485 



GRAVITATION. 



GRAVITATION. 



486 



distant from the sun must be g instead of o, and the line of apses must 

 have changed from 8 o to s g, or must have regressed. 



Fig. 15. 



(65.) Collecting these conclusions,* we see that, if a disturbing force 

 act perpendicularly to the radius vector, in the direction in which the 

 planet is moving, its action, while the planet passes from perihelion to 

 aphelion, causes the line of apses to progress ; and its action, while 

 the planet passes from aphelion to perihelion, causes the apses to 

 regress. 



(66. ) By similar reasoning, if the direction of the disturbing force is 

 opposite to that in which the planet is moving, its action, while the 

 planet passes from perihelion to aphelion, causes the line of apses to 

 regress, and while the planet passes from aphelion to perihelion causes 

 the apses to progress. 



(67.) (XI.) For the effect on the eccentricity : suppose the disturbing 

 force, increasing the velocity, to act for a short time at perihelion ; 

 the effect is the same as if the planet were projected from perihelion 

 with a greater velocity than that which would cause it to describe 

 the old orbit. The sun's attraction therefore will not be able to pull 

 it into so small a compass as before ; and at the opposite part of its 

 orbit, that is, at aphelion, it will go off to a greater distance than 

 before ; but as it is moving without disturbance, and, therefore, in 

 an ellipse, it will return to the same perihelion. The perihelion 

 distance therefore remaining the same, and the aphelion distance 

 being increased, the inequality of these distances is increased, and 



. the orbit therefore is made more excentric. Now, suppose the force 

 increasing the velocity to act at aphelion. Just as before, the sun's 

 attraction will be unable to make the planet describe an orbit so 

 small as its old orbit, and the distance at the opposite point (that is, 

 at perihelion) will be increased ; but the planet will return to the 

 same aphelion distance as before. Here, then, the inequality of 

 distances is diminished, and the excentricity is diminished. 



(68.) Thus we see that a disturbing force, acting perpendicularly to 

 the radius vector, in the direction of the planet's motion, increases the 

 excentricity if it acts on the planet near perihelion, and diminishes the 

 excentricity if it acts on the planet near aphelion. And, similarly, if 

 the force acts in the direction opposite to that of the planet's motion, 

 it diminishes the excentricity by acting near perihelion, and increases 

 it by acting near aphelion. 



(69.) (XII.) In all these investigations, it is supposed that the dis- 

 turbing force acts for a very short time, and then ceases. In future, 

 we shall have to consider the effect of forces, which act for a long 

 time, changing in intensity, but not ceasing. To estimate their 

 effect we must suppose the long time divided into a great number 

 of short times ; we must then infer, from the preceding theorems, 

 how the elements of the instantaneous ellipse (43) are changed in 

 each of these short time* by the action of the force, which is then 

 disturbing the motion ; and we must then recollect, that the instan- 

 taneous ellipse, at the end of the long time under consideration, will 

 be the same an the permanent ellipse in which the planet will move. 

 if the disturbing force then ceases to act (43), and that it will, at all 

 events, differ very little from the curve described in the next revo- 

 lution of the planet, even if the disturbing force continue to act. 



IV. On the Nature of the Pone duturbing a Planet or Satellite, 

 produced by the Attraction of other Boditt. 



(70.) Having examined the effect* of disturbing forces upon the 

 elements of a planet's or satellite's orbit, we have now to inquire into 

 the kind of the disturbing force which the attraction of another body 

 produces. The inquiry is much simpler than might at Erst sight be 

 expected ; and this simplicity arises, in part, from the circumstance 

 that (as we have mentioned in (6) ) the attraction of a planet upon the 

 sun is the same as its attraction upon another planet, when the sun 

 and the attracted planet are equally distant from the attracting planet. 



