Kinetic Theories of Gravitation/Challis, 1859

Professor James Challis, of the University of Cambridge, England, in the prosecution of a "Mathematical Theory of Heat," published in the Philosophical Magazine for March, 1859, advanced in November of the same year, to a " Mathematical Theory of Attractive Forces," based on the assumption " that all substances consist of minute spherical atoms of different but constant magnitudes, and of the same intrinsic inertia; and that the dynamical relations and movements of different substances, and of their constituent atoms, are determined by the pressures of the aether against the surfaces of the atoms, together with the reaction of the atoms against such pressure by reason of the constancy of their form and magnitudes. The aether is assumed to be a uniform elastic fluid medium pervading all space not occupied by atoms, and varying in pressure proportionally to variations of its density. The theory recognizes no other kinds of force than these two, the one an active force resident in the aether, and the other a passive reaction of the atoms."

After a formidable array of partial differential equations, the author concludes: "Having now shown that waves of large breadth attract a small spherical body toward their origin, and having previously shown that waves of small breadth may repel such a body in the contrary direction, the main difficulty in forming a theory of attractive and repulsive forces seems to be overcome." It is supposed by Professor Challis that by the disturbance of a material element, a series of undulations differing greatly in their order of magnitude and velocity maybe simultaneously propagated in the aetherial medium, giving rise to as many different manifestations of force ; and that according to their relative wavelength, some of these will produce a permanent motion of translation on molecules of determinate mass subjected to their influence. This is partly in accord with the striking experiments of Guyot previously referred to.

In a following paper the author undertook A Theory of the Force of Gravity;" remarking that, "As we have no conception from personal experience and sensation of any other species of force than pressure, the actio in distans does not admit of being explained by any previous or [248] concomitant knowledge, but if it be a reality, must forever remain to us incomprehensible." How the school-boy's personal experience of the strain exerted in drawing by a cord his winter's sled can be resolved into a sensation of "pressure" does not seem easy of discovery.

Assuming then an order of aetherial waves having a much larger range than those of light. Professor Challis endeavors to deduce the several laws, of action proportional to the number of atoms acted upon, of the inverse square of the distance of action, and of simultaneous action in different directions. He infers that if such waves encountered a slight retardation of propagation in passing through the earth, they would be refracted, so to speak, by the form of the large inequalities of the terrestrial surface, producing the observed deflection of the plumbline.

He also supposes that a small function involving r$4$ must be added to the usual formula m/r² increasing sensibly the action of the sun near its surface and diminishing its action notably through interstellar distances. He thinks that this alone will explain why the sidereal system does not collapse toward its common center of gravity. "According to the theory of gravity I have proposed, although the ordinary law may be exact through the solar system and far beyond, there must be distances at which the condition that the excursions of the vibrating particles of the aether are large compared to the dimensions of the atoms ceases to be fulfilled. In that case the attraction changes to repulsion."

Renewing the discussion of "A Theory of Molecular Forces," the following year. Professor Chailis contends: " It is a matter of demonstration that a theory of molecular forces cannot be constructed on the hypothesis that the forces vary according to some law of the distance from individual material particles, unless the law be such that the force changes sign with the distance so as to become attractive after being repulsive. But if force be a virtue resident in the particle, it must at its origin be either attractive or repulsive, and it seems impossible to conceive how by emanation to a distance it can change its quality. This difficulty, as will be shown, is not encountered in a theory of molecular forces which deduces their laws from the dynamical action of an elastic medium." While it is probably no more difficult to conceive an innate force or virtue which " at its origin " shall have a law of radial intensity whose value passes through zero, than to conceive any other mathematical law of increment or decrement, there is certainly no necessity for assuming such a law. If we should suppose the attraction by inverse squares to be absolute, with the superposition of a repulsion of much higher inverse power, and of far greater intensity, it is evident that the two curves of force would cross each other, and that at the intersection the resultant would involve a "change of sign." However difficult it may be to realize such a conception, the actual superposition of [249] opposing forces is daily exhibited to us in the behavior of the magnet. Another possible conception is that repulsion is a positive material or aetherial atmosphere of definite radius.

Indeed, the author's theory is really one of the superposition of two systems of waves, rather than one of a single system changing its sign. For he supposes that the attraction of gravity results from aetherial waves of great length and correspondingly large excursions or amplitudes, in which the diameter of the material atom is a vanishing quantity ("&mu;=0"), and there is no sensible difference between the velocities on its two hemispheres; while atomic repulsion results from such small waves (smaller even than those of light) that the atom is large in comparison, and the difference of the wave on its two hemispheres is very notable. " Thus the conditions assumed in the mathematical theory of heat are satisfied by supposing &mu; to be very large and q [the excursion of the wave] to be very small ; and the fulfillment of these conditions accounts for the great energy of calorific repulsion. . . Hence atoms of very small size, acting upon each other by the intervention of waves of which the excursions are very small, mutually repel with a very great force ; and at the same time, as was shown in the Theory of Heat, the force varies very rapidly with the distance."

