. 1 



GRAVITATIuK. 



GRAVITATION. 



|>rrijovo of the third, the effect of the disturbance on the MCpnd in 

 rather greater than at any other tune; and this produce* an irregu- 

 larity in the excentncity of the second, and in the motion ..f iu apses, 

 iU>lMi<liiix <>n the distance of the line of cunjuiictiun from the iutlejien- 

 dent |<erijove of the third. The departure from uniformity in the 

 angular motion of the third also produces a departur>> from uniformity 

 in the regression of the line of conjunction, and this contributes to the 

 same irregularity. 



(146.) the disturbing force in the direction of !the radius vector, 

 produced by an inner satellite, a sometimes directed to the central 

 bodr and sometimes from it ; but on the whole the former exceed* 

 the latter (86). Now the principal part of the effect really takes place 

 when the satellites are near conjunction ; consequently, when the lino 

 of conjunction panes near the independent perijove of the third 

 satellite, the force by which the third satellite is urged to the planet 

 is greater than at any other time ; and as the line of conjunction 

 revolves, the force alternately increases and diminishes. This pro- 

 duces an irregularity in the major axis, and consequently in thu 

 motion of tin- third satellite (47), depending on the distance of tin- line 

 of conjunction from the perijove of the third. 



(147.) The disturbing force in the direction of the radius vector 

 produced by an outer satellite U sometimes directed to the central 

 body and sometimes from it ; but on the whole the latter exceeds the 

 former (80). For the reasons therefore, in the lost article, there is in 

 the motion of the second satellite an irregularity depending on the 

 distance of the line of conjunction from the independent perijove of 

 the third, but opposite in its nature to that of the third satellite. 



ill-. I Koch of these irregularities in the motion of one of these 

 satellites produces an irregularity in the motion of the others; and 

 thus the whole theory becomes very complicated when we attempt to 

 take the minute irregularities into account. 



(149.) The motion of the fourth satellite is not related to the others 

 in the same way in which they are related among themselves. Its 

 |>criodic time is to the periodic time of the third nearly in the propor- 

 tion of 7 : 3. Some of the irregularities then which it experiences and 

 which it occasions are nearly similar to those in the motions of the 

 planets. These however are small : the most important ore those 

 depending on the changes in the elements which require many 

 revolutions of the satellites to go through all their various states, but 

 which nevertheless have been observed since the satellites were dis- 

 covered. We shall proceed with these. 



(150.) First, let us suppose that the third satellite has no exceu- 

 tricity independent of perturbation, and that the fourth satellite has a 

 sensible excentncity, its line of apses progressing very slowly, in 

 consequence principally of the shape of Jupiter (so slowly as not to 

 have gone completely round in eleven thousand revolutions of the 

 satellite). When each of the satellites bos revolved a few hundred 

 times round Jupiter, their conjunctions will have token place almost 

 indifferently in every part of their orbits. If the orbit of the fourth 

 as well as that of the third had no independent ellipticity, there would 

 be no remarkable change of shape produced by perturbation, as the 

 action of one satellite upon the other would be the same when in con- 

 junction in all the different parts of the orbit. But the orbit of the 

 fourth being excentric, the action of each satellite on the other is 

 greatest when the conjunction happens near the perijove of the fourth 

 satellite. We may consider then that the preponderating force takes 

 place at this part of the orbits ; and we have to inquire what form the 

 orbit of the third satellite must have, to preserve the some excentricity 

 at every revolution. It must lie remembered here that the effect of 

 Jupiter's shape is to cause a more rapid progress of the line of apses 

 of the third satellite, if its orbit be excentric, than of the line of apses 

 of the fourth, 



(151.) Considering then that the preponderating force on the third 

 satellite in the direction of the radius vector is directed from the 

 central body towards the perijove of the fourth, and that the prepon- 

 derating force perpendicular to the radius vector accelerates it as it 

 approaches that part, and retards it afterwards, it is plain from (51), 

 (65), and (66), that if the perijove of the third satellite were in that 

 position, the forces would cause the line of apses to regress; and this 

 regression, if the excentncity of the third Ito small, may i 

 siderable (though the prep , hich cause.* it in 



extremely small), and may overcome so inueh of the progression 

 caused by Jupiter's shape, as to make the real motion of the line >f 

 apses as nearly equal as we please to the motion of the line of apses of 

 the fourth. Hut the motion of thu line of apses of the fourth will 

 itself be affected (though very little) by the greater action of the third 

 satellite on it at the same place ; and the part in the radiu* 

 being directed at its perijove to the central body, and Hi 

 perpendicular to the radius vector retarding it before it reaches the 

 perijove, and accelerating it afterwards, will cause a small increase of 

 progression of iU apse. The state of things will be prnnaiirtit, so far 

 as dc|K..nd* on tlii-m- forces, when the incrroftoil proKrwaioii ,,f t' 

 of th. foniili lU-lliU- iseqiialto the dimin 

 apse of the third ; and thus th* progression ..: 



will 1.. increoN*). and the third Kitelhu-'s oil.it will Irive a 



iiiK indirection to th' ; the fourth, 



and an elongation iu the same direction as the apojove of the fourth. 

