609 



GRAVITATION. 



GRAVITATION. 



610 



draw it to the sun is greatest ; and here the mean distance is greater 

 than it would have been with the same periodic time, had there been 

 no disturbance. But so long as these general effects in the force 

 directed to the sun continue unaltered, the mean distances will not 

 alter (46), ic. Now, upon taking a very long peiiod (as several thou- 

 sand years), it is easy to see that, if we divide that period into two or 

 three parte, the two planets have in each of those two parts been in 

 conjunction indifferently in all parts of their orbita ; that they have 

 had every possible relative position, in every part ; and that (if we make 

 the periods long enough) the force which one planet has sustained in 

 any one point will be accurately the mean of all which it would sus- 

 tain, if we estimated all those that it could suffer from supposing the 

 other planet to go with its usual motion through the whole of its orbit. 

 As this mean will be the same for each of the periods, there will, in the 

 long run, be no alteration of the force in the direction of the radius 

 vector, and we may assert at once that the mean distance cannot be 

 altered by it. 



(181.) But with regard to the disturbing force acting perpendicu- 

 larly to the radius vector, the circumstances are different. The mere 

 existence of such a force, without variation, causes an alteration in the 

 mean distance (48) ; and it is necessary to show that the nature and 

 variations of the force are such that, in the long run, the velocity of 

 the disturbed planet is not affected by it. For this purpose, instead of 

 considering merely the disturbing force perpendicular to the radius 

 vector, we will consider separately the whole force which the disturbing 

 planet exerts on the sun, and the whole force which it exerts on the 

 I jed planet. Now, the force which it exerts on the sun tends to 

 pull the sun sometimes in one direction and sometimes in another, but 

 on the whole produces no permanent displacement : this force, then, 

 may at once be neglected. The force which one planet has exerted on 

 the other has acted when, for any arbitrary position of the disturbing 

 planet, the disturbed planet has been at every point of its orbit. Since 

 the whole acceleration produced in a long time is the sum of all the 

 accelerations diminished by the sum of all the retardations, we may 

 divide them into groups as we please, and sum each group. Let us, 

 then, group together all the accelerations and retardations produced in 

 one position of the disturbing planet. The disturbed planet having 

 been in every small part of its orbit, during a time proportional to the 

 time which it would occupy in passing through that small part in any 

 one revolution, the various accelerations and retardations will bear the 

 same proportion as if the disturbed planet had made one complete 

 revolution, and the disturbing planet had been fixed. Now, it is a 

 well-known theorem of mechanics, that when a body moves through 

 any curve, acted on by the attractions of any fixed bodies, its velocity, 

 when it reaches the point from which it started, is precisely the same 

 as when it started : the accelerations and retardations having exactly 

 balanced. Consequently, in the case before us, if the disturbing planet 



en fixed, and the disturbed planet had made one complete revo- 

 lution, the latter would, on the whole, have been neither accelerated 

 nor retarded ; and, therefore, in the long run, all the accelerations and 

 retardations of the disturbed planet, produced in any arbitrary position 

 of the disturbing planet, will exactly balance. The same may be shown 

 for every jmsition of the disturbing planet; and thus, on the whle, 

 there is no alteration of velocity. Since, then, in the long run, the 

 planet's velocity is not altered, and since (180) the force directed to 

 the sun is not altered, the planet's mean distance will not be altered. 

 ThU reasoning docs not prevent the increase or diminution of the velo- 

 city at particular parts of the orbit, and therefore the excentricity and 

 the line of apses may vary ; but it shows that, if there is an increase at 

 one part, there is a diminution that balances it at another ; and at the 

 p'>int where the orbit at the beginning of a long time and the orbit at 



1 of that time intersect (which will be at mean distance nearly) 

 the velocity will not be altered. 



( )ur demonstration supposes that the portions of the curves described 

 in different revolutions, for the same position of the disturbing planet, 

 are parts of one orbit, and therefore does not take account of the 

 alteration in the magnitude of the disturbing force produced by the 

 alteration of place which that force has previously caused. This has 



iken into account, to a certain degree, by several mathematicians; 

 and it appears, as far as they have gone, that this produces no alteration 

 in the conclusion. 



(182.) Secondly, as to the place of perihelion, or the position of the 

 line of apses. The motion of this will depend essentially on the excen- 

 tricity of the orbit of the disturbing planet. Suppose, for instance, that 

 the orbit of Venus was elliptical and the earth's orbit circular ; as the 

 distance of these pL-mets in conjunction is Jittle more than one-fourth of 

 the earth's distance from the sun, the ellipticity of the orbit of Venus 

 would bring that planet at aphelion so much nearer to the earth's orbit, 

 that tiy far the greatest effect would take place when in conjunction 

 there ; and this, by (54), would make Venus' line of apses progress. 

