RΣm,a'a, {ss, - s2 cos (v, - v) } B, cos (v,

l, sin (v, + pt + ^)

8 l. sin (v + pt + A); S= L'. sin (U+ pt + ^); Lsin (v + pt + A), ̧l, l1, l, lз, L' and L being the inclination of the orbits of the four satellites, of Jupiter's orbit and equator on the fixed plane, p and A, quantities on which the sidereal motions and longitudes of the nodes depend.

If the motion of only one satellite be considered at a time, then substituting for s, s, and S, also putting for t, and omitting p2,

v

n

the comparison of the coefficients of sin (v + pt + A) gives

0 = 1

+

(p = 10)

a2

a2

a,

2

{1 − 2a cos ß + a2 } = a‚3 ({B,+ B, cos ß + B2 cos 2ß + &c.,) which is identical with the series in article 446, and therefore the

formulæ for the planets give by article

0=1{p (0) 0 − (0.1)} + L (0) + L' [Q] + (0.1) 1,

but the action of the satellites m, and m, produce terms analogous to those produced by m,; so the preceding equation, including the disturbing action of all the bodies, and the compression of Jupiter, is

0 - (0.1) (0.2) - (0.3)} (276)

0 = 1{p (0) - O

+(0) L+ 0| L + (0.1) 4 + (0.2)+(0.3).

By the same process the corresponding equations for the other satellites are

0 = 4, {p (1) — 1] - (1.0) — (1.2) — (1.3)}

+ (1) L + 1] L' + (1.0) 7 + (1.2) ↳1⁄2 + (1.3) ↳1⁄2 ;

+ (2) L + [2] L' + (2.0) 1 + (2.1) ↳, + (2.3) 1⁄2 ;

0 = {p (3) 3 (3.0)

(3.1) — (3.2)}

+ (3) L + 3 L' + (3.0) 1 + (3.1) ↳, + (3.2) ↳.

862. These four equations determine the coefficients of the latitude; they include the reciprocal action of the satellites, together with that of the sun, and the direct action of Jupiter considered as a spheroid, but in the hypothesis that the plane of his equator retains a permanent inclination on the fixed plane: that, however, is not the case, for as neither the sun nor the orbits of all the satellites are in the plane of Jupiter's equator, their action on the protuberant matter causes a nutation in the equator, and a precession of its equinoxes, in all respects similar to those occasioned by the action of the moon on the earth, which produce sensible inequalities in the

motions of the satellites. Thus the satellites, by troubling Jupiter, indirectly disturb their own motions.

The Effect of the Nutation and Precession of Jupiter on the Motion of his Satellites.

863. The reciprocal action of the bodies of the solar system renders it impossible to determine the motion of any one part independently of the rest; this creates a difficulty of arrangement, and makes it indispensable to anticipate results which can only be obtained by a complete investigation of the theory on which they depend. The nutation and precession of Jupiter's spheroid can only be known by the theory of the rotation of the planets, from whence it is found that if 0 and y be the inclinations of Jupiter's equator and orbit on the fixed plane, & the retrograde motion of the descending node of his equator on that plane, and estimated from the vernal equinox of Jupiter, 7 the longitude of the ascending node of his orbit, it the rotation of Jupiter, and A, B, C, the moments of inertia of his spheroid with regard to the principal axes of rotation, as in article 177, the precession of Jupiter's equinoxes is

whence May sin (7+) is the action of the sun, and Emn2y' sin (7'+4)

The first of these equations, multiplied by sin, added to the second multiplied by cosy, gives

864. Now, in order to have some idea of the positions of the different planes, let JN be the orbit of Jupiter, QN the plane of his

would be the latitude of a satellite if it moved on the plane of Jupiter's equator, for the latitudes are all referred to the fixed plane FN; and if they are positive on the side FJ, they must be negative on the side FQ; but by the value assumed for s, in article 861, that latitude is equal to a series of terms of the form

865. Likewise, y = JNF, being the inclination of Jupiter's orbit on the fixed plane, y sin (U — 1) is the latitude of the sun above the fixed plane, by article 863; but by the value assumed for S, in article 861, it is easy to see that

In the same manner y mNF being the inclination of the orbit of a satellite on the fixed plane, its latitude is y' sin (v + 1), and by article 861

'sin 'El. sin (pt + A)

y' cos 7' 'l. cos (pt + A).

(282)

Σ' denotes the sum of a series, but Σ is the sum of the terms relating to the different satellites.

When these quantities are put in equations (279) and (278), a comparison of the coefficients of similar sines and cosines gives

0 = pL + 3

(2C-A-B)
4iC

{M2 (L' - L) + Σ mn2 (l - L)},

which equation determines the effect of the displacement of Jupiter's equator.

866. If Jupiter be an elliptical spheroid, theory gives

2C-A-B 2 (p-1)/P.R1 dR

=

As the celestial bodies decrease in density from the centre to the surface, P represents the density of a shell or layer of Jupiter's spheroid at the distance of R from his centre, the integral being between R = 0, the value of the radius at the centre to R = 1, its value at the surface.

$67. The equations in article 277 may be put under the form 0 = {p-(0)-- (0.1) - (0.2) - (0.3)} (L−1)

+(0.1)(L−1,)+(0.2) (L−1)+(0.3) (L−↳)+ [0] (L-L')-pL;

0 = {p-(1) --(1.0)-(1.2) −(1.3)} (L−1)

+(1.0)(L−1)+(1.2)(L−1)+(1.3) (L−1)+ |1|(L-L')-pL;

0 = {p−(2) − |2-(2.0)-(2.1) −(2.3)} (L−1)

(283)

+(2.0)(L−1)+(2.1)(L−l,)+(2.3) (L−1)+ |2| (L-L')−pL; 0={p-(3)-3-(3.0)-(3.1)-(3.2)} (L-4)

+(3.0)(L−1)+(3.1)(L−l,)+(3.2) (L−1,)+ [3] (L—L')—pL;

0 = pl

3(2C-A-B)

4iC

{M2 (L-L')+mn2 (L−1)+

m1n,2 (L-1) + m ̧n2 (L−l)+Mgn32 (L−l) } ;

which determine the positions of the orbits of the satellites, including the effects of Jupiter's nutation and precession.

Inequalities occasioned by the Displacement of Jupiter's Orbit.

868. The position of Jupiter's orbit is perpetually changing by very slow degrees with regard to the ecliptic, from the action of the planets. In consequence of this displacement the inclination of the plane of Jupiter's equator on his orbit is changed, and a correspond