moon, and ss' is the moon's parallax in altitude; the angle SPS' is her parallax in right ascension, and the difference between PS and PS', is her parallax in declination: also ZP is equal to the colatitude of the station. By Spherical Trigonometry (art. 61.) we have, in the triangle SPS', sin. PS sin. ss': sin. s': sin. SPS'; whence, sin. ss' sin. s' sin. SPS' ; sin. PS and in the triangle ZPS', sin. Zs': sin. PZ:: sin. ZPS': sin. s': Let 7 represent the geocentric latitude of the station. A the true right ascension of the celestial body. z the apparent geocentric zenith distance. P' the geocentric horizontal parallax. a the parallax in right ascension. & the parallax in declination. 7 the true horary angle. +a the apparent horary angle. and let the sine of the geocentric parallax in altitude (=sin. ss') be expressed by sin. P' sin. z. (Art. 158.) Then, by substitution in the last equation, or developing sin. (7+a) in order to eliminate a, and, approximatively, the second term in the denominator being very small, α= P' cos. I sin. T (III). Thus is found the parallax in right ascension; and, on putting any of the formulæ in numbers, logarithms with five decimals will suffice. T If for sin. 7 in the last formula there be put the sine of the equatorial interval between any wire and the mean wire in a transit telescope, the resulting value of a will be the effect of parallax in right ascension on the moon's limb at the time of an observed transit at the former wire; and being divided by 15, it will give, in time, the increase in the time of coming up to the mean wire, or the diminution in the time of passing away from the mean wire, according as the time of observation preceded or followed the transit at the latter wire. This is the correction alluded to in art. 93. ; in the former case it is additive, and in the latter subtractive. 162. In the spherical triangles PZS, PZS', from either of the formulæ (a), (b), or (c), art. 60., we have Equating these values of cos. z, and cancelling sin. PZ which is common, COS. PS-cos. PZ cos. ZS cos. P S'-cos. PZ cos. Z s' sin. zs hence, and by transposition, cos. PZ cos. zs sin. zs' cos. PS' sin. zs cos PZ cos. Zs' sin. zs; cos. Ps sin. zs'-cos. PZ (cos. zs sin. zs'-cos. zs' sin. zs)= or again, cos. Ps' sin. ZS; cos. PS sin. zs'-cos. Pz sin. (zs'—zs)=cos. Ps' sin. zs. But sin. (zs'-zs), or sin. ss', is equal to sin. P′ sin. zs' (art. 155.); therefore, cos. PS sin Zs'. cos. PZ sin. P' sin. zs' cos. Ps' sin. ZS, and, sin. Zs' (cos. PS-cos. PZ sin. P') =cos. Ps'. sin. zs Now, in order to eliminate zs and zs', we have (art. 61.), in the spherical triangles, ZPS, ZPS', sin. zs : sin. PS :: sin. ZPS : sin. z ( (= = sin. PS sin.Z PS sin. zs sin. Ps' sin. ZPS' sin. zs' and sin. zs′ : sin. Ps′ :: sin. ZPS': sin. Z (: Equating these values of sin. z, we obtain sin. zs' sin. Ps' sin. ZPS' sin. zs = therefore, by substitution, sin. PS sin. ZPS : Thus, D being known, we might obtain 8, the parallax in declination; but, as in the process, logarithms with seven decimals must be employed, it would be advantageous to have a formula for & alone. The equation for cotan. Ps' may be put in the form cotan. PS sin. ZPS'-cotan. Ps' sin. ZPS= cos. PZ sin. P'sin. ZPS' sin. PS and dividing the first member by cotan. PS-cotan. Ps', it may be put in the form {sin. ZPS'+ cot. PS' (sin. ZPS-sin. ZPS)} (cot. PS-cot. PS')= cotan. PS-cotan. cos. PZ sin. P ́ sin. ZPS' (ZPS'-ZPS) cos. (ZPS'+ZPS), a cos. (T+a), sin. (PS'-PS). sin. PS sin. PS' therefore, by substitution, the equation (A) becomes sin. (PS-PS) sin. ZPS'+cotan. Ps' 2 sin. a cos. (7+1a) = sin. PS sin. PS' cos. PZ sin. P' sin. ZPS' sin. PS or, transposing and subsequently multiplying by sin. PS sin. PS' sin. ZPS' sin. (PS' - PS) = cos. PZ sin. P' sin. PS' and in this fraction substituting for sin. a, its equivalent, sin. P' cos. 7 sin. (7 + a) (1) above, we have : therefore sin d = sin. P' cos. 7 sin. (7 + a) cos. D cos. α sin/sin. P'cos.(D-8)— sin. P'cos. cos. (T+a) sin. (D −8).. (IV). cos. a But again, cos. (D-8)= cos. D cos. 8 + sin. D sin d, sin. (D-8)= sin. D cos. d cos. D sin. &; therefore, substituting these values, and dividing by cos d, we and have and, approximatively, cos.a sin. p' cos. l cos. (T + 1α) (v) ; COS. D } 8P' {sin. l cos. D cos. 7 sin D cos (7 + 1⁄2 α)}. In finding the value of tan. 8, or the tangent of the parallax in declination, logarithms with five decimals will suffice. It is easy to perceive that, when a celestial body is on the eastern side of the meridian of a place, the parallax in right ascension increases the true right ascension of a body; and, when the body is on the western side of the meridian, the parallax diminishes the right ascension: the parallax in declination increases the polar distance of the celestial body in both situations. 163. When the altitude of the upper or lower limb of the sun, the moon, or a planet, is obtained from an observation, the altitude of the centre of the celestial body is found by adding to it, or subtracting from it the angular measure of its semidiameter: this element is given in the Nautical Almanac ; and when the celestial body is the moon, it is necessary that there should be applied to it a correction depending on her altitude above the horizon. A corresponding correction for other celestial bodies is scarcely necessary. In order to investigate it for the moon, let c (fig. to art. 154.) be the centre of the earth, s the place of a spectator on its surface, and let z'M'M be part of a vertical circle passing through the moon: let also м be the place of the moon when in the horizon of the spectator, and M' her place when elevated above it. Let fall CA perpendicularly on м's produced; then if MS, M'A, Z'C be each considered as equal to the distance of the moon from the earth's centre, SC and SA may represent the diminutions of the moon's distance from the spectator at s when she is at z' and at M' respectively, compared with her distance when at M. But in the triangle SCA, right angled at A, SC SA rad. : sin. SCA (: sin. M'S M, or the sine of the moon's altitude). Therefore the diminution of her distance from the spectator varies with the sine of her altitude. But the angle subtended by any small object increases as the distance from the spectator diminishes, and the augmentation of the angle varies with the diminution of the distance: hence, since the angle subtended by the moon's semidiameter is greater when she is in the zenith than when she is in the horizon, by a certain number of seconds, which may be represented by a; it follows that Rad. sin. moon's altitude :: a: a sin. moon's altitude, and the last term expresses the augmentation of the subtended angle, when the moon is between the horizon and the zenith. : The value of the angle subtended by the moon's semidiameter, when in the horizon, depends upon her distance from the earth, which is variable; and consequently the augmentation of the semidiameter when in the zenith experiences variations. The second line in the following table shows the values of those augmentations between the limits within which the moon's horizontal semidiameter may vary; and the fourth exhibits the augmentation in the zenith corresponding to different values of the moon's horizontal parallax. 14' 30" 14' 50" 15′ 10′′ 15′ 30′′ 15′ 50′′ 16′ 10′′ 16′ 30′′ 16′ 45′′ } 13".4 14" 14".6 15".2 15".8 16".3 16".8 17".4 |