1

gravitation

mass may be defined in two ways:

1o by the quotient of the force by the acceleration; this is the true definition of the mass, which measures the inertia of the body.

2o by the attraction the body exercises upon an exterior body, in virtue of newton’s law. we should therefore distinguish the mass coefficient of inertia and the mass coefficient of attraction. according to newton’s law, there is rigorous proportionality between these two coefficients. but that is demonstrated only for velocities to which the general principles of dynamics are applicable. now, we have seen that the mass coefficient of inertia increases with the velocity; should we conclude that the mass coefficient of attraction increases likewise with the velocity and remains proportional to the coefficient of inertia, or, on the contrary, that this coefficient of attraction remains constant? this is a question we have no means of deciding.

on the other hand, if the coefficient of attraction depends upon the velocity, since the velocities of two bodies which mutually attract are not in general the same, how will this coefficient depend upon these two velocities?

upon this subject we can only make hypotheses, but we are naturally led to investigate which of these hypotheses would be compatible with the principle of relativity. there are a great number of them; the only one of which i shall here speak is that of lorentz, which i shall briefly expound.

consider first electrons at rest. two electrons of the same sign repel each other and two electrons of contrary sign attract each other; in the ordinary theory, their mutual actions are proportional to their electric charges; if therefore we have four electrons, two positive a and a′, and two negative b and b′, the charges of these four being the same in absolute value, the repulsion of a for a′ will be, at the same distance, equal to the repulsion of b for b′ and equal also to the attraction of a for b′, or of a′ for b. if therefore a and b are very near each other, as also a′ and b′, and we examine the action of the system a + b upon the system a′ + b′, we shall have two repulsions and two attractions which will exactly compensate each other and the resulting action will be null.

now, material molecules should just be regarded as species of solar systems where circulate the electrons, some positive, some negative, and in such a way that the algebraic sum of all the charges is null. a material molecule is therefore wholly analogous to the system a + b of which we have spoken, so that the total electric action of two molecules one upon the other should be null.

but experiment shows us that these molecules attract each other in consequence of newtonian gravitation; and then we may make two hypotheses: we may suppose gravitation has no relation to the electrostatic attractions, that it is due to a cause entirely different, and is simply something additional; or else we may suppose the attractions are not proportional to the charges and that the attraction exercised by a charge +1 upon a charge ?1 is greater than the mutual repulsion of two +1 charges, or two ?1 charges.

in other words, the electric field produced by the positive electrons and that which the negative electrons produce might be superposed and yet remain distinct. the positive electrons would be more sensitive to the field produced by the negative electrons than to the field produced by the positive electrons; the contrary would be the case for the negative electrons. it is clear that this hypothesis somewhat complicates electrostatics, but that it brings back into it gravitation. this was, in sum, franklin’s hypothesis.

what happens now if the electrons are in motion? the positive electrons will cause a perturbation in the ether and produce there an electric and magnetic field. the same will be the case for the negative electrons. the electrons, positive as well as negative, undergo then a mechanical impulsion by the action of these different fields. in the ordinary theory, the electromagnetic field, due to the motion of the positive electrons, exercises, upon two electrons of contrary sign and of the same absolute charge, equal actions with contrary sign. we may then without inconvenience not distinguish the field due to the motion of the positive electrons and the field due to the motion of the negative electrons and consider only the algebraic sum of these two fields, that is to say the resulting field.

in the new theory, on the contrary, the action upon the positive electrons of the electromagnetic field due to the positive electrons follows the ordinary laws; it is the same with the action upon the negative electrons of the field due to the negative electrons. let us now consider the action of the field due to the positive electrons upon the negative electrons (or inversely); it will still follow the same laws, but with a different coefficient. each electron is more sensitive to the field created by the electrons of contrary name than to the field created by the electrons of the same name.

such is the hypothesis of lorentz, which reduces to franklin’s hypothesis for slight velocities; it will therefore explain, for these small velocities, newton’s law. moreover, as gravitation goes back to forces of electrodynamic origin, the general theory of lorentz will apply, and consequently the principle of relativity will not be violated.

we see that newton’s law is no longer applicable to great velocities and that it must be modified, for bodies in motion, precisely in the same way as the laws of electrostatics for electricity in motion.

we know that electromagnetic perturbations spread with the velocity of light. we may therefore be tempted to reject the preceding theory upon remembering that gravitation spreads, according to the calculations of laplace, at least ten million times more quickly than light, and that consequently it can not be of electromagnetic origin. the result of laplace is well known, but one is generally ignorant of its signification. laplace supposed that, if the propagation of gravitation is not instantaneous, its velocity of spread combines with that of the body attracted, as happens for light in the phenomenon of astronomic aberration, so that the effective force is not directed along the straight joining the two bodies, but makes with this straight a small angle. this is a very special hypothesis, not well justified, and, in any case, entirely different from that of lorentz. laplace’s result proves nothing against the theory of lorentz.

