Ill 



GRAVITATION. 



GRAVITATION. 



n 



whU. the disturbing planet has been at almost every put of iU orbifr 

 The disturbing force a always the difference of the fbrcei which act 

 on Ui sun and on the disturbed planet. A* the disturbing pUnet, in 

 thnas various positions, acts upon the ran in all direction* in the plane 

 of iU orbit, iU effect on the nan may be wholly neglected ; and then 

 it U easy to ee that, whether the disturbing planet be exterior ..r in- 

 terior to the other, the combined offset of the force* in all these point* 

 on the disturbed planet at one point U to pull it from it* orbit towards 

 the plan* of the disturbing planet'! orbit (Thin depend* upon the 

 circumstance that the force U greateat when the disturbing planet ia 

 nearest.) Consequently, by (192), the line of nodes of the disturbed 

 planet'* orbit on the disturbing planet's orbit, in the long run, always 

 regresses. If the orbiu are circular, there is no alteration of the in- 

 clination, since, at points equally distant from the highest point, there 

 is the same fore* on the disturbed planet ; and, therefore, by (192), the 

 inclination is increased at one time, and diminished as much at another. 

 If the orbits an elliptic, one point may be found where the effect of 

 the force on the inclination ia greater than at any other, and the whole 

 fleet on the inclination will be similar to that. 



(-11.) In stating that the nodes always regress in the long run, the 

 reader must be careful to restrict this expression to the sense of re- 

 gressing on the orbit of the disturbing planet. It may happen that 

 on another orbit they will appear to progress. Thus the nodes of 

 Jupiter's orbit are made to regress on Saturn's orbit by Saturn's dis- 

 turbing force. The nodes of these orbits on the earth's orbit are not 

 vary widely separated ; but the inclination of Saturn's orbit U greater 

 than that of Jupiter's. If we trace these on a celestial globe, we shall 

 hare such a figure aa fg. 46, where EC represents the plane of the 



\ '.; 



earth's orbit, J B the orbit of Jupiter, and ST that of Saturn. The 

 orbit of Jupiter, by regressing on Saturn's orbit, assumes the position 

 of the dotted line je; but it is plain that the intersection of this 

 orbit with the earth's orbit has gone in the opposite direction, or has 

 progressed. If the motion of the node on Saturn's orbit from J toj 

 is regression, the motion of the node on the earth's orbit from E to c 

 must be progression. 



(212.) There is a remarkable relation between the inclinations of all 

 the orbits of the planetary system to a fixed plane, existing through 

 all their secular variations, similar to that between their eccentricities. 

 The sum of the products of each mass, by the square root of the 

 major axis of its orbit, and by the square of the inclination to a fixed 

 plane, is invariable. 



(213.) The disturbance of Jupiter's satellites in latitude presents 

 circumstances not less worthy of remark than the disturbance in lon- 

 gitude. The masses are so small, and their orbits so little inclined to 

 each other, that the small inequalities produced in a revolution may be 

 neglected. Even that depending on the slow revolution of the line of 

 conjunctions of the first three satellites, so small is the mutual in- 

 clination of their orbits, does not amount to a sensible quantity. We 

 shall therefore consider only those alterations in the position of the 

 planes of the orbits which do not vary sensibly in a small number of 

 revolutions. For this purpose we must introduce a term which has 

 not been introduced before. 



(214.) If the moon revolved round the earth in the same plane in 

 which the earth revolves round the sun, the sun's attraction wmiM 

 never tend to draw the moon out of that plane. But (taking the cir- 

 cumstances as they really exist) the moon revolves round the earth in 

 a plane inclined to the plane in which the earth revolves round the sun ; 

 and the consequence, aa we have seen is, that the line of nodes upon 

 the latter plane regresses, and the inclination of the orbit to the latter 

 plane remains, on the whole, unaltered. The plane of the earth's orbit, 

 then, may be considered a fundamental plane to the moon's motion ; by 

 which term we mean to express, that if the moon moved in that plane, 

 the disturbing force would never draw her out of it ; and that if she 

 moved in an orbit inclined to it, the orbit would always be inclined at 

 nearly the same angle to that plane, though its line of nodes had 

 sensibly altered. The latter condition will, in general, be a consequence 

 of the former. 



(215.) In order to discover what will be the fundamental plane for 

 one of Jupiter's satellites, we must consider that, besides the sun's 

 attraction, there is another and more powerful disturbing force acting 

 on these bodies, namely, the irregularity of attraction produced by 

 Jupiter's flatness. The effect of this (as we shall show) is always to pull 

 the satellites towards the plane of Jupiter's equator. If Jupiter were 

 spherical, the only disturbing force would be the sun's attraction, tend- 

 ing on the whole to draw the satellite towards the plane of Jupiter ' 

 orbit, and that plane would be the fundamental plane of the satellite. 

