GRAVITATION. 



GRAVITATION. 



HI 



true : for different planet*, or even far different bodies revolving round 

 different centre* of force, the cubes of the mean distances are in the 

 am* proportion as the products of the square of the periodic time by 

 the sum of the mum of the attracting and attracted body. 



Sicnos UL General ffotiont of Perturbation ; and Perturbation of 



Ote Elemmtt of Orbiu. 



(41.) We have spoken of the motion of two bodies (as the sun and a 

 planet) as if no other attracting body existed. But, as we have stated 



. MM !-.< 



is perpetually varying, the motion i's no longer the same as if it was 

 only attracted by the sun. The planets therefore do not move exactly 

 in ellipses ; the radius vector of each planet does not pass over areas 

 exactly proportional to the times ; and the proportion of the cube of 

 the mean distance to the product of the square of the periodic time 

 by the sum of the mssmm of the sun and the planet, is not strictly the 

 same for all. Still the disturbing forces of the other planets are so 

 mall in comparison with the attraction of the sun, that these laws are 

 very nearly true ; and (except for our moon and the other satellites) it 

 is only by accurate observation, continued for some years, that the 

 effect* of perturbation can be made sensible. 



(42.) The investigation of the effects of the disturbing forces will 

 consist of two parts : the examination into the effects of disturbing 

 forces generally upon, the motion of a planet, and the examination into 

 the kind of disturbing force which the attraction of another planet 

 produce*. We shall commence with the former; we shall suppose 

 that a planet is revolving round the sun, the sun being fixed (a sup- 

 position made only for present convenience), and that some force acts 

 on the planet without acting on the sun (a restriction introduced only 

 fur convenience, and which we shall hereafter get rid of ). 



(43.) The principle upon which we shall explain the effect of this 

 force is that known to mathematicians by the name of variation of 

 element*. The planet, as we have said, describes some curve which is 

 not strictly an ellipse, or indeed, any regularly formed curve. It will 

 not even describe the same curve in successive revolutions. Yet its 

 motion may be represented by supposing it to have moved in an ellipse, 

 provided we suppose the elements of the ellipse to have been per- 

 petually altering. It is plain that by this contrivance any motion 

 whatever may be represented. By altering the major axis, the excen- 

 tricity, and the longitude of perihelion, we may in many different ways 

 make an ellipse that will pass through any place of the planet ; and by 

 altering them in some particular proportions, we may, in several ways, 

 make an ellipse in which the direction of motion at the place of the 

 planet shall be the same as the direction of the planet's motion. But 

 there U only one ellipse which will pass exactly through a place of the 

 planet, in which the direction of the motion at that place shall be 

 exactly the same as the direction of the planet's motion, and in which 

 the velocity (in order that a body may revolve in that ellipse round 

 the sun) will be the same as the planet's real velocity. The dimen- 

 sions and position of this ellipse may be conceived as follows ; if at 

 any instant we suppose the disturbing force to cease, and conceive 

 the planet to be as it were projected with the velocity which it 

 happens to have at that instant, the attraction of the sun or central 

 body will cause it to describe the ellipse of which we are speaking. 

 We shall in future mention this by the name of the inttantancoui 

 tUiptc. 



(44.) If the disturbing force ceases, the planet continues to revolve 

 in the same ellipse, and the permanent ellipse coincides with the in- 

 stantaneous ellipse corresponding to the instant when the disturbing 

 force cos sos. 



(45.) If the disturbing force continues to act, the dimensions of the 

 instantaneous ellipse are continually changing : but in the course of a 

 single revolution (even for our moon), the dimensions alter so little, 

 that the motion in the instantaneous ellipse corresponding to any 

 instant during that revolution will very nearly agree with the real 

 motion during that revolution. 



We shall now consider the effects of particular forces in altering the 

 elements. 



(46.) (I.) Suppose that the disturbing force is always directed to the 

 central body. The effect of this would be nearly the same as if the 

 attraction, or the mass of the central body, was increased. The 

 result of this on the dimensions of the orbit will be different accord- 

 ing to the part of the orbit where it begins to act, and may be 

 gathered from the cases to be mentioned separately hereafter (we 

 do not insist on it at present, as there is no instance in the planetary 

 system of such sudden commencement of force). But at all events 

 the relation between the mean distance and the periodic time will 

 not be the same as before ; the time will be leaf for the same mean 

 distance, or the mean distance greater for the same periodic time 

 than if the disturbing force did not act (88). If the disturbing force 

 is always directed from the central body, the effect will be exactly 

 opposite. If the disturbing force does not alter, except with the 

 planet's distance, the planet will at every successive revolution 

 describe an orbit of the same size. For, as we have stated (29), 

 the radius vector will in equal times pass over equal areas ; and 



mathematicians have proved that, if the variation of force depends 

 only on the distance, the velocity of the plniii-t will depend 

 only on the distance; and the consideration which determine! 

 the greater or least distance of the planet is, that the planet, moving 

 with the velocity which is proper to the distance, cannot describe 

 the proper area in a short time, unless it move in the direction per- 

 ) n.lieular to the radius vector. This consideration will evidently 

 give the same values for the greatest and least distances at I-MTV 

 revolution. It may happen that all the greatest distances will nut 

 be at the same place ; the body may describe such an orbit as that 

 in .fa. 6. 



