Cuar. II, § 1.] 
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 are (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- 
PHYSICAL ASTRONOMY——-LAGRANGE, 
811 
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, 
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 
Few of thesecular inequalities have been detected by Cicer of 
observation throughout the entire records of Astro- apsides and 
(46.) 
Tae —T 
the elliptic elements to analytical calculation, and to 
ascribe to each planet its influence in perturbirg 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 
nomy. It is known, however, that the apsides of the &*¢e™tri- 
planetary orbits (at least in the case of the old 
planets) all progress, with the exception of those of 
Venus, which retrograde, and that the inclination of 
allis 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. Theplace of a planet is completely determined 
by three co-ordinates—its longitude, latitude, and 
radius vector; whilst theelements of the orbitare 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.— Seki , 
After it had been clearly recognised, principally by the solar 
the labours of Lagrange, that the elements of the pla- system. 
revolution (and on account of the independent excen- 
tricities of the orbits during many successive revo- 
lutions) of Mars and the Harth, the attraction of the 
. aa 2 
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 re- 
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- 
