or, if express the true zenith distance,

sin. p = sin. P sin. (z′ + p).

It may be observed that the greatest parallax of a celestial body takes place when the body is in the horizon; and, both above and below the horizon, the parallax diminishes with the sine of the distance from the zenith. The apparent place of a celestial body is always given by observation; but, in seeking the effect of parallax when the true place of the body is obtained by computation from astronomical tables, it will be convenient first to find the parallax approximatively from the formula p' P sin. 2' (where is the true zenith distance), and then to substitute that approximate value in the formula sin. p = sin. P sin. (z' + p'); the result will in general be sufficiently near the truth. Or a second approximative value may be found from the formula p' P sin. (z+p'); and the value of p" being substituted in the formula sin. P sin. P sin. (z+p") will give a still more correct value of the parallax in altitude. The following series for the parallax in altitude in terms of the horizontal parallax and the true zenith distance is given in several treatises on astronomy:

p (in seconds)=

sin. P

sin. 1′′sin. z′+

sin.2 P
sin. 2′′

sin.3 P

sin. 2 z+

sin. 3z

sin. 3"

+ &c.

and the three first terms are sufficient.

156. If it be assumed that the earth is of a spheroidal figure, and such as would be produced by the revolution of an ellipse about its minor axis, it must follow that the horizontal parallaxes, which are the angles formed at a celestial body by a semi-diameter passing through the place of the observer, will be the greatest at the equator and the least at the poles; and that they will vary with the distance of the spectator from the centre of

the earth. The value of the geocentric horizontal parallax at any station whose latitude is given may be investigated in the following manner. Let PSQ be one quarter of the terrestrial meridian passing through any stations; also let P be the pole, and Q the point in which the meridian cuts the equator: again, let м and M' be places of any celestial body,

M#

MI

M

as the moon, when in the apparent horizon of an observer at the equator and at any other point s on the earth's surface. Then, since CM may be supposed to be equal to CM' (and, on account of the smallness of the ellipticity of the earth, cs may be considered as perpendicular to Sм', as QM is perpendicular to CQ, so that CQ and cs may represent the sines of the horizontal parallaxes at Q and s) we have, by proportion (P, w, and p being the horizontal parallaxes at the equator, the pole, and at any place s respectively)

CQ-CP CQ CS:: sin. P

sin. : sin. P— sin. p.

But by conic sections, in an ellipse, the semi-transverse axis CQ being equal to unity, the semi-conjugate axis = c, the excentricity=e, and scQ the geocentric latitude (art. 151.) of s = 7; we have, as in art. 149., e being very small (= 0.08 nearly when cQ = 1),

cs=1-ce2 sin.2 l, and cQ-cs = ce2 sin.27 ;

or the decrements of cQ from the equator towards the pole vary with sin.2 l.

or,

157. Hence, rad.2 : sin.:: C Q ·

CP CQ-CS,

1 : sin.27 :: sin. P - sin. : sin. P — sin. p.

Let the ratio of CQ to CP be as 305 to 304 as in art. 152.; then

305 304 :: sin. P: sin. P (sin. w, the sine of the

304
305

sin.,

polar horizontal parallax);

1

sin. P, or sin. P expresses the difference

305

between the sines of the equatorial and polar horizontal parallaxes. Consequently,

and this last term expresses the difference between the sines of the horizontal parallaxes at the equator and at any places;

also sin. P

1

305

sin. P sin.27 is equivalent to the sine of the geocentric horizontal parallax at such place; let it be represented by sin. P'.

The following table, which depends on the equatorial horizontal parallax of the moon and upon the latitudes of stations, will show by inspection the value of the second term in the

expression; the ellipticity of the meridian, that is

being

=

1

305

The terms are to be subtracted from the equa

torial horizontal parallax of the moon (in the Nautical Almanac) in order to reduce it to the geocentric horizontal parallax at any station whose latitude is given.

158. If the celestial body be at any point M" elevated above the horizon of the spectator at s, the perpendiculars ca and cb let fall from the centre of the earth upon M's, M's produced may represent the sines of the horizontal parallax and of the parallax in altitude; and the hypotenuse Cs of the right-angled triangles sac, sbc being common, we have

ca: cb sin. csa: sin. csb;

that is, the sines of the parallaxes are to one another as the sines of the distances of the celestial body from the geocentric zenith. But sin. csa may be considered as equal to radius (1) without sensible error; therefore, if the apparent geocentric zenith distance ZSM" be represented by z, and the horizontal parallax by P', we have

rad. sin. z :: sin. P' : sin. z sin P',

:

and the last term expresses the sine of the geocentric parallax in altitude.

