity is necessary to accord with the observed facts of gravity. One velocity cannot be said à priori to be more likely than another. We must just be guided by the teaching of facts as to what the velocity is.

8. It is an interesting fact pointed out by Sir William Thomson (Phil. Mag. May 1873) that the distance through which gravity is effective would depend on the distance through which the gravific particles move before being intercepted by collision with each other (which is equivalent to the mean length of path of the particles). By assuming the distance of the stars to be a multiple of the mean length of path of the particles, it would therefore follow that the stars do not gravitate towards each other — this satisfying the condition for the stability of the universe. The assumption of all the bodies of the universe gravitating towards each other is evidently quite inconsistent with stability (as already pointed out by Professor Challis). All that we require to admit is that the effects of gravity hold through as great distances as we have observed them.

9. The distance through which gravity has been observed to act is well known to be but an infinitesimal fraction of the distance of the stars. It may therefore well be that the mean length of path of the particles of the medium producing gravity may be but an infinitesimal fraction of this distance. The column of the gravific medium intercepted between two stars would therefore on the whole be at rest, just as a column of gas is at rest between two bodies a visible distance apart (i. e. a distance which is a large multiple of the mean length of path of the particles of gas). Le Sage appears to have assumed that the mean length of path of the gravific particles swept through the universe ; or he assumed that streams of matter came from the depths of space and passed entirely through the visible universe into space beyond.

This [211] assumption cannot but be regarded as fantastic, and, as we observe, is by no means necessary. The mean length of path of the particles, so far from being comparable to the dimensions of the visible universe, may be but an infinitesimal fraction of the distance of two of its primary components. All we require to admit is that the mean length of path of the particles of the medium is at least as great as the very limited range through which gravity has been observed to act ; or, in order to explain all the observed facts, it is sufficient to admit that the universe is immersed in a gas (or medium constituted according to the kinetic theory) the mean length of path of whose particles is so adjusted as to cause the minor or secondary portions of the universe to gravitate towards each other. Under the simple conception of a variation in the diameter of the particles of a medium, the mean length of path of the particles (and with it the range of gravity) is capable of adjustment with precision to any range. It would probably be difficult to imagine any more simple condition as a mechanical means to an end than this.

10. It is a necessary condition to Le Sage's theory (in order [212] that gravity may be sensibly proportional to mass) that the total volume of free space in a substance, in the form of interstices between the molecules or in their structure, must be great compared with the total volume of matter contained in the molecules themselves. Le Sage assumed the molecules of substances to have a sort of open structure in the form of cages with wide interstices. This condition of free interstices would be equally satisfied by assuming the molecules to be small relative to their mean distance, or on the condition of the vortex-ring atom theory, without any necessity for making the above somewhat fantastic assumption of cage-structure.

11. It is necessary to assume that the particles producing gravity are in very close proximity compared with molecules, otherwise the particles would be unable by their motion to produce a perfectly equable pressure upon the molecules of matter. It might be thought that, because the particles of the gravific medium are so close, and the molecules of ordinary matter relatively far apart, therefore the quantity of matter in the form of gravific particles enclosed in a given volume of space must be very great compared with the quantity of ordinary matter that that same volume of space would contain — or, in other words, that there must be a relatively enormous quantity of matter in the form of gravific particles. This by no means follows; for although the gravific particles may be very close, Ihe relative quantity of matter in them may be very small, provided the particles themselves are small. Indeed by simply conceiving an extreme degree of subdivision, the particles pervading a given volume of space may by continued subdivision be conceived to be brought into as close proximity as we please ; and though the space itself is large, the total quantity of matter thus used may be conceived as small as we please. No consequence how minute the size (or mass) of a particle may be, the effect produced by its motion remains as great, provided its velocity be adequately augmented. The minute size is the very condition adapted to a high velocity; and this minute size is at the same time the necessary condition for a long mean path. Thus we may observe that the mechanical conditions of the problem fit into each other. The matter of the gravific medium is in such a finely subdivided state, and its motion so rapid, that its presence necessarily eludes detection. The pressure (termed "gravity") due to the motion of the particles of the gravific medium is no more difficult of realization than the pressure due to the motion of the molecules of air. If the motion of the molecules of air be unrecognized by the senses, how much more must this be the fact with the minute gravific particles; indeed it is difficult to see what mechanical objection can be urged against this realization of the problem, which is extremely simple.

