Phases of Venus.

On the 1st of this month, this beautiful planet will shine with great brilliancy, and the breadth of her illuminated part will be 3·2345, and of her dark part 8.7655 digits.

By comparing the digits of her illumination with the result obtained for her greatest brilliancy by the following solution, it will be seen that, on account of her nearness to the Earth at this time, she will display a greater brightness than in any other part of her orbit; she has, indeed, under similar circumstances, frequently been seen in the day-time, and even when the Sun was shining with great brightness.

Eclipses of Jupiter's Satellites.

There will be several eclipses of Jupiter's satellites this month; but none of these will be visible at the Royal Observatory, or its neighbourhood.

Other Phenomena.

The Moon will be in conjunction with ẞ in Taurus at 26 m. after 7 in the evening of the 4th of this month; with Pollux at 23 m. after 6 on the evening of the 7th; and with Saturn at 7 m. after 11 in the morning of the 25th. Mars and Venus will be in conjunction at 1 m. past 5 in the evening of the 11th, at which time Mars will be 8' south of Venus. Mars and Jupiter will also be in conjunction at 41 m. after 12 on the 13th, when Mars will be 36' south of Jupiter.

On the PHASES of MERCURY and VENUS.

[Concluded from p. 23.]

As the planets are at very different distances from the Earth during different periods of their revolutions, they do not always appear the most brilliant when the whole of the enlightened disc is turned towards the Earth; for the intensity of

light varies inversely as the square of the distance of the planet from the Earth, and the diminution from this cause frequently more than counterbalances the increase of brilliancy arising from the increased area of the illuminated part of the disc. On this account, to determine the situation of a planet at the moment of its greatest brilliancy becomes an interesting problem. No. 349 of the Philosophical Transactions contains a solution of this problem, by Dr. Halley, as it applies to Venus. But as it necessarily requires the aid of fluxions, in order to determine the maximum, and most of our readers are unacquainted with that science, we shall only give the result of his solution.

Denoting the distance between the Sun and the Earth by a, that between the Sun and Venus by b, and between the Earth and Venus by x; the first two being of course known, and the last an unknown quantity; then the solution of the problem gives

x=(3a2+b1)*—2b.

In applying this result to the case in question, we may take a=1, and then b≈72333; and these values subsituted in the above formula, give x=43036. These numbers therefore express the ratio of the respective sides of the triangle ESV, in Fig. 1; whence the angles are easily found by the elements of plane trigonometry. This gives the angle ESV 22° 21', and SEV=39° 44'; and consequently their sum is 62° 5', the natural versed sine of which is 5318; and multiplying this by 6, we have 3.1908 digits for the breadth of the enlightened part, and which is therefore a little more than a fourth part of the whole apparent diameter of the planet.

If we apply the same solution to Mercury, taking, as before, a=1, we have b=3871, and by substituting these, x=1-0006. Then the angle

ESV, found as in the former case, is 78° 55'; and SEV=22° 19′; and consequently their sum is 101° 14', the versed sine of which is 1.1948; which being multiplied by 6, gives 7.1688 digits. Mercury therefore appears the brightest when about ths of his apparent diameter is illuminated. This difference between the enlightened phase of Mercury and that of Venus, at the time of their greatest brilliancy, arises from the circumstance, that the distance of Mercury from the Earth varies much less than that of Venus; and consequently the intensity of his light is more constant.

On the FORM of SATURN'S RING, and the ORbits of his first four SATELLITES.

We have already stated some of the principal circumstances respecting Saturn's ring, and shall here only add the method of calculating its form for any given time. This ring sometimes appears like a dark line crossing the disc of the planet, and at others it appears to be elliptical, and to surround the body of the planet; so that the opening between the planet and the external part of the ring may easily be seen by means of a good telescope. The greatest value of the conjugate diameter is about half that of its transverse, and it varies from this to nothing. The ring of Saturn, like the Earth's equator, being considerably inclined to the plane of the Earth's orbit, sometimes one side, and sometimes the other, will be illuminated by the solar rays; by which means, they both become alternately visible to the inhabitants of the Earth. During the change of illumination from the north to the south side of the ring, its plane coincides with that of the Earth's orbit, and then it becomes altogether invisible to us.

In order to find the figure of Saturn's ring at any given time, add 13° 43′ 30′′ to his geocentric

longitude for the time required; and with this sum, as an argument, enter the following small table, in which the signs are inserted at the top or bottom, and the degrees on the sides, and the corresponding numbers will be the conjugate axis of the ring, its transverse being 1.000. This result, however, requires a correction, depending upon the geocentric latitude of the planet. To obtain this correction, the latitude must be reduced to minutes, and 4th part applied to the number above found, with the sign+, if his latitude be south; but with the signif it be north, and the result will give the apparent conjugate diameter of the ring.

TABLE

For finding the Form of Saturn's Ring, and the Orbits of his first four Satellites.

As an example of the preceding method of finding the ratio of the diameters of Saturn's ring,

let it be required to find his conjugate axis on the 1st of January 1819; then we have Saturn's geocentric longitude

Add the constant quantity

11. 130 23' 13 43

Hence, as the ratio of the two axes of the ring at this time are 058 to 1, it will appear only like a line. When the sign + is prefixed to the result, it indicated that it is the upper surface or northern side of the ring that is then visible; but when - is employed to characterise the result, it shows that it is the southern face of the ring that is then seen. The result thus obtained also applies to the orbits of the first four of Saturn's satellites. When the result is equal 0, Saturn's ring totally disappears, the edge being too thin to reflect sufficient light to render it visible.

The Naturalist's Diary

For FEBRUARY 1819.

Reviving nature seems again to breathe,

As loosened from the cold embrace of death.

IN the course of this month all nature begins, as it were, to prepare for its revivification. God, as the Psalmist expresses it, renews the face of the earth; and animate and inanimate nature seem to vie with each other in opening the way to spring. About the 4th or 5th, the woodlark'