605 



GRAVITATION. 



GRAVITATION. 



606 



we consider the perturbations of two planets which are nearer to each 

 other, we are obliged to alter our conclusions considerably. The dis- 

 turbing force becomes so much greater where the planets are near 

 conjunction than at any other part, that the orbit is much more 

 changed there than at any other part. However, the reasoning upon 

 which, in (91), we determined the form of the moon's orbit, laying 

 aside the consideration of independent eccentricity, will, to a certain 

 extent, apply here. The orbit in several cases will be flattened on the 

 side where conjunction takes place, and on the opposite side, but 

 generally moat so on the latter ; and will be made protuberant at the 

 parts where the disturbing force tends wholly to increase the gravi- 

 tation towards the sun. The same general reasoning will, in many 

 cases, help us to find the form of the orbit which is influenced by the 

 attraction of an interior planet. 



(159.) A consideration, however, of particular cases will show how 

 utious we must be in applying this conclusion. Suppose, for in- 



stance, we consider the reciprocal perturbations of the Earth and Mars. 

 The periodic time of Mars is nearly double that of the Earth. Here, 

 then, we fall upon an inequality of the same kind as that discussed 

 in (122), ILC., for the satellites of Jupiter. And though the periodic 

 time of Mars is not very nearly double that of the Earth, so that the 

 distortions produced in the orbits of the Eirth and Mars are not very 



c l that of Saturn, Jupiter will have gone quite round, and also as far 

 in the next revolution as Bj, while Saturn has described part of a revo- 



Fig. 38. 



lution only to c a : then Jupiter will again have gone quite round, and 

 also as far in the next revolution as B 3 , while Saturn has described part 

 of a revolution to c a : then Jupiter will have performed a whole revo- 

 lution, and part of another to B,, while Saturn has performed part of a 

 revolution to c, : and then the same order of conjunctions will go on 

 again. If, then, the periodic times were exactly in the proportion of 

 2 to 5, the conjunctions would continually take place in the same three 

 points of the orbits. This conclusion will not be altered by supposing 



might suppose analogous to the variation of the Moon, becomes, from on ;J the same order, and happen at the same places as before. 



the small disproportion of distances, and the near commensurability of (165.) But the periodic times are not exactly in the proportion 



the periodic times, much more nearly similar to the slow variation of j o { 3 to 5, but much more nearly in the proportion of 29 : 72. This 



the dements of orbits. 



(160.) It seems quite hopeless to attempt to give a notion of the 

 calculations by which, in all the different cases", the disturbances inde- 

 pendent of the eccentricities can be computed. It is sufficient to state, 

 that the same methods apply to all, and that they are much more 

 simple than those relating to other pointa, of which an idea may be 

 given by general explanation. 



( Itjl.) Let us now consider the inequalities of motion which depend 

 on the eccentricities and inclinations of the planets' orbits. The idea 

 that will probably first occur to the reader is this. " If the distur- 

 bances of the planets, supposing their orbits to have no independent 

 eccentricities, amount only to a few seconds, how is it likely that the 

 small alterations of place, which are produced by the trifling eccen- 

 tricities and inclinations of their orbits, will so far alter their forces 

 upon each other as to produce any sensible difference in the magnitude 

 of irregularities which are already insignificant ? " In answer to this 

 we must say, " It is true that these forces, or alterations of forces, are 

 exceedingly small, and those parts of them which act in the same 

 direction for a short time only (as for a fraction of the periodic time of 

 a planet) do not produce any sensible effect. But we can find some 

 parts of them which act in the same manner during many revolutions ; 

 and thin in many cases where no disturbance can be found, independent 

 of the eccentricities, similar to those discussed in (122), &c. ; the effects 

 of these may grow up in time to be sensible ; and those in particular 

 which alter the mean distance and the periodic time may produce in 

 time an effect on the longitude of the planet (49), very much more 

 conspicuous than that in the alteration of the orbit's dimensions." 



(162.) In this consideration is contained the whole general theory of 

 those inequalities known by the name of inequalities of limy period. 

 They are the only ones depending on the eccentricities (besides those 

 similar to the moon's evection) which ever become important. 



(163.) To enter more minutely into the explanation, let us take the 

 instance of the long inequality of Jupiter and Saturn : the most re- 

 markable for its magnitude, and for the length of time in which the 

 forces act in the same manner, as well as for the difficulty which it had 

 given to astronomers before it was explained by theory, that hag been 

 noticed since the first explanation of the Moon's irregularities. 



