masses of all the bodies in the system, and of the elements of their orbits. Approximate values of these are only known with regard to the planets, but of the masses of the comets we are in total ignorance; however, as the mutual gravitation of the planets is sufficient to represent all their inequalities, it shows that, hitherto at least, the action of the comets on the planetary system is insensible. Besides, the comet of 1770 approached so near to the earth that its periodic time was increased by 2.046 days; and if its mass had been equal to that of the earth, it would have increased the length of the sidereal year by nearly one hour fifty-six minutes, according to the computation of La Place; but he adds, that if an increase of only two seconds had taken place in the length of the year, it would have been de tected by Delambre, when he computed his astronomical tables from the observations of Dr. Maskelyne; whence the mass of the comet must have been less than the 500 part of the mass of the earth. The same comet passed through the Satellites of Jupiter in the years 1767 and 1779, without producing the smallest effect. Thus, though comets are greatly disturbed by the action of the planets, they do not appear to produce any sensible effects by their reaction.

527. If the position of the ecliptic in the beginning of 1750 be assumed as the fixed plane of the co-ordinates x and y, and if the line of the equinoxes be taken as the origin of the longitudes, it is found that at the epoch 1750 the longitude of the ascending node of the invariable plane was 102° 57′ 30′′, and its inclination on the ecliptic I 1°35′ 31′′; and if the values of the elements for 1950 be substituted in the preceding formulæ, it will appear that in 1950 N = 102° 57' 15′′; I = 1° 35′ 31′′;

=

which differ but little from the first.

528. The position of this plane is really approximate, since it has been determined in the hypothesis of the solar system being an assemblage of dense points mutually acting on one another, whereas the celestial bodies are neither homogeneous nor spherical; but as the quantities omitted have hitherto been insensible, the position of the plane as it is here given, will enable future astronomers to ascertain the real changes that may have taken place in the forms and positions of the planetary orbits.

CHAPTER VII.

PERIODIC VARIATIONS IN THE ELEMENTS OF THE
PLANETARY ORBITS.

Variations depending on the first Powers of the Eccentricities and Inclinations.

529. THE differential dR relates to the arc nt alone, consequently the differential equation da = 2a2. dR in article 439 becomes da + m'a2. in. ZA, sin i (n't nt + e − e)

+ m'a3en (i-1). M, sin {i(n't-nt + e' − e) + nt +e-w}

+ m'ae'n (i-1). M sin {i(n't - nt + e − e) + nt + e-w'}. The integral of this equation is the periodic variation in the mean distance, and if represented by da, then

(i-1)n
m'a2e'
i(n'-n)+n

M。 cos {i(n't―nt+e'—e)+nt+e-w}

M, cos {i(n't-nt+e'—e)+nt+e—w'}.

In a similar manner it may be found that the periodic variation in the mean motion dy =

+ m'ae'

n2

3fandt dR is,

{i(n' — n) +n}2

(i-1)n2
{i(n'−n)+n)}2

M, sin{i(n't-nt+e'-e)+nt+e-w}

M ̧ sin{¿(n't—nt+e'-e)+nt+e-w'}.

From the other differential equations in article 439 it may also be found that the periodic variation in the ec

+m'ae'

+m'ae'

i(n'-n)

·M, sin { i (n't—nt+e'—c)+nt+e—w}

No sin {i (n't-nt +e' − e)+2nt+2€−2☎ }

N, sin i (n't nt + e' — €)

- N1 sin {i(n't-nt+e'— €)+2nt + 2e −w—w'} i (n'-n)+2n

n

ï(n'—n) N ̧ sin { i (n't — nt + e' − e) + w — w' }

in ar

When e2, ey, e'y, are omitted, the differentials of P and q

When the orbit of m at the epoch is assumed to be the fixed plane, z = 0, and z' = a'y sin (n't + ť

I).

the products of the inclination by the eccentricities being omitted.

dR

Now although z be zero, its differential is not, therefore

must

dz

m'
2

m'z'
al3

・m'

a/2

m'z'
2

Σ B, cos i (n't—nt + e'e),

y sin { (n't + ' II)

+ a' B(-) y sin {i (n't — nt + e' − e)+nt+e—II}

where i may be any whole number, positive or negative, except When this quantity is substituted in dp, dq, their integrals are

m' a2n

sin (n't nt + e

n

n' + n

sin {i (n't-nt + e − e) + 2nt + 2e-II}}

a2a'nΣ B (i-1) Y

1

i (n' — n)+2n

[1

n'+n

1

n'-n

cos (n't + nt + e' + e − II)

cos {n't

nt + e' − e) – II} }

cos (i (n't - nt + 6' − e) + II)

cos { i (n't — nt + e' − e) + 2nt + 2e − II } }.

530. The equations which determine the variations in the greater axes and mean motion show that these two elements are subject to very considerable periodic variations, depending on the con

figurations of the bodies, when the divisor i (n' — n) + n or i'n'in is very small.

There is no instance of the mean motions of any two of the celestial bodies being so exactly commensurable as to have i'n'—in=0, therefore the greater axes and mean motions have no secular inequalities, but in several instances this divisor is a very small fraction, and as a quantity is increased in value when divided by a fraction, the divisor i'n' in, and still more its square, increases the values of these periodic variations very much. For this reason the periodic variation in the mean motion is much greater than that in the greater axis, evidently arising from the double integration in the former.

531. It is unnecessary to add constant quantities to the preceding integrals, for they may be included in the elements of elliptical motion, which then become

a+a, e + e, ☎ + w1, € + 1, P + P1 & + qi

and in the troubled orbit they are

a + a + da; e + e, + de; w + w, + dw;

e + e + de; p + P1 + dp ; q + q, + Sq.

Since a,, e,, &c., da,, de, &c., are very small quantities of the order m', a + a,, e + e,, &c., may be substituted in the latter quantities instead of a, e, &c., they will then be functions of the time and of the six constant quantities a + a,, e + e,, &c.: so that the formula of troubled motion in reality contain but six arbitrary constant quantities, as they ought to do. In order to determine a,, e,, &c., suppose the perturbations of the planet m were required during a given interval of time. The quantities a, e, &c., are given by observation at the epoch when t0 in the elliptical orbit, that is, assuming the disturbing force to be zero; but as a, + da, e, + de, &c., arise entirely from the disturbing force, they must also be zero at the epoch; therefore, values of the arbitrary constant quantities a, e,, &c., are obtained from the equations.

&c.,

a, + da = 0, e, † de = 0, ∞, + dã = 0, da, de, &c., being the values of da, de, &c., at the epoch.

The effect of the disturbing forces upon each of the elliptical elements will be completely expressed by a, + da, e, + dé, &c. during the time under consideration. Thus both the periodic and secular variations of the elements of the orbits are determined.