OKAVITATION. 



GRAVITATION. 



47f 



move in *n ellipse ; but if it U much greater, the body will more in a 

 parabola or a hy perbola. 



If A it in oblique U. s A, and the Telocity of projection U mull, the 

 body will more in an clli|e ; but U the velocity U gnat, it may move 

 in a parabola or hyperbola, but not in a circle. 



If tin- body describe a circle, the gun is the centre of the circle. 



If the Uxly describe an ellipse, the iuo U not the centre of I lie 

 vlli|Mv. but one focus. (Tho method of describing an ellipse U t.. i,\ 

 t .< pin* in a board, a* at s and n,jig.S; to fanteu a thread s r n to 

 them, and to keep this thread stretched by tin- point of a pencil, as 

 at r ; the pencil will trace out an ellipw, and the places of the pins 

 a and u will bo the two focuses.)' 



l .. . 



If the body describe a parabola or hyperbola, the sun is in the 

 focus. 



(19.) The planets describe ellipses which are very little, flattened, 

 and differ very little from circles. Three or four comets describe very 

 long ellipses ; and nearly all the others that have been observed are 

 found to move in curves which cannot be distinguished from parabolas. 

 There is reason to think that two or three comets which have been 

 observed move in hyperbolas. But as we do not propose, in this 

 treatise, to enter into a discussion on the motions of comets, we shall 

 confine ourselves to the consideration of motion in an ellipse. 



(20.) Everything that has been said respecting the motion of a 

 planet, or body of any kind, round the sun, in consequence of the sun's 

 attraction according to the law of gravitation, applies equally well id 

 the motion of a satellite about a planet, since the planet attracts with 

 a force following the same law (though smaller) as the attraction of the 

 sun. Thus the moon describes an ellipse round the earth, the earth 

 being the focus of the ellipse ; Jupiter's satellites describe each an 

 ellipse about Jupiter, and Jupiter is in one focus of each of those 

 ellipses ; the same is true of the satellites of Saturn and Uranus. 



(21.) In stating the suppositions on which the calculations of orbits 

 are made, we have spoken of a force of attraction, and a force by which 

 a pi. met is projected. But the reader must observe that the nature of 

 these forces U wholly different The force of attraction is one which 

 acts constantly and steadily without a moment's intermission (as we 

 know that gravity to the earth is always acting) : the force by which 

 the body is projected is one which we suppose to be necessary at some 

 past time to account for the planet's motion, but which acts no more. 

 The planets art in motion, and it is of no consequence to our inquiry 

 how they received this motion, but it is convenient, for the purposes 

 of calculation, to suppose that, at some time, they received an impulse 

 of the same kind as that which a stone receives when thrown from the 

 hand ; and this is the whole meaning of the term " projectile force." 



(22.) From the same considerations it will appear that, if in any 

 future investigations we should wish to ascertain what is tl>. 

 described by a planet after it leaves a certain jioint where the velocity 

 and direction of its motion are known, we may suppose the planet to 

 be projected from that point with that velocity and in that direction. 

 For it is unimportant by what means the planet acquires its velocity, 

 provided it has such a velocity there. 



(23.) We shall now allude to one of the points which, u|>on a cursory 

 view, has always appeared one of the greatest difficulties in the 

 theory of elliptic revolution; but which, when duly considered, will be 

 found to be one of the most simple and natural consequences of the 

 law of gravitation. 



(24.) The force of attraction, we have said, is inversely proportional 

 to the square of the distance, and is therefore greatest when the distance 

 is least. It would seem then, at first sight, that when a planet has 

 approached most nearly to the sun, as the sun's attraction is then 

 greater than at any other time, the planet must inevitably fall t . the 

 sun. But we assert that the planet begins then to recede from the 

 sun, and that it attains at length as great a distance as before, and goes 

 on continually retracing the same nrl.it. How is this receding from 

 the sun to be accounted for ? 



< .'.',. ) The explanation depends on the increase of velocity as the 

 plain -I Approaches to the point where its distance from the sun is leant, 

 and on the considerations by which we determine the form of the 

 which a certain attracting force will cause a planet to dem-ribo. 

