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sun in right ascension, or those by which the luminary appears to advance from west to east; and the difference, on any day, between the times of the transit of the sun and of the fixed star, will be the difference between the true right ascensions of the sun and star at the instant of the sun's transit.

S'

172. Now, for several days before and after the 21st of March or the 23rd of September, let the daily transits and declinations of the sun be observed, and let the daily differences of each be taken. Let ss' be part of a great circle representing the sun's path in the ecliptic for one day, and mm' be part of the equator in the celestial sphere; also let sm, s'm' be portions of de- R clination circles, or perpendiculars let fall from s and s' on the equator:

S

&

m'

E

n

m

then, sm being the declination at the noon preceding that in which the sun ascends above the equator, and s'm' the declination at the following noon, also mm' expressed in sidereal time, being the difference between the sun's right ascensions at the two consecutive noons; if these arcs be considered as straight lines, on account of their smallness, and s'n be drawn parallel to mm', sn (= sm + s'm') will be the difference between the declinations at the consecutive noons, and the rightangled triangles sns' and smy, considered as plane triangles, will be similar to one another: therefore we shall have

(:

sn : s'n (= mm') :: sm: mv.

Thus mv, in time, being added to the time expressed by the clock at the preceding noon, will give the time which would be shown by the clock when the sun is precisely in the plane of the equator. If a like computation be made with the differences of right ascension and declination for the several days before and after the noon of March 21st, and a mean of all the times at which the sun is in the equator be taken, that time may be considered as correctly ascertained. Then the difference between the right ascensions of the sun and of the fixed star being found by observation, the distance of the star, in right ascension, from the equinoctial point becomes known; and if the clock be set to indicate that distance, at the moment when the star next comes to the meridian of the station, the clock will indicate the zero of time (twenty-four hours) when the equinoctial point comes to the meridian: from that time, if well regulated, it will show the absolute right ascension of any celestial body at the moment of culmination.

173. The place of the equinoctial point may, however, be

obtained more correctly by the following process: -- in the right-angled spherical triangles vms, vm's' we have (art. 62. (e'))

sin. m = cotan. Y tan. ms, or tan. Y sin. Y m = tan. ms;

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Equating these values of tan. Y we obtain

sin. Ym tan. m's′ = sin. Y m' tan. m s ;

whence, sin. vm: sin. r m' :: tan. ms: tan. m's',

and by proportion,

sin. Y m+sin. Y m': sin. vm-sin. Ym' :: tan. ms+tan. m's'

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But (Pl. Trigo., art. 41.) the first and second members of this equation are respectively equal to

tan. (m vm') sin. (ms-m's').

and

tan. (vm + vm') sin. (ms+m's')

:

therefore, since ms, m's' and vm + vm' are known from the observations, the value of vm. m' is found, and consequently m and m' are separately determined.

It follows from what has been said that the sidereal clock will express in time, at any instant, the angle contained between the plane of the meridian of a station and a plane passing through the axis of the earth and the equinoctial point : that is, the time indicated by the sidereal clock at any instant expresses the right ascension of the mid-heaven or of the meridian.

be

174. Computations similar to those which are supposed to have been made in March may be made in September when the sun is again near the place at which it crosses the equator; and thus the situation of the other equinoctial point may determined. If the sidereal clock were correct it would be found to indicate 12 hours when the sun is in this second point; from which it may be inferred that the two equinoctial points are distant from one another 180 degrees, that is, half the circumference of the heavens, or 12 hours in right sion, or that the intersections of the planes of the celestial

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equator and ecliptic are diametrically opposite to one another in the heavens, in a line passing through the earth.

On comparing together, with respect to a fixed star, the positions of the equinoctial points determined in the manner above described, in different years, it is found that the apparent places of the points are not fixed in the heavens; and the phenomena indicate either that the points have a continual motion from east to west about the earth, or that all the stars called fixed have a general movement in a contrary direction. The movement, in either sense, is called the precession of the stars, or of the equinoctial points; and physical considerations prove that, in reality, those points are in movement on the celestial equator. From the comparisons of observations made at intervals of several years it has been found that the retrogradation, or movement from east to west, takes place at a mean rate of about 50".2 annually; and M. Bessel expresses the mean precession for one year by the formula 50.211290.0002443 t, t denoting the number of years which have elapsed since 1750. The interval between the times at which the sun arrives at the point v is 365.242217 days, which is the length of a tropical year; also the interval between the times at which the sun has the same distance in right ascension from a fixed star is 365.256336 days, and this is the length of a sidereal year.

