NAVIGATION NAUTICAL ASTRONOMY. 



[LONGlTriiE TIME. 



the angle Z S P, because 8' being right angle, S 1 S 8' is 

 the complement of each ; and then-fore, from the rela- 

 otween the sides and angle* of a iphorical triangle, 

 we have 



sin. S" :sin. SZP : : sin. PZ :sin 1'S 



.'. sin. S'ain. PS -sin. PZ sin. 8PZ . . .(2) 



Substituting the second member of (2) in (1), wo there- 

 fore have 



a , 



~ 



8 8 



error i i 

 cos. Int. .sin. azimiith" 



. 

 "' ( ' 



Vlf. 3D. 



This expression for the error in time will obviously be 

 the least possible, when the sine of the azimuth is thu 

 greatest possible ; hence, if the co-declination be suffi- 

 i iently great, or the co-latitude sufficiently small for the 

 olject to oross the prime vertical 7, N, above the horizon, 

 in its progress towards the meridian or towards the hori- 

 zon, the most favourable time for an observation of the 

 altitude will be when the prime vertical is reached that 

 is, when the object is due east or due west the azimuth 

 being then 00. If other circumstances weather, prox- 

 imity to the horizon, &c. be unfavourable, then the 

 nearer the position of the object to the prime vertical the 



I'.nt if the co-latitude PZ exceed the co-declination 

 P s', then the object cannot arrive at the prime vertical ; 

 the azimuth P / N will then be the greatest, and have 

 the greatest aim-, when the object is at S, the point in 

 which the azimuth circle Z S N touches the parallel of de- 

 ,-ion ss'. It is plain that in this position the appa- 

 rrnt motion of the object is nearly perpendicular to the 

 horizon. (See Fig. 30). 



The triangle P S Z (Fig. 29) is right-angled at S, for 

 P S is the shortest arc from P to Z N, and is, therefore, 

 at right angles to Z N ; in this triangle the two sides P S, 

 P Z, the co-declination and the co-latitude, are supposed 

 to be given : we may, therefore, find the angle Z, or the 

 bearing the object ought to have, so that the error in 

 altitude may affect the time as little as possible ; or we 

 may find the angle P, 

 the time from apparent 

 noon, when the obser- 

 vation should be made. 

 If it be not practicable 

 to take the altitude 

 when the object is in 

 the most favourable po- 

 sition, then a position 

 as near to it as possible 

 should be chosen ; and 

 it must be remembered 

 that, throughout the 

 interval between the 

 best position and the 

 meridian, the nearer 

 the object is to the me- 

 ridian, the more unfavourably is it situated for the 

 purpose of computing the time from an altitude ; the 

 f -nuiila (3) sufficiently shows this. 



This formula (3) will of course serve for determining 

 the error in the time consequent upon any assumed error 

 in the altitude, at any observed azimuth ; but as the 

 angle S PS' is expressed in minutes of the equinoctial, as 

 appears from (1) above, it is necessary to divide by 15, 

 to bring the measure into minutes of time : so that 



Error of app. time in minutes - L error of altitu.l.- 



15 cos. lat. sin. azimuth 



For example. Suppose the latitude to be 51' 31' N.' 

 the azimuth of the object 8. 48 10 E , and that the 

 error of altitude is estimated at 10'. Required the error 

 in the apparent time. 



The above formula, put into logarithms, is, 

 log. error of time lo^. error of alt. - log. cos. lat. - 

 log. sin. azimuth - log. 15 + 20. 



10 1-0000 



po. 51" 31' Arith. Comp. -2000 

 ain. 48 10* Arith. Comp. 1 



15 Arith. Comp. 8 B 



i'-4:i8 -1577 



Hence the error in time is l'~438 1' L 

 It will, of course, be observed that if the latitude and 

 azimuth remain the same, as also the error in altitude, 

 the error in time will remain the same, however the alti- 

 tude may vary. 



TlMK AT SHIP DKDCCED FROM A SlfJGLB ALTITUDE. 



If the object observed be the sun, the hour-angle which 

 its circle of declination, at the instant of observation, 

 makes with the meridian, is the apparent time from that 

 meridian. By applying the equation of time, ^iven in 

 the Nautical A Imatiac, and reduced by means of the lon- 

 gjtude by account, to the estimated instant of observa- 

 tion, the apparent may be converted into mean time. 



If the object be a star, the hour-angle its circle of de- 

 clination makes with the meridian is a portion of 

 sidereal time ; it is the difference between the right as- 

 cension of the star, and the right ascension of the meri- 

 dian. When the star is to the west of the meridian, its 

 R. A. (right ascension) must be increased by the hour- 

 angle ; when it is to the east, its R. A. must be dimin- 

 ished by the hour-angle : the result is the R. A. of the 

 meridian ; and the difference between this and the sun's 

 K. A. at the time of observation, is the time from the 

 same meridian, and which is apparent, or mean time, 

 according as the sun's 11. A. is taken from p. 1 or p. 2 

 of the Naut. Aim. 



If the object be the moon or a planet, the apparent 

 and mean time aro obtained just as in the case of a 

 star. 



To determine the hour-angle.P from which the mean 

 time at the ship is thus deduced, there are given in the 

 spherical triangle P Z S, the co-latitude P Z, the co-de- 

 clination P S, and the co-altitude Z o ; hence, putting 

 for half the sum of these three sides, we have for 

 cos. 4 P. 



/ sin. s sin. ( Z S) (TBIGONOMETRY, 

 - * P = V sin.PZsin.FS~ P. 658). 



The co-altituJe Z S, used in working this formula, is 

 usually deduced from a set of altitudes taken within a 

 minute or two of each other, as in the following 

 examples : 



Examplet. 



1. Time deduced from the Sun. On September 23, 

 1845, in latitude 50 30' N., and longitude by account 

 110 W. , a set of altitudes of the sun were taken as 

 below ; the index correction was 3' 20", and the 

 height of the eye 20 feet. Required the mean time at the 

 ship. 



NOTE. Besides a chronometer or two a ship always 

 carries a good common watch, marking seconds, by 

 which, what may be called the time at ship by account, 

 or the estimated time, is kept. This may be regulated 

 by the meridian altitudes, or by the double altitudes, or, 

 as already noticed, by means of the chronometer ami 

 estimated longitude. For the purposes to which the 

 watch is applied, an error of a few minutes is of no con- 

 sequence ; in the present problem the estimated time is 

 used to get the declination and equation of time. Thu 

 times recorded below arc the mean times at the ship, as 

 shown by the watch. 



A Ititudes of the Sun. Times per Watch. 



11 4' 40" ... . 4h. 43m. 4_'s. 



11 2' 25" 4h. 44m. 35s. 



lo r.'.r 4.-," 4h. 45m. 24s. 



10 56' 30" 4h. 4Cm. 19s. 



4)44 3' 20" 

 11" 0- 60* 



4)1%. Om. Os. 

 4h. 46m. OB. 