159. When the moon and the sun, the sun and a planet, or the moon and a planet, are very near one another, as in eclipses and occultations, instead of computing separately the parallaxes of the two luminaries in altitude, it is more convenient to substitute in the last expression the difference between their geocentric horizontal parallaxes at the station of the observer: the result will be the difference between the parallaxes of the luminaries in altitude. Thus, if at any place, as s, P' be the geocentric horizontal parallax of the moon, and p' that of the sun (the luminaries being near one another), the difference between their parallaxes in altitude will be expressed by sin. z sin. (P'-p'); where P'-p' is

called the moon's relative horizontal parallax. The like is to be understood of the difference between the parallax of the sun and a planet, and of the moon and a planet.

Z

S

m'

m

160. The horizontal parallax of the moon or of a planet may be found from the difference between the zenith distances of the moon or planet and of a fixed star, observed at two places on or near the same terrestrial meridian. Thus M being the place of the body whose parallax is to be found, E the centre of the earth, and s the place of any fixed star, let A and B be the stations of two observers, which for simplicity may be supposed to be on the same meridian, and let them be so situated that the moon or planet, when on the meridian, may be southward of one observer and northward of

E

M

B

the other also let it be supposed that the latitudes of the stations A and B are known. Then, by previous agreement between the observers, the zenith distances ZAS and Zam, Z'BS and Z'Bm' being taken when the star, and the moon, or a planet, respectively culminate or come on the meridian; the arc sm or the angle sam, and the arc sm' or the angle SBM' will represent the differences between the zenith distances at the respective stations. But the fixed star s being incalculably remote, BS may be considered as parallel to AS; hence the angle sa'm may be taken for sam, and the difference between sa'm or Sam and SBm', will be equal to the angle m'Mm or AMB, which is therefore found.

Now, joining the points E and M, the angles A ME, BME, are the parallaxes of the body at M, in altitude, for the observers at A and B respectively, and we have

rad. : sin. zam :: sin. horizontal parallax: sin. A M E, also, rad. : sin. Z'Bm' :: sin. horizontal parallax: sin. BME: therefore, by proportion

rad. : sin. zam+sin. Z'B m' :: sin. hor. par. sin. AME + sin. BME.

If the arc which measures the horizontal parallax, and those which measure the angles AME, BME, be substituted for their sines, then AME+BME being equal to the angle AMB, which was obtained from the observation, we have from the last proportion,

sin. ZAM+ sin.Z' BM: rad.:: AMB (in arc, or in seconds): hor. par. (in arc or seconds):

and thus the horizontal parallax of the moon or planet at the time of observation is found. It should be observed that this method is not applicable to planets beyond the orbit of Mars; and if the observers be not situated precisely on the same terrestrial meridian, it would be necessary to correct the observed zenith distance of the moon or planet, at one of the stations, on account of the variation in its declination during the time in which it is passing from one meridian to the other. Since the horizontal parallax of a celestial body may be represented by the angle SMC (arts. 154. 156.), in which the angle at s is, or may be, considered as a right angle, and that SC=MS tan. SMC; it follows that, for different celestial bodies, the tangents of the horizontal parallaxes vary inversely with the distances of the bodies from the earth.

161. The parallax of a celestial body in altitude being obtained, the deviation of the apparent from the true place of the body in any other direction (as far as it depends upon the place of the observer) may be readily found by Plane Trigonometry, when the angle at the celestial body between a vertical circle passing through it and a circle of the sphere, also passing through it, in the direction for which the deviation is required, is known; since that deviation may be considered as one side of a right-angled triangle, of which the hypotenuse is the parallax in altitude: but this is not always the most convenient method of determining the deviation in a direction oblique to the vertical circle; and the following are investigations of formula, by which the parallaxes of a celestial body in right ascension and declination (that is, in the directions which are particularly required for the solution of problems relating to practical astronomy), are generally

obtained.

ရာ

Let YHQ (fig. to art. 147.) be part of the celestial equator, stereographically projected, P its pole, and let z be the geocentric zenith of the spectator's station. Let s' be the true place of the celestial body, suppose the moon, and the equinoctial point, so that at a given time H' is the moon's true right ascension (in arc) and PS' her true north polar distance. At the same instant let s be the apparent place of the moon, so that H is the moon's apparent right-ascension (in arc), and PS her apparent polar distance. Then PQ being the meridian of the station, QPS' is the moon's true, and QPS her apparent, horary angle at the given time. Again, zs' is the true, and zs is the apparent, zenith distance of the