CHAP. II., 1.] 



PHYSICAL ASTRONOMY LAGKANGE. 



13 



(46.) 



Plauetary 

 theory. 



(47.) 

 Periodic 

 and secular 

 inequali- 

 ties. 





a single example; the planet Uranus has not yet 

 completed one revolution since the time of its disco- 

 very in 1781, yet its observed path differed so much 

 from a true elliptic arc (even when we allow for the 

 perturbation of Jupiter and Saturn), that the orbit 

 which satisfied the observations from 1781 to 1800 

 would not satisfy those from 1800 to 1820; and 

 since 1820 a new orbit had to be computed for every 

 few years, so great were the variations of the instant- 

 aneous from any permanent ellipse. These varia- 

 tions led to the discovery of the planet Neptune. 



To adapt the notion of the perpetual variation of 

 the elliptic elements to analytical calculation, and to 

 ascribe to each planet its influence in perturbing the 

 elliptic motion of the others, was the great problem 

 mainly solved by Lagrange. In the planetary theory, 

 where the perturbations are all very small, on account 

 of the excessive preponderance of the mass of the 

 sun, the motion of each planet may be considered as 

 under the separate disturbing influence of every 

 other, and the whole perturbation is the sum of the 

 separate perturbations. 



Now, these perturbations of elliptic motion may 

 be divided into two great classes, which Lagrange 

 first, in 1782, included in a common analysis, which 

 expressed the disturbed elements of planetary motion 

 by two sets of terms : those which include the rela- 

 tive positions (or configuration) of the disturbing and 

 disturbed planets being the one set, and those which 

 included only the masses and elements being the 

 other. The former are called periodic, the latter 

 secular inequalities. The distinction is important, 

 since, after a sufficiently long time, two planets (sup- 

 pose the Earth and Mars) will have been presented 

 to one another in space in every conceivable posi- 

 tion of which, by the form and position of their or- 

 bits, they are susceptible, a like recurrence of confi- 

 gurations will recommence, and like perturbations 

 will result. Such influences, though running through 

 long periods, will be evidently recurring. But there 

 is another class of disturbances, which may in thought 

 be entirely separated from the former, being the ulti- 

 mate or average effect of the influence of one planet 

 on another, arising, not from the position of the pla- 

 nets in their orbits at any one time, but from the po- 

 sition of the orbits themselves. Thus in a single 

 revolution (and on account of the independent excen- 

 tricities of the orbits during many successive revo- 

 lutions) of Mars and the Earth, the attraction of the 

 former on the latter sometimes conspires with the 

 sun's attraction, sometimes opposes it, sometimes 

 urges the Earth forward in its path, and sometimes 

 pulls it back, producing numerous periodic inequali- 

 ties ; but it is quite evident that, in the long run, 

 the attraction of Mars on the Earth tends to pull it 

 away from the sun, and to diminish the effect of the 

 solar attraction in fact, to increase the length of 

 our year ; and that this influence will be precisely 

 the same if we take the average of a great many rer 



(48.) 



volutions now, and compare them with a similar ave- 

 rage hereafter, provided that the orbits undergo no 

 permanent change. This, therefore, though not strictly 

 an inequality, because the length of the year is per- 

 manently changed by it, shows an average effect in- 

 dependent of the configuration of the planets. An 

 example of a true secular inequality is the revolution 

 of the line of apsides or major axis of any orbit, by 

 the influence of the disturbing forces of the planets, 

 whether interior or exterior to the one considered. 



Few of the secular inequalities have been detected by change'of 

 observation throughout the entire records of Astro- apsides and 

 nomy. It is known, however, that the apsides of the e fcentri- 

 planetary orbits (at least in the case of the old 011 

 planets) all progress, with the exception of those of 

 Venus, which retrograde, and that the inclination of 

 all is at present diminishing. The excentricity of the 

 Earth's orbit is decreasing at the rate of 40 miles per 

 annum. The exclusive dependence of the secular ine- 

 qualities on the orbits, not on the places of the planets, 

 may be well illustrated by a method actually employed 

 by Gauss for computing them (though it does not 

 appear to be attended by any special advantage). 

 He conceives the orbit of the disturbing planet to be 

 strewed with attractive matter, whose thickness at 

 any point is inversely as the planet's velocity there, 

 or directly as the time of its sojourn in any small 

 length of the orbit. . 



The method of variations of the elements is evidently 

 most applicable to the determination of Secular Per- 

 turbations ; for to compute by means of it the ordi- 

 nary inequalities involves an apparently unnecessary 

 labour. The place of a planet is completely determined 

 by three co-ordinates its longitude, latitude, and 

 radius vector; whilst the elements of the orbit are six in 

 number, and when found, a further calculation must 

 be made to find the co-ordinates of position. The 

 more direct method of deducing the co-ordinates at 

 once from the conditions of perturbation, was gene- 

 rally followed until 1808, when Lagrange and 

 Laplace almost simultaneously devised methods of 

 using the variation of the elements with directness 

 and despatch in the calculation of Planetary Pertur- 

 bations. In the estimation of the second and higher 

 orders of disturbance, it has even the advantage in 

 these respects over the other method. 



Stability and Permanence of the Solar System. stability of 

 After it had been clearly recognised, principally by the solar 

 the labours of Lagrange, that the elements of the pla- system, 

 netary orbits are in a condition of perpetual change, 

 it came to be a most interesting question how far 

 such variations were likely to be continuous, and ulti- 

 mately so great as to modify altogether the forms of 

 the orbits, and even endanger the separate existence 

 of the planets. This is a question which has excited 

 a very general, as well as scientific interest. It is 

 evident that the variations of the different elements 

 are not all equally important in affecting the perma- 

 nence of an orbit. The five properly orbital ele- 



