5. The apparent velocities of the planets, whether direct or retrograde, are accelerated from one of the stationary points, to the midway between that and the following stationary point; from thence they are retarded till the next station.

6. Their greatest direct velocity is in their conjunction, and their greatest retrograde velocity is in the opposition of the superior planets, and in the lower conjunction of the inferior planets.

The greatest apparent motion of a planet when in opposition or conjunction with the Sun, is owing to the parallel motion of the Earth and planet in these points of its orbit. 'The lower conjunction of an inferior planet, is the same as the inferior conjunction.

7. The shorter the periodic time of an inferior planet, the more frequent are its stations and retrogradations, the shorter time they continue, and the less they are in quantity.

This is well known to be the case, both from observation and calculation. For instance, in the year 1819, Mercury was stationary no less than six times, and retrograded four times, whilst Venus was stationary only once, and retrograde only once. The mean arc of Mercury's retrogradation is about 13° 30', and its mean duration about 23 days; whilst that of Venus is about 16° 12', and its mean duration about 42 days..

8. The longer the periodic time of a superior planet, the more frequent are its stations and retrogradations, but they are less in quantity, yet continue a longer time.

The greater the relative motion of the Earth and a superior planet is, the more frequent will a given situation of the two bodies occur; and the less it is, the longer time it will be before similar situations of the two bodies take place.

The mean arc of Mars, is 160 12, and its mean duration about 78 days; whilst the mean arc of retrogradation of Jupiter is only 90 54', but its mean duration is about 121 days.

9. When the planets are in their syzygies, their longitude, seen from the Earth is the same as their longitude seen from the Sun, except in the lower • conjunction of an inferior planet, when its longitude seen from the Earth, differs 180 degrees from. its longitude as seen from the Sun.

Obs. 1. That the superior planets have the same longitude as seen from the Earth and Sun, when in conjunction or opposition, will readily appear, (see fig. 3, page 264,) for when the planet is at I, and the Earth at e, in opposition, it will have the same longitude as seen from the Earth or Sun, the three bodies being in the same vertical plane, or right line directed to the same part of the heavens. The like will be the case when the Earth is at E, or the planet is in conjunction at I.

2. When an inferior planet is in its superior conjunction at N, (fig. 1, page 258) it will have the same longitude, whether observed from the Sun or Earth; but when the planet is in its inferior conjunction at a, it will appear from the Sun to be in the opposite part of the heavens, or 180 degrees from its place, as seen from the Earth, the planet being at the time between the Earth and the Sun.

3. To find the geocentric latitude of a planet, we have the proportion; as the sine of the difference of longitudes of the Earth and planet: the sin. of elongation in longitude :: to tang. of the heliocentric latitude: the tang. of the geocentric latitude.

For example, to find the geocentric latitude of Mars, December 1st, at noon, 1819. Sun's long. 8s. 8° 30' 7", hence the Earth's place, 2s. 8° 30′ 7"; heliocentric long. of planet, 3s. 40 7', geocentric longitude, 4s. 50 29', heliocentric latitude of the planet 10 20. Then, 2s. 8° 30′.7" subtracted from 3s. 4° 7', gives 25° 36′ 53′′ diff. long. of Earth and planet. Again, 4s. 50 29′ taken from 8s. 8° 30′ 7", leaves 123° 1' 7", elongation in longitude. Hence, sin. 250 36′ 35′′: sin. 123° 1' 7" :: tang. 10 20': tang. 2° 35' 5" the geocentric latitude, as required.

4. It may not be improper to observe, that by knowing the longitude of the Earth, its distance from the Sun, the heliocentric long. of the planet, and its distance from the Sun when referred to the ecliptic, there are given two sides of a plane triangle, and the included angle, to find the angle at the Earth, or elongation in longitude: so that by knowing the heliocentric place of a planet, its geocentric place may be found; and on the contrary, if its geocentric place be known, its heliocentric may be found.

10. The periodic times of the inferior planets can be deduced nearly, from observing the time between two conjunctions, their orbits being supposed circular.

