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NATURE 



[June 23, 1898 



that these elements will not only regain their original values, but 

 that they will never deviate much from them. 



This last demonstration has never been given in a definite 

 manner, and it is even probable that the proposition is not strict^ 

 true. The statement that is true, is that the elements can only 

 deviate extremely slowly from their original values, and this 

 after a long interval of time. To go further, and affirm that 

 these elements will remain not for a very long time, but always 

 confined within narrow limits, is what we cannot do. 

 But the problem does not take this form. 

 The mathematician only considers fictitious bodies, reduced 

 to simple material points, and subject to the exclusive action of 

 their mutual attractions, which rigorously follows Newton's law. 

 How would such a system behave, would it be stable ? This is 

 a problem which is as difficult as it is interesting for an analyst. 

 But it is not one which actually occurs in nature. Real bodies 

 are not material points, and they are subject to other forces 

 than the Newtonian attraction. These complementary forces 

 ought to have the effect of gradually modifying the orbits, even 

 when the fictitious bodies, considered by the mathematician, 

 possess absolute stability. 



What we must ask ourselves then is, whether this stability 

 will be more easily destroyed by the simple action of Newtonian 

 attraction or by these complementary forces. 



When the approximation shall be pushed so far that we are 

 certain that the very slow variations, which the Newtonian 

 attraction imposes on the orbits of the fictitious bodies, can 

 only be very small during the time that suffices for the comple- 

 mentary forces to destroy the system ; when, I say, the 

 approximation shall be pushed as far as that, it will be useless 

 to go further, at least from the point of view of application, and 

 we must consider ourselves satisfied. 



But it seems that this point is attained ; without quoting 

 figures, I think that the effects of these complementary forces 

 are much greater than those of the terms neglected by the 

 analysts in the most recent demonstrations on stability. 



Let us see which are the most important of these complementary 

 forces. The first idea which comes to mind is that Newton's 

 law is, doubtless, not absolutely correct ; that the attraction is not 

 rigorously proportional to the inverse square of the distances, 

 but to some other function of them. In this way Prof. Newcomb 

 has recently tried to explain the movement of the perihelion of 

 Mercury. But it is soon seen that this would not influence the 

 stability. It is true, according to a theory of Jacobi, that there 

 would be instability if the attraction were inversely proportionate 

 to the cube of the distance. It is easy by rough reasoning to 

 account for this ; with such a law, the attraction would be great 

 for the small distances and extremely feeble for great distances. 

 If therefore, for any reason, the distance of one of the planets 

 from the central body were to increase, the attraction would 

 diminish rapidly until it would not be capable of retaining the 

 planet in its orbit. But that only takes place with laws very 

 different from that of the square of the distances. All laws, 

 near enough to that of Newton's to be acceptable, are equivalent 

 from the stability point of view. 



But there is another reason which opposes the theory that 

 bodies move without ever deviating much from their original 

 orbits. According to the second law of thermodynamics, known 

 by the name of Carnot's Principle, there is a continual dissipa- 

 tion of energy, which tends to lose the form of mechanical work 

 and to take the form of heat. There exists a certain func- 

 tion called entropy, which it is unnecessary to define here ; 

 entropy, according to this second law, either remains con- 

 stant or diminishes, but can never increase. When once it has 

 deviated from its original value, which it can only do by 

 diminishing, it can never return again, as it would have to 

 increase. The world consequently could never return to its 

 original state, or to a slightly different state, so soon as its 

 entropy has changed. It is the contrary of stability. 



But the entropy diminishes every time that an irreversible 

 phenomenon takes place, such as the friction of two solids, the 

 movement of a viscous liquid, the exchange of heat between 

 two bodies of different temperatures, the heating of a conductor 

 by the passage of a current. If we observe, then, that there is not 

 in reality a reversible phenomenon, that the reversibility is only 

 a limiting case — an ideal case which nature can more or less 

 approach but can never attain — we shall be led to conclude that 

 instability is the law of all natural phenomena. 



Are the movements of the heavenly bodies the only ones to 

 escape ? One might believe it by seeing that they move in a 



NO. 1495, VOL. 58] 



vacuum, and are thus free from friction. But is the inter- 

 planetary vacuum absolute, or do the bodies move in an ex- 

 tremely attenuated medium of which the resistance is extremel) 

 feeble, but nevertheless is capable of offering resistance ? 



