GRAVITATION. 



GRAVITATION. 



the two oases ; with contra t describe the circle cdt. Then it i- 

 rident that the TOT; exoentrie orbit c B in fa 8 u widely separated 



. from the circle cdt, and therefore, when it ii beat through a given 

 angle to the position cd, it will internet the circle at a point d not 

 distant from c. In > /. 9, on the contrary, the orbit c B U not widely 

 MparaUd from the circle, and therefore, when it u brat through a 

 given angle, its iatereection d will be distant from c. Mow the new 

 perihelion will be found, in both cases, by bisecting erf; and, 

 therefore, iU change of position in fg. 8, where the orbit u Terr 

 xeentrie, is much leee than inty, 9, where the exoentrioity U amalL 

 Or we may itate it thu* : The alteration of the place of perihelion, 

 or aphelion, depend* on the proportion which the alteration in the 

 approach or noses produced by the disturbing force bean to the 

 whole approach or recess; and is therefore greatest when the 

 whole approach or reoeee ia least; that u, when the orbit U little 

 exoentric. 



(57.) (IX. > To judge of the effect which a disturbing force, directed to 

 the sun, will produce on the excentricity of a planet's orbit, let us 

 suppose the planet to have left its perihelion, nd tj be moving 

 towards aphelion, and, consequently, to be receding from the sun, 

 and now let the disturbing force act for a short time. This will 

 eause it to recede from the sun more slowly than it would have 

 receded without the action of the disturbing force; and conse- 

 quently, the planet, without any material alteration in its velocity, 

 and, therefore, without any material alteration in the major axis of 

 its orbit (28), will be moving in a path more inclined to the radius 

 rector than if the disturbing force had not acted. The planet may, 

 therefore, be considered as projected from the point A, fg. 10, in the 



Fig. 10. 



direction A I instead of A B, in which it was moving ; and, therefore, 

 instead of describing the orbit A c o, in which it was moving before, 

 it will describe an orbit A c y, more resembling a circle, or less excen- 

 tric than before. The effect, therefore, of a disturbing force directed 

 to the centre, while a planet is moving from perihelion to aphelion, 

 is to diminish the excentricity of the orbit. 



(58.) If we suppose the planet to be moving from aphelion to 

 perihelion, it is approaching to the sun ; the disturbing force directed 

 to the sun makes it approach more rapidly ; its path is therefore less 

 inclined to the radius vector than it would have been without the 

 disturbing force ; and this effect may be represented by supposing that 

 at z,/fji. 11, instead of moving in the direction EF in which it was 



Fig. 11. 



irojecUd, if it is projected in a direction perpendicular to the radius 

 ector (which is implied in our supposition, that the plaoe where the 

 orce acts was an apse in the old orbit), the pi iee ef projection will 

 nfallibly be an apse in the new orbit ; and the line of apses, which 



is the line drawn from that point through the sun, will I* the same 



as before. 



(62.) But if the force act for a short time before the planet reaches 

 he perihelion, its principal * effect will be to increase its velocity ; 

 he sun's attraction will therefore have lees power to curve iU path 

 25) ; the new orbit will be, in that part, exterior to the old one. In 

 fy. 12, we must therefore suppose that the planet, after leaving A, 



Fij. U. 



moving, the planet is projected in the direction K /. Instead therefore 

 of describing the ellipse x o B, in which it was moving before, it wil 

 describe such an ellipse as E g h, which is more excentric than the 

 former. The effect therefore of a disturbing force directed to the 

 centre, while a planet is moving from aphelion to perihelion, is to 

 increase the excentricity of the orbit. 



(59.) In a similar manner it will appear, that the effect of a dis 

 turbing force, directed from the centre, is to increase the excentricity 

 as the planet is moving from perihelion to aphelion, and to dimmish ii 

 a* the planet moves from aphelion to perihelion. 



(80.) (X.) Let us now lay aside the consideration of a force acting in 

 the direction of the radius vector, and consider the effect of a force 

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

 the planet is moving. And first, its effect on the position of the 

 line of 



(81.) If such a force act at one of the spies, either perihelion o 

 aphelion, for a short time, it is clear that its effect will be represents 

 by supposing that the velocity at that apse is suddenly increased, o 

 that the velocity with which the planet is projected from perihelion i 

 greater than the velocity with which it would have been projected i 

 no disturbing force had acted. This will make no difference in th 

 position of the line of apses ; for with whatever velocity the planet is 



where the force has acted to accelerate its motion, instead of describing 

 the orbit AGO, proceed* to describe the orbit A cd, which at A has the 

 same direction (or has the same tangent A B) as the orbit AGO. It i 

 plain now that c is the part nearest to the sun, or c is the perihelion : 

 and it is evident here, that the line of apses has altered its position 

 from s c to s e, or has twisted in a direction opposite to the ^ng"lr 

 motion of the planet, or has regressed. 



(63.) If the force act for a short time after the planet has passed 

 perihelion, as at D, in Jiy. 13, the planet's velocity is increased there, 



Fig. 13. 



' 



and the path described by the planet is D/, instead of D r, having the 

 same direction at D (or having the same tangent D E), but less curved, 

 and therefore exterior to D r. If now we conceive the planet to have 

 received the actual velocity with which it is moving in D/, from 

 moving without disturbance in an elliptic orbit CD/ (which is the 

 orbit that it will now proceed to describe, if no disturbing force con- 

 tinues to act), it is evident that the part c D must be described with a 

 greater velocity than c D, inasmuch as the velocity at D from moving 

 in e D is greater than the velocity from moving in c D ; < D is there- 

 fore less curved than c D, and therefore exterior to it (since it has the 

 same direction at D) ; and then the perihelion is some point in the 

 position of e, and the line of apses has changed its direction from 8 c 

 to s c, or has twisted round in the same direction in which the planet 

 is moving, or has progressed. 



(64.) If the force act for a short time before passing the aphelion, it 

 will be seen in the same manner that the line of apses is made to pro- 

 gress. It is only necessary to consider that (as before) the new orbit 

 has the same direction at the point H, fig. U, where the force has acted, 



Fig. 14, 



as the old one, but is less curved, and therefore exterior to it ; and the 

 ..[In li"ii, or points raot distant from the sun, is g intH O f o, and the 

 position of the line of apse* has shifted from t a to 8 g. U the force 

 act after the planet has passed the aphelion, as at K, Jig. 15, the orbit 

 in which we must conceive the planet to have come, in order t<> have 

 the increased velocity, must be y K exterior to o K ; the point most 



It ii luppoMd here, and in all our inmtigatioM, that the cxceni;. 

 tbc orbit U immll, and, consequently, that a force p< rpcndicular to tba radiui 

 rector produco nearly th same effect a* a force acting in the direction of a 

 Ungent to the cllipM. 



