: 



GRAVITATION. 



GRAVITATIOX. 



nut the Brno. It U true that, in the figure, the distance at 4. c, U leas 

 than at i;,c,, while that at l> t r, it greater than at ,*.',; and thus there 



I is. 10. 



U a partial compensation in the changes of the effects produced in 

 different point* of the orbit. Hut it can be discovered only i 

 complete calculations, whether the compensation U perfect or not. 

 The calcuUtionn necessary for this purpose are probably the most 

 complicated that physical science has ever given occasion for ; and the 

 reader must not here expect the smallest account of them. This only 

 can be stated as a result, that in no instance in the planetary system 

 is the compensation perfect, and that the chances for its being perfect 

 in any case are infinitely small 



(17*1.) We have here considered the varying influence of one body 

 on the other at conjunction, as depending entirely on the eccentricities 

 of the two orbits. But there is another circumstance which may also 

 cause the influence to vary. The orbits may be inclined, and this will 

 affect both the distance of the bodies .and the direction in which they 

 attract each other. 



(172.) In the case, then, of Jupiter and Saturn, we have the two 

 planets acting on each other with forces which are nearly the same at 

 every third conjunction, but are not exactly the same, and whose 

 variations occupy a period of 850 years. Of these forces, parU are in 

 the direction of the radius vector, and these tend directly to affect the 

 major axes of the orbits described : other parts are perpendicular to the 

 radius vector, sometimes accelerating -and sometimes retarding ; and 

 these tend (though in opposite ways) to affect the major axes of the 

 orbits. There are, therefore, forces tending to alter the major axes of 

 the orbits, which go through all their changes only in 850 yeans. 

 During half of this time they tend to make the major axis of Jupiter's 

 orbit less, and that of Saturn's orbit greater ; and during the other 

 half they tend to make the major axis of Jupiter's orbit greater, and 

 that of Saturn's orbit less. This coincidence, in time, of the increase 

 of one major axis with the decrease of the other, is the result of 

 investigations that we cannot explain here. 



(173.) After the partial compensation that we have mentioned, it 

 will readily be understood that the varying force which produces these 

 effects is small : -so small, indeed, is it, that after acting more than 400 

 years, it has increased (or diminished) the major axis of Saturn's orbit 

 only by jj^th part, and diminished (or increased) that of Jupiter's orbit 

 only by j.^th part. These alterations would hardly be discoverable 

 with our best instruments. But during 400 years the major axis of 

 each orbit differs from the major axis during the next 400 years by a 

 part of these quantities. The planet's rate of annual angular motion 

 is, for 400 years, constantly less than its average rate ; and for the next 

 400 yean it is constantly greater than its average rate ; and in this 

 length of time the inequality in longitude may (49) grow up into a 

 most formidable quantity. In fact, the inequality thus produced in 

 Saturn's longitude amounts to about 48', by which its true place is 

 sometimes before and sometimes behind its mean place : that in 

 Jupiter's longitude amounts to about 21'. (The greatest inequality of 

 any other planet does not exceed 3', and the greatest of the planets 

 inferior to Jupiter does not exceed 25".) The theoretical explanation 

 of these inequalities was first given by Laplace in 1785. 



U74.) The magnitude of these inequalities in the motions of Jupiter 

 and Saturn, as we have seen, depends principally on the length of time 

 during which the forces act in the same manner ; first, because in Un- 

 iting time they can produce a sensible alteration in the major axis and 

 annual angular motion ; secondly, because the two planets move for BO 

 long a time with this altered angular motion. But it must also be 

 borne in mind that these two planet* are by far the largest in the sys- 

 tem ; the man of Jupiter being 300 time* that of the earth, and the 

 man of Saturn being 100 times that of the earth (the next of the 

 planU in the order of magnitude, except Uranus). 



(175.) The same general reasoning, by which we have shown that 

 there is a periodical inequality of the major axis of either of these 

 orbits, will also show that there is a periodical inequality in the excen- 

 tricity and in the place of the perihelion. It will also appear, in the 

 same way, that these effects are the remainder, after partial compen 

 sation of effects in different parU of the orbit. Thus if one conjunctio 

 happen when Jupiter is going towards aphelion, the effect of .Saturn's 

 disturbing force is to pull Jupiter from the sun; and, therefore, by 

 (59), to increase the eccentricity of Jupiter's orbit. But it U then per- 

 fectly certain that either the next conjunction, or the next but one, or 

 perhaps both these, will happen at a part where Jupiter is going 

 towards perihelion ; and then, by (69), the excentricity of .1 

 orbit U diminished. Similar reasoning uppliw to the excentri.-ity ,.f 

 Saturn's orbit. It becomes, then, a matter of calculation, whether* the 



compensation is perfect or not. Now it appears, upon investigation, 



diminish it. It appears, also, that there is no necessary connection 

 between the time at which the exoentricity is greatest or least, and 

 that when the major axis U greatest or least ; so that we cannot aiwert 

 that when the major axis is greatest the excentricity is greatest, or the 

 contrary, or that the excentrioity of one in greatest when that < 

 other is greatest : all that we can assert is, that the excentricity of each 

 orbit occupies the same time in going through its changes from greatest 

 to least, as the major axis occupies in going through its change from 

 greatest to least. The effect on the planet's distance from the sun, pro- 

 iuced by the change of exceutricity, is much more considerable than 

 that from the change in the major axis ; bein : ,,',, of his 



distance, aii.l for Saturn ,;. of liin whole distance. 