 This would be the case U the third satellite had no excel 



independent of perturbation ; but we may, as in other cases, consider 

 that the same kind of distortion will be produced in the orbit if it has 

 .in independent excentricity. 



.-. let us suppose the fourth satellite to have no excen- 

 :! pendent of perturbation, and the third satellite to have an 

 :)!! !, ii.l. ut exeentricity. The greatest action w.ll now be at the 

 apojove of the third satellite, and this will (though in a small degree) 

 cause the line of apses of the third satellite to progress ; that is, it will 

 increase the rapidity of progression which Jupiter's shape gives it. If 

 now we wish to discover the form of orbit r b satellite which 



will at every revolution preserve the same excenti ieity, and h 

 line of apses always corresponding ith that of the third satellit 



re progressing more rapidly than the shape of Ji 

 would make it progress, we must evidently suppose the perijove 

 fourth satellite turned towards the apojove of the third, and, by sup- 

 posing the excentricity small enough, the progression may be mode as 

 rapid as we please. Thus the effect of excentricity in the orl>ii 

 third satellite is, that its line of apses is made to progress rather ni.>re 

 rapidly, and that the orbit of the fourth satellite is compressed on tin' 

 -i<l- next the apojove of the third satellite, and elongated on the 

 site side. We have supposed for this investigation that the fourth 

 satellite hod no excentricity independent of perturbation, but th 

 elusion as to the distortion of the orbit may be applied if we suppose 

 it to hi lent exceutricity. 



(153.) In fact, the orbits of both the third and fourth satellites 

 have independent excentricities, and both our conclusions apply to 

 them. The fourth satellite, besides its independent exceutricity, bos 

 an excentricity impressed upon it, opposite in kind to that of the 

 third ; and the third satellite, besides its indc|>vudeut excentricity, 

 has an excentricity impressed upon it of the same kind as that of the 

 fourth. In the some manner, the orbits of the first and - 

 lites have small excentricities impressed on them, similar in tin : 

 to those of the third and fourth. 



(154.) It will readily be conceived that the excentricities of th> 

 of the third satellite will affect the great inequality <l:i7l which it 

 produces in the motion of the second ; and on the contrary, that the 

 inequality in the motion of the third produced by the attract 

 the second, will influence the effect of the third on the fourth. We 

 shall not however notice these further than to state that their effects 

 are small. 



(155.) We have now gone over the principal inequalities of the 

 motions of Jupiter's satellites. They are so much connected, and (as 

 we may say) so completely entangled, that though they may be ex- 

 plained in the way in which we have considered them, it would hardly 

 be possible to calculate them in that way. A mathematical process of 

 the most abstruse kind, which will at the some time embrace the 

 motions of all, is alone competent to this object. We shall however 

 have attained our end if we have given the reader a genei.il 

 the explanation of disturbances in the most curious and couipl 

 system that has ever been reduced to calculation. 



SECTION VII. Theory of Planett. 



(156.) The theory of the planets may be considered as holding a 

 middle place between that of our moon and that of Jupiter's satellites. 

 In our moon, the principal inequalities are those that exhibit them- 

 selves in nearly the same order at every revolution, or, at longest, iu 

 the earth's revolution round the sun, depending entirely upon the 

 position of the moon, the the lines of ajises. In 



Jupiter's satellites, some of the principal inequalities (as those of the 

 third and fourth satellites) do not depend at all upon the relative 

 position of the bodies, but depend on the position of the lines of 

 apses, whose revolutions, though slow, may yet be completely observed. 

 But in the planets the terms analogous to those which we have men- 

 tioned in the moon's motions are small ; the changes of elements are 

 so slow, that though they may be in some degree observed, many 

 thousands of years would be necessary to observe them completely. 

 The most remarkable irregularities ore those produced by changes in 

 > 1 lying several revolutions of the planets, and more, 

 nearly analogous to the mutual perturbations of the three first natel- 



.lupiter tlian to any other that we ha\ e wen; dilli riii 

 them however in this rcnpect, that for most, of them Independent 

 excentricities arc quite essential. 



(157.) There are, however, some terms very nearly similar t" 

 i the theory of the moon. Suppo.-e, for instance, v 

 ;h pcrtm Kitions of Mercury by Jupiter (whose distance from 

 the sun is more than thirteen times as great). This case is 

 exactly analogous to the cose of the moon disturbed by the .-un. And 

 in consequence, Mercury's orbit is flattened a little on nearest 



to and farthest from Jupiter ; but this effect is much disguised Ly the 

 effect of forces analogous to those mentioned in (94), which hen pie 



itea greatly : his line of apses progresses a lit 



liition. when Jupiter ii nearly in th.it. line and regresses a little whan 

 Jupiter IM in the line pci-|H-ndieii! little more 



excentric in thu former case, and a Uttle less so in the latter ; and hi., 

 orbit is a little larger when Jupiter is at (wrihelion than when at 

 aphelion. The same thing applies very nearly to thu disturbances of 

 Venus, the Earth, and Mars, produced by Jupiter. 



.) The instance taken above is almost an extreme one. \\l. ,L 



