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time, or approximate time of noon, in order to give the mean time of apparent noon by the watch.

348. The problem for determining the error of the watch by equal altitudes presents some difficulties when put in practice at sea on account of the displacement of the ship during the interval between the observations: yet, as a means of approximating to the error of the watch, it may sometimes be convenient to employ it, and in so doing the error due to the variation of the sun's declination is usually neglected. In order to ascertain the correction on account of the displacement of the ship, let z be the

zenith of the latter when the sun is at s, that is when the first observation is made: at the second observation let z' be the zenith, and let the sun be at s'; then the interval in time between the observations is expressed by SPS'.

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In the figure, the ship is supposed to sail towards the sun, in which case the zenith of the ship approaching the luminary, the altitude of the sun does not diminish so rapidly as if the ship had remained at Z, or had changed its longitude without changing its latitude; consequently the motion of the ship in the direction zz' must continue longer, before the sun descends to an altitude equal to that of the morning observation, than if the latitude had not changed. Let z" be the place of the zenith at the time of the second observation if the latitude should not change; then s", the place of the sun at that time, will be so situated that S'PZ" SPZ. Bisect SPS" or ZPZ" by PM; then the middle time, or the mean of the times when the morning and afternoon observations are in that case made, is the time by the watch, of apparent noon for the meridian PM, on which, at that middle time, is the zenith of the ship. Here no correction of the middle time is neces

sary.

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But the ship changing its latitude, and its zenith being at z' when the sun is at s', the angle z'PS' (in consequence of PZ' being greater than PZ") will exceed z"PS" by an angle S"Ps', which represent by dr. If then SPS' be bisected by PN we shall have MPN dP; and this is the correction to be subtracted (in the above figure) from the mean between the times of observation, in order to give the time, by the watch, of apparent noon for the place of the ship at the middle of the interval, By differentiating the equation for cos. zs (art. 346.), considering PS and zs as constant, and proceeding

as in that article, there will be obtained a formula for dr (dzPs) corresponding to that in equation (II); or the latter may be transformed into the required formula on substituting in it PZ for PS, and PZS for PSZ.

349. The determination of the error of a watch regulated according to mean solar time, by equal altitudes of a fixed star or of a planet, may be made in a manner similar to that which has been above explained. With a fixed star, which does not change its declination sensibly in a few hours, half the interval measured by the watch is to be added to the time of the first observation, and the sum will be the instant at which the star culminates, in solar time. With a planet, the variation of the hour angle, in consequence of the change of declination, must, when accuracy is required, be found and applied as a correction to such sum in order to have the time, by the watch, when the planet was on the meridian; and in either case, this time must be compared with the computed solar time (art. 313.) of culmination in order to obtain the error of the watch. The method of observing equal altitudes of a star on opposite sides of the meridian is, however, not often put in practice at sea, in consequence of the officer who should make the observations not being on duty in an evening and on the following morning; and, in order to obtain the end, both Capt. Thomson and Mr. Riddle have proposed to observe equal altitudes of a star on different nights, on the same side of the meridian. Now, a fixed star attains equal altitudes on the same side of the meridian, at any one station, at intervals equal to one sidereal day, or at intervals less than one solar day by 3'55".9; hence, if the difference between the two consecutive times at which a star attains the same altitude, at any place, on the same side of the meridian, be less than 3′55′′. 9, the watch will have gained; if more, it will have lost the excess, in a sidereal day. Therefore, if the observed error be multiplied by 1.0027, the result will be the error, or daily rate, in a mean solar day. Mr. Riddle has also shown in the Memoirs of the Astronomical Society (Vol. III. pt. 2.) that the formula for correcting the mean time between the observations on account of a difference (caused by a change of temperature, &c.) between the refractions at the two times of observation is where a is an observed change in the double altitude of a celestial body in a time t (a few minutes), and r is the difference between the refractions at the two times of observation. The correction is additive when the refraction at the

tr

a

,

24

23.93

or by

second observation is the smaller of the two, and subtractive when it is the greater, if the object be east of the meridian; the contrary, if west.

350. When it is impossible to obtain by observation the meridional altitude of a celestial body, the latitude of a station or ship may be determined by means of an observation made when the body is at a distance from the meridian; and an altitude of the sun, the moon or a star, with a knowledge of the time at the station, or of the azimuth of the celestial body, together with the declination of the latter, will suffice for the purpose.