Let Tequal the time between two successive inferior or superior conjunctions. E equal to the periodic time of the Earth. P equal to the periodic dic time of the planet. Then, considering the planet's angular motion as uniform, P: E:: 4 right angles: angle described about the sun in time of Earth's revolution=4 right angles plus angle gained by planeton Earth, in time of the Earth's revolution. But as the angles gained are as the times of gaining them, therefore 4 right angles: 4 right angles+angle gained by planet on Earth in time of Earth's revolution:: T:T+E.

Hence, P: E::T:T+E; therefore, P

TXE

T+E

: conse

quently knowing the time between two inferior conjunctions, which can be readily observed, we obtain the periodic times of the planets Mercury and Venus.

The interval between the inferior conjunctions of Mercu

115×365

ry is 115 days, therefore its periodic time days.

87

115+365

The interval for Venus is 584 days, and consequently its

584 × 365

11. The periodic times also of the superior planets can be obtained, from observing the time between two successive oppositions.

Let T, E and Prepresent as before. Then P: E:: 4 right angles: angle described by planet in time of Earth's revolution, equal to 4 right angles minus angle gained by Earth or planet in time of Earth's revolution. Also 4 right angles: 4 right angles-angle gained by Earth in time E::T: T-E; hence P:E::T:T-E; therefore

P

T+E

The interval between two oppositions of Uranus or Herschel is 3693 days; hence the periodic time of Uranus 369.75×365.25 369.75×365.25

369.75-365.25

4.5

-=82 × 3651=82 years

For Saturn, the interval is 378 days, and consequently the

378×365

periodic time of Saturn 29×365-29

378-365

years. In like manner, the periodic times of the other superior planets may be nearly determined.

12. When an inferior planet is near one of its nodes at inferior conjunction, it appears a dark spot on the Sun's surface, and thereby is shown that the inferior planets receive their light from the Sun.

Obs. 1. When Venus is in superior conjunction, at a con siderable distance from its node, it may be seen, by help of a telescope, to exhibit an entire circular disc. Indeed all the different appearances of the inferior planets, as seen through a telescope, are consistent with their being opaque bodies, illuminated by and moving about the Sun in orbits, nearly circular. Near inferior conjunction they appear crescents, exhibiting the same appearances as the Moon a few days old.

At the greatest elongation they appear like the Moon when halved, and between the greatest elongation and superior conjunction they appear gibbous, or like the Moon between being halved and full: these appearances are usually called the phases of the Moon or planets.

2. These appearances are easily explained. The planet being a spherical body, body, the hemisphere turned towards the Sun is illuminated. A small part only of this hemisphere is turned towards the Earth, when the planet is near inferior conjunction. Half the enlightened hemisphere is turned towards the Earth, when the planet is at its greatest elongation. More than half, when the planet is between its greatest elongation and superior conjunction. For, generally, both with respect to inferior and superior planets, the greatest breadth of the part of the illuminated hemisphere turned towards the Earth, is proportional to the exterior angle at the planet, formed by lines drawn from the planet to the Sun and

Earth.

illuminated part seen from

B

the Earth. The measure of this is the angle IPH=IPS+ SPH=HPG+SPH=SPG the exterior angle at the planet. Now near inferior conjunction the exterior angle is less than a right angle; at the greatest elongation it is a right angle; and afterwards greater than a right angle. Therefore the breadth of the illuminated part is respectively less than a quadrant, equal to a quadrant, and greater than a quadrant

3. It is easy to see that as the planets appear flat discs on the concave surface, so their illuminated parts will be projected on the flat surface, and the greatest breadth will be projected into its versed sine, as in figures 5, 6, 7, where IH is projected into its versed sine AB.

Fig. 6,
P

A

Because the projection of a circle, inclined to a surface, by right lines, perpendicular to that surface, is an ellipse, the inner termination of the enlightened part appears elliptical, and the enlightened surface: surface of planet :: Ab: AC:: versed sine of exterior angle: diameter.

4. With respect to the superior planets; the exterior angle of the planet is least when the planet is in quadrature; for when the exterior is least the interior is greatest. Now it is evident that SDE, (see fig. 5, page 264) when DE is

a