12. The theory of "action at a distance" being rejected, which is necessary in order to explain the facts at all, the effects of gravity can in principle be referred to only two conceivable causes. The tendency of two molecules of matter to approach each other can be referred (1) to a motion possessed by the molecules themselves disturbing the equilibrium of pressure of the medium between them ; (2) to a motion possessed by the medium itself (in the form of streams or currents) acting upon the molecules. The first of these two conditions appears to be inadmissible, from the fact that we cannot interfere with or modify gravity at will, whereas we can very readily interfere with or modify the motion of the molecules of matter (as by adding or subtracting heat, for example). It therefore would appear that gravity must be due to some motion that we cannot interfere with, i. e. to a motion in the external medium which we cannot handle or which is beyond our control. Only one conclusion appears therefore to be possible here ; and therefore it would seem that the theory of Le Sage can scarcely be regarded as a mere hypothesis, but rather as an irresistible deduction which is forced upon us in the absence of any other conceivable inference. Certainly, if simplicity be a recommendation, the theory needs no recommendation on that ground.

London, July 1877.

II

No. II

1. THE explanation of gravitation has now risen to the rank of one of the foremost problems of modern science; indeed the day for the ascription of occult qualities to matter is now gone, and phenomena, demand an explanation by the reason. The ascription of an occult quality, so far from throwing light upon a phenomenon, only serves to darken it. The effects of gravity (like all other physical effects) being effects of motion, have, like other physical effects, to be explained. A rational explanation has to be given for the motion of approach of two masses. If we were to make an exception of this case, it might be argued that we might make an exception of other cases; and since all physical effects are effects of motion, we should thus in principle have nothing to explain at all.

2. The absolute necessity for giving an explanation of gravity being admitted, we may inquire what has hitherto been done in this respect. The only theory worthy of serious consideration, or which has stood any test at all, is the theory put forward by Le Sage of Geneva. An immense advance in dynamics has been made since his day. It therefore behoves any one to ask how far the principles put forward by him admit of being improved and modified according to modern advances in dynamics.

3. To any one who has carefully read Le Sage's theory, it will be evident that the theory consists mainly in a series of postulates or conditions arbitrarily assumed so as to be adapted to produce the results required. Le Sage assumed (1) the movement of streams of particles coming from an indefinite distance in space and converging towards the visible universe. He even calculates (by a given velocity of motion) the distance those particles would require to have come which produce gravitation, at a remote epoch of 10,000 years (page 22). He therefore calls the particles, from the enormous distance he supposes them to come, "ultramundane particles." Next he assumes, quite arbitrarily, that the particles move uniformly or equally in all directions, this being necessary in order that the action [365] of gravity may be equal in all directions. He computes (roughly at about 3,000,000) the number of different directions in which separate streams of particles would require to be moving in order to produce everywhere that sensible uniformity of pressure which is the characteristic of gravity (page 25). It will be noted that these are all assumptions in themselves entirely arbitrary. He next assumes that the mean velocity of the streams of particles is everywhere the same, and the density everywhere the same.

4. It will be observed that this theory gives no possible idea as to how such a motion of streams of particles among themselves could be kept up, or naturally maintained. Le Sage attempts to evade the difficulty of the particles encountering each other by assuming them to be so small that "not more than one out of every hundred of the particles meets another during several thousands of years." This only removes the difficulty a step further on, without avoiding it. Indeed it may be observed that the theory, in the state in which Le Sage left it, is little more than a series of postulates, some of them almost as unrealizable as gravity itself. This does not detract from a distinct merit in the origination of the theory; for it must be remembered how little dynamical principles were advanced at Le Sage's time, and how few resources he had to draw upon.