Attributing to the spherical hard atom of matter only inertia, "it would be contrary to these principles to ascribe to an atom the property of elasticity, because from what we know of this property by experience it is quantitative, and being most probably dependent on an aggregation of atoms, may admit of explanation by a complete theory of molecular forces." Of this fundamental property however, — necessarily precedent to all theory of wave action, — no explanation is given.

That the author did not feel entirely satisfied with his vibratory theory of molecular forces, would appear from his returning to the subject two years later with the remark : " Such vibrations, when we calculate their effect only to the first power of the velocity, are found to produce simply oscillations of small spherical bodies submitted to their action, and not motion of translation. To account for the latter, it is necessary to proceed to the consideration of effects due to the second power of the velocity Lastly, there is yet another physical force, the relations of which to an aetherial medium and to other modes of force are not readily made out : I mean the force of gravity. If however, all the other forces are modifications of aetherial pressure, it is reasonable to suppose that this one is of the same kind. I have ventured to reason on this supposition, and have attempted to deduce (I think with success) the known laws of gravity from the dynamical action of aetherial waves of much larger magnitude than those which correspond to molecular forces."

While it is comparatively easy to explain the origin of heat-waves [250] by the theory of the functions of antecedent forces, the converse problem, to explain forces by the hypothesis of antecedent waves, is by no means so simple. Accordingly, a very weak point in all undulatory theories of gravitation has been an account of the origin (to say nothing of the continuance) of the primum mobile. "On the source and maintenance of the sun's heat," Professor Challis remarks as follows: "We arc led to the conclusion that the undulations which emanate equally in all directions from each atom, and constitute by their dynamic action the repulsion of heat, are mainly produced by the reaction of the atoms due to their inertia and impenetrability. It is plain however, from this reasoning, that there must be an original and independent source of undulations. Now obviously such a supply may be conceived to be furnished to the sun by undulations emanating from the stars. We have ocular evidence that stars transmit light-undulations, and it is quite possible that they originate others not sensible to the sight." As our sun is one of the stellar fraternity, surely this device of mutual borrowing is an extraordinary method of production.

" That action at a distance is not a universal condition of force is proved by the modern discovery that light and heat, which are modes of force, are transmitted through space by the intervention of a medium. If one kind of force requires a medium of transmission, why not another ? Again, it is found by experience that the same portion of matter may attract or repel, according to circumstances. But inherent force cannot possibly be so changed by circumstances. In the same matter it must continue to be always the same."

"All physical force being pressure, there must be a medium by which the pressure is executed." And the origin of this pressure is therefore, the origin of "all physical force." The fundamental postulate of this medium is that it is a uniformly continuous elastic fluid, exerting a pressure always proportional to its density. As it is contrary to principle to ascribe elasticity to atomic matter, the question might arise, why is it more proper to ascribe this occult property to the aether ? In the case of air (of uniform temperature) the constant ratio of pressure to density results we are told, from a vis a tergo, the vibratory action of the interstitial aether. The author perceives the incongruity of denying to the air a quality attributed to the aether, when the law is implicitly the same in both ; and he suggests in extenuation, " we can conceive of the existence of another order of aether having the same relation to the first as that has to air, and so on ad libitum.'"

The very key-note of the hypothesis is dynamic aetherial "pressure." "All the different kinds of physical force detected by observation and experiment are modifications of pressures of the aether." But when we [251] seek the cause of this pressure, it forever eludes us. Here then the system stands self-convicted of impotence to exercise its prime prerogative. At whatever point in the infinite series of descending orders of aether we stop, the secret of its power is ever one step backward. We must still "conceive of the existence of another order of aether having the same relation to it" that it had to the preceding. And that no possible, element of embarrassment may be wanting to our conception, the first tether is absolutely "continuous," without atoms and without interstices !

In 1869, Professor Challis published an elaborate extension of his mathematical discussion of kinetic theories of the physical forces, in a large octavo volume of some 750 pages ; the first half of the work being devoted to a general mathematical treatise, of high merit and value, under the title of " Notes on the Principles of Pure and Applied Calculation." In the latter portion of the work, (on theoretical physics,) the subjects treated of" are those of light, heat, and molecular attraction, force of gravity, electricity, galvanism, and magnetism, respecting which I make the general hypothesis that their phenomena all result from modes of action of an elastic fluid, the pressure of which is proportional to its density." And in the "introduction" to the work, he has more explicitly stated: "The hypothesis respecting the tether is simply that it is a continuous elastic medium, perfectly fluid, and I hat it presses proportionally to its density." The forces of elasticity, and of chemical affinity, are excluded as beyond the present reach of analysis.

A distinction is made between the aetherial radiations of light and of heat, not justified by any observed phenomenon. " Since in the theory I have proposed, the transverse vibrations of rays always accompany direct vibrations, and it was concluded that the sensation of light is entirely due to the former, we are at liberty to refer the action of heat, or other modes of force, to the direct vibrations." This would leave the polarization of heat quite inexplicable ; as obviously vibrations of the acoustic type are incapable of polarization.

