497 



GRAVITATION. 



GRAVITATION. 



433 



(115.) Now the earth moves round the sun, and the sun therefore 

 appears to move round the earth, in the order successively represented 

 by the Jigs. 31, 32, and 33. Hence; then ; when the sun is in the line 

 of the moon's apses, the exeentricity does not alter (110) ; after this it 

 diminishes till the sun is seen at right angles to the line of apses (112) ; 

 then it does not alter (111); and after this it increases till the sun 

 reaches the line of apses on the other side. Consequently, the excen- 

 tricity is greatest when the line of apses passes through the sun, and is 

 least when the line of apses is perpendicular to the line joining the 

 earth and sun. 



The amount of this alteration in the excentricity of the moon's orbit 

 is more than ^th of the mean value of the excentricity; the excentricity 

 being sometimes increased by this part, and sometimes as much dimi- 

 nished ; so that the greatest and least excentricities are nearly in the 

 proportion of 6 : 4 or 3 : 2. 



(116.) The principal inequalities in the moon's motion may therefore 

 be stated thus : 

 1st. The elliptic inequality, or equation of the centre (31), which would 



exist if it were not disturbed. 

 2nd. The annual equation (90), depending on the position of the 



earth in the earth's orbit. 

 3rd. The rariation (93), and parallactic inequality (94), depending on 



the position of the moon with respect to the sun. 

 4th. The general progression of the moon' i perigee (104). 

 5th. The irregularity in the motion of the perigee, depending on the 



position of the perigee with respect to the sun (109). 

 6th. The alternate inmate and diminution of the excentricity, depending 



on the position of the perigee with respect to the sun (115). 



These inequalities were first explained (some imperfectly) by New- 

 ton, about 1680. 



(117.) The effects of the two last are combined into one called the 

 erection. This is by far the largest of the inequalities affecting the 

 moon's place : the moon's longitude is sometimes increased 1 15' and 

 sometimes diminished as much by this inequality. It was discovered 

 by Ptolemy, from observation, about A.D. 140. 



(118.) It will easily be imagined that we have here taken only the 

 principal inequalities!. There are many others, arising chiefly from 

 small errors in the suppositions that we have made. Some of these, 

 it may easily be seen, will arise from variations of force which we 

 have already explained. Thug the difference of disturbing forces at 

 conjunction and at opposition, whose principal effect was discussed in 

 (94), will also produce a sensible inequality in the rate of progression 

 of the line of apses, and in the dimensions of the moon's orbit. The 

 alteration of disturbing force depending on the excentricity of the 

 earth's orbit will cause an alteration in the magnitude of the <" 

 and the erection. The alteration of that part mentioned in (94) 

 produces a sensible effect depending on the angle made by the moon's 

 radius vector with the earth's line of apses. All these, however, are 

 very small : yet not so small but that, for astronomical purposes, it is 

 necessary to take account of thirty or forty. 



(119.) There is, however, one inequality of great historical interest, 

 affecting the moon's motion, of which we may be able to give the 

 reader a general idea. We have stated in (89) that the effect of 

 the disturbing force is, upon the whole, to dimmish the moon's gravity 

 to the earth : and in (90) we have mentioned that this effect is greater 

 when the earth is near perihelion, than when the earth is near aphelion. 

 It is found, upon accurate investigation, that half the sum of the 

 effects at perihelion and at aphelion is greater than the effect at mean 

 distance, by a small quantity depending on the excentricity of the 

 earth's orbit : and, consequently, the greater the excentricity (the 

 mean distance being unaltered) the greater is the effect of the sun's 

 disturbing force. Now, in the lapse of ages, the earth's mean distance 

 ig not sensible altered by the disturbances which the planeta produce 

 in its motion; but the excentricity of the earth's orbit is sensibly 

 diminished, and has been diminishing for thousands of years. Con- 

 sequently the effect of the sun in disturbing the moon has been 

 gradually diminishing, and the gravity to the earth has therefore, on 

 the whole, been gradually increasing. The size of the moon's orbit 

 has therefore, gradually, but insensibly, diminished (47) : but the 

 moon's place in ita orbit b,T sensibly altered (49), and the moon's 

 angular motion has appeared to be perpetually quickened. This phe- 

 nomen was known to astronomers by the name of the acceleration of the 

 MWH'I mean motion, before it was theoretically explained in 1787, by 

 Laplace : on taking it into account, the oldest and the newest obser- 

 vations are equally well represented by theory. The rate of progress 

 of the moon's line of apses has, from the same cause, been somewhat 

 diminished. 



SECTION VI. Theory of Jupiter" Satellite!. 