PROB. VII.

To find the latitude of a station or ship by means of an altitude of the sun observed at a given time in the morning or afternoon.

Let it be supposed that the usual corrections have been applied to the observed altitude, so that the true zenith distance of the sun's centre is found, and that the time of the observation is given by a watch whose error is known, so that the apparent time of day, and consequently the sun's hour angle, in degrees, is ascertained: also that the longitude of the station is known, at least approximatively; and therefore, that the sun's declination can be found.

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In the diagram, the primitive circle PZQ represents the meridian of the station, and P is the pole of the equator. The angle ZPQ is made equal to the sun's hour angle, and about P as a pole the parallel Dd of declination is described: its intersection s with the circle PSQ is the place of the sun.

About s as a pole,

the circle zaz' is described at a distance equal to the sun's distance from

Q

d

the zenith: therefore z or z' is the zenith of the station, and PZ or PZ' is the required colatitude. The case is ambiguous unless the place of the sun with respect to the prime vertical be also known.

If st be let fall perpendicularly on PZQ there will be formed the two right angled spherical triangles Pst and zst; and in the triangle P St,

Rad. cos. ZPS = cotan. PS tan. Pt (art. 62. (f')), also Rad. sin. st sin. PS sin. ZPS (art. 60. (e) ), again, in the triangle z st,

Rad. cos. zs cos. t s cos. zt (art. 60. (d) ).

In using these rules, six logarithms must be taken from the tables, and the method requires that the apparent time at the station should be known with considerable precision. A similar process must be used when the altitude of a planet or fixed star is observed at a distance from the meridian.

If the latitude of a ship or station were already known approximatively, and such approximation is obtained by the dead reckoning, a more correct value of it might be easily obtained by means of the equation

or

vers. sin. ZPS=

COS. ZD-cos. ZS
sin. PS sin. PZ

(art. 337. (c)),

vers. sin. ZPS sin. PS sin. PZ cos. ZD-cos. ZS; for the angle ZPS being given, in time, the number corresponding to it in the table denominated log-rising (Requisite Tables, tab. XVI.) is equivalent to the logarithm of the second member in the first of these equations: let this number be represented by M; then the natural number corresponding to M+ log. sin. PS + log. sin. Pz will be equal to cos. ZDzs: let N be this natural number; then

N+nat. cosin. zs nat. cosin. ZD,

COS.

and the last term is the natural cosine of the sun's meridional zenith distance: therefore PD-ZD or PS-ZD is the required colatitude. For Pz in the value of N there must be put the given approximate value of the colatitude.

Ex. 1. April 21. 1842, at 9 ho. 50' 11".5 by the watch, the double altitude of the sun's upper limb, by reflexion from mercury, was found at Sandhurst to be 86° 45′ 40′′: the index error of the sextant was 6' 40" subtractive, and the watch was 3′ 39′′.2 too slow.

On applying the corrections for the index error, refraction, and the sun's parallax in altitude, and subtracting the sun's semidiameter, the correct altitude of the sun's centre was 43° 2′ 39′′; consequently the zenith distance = 46° 57′ 21′′.

N.B. The longitude of the place being by estimation 3' (in time) west of Greenwich; the time of the observation from apparent noon at Greenwich is 2 ho. 1' 52".

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PZ=38 39 57, the required colatitude of the station.

By the rule in the Requisite Tables; having the latitude by account, or having an approximate knowledge of the latitude, the process is as follows:

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The sun's hour angle in time, as above (= 2 ho. 4′52′′) log. rising=4.16073 log. sin. PS (78° 12′ 45′′)

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log. sin. Pz (approximatively, suppose 38° 30')

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Ex. 2. At Sandhurst, November 17. 1843, there was observed, by reflexion from mercury, the double altitude of a Polaris, equal to 105° 31' 10", the index error of the sextant being 50" additive: at the same instant the time given by the watch was 10 ho. 44′ 48′′ P. M., the error of the watch being 1' 36" too slow.

After applying the corrections for the index error and the star's refraction, the true altitude of the star was found to be 52° 45′ 16′′; consequently its true zenith distance was 37° 14′ 44′′.

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