110 



. ITATloX. 



GRAVITATION. 



effective attraction on a than if they were collected at A ; but tho 

 whole cnhcro produce* the nine effect M if iU whole mass wen 

 collected at A ; and, therefore, the part left after cutting away the 

 lump* at D and K produce* a greater attraction than if it* whole maa* 

 were collected at A. 



i Hut it in important to inquire, whether thU attraction i- 

 greater than if the nutter of the spheroid were collected at the centre, 

 in the same ].r..j...ui..n at all distances of the satellite. Fur thin 

 purpose, suppose tho distance of the satellite to be 20. The same 

 reasoning would show, that the attraction of the lump D, if placed at 



A, miiHt now be represented by jjjj = 0-0025 ; but that, if placed at 



20 

 D, iU effective attraction U repreeented by - "-^ = 0-002490058. 



The difference now is 0-000009347, or nearly louooo of the whole 



attraction of D. Consequently by removing the satellite to twice tho 

 distance fr.>m A, the difference between the effective .ittr.u-ti.ni of the 

 lump at A and at n bears to the whole attraction of the lump at A a 

 proportion four times smaller than before. Ami. therefore, the 

 attraction of the spheroid, though still greater than if its whole matter 

 illected at A, differs from that by a quantity, whose proportion 

 to the whole attraction U only one-fourth of wh.it it wan before. 1 f 

 we tried different distances in the same manner, wo should find, .is a 

 general law, that the proportion which the difference (of the actual 

 attraction, and the attraction supposing all the matter collected at tho 

 centre) bears to the latter, diminishes as the square of the distance 

 from A increases. 



1 1. The attraction of an oblate spheroid upon a satellite, or other 

 body, in the plane of its equator, may therefore, be stated thus : 

 There is tho same force as if .ill the matter of the spheroid were 

 collected at its centre, and, besides this, there is an additional force 

 depending upon the oblateness, whose proportion to the other force 

 diminishes as the square of the distance of the satellite is increased. 



(233.) Now, let us investigate the law according to which an oblate 

 spheroid attracts a body, situate in the direction of its axis. 



Proceeding in the same manner as before, and supposing the 

 distance A B to, be 10, the attraction of the lump, which at A would be 

 represented by '. will at D be represented by ,',. and will at K 

 be represented by -,},, (since the distances of D and E from B are 

 respectively 9 and 11). Hence, if the two equal lumps, D and E, were 



fig. 49. 



collected at the centre, their attraction on B would be rjrjr + T7wi = I<\ 



1UU 1UU O'l 



= 0-02. In the positions D and I the sum of their attractions on B is 

 1 1 

 gj + j2i = 0-0206100. The difference ia O'OOOOIOO, by which the 



attraction in the latter case is the greater. Consequently, the attraction 

 of the lumps in the positions n and E is greater than if they were col- 

 lected at the centre by nearly ^ths of their whole attraction ; but tho 

 attraction of the whole sphere is the same as if all the matter of tho 

 sphere were collected at the centre ; therefore, when these parts are 

 removed, they must leave a mass, whose attraction is less than if 

 its whole matter were collected in tho centre. With regard to the 

 alteration depending on the distance of B, it would be found, on trial, 

 to follow the same law as before. 



(234). The attraction of a spheroid on a body in the direction of its 

 axis may, therefore, be represented, by supposing the whole matter 

 collected at the centre, and then supposing the attraction to be 

 diminished by a force depending on the obbteness, whose proportion 

 to the whole force diminishes as the square of the distance of the body 

 is increased. 



(235). Since the attraction on a body, in the pi me of the equator, 

 is greater than if the mam of the spheroid were collected at its centre, 

 and the attraction on a body in the direction of the axis is lean, it "ill 

 readily l>c understood, that in taking directions, successively more and 

 more inclined to the equator, on both sides, the attraction successively 

 diminishes. And there is one inclination, at which the attraction in 

 exactly the same as if the whole mass of the spheroid wore, collected at 

 its centre. 



(236). Now, suppose that a satellite revolves in an orbit, which 

 ' - with the plane of the equator, or make* a small angle with 



it ; what will be the nature of iU orbit f For this investigation we have 

 only to consider, that there is acting upon the satellite n for 

 same as if all the matter of the spheroid were collected at its centre, 

 and, consequently, proportional inversely to the square of the distance, 

 ;m.l that, with this force only, the satellite would move in an elli|e, 

 whose focus coincided with the centre of the npherid. I'.ut beside* 

 this, there is a force always directed to tho centre, depending on the 

 oblaUme**. One effect of it will be, that the periodic time will U- 

 shorter with the same mean distance, or the mean distance greater 

 with the same periodic time, than if the former were the only force. 