2

comparison with astronomic observations

can the preceding theories be reconciled with astronomic observations?

first of all, if we adopt them, the energy of the planetary motions will be constantly dissipated by the effect of the wave of acceleration. from this would result that the mean motions of the stars would constantly accelerate, as if these stars were moving in a resistant medium. but this effect is exceedingly slight, far too much so to be discerned by the most precise observations. the acceleration of the heavenly bodies is relatively slight, so that the effects of the wave of acceleration are negligible and the motion may be regarded as quasi stationary. it is true that the effects of the wave of acceleration constantly accumulate, but this accumulation itself is so slow that thousands of years of observation would be necessary for it to become sensible. let us therefore make the calculation considering the motion as quasi-stationary, and that under the three following hypotheses:

a. admit the hypothesis of abraham (electrons indeformable) and retain newton’s law in its usual form;

b. admit the hypothesis of lorentz about the deformation of electrons and retain the usual newton’s law;

c. admit the hypothesis of lorentz about electrons and modify newton’s law as we have done in the preceding paragraph, so as to render it compatible with the principle of relativity.

it is in the motion of mercury that the effect will be most sensible, since this planet has the greatest velocity. tisserand formerly made an analogous calculation, admitting weber’s law; i recall that weber had sought to explain at the same time the electrostatic and electrodynamic phenomena in supposing that electrons (whose name was not yet invented) exercise, one upon another, attractions and repulsions directed along the straight joining them, and depending not only upon their distances, but upon the first and second derivatives of these distances, consequently upon their velocities and their accelerations. this law of weber, different enough from those which to-day tend to prevail, none the less presents a certain analogy with them.

tisserand found that, if the newtonian attraction conformed to weber’s law there resulted, for mercury’s perihelion, secular variation of 14′′, of the same sense as that which has been observed and could not be explained, but smaller, since this is 38′′.

let us recur to the hypotheses a, b and c, and study first the motion of a planet attracted by a fixed center. the hypotheses b and c are no longer distinguished, since, if the attracting point is fixed, the field it produces is a purely electrostatic field, where the attraction varies inversely as the square of the distance, in conformity with coulomb’s electrostatic law, identical with that of newton.

the vis viva equation holds good, taking for vis viva the new definition; in the same way, the equation of areas is replaced by another equivalent to it; the moment of the quantity of motion is a constant, but the quantity of motion must be defined as in the new dynamics.

the only sensible effect will be a secular motion of the perihelion. with the theory of lorentz, we shall find, for this motion, half of what weber’s law would give; with the theory of abraham, two fifths.

if now we suppose two moving bodies gravitating around their common center of gravity, the effects are very little different, though the calculations may be a little more complicated. the motion of mercury’s perihelion would therefore be 7′′ in the theory of lorentz and 5′′.6 in that of abraham.

the effect moreover is proportional to n3a2, where n is the star’s mean motion and a the radius of its orbit. for the planets, in virtue of kepler’s law, the effect varies then inversely as √a5; it is therefore insensible, save for mercury.

it is likewise insensible for the moon though n is great, because a is extremely small; in sum, it is five times less for venus, and six hundred times less for the moon than for mercury. we may add that as to venus and the earth, the motion of the perihelion (for the same angular velocity of this motion) would be much more difficult to discern by astronomic observations, because the excentricity of their orbits is much less than for mercury.

to sum up, the only sensible effect upon astronomic observations would be a motion of mercury’s perihelion, in the same sense as that which has been observed without being explained, but notably slighter.

that can not be regarded as an argument in favor of the new dynamics, since it will always be necessary to seek another explanation for the greater part of mercury’s anomaly; but still less can it be regarded as an argument against it.

