i'n'

in is in no case exactly zero; consequently the greater axes of the celestial bodies are not subject to secular inequalities; a, their mean motions are

and on account of the equation n = a uniform.

Thus, when the squares and products of the masses m, m' are omitted, the differential dR does not contain any term proportional to the element of the time, however far the approximation may be carried with regard to the eccentricities and inclinations of the orbits, does not contain a constant term;

or, which is the same thing,

dR ndt

for if it contained a term of the form m'k, then would

a = 2fa2. dR = 2a3m'knt, and § = − 3 ffandt. dR

= 1

would become so that the greater axes would increase with the time, and the mean motion would increase with the square of the time, which would ultimately change the form of the orbits of the planets, and the periods of their revolutions. The stability of the system is so important, that it is necessary to inquire whether the greater axes and mean motions be subject to secular inequalities, when the approximation is carried to the squares and products of the masses.

=3ffan m'kdt 3an m'kt®,

463. The terms depending on the squares and products of the masses are introduced into the series R by the variation of the clements of the orbits, both of the disturbed and disturbing bodies. Hence, if da, de, &c. be the integrals of the differential equations of the elements in article 439, the variable elements will be a + da, ede, &c. for m, and a' + da', e' + de', &c. for m'; and when these are substituted for a, e, a', e', &c. in the series R, it takes the form R = R + SR + S'R;

and from what has been said, the greater axis and mean motion of m will not be affected by secular inequalities, unless the differential

dR, = dR + d.♪R + d.d'R

contains a term that is not periodic.

dR is of the first order relatively to the masses, and has been proved in the preceding article not to contain a term that is not periodic. d.&R and d.'R include the squares and products of the masses; the first is the differential of R with regard to the clements of the

troubled planet m, and d. d'R is a similar function with regard to the disturbing body m'. It is proposed to examine whether either of these contain a term that is not periodic, beginning with d.R.

464. The variation &R regards the elements of m alone, and is

If the values in article 439, be put for da, de, &c. this expression

And its differential, according to the elements of the orbit of m alone, is obtained by suppressing the signs introduced by the integration of the differential equations of the elements in article 439, which reduces this expression to zero; therefore to obtain d. JR, it is sufficient to take the differential according to nt of those terms in R that are independent of the signs.

When the series in article 449 is substituted for R, SR will take the form P.f. Qdt Q. f. Pdt.

Where P and Q represent a series of terms of the form

i' and i being any whole numbers positive or negative.

belong to P, and let k' cos (i'n't- int + c') be the corresponding

term of Q, k, k', c, c', being constant quantities.

A term that is not periodic could only arise in

d&R= d{PfQdt

if it contained such an expression as

kk' cos {i'n't-int+c} cos {i'n't int + c'} = kk' cos (c— c')

+ kk' cos {2i'n't- 2int + c + c'};

or a similar product of the sines of the same angles. But when

k cos (i'n't int+c) is put for P, and k' cos (i'n't — int + c') for Q, d&R becomes

d.&R= kindt. sin (i'n't— int + c). fk'dt. cos (i'n't — int + c') -k'indt. sin (i'n't - int+c). fkdt. cos (i'n't- int + c), which is equal to zero when the integrations are accomplished. Whence it may be concluded that d.§R is altogether periodic.

465. It now remains to determine whether the variation of the elements of the orbit of m' produces terms that are not periodic in d. d'R. This cannot be demonstrated by the same process, because the function R, not being symmetrical relatively to the co-ordinates of m and m', changes its value in considering the disturbance of m' by m. Let R' be what R becomes with regard to the planet m' troubled by m;

m'

and

S'R=

m

¿'R' + m'd' {(xx' ++ yy' + z=') ( +/+ − −)}·

If the differential of this equation according to d be periodic, so will d. R. Now in consequence of the variations of the elements of

the orbit of m,

And as this expression with regard to the planet m' is in all respects similar to that of SR in the preceding article with regard to m, by the same analysis it may be proved, that d. JR' is altogether periodic. Thus the only terms that are not periodic, must arise from the differential of

m'd'{xx' + yy' + zz'

The co-ordinates y z, y' z', furnish similar equations. Thus,

m' (d (rde madr+ydy – y'dy + zdz' - z'dz)

L=

S

dt2

- z'dz)}+ N, where

m" -
'xx' + yy' + zz' mm' (xx' + yy' + zz`

N =

+

S

(MP) a

If N be omitted at first,

m'

d. L=- d { d (x'dx

S

466. The elliptical values of the co-ordinates being substituted, every term must be periodic. For example, if

dt

0, which never can

a quantity that must be periodic unless n't happen, because the mean motions of no two bodies in the solar system are exactly commensurable; but even if a term that is not periodic were to occur, it would vanish in taking the second differential; and as the same thing may be shown with regard to the other products y'dy ydy' z'dz — zdz',

dL is a periodic function. With regard to the term dL = dN, if the elliptical values of the co-ordinates of m and m' be substituted, it will readily appear that this expression is periodic, for the equations of the elliptical motion of m and m', in article 365, give

467. From what has been said, it will readily appear that the terms of this expression, consisting of the products a'd2x, xď2x', &c. &c., are periodic when the elliptical values are substituted for the coordinates, and their differentials.

468. The last term of the value of N is also periodic; for, if the elliptical values of the co-ordinates of m and m' be put in R, it may be developed into a series of cosines of the multiples of the arcs nt and n't, and the differential may be found by making R vary with regard to the quantities belonging to m alone; hence this differential may contain the sines and cosines of the multiples of nt, but no sine or cosine of n't alone; and as

quently periodic; and as the same may be demonstrated for each of

not only N but its differential are periodic, and consequently d. ♪'R. Thus it has been proved that when the approximation is carried to the squares and products of the masses, the expression

dR, = dR + d. SR + d. d'R

relatively to the variations of the mean motions of the two planets m and m' is periodic.

469. These results would be the same whatever might be the number of disturbing bodies; for m" being a second planet disturbing the motion of m, it would add to R the term

The variation of the co-ordinates of m' and m" resulting from the reciprocal action of these two planets, would produce terms multiplied by mm" and m" in the variation of R; and by the preceding analysis it follows that all the terms in d. "R are periodic. lates to the variation of the elements of the orbit of m".

"R re

The variations of the co-ordinates of m' arising from the action of m" on m', will cause a variation in the part of R depending on the action of m' on m represented by