(71.) First, then, we have to remark, that the disturbing force is not 

 the whole attraction. The sun, for instance, attract* the moon, and 

 disttnbs it* elliptic motion round the earth ; yet the force which dis- 

 turbs the moon's motion is not the whole attraction of the sun upon 

 the moon. For the effect of the attraction u to move the moon from 



e place where it would otherwise have been ; but the sun's attraction 

 upon the earth also moves the earth from the place where it would 



* These condemn., and those that follow, will be easily Inferred from 

 iwtoa'i con.tructlon, I'top. XVII., by observing, that n IncnaM of the 

 i it r HWMMi the lenjthof tu ta Newton's figure without ulturuig its 



fJS.l.'/n 



otherwise have been ; and if the alteration of the earth's place is 

 exactly the same as the alteration of the moon's place, the relative 

 situation of the earth and moon will be the same as before. Thus, if, 

 in fg. 16, any attraction carries the earth from E to e, and carries the 



Fig. 10. 



moon from M to m, and if E e is equal and parallel to M m, then e m, 

 which is the distance of the earth and moon, on the supposition that 

 the attraction acts on both, is equal to E M, which is their distance, on 

 the supposition that the attraction acts on neither ; and the line e m, 

 which represents the direction in which the moon is seen from the 

 earth, if the attraction acts on both, is parallel to E M, which represents 

 the direction in which the moon is seen from the earth, if the attrac- 

 tion acts on neither. The distance therefore of the earth and moon, 

 and the direction in which the moou is seen from the earth, being 

 unaltered by such a force, their relative situation is unaltered. An 

 attraction, therefore, which acts equally, and in the same direction, on 

 both bodies, does not disturb their relative motions. 



From this we draw the two following important conclusions : 



(72.) Firstly. A planet may revolve round the sun, carrying with it a 

 satellite ; and the satellite may revolve round the planet in nearly 

 the same manner as if the planet was at rest. For the attraction of 

 the sun on the planet is nearly the same as the attraction of the sun 

 on the satellite. It is true that they are not exactly the same, and 

 the effects of the difference will soon form an important subject of 

 inquiry ; but they are, upon the whole, very nearly the same. The 

 moou is sometimes nearer to the sun than the earth is, and some- 

 times farther from the sun ; and, therefore, the sun's attraction on 

 the moon is sometimes greater than its attraction on the earth, and 

 sometimes less ; but, upon the whole, the inequality of attractions is 

 very small. It is owing to this that we may consider a satellite as 

 revolving round a planet in very nearly the same manner (in respect 

 of relative motion) as if there existed no such body as the sun. 



(73.) Secondly. The force which disturbs the motion of a satellite, or 

 a planet, is the difference of the force's (measured, as in (4), by the 

 spaces through which the forces draw the bodies respectively) which 

 act un the central and the revolving body. Thus, if the moon is 

 between the sun and the earth, and if the sun's attraction in a 

 certain time draws the earth 200 inches, and in the same time draws 

 the moon 201 inches, then the real disturbing force is the force which 

 would produce in the moon a motion of one inch from the earth. 



(74.) In illustrating the second remark, we have taken the simplest 

 case that can well be imagined. If, however, the moon is in any other 

 situation with respect to the earth, some complication is introduced. 

 Not only is the moon's distance from the sun different from the 

 earth's distance, (which according to (9) produces an inequality in tho 

 attractions upon the earth and moon,) but also the direction in which 

 the attraction acts on the earth is different from the direction- in 

 which it acts on the moon, (inasmuch as the attraction always acts in 

 the direction of the line drawn from the attracted body to the attract- 

 ing body ; and the lines so drawn from the earth and moon to the sun 

 are in different directions.) The same applies in every respect to the 

 perturbation which one planet produces in the motion of a second 

 planet round the sun, and which depends upon the difference in the 

 tirst planet's attractions upon the sun and upon the second planet. To 

 overcome this difficulty we must have recourse to geometrical con- 

 siderations. In Jig. 17, let B, be a body revolving about A, and let be 



; 





another body whose attraction disturbs the motion of B, round A 

 The attraction of c will in a certain time draw A to a ; it will in the 

 same time draw B,to 6,. Make B, rf t equal and parallel to A a ; then 

 a d will be equal and parallel to A B,. Now if the force upon B r were 

 such as to draw it to rf,, the motion of B, round A would not be dis- 

 turbed by that force But the force upon B, is really such as to draw 

 it to 6,. The real disturbing force then may be represented as a force 

 which draws the revolving body from rf, td &,. If, instead of sup- 

 posing the revolving body to be at B, we suppose it at B s , and if the 

 attraction of would draw it through B 2 b. t while it draws A through 

 A a, then (in the name manner, making B a (/ equal and parallel to A o) 

 the real disturbing force may be represented by a force which in the 

 same time would draw B, through d. t 6,. 



