ITJNAB DISTANCES.] 



NAVIGATION NAUTICAL ASTRONOMY. 



1103 



n few seconds may always be safely added to or subtracted 

 from the observed altitudes, if such a modification be 

 found to facilitate the calculation ; the seconds may, for 

 instance, be always made a multiple of 10. The observed 

 altitudes may, indeed, without occasioning any appre- 

 ciable error in the result, be taken to the nearest minute ; 

 and even an imperfect altitude, that may err from the 

 truth by so much as two or three minutes, may still be 

 em ployed, with safety, provided the time is not to be 

 computed as well as the distance. In most cases an error 

 of altitude to the extent of 10 minutes will affect the 

 resulting distance by less than that number of seconds. 

 It is also worthy of notice that the apparent distance 

 itself may be so modified as to be rendered free from 

 seconds, provided that when D is deduced, the seconds of 

 error, whether in defect or excess, be applied the reverse 

 way to D. 



But except in the merely approximative methods of 

 solution, seconds cannot be wholly dispensed with 

 throughout the work : the corrections for deducing the 

 observed altitudes to the true will always introduce them. 

 By means, however, of the tables already referred to, in 

 which the trigonometrical quantities are computed to 

 every ten seconds of the arcs or angle, the proper al- 

 lowance for the odd seconds may always be made with 

 but comparatively little trouble. 



Examples for Exercise. 



1. The apparent distance between the moon's centre 

 and a star is 64^ 36' 40" ; the apparent altitude of the 

 moon's centre 44 33' ; the apparent altitude of the star 

 ] 1 ">!' ; the true altitude of the moon's centre 45 15' 38* ; 

 and the true altitude of the star 11 46' 33'. Required 

 the true distance. Ans. D = 04 40' 14". 



2. The apparent distance between the centres of the 

 sun and moon is 108 14' 34" ; the apparent altitude of 

 the moon's centre 24 60' ; the apparent altitude of the 

 sun's centre 36 25'; the true altitude of the moon's 

 centre 25 41' 39"; and the true altitude of the sun's 

 centre 36' 23' 50'. Required the true distance. 



Ans. D = 107 32' 1". 



3. The apparent distance between the centre of the 

 moon arid a star is 51 28' 30* ; the apparent altitude of 

 the moon's centre 12 30' 4" ; the apparent altitude of 

 the star 24 48' 17" ; the true altitude of the moon's 

 centre 13 20' 40' ; and the true altitude of the star 

 24 46' 14'. Required the true distance. 



Ans. D = 51 y 48". 



4. The apparent distance between the moon's centre 

 and a star is 31 13' 26" ; the apparent altitude of the 

 moon's centre b 26' 13' ; the apparent altitude of the star 

 35 40' ; the true altitude of the moon's centre 9 20 7 45 ; 

 and the true altitude of the star 35 38' 49'. Required 

 the true distance. Ans. D = 30 23' 56". 



6. The apparent distance of the centres of the sun and 

 moon is 90 21' 17" ; the apparent altitude of the moon's 

 centre 6 17' 9 ; the apparent altitude of the sun's 

 centre 84 7 20' ; the true altitude of the moon's centre 

 6 9' 14" ; and the true altitude of the sun's centre 

 84 7' 15". Required the true distance. 



Ans. D = 89 29' 18". 



NOTE. The method of finding the true distance be- 

 tween the centre of the moon and a star, is the same as 

 that for the distance between the moon's centre and a 

 planet : in the case of a star, the true altitude is deduced 

 from the apparent altitude, by applying the correction 

 for refraction merely ; but in the case of a planet, a 

 correction may be necessary for the parallax in altitude. 

 Hut in general this correction may be neglected, as it is 

 n -n -illy too small to be of much importance. 



'MITTINO TUB ALTITUDES. It sometimes hap- 

 pens that, though circumstances may be favourable for 

 taking a lunar distance, yet the obscurity of the horizon 

 may present au obstacle to the observations for altitude; 

 in such a case the true altitudes will have to be com- 

 puted, and from these the apparent altitudes may be 

 deduced, by applying the corrections the contrary way. 