 In explaining the motion of a stone thrown from the hand, to whirh 

 the motion of a planet for a very small time U exactly similar, we have 

 seen that the deflection of the stone from the straight line in which it 

 began to move is exactly equal to the space through which gravity 

 could have made it fall in the same time from rest, whatever v 

 velocity with which it was thrown. Consequently, when the stone is 

 thrown with very great velocity, it will have gone a great distance 

 before it is much deflected from the straight Tine, and 

 path will be very little curved ; a fact familiar to the ex|>eiionce of 

 every one. The same thing holds with regard to the motion of a planet ; 



and thus the curvature of any part of the orbit which a planet describes 

 ill not deiwnd simply upon the force of the sun's attraction, but will 

 also depend on the velocity with which the planet is moving. Tho 

 greater U the velocity of the planet at any point of its orbit, the less 

 will the orbit be curved at that part Now if we refer to .>/. 2, we 

 shall see that, supposing the planet to hare passed the point o with so 

 small a velocity th.a the attraction of the sun bends its path very 

 much, and causes it immediately to begin to approach toward* the 

 sun : the sun's attraction will necessarily increase its velocity as it 

 moves through D, E, and r. For the sun's attractive force on the 

 planet, when the planet is at D, is acting in the direction D s, and it i.- 

 plain that (on account of the small inclination of D E to D 8) the force 

 pulling in the direction D s helps the planet along in its path D B, and 

 thereby increases it* velocity. Just as when a ball rolls down a sloping 

 bank, the force of gravity (whose direction is not much inclined to the 

 bank) helps the ball down the bank, and thereby increases its Telocity. 

 In this manner, the Telocity of the planet will be continually increasing 

 as the planet passes through D, B, and r ; and though the sun's at- 

 tractive force (on account of the planet's nearness) is very mu.-li 

 increased, and tends, therefore, to make the orbit more curv.-l. \ 

 velocity is so much increased that, on that account, the orl.it i- not 

 more curved than before. Upon making the calculation more accu- 

 rately, it is found that the planet, after leaving o, approaches to the 

 sun more and more rapidly for about a quarter of its time of revo- 

 lution ; then for about a quarter of its time of revolution the velocity 

 of its approach is constantly diminishing: and at half tin- periodic 

 time after leaving o, the planet is no longer approaching to the sun ; 

 and its Telocity is so great, and the curvature of the orbit in conse- 

 quence so small (being, in fact, exactly the same as at c), that it begins 

 to recede. After this it recedes from the sun by exactly the same 

 degrees by which it before approached it. 



(26.) The same sort of reasoning will show why, when the planet 

 reaches its greatest distance, where the sun's attraction is least, it does 

 not altogether fly off. As the planet passes along u, K, A, the nun's 

 attraction (which is always directed to the sun) retards the planet in 

 its orbit, just as the force of gravity retards a ball which is bowled up 

 a hill ; and when it has reached c, its velocity is extremely small ; and, 

 therefore, though the sun's attraction at c is small, yet the del 

 which it produces in the planet's motion is, (on account of the planet's 

 slowness there) sufficient to make its path very much curved, and the 

 planet approaches the sun, and goes over the same orbit as before. 



(27.) The following terms will occur perpetually in the rest of this 

 article, and it is therefore desirable to explain them now. 



Let 8 and H, fa. 4, be the focuses of the ellipse A E B D; draw the 



Fig. 4. 



line A D through 8 and n ; take c tlie middle point between 8 and ir, 

 and draw D c K perpendicular to A c B. Let B be that focus which is the 

 place of the sun, (if we are speaking of a planet's orbit,) or the place of 

 tin planet, (if we are speaking of a satellite's orl.it>. 



Then A B is called the major ajcii of the ellipse. 



c is the entire. 



A O or c B is the irmi-majnr run*. This is equal in length to s D ; it 

 is sometimes called the mean di$ta*ce, because it is half-way between 

 A 8 (which is the planet's smallest distance from s) and B B (which i* 

 the planet's greatest distance from s). 



i - the minor nrit, and o c or o I the *nm -minor am. 



A is called the perihelion (if we are speaking of a planet's orbit) ; the 

 perigee (if we are speaking of the orbit described by our moon about 

 the earth) ; the prrijore (if we are speaking of the orbit dwrib, .! by 

 one of Jupiter's satellites round Jupiter); or the pcruat*mim (ii we 

 are speaking of the orbit described by one of Saturn's satellites about 

 Saturn.) 



B, in the orbit of a planet, is called the a/JifUnn ; in the i- 

 it is called the apogee; in the orbit of one of Jupiter's satellites, we 

 hall call it the ajtajort. 



A and B are both called apt* ; and the line A B, or the major axis, is 

 sometimes called the War ofajaet. 



8 c is sometimes called the linear nceentririty : but it i- < 

 to speak only of the proportion which g o bears to A o, and thin pro- 

 portion, expressed by a number, is called the txcenlririty. Thus, if 

 R c were one-third of A c, we should say, that the exoentricity of the 

 nrl.it was J. or 0*3888. 



If s <Y> is drawn towards a certain point in Hie heavens, called the 

 firit point of Aria, then the angle V 8 A is called the longitude of 

 /H-rilnlii:.! <.>r of perigee, or of per 



i the place of the planet in it* orbit nt any particular time, 

 t hen the angle T s P is its lonyttwlt at that time, and the angle A B r 



