GRAVITATION. 



ORAVITATION. 



of the first atelliu- cannot be in the direction of the points of con. 

 junction. 



(188.) But if we nippoM the perijove of the fint satellite to be in 

 the direction of the potato of conjunction, every thing become* con- 

 {tent. The disturbing force, in the direction of the radius vector, 

 from the centra) body, will, by (51), cause the lino of apses to regress. 

 The force perpendicular to the radius vector, which accelerates the 

 first satellite before it has reached conjunction, that is, before it has 

 reached the perijove, and retards it after that time, will also, 

 and (86), cause the line of apses to regress. Also, as in (5<ii. tlii* 

 regression witt be greater as the excentricity of the orbit is 1cm, 

 because the disturbing force, which acts here, does not depend on the 

 excentricity. By proper choice, therefore, of a value of the eccen- 

 tricity, we can make an orbit, whose line of apses will always regress 

 exactly as fast as the line of conjunction, and will, therefore, always 

 coincide with it ; whose exceutricity, in consequence, will never alter, 

 by (59) and (68) ; and whose general shape, therefore, will be the same 

 at every successive revolution. 



(129.) We shall mention hereafter, that the form of Jupiter is such 

 as would cause the perijove of the first satellite, if it were not dis- 

 turbed by the second satellite, to progress with a velocity not depend- 

 ing upon the excentricity of the orbit. The only alteration which 

 this makes in our conclusions is, that the excentricity of the orbit 

 must be so chosen, that the perturbation of which we have epoken will 

 cause a regression equal to the turn of the progression which Jupiter's 

 shape would occasion, and the regression of the line of conjunction. 

 As this is greater than the regression of the line of conjunction alone, 

 the excentricity of the orbit must be less. So that the only effect 

 of Jupiter's shape is to diminish, in gome degree, the excentricity of 

 the orbit. 



(180.) Now let us inquire what must be the form and position of 

 the orbit of the second satellite. AH before, the prmci[>al part of the 

 perturbation is near conjunction. At and near the conjunction, the 

 disturbing force, in the direction of the radius vector, is directed to 

 the central body. Before conjunction, when the first satellite is less 

 advanced than the second, the disturbing force, perpendicular to the 

 radius vector, retards the second, by (86). For, the periodic time of 

 the second being nearly double that of the first, the mean distances 

 from the planet will be nearly in the proportion of 7 to 11 (as the 

 proportion of the cube of 7 to the cube of 11 is nearly the same as the 

 proportion of the square of 1 to the square of 2, see (84) ), and, there- 

 fore, near conjunction, the distance of the first from the second is 

 less than the distance of the first from the central body. After con- 

 junction, the disturbing force accelerates the second body. Now, 

 without going through several cases as before, which the reader 

 will find no trouble in doing for himself, we shall remark, at once, 

 that if the apojove of the second satellite is in the direction of 

 the point* of conjunction, both the disturbing force, directed to the 

 central body at apojove, and that perpendicular to the radius vector, 

 retarding it before it reaches apojove, and accelerating it afterwards, 

 by{53), (65), and (66), will cause the line of apses to regress; and 

 that, by proper choice of excentricity, the regression of the line of 

 apses may be made exactly equal to the regression of the line of 

 conjunction. 



(131.) Our conclusion, therefore, is : If two satellites revolve round 

 a primary, and if the periodic time of one is very little greater than 

 double the periodic time of the other, and if we assume that the 

 orbits described have always the same form ; (that is, if they have no 

 excentricity independent of perturbation) ; then the orbits* will not 

 sensibly differ from ellipses, the lines of apses of both orliits must 

 always coincide with the line of conjunctions, and the perijove of the 

 first orbit, and the apojove of the second, must always be turned 

 towards the points of conjunction. It appears also, that these con- 

 ditions are sufficient, inasmuch as the rate of regress of the lines of 

 apses will (with proper values for the excentricitics) be the same as 

 the rate of regress of the line of conjunctions, and the excen; 

 tli. M will not change. The excentricitics of the orbits will be greater 

 as the regress of the line of conjunction* is slower, or as the pro- 

 n of the periodic times approaches more exactly to the propor- 

 tion of 1 : U'. 



(182.) In the same manner it would bo found, that if the periodic 

 time of one satellite were very little less than double that 

 other, the lines of apses (in order that similar orbits may be tm 

 at each revolution) must always coincide with the line of conj. 

 and the apojove of the first satellite and tho perijovo of the second 

 must always be turned towards the points of conjunction; and 

 tho excentricitics of the orbits must be greater, as tin- p 

 of tho periodic times approaches more exactly to the prop. 



(138.) The same thing exactly would hold, if tho periodic times 

 were very nearly in the ratio of 2 : 8, or of 8 : 4, Ac., but these sup- 

 position, do not apply to Jupiter's satellite*. 