It is now familiar to opticians that fine rulings on glass, whose distance apart is less than a half of the wave-length of light, are readily resolvable with optical distinctness by our modern microscopes, while the intimate texture of the glass is apparently as far removed from resolution as with the unarmed eye. What part can aetherial vibrations play in giving cohesion to the ultimate molecules of the glass? Here then is apparently a new difficulty for the undulatory theory of force; for not only are the ultimate molecules of the silicate bound together with a powerful force, (giving seeming continuity of substance to our highest artificial vision,) but they are also held apart with a still more potent stress. Professor Challis does not shrink from the solution. [252] " Making the only hypothesis which is consistent with the theoretical principles advocated in this work, namely, that the ultimate atoms of the glass are kept asunder by the repulsive action of aetherial undulations which have their origin at individual atoms, it may be presumed that this atomic repulsion is attributable to undulations incomparably smaller than those which cause the sensation of light. The only additional hypothesis that will now be made is that there are undulations of the aether for which the values of &lambda; are very much inferior in magnitude to those of the undulations which produce the phenomena of light. The origin of this class of undulations may, as well as that of all others, be ascribed to disturbances of the aether by the vibrations and motions of atoms. Although the periods of the aetherial vibrations may, under particular circumstances, be determined by the periods of the vibrations of the atoms, this is not necessarily the case. . . . "

"However small may be the condensation propagated from a single atom, the resulting condensation from an aggregation of atoms contained in a spherical space will be of sensible magnitude at distances from the center of the space very large compared to its radius, provided the space be not less than a certain finite magnitude, and the atoms contained in it be not fewer than a certain finite number. We have hitherto had under consideration the waves of atomic repulsion and the waves of molecular attraction, and it was argued that the latter might result from compositions of the former, and that in that case the values of &lambda; would be much larger for the composite waves than for the components."

In 1872, the author again writing "On the Hydrodynamical Theory of Attractive and Repulsive Forces," says in regard to the discussion of the first and second orders of small quantities, " Having in fact succeeded in overcoming the mathematical difficulty of effecting a second approximation by this means, [starting from the first approximation,] I have ascertained that the solution contains terms of indefinite increase, whence it must be concluded that the logic of the process is somewhere at fault Both in this Magazine and in my work on the Mathematical Principles of Physics, I have in various ways attempted to solve to the second approximation the problem of the motion of a small sphere acted upon by the vibrations of an elastic fluid. But I must confess that owing to the difficulty of including the effect of the spontaneous vibrations, my efforts have been only partially successful." He remarks that as his equations involved two unknown constants H$1$ and H$2$ (representing the amounts of wave condensation on the nearer and on the further hemispheres of the atom,) " on this account the theories which attribute the forces of heat, molecular attraction, and gravity to action on the atoms by pressure of the aether in vibration, are incomplete." And he admits that for any purpose of quantitative [253] determination or verification, " The whole theory however of attractive and repulsive forces, regarded as due to vibrations of the aether, is incomplete for want of an a priori determination of the composition and value of the quantity H."

In 1876, the author returns to the discussion with a view "to correct in certain respects that investigation, [the one last cited,] and to carry it a step further." And in dealing with a residual effect of vibration, " which in fact is the attraction of gravity, and determining that the accelerative force is independent of the size of the atom or molecule, he says : " This result expresses the force of gravity as due to the attractive action of a molecule of a higher order as to magnitude than the molecule of molecular attraction. For distinction, a molecule of this superior order might be called a gravity-molecule. Its magnitude may still be considered to be so small that in comparison with the magnitudes of terrestrial and cosmical masses, it may be treated as an infinitesimal quantity, Lastly, it is to be noticed that on account of the large value of &lambda; for gravity-waves, they do not suffer sensible retardation or refraction in passing through gross bodies. I have on several previous occasions, treated of the problem of gravitating force theoretically, and by slow steps have approximated to its solution ; but before the present attempt, I had not succeeded in exhibiting satisfactorily the rationale of this kind of attraction by vibrations.

Such is an imperfect abstract of the most carefully-studied and the most diligently-elaborated exposition of the wave-theory of attraction which has yet been proffered to the scientific world. Discussed with the earnestness and candor of a fervent conviction, and with all the resources of a high order of mathematical culture and ability, the result yet fails, sadly and fatally, to satisfy the conditions of the problem. Strong as is the author's assurance that he has successfully grasped by his formulas and equations the several functions embraced in the first three of our propositions, this is by no means mathematically established; and the last three propositions are hopelessly ignored and violated. We have seen that Elasticity, that puzzling " occult quality," driven out from the sober presence of the purely rational atom, has in an inexplicable manner, slipped in by the back-door of aetherial pressure.

Naturam expelles furca, tamen usque recurret,

Et mala perrumpet furtim fastidia victrix.

With the multitudinous duties imposed upon the much-suffering aether, in all the varying ranges and orders of undulations derived from atoms and from molecules with ever-changing motions, amid all the perturbations and transformations of the mechanical energy of matter, [254] there is one resultant alone which never by any accident incurs a composition or experiences a commutation, the constant and unchangeable undulation of gravitation.