3

the theory of lesage

it is interesting to compare these considerations with a theory long since proposed to explain universal gravitation.

suppose that, in the interplanetary spaces, circulate in every direction, with high velocities, very tenuous corpuscles. a body isolated in space will not be affected, apparently, by the impacts of these corpuscles, since these impacts are equally distributed in all directions. but if two bodies a and b are present, the body b will play the r?le of screen and will intercept part of the corpuscles which, without it, would have struck a. then, the impacts received by a in the direction opposite that from b will no longer have a counterpart, or will now be only partially compensated, and this will push a toward b.

such is the theory of lesage; and we shall discuss it, taking first the view-point of ordinary mechanics.

first, how should the impacts postulated by this theory take place; is it according to the laws of perfectly elastic bodies, or according to those of bodies devoid of elasticity, or according to an intermediate law? the corpuscles of lesage can not act as perfectly elastic bodies; otherwise the effect would be null, since the corpuscles intercepted by the body b would be replaced by others which would have rebounded from b, and calculation proves that the compensation would be perfect. it is necessary then that the impact make the corpuscles lose energy, and this energy should appear under the form of heat. but how much heat would thus be produced? note that attraction passes through bodies; it is necessary therefore to represent to ourselves the earth, for example, not as a solid screen, but as formed of a very great number of very small spherical molecules, which play individually the r?le of little screens, but between which the corpuscles of lesage may freely circulate. so, not only the earth is not a solid screen, but it is not even a cullender, since the voids occupy much more space than the plenums. to realize this, recall that laplace has demonstrated that attraction, in traversing the earth, is weakened at most by one ten-millionth part, and his proof is perfectly satisfactory: in fact, if attraction were absorbed by the body it traverses, it would no longer be proportional to the masses; it would be relatively weaker for great bodies than for small, since it would have a greater thickness to traverse. the attraction of the sun for the earth would therefore be relatively weaker than that of the sun for the moon, and thence would result, in the motion of the moon, a very sensible inequality. we should therefore conclude, if we adopt the theory of lesage, that the total surface of the spherical molecules which compose the earth is at most the ten-millionth part of the total surface of the earth.

darwin has proved that the theory of lesage only leads exactly to newton’s law when we postulate particles entirely devoid of elasticity. the attraction exerted by the earth on a mass 1 at a distance 1 will then be proportional, at the same time, to the total surface s of the spherical molecules composing it, to the velocity v of the corpuscles, to the square root of the density ρ of the medium formed by the corpuscles. the heat produced will be proportional to s, to the density ρ, and to the cube of the velocity v.

but it is necessary to take account of the resistance experienced by a body moving in such a medium; it can not move, in fact, without going against certain impacts, in fleeing, on the contrary, before those coming in the opposite direction, so that the compensation realized in the state of rest can no longer subsist. the calculated resistance is proportional to s, to ρ and to v; now, we know that the heavenly bodies move as if they experienced no resistance, and the precision of observations permits us to fix a limit to the resistance of the medium.

this resistance varying as sρv, while the attraction varies as s√(ρv), we see that the ratio of the resistance to the square of the attraction is inversely as the product sv.

we have therefore a lower limit of the product sv. we have already an upper limit of s (by the absorption of attraction by the body it traverses); we have therefore a lower limit of the velocity v, which must be at least 24·1017 times that of light.

from this we are able to deduce ρ and the quantity of heat produced; this quantity would suffice to raise the temperature 1026 degrees a second; the earth would receive in a given time 1020 times more heat than the sun emits in the same time; i am not speaking of the heat the sun sends to the earth, but of that it radiates in all directions.

it is evident the earth could not long stand such a régime.

we should not be led to results less fantastic if, contrary to darwin’s views, we endowed the corpuscles of lesage with an elasticity imperfect without being null. in truth, the vis viva of these corpuscles would not be entirely converted into heat, but the attraction produced would likewise be less, so that it would be only the part of this vis viva converted into heat, which would contribute to produce the attraction and that would come to the same thing; a judicious employment of the theorem of the viriel would enable us to account for this.

the theory of lesage may be transformed; suppress the corpuscles and imagine the ether overrun in all senses by luminous waves coming from all points of space. when a material object receives a luminous wave, this wave exercises upon it a mechanical action due to the maxwell-bartholi pressure, just as if it had received the impact of a material projectile. the waves in question could therefore play the r?le of the corpuscles of lesage. this is what is supposed, for example, by m. tommasina.

the difficulties are not removed for all that; the velocity of propagation can be only that of light, and we are thus led, for the resistance of the medium, to an inadmissible figure. besides, if the light is all reflected, the effect is null, just as in the hypothesis of the perfectly elastic corpuscles.

that there should be attraction, it is necessary that the light be partially absorbed; but then there is production of heat. the calculations do not differ essentially from those made in the ordinary theory of lesage, and the result retains the same fantastic character.