 To compute an altitude it is necessary to know the hour- 

 angle, or angular distance, of the object from the me- 



ridian. If the object be the sun, this angle is tha 

 apparent time for the meridian ; and as the altitude, as 

 already remarked, is not required with precision, tbe 

 estimated time at the ship will answer for the purpose. 

 If the object be the moon or a star, the sun's right ascen- 

 sion, at the instant, must be increased or diminished 

 by the apparent time, according as this time is P.M. or 

 AM., to get the right ascension of the meridian ; the 

 difference between this and the right ascension of the 

 object, is the hour-angle or the meridian distance of the 

 object. The hour-angle being thus found, a formula for 

 the altitude may be investigated as follows : 



Let P be the hour-angle, I the co-latitude, z the co- 

 altitude, and p the polar distance or co-declination. 

 Then the fundamental formula of spherical trigonometry 

 gives, 



_, COS. 2 -COS. I COS. p 

 COS. P = -- ; - i : - 



sin. I sin. p 



.'. cos. 2 = cos. I cos. p -f- sin I sin. p cos. P ; 

 but cos. P=l-2sin. 2 P 



.'. cos. z = cos. I cos. p -f- sin. I sin. p 2 sin. I 



sin. p sin. 2 P 

 .'. cos. z cos. (I t p) 2 sin. / sin. p sin. 3 P . . . (1) 



By means of these tables of natural and logarithmic 

 sines and cosines, the co-altitude z may be obtained from 

 this formula ; but to adapt it wholly to logarithms, let 1 

 be added to each side of the equation ; then remembering 

 that 1 + cos. A = 2 cos. 2 $ A, we have, upon dividin" 

 by 2, 



cos. 2 $ z = cos. 3 % (I co p) sin. I sin. p sin. HP 

 = cos. 2 i (I m p) j 1 sin. I sin. p sin. ! $ P 



X sec. * 



p ) 



1'ut cos. ! M for the expression within the braces ; then 

 for computing the altitude a, that is, the complement of 

 z, we have the following formula namely, 



sin. M=sin. i P pec. ^ (I </> p) J sin. I sin. p ) 

 Bin. (a -f 90)=cbs. (I c/> p) cos. M j 

 or, which is perhaps a little more convenient, 



sin. iP . . 



Sm - M= c^W^p) ^ sin ' l 8ln ' f] ..... (2) 

 sin $(a + 90 3 ) = cos. (I v, p) cos. M ) 



Suppose, for example, that by means of the estimated 

 time at the ship, and the longitude by account, the 

 moon's angular distance from the meridian is found to bo 

 33 SO 1 , at the time of taking a lunar distance in latitude 

 38 14', the moon's co-declination reduced to the time, 

 being 64 13' 13". Required the apparent altitude by 

 computation, the obscurity of the horizon preventing au 

 observation. Working by the formula (2) the operation 

 is as follows, where P=33 30'p = 04 13' Z=51 46': 

 J5140' . . i sin. 4-947572 

 p 64 13' . . J sin. 4 977228 

 iP1645' . . " sin. 9-459088 

 (l <^p) 6 13^' . comp. cos. -003503 . cos. 9-997432 



sin. 9-38705C . cos. 9-986683 



i (a + 90) = 74 36' 

 2 



sin. 9 934115 



/. a=59 12' the true altitude 

 cor. for pnr. and refraction 23' 



68 44' the apparent alt. 



It is obvious, that in strictness, -the correction for 

 parallax and refraction should be taken out of the tables 

 in accordance with the apparent altitude, not the true 

 altitude : the correction above, therefore, belongs to an 

 altitude somewhat too great, so that only an approximate 

 apparent altitude is in reality deduced from the true 

 altitude. The proper correction may, however, be found 

 by again entering the table with this approximate ap- 

 parent altitude ; we shall thus get 27' 64" for the curreu- 