Fig. 6. 



(47.) (II.) If, however, the disturbing force directed to the central 

 body increases gradually and constantly during many : 

 there is no difficulty in seeing that the planet will at every revolution 

 be drawn nearer to the central body, and thus it will move, at 

 succeeding revolution, in a smaller orbit than at the preceding one ; 

 and will consequently perform its revolution in a shorter time. If 

 the disturbing force directed to the central body diminishes, the 

 orbit will become larger, and the periodic time longer. In the same 

 manner, if the disturbing force is directed from the central body, a 

 gradual increase of the disturbing force will increase the dim. 

 of the orbit and the periodic time, and a gradual diminution of the 

 disturbing force will diminish the dimensions of the orbit and the 

 periodic time. 



(48.) (III.) Suppose that the disturbing force acts always in the 

 direction in which the planet is moving. The reader might imagine 

 at first sight that this would shorten the time of revolution. The 

 effect, however, is exactly opposite ; for in fig. 2, if the planet be 

 projected from A, the reason that the sun's attraction is able to pull 

 the planet in at o and make it approach to itself is, that the velocity 

 of the planet is so small as to allow the force to curve the orbit 

 much. If the velocity were greater, the orbit as we have 

 (25) would be let* curved in every part, and would therefore pass on 

 the outside of the orbit A. o D K P. The effect then of a force in the 

 direction of the planet's motion, which increases the planet's velo- 

 city, is to increase the size of its orbit ; and the bigger the orr.it is, 

 the longer is the time of revolution. If the force acts continually, 

 the time of revolution lengthens continually. If the disturbing 

 force acts in the direction opposite to that in which the planet is 

 moving, the effect is to make the orbit smaller, and to make the time 

 of revolution shorter. The retardation produced by motion through 

 extremely thin air is of this kind : it i* found that a comet (known 

 by the name of Encke's comet) which moves in an ellipse, whose 

 length is not much greater than the diameter of Mars' orbit, per- 

 forms every new revolution in a shorter time than the preceding 

 one ; and we infer from this circumstance that it experiences some 

 resistance in its motion. 



(49.) (IV.) There is one consideration of great importance in the 

 estimation of the effects mentioned in (II.) -and (III.). The altera- 

 tion of the dimensions of the orbit produces an alteration in the 

 periodic time, and this alters the planet's mean motion, or the num- 

 ber of degrees by which the mean longitude is increased in a given 

 time (suppose one year). The effects of these alterations are added 

 together at every successive revolution, and thus may produce an 

 alteration in the planet's mean longitude (which differs from tlio 

 true longitude only by the equation of the centre) that is vastly 

 mote conspicuous than the alteration in the dimensions of the orbit. 

 Suppose, for instance, that a disturbing force acted on a planet 

 (either a constant force in the direction of its motion, or a variable 

 force in the direction of the radius vector), such as to increase the 

 mean distance by n^jth part in 100 revolutions of the planet. This 

 alteration of the planet's distance from the sun could hardly be dis- 

 covered by the nicest observations. But as the mean distance has 

 been altered in the proportion of 10000 : 10001, the periodic time 

 will have been altered in the proportion of 10000: 100014 nearly, 

 or the mean motion will have been altered in the proportion of 

 100014 to 10000, or 1 : 0-99985 nearly. If this alteration has gone 

 on uniformly, we may suppose the whole motion in the 100 revo- 

 lutions to have been nearly the same as if the planet had moved 

 with a mean motion, whose value is half way between the values of 

 the first and the last, or 0-999925 x the original mean motion. 

 Therefore, at the time when we should expect the planet to have 

 made 100 revolutions, it will only have made 92'99'25 revolutions, or 

 will be behind the place where we expected to see it by 0-0075 revo- 

 lution, or nearly three degrees ; a quantity which could not fail t 

 be noticed by the coarsest observer. To use a borrowed illustration, 

 the alteration of the mean distance in an orbit produces the same 

 kind of effect as the alteration of the length of a clock pendulum 



