GRAVITATION. 



GRAVITATION. 



attim than the 6nt conaideration would lead u* to think, for the 



feUowing raMcnu : 



(10.) Firstly. The earth U revolving round the nun, or tin- mm appears 

 to move round the earth, in the none direction in which the moou 

 is going. Thi lengthen* the time fur which the mm .tn in any '!' 

 manner upon the inoou, but it lengthen! it more for the linn- in 

 which the moon u moving slowly, than for that in which it in moving 

 quicklj. Thus, suppose that the moon's angular motion when she 

 is near perigee u fourteen time* the sun's angular motion ; and when 

 near apogee, only ten time* the sun's motion. Then ahe passes the 

 aun at the former time (u Been from the earth), with ftth* of II.T 

 whole motion, but at the Utter with only 5,ths; consequently, wli.-n 

 near perigee, the time in which the moon passes through n givi -n 

 angle from the moving line of syzygies (or the time in which the 

 angle between the un and moon increase* by a given quantity). 

 U Uths of the time in which it would have paused through the same 

 angle, had the sun been stationary ; when near apogee, the number 

 expressing the proportion is ftia. The latter number is greater 

 than the former ; and, therefore, the effect of the forces acting near 

 apogee is increased in a greater proportion than that of the forces 

 acting near perigee. And as the effective motion of the line of apses 

 U produced by the excess of the apogeal effect above the perigeal 

 effect, a very small addition to the former will bear a considerable 

 proportion to the effective motion previously found ; and thus the 

 effective motion will be sensibly increased. 



(107.) Secondly. When the line of apses is directed toward the sun, 

 the whole effect of the force is to make it progress, that is, to move 

 hi the same direction as the sun : the sun passes through about IT 

 in one revolution of the moon, and therefore departs only 16 from 

 the line of apses ; and therefore the apse continues a long time near 

 the sun. When at right angles to the line joining the earth and sun, 

 the whole effect of the force is to make it regress ; and, therefore, 

 moving in the direction opposite to the sun's 'motion, the angle 

 between the sun and the line of apses U increased by 36" in < -li 

 revolution, and the line of apses soon escapes from this position. 

 The effect of the former force is therefore increased, while that of 

 the latter is diminished ; and the preponderance of the former is 

 much increased. It is in increasing the rapidity of progress at one 

 time, and the rapidity of regress at another, that the force perpen- 

 dicular to the radius vector indirectly increases the effect of the 

 former in the manner just described. . 



(108.) From the combined effect of these two causes, the actual pro- 

 gression of the line of apses is nearly double of what it would have been 

 if, in different revolutions of the moon, different parts of its orbit had 

 been equally subjected to the disturbing force of the sun. 



(109.) The line of apses, upon the whole, therefore, progresses; and 

 (as calculation and observation agree in showing) with an angular 

 velocity that makes it (on the average) describe 8 in each revolution 

 of the moon, and that carries it completely round in nearly nine years. 

 But as it sometimes progresses and sometimes regresses for several 

 months together, its motion its extremely irregular. The general 

 motion of the line of apses has been known from the earliest agea of 

 astronomy. 



(110.) V. For the alteration of the exceiitricitv of the moon's orbit : 

 first, let us consider the orbit in the position in which the line o( 

 apses passes through the sun, Jiy. 31. While the mnon moves from 

 B, (the perigee) to B, (the apogee), the force in the direction of the 

 radius vector is sometimes directed to the earth, and sometimes from 

 the earth ; and, therefore, by (57) and (59), it sometimes diminishes 

 the excentricity and sometimes increases it. But while the moon 

 moves from B, to B,, there are exactly equal forces acting in the same 

 manner at corresponding parts of the half-orbit, and these, by (58) 

 will produce effect* exactly opposite. On the whole, therefore, the 

 disturbing force in the direction of the radius vector prod i 

 effect on the excentricity. The force perpendicular to the radiui 

 vector increases the moon's velocity when moving from B, to ,, an< 

 diminishes it when moving from B, to B t ; in moving, therefore 

 from B. to B,, the excentricity is increased (65) ; and in moving from 

 B, to B,, it in as much diminished (66). Similarly, in moving from 

 B to B,, the excentricity is diminished ; and in moving from B. to 

 B), it is as much increased. This force, therefore, produces no effect 

 on the excentricity. 



On the whole, therefore, while the lino of apses passes through 

 the sun, the disturbing forces produce no effect ou the excentricity 

 of the moon's orbit. 



(111.) When the line of apses is perpendicular to the line joining 

 the earth and sun, the same thing is true. Though the forces near 

 perigee and near apogee are not now the same as in the last case, thei 

 effect* on different sides of ]rigee and apogee balance each other in 

 the same way. 



