These inequalities in the circular orbits are independent of their positions.

Determination of the Masses of the Satellites and the Compression of Jupiter.

889. Approximate values of the masses of the satellites, and of the compression of Jupiter, are sufficiently accurate for calculating the periodic inequalities in the circular orbit; but it is necessary to have more correct values of these quantities for computing the secular variations. The periodic and secular inequalities determined by theory, when compared with their observed values, furnish the means of finding the true values of these very minute quantities. The principal periodic inequality in the longitude of the first satellite is, by observation, 1636".4 at its maximum; but by article 888 this inequality is, by theory, 7042".6m,, whence

m10.232355.

The greatest periodic inequality in the longitude of the second satellite is, by observation, 3862".3 at its maximum; the same, by (298), is

m . 2252′′.28 + m2 . 3923′.3,

which arises from the combined action of the first and third satellites, hence

m = 1.714843

mg. 1.741934.

(300)

The other unknown quantities must be computed from equations (271) and (290). For that purpose let

μ being an indeterminate quantity depending on the compression of Jupiter's spheroid. Then from the expressions

and the formulæ in article 474, it will be found that

The numerical values of F, G; F, G', are determined from articles 825, 826, and 827, to be

and with the same quantities the coefficients Q, Q1, Q2, of the equations in article 839 are found to be

Not only these quantities, but several data from observation are requisite for the determination of the unknown quantities from equations (271) and (290).

890. The eclipses of the third satellite show it to have two distinct equations of the centre; the one depending on the apsides of the fourth satellite is 2h, = 245.14. The other datum is the equation of the centre of the fourth satellite, which is, by observation, equal to 3002".04 2h. Again, observation gives the annual and sidereal motion of the apsides of the fourth satellite equal to 2578".75, which, by article 831, is one of the four roots of g in equation (271), so that g=2578".75. And, lastly, observation gives 43374" for the annual and sidereal motion of the nodes of the orbit of the second satellite on the fixed plane, which is one of the roots of p in equation (290), so that

P, 43374".

891. If the values of m, and m, as well as all the quantities that precede, be substituted in equations (271) and (290), they become, when the first are divided by h,, and the last by l1,

{80109 +179457 μ. + 51581′′.5 m + 1686′′.44 m,

+248".55 m,}

+(4977".22 + 18729". m - - 16020".3 m2}

+ 544".86 m + 69′′.16 m3.

0 (18305".3 m+72999".2 m2-63180".4 mm,}

+

-26505".7 m2 + 45344".8 mm.

2511.6-35317". μ-14128 m-13455". m,-584".554 m.] h1

+ 594".41 m". + 256′′.12 m。 — 677′.04 mm, + 592′′.6 m ̧3.

h

h1

h

0=4831".9 m · 1/2 + (1352"8 - 1569". m+1342′′, m2}. (306)

.1

h3

+89'.7562′′.6 μ - 86′′.44 m —

h3

40". m2 + 1138′′.7 m2.

0=2998".23+ (40342".3-179457". μ-1686".44m,-248".57 m2).

+ {42976′′.3 —954′′.82 μ — 117′′.64 m — 1438′′.2 m.}

892. These are the particular values of equations (271) and (290)