5. I have pointed out (Phil. Mag. Sept. 1877) what (whether already observed by others or not) cannot but be regarded as a somewhat startling fact, viz. that no postulates whatever are required for a dynamical theory of gravitation, but that it may be shown that particles of matter in free motion in space must inevitably of themselves arrange their motions so as to produce the effects of gravity — or the special effects of gravity (variation as the square of the distance &c.) must be produced from pure dynamics in the case of a system of particles in free motion in space, without any necessity for postulates as to the character of the motion at all. This follows from the principles which have been investigated in connexion with the modern kinetic theory of gases, of which Le Sage was ignorant. For it has been demonstrated by Professor Maxwell, in connexion with the kinetic theory of gases, that particles of matter in free collision among each other in space will automatically arrange their motions so as to move uniformly in all directions, i. e. so that an equal number of particles are moving in any two opposite directions (this being the necessary condition for equilibrium of pressure in a gaseous medium). This character [366] of motion, it may be observed, is the first important condition required by Le Sage's theory — which condition therefore follows as a rigid dynamical fact, not as an arbitrary postulate, as he made it. This motion of the particles uniformly or equally towards all directions is not a mere chance fact, but a rigid adjustment, of such a character that when by any artificial means this mode of motion of the particles is disturbed they will automatically, of themselves, return back to this regular form of motion (i. e. so that an equal number of particles are moving in all directions'). The other conditions put forward as postulates by Le Sage, viz. that the density of the streams of particles should be the same in all parts, and the mean velocity the same in all parts, are equally necessary results following from the kinetic theory of gases — not, therefore, postulates at all.

6. The only further condition required is that the mean length of path of the particle, before being intercepted by collision with each other, should be great enough to produce the effects of gravity — i. e. so that the particles of the medium may act as streams upon masses immersed in the medium, or may stream past two opposed masses, which by their mutual screening or sheltering action produce the observed effects of gravity. The mean length of path of a particle depends (as is known) upon its size. One size of particle is not à priori more likely than another. By simply, therefore, making the particles small enough, any mean path, however great, may be attained. We thus observe that all the arbitrary postulates of Le Sage's theory, together with all the effects of gravity, naturally and inevitably follow from the simple admission of the existence of matter in space whose normal state is a state of motion, or the existence of a medium in space constituted according to the kinetic theory of gases. In this way the mode in which the motion of the streams of particles through each other is naturally kept up in a state of dynamical equilibrium, is quite easily explained.

7. Here we do not want an indefinite waste of matter, or an indefinite supply of matter from ultramundane space (as Le Sage imagined) to produce gravity, but gravity is produced by matter or a medium which as a whole is stationary, and whose internal motion is kept up and perfectly naturally maintained by the rigid laws of dynamics. The difficulty of the collisions is here completely got over; for the collisions of the particles among each other, so far (as Le Sage supposed) [367] from interfering with the uniformity of their motions, is the very condition which corrects and maintains the uniformity of motion in opposition to external disturbing causes. This is the main point which was utterly inexplicable in Le Sage's theory; for it was impossible to see, in the way he put it, how such a motion of streams of particles should be kept up uniformly in all directions, under continual collisions with themselves and with mundane matter. By the application of the principles of the modern kinetic theory of gases to the case, this point is completely solved. As before remarked, it is almost startling that the particular form of motion which the particles themselves automatically keep up should be precisely that one which is required to produce gravity (or an effect varying as the square of the distance &c.), — also that it should make the density and mean velocity equal in all parts, which is necessary for the effects of gravity.

