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at M, at the time T, there may evidently be obtained the times of the immersion and emersion.

377. When the two inferior planets happen to be in conjunction with the earth and sun near the nodes of their orbits they appear to pass across the sun's disk: this phenomenon is frequently presented by Mercury, but the last time at which Venus was so seen was in the year 1769; and this planet will not be again in a like position with respect to the sun till the month of December in 1874. On all such occasions the observation of the phenomenon may be made subservient to the determination of the difference between the parallaxes of the planet and the sun; and subsequently, the ratio between the distances of the planet and sun from the earth being known, to the determination of the absolute parallaxes and distances.

Of the two planets, Venus is that which is the most favourably situated for affording precise values of those elements; and the computation of the time at which, for a particular station, the phenomenon will occur may be briefly indicated in the following manner. The process consists in finding the times of ingress or egress, or those at which the planet enters and quits the sun's disk, as if the spectator were at the centre of the earth; and then, in finding with the present knowledge of the relative parallaxes, the corrections which are to be made to that time on account of his position on the earth's surface.

Comparing together the geocentric right ascension of the planet and the right ascension of the sun, as they are given in the Nautical Almanac for the noons of the two days between which noons a conjunction in right ascension must take place, there may be found approximatively (by means of the hourly motions which in this case may be considered as uniform) the time of the conjunction of the sun and planet in right ascension. This will be expressed in Greenwich mean time, and will be within one or two minutes of the true time of conjunction: let it be represented by T.

For this time find from the Nautical Almanac, with an attention to second differences, the following elements:- 1. The right ascension of the sun and the geocentric right ascension of the planet: take the latter from the former, and let the difference be represented by A'.-2. The declination of the sun and the geocentric declination D of the planet: take the latter from the former, and let the difference be represented by D'; these last elements are to be considered as positive if north, and negative if south. 3. Find the semidiameters of the sun and planet.

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Let s (fig. to art. 369.) be the place of the sun, and M the place of the planet at the time T: let XY be part of the planet's orbit, and PS a horary circle passing through the sun: also let мp be an arc of a great circle perpendicular to PS and Mg an arc of a parallel of declination passing through the planet. Then Mq or Mp A'cos. D; and, neglecting pq, which is very small, sp = D' (in seconds).

From the Nautical Almanac, with an attention to second differences, take the relative geocentric hourly motion of the planet in right ascension; which, since at the time of transit the movement of the planet in its orbit is retrograde, will be equal to the sum of the hourly motion of the sun and the geocentric hourly motion of the planet: let this be expressed in minutes of a degree and be designated by a. Then a cos. D will be the relative horary motion of the planet on the parallel Mq of declination. Imagine mn to be drawn parallel to PS; then a cos. D may be represented by Mm. Take in like manner, with second differences, the relative hourly motion of the planet in declination, which is either the difference or sum of the hourly motion of the sun and the geocentric hourly motion of the planet according as the variations of declination take place in the same, or in contrary directions: let this hourly motion be represented by mn in the figure, and designated d.

378. Proceeding next as in the investigations concerning eclipses of the moon and sun (arts. 369, 373.) we have

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and representing the angle mмn or Qsv by I,

;

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Now, to find sv and Mv:

In the triangle MpQ, cos. I rad. :: Mp: MQ

cos. I sin.I :: Mp: Qp (

and

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(=

(

sin.

A'cos. D

COS.

sin. I

COS. I

and in the triangle sQv, SQ cos. I=Sv;

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Sv=D' cos. I- A' cos. D sin. I;

A'cos. D),

COS.

1):

therefore

sin.2 I

also,

Qv (= SQ sin. 1) = D' sin. I— A' cos. D

COS. I

:

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and Mv (=Qv+MQ) = D′ sin. I — A' cos. D

or

D' sin. I+A' cos. D cos. I.

COS. I COS. I

Let X and Y be the places of the planet when, as seen from the centre of the earth, the limbs of the sun and planet are in contact at the ingress and egress; so that, at the ingress, sx and Sy are, each, equal to the sum of the semidiameters of the sun and planet: then

sx:sv:: rad.: cos. x sv, (= 6),

S

this angle is therefore found; and from it there may be obtained the value of xv or vY (XS sin. XSv).

Now if, as in the figure, the time T, at which the planet is at M, precedes the instant of conjunction in right ascension, Xм=Xv-Mv and MY = Xv + Mv; and xv+Mv may be expressed by xs sin. XS v + (D′ sin. I + A' cos. D cos. I).

a cos. D

COS. I

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the relative

This quantity being divided by horary motion in the orbit, will give, in decimals of an hour, or (after such division) being multiplied by 3600, it will give in seconds, the time t, which must be subtracted from T in order to give the instant of ingress, and added to T to give the time of egress; we have, therefore, for a spectator at the centre of the earth,

TFt=TF

3600 cos. I

a cos. D

{xssin. xsv (D'sin. I + A' cos. D cos. I)}.