 But if the earth's orbit were more elliptic than that of Venus, and if 

 fie earth's perihelion were on the same side of the sun as the perihelion 

 .t, happen that the principal action would take place 

 at perihelion, m;<l then, by (51), the line of a).-. uress. These 



effects would continue to go on, while the relative position of the lines 

 of aps<, and the proportion of the excentricities, remained nearly the 

 name. As, in the long run, conjunctions would happen everywhere, the 

 preponderating effect would be similar to the greatest effect ; and thus 



the secular motion of the line of apses will be constant (till the positions 

 of the lines of apses, &c., shall have changed considerably) ; its magni- 

 tude and direction will depend on the excentricities of both orbits ; but 

 if the disturbed planet is the interior, and if the orbit of the other be 

 not excentric, the line of apses will progress. The same is true if tho 

 disturbed planet is exterior (the greatest action being then at the peri- 

 helion, if the interior orbit have no excentricity, and being directed to 

 the sun). 



(183.) Thirdly, as to the excentricity. If the orbit of the disturbing 

 planet were circular, the effect on the excentricity produced by con- 

 junction at the place where the orbits are nearest, would be of one 

 kind before conjunction, and of the opposite kind after conjunction, 

 from the disturbing force in the radius vector ; and thus the excen- 

 tricity would not be altered. The same would happen if both orbits 

 were excentric, provided their lines of apses coincided. Thus it appears 

 that there is no variation of excentricity, except the orbit of the dis- 

 turbing planet is excentric, and its line of apses do not coincide with 

 that of the disturbed planet. When these conditions hold (as they do 

 in every planetary orbit), a general idea of the effect may be obtained 

 by finding where the orbits approach nearest ; then, if we consider the 

 disturbance of the interior planet, since the force draws it from the 

 sun , the excentricity will be increased if it is moving from perihelion, 

 or diminished if it is moving towards perihelion. For the exterior 

 planet, as the force draws it towards the sun, the conclusion will be 

 of the opposite kind. These effects are constant, till the excentricities 

 and the positions of the lines of apses have changed sensibly. The 

 place where the force at conjunction produces the greatest effect on 

 the excentricity may not be strictly the place where the orbits are 

 nearest, but probably will not be far removed from that place. 



At the place where the orbits approach nearest, both planets in 

 general are moving from perihelion, or both towards perihelion, so that 

 when one excentricity is increased, the other is diminished. 



(184.) For the general stability of the planetary system, the posi- 

 tions of the lines of apses are not important, but the permanency of 

 the major axes and the excentricities are of the greatest importance. 

 The conclusion which we have mentioned as to the absence of secular 

 variation of the major axis, from the action of one planet, applies also 

 to the disturbances produced by any number of planets, and thus we 

 can assert that the major axes of the orbits of the planets are not 

 subject to any secular variation. The excentricities are subject to 

 secular variation, but even this corrects itself in a very long time : 

 whefl the investigation is fully pursued, it is found that each of the 

 excentricities is expressed by a number of periodic terms, the period 

 of each being many thousands of years. Thus the major axis of the 

 earth's orbit, notwithstanding its small and frequent variations, has 

 not sensibly altered in many thousands of years, and will not sensibly 

 alter; the excentricity, besides suffering many small variations, has 

 steadily diminished for many thousands of years, and will diminish for 

 thousands of years longer, after which it will again increase. 



(185.) A remarkable relation exists between the variation of the 

 excentricities (of which that mentioned in (183) is a simple instance), 

 the result of which, as to the state of the excentricities at any time, is 

 given thus : The sum of the products of the square of each excentricity 

 by the mass of the planet, and by the square root of the major axis, is 

 always the same. 



SECTION Vlll.Per/uti/ation of Inclination and Place of Node. 



(186.) We have hitherto proceeded as if the sun, the moon, and all 

 the planets, revolved in the same plane as if, for instance, the sun 

 were fixed in the centre of a table, and all the planets, with their 

 satellites, revolved on the surface of the table. But this supposition 

 is not true. If we suppose the earth to revolve on the surface of the 

 table, the moon will, in half her revolution (we mean while she 

 describes 180, not necessarily in half her periodic time), rise above 

 the table, and in the other half she will go below it, crossing the 

 surface at two points which, as seen from the earth, are exactly 

 opposite. Venus will, in half her revolution, rise above the table, and 

 iu half will sink below it, crossing the table at two points which, as 

 seen from the sun, are exactly opposite ; each of the other planets and 

 satellites in like manner crosses the plane at points which, as seen 

 from the central body, are exactly opposite. In different investigations 

 it is necessary to consider the inclination of the plane of revolution or 

 the plane of the orbit to different planes of reference : the line in 

 which the plane of revolution crosses the plane of reference is called 

 the line of nodes on that jjlane ; and the angle which the plane of 

 revolution makes with the plane of reference is called the inclination 

 > /ilane. The plane of reference must always be supposed to pass 

 through the central body. 



(187.) The inclinations of all the orbits, except those of the small 

 planets, are so trifling (the largest namely, that of the moon's orbit 

 to the earth's orbit being, at its mean state, only 5) that they may 

 in general be wholly neglected in estimating the disturbance which 

 one planet produces in the motion of another in its own plane. In 

 some cases, however, as in the inequalities of long period, where the 

 effective force is only the small part which remains after a com- 

 pensation more or loss perfect, no alteration of the forces must be 

 neglected; and here, as we have hinted in (171), the inclinations 

 must be taken into account. 