on the other hand, attraction is not absorbed by the body it traverses, or hardly at all; it is not so with the light we know. light which would produce the newtonian attraction would have to be considerably different from ordinary light and be, for example, of very short wave length. this does not count that, if our eyes were sensible of this light, the whole heavens should appear to us much more brilliant than the sun, so that the sun would seem to us to stand out in black, otherwise the sun would repel us instead of attracting us. for all these reasons, light which would permit of the explanation of attraction would be much more like r?ntgen rays than like ordinary light.

and besides, the x-rays would not suffice; however penetrating they may seem to us, they could not pass through the whole earth; it would be necessary therefore to imagine x′-rays much more penetrating than the ordinary x-rays. moreover a part of the energy of these x′-rays would have to be destroyed, otherwise there would be no attraction. if you do not wish it transformed into heat, which would lead to an enormous heat production, you must suppose it radiated in every direction under the form of secondary rays, which might be called x′′ and which would have to be much more penetrating still than the x′-rays, otherwise they would in their turn derange the phenomena of attraction.

such are the complicated hypotheses to which we are led when we try to give life to the theory of lesage.

but all we have said presupposes the ordinary laws of mechanics.

will things go better if we admit the new dynamics? and first, can we conserve the principles of relativity? let us give at first to the theory of lesage its primitive form, and suppose space ploughed by material corpuscles; if these corpuscles were perfectly elastic, the laws of their impact would conform to this principle of relativity, but we know that then their effect would be null. we must therefore suppose these corpuscles are not elastic, and then it is difficult to imagine a law of impact compatible with the principle of relativity. besides, we should still find a production of considerable heat, and yet a very sensible resistance of the medium.

if we suppress these corpuscles and revert to the hypothesis of the maxwell-bartholi pressure, the difficulties will not be less. this is what lorentz himself has attempted in his memoir to the amsterdam academy of sciences of april 25, 1900.

consider a system of electrons immersed in an ether permeated in every sense by luminous waves; one of these electrons, struck by one of these waves, begins to vibrate; its vibration will be synchronous with that of light; but it may have a difference of phase, if the electron absorbs a part of the incident energy. in fact, if it absorbs energy, this is because the vibration of the ether impels the electron; the electron must therefore be slower than the ether. an electron in motion is analogous to a convection current; therefore every magnetic field, in particular that due to the luminous perturbation itself, must exert a mechanical action upon this electron. this action is very slight; moreover, it changes sign in the current of the period; nevertheless, the mean action is not null if there is a difference of phase between the vibrations of the electron and those of the ether. the mean action is proportional to this difference, consequently to the energy absorbed by the electron. i can not here enter into the detail of the calculations; suffice it to say only that the final result is an attraction of any two electrons, varying inversely as the square of the distance and proportional to the energy absorbed by the two electrons.

therefore there can not be attraction without absorption of light and, consequently, without production of heat, and this it is which determined lorentz to abandon this theory, which, at bottom, does not differ from that of lesage-maxwell-bartholi. he would have been much more dismayed still if he had pushed the calculation to the end. he would have found that the temperature of the earth would have to increase 1012 degrees a second.

4

conclusions

i have striven to give in few words an idea as complete as possible of these new doctrines; i have sought to explain how they took birth; otherwise the reader would have had ground to be frightened by their boldness. the new theories are not yet demonstrated; far from it; only they rest upon an aggregate of probabilities sufficiently weighty for us not to have the right to treat them with disregard.

new experiments will doubtless teach us what we should finally think of them. the knotty point of the question lies in kaufmann’s experiment and those that may be undertaken to verify it.

in conclusion, permit me a word of warning. suppose that, after some years, these theories undergo new tests and triumph; then our secondary education will incur a great danger; certain professors will doubtless wish to make a place for the new theories.

novelties are so attractive, and it is so hard not to seem highly advanced! at least there will be the wish to open vistas to the pupils and, before teaching them the ordinary mechanics, to let them know it has had its day and was at best good enough for that old dolt laplace. and then they will not form the habit of the ordinary mechanics.

is it well to let them know this is only approximative? yes; but later, when it has penetrated to their very marrow, when they shall have taken the bent of thinking only through it, when there shall no longer be risk of their unlearning it, then one may, without inconvenience, show them its limits.

it is with the ordinary mechanics that they must live; this alone will they ever have to apply. whatever be the progress of automobilism, our vehicles will never attain speeds where it is not true. the other is only a luxury, and we should think of the luxury only when there is no longer any risk of harming the necessary.