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the disc of Jupiter when he is near opposition. The eclipses only happen when the satellites are moving towards the east, the transits only when they are moving towards the west; their motion round Jupiter must therefore be from west to east, or according to the order of the signs. The transits are real eclipses of Jupiter by his moons, which appear like black spots passing over his disc.
924. It is important to determine with precision the time of the disappearance of a satellite, which is however rendered difficult by the concurrence of circumstances: a satellite disappears before it is entirely plunged in the shadow of Jupiter; its light is obscured by the penumbra its disc, immerging into the shadow, becomes invisible to us before it is totally eclipsed, its edge being still at a little distance from the shadow of Jupiter, although we cease to see it. With regard to this circumstance, the different satellites vary, since it depends on their apparent distance from Jupiter, whose splendour weakens their light, and makes them more difficult to be seen at the instant of immersion. It also depends on the greater or less aptitude of their surfaces for reflecting light, and probably on the refraction and extinction of the solar rays in the atmosphere of Jupiter. By comparing the duration of the eclipses of all the satellites, an estimate may be formed of the influence of the causes enumerated. The variations in the distance of Jupiter and the sun from the earth, by changing the intensity of the light of the satellites, affects the apparent durations. The height of Jupiter above the horizon, the clearness of the air, and the power of the telescope employed in the observations, likewise affect their apparent duration; whence it not unfrequently happens that two observations of the same eclipse of the first satellite differ by half a minute for the second satellite the error may be more than double; for the third, the difference may exceed 3', and even 4' in the fourth satellite. When the immersion and emersion are both observed, the mean is taken, but an error of some seconds may arise, for the phase nearest the disc of Jupiter is liable to the greatest uncertainty on account of the light of the planet; so that an eclipse may be computed with more certainty than it can be observed. Although the eclipses of Jupiter's satellites may not be the most accurate method of finding the longitude, it is by much the easiest, as it is only requisite to reduce the time of the observation into mean time, and compare it with the time of the same eclipse computed for Greenwich in the
Nautical Almanac, the difference of time is the longitude of the place of observation. The frequency of the eclipses renders this method very useful. The first satellite is eclipsed every forty-two hours; eclipses of the second recur in about four days, those of the third every seven days, and those of the fourth once in seventeen days. The latter is often a long time without being eclipsed, on account of the inclination of its orbit. Of course, the satellites are invisible all the time of Jupiter's immersion in the sun's rays.
925. Let mn, fig. 113, be the orbit of the satellite projected on the plane of Jupiter's orbit, then Jn will be the curtate distance of the satellite at the instant of conjunction, and mm' the projection of the arc described by the satellite on its orbit in passing through the shadow. In order to know the whole circumstances of an eclipse, the form and length of the shadow must first be determined; then its breadth where it is traversed by the satellite, which must be resolved into the polar co-ordinates of the motion of the satellite; whence may be found the duration of the eclipse, its beginning and its end. These are functions of the actual path of the satellite through the shadow, and of its projection mm'. If Jupiter were a sphere, the shadow would be a cone, with a circular base tangent to his surface ; but as he is a spheroid, the cone has an elliptical base; its shape and size may be perfectly ascertained by computation, since both the form and magnitude of Jupiter are known.
926. The whole theory of eclipses may be analytically determined, if, instead of supposing the cone of the shadow to be traced by the revolution of the tangent AV, we imagine it to be formed by the successive intersections of an infinite number of plane surfaces, all of which touch the surfaces of the sun, and Jupiter in straight lines AaV.
927. A plane tangent to a curved surface not only touches the surface in one point, but it coincides with it through an indefinitely small space; therefore the co-ordinates of that point must not only have the same value in the finite equations of the two surfaces, but also the first differentials of these co-ordinates must be the same in each equation. Let the origin of the co-ordinates be in the centre of the sun; then if his mass be assumed to be a sphere of which R' is the radius, the equation of his surface will be
x/2 + y12 + ~/2 = R12.
The general equation of a plane is
x = ay + bz + c,
a and b being the tangents of the angles this plane makes with the co-ordinate planes. In the point of tangence, x, y, z must not only be the same with x', y', z', but dx, dy, dz must coincide with dx', dy', dz'; hence the equation of the plane and its differential become x' = ay' + bx + c
dx' ady' + bdz'.
If this value of dx' be put in
x'dx' + y'dy' + z'dz' = 0,
which is the differential equation of the surface of the sun, it becomes ax'dy' + bx'dz' + y'dy' + z'dz' = 0,
whatever the values of dy' and dz' may be. But this equation can only be zero under every circumstance when
ax' + y' = 0
bx' + z' = 0.
Thus the plane in question will touch the surface of the sun in a point A, when the following relations exist among the co-ordinates. x2 + y12 + 2′′2 = R12
ax' + y' = 0, bx+z = 0
x' = ay' + b≈' + c.
928. This plane only touches the surface of the sun, but it must also touch the surface of Jupiter, therefore the same relations must exist between the co-ordinates of the surface of Jupiter and those of the plane, as exist between the co-ordinates of the plane, and those of the surface of the sun. So the equations must be similar in both cases. Without sensible error it may be assumed that Jupiter's equator coincides with his orbit. Were he a sphere, there would be no error at all, consequently it can only be of the order of his ellipticity into the inclination of his equator on his orbit, which is 3° 5' 27".
The centre of the sun being the origin of the co-ordinates, if SJ, the radius vector of Jupiter, be represented by D, the equation of Jupiter's surface, considered as a spheroid of revolution, will be
(x,D)2 + y2 + (1 + p)2 (z,2 — R,°) = 0, R, being half his polar axis, and p his ellipticity. The equations of contact are, therefore,
929. These eight equations determine the line Aav, according to which the plane touches the sun and Jupiter; but in order to form the cone of the shadow, a succession of such plane surfaces must touch both bodies. The equations
x = ay + bz + c, and dx = ady + bdz,
both belong to the same plane, but because one plane surface only differs from another by position, which depends on the tangents a and b, and on c, the distance from the origin of the co-ordinates; these quantities being constant for any one plane, it is evident they must vary in passing to that which is adjacent, therefore
in which b and c are considered to be functions of a.
If values of b, c,
substituted in this equation, and in that of the plane, they will only contain a, the elimination of which will give the equation of the shadow; hence, if to these be added
they will determine the whole theory of eclipses. If the bodies be
spheres, it is only necessary to make p = 0.
930. In order to determine the equation of the shadow, values of
must be found. The three first of equations (325) give
x12 (1 + a2 + b2) = R12,
and the three last give
x' (1 + a2 + b2) = c;
c=R' √1+ a2 + b2,
c - D = R' √1 + a2 + b2 − D;
but from equations (326) and (327)
c− D = (1 + p) R, √√1+a2+b2,
the square of p being neglected.
In order to have the equation of the shadow, a value of a must be
found from this equation;
tion (328) of the plane.
by making p = 0 in the preceding expression;
which, with b and c, must be put in equaThis will be accomplished with most ease
is the value of a in the spherical hypothesis; but as Jupiter is a