379. Now, for a spectator on the earth's surface, let the relative parallaxes in right ascension and declination be represented by a and 8: then Mp considered as a parallel of declination when affected by parallax will become, as in art. 372., (A'-a) cos. D; and, neglecting pq, sp will become D'-d: hence, substituting A'-a for A', and D'-8 for D' in the above expressions for sv and Mv,

we have sv (D' — 8) cos. I and

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(A'-a) cos. D sin. I,

Mv= (D' — d) sin. I + (A'— a) cos. D cos. I.

Now the values of a and 8 (the parallaxes in right ascension and declination) may be obtained from the formula (II) art. 161. and (v) art. 162., on putting for D the geocentric declination of the planet, for the sun's hour angle at the time T, and for P' the difference (supposed to be known) between the geocentric horizontal parallaxes of the sun and planet. This last value of sv being divided by sx (the known sum or difference of their semi-diameters according as the

time of an exterior or of an interior contact is required), there will be obtained a value of cos. Xsv, from which the angle and consequently its sine may be found. This sine, together with the last value of Mv, may then be substituted in the second member of the equation for Tt; and the result will be the time of either contact, at ingress or egress, for the given station on the surface of the earth.

380. If it be required to determine from the instants of ingress and egress observed at places on the earth's surface, the value of P', the difference between the parallaxes of the Sun and Venus, the process may be as follows. Let t and t' be the intervals of time between the instants of ingress and egress and the time T of conjunction in right ascension; then, neglecting the effects of parallax on the horary motions, t.a cos. D will express the relative motion of Venus and the Sun on the parallel of declination, from the hour circle passing through P and X (fig. to art. 369.) to that which passes through P and M, while t.d will be the relative motion in declination from the parallel passing through x to that which passes through M. These being added to the above values of мр and sp give (A'-a) cos. D + t.a cos. D and D'-d+t.d; of which the former may be represented by xv and the latter by sv: in like manner, t'.a cos. D― (A'—a) cos. D and t'.d-(D'-8) may be represented by v'y and sv'. Then, neglecting the errors in the tabular values of the right ascensions and declinations and in the semidiameters of the Sun and Venus,

{(A'— a) cos. D + t.a cos. D}2 + {D' — d + t.d}2 =sx', and a corresponding equation may be found for sy2. But the parallaxes and horary motions being small quantities, in developing the first members of these equations, the second powers and the products of a, d, a and d may be neglected, and we shall have

{”—2▲′(a—ta) } cos.2 D=Xv2, whence xs2 =

D'2-2 D'(8-td) = sv2;

A2 cos.2 D + D'2 + 2 A'ta cos.2 D + 2D'td-2 A'a cos.2 D-2 D'd, and a corresponding equation containing t' may be obtained for Ys2.

In these equations substituting the values of t and t', or the differences between the observed times of ingress and egress and the computed time T of conjunction in right ascension, with the values of xs or Ys, the sum or difference of the semidiameters, the numerical equivalents of a and d

may be found; and from these, by the formulæ in arts. 161, 162. the value of P' may be obtained.

Now, by Kepler's law (art. 254.), the ratio between the distances of the Earth and Venus from the Sun is known; let this be as 1 to r: then the ratio between the distances of the Sun and Venus from the Earth will be as 1 to 1-r. But the horizontal parallaxes being angles subtended at the centres of the Sun and Venus by the semidiameter of the Earth, this ratio is the reciprocal of the ratio of the parallaxes: hence, if p represent the horizontal parallax of the Sun and p' that of which is therefore known.

Venus,

1

1 -r

is equal to

P'.
Р

Now, the value of P', supposed to have been found above, is the equivalent of p'-p; and from these two equations the separate values of p' and p may be found.

From the last transit of Venus, the parallax of the Sun, at his mean distance from the Earth, was found to be 8".702.

381. Eclipses of the moon are of no value as means of determining the longitude of a station, it being impossible to observe the commencement or end of the obscuration with sufficient accuracy on account of the ill-defined edge of the earth's shadow on the moon; were it otherwise, the occurrence of either of these phenomena expressed in mean time at the station, and compared with the Greenwich time found by computation, would show at once the longitude of the station. With respect to an eclipse of the sun, the time of the commencement or end may be observed with considerable precision: hence, though the operation of correcting the right ascension and declination of the luminaries, or the longitude and latitude of the moon, on account of parallax, is laborious, the longitude of a station on the earth may be determined by it with accuracy, and the following process may be employed for the purpose.

382. The instant at which the phenomenon occurs may be expressed in sidereal or in mean solar time; for example, let it be the latter, and let such mean time be represented by T. The difference between the longitudes of Greenwich and the station being approximatively known by estimation or otherwise, this difference (in time) must be added to, or subtracted from the mean time of the observation, according as the station is westward or eastward from Greenwich, and the result will be the approximate mean time which would be reckoned at Greenwich at the instant of the observation being made at the station. Let this be represented by T'.

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