(112.) But if the line of apses is inclined to the line joining the earth 

 and sun, as in fig. 32, the effects of the forces do not balance. \Vhil 

 the moon is near B, and near B, the disturbing force in the radiu 

 vector is directed to the earth ; at B, therefore, (58), as the moon is 

 moving towards perigee, the excentricity is increased ; and at B,, as th 

 moon a moving from perigee, the excentricity is diminished. From 



the *lowne of the motion at B, (which give* the disturbing force 

 more time to produce it* effects), and the greatness of the force, the 



B4. 



Fig. 52. 



,/ 



effect at B, will preponderate, and the combined effect* at i), and B, u ill 

 liminish the excentricity. This will appear from reasoning of the same 

 and as that in (98). At B, and u, the force in the radius vector is 

 lirected from the earth : at B,, therefore, by (59), as the moon U 

 moving from perigee, the excentricity is increased, and at B, it is dimi- 

 nished ; but from the slowness of the motion at B, and the magnitude 

 of the force, the effect at n, will preponderate, and the combined effect* 

 at B, and B, will diminish the excentricity. On the whole, therefore, 

 he force in the direction of the radius vector diminishes the excen- 

 ricity. The force perpendicular to the radius vector retards the moon 

 rom B, to B,, but the first part of this motion may be considered near 

 wrigee, and the second near apogee, and, therefore, in the first part, it 

 liinim'shes the excentricity, and in the second increases it ; and the 

 vholc effect from B, to B, is very small. Similarly, the whole effect 

 rom B, to B, is very small. But from B 4 to B, the force accelerates 

 he moon, and, therefore, by (68), (the moon being near perigee) 

 ncreases the excentricity ; and from B, to B, the force also accelerates 

 he moon, and by (68) (the moon being near apogee) diminishes the 

 excentricity; and the effect is much* greater (from the slowness of 

 he moou and the greatness of the force) between B, and n v than 

 ictwecn B, and B,, and therefore the combined effect of the forces iu 

 ;he?e two quadrants is to diminish the excentricity. 



On the whole, therefore, when the line of 'apses is inclined to the 

 ine joining the earth and sun, in such manner that the moon passes 

 ;he line of apses before passing the line joining the earth and sun, the 

 excentricity is diminished at every revolution of the moou. 



(113.) In the same manner it will appear that if the line of apses is 



Hg. 33. 



BO inclined that the moon passes the line of apses after passing the line 

 joining the earth and sun, the excentricity is increased at every revo- 

 lution of the moon. Here the force in the radius vector is directed to 

 the earth, as the moon moves from perigee and from apogee ; and is 

 directed from the earth as the moon moves to perigee and to apogee ; 

 which directions are just opposite to those- in the case already con- 

 sidered. Also the force perpendicular to the radius vector retards the. 

 moon both near perigee and near apogee; and this U opposite to tin- 

 direction in the case already considered. On the whole, therefore, the 

 excentricity is increased at every revolution of the moou. 



(114.) In every one of these cases the effect is exactly the same if 

 the sun be supposed on the side of the moon's orbit, opposite to that 

 represented in the figure. 



To the reader who is acquainted with Newton's 3rd section, the following 

 demiiM-lriitinii of this point will be u(Bcicnt : Four times the reciprocal of the 

 iniin i-rrlum if equal to the sum of the reciprocal! of the apogeal and perigeal 

 I, The effect of an increase of velocity at perigee in a given pro|Kirii<m 

 i* tip alter the area described in a given time in the name proportion, anil there. 

 fore to alter the latut rrrlum in n corresponding proportion. Conc<|iiently an 

 increase of velocity at perigee in a given proportion alters the reciprocal of the 

 apogeal distance tiy n given quantity, and then-fore alters the apogeal r 

 by a quantity nearly proportional to the square of the apogeal distance ; and 

 . i- the ratio of the alteration of apogral distance to apogcal distance (on 

 which the alteration of excentricity depends) is nearly proportional to the 

 apogeal distance. Similarly, if the velocity at apogee it increased in 



,n, the ratio of the alteration of perigeal distance to perigeal ditancc 

 (on which the alteration of cxccntricHy depends) is nearly proportional to the 

 perigeal distance. Thus if the velocity were Increased in the same proportion 

 at perigee nml at apogee, the Increase of excentricity at the former wo. ild lie 

 greater than the diminution at the latter, In the proportion of apogeal distance 

 tn perigeal distance. Hut in Die case before u, the proportion of Increase of 

 Telocity is much greater at apogee than at perigee. First, because the force is 

 greater (being in the same proportion as the distance). Second, because the 

 time In which the moon describes a given angle U greater (being in the name 

 proportion as the square of the distance), so that the increase of velocity i in 

 the proportion of the cube of the distance. Third, becauM the actual velocity is 

 leu (being Inversely as the distance), so that the ratio of the increase to the 

 actual reloctty Is proportional to the fourth power of the distance. Combining 

 thl proportion with that above, the alterations of excentricity in the case before 

 i ., produced by the" forces acting at apogee and at perigee, arc in the proportion 

 01 the cube* of the apogeal and perigeal distances respectively. 