Astronomers have only been able to explain the movement of 

 Encke's comet by supposing the existence of such a medium. But 

 the resisting medium which would account for the anomalies 

 of this comet, if it exists, is confined to the immediate neigh- 

 bourhood of the sun. This comet would penetrate it ; but at the 

 distances at which the planets are, the action of this medium 

 would cease to make itself felt, or would become much more 

 feeble. As an indirect effect, it would accelerate the move- 

 ments of the planets ; losing energy, they would tend to fall on 

 the sun, and by reason of Kepler's third law the duration of the 

 revolution would diminish at the same time as the distance to 

 the central body. But it is impossible to form an idea of the 

 rapidity with which this effect would be produced, as we have 

 no notion of the density of this hypothetical medium. 



Another cause to which I am now going to refer must have, 

 it seems, a more rapid action. It had for some time been 

 imagined, but was first more especially brought to light by 

 Delaunay, and afterwards by G. Darwin. 



The tides, which are direct consequences of celestial move- 

 ments, could only stop if these movements ceased. But the 

 oscillations of the seas are accompanied by friction, and conse- 

 quently produce heat. This heat can only be borrowed from the 

 energy which produces the tides — that is to say, to the vis viva 

 of the celestial bodies. We can therefore foresee that, for this 

 reason, Xkix?, vis viva is gradually dissipated, and a little reflec- 

 tion will enable us to understand by what mechanism. The 

 surface of the seas, raised by the tides, presents a kind of wave. 

 If high tide took place at the time of the meridian passage of 

 the moon, this surface would be that of an ellipsoid, the axis of 

 which would pass through the moon. Everything would be 

 symmetrical in relation to this axis, and the attraction of the 

 moon on this wave could neither slow down nor accelerate the 

 terrestrial rotation. This is what would happen if there were 

 no friction ; but in consequence of this friction, high tide is late 

 on the moon's meridian passage ; symmetry ceases ; the attrac- 

 tion of the moon on the wave no longer passes through the 

 centre of the earth, and tends to slow down the rotation of our 

 globe. 



Delaunay estimated that, for this cause, the length of the 

 sidereal day increases by one second in a hundred thousand 

 years. It is thus he wished to account for the secular acceler- 

 ation of the moon's motion. The lunation would seem to us to 

 become shorter and shorter, because the unit of time to which 

 we ascribed it, the day, would become longer and longer. 



Whatever we may think of the figures given by Delaunay, 

 and the explanation which he proposes for the anomalies 

 of the moon's movement, it is difficult to dispute the effect 

 produced by the tides. 



It is just this that may help us to understand a well-known 

 but very surprising fact. It is known that the period of rotation 

 of the moon is exactly equal to that of its revolution ; in such a 

 way that, if there were seas on this body, they would have no 

 tides — at least, tides due to the attraction of the earth ; because 

 for an observer situated at a point on the surface of the moon, 

 the earth would be always at the same height above the horizon. 

 It is also known that Laplace tried to explain this curious 

 coincidence. How can the two velocities be exactly the same ? 

 It is exceedingly improbable that this strict equality is due to 

 mere chance. Laplace supposes that the moon has the form of 

 an elongated ellipsoid ; this ellipsoid behaves like a pendulum, 

 which would be in equilibrium when the major axis is directed 

 along the line joining the centres of the two bodies. 



If the initial velocity of rotation differs slightly from that 

 of revolution, the ellipsoid will oscillate about its position of 

 equilibrium without ever deviating much from it. A pendulum 

 which has received a slight impetus behaves in this way. The 

 mean velocity of rotation is then exactly the same as that of 

 the position of equilibrium round which the major axis oscillates ; 

 it is, therefore, the same as that of the straight line which joins 

 the centres of the two bodies. It is therefore strictly equal to 

 the velocity of revolution. 



If, on the contrary, the initial velocity differs considerably 

 from the velocity of revolution, the major axis will not oscillate 

 any more round its position of equilibrium, like a pendulum 

 which under a strong impulse describes a complete circle. 



It suffices, therefore, that the velocity of revolution should be 



