181 



GRAVITATION. 



GRAVITATION. 



482 



which, though so small as to be insensible to the eye, will in a few 

 days produce a very great effect on the time shown by the clock. 

 (50.) (V.) Now, suppose the orbit of the planet or satellite to be an 

 ellipse ; and suppose a disturbing force directed to the central body 

 to act upon the pknct, &c., only when it is near its perihelion or 

 perigee, &c. In Jig. 7, let AB be the curve in which the planet is 



Fig. 1. 



moving, and let the dotted line B c D A represent the orbit in which 

 it would have moved if no disturbing force had acted, c being the 

 place of perihelion. At B let the disturbing force, directed towards 

 s, begin to act, and let it act for a little while, and then cease. The 

 planet is at that place approaching toward the sun, and the direction 

 of its motion makes an acute angle with s B. It is evident that the 

 disturbing force, which draws the planet morn rapidly towards the 

 Bun without otherwise affecting its motion, will cause it to move in 

 a direction that makes a more acute angle with 8 B. The part of the 

 new path, therefore, which is nearest to the sun (that is, the new 

 perihelion) will be farther from B than the perihelion c of the orbit 

 in which the planet would have moved. The reader's conception of 

 this will be facilitated by supposing the orbit, instead of a curve, to 

 be a straight line, as B c in Jig. 7*, and the place of perihelion to be 

 determined by letting fall a perpendicular s c from the sun upon the 

 line, when it will be seen that if the disturbing force, acting towards 

 8 for a short time at B, changes the path of the planet from the 

 direction B c to B c, the distance of the foot of the perpendicular s c 

 from B is greater than that of s c. With a curved orbit the result is 

 just the same. In other words, the planet, instead of describing B c 

 Jig. 7, will, in consequence of the action of the disturbing force, 

 describe B c ; and the place of perihelion, instead of c, will be c, a 

 point more distant from B than c is. Now, if the disturbing force 

 should not act again, the planet would move in an ellipse cdb, and 

 the line of apses, instead of c s D, would be c s </. The line of apses 

 has, therefore, twisted round in the same angular direction as that 

 in which the planet was going ; and this is expressed by saying that 

 tlu line of apses program. If, after passing c, the disturbing force 

 should again act for a little while, at e for instance, the recess of 

 the planet from the sun would be diminished ; its path would be 

 more nearly perpendicular to the radius vector ; and, therefore, the 

 inclination of the path would be such as corresponds to a smaller 

 distance from perihelion than the planet really has ; that is, when 

 the planet leaves e, the inclination of its path to the radius vector is 

 greater than it would have been if the planet had continued to move 

 in the orbit c d b, but i the same as if its perihelion had been at 

 gome such situation as /, supposing no disturbing force to act. Now 

 let the disturbing force cease entirely to act, and the planet, which 

 at e is moving as if it had come from the perihelion /, will continue 

 to move as if it had come from the perihelion/; it will proceed, 

 therefore, to describe an elliptic orbit in which fsg is the line 

 of apses : the line of apses has been twisted round in the same 

 direction ag before, or the line of apses hag still progressed. The 

 effect, then, of a disturbing force directed to the central body before 

 and after passing the perihelion, is to make the line of apses 

 progress.* 



(51.) In the same manner, it will be seen, that the effect of a dis- 

 turbing force, directed from the central body before and after passing 

 the perihelion, is to make the line of apses regress. 



(52.) The motion of the planet, subject to such forces an we have 

 mentioned, would be nearly the same as if it was revolving in an 

 elliptic orbit, and this elliptic orbit was at the same time revolving 

 round it* focus, turning in the same direction as that in which the 

 planet goes round, ami always carrying it on its circumference. And 

 this is the easiest way of representing to the mind the general effect of 

 this motion ; the physical cause is to be sought in such explanations as 

 that above. 

