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846. As the three first satellites move in orbits that are nearly circular, the error would be very small, in assuming
nt + e, n,t + €,, nt2 + €2,
to be their true longitudes.
The preceding inequalities in the mean motions of the three first satellites are therefore
the result will be
3n mm, F'G
847. In order to abridge, let v — 3v, + 2v,; whence
If the preceding values be put in this, and if to abridge, 3nF'G
In3. mm. F'G 32(n- n,—N,)
sin (v−3v, + 2v2)
2 sin (v−3v,+2v1).
m ̧m2 + 2 mm, +
= K.n2. sin Q.
K and n2 may be assumed to be constant quantities, their variations are so small; hence the integral of this equation is
c is a constant quantity introduced by integration, the different values of which give rise to the three following cases.
848. 1st. If c be greater than 2Kn, without regard to the sign, it must be positive; and the angle ± will increase indefinitely, and will become equal to one, two, three, &c., circumferences.
2d. If K be positive, and c less than 2n K, abstracting from the sign, the radical will be imaginary when is equal to zero, or to one, two, three, &c. circumferences. The angle must therefore oscillate about the semicircumference, since it never can be zero, or equal to a whole circumference, which would make the time an imaginary quantity. Its mean value must consequently be 180°.
3d. If c be less than 2Kn2, and K negative, the radical would be
imaginary when the angle is equal to any odd number of semicircumferences; the angle must therefore oscillate about zero, its mean value, since the time cannot be imaginary. However, as it will be shown that K is a positive quantity, the latter case does not exist, so that must either increase indefinitely, or oscillate about 180°. In order to ascertain which of these is the law of nature, let φ = π Ι π,
being 180° and any angle whatever; hence
√c+2Kn2 cos @
If the angles and increase indefinitely, c is positive, and greater than 2Kn2; hence, in the interval between increase to 90°, dt is less than
= 0, and its
√2K Thus the time t that the angle equal to 90°, will be less than
This time is less than two years: but from the discovery of the satellites the libration or angle has always been zero, or extremely small; therefore this angle does not increase indefinitely, it can only oscillate about its mean value of zero.
The second case, then, is what really exists, and the angle
3,v1 + 2v2, must oscillate about 180°, which is its mean value.
849. Several important results are given by the equation v- 3v1 + 2v2 = π + w.
If the insensible part be omitted, 3nt2nt + e
employs in increasing till it be
3n1 + 2n2 = 0
3€ + 2€2 = 180°.
These two equations are perfectly confirmed by observation, for Delambre found, from the comparison of a great number of eclipses of the three first satellites, that their mean motions in a hundred Julian years, with regard to the equinox, are
1st satellite 7432435°.46982
whence it appears, that the mean motion of the first, minus three times that of the second, plus twice that of the third, is equal to 9".0072, so small a quantity, that it affords an astonishing proof of the accuracy both of the theory and observation. Delambre determined also, from a great number of eclipses, that the epochs of the mean motions of the three first satellites, at midnight, on the first of January 1750, were
€ - 36, +26,180° 1' 3",
a result that is less accurate than the preceding; but it will be shown, in treating of the eclipses of the satellites, that it probably arises from errors of observation, depending on the discs of the satellites, which vanish to us before they are quite immersed in the shadow.
850. The same laws exist in the synodic motions of the satellites ; for in the equation
nt 3nt + 2nt + €
the angles may be estimated from a moveable axis, since the position of the axis would vanish in this equation: we may therefore suppose that
nte, n1t + 1, not + €g,
are the mean synodic longitudes. This has a great influence on the eclipses of the three first satellites, as will appear afterwards.
851. On account of these laws the actions of the first and third satellites on the second are united in one term, given in article 826, which is the great inequality in that body indicated by observations. These inequalities will never be separated.
852. Without the mutual attraction of the satellites the two equa
would be unconnected. It would have been necessary in the beginning of their motions that their epochs and mean motions had been so arranged as to suit these equations, which is most improbable;
and in this case the slightest action from any foreign cause, as the attraction of the planets and comets, would have changed the ratios. But the mutual action of the satellites gives perfect stability to these relations, for, at the origin of the motion, when t = 0,
dv, 3 n.dt
−2K cos (e−2¤ ̧ + 3€)
c being less than 2Kn2. It would be sufficient for the accuracy of the preceding results that the first member of this equation had been comprised between the limits.
+2K sin ( §€¡ + €2)
2K sin (e¤, + €2)
at the origin of their motions, and it is sufficient for their stability that no foreign force disturbs it.
853. It appears then, that if the preceding laws among the mean motions of the three first satellites had only been approximate at their origin, their mutual attraction would ultimately have rendered them exact.
854. The angle
is so small, that we may make
cos = 1 - @2;
and if to abridge
being arbitrary, on account of the arbitrary constant quantity c that it contains, equation (273) becomes
6 sin (nt √K + 4),
A being a new arbitrary quantity.
855. As the motions of the four satellites in longitude, latitude, and distance, are determined by twelve differential equations of the second order, their integrals must contain twenty-four arbitrary quantities, which are the data of the problem, and are given by observation. Two of these are determined by the equations
they are, however, replaced by 6 and A, the first determines the extent of the libration, and A marks the time when it is zero: neither are determined, since the inequality @ has as yet been insensible.
856. The integrals of the three equations (272) may now be found, for as
v − 3v1 + 2v2 = # + ∞ = x + 6 sin (nt √K+ 4),
=-6. sin (nt √K + A) ;
or, if to abridge,
sin (nt K+ A)
6 sin (nt √K + A),
6 sin (nt √K+4).
which are the three equations of the libration. They have hitherto been insensible, but they modify all the inequalities of long periods in the theory of the three first satellites.
857. For example, the inequality
am 6 sin (nt √K+4)
H sin (Mt+E — II),
But the differential of the first of the equations of libration is
= + H sin (Mt+E — I);