617 



GRAVITATION. 



GRAVITATION. 



618 



litea will regress more rapidly than those of the exterior ones. Their 

 annual regressions (beginning with the second) are, in fact, 12, 2, 32', 

 and 41'. 



(221.) But the disturbing force of one satellite upon the others will 

 be altered by the circumstance of its orbit not coinciding with its 

 fundamental plane ; and the orbit remains long enough in nearly the 

 same position to produce a Tery sensible irregularity. To discover the 

 nature of this, we must observe that the force of one satellite, per- 

 pendicular to the orbit of another, depends wholly upon the incline 

 . tion of the two orbits, so that, upon increasing the inclination, the 

 disturbing force is affected. Suppose now, to fix our ideas, the second 

 satellite moves in an orbit inclined to its fundamental plane ; what is 

 the kind of disturbance that it will produce in the latitude of the first 

 satellite ? First, it must be observed, that when moving in the funda- 

 mental planes, the forces depending upon the inclination of those 

 planes were taken into account in determining the position of those 

 planes : so that here we have to consider only the alteration produced 

 by the alteration in the second satellite's place. Next we shall proceed 

 in the same manner as in several preceding instances, by finding what 

 is the motion of the first satellite, related to the motion of the second 

 satellite, which can exist permanently with this inclination of the 

 second satellite. Now, in whatever part the actual orbit of the second 

 is higher above, or less depressed below, the orbit of the first, than the 

 fundamental plane of the second was, at that part there will be a 

 greater force drawing the first satellite up, or a smaller force drawing 

 it down (in the conjunctions at that part), than before. The alteration 

 of force, then, will be generally represented by supposing a force to 

 act on the first satellite, at different points of its orbit, towards the 

 same side of ita orbit as the side on which the second satellite's orbit 

 is there removed from its fundamental plane, and proportional to the 

 magnitude of that removal. Now, conceiving the inequality intro- 

 duced into the motion of the first satellite to be a small inclination of 

 ita orbit to its fundamental plane (which is the only inequality of 

 Jupiter's satellites that we consider), the nodes of this orbit cannot 

 correspond to the places where the second satellite is furthest from its 

 fundamental plane ; for then, at one node of the first satellite, the 

 disturbing force, before and after passing that node, being great, and 

 not changing its direction, would not alter the place of the node, but 

 would greatly alter the inclination : and at the opposite node, the 

 force acting in the opposite direction would produce the same effect ; 

 and thus the permanency of the inequality would be destroyed. We 

 must then suppose the nodes of the orbit of the first satellite on its 

 fundamental plane to coincide with those of the orbit of the second 

 satellite on its fundamental plane. But is the inclination to be the 

 same way, or the opposite way ? To answer this, we must consider 

 that the action of Jupiter's shape would tend to make the nodes of the 

 first satellite regress much more rapidly than those of the second ; but 

 as our orbit of the first satellite is assumed to accompany the second 

 in its revolution, the disturbing force depending on the second must 

 be such as to destroy a part of this regression, or to produce (separately) 

 a progression of the nodes of the first ; consequently, the disturbing 

 force produced by the second must tend to draw the first from ita 

 fundamental plane (193). But the disturbing force produced by the 

 second is in the same direction as the distance of the second from the 

 fundamental plane of the second ; consequently, the orbit of the first 

 must lie in the same position, with regard to the fundamental plane 

 of the first, in which the orbit of the second lies with regard to the 

 f undaincnt.il plane of the second. The same reasoning applies to every 

 other case of an interior satellite disturbed by an exterior ; and thus 

 we have the conclusion : If the orbit of one of Jupiter's satellites is 

 inclined to ita fundamental plane, it affects the orbit of each of the 

 satellites interior to it with an inclination of the same kind, and witli 

 the same nodes. 



(222.) Let us now inquire what will l>e the nature of the inequality 

 produced in the latitude of the third satellite. The same reasoning 

 and the same words may, in every part, be adopted, except that the 

 regression of the nodes of the third satellite, aa produced by Jupiter's 

 shape, will be slower than that of the second satellite, and therefore 

 tin- di.itnrliing force which acts on the third must now be such aa to 

 quicken the regression of ita nodes, and must therefpre be directed 

 towards its fundamental plane. From this consideration we find, as a 

 general conclusion, if the orbit of one of Jupiter's satellites is inclined 

 to its fundamental plane, it affects the, orbit of each of the satellites 

 exterior to it, with an inclination of the opposite kind, but with the 



S..1I,.' nodi -. 



(223.) The first satellite's orbit appears to have no sensible inclina- 

 tion to ita fundamental plane; but thoae of the second, third, and 

 fourthj are inclined to their fundamental planes (the second 25', and 

 nrd and fourth about 12'), and these are found to produce in the 

 others inequalities such as we have investigated. 



