(13 



GRAVITATION. 



GRAVITATION. 



611 



the nodes. Consequently, while the moon moves from B, through 

 B, to B 2 , she is nearer to the sun than the earth is, and therefore the 

 disturbing force, by (197), tends to pull her downwards from the 

 plane of her orbit : while the moon moves from B,,, through B.,, to B t , 

 she is farther from the sun than the earth is, and therefore the dis- 

 turbing force tends to pull her upwards from the plane of her orbit. 

 In the case before us, then, the disturbing force is always directed 

 towards the plane of reference. Consequently, by (192), while the 

 moon moves from B 4 to B,, the line of nodes is made to regress, and 

 the inclination is diminished ; while the moon moves from B, to B s , 

 the line of nodes regresses, and the inclination is increased ; while the 

 moon moves from B,; to B s , the line of nodes regresses, and the incli- 

 nation is diminished : and while the moon moves from B 3 to B ( , the 

 line of nodes regresses, and the inclination is increased. The in- 

 clination, therefore, is not sensibly altered in a whole revolution, but 

 the line of nodes regresses during the whole of the revolution. 



(201.) Thirdly : suppose the line of nodes to be in such a position 

 that the moon passes the line of nodes in going from quadrature to 

 syzygy, as in f<j. 44. Here the sun is to be considered as below the 

 moon's orbit, and, therefore, while the moon moves from B t , through 



Fig. 44. 



B,, to B,, the disturbing force tends to pull her down from the plane 

 of the orbit, and while she moves from B 2 , through B,, to B 4 , the force 

 tends to pull her up from the plane of her orbit. Therefore, in going 

 from B, to N, the force pulls the moon from the plane of reference ; 

 and causes thereby a progression of the line of nodes and a diminution 

 of the inclination (193) : in going from N to the highest point o, the 

 force pulls the moon towards the plane of reference ; and, therefore, 

 causes the nodes to regress, and the inclination to diminish (192) ; in 

 going from the highest point o to B,, the force still pulls the moon 

 towards the plane of reference ; and, therefore, still causes the nodes 

 to regress, but causes the inclination to increase. Thus while the 

 moon moves from B, to N, the force causes the line of nodes to pro- 

 gress, and while she moves from N to B,, it causes the line of nodes to 

 regress ; and, similarly, while she moves from B, to M, the force causes 

 the line of nodes to progress ; and while she moves from M to B , it 

 causes the line of nodes to regress. On the whole, therefore, the line 

 of nodes regresses, but not so rapidly as in the second case. Also, 

 while the moon moves from B, to o the inclination is diminished, and 

 while she moves from o to B, the inclination is increased ; and, 

 similarly, while she moves from B, to p the inclination is diminished ; 

 and while she moves from P to B, the inclination is increased. On the 

 whole, therefore, the inclination is diminished. 



(202.) Fourthly : suppose the line of nodes to be in such a position 

 that the moon passes it in going from syzygy to quadrature, as in 

 fg. 45. Here, also, the sun is below the plane of the orbit produced ; 

 and, therefore, from B 4 to B, the force tends to pull the moon down 

 from her orbit ; and from B, to B, it tends to pull her up from it. Aa 

 in the last cage it would be seen, that while the moon mores from 



B 4 to M, the line of nodes regresses ; while from M to B., the line of 

 nodes progresses ; while from B, to y, the line of nodes regresses ; and 

 while from s to B,, the line of nodes progresses. On the whole, there- 

 fore, the line of nodes regresses. -Also, it will be seen, that while the 

 moon moves from B, to o, the inclination is diminished ; while from 

 o to 3,, the inclination is increased ; while from B. to p, the inclination 

 is diminished ; and while from p to B,, the inclination is increased. 

 On the whole, therefore, the inclination is increased. 



The same reasoning would apply, and lead to the same conclusions 

 in every respect, if we supposed the moon's orbit inclined in the 

 opposite direction. 



)3.) Now the earth moves round the sun, and, therefore, the sun 

 appears to move round the earth, and in the same direction in which 

 10 moon moves round the earth. If then we begin with the state in 

 which the line of nodes is passing through the sun (and in which 

 neither the node nor the inclination undergoes any change, by the first 

 se), we come next to the state in which the moon passes the line of 

 nodes in going from quadrature to syzygy (in which the node regresses 

 and the inclination diminishes, by the third case) ; then we come to 

 the state in which the line of nodes coincides with the line of quad- 

 ratiiFes (in which the node regresses rapidly, and the inclination is not 

 Hered, by the second case) ; then we come to the state in which the 

 >n pomes the line of nodes in going from syzygy to quadrature (in 

 le node regresses and the inclination is increased, by the 

 irth case) ; and then we come to the state in which the line of nodes 

 again pane* through the sun. This is when the sun has described 

 apparently, half a revolution round the earth (or rather less in con- 

 nee of the regression of the node), and in the other half revolu- 

 tion, the name changes in every respect take place in the same order 

 AKTS AND SCT. DIV. VOL. IV. 



