that altitude, should be registered: the instrument should then be laid aside till the afternoon, when the times must be marked at which the lower limb, the centre, and the upper limb have, respectively, that same altitude. Now, if it were not that the sun changes his declination in the interval between the observations, half the interval, in time, between each pair of observations, added to the time of the morning observation for that pair, would express the hour, by the watch, when the sun was on the meridian of the station: but the declination does change, and a correction of the half interval must therefore be made on that account, as in the following problem. PROB. VI. To determine the hour of the day and the error of a watch by equal altitudes of the sun. Let the primitive circle NQD represent the horizon of the observer; ABC the parallel of the sun's altitude, and ABD a parallel of declination supposed to pass through the sun when on the meridian of the station. Then, supposing the time of year to be between mid-winter and mid-summer, s and s' (the former below, and the latter above the parallel circle BAD) may represent the places of the sun's centre at the two times of observation. Let the dia D B N D' meter NQ be the meridian, in which z, the centre of the primitive, is the zenith, and P the pole of the equator; PZ being equal to the colatitude of the station: imagine the hour circles to be drawn from P as in the figure through A, S, B, S', and Pm to be drawn bisecting the angle SPS'; the meridian NQ evidently bisects the angle APB. Now ts and t's', in the figure, represent respectively the increase of the sun's declination from the time of the morning observation till noon, and from noon till the time of the afternoon observation: these changes may be considered as equal to one another, because the intervals of time are nearly equal, and, for a few hours, the sun's declination varies almost uniformly; therefore sts't'. The angles SAt and S'B' are equal to one another, and those at t and t' are right n m Q angles; consequently At Bt, and the angle APSBPS': adding SPB to both of these angles, APB SPS', or taking the half of each, APZ-SPM; and subtracting the common angle SPZ, the angle APS=ZPm. This last angle is manifestly the correction which, in the present case, must be subtracted from the mean of two corresponding times of observation (at which middle time the sun is on the circle Pm), to give the hour which the watch would show when the sun is on the meridian PZ of the station. But the time which a correct watch would show when the sun is on the meridian is equal to twelve hours equation of time (mean time of apparent noon); consequently, comparing the time of noon found by the watch from the observation, with this last, the difference will be the error of the watch. 346. The relation between the angle APS or ZPM and the variation st of the sun's declination between noon and the time of either observation may be found in the following manner: On substituting Z, P and s for A, C and B respectively, in the equation preceding (a) in art. 60., and transposing, we have in the triangle ZPS, cos. zs cos. ZPS sin. PZ sin. PS + cos. PZ cos. PS: differentiating, considering PZ and zs as constant, o = - sin. ZPS dzPS sin. PZ sin. PS+cos. PS dps sin. PZ cos. ZPS cos. PZ sin. Ps dps (1), cos. PS sin. PZ cos. ZPS cos. PZ sin. PS or dzPs dps sin. PZ sin. PS sin. ZPS =des (cotan. PS cotan. ZPS — cotan. PZ cosec. ZPS). And by this formula Mr. Wales' tables of equations to equal altitudes were computed. Here PS is the sun's polar distance at the morning observation, and the angle ZPS may be found in degrees on multiplying by 15 half the interval in time between the observations. But we have (art. 60., (a), (b) or (c)), then, putting cos. PZ sin. PS in the form and subtracting the last of these equations from the first, we shall have for the equivalent of the numerator in the first value of dzPS above, cos. ZS cos. PS. COS. PZ sin. PS But (art. 60.) this fraction is evidently equivalent to therefore we obtain -cos. PSZ sin. ZS; The negative sign indicates that if PS increase, the angle ZPS diminishes, or if the declination increase or decrease, the hour angle increases or decreases. In using equal altitudes of a fixed star for the purpose of finding the time at night, no correction is required on account of the change of declination; and the mean between the two times of observation must be compared with the computed mean time at which the star culminates, in order to have the error of the watch. 