(ISO.) Jupiter has four satellites revolving round him in the same 

 manner in which the moon revolves round the earth ; and it might 

 seem, therefore, that the theory of the irregularities in the motion 

 of these satellites is similar to the theory of the irregularities in the 

 moon'* motion. But the fact is, that they are entirely different. The 

 fourth satellite (or that revolving in the largest orbit) has a small irre- 

 gularity analogous to the moon's variation, a small one similar to the 

 evection, and one similar to the annual equation : but the last of 

 these amount* only to about two minutes, and the other two are very 



ARTS AND SCI. DIV. VOL. IV. 



much less. The corresponding inequalities in the motion of the other 

 satellites are still smaller. But these satellites disturb each other's 

 motions, to an amount and in a manner of which there is no other 

 example in the solar system ; and (as we shall afterwards mention) 

 their motions are affected in a most remarkable degree by the shape of 

 Jupiter. 



(121.) The theory, however, of these satellites is much simplified by 

 the following circumstances : First, the disturbances produced by the 

 sun may, except for the most accurate computations, be wholly 

 neglected. Secondly, that the orbits of the two inner satellites have 

 no excentricity independent of perturbation. Thirdly, that a very 

 remarkable relation exists (and, as we shall show, necessarily exists) 

 Between the motions of the three first satellites. 



Before proceeding with the theory of the first three satellites, 

 we shall consider a general proposition which applies to each of them. 



(122.) Suppose that two small satellites revolve round the same 

 planet; and that the periodic time of the second is a very httle 

 greater than double the periodic time of the first ; what is the form of 

 the orbit in which each can revolve, describing a curve of the same 

 form at every revolution ? 



(123.) The orbits will be sensibly elliptical, as the perturbation 

 produced by a small satellite in one revolution will not sensibly alter 

 the form of the orbit. The same form being supposed to be described 

 each time, the major axis and the excentricity are supposed invariable, 

 and the position of the line of apses only is assumed to be variable. 

 The question then becomes, What is the excentricity of each orbit, 

 and what the variation of the position of the line of apses, in 

 order that a curve of the same kind may be described at every 

 revolution ? 



(124.) In fg. 34, let B 3 , B,, B.,, represent the orbit of the first, and 



Fig. 31. 



c a , c u c,,the orbit of the second. Suppose that when B was at B J( c was 

 at| c i> 8O that A, B,, c,, were in the same straight line, or that B and c 

 were in conjunction at these points. If the periodic time of c were 

 exactly double of the periodic time of B, B would have made exactly 

 two revolutions, while o made exactly one; and, therefore, B and c 

 would again be in conjunction at B,, and c,. But as the periodic time 

 of c is a little longer than double that of B, or the angular motion of o 

 rather slower than is supposed, B will have come up to it (in respect 

 of longitude as seen from A) at some line B 2 c a , which it reaches 

 before reaching the former line of conjunction B o,. And it is plain 

 that there has been no other conjunction since that with which wo 

 started as the successive conjunctions can take place only when one 

 satellite has gained a whole revolution on the other. The first con- 

 junction then being in the lineAB.c,, the next will be in the line 

 A B 2 c,, the next in a line A B, Cj, still farther from the first, &c. ; so 

 that the line of conjunction will regress slowly; and the more nearly 

 the periodic time of one satellite is double that of the other, the more 

 slowly will the line of conjunction regress. 



(125.) As the principal part of the perturbation is produced when the 

 satellites are near conjunction (in consequence of the smallness of their 

 distance at that time), it is sufficiently clear that the position of the 

 line of apses, as influenced by the perturbation, must depend on the 

 position of the line of conjunction ; and, therefore, that the motion 

 of the line of apses must be the same as the motion of the line of con- 

 junction. Our question now becomes this : What must be the 

 excentricities of the orbits, and what the positions of the perijoves, 

 in order that the motions of the lines of apses, produced by the 

 perturbation, may be the same as the motion of the line of con- 

 junction ? 



(126.) If the line of apses of the first satellite does not coincide 

 with the line of conjunction, the first satellite at the time of con- 

 junction will either be moving from perijove towards apojove, or from 

 apojove towards perijove. If the former, the disturbing force, which 

 in directed from the central body, will, by (59), cause the excentricity 

 to increase ; if the latter, it will cause it to decrease. As we have 

 started with the supposition, that the excentricity is to be supposed 

 invariable, neither of these consequences can be allowed, and, therefore, 

 the line of apses must coincide with the line of conjunction. 



(127.) If the apojove of the first satellite were in the direction of 

 the points of conjunction, the disturbing force in the direction of 

 the radius vector, being directed from the central body, would, by (54), 

 cause the line of apses to progress. Also the force perpendicular to 

 the radius vector, before the first satellite has reached conjunction (and 

 when the second satellite, which moves more slowly, is nearer to the 

 point of conjunction than the first), tends to accelerate the first satel- 

 lite ; and that which acts after the satellites have passed conjunction, 

 tends to retard the first satellite ; and bpth these, by (05) and 

 (66), cause the line of apses to progress. But wo have assumed, 

 that the line of apses shall move in the same direction as the 

 line of conjunction, that is, shall regress; therefore, the apojovo 