8. It is an interesting fact to observe that the distance through which gravity will act will depend on the range through which the streams of particles are comparatively unimpeded, i. e. on the mean length of path of the particles. By making, therefore, the mean length of path of the particles loss than the average distance of the stars, it would follow that the stars do not gravitate towards each other, which satisfies the condition for the stability of the, universe. It is evident (as has been already pointed out by others) that the assumption of all the bodies of the universe gravitating towards each other is quite inconsistent with stability; and to the truly philosophical mind any theory which rendered such an assumption of instability necessary would be in itself improbable. It is only necessary that the mean length of path of the particles be great enough to produce effects of gravity throughout [368] the greatest range in which we have observed them, which is but an infinitesimal fraction of the distance of the stars.

9. It may perhaps be well just to sketch here the mode of action of the medium in producing gravity, the manner in which the intensity is made to vary as the square of the distance, &c. Let A (in the annexed diagram) represent a molecule or mass; let C represent the bounding surface of an imaginary hollow sphere described about A. Then, since the particles of the medium are moving uniformly in all directions, a number of them will be passing in all directions through the imaginary spherical surface C. Only those particles which are passing (sensibly) along the radii of the spherical surface will strike A; and therefore we need only regard those special particles which radiate towards A. The molecule A being therefore struck equally on all sides, will accordingly remain at rest. But if now we suppose a second molecule to be placed at B, then out of the whole number of particles which are directed towards A, the molecule B will intercept a number which is proportional to the area which B cuts off from the whole spherical area C. The molecule A will therefore now, owing to the sheltering power of B, be struck with a fewer number of particles in the direction B A. The balance of the pressure being thus upset, A will be propelled towards B. The same holds true of B relatively to A (on drawing the imaginary spherical surface C'). The two molecules A and B are therefore propelled towards each other mutually. It now remains to illustrate how the impulsive action varies as the square of the distance. It will be at once evident that since the area of a spherical surface is as the square of the radius, therefore, if B [369] were removed to a double distance, the imaginary spherical surface described through B with A as a centre, would have four times the area; but the area of the molecule B remaining constant, B would therefore only shelter A one fourth the amount it did before; and accordingly A would be impelled towards B with one fourth the force, — the same being true of B relatively to A.

10. It is found that gravity is proportional to mass. It must therefore be assumed that, owing to the porosity of bodies, or open structure of the molecules, the gravific medium ( whose particles are extremely minute) can penetrate freely into the interior of bodies, and thus act upon the internal molecules, so that the total effect is proportional (sensibly) to the number of molecules, or gravity is proportional to mass. Of course this could not hold true with an infinite mass; but it is rigidly demonstrable (by a given degree of porosity) that it could hold true with as near a degree of approximation as experience has shown, and even nearer if necessary. Independent physical reasons tor inferring this extreme porosity or permeability of matter will be given further on.

11. My main object in this paper is to meet all possible objections which have been or might be urged against this theory, as, if it be true, it ought to stand against all criticism; and if, on the other hand, it be erroneous, the sooner it is demonstrated so the better. It may just be remarked, in the first place, that in principle there appears to be no other theory conceivable which at all would satisfy the conditions of gravity. Gravity can be referred to two conceivable causes : — (1) to a motion possessed by the molecules of matter themselves, disturbing the equilibrium of pressure of the surrounding medium; (2) to an independent motion of the medium itself acting upon the molecules. The first of these two conditions appears to be inadmissible; for if gravity were due to a motion of the molecules of matter, then since we can readily modify or interfere with the motion of the molecules of matter (as by heat), we could thereby interfere with gravity. The fact, therefore, that it has been found impossible to interfere with gravity, points to a motion in the external medium (which is beyond our control) as the cause. It would be well to keep this fact in view before lightly regarding Le Sage's theory as a mere hypothesis. To the careful observer it will appear to contain rather the essence of a necessary fact, from the absence of (in principle) any other conceivable cause; and if the theory can be shown to be a practical one, consistent with admitted mechanical principles, it will have every condition for acceptance. To us it seems that a closer study of the theory only serves to show its [370] many mechanical beauties and extreme simplicity as a means to an end, satisfactory, not only in the absence of any other conceivable cause, but as affording a perfectly rational conception to the mind of the processes by which the effects are brought about.