known, serves for the conversion of mean time into apparent, and the reverse. 322. To find the equation of time. The hour angle of the sun (p. 11, def. 16) varies at the rate of 360° in a solar day, or 15° per solar hour. If, therefore, its value at any moment be divided by 15, the quotient will be the apparent time at that moment. In like manner, the hour angle of the mean sun, divided by 15, gives the mean time. Now, let the circle V S D (Fig. 48) represent the equator, V the vernal equinox, M the point of the equator, which is on the meridian, and V S the right ascension of the sun, and we shall have, Again, if we suppose the circle V M D to represent the mean equator, V' the mean equinox, and S' the position of the mean sun, (V S' being equal to the mean longitude of the sun,) we shall have, V M — V S' _ V M—(V'S' + V V') . mean time M S' = or, the equation of time is equal to the difference between the sun's true right ascension and mean longitude, corrected by the equation of the equinoxes in right ascension, and converted into time. A formula has been investigated, and reduced to a table, which makes known the equation of time by means of the sun's mean longitude as an argument. (See Table XII.) The value of the equation of time at noon, on any day of the year, is also to be found in the ephemeris of the sun, published in the Nautical Almanac and other works. If its value for any other time than noon be desired, it may be obtained by simple proportion. The value of the equation of time, determined from formula (74), is to be applied with its sign to the apparent time to obtain the mean, and with the opposite sign to the mean time to obtain the apparent. 323. The equation of time is zero, or mean and true time are the same four times in the year, viz: about the 15th of April, the 15th of June, the 1st of September, and the 24th of December. Its greatest additive value (to apparent time) is about 14 minutes, and occurs about the 11th of February; and its greatest subtractive value is about 16 minutes, and occurs about the 3d of November. 324. To convert sidereal time into mean time, and vice versa. Making use of Fig. 48 already employed, the arc V M, called the Right Ascension of Mid-Heaven, expressed in time, is the sidereal time; V S' is the right ascension of the mean sun, estimated from the true equinox, or the mean longitude of the sun corrected for the equation of the equinoxes in right ascension (Art. 322); and M S' expressed in time, is the mean time. Let the arcs V M, M S' and V S', converted into time, be denoted respectively by S, M, and L. Now, or, VM = MS' + V S'; S = M + L.. (75); and M = S — L . . (76). If M+L in equation (75) exceeds 24 hours, 24 hours must be subtracted; and if L exceeds S in equation (76), 24 hours must be added to S, to render the subtraction possible. This problem may in practice be solved most easily by means of an ephemeris of the sun, which gives the value of S, or the sidereal time at the instant of mean noon of each day, together with a table of the acceleration of sidereal on mean solar time, and the corresponding table of the retardation of mean on sidereal time. 325. The conversion of apparent time into sidereal, or sidereal time into apparent, may be effected by first obtaining the mean time, and then converting this into sidereal or apparent time, as the case may be. Determination of the Time and Regulation of Clocks by 326. The regulation of a clock consists in finding its error and its rate. 327. The error of a mean solar clock is most conveniently determined, from observations with a transit instrument of the time, as given by the clock, of the meridian passage of the sun's centre. The time noted will be the clock time at apparent noon, and the exact mean time at apparent noon may be obtained by applying to the apparent time the equation of time with its proper sign, which may for this purpose be taken from the Nautical Almanac by simple inspection. A comparison of the clock time with the exact mean time will give the error of the clock. 328. The daily rate of a mean solar clock may be ascertained by finding as above the error at two successive apparent noons. If the two errors are the same and lie the same way, the clock goes accurately to mean solar time; if they are different, their difference or sum, according as they lie the same or opposite ways, will be the daily gain or loss, as the case may be. 329. To find the error of a sidereal clock, compute the true right ascension of some one of the fixed stars, (see Prob. XXI,) and note the time of its transit; the difference between the time observed and the right ascension in time will be the error. The error of the daily rate is determined by observing two successive transits of the same star. The variation of the time of the second transit from that of the first will be the error in question. The error and rate may be determined more accurately from observations upon several stars, taking a mean of the individual results. Stars at a distance from the pole are to be selected, for reasons which have been already assigned. 330. In default of a transit instrument, the time may be obtained, and time-keepers regulated, by observations made out of the meridian. There are two methods by which this may be accomplished, called, respectively, the method of Single Altitudes, and the method of Double Altitudes or of Equal Altitudes. These we will now explain. 1. To determine the time from a measured altitude of the sun, or of a star, its declination and also the latitude of the place being given. = Let us first suppose that the altitude of the sun is taken ; correct the measured altitude for refraction and parallax, and also, if the sextant is the instrument used, for the semidiameter of the sun. Then, if Z (Fig. 13) represents the zenith, P the elevated pole, and S the sun; in the triangle ZPS we shall know ZP co-latitude, PS co-declination, and ZS= co-altitude, from which we may compute the angle ZPS (= P), which is the angular distance of the sun from the meridian, or, if expressed in time, the time of the observation from apparent noon, by the following equations (App., Resolution of oblique-angled spherical triangles, Case I), = 2 k=ZP+PS+ZS co-lat. + co-dec. + co-alt. . . . – P S) ... (78). sin (k-Z P) sin (k - PS) sin Z P sin PS (77). sin P= = The value of P being derived from these equations and converted into time (see Prob. III), the result will be the apparent time at the instant of the observation, if it was made in the afternoon; if not, what remains after subtracting it from 24 hours will be the apparent time. The apparent time being found, the mean time may be deduced from it by applying the equation of time. A more accurate result will be obtained if several altitudes be measured, the time of each measurement noted, and the mean of all the altitudes taken and regarded as corresponding to the mean of the times. The correspondence will be sufficiently exact if the measurements be all made within the space of 10 or 12 minutes, and when the sun is near the prime vertical. If an even number of altitudes be taken, and alternately of the upper and lower limb, the mean of the whole will give the altitude of the sun's centre, without it being necessary to know his apparent semi-diameter. In practice, the declination of the sun may be taken for the solution of this problem from an ephemeris of the sun. For this purpose the time of the observation and the longitude of the place must be approximately known.. Example. On the 1st of June, 1838, at about 10h. 45m. A. M., the altitude of the sun's lower limb was measured at New York with a sextant, and found to be 64° 55' 5". What was the correct time of the observation? Measured alt. of sun's lower limb, 64° 55' 5" 15 48 |