substitutions in the second member of the equation (art. 235.) the result will be the difference between the geocentric and heliocentric declinations of the star, or its parallax in declination. From numerous observed declinations of a Lyræ which were made by the Astronomer Royal in 1836, the computed parallax of that star was found to be in some cases positive, in others negative; and the greatest value was 0.505. Mr. Henderson, from observations made at the Cape of Good Hope in 1832-3, concludes that the star a Centauri has a parallax equal to about one second; and the same astronomer found that the parallax of Sirius does not exceed half a second (Mem. Astr. Soc., vol. xi). 299. The efforts to arrive at a satisfactory value of the parallax of a fixed star by means of meridional observations having failed, M. Bessel, in 1837-8, attempted the problem by observations on double stars. For this purpose, by means of a heliometer (art. 121.) he measured the distance of the principal star from a small star designated a in its neighbourhood (the latter from its smallness being presumed to have no sensible parallax) and also the angle of position between the hour circle passing through the first star and a line supposed to connect it with the star a; and from these observations he computed, by means of formulæ, the investigations of which are given by Mr. Main in the twelfth volume of the Astronomical Society's Memoirs, the effects of parallax on the distance between the same stars and also on their angle of position. 300. In these investigations the first step is to obtain an expression for the parallax of a star, in terms of its maximum value, in the direction of an arc of a great circle supposed to pass through the star, the sun and the earth. Thus, let s be the sun and E the earth; also let σ and s be the places of two stars supposed to be at equal distances from the sun, σ being in the pole of the ecliptic or in the line so drawn perpendicularly to SE. Then the angles So E and SSE will be the annual parallaxes of the two stars. Produce s E till it meets sm let fall perpendicularly on it from s; then SE, sm may be considered as the sines of the parallaxes, or, on account of the smallness of the angles at and s, as the parallaxes themselves (in seconds). But m and the angle sEm may be considered as equal to sSE: there fore if p represent the angle so E, or the maximum, commonly called the constant of parallax, the parallax for any other star, as s, is equal to p sin sse. P 301. Now let Y MF be a projection of the celestial equator, Y the equinoctial point, and Y NE a projection of the ecliptic: let P and p be the poles of these circles, and let s be the place of a star. Imagine a plane to pass through the sun, the earth, and the star s, and to cut the trace of the ecliptic in E; then by the effect of parallax the star will appear, to a spectator on the earth, to be at some point s' in the great N M F circle Es produced; and from what was said above (art. 300.) we have ss'p sin. s E. Let a be a small star in the neighbourhood of s, and let fall s'n perpendicularly on the arc as; then sn will denote the effect of the parallax of s in the direction sa draw also s I perpendicular to sa. In the right-angled triangle ss'n considered as plane, snss'sin. ss'n, ss'sin. Is E, or =p sin. SE sin. ISE.... But in the spherical triangle Is E (art. 61.), therefore sin. sE : sin. IE :: sin.SIE : sin.ISE; sin. SE sin. ISE sin. IE sin. SIE. · (A). Now IE is equal to the difference between the longitude of the earth and that of the point I; and the longitude of the earth is equal to that of the sun + 180°; therefore if L represent the longitude of the sun, we have IE=L+ 180° — long. I, and sin. IE sin. (L-long. 1); therefore sn, or p sin. sE sin. ISE, becomes equal to -p sin. (L-long. 1) sin. SI E. Again, in the spherical triangle ps P (art. 61.), sin. Ps: sin. s p P :: sin. Pp: sin. p s P (= sin. M s N): but Ps is the star's north polar distance, the angle sp P is the complement of the star's longitude, and Pp is equal to the obliquity of the ecliptic; therefore the value of the angle MSN may be found from the proportion. The arc sI being at right angles to sa, the angle Is N is equal to the complement of the sum of the angles as M and MSN: and since psa is the observed angle of position for the star a, its supplement as M is known; therefore the angle ISN may be found. Lastly, in the triangle ISN, we have sN, the star's latitude, the angle ISN, and the right angle at N; therefore (arts. 62. (e') and 60 (ƒ)) and r sin. SN tan. IN cotan. Is N, r cos. NIs sin. ISN cos. s N. Thus there are obtained the value of IN and of the angle NIS or its supplement SIE: the arc TI, or the longitude of I, is equal to IN+YN; that is, to the sum of IN and the star's longitude; and substituting these values in the formula (A) we have the value of sn, the star's parallax in the direction sa. 302. The double star 61 Cygni was employed by M. Bessel because, consisting of two stars very near together, he was enabled, in using the heliometer (art. 121.), to bring the other star a seen through one half of the divided object glass correctly into the direction of a line joining the two stars composing the double star, which was seen through the other half. Numerous measures of the distances between the star a and the middle point between the two which compose the double star were taken during more than a year (1837-8), and these were reduced to the distance for one epoch (Jan.1. 