(176.) Similar remarks apply, in every respect, to the motion of the 

 IIIM iliclinn of each orbit Each is made to progress during 425 yean 

 uxl to regress during 425 years; but there is no necessary r 

 bctw cen the time when one has progressed furthest and the time when 

 the other has progressed furthest. There is, however, a necessary 

 relation between the change of excentricity and the motion i :i. 

 lieliou of each orbit: the u.v i ritlirr orbit has its 



value when the perihelion of that orbit has progressed furthest or 

 regressed furthest ; and when the excentricity is either greatest or 

 least, the perihelion is at its mean place. 



(177.) We have taken the long inequality of Jupiter and Saturn as 

 the most imposing by its magnitude, and the most celebrated for its 

 history (as, before it was explained theoretically, astronomers were 

 completely bewildered by the strange irregularity in the mot: 

 these planets). But there are several others which, in theory, are 

 as curious. Eight times the periodic time of the earth is very 

 nearly equal to thirteen times the periodic time of Venus; anil thU 

 produces, in the motions of the earth and Venus, a small inequality, 

 which goes through all its changes in 239 years. Four times the 

 periodic time of Mercury is nearly equal to the periodic time of the 

 earth, and this produces an inequality whose period is nearly 7 years. 

 The periodic time of Mars is nearly double of the earth's, and this pro- 

 duces a considerable inequality, depending on the excentricities, Ac., 

 besides that mentioned in (159), which was independent of the < 

 tricities. Twice the periodic time of Venus is nearly equal t 

 times that of Mercury; three times the periodic t mix is 



nearly equal to that of Mars ; three times the periodic time of Saturn 

 does not much differ from that of Uranus. Each of these a)'; 

 mations to equality gives rise to an equation of sensible magnitude, ;m< ! 

 of long period, in the motion of both planets. 



(178.) But it will easily be seen that the defect of compensation, on 

 which the effects depend, is much greater in some cases than in others. 

 The conjunctions of the earth and Mars take place at only one point, 

 and tlio points near it, for several revolutions ; those of Venus and 

 Mars take place only at two opposite points and their neighbourhood 

 (as each successive conjunction takes place when Mars has described 

 half a revolution, and Venus H revolution) ; those of Jupiter and 

 Saturn, as we have seen, at three points ; those of Venus and the earth 

 at five points. It is evident that, in the first of these, the win ; 

 of the change of one point of conjunction has it* influence in altering 

 the orbit's dimensions ; that in the second there is only the <lit! 

 between two effects ; that in the third there is the mixture of three, 

 which tend to balance ; that in the next there is the mixtu' 

 the same way. The smaller, then, is this number of points, the more 

 favourable are the circumstances (supposing the same length of period 

 for the inequality) for producing a large inequality. Thin number of 

 points is always the same as the difference between the two least num- 

 bers, expressing nearly the proportion of the periodic tinn -. T 

 may expect to find a large inequality when the periodic times of two 

 planets are very nearly in the same proportion as two numbers, whose 

 difference is small 



(179.) We shall now proceed to mention the netular variations 

 elements of the orbita of planets. By this term is meant 

 tions which do not depend upon the position* of the planets in their 

 orbits, or the places of conjunction, but merely upon their r 

 distances and excuntricitiea, and tlie positions of their lines of apses. 

 They are, therefore, the variations which dc|'iid upon the iin 

 average action of one planet upon another in the long run : all the 

 sensible departures from the secular v:ui;>ti..n, produced l>. 

 larity of the action of one planet upon another, beiug supposed to bo 



11 d in the inequalities already discussed. 



(180.) Kir.it, then, with regard to the mean distance of a planet. If 

 we consider t planet disturbing an interior one (as S 



i ), the disturbing force ( the radius 



vector, by (77), Ac., tends sometimes to draw it from the sun, 

 tunes to draw it towards the sun ; but : greater, and 



we may therefore consider the force as, upon th.- whole, aimiuuhiog 

 the sun's attraction. This, by (46), alters the relation between the 

 !! ii'"lie time and the mean distance, so that the mean distance is leu 

 than it would have been with the same periodic time, had there been 

 no disturbance. If we Intel ir planet disturbing an exte- 



rior one (as Jupiter disturbing Saturn), the disturbing force tending to 