The inclination, therefore, is greatest when the line of nodes passes 

 through the sun, or coincides with the line of syzygy ; and is least 

 when the line of nodes coincides with the line of quadratures ; since 

 it is constantly diminishing while we are going from the former state 

 to the latter, and constantly increasing while we are going from the 

 latter state to the former. This is the principal irregularity in the in- 

 clination of the moon's orbit ; all the others are very small. 



(204.) The line of nodes is constantly regressing at every revolution 

 of the moon, except when the line of nodes passes through the sun. 

 The annual motion which we might at first expect it to have is some- 

 what diminished by the circumstance that the rapid regression of the 

 line of nodes, when in the position in which the greatest effect is pro- 

 duced, carries it from the line of quadratures more swiftly than the 

 sun's progressive motion only, by making the line of quadratures to 

 progress, would separate them. But as the line of nodes never pro- 

 gresses, the diminution of the motion of the line of nodes thus occa- 

 sioned is very much less than the increase of the motion of the line of 

 apses (107). Also, as the force acting on opposite points of the orbit 

 tends to produce effects of the same kind, there is no irregularity 

 similar to that explained in (106). Hence the actual regression of the 

 line of nodes, though a little less than might at first be expected, 

 differs from that regression by a much smaller quantity than that by 

 which the actual motion of the line of apses differs from the motion 

 which at first we might expect it to have. The line of nodes revolves 

 completely round in something more than nineteen years. 



(205.) The effect of the irregularity in the regression of the nodes, 

 and the effect of the alternate increase and diminution of the incli- 

 nation, are blended into one inequality of latitude, which depends on 

 the sun's longitude, the longitude of the moon's node, and the moon's 

 longitude. This inequality was discovered (from observation) by 

 Tycho Brah<5, about 1590. It may be considered to bear the same 

 relation to the inclination which the evection bears to the excentricity ; 

 and, like the evection in longitude, it is the greatest of the inequalities 

 in latitude. It is, however, much less than the evection ; its greatest 

 effect on the moon's latitude being about 8', by which the mean in- 

 clination is sometimes increased and sometimes diminished. 



(206.) There are other small inequalities in the moon's latitude, 

 arising partly from the changes in the node and inclination, which take 

 place several times in the course of each revolution (200), Ac. ; partly 

 from the excentricity of the orbits of the moon and the earth, partly 

 from the distortion accompanying the variation, and partly from the 

 variability of the inclination itself. We shall not, however, delay 

 ourselves with the explanation of all these terms. 



(207.) We shall now proceed with the disturbance of the planete in 

 latitude. 



In this inquiry it is always best to take the orbit of the disturbing 

 planet for the place of reference. Now let us first consider the case 

 of Mercury or Venus disturbed by Jupiter. In this case Jupiter 

 revolving in a long time round the sun, which is the central body to 

 Mercury or Venus, produces exactly the same effect as the sun 

 revolving (or appearing to revolve) round the earth, which is the 

 central body to the moon. The disturbing force of Jupiter therefore 

 produces a regression of the nodes of the orbits of Mercury and Venus 

 on Jupiter's orbit ; and an irregularity in the motion of each node, and 

 an alteration in the inclination, whose effects might be combined into 

 one : and this is the only inequality in their latitude, produced by 

 Jupiter, whose effects are sensible. 



(208.) The other inequalities in latitude, depending on the relative 

 position of the planets, possess no particular interest ; and a general 

 notion of them may be formed from the remarks in the discussion of 

 the motion of the moon's node. One case, however, may be easily 

 understood. When an exterior planet is disturbed by the attraction 

 of an interior one, whose distance from the sun is less than half the 

 distance of the exterior planet, and whose periodic time is much 

 shorter, then the exterior planet is always further from the interior 

 planet than the sun is ; and therefore, by (195), there is a disturbing 

 force urging it from the plane of reference when the planets are in 

 conjunction, and to it when they arc in opposition ; and thus the ex- 

 terior planet is pushed up and down for every conjunction of the two 

 planets. The disturbance, however, is nothing when the exterior 

 planet is at the line of its nodes (195). 



(209.) The near commensurability of periodic times, which so 

 strikingly affects the major axis, the excentricity and the place of peri- 

 helion, produces also considerable effect* on the node and the incli- 

 nation. The reasoning of (175) and (176) will in every respect apply 

 to this case ; the greatest effect is produced, both on the motion of the 

 node and on the change of inclination, when the planets are in con- 

 junction : the gradual alteration of the point of conjunction produces 

 a gradual alteration of these effects, which, however (in such a case as 

 that of Jupiter and Saturn), is partially counteracted by the gradual 

 change of the other points of conjunction : the uncompensated part, 

 however, may, in many years, produce a very sensible irregularity in 

 the elements. If we put the words line of noda for line nf apses, and 

 inclination for eccentricity, the whole of the reasoning in (175), &c., will 

 apply almost without alteration. 



(210.) For the secular variation of the position of the orbit, the 

 following considerations seem sufficient. In the long run the disturbed 

 planet has been at every one point of its orbit a great number of times, 



L L 