(164.) The periodic times of Jupiter and Saturn are very nearly in 

 the proportion of 2 to 5 (the periodic tunes being 4332 days, 17 hours, 

 and 10,759 days, 5 hours), or the number of degrees of longitude that 

 they will describe in the same time, omitting all notice of their eccen- 

 tricities, will be in the proportion of 5 to 2 nearly. Suppose, now, 

 that they were ecactly hi the proportion of 2 to 5 ; and suppose that 

 Jupiter and Saturn started from conjunction ; when Saturn has de- 

 scribed 240 degrees, Jupiter will have described 600 degrees (as these 

 numbers are in the proportion of 2 to 5) ; but as 360 degrees are the 

 circumference, Jupiter will have gone once round, and will besides 

 have described 24U degrees. It will, therefore, again be in conjunction 

 with Saturn. When Saturn has again described 240 degrees, that is, 

 when Saturn has described in all 480 degrees, or has gone once round 

 and has described 120 degrees more, Jupiter will have described 1200 

 degrees, or will have gone three times round and described 120 degrees 

 more, and, therefore, will again be in conjunction with Saturn. When 

 Saturn li.-w again described 240 degrees, that is, when it has gone 

 ly twice round, Jupiter will have gone ecactly five times round, 

 and they will again be in conjunction. So that, if the periodic time* 

 were exactly in the proportion of 2 to 5, there would be a continual 

 succession of conjunctions at the points whose longitudes exceeded the 

 longitude of the first pkce of conjunction by 210, 120, 0, 240, 

 120, 0, 4c. Thus, iajiy. 38, if B, is the place of Jupiter at first, and 



alters the distance of the places of conjunction. We must now 



through _ 



conjunction again. The next conjunction will take place when Saturn 

 has moved through double this angle, or 485 '58, or when Saturn has 

 performed a whole revolution, and 125'58 of the next revolution : and 

 the following conjunction will take place when Saturn has moved 

 I through 728'37, or when Saturn has gone twice round, and has 

 described 8'37 more. Now, then, the same order of conjunctions will 

 not go on again at the same places as before, but the next three after 

 this will be shifted 8'37 before the former places, the three following 

 the last-mentioned three will be again shifted 8'37, and so on. The 

 places of successive conjunction, in fg. 33, will be at B 1 c lt J 2 c a) 

 ^ r v lj t C 4> * c v * c "> *- The shifting o tne place" f conjunction 

 will take place in nearly the same manner, whether the orbits are 

 eccentric or not. 



(166.) From this the following points are evident : 

 First. In consequence of the periodic times being nearly in the pro- 

 portion of 2 to 5, many successive conjunctions happen near to three 

 equidistant points on the orbits. 



Secondly. In consequence of the proportion being not ecactly that 

 of 2:5, but one of rather less inequality, the points of conjunction 

 shift forward, so that each successive set of conjunctions is at points 

 of the orbits more advanced, by 8'37, than the preceding one. 



(167.) Let us now inquire how long it will be before the conjunc- 

 tions happen at the same parts of the orbits as at first. 



This will be when the series of points b t , b,, 4 10 , &c., ectends to B 3 . 

 For then the series b u b,, &, &c., will extend to B 1( and the series 6 3 , 

 6 e , b,, &c., will ectend to B,. -The time necessary for this will be 

 gathered from the consideration, that in three conjunctions the points 

 are shifted 8'37 : and that the points must shift 120" from B,, before 

 they reach B a : and that we may, therefore, use the proportion, As 8-37 

 is to 3, so is 120 to 43 nearly, the number of conjunctions that must 

 have passed before the points of conjunction are again the same. And 

 as Saturn advances 242'79 between any conjunction and the next, he 

 will, at the forty-third conjunction from the first, have described 

 10440, or 29 circumferences ; and Jupiter, therefore (by the propor- 

 tion of their periodic times), will have described 72 circumferences. 

 The time, then, in which the conjunctions return to the same points 

 is twenty-nine times Saturn's periodic tune, or seventy-two times 

 Jupiter's periodic time, or about 855 years.* 



(168.) Now let us examine into the effects of this slow motion of 

 the points of conjunction upon the forces which one body exerts to 

 disturb the other. 



(169.) If the orbits had no independent eccentricity, it would afl'ect 

 them no further than by the periodical distortion which would take 

 place at every conjunction. There would be nothing in one set of 

 conjunctions, more than in another, which could affect the dimensions 

 of the orbits. 



(170.) But if the orbits are not circular, this is no longer true. It 

 is not the same thing whether the conjunctions take place at B : c,, 

 B, o,, and B, c s , fy. 39, or at 6 t e v b, c v and 6 3 c,. The distances of 

 the planets are not the same, and consequently the forces which they 

 ecert on each other are not the same ; also thuir velocities are different 

 in different parts of their orbits, or at different points of conjunction, 

 and therefore the times during which they can act on each other are 



These numbers arc not quite exact : tlie proportion of 20 : 72 not being (ante 

 accurate. 



