must be subtracted from equation (266).

841. The same analysis applied to d(e, cos ) will determine the increment of the first of equations (267), with regard to the second satellite. But, in this case,

+ m‚A ̧(1.2) cos (v−v1)+m,A,(3·o) cos 2(v,—v2),

and equations (269) and (270) must be employed. The result is,

must be added to the second of equation (267).

As these quantities only arise from the ratios among

the mean mo

tions of the three first satellites, the secular variations of the fourth are not affected by them. In consequence of these additions, equations (266) and (267) become

0=h{g-(0)--(0.1)-(0.2)-(0.3)}+[0.1]h2+[0.2h2+0.3|hg

0=h1{g-(1)-[1]-(1.0)-(1.2)−(1.3)}+[1.0]h+|1.2|h2+1.3|h ̧

0=h,{g-(2)-[2-(2.0)-(2.1)-(2.3)}+2.0]h+2.1h,+2.3]h2

0=h2{g-(3)-[3]-(3.0)–(3.1)-(3.2)}+|3.0]h+|3.1 h ̧+|3.2]h2

842. An inequality which is only sensible in the theory of the second satellite may now be determined; for, by (260),

+sin(nt-2nt +ε-2€). cos (nt+e)};

but as v 2e sin (nt + e - ), and for the variable ellipse which

we are now considering,

Sv=28. (e cos ☎) . sin (nt+e)—2♪. (e sin w).cos (nt+€).

By comparing these two values,

If e sin + d(e sin w), and e cos ☎ + d(e cos ₪) be put for e sin, and e cos w, it becomes

=

dvd. e cos w) (♪. e sin w)} sin 2(nt + €)

and in consequence of the preceding values of (e cos ☎), d(e sin ☎), there is the following inequality in the longitude of the first satellite,

By the same process the corresponding inequalities in the second and third satellites are found to be

Librations of the three first Satellites.

843. Some very interesting inequalities arising from the equation 3n t2nt +€ — 36, +26,180°,

nt

are found among the terms depending on the squares of the disturbing forces, that affect the whole theory of the satellites, in consequence of the very small divisor (n−3n, + 2n,) which they acquire by double integration. If the orbits be considered as variable ellipses, and if , S1, S2, be the mean longitudes of the three first satellites, it is clear that the terms having the square of n −3n, + 2n2 for divisor, can only be found from

d3andt. dR

d3a,n,dt. dR,

d2¿2 = 3a,n,dt. dR

which are the variations in the mean motions by article 439.

844. With regard to the action of m, on m, the series R in article 815 only contains the angle n,tntee and its multiples, it is evident therefore, that the angle nt − 3n ̧t + 2n ̧t can only arise from the substitution of the perturbations (262) which depend on the angle 2n,t2nt. By article 814, dv, contains both the elliptical part of the longitude and the perturbations, and if the latter be expressed by δε, then

If then du, and

a

is

R = m

or

Sribe put for 2d(r8r,) and or the part of R required

a.ndt

a,

(14,). dr, . cos (n,t — nt + e,− e)

- m,. do,. A,. sin (n,t-nte,- €),

+

dRm, Adv. cos (n,t-nt +,-e), ndt

(dd,). dr,. sin (n,t—nt +e,—c), rdt.

for in this case dir, and de, are zero, since equations (262), or

do not contain the arc nt. If these quantities be substituted in dR, it will be found, in consequence of

dA

G = 2a,A,— a2

and n = 2n,,

da,

that

3nm,m,FG a sin (nt - 3n,t+2n,t+ e − 3€, + 2€); 8(n-n-N) a,

The variation in the mean motion of the second satellite consists of two parts; one arising from the action of m, and the other from that of me.

The value of R for the first is

R = m. A. do, sin (nt nt + e − e)

If the differential of R be taken with regard to n,t, making dʊ, and Sr, vary, by the substitution of the preceding values of du,, dr,, and their differentials, it will be found, in consequence of

and non, that the variation in the mean motion of the second satellite from the action of the first must be

from article 826, be substituted in the differential of

Rm2

dA (3.2)

da

Sr, cos (2n,t 2n ̧t+2€,- €)

- 24(3.2). Su, sin (2n,t2nt + 2€,- 2€)},

which is the value of R with regard to m, and m1, observing that n = 2n,; and, by article 826,

arising from the action of m, on m1, will be found

32(n-n, - N)

FG sin (nt-3nt+2n ̧t+e−3¢ ̧+2€1⁄2) ;

and the whole variation in the mean motion of m,, from the combined action of m and m2, is

With regard to the action of m, on m1⁄2

R = m1 {−2A,(3.2), dv, sin 2(n ̧t-n ̧t + € ̧ − €)

If the same values of do, and dr, be substituted in the differential of this with regard to not, it will be found that the action of m, and mg produces the inequality

by comparing the values of these three quantities in the last article

the result is

mdR+m,dR, = 0,

and m,dR, + m ̧dR2 = 0,

which is conformable with what was shown in article 573, with re

gard to the planets.