I Having thus found the distortion produced by the disturbing 

 force in orl.iti which have no cxcrn -nt of JK-HH 



it will easily be imagined tli.it tl,,> SUM Und of distortion will 

 lucedif the orbits have nn original exccntri. itiv. If wr DM! 



orbit, the same kind of alteration which must b* made In a 

 circular orbit, in order to form the figure found abore, wo shall Imvp 



nearly the orbit that will be described from the combined effects of 

 perturbation and of excentricity independent of perturbation.* 



The truth of thii propotition may be shown more fully In the fo'lowing 

 m.nncr : It A (ftf. SS) bo the place of the prinury, A c the lino of conjip 

 of the first and second satellite, BOS the elliptic orbit. In which the fint 

 -.itdlile would move If undisturbed, D It. person. Suppose (to linplily the 

 figure) that the attraction of the stoond satellite seta only space ; 



fur Instance, while the flrit satellite posse* from r tn ii. Tim 

 the investigations from (US) to (131) is, that the first] satellite will be drawn 



C, 

 11 

 //* ^ 



rig. sj. 



s from the orbit In which it would have moved, so u to describe a curve 

 ran; and when the disturbing force ceases at n, it will proceed to dneribe an 

 ellipse, Hfkrf, similar to BDK, but with this difference, that the perljov . 



1 of D. The conclusion, however, now that it has been securely obtained 

 from the reasoning above, may be stated as the reult of the following 

 reasoning: In consequence of the disturbing force, which has drawn r 

 satellite outwards, without, upon the whole, altering In velocity (accelerating 

 it before conjunction, and retarding It aftcrwai lite has moved In 



curve, p o a, external to the ellipse D, in which it would have moved ; and 

 after the disturbing force hu ceased at it, the satellite (which is moving In a 

 path inclined externally from the old orbit) eontinues to n c. clc from 

 orbit till the diminution of velocity (26) allows ita path to be so much cm vc.l, 

 that at e it begins to approach, and at i. the new orbit intersect* the old one ; 

 and after this, the path i inclined internally from the old orbit, till the i: 

 of velocity (2S) makes Its path so little curved that it approaches the old orbit 

 again, end again crosses it between d and D. In like manner, if, an in ftg. SO, 

 the orbit B r E have an excentricity Independent of perturbation (the pevljove 

 being at any point ,), nevertheless, we may stnte that, in consequence of the 

 disturbing force, the satellite will move in a carve F o H external to F E ; but 

 when the disturbing force ceases at n, the satellite (which is moving in a path 



Fig. SO. 



inclined externally from the old orbit) continues to recede from the old orbit 

 till the diminution of velocity (20) allows its path to be so much curved, that 

 it begins to approach at some point e ; that at some point i, nearly opposite to 

 e, the new orbit intersect!, the old one ; and that, after this, the path i inclined 

 internally from the old orbit, till tin increase of velocity (25) makes its path 

 so little cnrved that it approaches the old orbit again, and again crosses it 

 between F and n. Thus, the alteration of the radius vector, drawn in any given 

 direction, as AK (which in the new orbit is altered to A*) U nearly the same in 

 the second case as in the first. This, however, is the alteration produced in a 

 single revolution of the satellite ; but as the fame applies to every successive 

 revolution, it follows that the inequality or variation of the radius vector in tho 

 second case is nearly the fame as in the first ease ; and thus the proposition of 

 the text is proved. 



The inequality of the radius vector would be somewhat different If the excen- 

 tricity of the orbit in the second case were considerable, partly becaisc tho 

 places of conjunction would not he at equal angular distances, partly because 

 ;'n .li-tiiil.ing forces would be different (as the distance between the >..- 

 in conjunction would not always bo tho same), and partly because the effect of 

 a given force is really diftVirni, according to the part of the orbit at which it 

 ut where tli< . is so small, as in the orbit of Jupiter's ihird 



atcllite, or in those of the old planets, the alteration of the Inequality of the 

 radius vector produced by these differences is hardly sensible. 



The reasoning of this note' may be applied, with the proper alterations, to 

 every case of perturbation, produced by a disturbing force whirl, is nearly 

 independent of the form of the orbit ; and as this will apply success 

 neh of the causes producing ili-tiirbance, we shall at last arrive at the following 

 general proposition : If several disturbing forces act on a planet or s > 

 and if we estimate the inequality In the radius vector, which each of these 



. supposing the orbit to have no cxccntri< 



it ion ; then tlie inequality really produced, supposing the orbit to have 

 nn in-lcpcndent cxcontrlclty, will be nearly the same u the sum of all the 

 Inequalities so estimated. 



It is to he remarked, that If an orbit have on independent excentricity, and 

 if the orbit receive an alteration similar to an elliptic inequality (that is, If it 

 be elongated on one side and flattened on the oilier), the orbit Is still scn-ibly 

 an ellipse, of which the original focus is still the focus. Thus, In the Instance 

 occupying the first part of this note, as the Inequality impressed on the elliptic 

 orbit In the second case Is the same ns the in. quality In the first case, that is, 

 Is similar to an elliptic inequality, the orbit so altered will still be an ellipse, 

 uhn-e excentricity and line of apse* are altered. We might, therefore, have 

 obtained our results by at once Investigating the alterations of the excentricity 

 and line of apses produced by the disturbing forces; but the method adopted In 

 the text Ii simpler. 