175. The absolute right ascension and declination of the sun being obtained for the noon of any day, the rules of spherical trigonometry enable the astronomer to compute the inclination of the plane of the ecliptic to that of the equator, or the obliquity of the ecliptic; for s' (fig. to art. 172.) being the place of the sun at the noon of any day, m' is its right ascension and m's' its declination; then (art. 62. (e')) in the spherical triangle s' m' Y, right angled at m',

r sin. m'y tan. m' s' cotan. s'y m';

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and the last factor is the required obliquity.

In order to obtain from observation the obliquity of the ecliptic, or the sun's declination at the time when that element is the greatest, that is, at the time when the sun is in either of the solstitial points (June 21. or December 21.), it is convenient to observe the declination at the noon of each day for several days before or after such epoch; and the obliquity of the ecliptic being known approximatively, let the corresponding longitudes of the sun be computed. Then if s be the place of the sun at the noon preceding (for example) the day of the solstice, or that on which the declination is the greatest, sr the observed declination and Y the equinoctial point, we

shall have, in the spherical triangle sr v right angled at r, representing the angle at v by q (art. 60. (e)),

r sin. sr = sin. 9 × sin. v s.

The arcs is the sun's longitude at the time of the observation; and if s" be the solstitial point, so that y s" is a quadrant, and s'e the solstitial declination, r s and sr will differ but little from v s" and from s"R or q, respectively. Let ss" be represented by 7 and the difference between s r and s"R by d; then srq-d and v s =

π

2

7; therefore, on substi

tuting these values in the above equation (radius being = 1),

= sin. q sin. (1⁄2 − 1);

sin. (g - d) =

and (Pl. Trigo., arts. 32. 36.

sin.

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2

cos. q sin. d = sin. q cos. l,

cos. d = 1 2 sin.2 } d, and cos. 1 = 1 2 sin. therefore 2 sin.2 d sin. q + cos. q sin. d = 2 and, dividing by cos. q,

l;

sin.2 17 sin. q;

2 sin.2d tan. q + sin. d = 2 sin.2 1 7 tan. q: rejecting the term containing sin.2 } d because of its smallness, sin. d = 2 sin.2 } l tan. g.

Thus d may be found, and its value being added to that of sr, the sum will be the greatest declination, or the obliquity of the ecliptic.

A like computation is made with the observation of each day, and a mean of all the results is taken for the correct obliquity.

The mean value of the obliquity, according to M. Bessel, January 1. 1843, is equal to 23° 27′ 35′′.15: but comparisons of the computed values of that element have shown that it has been continually diminishing from the epoch of the earliest astronomical observations; the annual diminution, however, is very small, being, in the present age, equal to 0".457, or about half a second.

CHAP. VII.

TRANSFORMATION OF THE CO-ORDINATES OF CELESTIAL BODIES FROM ONE SYSTEM TO ANOTHER.

176. THE relative positions of the celestial bodies with respect to one another, and to the earth or sun, are most conveniently determined by means of rectangular co-ordinates; that is, by the mutual intersections of three planes imagined to pass through the centre of a body perpendicularly to one another and to three other planes also perpendicular to one another, the latter system of planes being supposed to have some given position; or, in other words, the relative positions may be determined by three lines imagined to be let fall from the centre of the body perpendicularly on three given planes, also perpendicular to one another, like those which are about one angle of a rectangular parallelepiped.

177. The system of co-ordinate planes to which a celestial body may be referred, may, in the process of an investigation, be supposed to change its position in space either by a general motion in which the several planes remain parallel to themselves, or by a movement about the angular point in which they intersect one another; or again, about one of the lines of section as an axis; it is therefore necessary here to explain the method of determining the co-ordinates of a point with respect to a system of planes in one position, from those of the same point with respect to the system in another position; and the rules of plane and spherical trigonometry may be advantageously employed for this purpose. Let p be any point space, and EX, EY, Ez be three co-ordinate axes at right angles to one another; imagine a plane to pass through P perpendicularly to the planes XEY, XEZ, cutting the former in AM, which is therefore perpendicular to EX; and imagine another plane to pass through P perpendicularly to XEY, ZEY, cutting the former in MB, which is therefore perpen

in

E

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B'

P

F

M

A/

m

XI

B

Y

dicular to EY; the intersection of the cutting planes with each other will be a line PM perpendicular to XEY.

Then

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