 (53.) (VI.) Suppose a disturbing force directed to the centre, to act 



* This result, ami those which follow immediately, may be inferred from the 

 construction in Newton's ' Principia,' book i., sect. 3, prop. XTii. If we assume 

 (as we suppose in all these investigations) the exccntricity to be small, the 

 distn/bing force directed to the sun will not sensibly alter the planet's velocity, 

 bat will change the direction of its path at p, the place of action (in Newton's 

 figure) ; the length of p n, therefore, will not be altered (since that length 

 'It only on the Telocity), but its position will be altered, the position of 

 p B being determined by making the angle urn equal to the supplement of 

 RPR. On trying the effects of this in different positions of p, and observing that 

 the immediate effect of a disturbing force directed to the centre is to increase 

 the rate of approach, or to diminish the rate of receding, and tbat the effect of a 

 force directed from the centre is the opposite, nil the cases in the text will be 

 fully explained. 



ARTS AXD SCI. DIV. VOL. IV. 



upon the planet when it is near aphelion. As the planet is going 

 towards aphelion it is receding from the sun. The effect of the dis- 

 turbing force is to diminish the rate of recess from the sun ; and, 

 therefore, to increase the inclination of the planet's path to the 

 radius vector. The aphelion is the place where the planet's path is 

 perpendicular to the radius vector. The effect of the disturbing 

 force, then, which increases the inclination of the planet's path to 

 the radius vector, will be to make that path perpendicular to the 

 radius vector sooner than if the disturbing force had not acted. 

 That is, the planet will be at aphelion sooner than it would have 

 been if no disturbing force had acted. The aphelion has, as it were, 

 gone backwards to meet the planet. If the disturbing force should 

 entirely cease, the planet will move in an elliptic orbit, of which 

 this new aphelion would be the'permanent aphelion. The line passing 

 through the aphelion has, therefore, twisted in a direction opposite 

 to the planet's motion, or the line of apses has regressed. After 

 passing aphelion, if the disturbing force still continues to act, the 

 planet's approach to the sun will be quickened by the disturbing 

 force ; and, therefore, after some time, the planet's rate of approach 

 will be greater than that corresponding, in an undisturbed orbit, to 

 its actual distance from aphelion, and will be equal to that corre- 

 sponding in an undisturbed orbit to a greater distance from aphelion. 

 If, now, the disturbing force ceases, the planet, moving as if it came 

 in an undisturbed orbit from an imaginary aphelion, will continue to 

 move as if it came from Uiat imaginary aphelion ; and that imaginary 

 aphelion having been at a greater distance behind the planet than 

 the real aphelion, its place will be represented by saying that the 

 line of apses has still regressed. The effect, then, of a disturbing 

 force directed to the central body, before and after passing aphelion, 

 is to make the line of apses regress. 



(54.) In the same manner it will be seen, that the effect of a dis- 

 turbing force, directed from the central body, before and after passing 

 the aphelion, is to make the line of apses progress. 



(55.) (VII.) Since a disturbing force, directed to the central body, or 

 one directed from the central body, produces opposite effects with 

 regard to the motion of the line of apses, according as it acts near 

 perihelion or near aphelion, it is easy to perceive that there must be 

 gome place between perihelion and aphelion, where the disturbing 

 force, directed to the central body, will produce no effect on the 

 position of the line of apses. It is found by accurate investigation, 

 that this point is the place where the radius vector is perpendicular 

 to the line of apaes.* 



(56.) (VIII.) The effects mentioned above are greatest when the excen- 

 tricity is small. Let us compare the two orbits A C B in fig. 8 and 

 A c B in Jig. 9, in one of which the excentricity is great and in the 

 C 



Fig. 8. 



Fig. 9. 



other small : suppose the disturbing force to act for a short time at 

 the perihelion c, and to be nearly equal in the two orbits, so as to 

 deflect the new path c d from the old orbit c B by equal angles in 



To the reader who is familiar with Newton's ' Principia,' sect. 3, the fol- 

 lowing demonstration will be sufficient : The disturbing force, which is entirely 

 in the direction of the radius vector, will not alter the area described in a given 

 time, and, therefore, will not alter the latus rectvm (lo the square root of which 

 the area is proportional). But half the latus rectum of the undisturbed orbit is 

 the radius vector at the supposed place of action of the disturbing force (since 

 that radius vector is supposed perpendicular to the major axis). 'Iherefore, 

 half the lotus rectum of the new orbit is the radius vector at the point in 

 question ; and, consequently, the radius vector, at the point in question, is per- 

 pendicular to the major axis in the new orbit ; but it was so in the undisturbed 

 orbit ; and, therefore, the major axes in the new orbit and the undisturbed orbit 

 coincide. 



I I 