(224.) It is only necessary to add, that the disturbance of the first 

 satellite by the second produces an alteration in the action of the first 

 on the second, tending to draw the second from ita fundamental plane, 

 and therefore to diminish, by a small quantity, the regression of ita 

 nodes. In the same manner, the altered action of the third on the 

 econd tends to draw the second towards ita fundamental plane, and 

 therefore to increase, by a small quantity, the regression of its nodes. 

 There is exactly the same kind of complication with regard to the 



disturbances of these bodies in latitude as with regard to those in 

 longitude, explained in (150), 4c. 



(225.) The only other inequality in latitude, which is sensible, is 

 that depending on the position of the sun, with regard to the nodes of 

 the orbits on the plane of Jupiter's orbit (that is, with regard to the 

 node of Jupiter's equator on Jupiter's orbit), and this amounts to only 

 a few seconds. It is exactly analogous to that of the moon, explained 

 in (205). 



SECTION IX. Effects of the Oblateness of Flatlets upon the Motions of 

 their Satellites. 



(226.) In the investigations of motion about a central body, we have 

 supposed that central body to be a spherical ball. This makes the 

 investigation remarkably simple ; for it is demonstrated by mathe- 

 maticians, that the spherical form possesses the following property : 

 the attraction of all the matter in a sphere upon another body at any 

 distance external to it is exactly the same as if all the matter of the 

 sphere were collected at the centre of the sphere. In the investigation 

 of motion about a centre we may therefore lay aside (as we have usually 

 done) all consideration of the size of the attracting body, if that body 

 is spherical. 



(227.) But the planets are not spherical. Whether or not they have 

 ever been fluid, still they have (at least, the earth has) a great extent 

 of fluid on its surface, and the form of this fluid will be affected by 

 the rotation of the planet. The fluid will spread out most where the 

 whirling motion is most rapid, that is, at the equator. Thus it appears 

 from theory, and it is also found from measures, that the earth is not 

 a sphere, but a spheroid, flattened at the north and south poles, and 

 protuberant at the equator. The proportion of the axes differs little 

 from the proportion of 299 : 300 ; so that a line drawn through the 

 earth's centre, and passing through the equator, is longer than one 

 passing through the poles, by 27 miles. 



(228.) The flattening of Jupiter is still more remarkable. The pro- 

 portion of his axes differs little from that of 13 : 14, and thus the 

 difference of his diameters is nearly 6000 miles. In fact, the eye is 

 caught by the elliptic appearance of Jupiter, on viewing him for a 

 moment in a telescope. 



(229.) It is our business, in the present section, to point out the 

 general effects of this shape upon the motion of satellites. The agree- 

 ment of observation with calculation on this point is certainly one of 

 the most striking proofs of the correctness of the theory, " that every 

 particle of matter attracts every other particle, according to the law of 

 Universal Gravitation." 



(230.) We will begin with explaining the law according to which an 

 oblate planet attracts a satellite in the plane of ita equator. 



The spheroid represented by the dark line in fi'j. 48 may be supposed 

 to be formed from the sphere represented by the dotted line, by cutting 

 off a quantity of matter from each pole. To simplify our conception, 

 let us suppose that all the matter cut off was in one lump at each pole ; 



D 



Kig. 48. 



that is, at the points D and E. The attraction of the whole sphere on 

 the satellite B, as we have remarked, is the same as if all the matter 

 of the sphere were collected at A. But the attraction of the part cut 

 off is not the same as if it were collected at A, inasmuch as its distance 

 from B is greater, and as the direction of the attraction to D, or to K, 

 is not the same as that to A. Thus, suppose A D is called 1, and A B 

 called 10. Since the forces are inversely as the squares of the 

 distances at which the attracting mass is situate, the attraction of the 

 lump D, if at the point A, where its distance from B is 10, may be called 

 jJjj ; but if at D, it must be called jjj, since th<! square of B D is equal to 

 the sum of the squares of B A and A D, that is, to the sum of 100 and 1 . 

 Also the direction of attraction is not the same ; for, if the attraction of 

 D should draw the satellite through B b, and if 6 c be drawn perpendicular 

 to A B, the only effective approach to A is the distance B c, which is less 

 than B b in the proportion of B A to B D, or of 10 to V 101 ; and, there- 

 fore, the effective attraction of D, estimated by the space through which 



it draws the satellite towards A must be called . And this is 



101 x v'lOl 



the whole effect which its attraction produces ; for though the 

 attraction of D alone tends to draw the satellite above A B, yet the 

 attraction o E will tend to draw it as much below A B ; and thus the 

 parts of the force which act perpendicular to A B will destroy each 

 other. We have, then : the attraction of the lump D, if placed at A, 



would be represented by ^ = O'Ol ; but as placed aa D, its effective 



attraction is represented by - = 0'0098518. The difference 



101 x VlOl 



is 0-0001482, or nearly 1 jooooo tn o the wnole attraction of D, and the 

 same for E. Consequently, the lumps at D and E produce a smaller 