 If Jupiter were flattened, and if the sun did not disturb the satellite, the 

 irregularity in Jupiter's shape would always tend to draw the satellite 

 towards the plane of his equator, and the plane of his equator would 

 be the fundamental plane of the satellite. As both causes exist, the 



position of the actual fundamental plane must be found by the following 

 consideration. We must discover the position of a plane from win. li 

 the son'* disturbing force tends, on the whole, to draw the satellite 

 downwards, and the disturbing force depending upon Jupiter's shape 

 tends to draw it upwards (or rt'ee vend), by equal quantities ; and that 

 plane will be the fundamental plane. This plane must lie Ix-lirtm the 

 planes of Jupiter's orbit and Jupiter's equator, because thus only can 

 the disturbing forces act in opposite ways, and therefore balance each . 

 other : and it must pass through their intersection, aa otherwise it 

 would at that part be above both or below both, and the forces depending 

 on both causes would act the same way. 



(216.) The disturbing force of the sun, as we have seen (88), Ac., U 

 greater as the satellite is more distant ; the disturbing force de] 

 on Jupiter's shape la then less, as we shall mention hereafter, 

 sequently, as the satellite is more distant, the effect of the sun's disturb- 

 ing force is much greater in proportion to that depending on Jupiter's 

 shape. Thus, if there were a single satellite at the distance of Jupiter's 

 first satellite, its fundamental plane would nearly coincide with the 

 plane of Jupiter's equator; if at the distance of Jupiter's second satel- 

 lite, its fundamental plane would depart a little farther from coin- 

 cidence with the plane of the equator; and so on for other distances; 

 and if the distance were very great, it would nearly coincide with the 

 plane of Jupiter's orbit. If, then, Jupiter's four satellites did not 

 disturb each other, each of them would have a separate fundamental 

 plane, and the positions of these planes would depend only upon each 

 satellite's distance from Jupiter. 



(217.) In fact, the satellites do disturb each other. In speaking <>f 

 the planets (210), we have observed that the effect of theattra. 

 one planet upon another, in the long run, is to exert a disturbing force 

 tending to draw that other planet (at any part of it orbit) towai 

 plane of the first planet's orbit. The same thing is true of Jupitrr's 

 satellites. Now, though each of them moves generally in an orbit 

 inclined to its fundamental plane, yet in the long run (when the nodes 

 of the orbit have regressed many times round), we may consider the 

 motion of each satellite as taking place in its fundamental plane. The 

 ii, therefore, must now be stated thus. The four satellites are 

 revolving in four different fundamental planes ; and the position of 

 each of these planes is to be determined by the consideration that the 

 satellite in that plane is drawn towards the plane of Jupiter's orbit by 

 the sun's disturbing force, towards the plane of Jupiter's equator by 

 the force depending on Jupiter's shape, and towards the plane of each 

 of the other three satellites, by the disturbing force produced by each 

 satellite : and those forces must balance in the long run. 



(218.) The determination of these planes is not very difficult, when 

 general algebraical expressions have been investigated for the magnitude 

 of each of the forces. The general nature of the results will be easily 

 seen; the several fundamental planes will be drawn nearer t< 

 (that of the first satellite, that of the second, and that of the third, will 

 be* drawn nearer to Jupiter's orbit, while that of the fourth will lw 

 drawn nearer to Jupiter's equator). The four planes will still pass 

 through the intersection of the plane of Jupiter's equator with that of 

 Jupiter's orbit. Thus, if we conceive the eye to be placed at a great 

 distance, in the intersection of the planes of Jupiter's orbit and Jupit. T'R 

 equator, and if the dotted lines in fg. 47 represent the appearance of 



rig. 47. 



the fundamental planes which would exist if the satellites did not 

 disturb each other, then the dark lines will represent the positions of 

 these planes as affected by the mutual disturbances. The inclination < f 

 Jupiter's equator to Jupiter's orbit is about 8 6'; and so great i* the 

 effect of his shape, that the fundamental plane of the first satellite in in- 

 clined to his equator by only 7" ; that of the second satellite by about 

 1 ' ; that of the third by about 6' ; and that of the fourth by about 24 i'. 

 Without mutual perturbation, the inclinations to Jupiter's equator 

 would have been about 2", 20", 4', and 48'. 



(219.) Having considered the positions of the fundamental planes, 

 we shall now consider the motion of a satellite, when moving in an 

 orbit inclined to its fundamental piano. 



(220.) The general effect will be of the same kind as that fr the 

 moon. Since the disturbing force which then tends to pull it from the 

 plane of its orbit, tends to pull it towards the fundamental plane (as, 

 supposing the satellite to be on that side of the fundamental plane next 

 the plane of Jupiter's equator, the sun's disturbing force toward* 

 Jupiter's orbit is increased, that towards Jupiter's equator is diminished, 

 ana so for the others), the line of nodes will regress on the fun. I 

 plane. The inclination on the whole will not be altered. That part of 

 the regression of the nodes which depends on the sun's disturbing force 

 will be greater for the distant satellites than for the near OUCK ; but 

 that which depends on the shape of Jupiter (and which in much more 

 important) will be greater for the near satellites than for the distant 

 ones. On the whole, therefore, the lines of nodes of the interior satel- 