 (46). Another effect will be, that when tho satellite is at its greatest 

 distance, this force will cause the line of apses to regress, and when at 

 its smallest distance, this force will cause the line of apses to progress, 

 (50) and (58). If this force, at different distances, were in the same 

 proportion as the other attractive force, it would, on the whole, 

 cause no alteration in the position of the line of apses, (for it would 

 amount to the same as increasing the central mass in a certain |i- 

 ]...iti..ii, in which case an ellipse, with invariable line of apses, would 

 be described ; that is, the regression at the greatest distance would IN- 

 equal to the progression at the least distance. (See tic ./- i 

 Hut (!.)!) the proportion of this force to the other diminishes as the 

 distance is increased. Consequently, the regression at the greatest 

 distance is less than the progression at the least distance, and, there- 

 fore, on the whole, tho line of apses progresses. Also, the nearer the 

 satellite is to tho planet, the greater is the proportion of this force to 

 the other attraction ; and, therefore the more rapid is the progression 

 of the line of apses at every revolution. The progress, ui ..f the line of 

 apses of the moon's orbit, produced by the earth's oblateness, is so 

 small in comparison with that produced by the sun's disturbing 

 that it can hardly be discovered; but the progress!..]! . t the . 

 apses in the orbit* of Jupiter's satellites, produced by the oblateness of 

 Jupiter, is so rapid, especially for the nearest satellites, that the part 

 produced by the sun's disturbing force is small in comjiarison with it. 



(287.) \V e shall now proceed with the investigation of the disturbance 

 in a satellite's latitude, produced by the oblateness of a planet. 



(288.) First, It is evident that if the satellite's orbit coincides with 

 the plane of the planet's equator, there will be no force tending to pull 

 it up or down from that plane ; and, therefore, it will continue to 

 revolve in that plane. In this case, then, there is no disturbance in 

 latitude ; we must,, therefore, in the following investigation, suppose 

 the orl.it inclined to the plane of the equator. 



In j!<j. 50, then, let us consider (as before) the effect of tho attractions 

 B 



Fig. SO. 



of the two lumps at D and r., in pulling the satellite B perpendicularly 

 to the line A B. Now D is nearer to B than E is ; also the line o n in 

 more inclined than K n to A it. If the attraction of D alone acted, it 

 would in a certain time draw the satellite to rf ; and /rf would lie the 

 part of the motion of B, which is perpendicular to A B ; and this motion 

 is upwards. In like manner, if the attraction of E alone drew B to t in 

 the same time, g e would be the motion perpendicular to A B, and this 

 motion is downwards. When both attractions act, these effects are 

 combined; the question then is, which is greater, /d or gel Now, 

 since o is nearer than K, the attraction of D is greater than that of K, 

 therefore B d is greater than B c ; also B d is more inclined than B e to 

 B A ; therefore rf/ is much greater than g t. Hence, the force which 

 tends to draw B upwards is the preponderating force ; and therefore, 

 on the whole, the combined attractions of D and K will tend to draw 

 the satellite upwards from the line B A. But the attraction of the 

 whole sphere would tend to draw it along the line BA. 

 when D and E are removed, the attraction of the remaining mass (that 

 is, the oblate spheroid) will tend to draw B below the line B A. In 

 estimating the attraction of an oblate spheroid, therefore, we must 

 consider, that besides the force directed to the centre of the spheroid, 

 there is always a force perpendicular to the radius vector directed 

 towards the plane of the equator, or tending to draw a Hat. 'lln 



ue of its orbit towards the plane of tin- planet's equator. If the 

 satellite is near to the planet, the inequality of the proportion of tho 

 distances D B and E B is increased, and the inequality of tin' inclination)* 

 to B A is also increased ; and the disturbance is, therefore, much greater 

 for a near satellite than for a distant one. 



(239.) We have seen (215) the effect of this disturbing force in 

 determining the fundamental planes of the orbits of Jupiter's satellites. 

 And from (192), Ac., we can infer, at once, that this force will cause 

 the line of nodes to regress, if the orbit is inclined to the fundamental 

 plane, and the more rapidly as the satellite is nearer to the planet . 1 t 

 there were no other di-t mbing force, the inclination of those orbits to 

 the plane of Jupiter's equator would be invariable, and their nodes 

 would regress with diffurvnt velocities, those of the near satellites 