1838) by corrections on account of the annual proper motion of the star and the effects of the aberration of light on the distance between the star a and 61 Cygni. Then, considering the differences between the measured distances and the distance at the epoch as so many values of sn, there were obtained as many equations from which the value of p, the constant of parallax, was found. M. Bessel determined the amount of this constant from the stars 61 Cygni and a to be 0.369, and from the first star and another designated b, 0".261: the difference is ascribed to a sensible parallax in the latter star. CHAP. XIV. TIME. SIDEREAL, SOLAR, LUNAR, AND PLANETARY DAYS. EQUATION OF REDUCTIONS OF TIME. EQUINOCTIAL TIME. ASTRONOMICAL HOUR ANGLES. 303. SINCE time enters as an element in most of the problems relating to practical astronomy, it will be proper now to explain the kinds which are employed, and the manner of reducing a given interval in time from one denomination to another. The most simple unit of time is the length of the interval in which the earth makes an exact revolution on its axis, or in which a point on its surface describes exactly the circumference of a circle by the diurnal rotation: this movement of rotation is known to be uniform, and the interval in which it is performed, invariable. The time of such rotation is frequently considered as the length of a sidereal day, but it is not exactly that which is so designated by astronomers, the latter being the interval of time between the instant at which the plane of any meridian of the earth passes through the true place of the vernal equinox, or the first point of Aries, and that at which, in consequence of the diurnal rotation, it passes next through the same point. Now the equinoctial point is subject to a displacement in the heavens in consequence of the general precession and the effects of nutation; therefore this interval is not precisely the same as that of a revolution of the earth on its axis, being constantly less by about one hundredth part of a second, on account of the general precession merely it is moreover not invariable, since the effect of nutation on the equinoctial point is to make that point oscillate within certain small limits about the place which it would occupy by the effects of precession alone. This last inequality would not, however, in nineteen years, cause an error exceeding 2".3 in the going of a clock; and therefore the sidereal day, assumed as above, may be considered invariable. Hence sidereal time at any instant is represented by the angle at that instant, between the plane of the geographical meridian of a place and a plane passing through the axis of the earth and the true equinoctial point, or by an arc of the equator between the meridian and that point. 304. The sidereal day is divided into twenty-four hours, and the hour-hand of a clock, regulated according to sidereal time, should indicate Oho. Omin. O sec. every time that the equinoctial point is on the meridian. Now if a fixed star were situated precisely in the plane of the hour-circle (the equinoctial colure) passing through the two opposite equinoctial points, the meridian of the station would pass through the star every time that, by the diurnal rotation, it came in coincidence with that colure: thus the right ascension of such star would be zero or 180 degrees, according to its position with respect to the axis of the diurnal rotation, and the sidereal clock would indicate O ho., or 12 ho., when the star is observed on the meridian wire of the transit telescope. Therefore, on any other star being so observed, it is evident that the time indicated by the sidereal clock should express, in time, the apparent right ascension of that star, or the angle between the plane of the meridian and that of the equinoctial colure; and this sidereal time of the star's transit ought to agree with the apparent right ascension which is obtained by computation from astronomical tables. In the Nautical Almanacs there are now given the computed right ascensions of one hundred principal fixed stars; therefore if the transittelescope be well adjusted in the plane of the meridian, and the movements of the clock-hands be perfectly equable, a comparison of the observed time of the transit of one of these stars with the computed right ascension, will immediately show the error of the clock by the difference which may exist between them. Hence the daily rate of the sidereal clock may be found by observing the transit of a star each night, or at intervals not exceeding five or six nights; in the latter case the difference, if any, being divided by the number of nights in each interval, will give the gain or loss of the clocktime with respect to true sidereal time during twenty-four hours. 305. Besides the time which is regulated by the diurnal rotation of the earth with respect to a fixed star, astronomers also employ, for the sake of simplicity in calculation, a diurnal rotation of the earth with respect to the sun as a measure of time. Now the earth being supposed to revolve about the sun with its true or proper annual motion, each of the intervals between the instants at which, successively, by the earth's rotation on its axis, the plane of a terrestrial meridian when produced passes through the centre of the sun, is considered as the length of the apparent solar day; and it is |