347. If the sun's azimuths were observed at the two times when the altitudes are equal to one another, the position of the true meridian, on land, might be accurately found; but it would be necessary, previously, to find the change produced in the azimuth by the change of the sun's declination in the interval between the observations. Now, in the preceding figure, the angle szs' will represent the sum of the observed azimuths if the places of the sun in the morning and afternoon be s and s'; therefore, substituting P, Z and s for A, C and B respectively in the equation preceding (a) in art. 60., we have cos. PS cos. PZS sin. ZP sin. zs + cos. ZP cos. ZS; and differentiating, considering PZ and zs as constant, sin. Ps dps sin. PZs dezs sin. ZP sin. zs; Thus st, or the change in the sun's declination between noon and the time of either observation, being the value of drs; and the angle PZS (or its supplement szQ, in the figure) being considered as half the sum of the observed azimuths, we obtain the corresponding value of dPZs, which is repre sented by Q zn in the figure. This angle being subtracted from s zn, or half the sum of the observed azimuths, when the distance PS is diminishing, leaves the angle szą for the bearing of the sun from the true meridian at the time of the morning observation: consequently, the place of the observer being vertically under z, if a picket were driven in the ground in the direction zs at the time last mentioned, the angle szQ being subsequently set out with a theodolite, the line sq would be the direction of the true meridian. The most favourable season for determining the time, or the direction of the meridian, by equal altitudes, is at or near the solstices, when the changes of the sun's declination are so small that they may be disregarded. Ex. Sept. 4. 1843, at Sandhurst there were observed the times shown by a watch in the morning and afternoon when, by reflexion from mercury, the upper limb, the centre and the lower limb of the sun had equal altitudes, as follow: 11 57 58.5 11 57 57 Approximate time of noon 11 58 0.5 If Wales's tables of equations to equal altitudes be used, the sun's longitude (= 161° 16′ or 5s. 11° 16') must be taken from the Nautical Almanac, and the natural tangent (= 1.25) of the known, or estimated latitude (51° 20') of the station must be found: then, from Table I. we have, for the corrections corresponding to the first half interval,+ 15′′.05 × 1.25 (+18".81); to the second,+ 15".03 x 1.25 (= +18".79), and to the third, + 15′′.0 × 1.25 (= + 18.75). The corrections from Table II. are respectively, - 1.58, 1".59 and hence the complete corrections are - 1".6: + 17′′.23, + 17".2 and + 17”.15. These being applied, in order, to the approximate times of noon, give, for the mean times of noon by the watch, 11ho. 58' 17".73, 11ho: 58' 17".7 and 11 ho. 58' 14".15; and the mean of these is 11ho. 58′ 16′′.53. Now the equation of time (0′ 56′′.77) being subtracted (as the title in the Nautical Almanac directs) from 12 hours, gives 11ho. 59' 3".23 for the true mean time of apparent U noon therefore, taking the difference between this and the time found from the observation, the watch appears to be 46".7 too slow. Computation of the correction to the approximate time of noon by the above formula (11). At the time of making the observations, the index of the sextant gave for the apparent double altitude of the sun's upper limb, centre, and lower limb, 74° 11'45"; the index error being 45" subtractive. Then, on applying the corrections for the index error, the sun's refraction and his parallax in altitude (if the altitude of the sun's centre be required, the correction for the semidiameter is not, of course, to be applied), the correct altitude was found to be 37° 4' 20"; consequently his zenith distance = 52° 55′ 40′′: these values relating to the sun's centre, they belong to the middle time (9ho. 35' 7") above, and this being considered as correct mean time, the corresponding apparent time (adding the equation) is 9ho. 36′ 4′′, or 2 ho. 23′ 56′′ before noon, at the place. But the latter being distant in longitude from Greenwich 3 min. (in time) westward, the apparent Greenwich time is 9ho. 39' 4", or 2 ho. 20' 56" from Greenwich noon. Sun's declination at apparent noon Greenwich = = 20 44.2 N. (Na. Al.) 2 10.2 Then (art. 61.), in the spherical triangle ZPS (fig. to the prob.), log. sin. zs (52° 55′ 40′′) co. ar. 0.098064 log. sin. zPS (2 ho. 22′ 51′′ or 35° 42′ 45′′, the half interval) 9.766193 log. sin. PZ (38° 39′ 27′′ the colatitude of the station) log. sin. Psz (the angle of position) = 27° 11′ 40′′ 9.795654 9.659911 The variation of the sun's declination in 2 ho. 23′ 56′′ is equal to 133" (= drs); therefore, by substitution in the formula (11), The day of the observation being between mid-summer and mid-winter, this correction is to be added to the middle |