LATITUDE.] 



NAVIGATION NAUTICAL ASTRONOMY. 



1085 



the Litter being the case when S is below the pole (s 

 Fig. 26, at page 1078). Consequently 



cos. z cos. 



2 sin. HP E'PZBinTPS" 



cos, z-cos. z cos, zf -f- sin, z sin, z^ 

 sin. PZsin. PS 



If the difference z', between the meridian zenith dis- 

 tance and that actually observed, be so small that cos. 

 t! may be regarded as equal to 1, then we shall have 



sin. z sin. z 7 



.'. sin. z'=2 sin. PZ sin. PS cosec. z sin. 2 JP. 



The number of seconds in the arc z' is very nearly 

 equal to the number of times sin. z' contains sin. 1* : 

 consequently, No. of seconds in 



2 



sin. 1* 



sin. PZ sin. P S cosec. z sin. s i P 



=- p cos. lat. cos. dec. cosec. z sin. 2 \ hour-angle. 



To apply this formula, we must, of course, know z 

 approximately : this is deduced from the latitude by 

 account. By the aid of this approximate latitude and 

 the small hour-angle P, we may, therefore, discover what 

 correction z' must be applied to the zenith distance 

 actually observed, to reduce it to the meridian zenith 

 distance, from which the corrected latitude is easily ob- 

 tained, as in the examples already given. 



A result still more perfect may in general be arrived 

 at, by proceeding anew with this corrected latitude, 

 writing it in the formula in place of the latitude by 

 account, and thus getting a more accurate correction for 

 the reduction of the observed, to the true meridian zenith 

 distance. The additional work for this purpose will be 

 but trifling. 



The formula just deduced, furnishes the following rule 

 for deriving the latitude by aid of the latitude by account, 

 and from the observed altitude when near the meridian 

 of a celestial object whose declination is known. The 



number 5 '615455 is the logarithm of 



LATITUDE FROM AN ALTITUDE NEAR THE MERIDIAN, 

 THK DECLINATION, THE HOUR-ANCLE, AND THE LATITUDE 

 BY ACCOUNT. Rule 1 . Take the declination of the object 

 for the Greenwich time by account, and add it to the 

 latitude by account when they are of different names ; 

 otherwise, take the difference of the two ; the result is 

 the meridian zenith distance by account. 



2. If the object be the sun, the apparent time from 

 noon is the hour-angle : for any other object, add the 

 sun's right ascension to the apparent time, since preced- 

 ing noon. The difference between this sum and the 

 object's right ascension is the hour-angle. 



3. Add together the following logarithms : 



The constant logarithm 5 '615455, 



log. cosine of the latitude by account, 



log. cosine of tbe declination, 



log. cosine of the mer. zenith dist., as deduced 



from the two latter, 

 twice log, sine of half the hour-angle. 



The sum, rejecting the tens from the index, is the 

 logarithm of a number of seconds, which, subtracted from 

 the true zenith distance, deduced from the observation, 

 give* the meridian zenith distance. If this and the de- 

 clination are of the same name, their sum, otherwise 

 their difference, is the latitude, of the same name as the 

 greater. 



Examples. 



1. In latitude 48 12' N". by account, when the sun's 

 declination was 16 10' S., at Oh. 16m. P.M., apparent 



time, the sun's true zenith distance was 64 40' N. Re- 

 quired the latitude. 

 Constant log. ... ... 5-615455 



Latitude by acct. . . 48 12' N. cos. . 9-823821 

 Declination . . 16 10' S. cos. . 9 -982477 



/. Mer. zen. dist. acct. . 64 22' cosec. . 10-041995 

 Half hour-angle in deg. 2 0' 2 sin. . 17 '085638 



log. 2-552386 

 60)357* 



Reduction . . . 5' 57* 

 Zen. dist. from observation 64 4ff 0" 



N. 



True mer. zen. dist 

 Declination 



64 34' 3" N. 

 16 lO' 0"S. 



Latitude . . . 48 24' 3* N. 



This example is from Mr. Riddle's Treatise on Navi- 

 gation and Nautical Astronomy. We shall now solve it 

 anew by putting the latitude here deduced, in place of 

 the latitude by account. 



Constant log ; . . 5 '61 5455 



Latitude . . 48 24' 3* N. cos. . . 9-82211? 

 Declination . . 16 Iff 0" S. cos. . . 9-982477 

 Mer. zen. dist. . 64 34' 3* cosec. . 10 '044268 

 Half hour-angle 2 0* 0* 2 sin. . 17 '085638 



Reduction . . 355* log. . 2'549!:>5 



The difference between this and the former reduction 

 is only 2*, so that the corrected latitude is 48 24' 5" N. 



It may here be remarked, that as fractions of a second 

 are disregarded in the Reduction, the logarithms used in 

 finding it, need be taken out of the table only to the 

 nearest minute of each of the angular quantities. 



If the formulae marked (A), at page 1084, be applied 

 to the preceding example, the work will be as follows : 



Referring to Fig. 27, at page 1084, in connection with 

 the formulae (A), we have 



1. tan. P M = cos. hour ang. cotan. dec. 



cos. 4 0' . 9-908941 



cotan. . 16 Itf 



tan. P M 73 47' 45* 

 ZM . 64 36' 20* 



90 



. 10-537753 

 . 10-530699 



= 90 + lat., since lat. and dec. 

 have different names. 



48 24' 5*=lat. N., the same as before, 



2. oos. Z M = cos. P M cosec. dec. sin. alt . 



cos. . 73 47' 45" . . 9-445699 



cosec. . 16 10 7 . . 10-555280 



sin. . 25 2tf . . 9-631326 



cos. Z M 64 3ff 20* 



. 9-632305 



In this example the latitude is north, and the declina- 

 tion south, so that P M, in the general investigation of 

 the forinnltc, is in this particular case P' M, and there- 

 fore P 7 M + Z M 90 is the distance of Z above the 

 equinoctial ; that is, it is the latitude of the ship. 



In finding the latitude by the above formulas (A), it is 

 of course necessary to ascertain on which side of Z tho 

 foot M of the perpendicular from S falls ; that, is, whether 

 Z M is to be added to or subtracted from P M ; but 

 whether the correct co-latitude isPM ZMorPM 

 -f- Z M, can be matter of doubt only when Z M is so 

 small as to make it of little consequence which be taken. 

 But as the method by the rule is free from all ambiguity, 

 it is to be preferred when the object is near the meri- 

 dian. 



2. In latitude 50 50' N. by account, when the sun's 

 declination was 11 41' 58" N. at 12m. 3s. from apparent 

 noon, the sun's true altitude was 50 52' 29". Required 

 the latitude. Ans. Latitude, 50 47' 49" N. 



3. At 3iu 5m. 36s, apparent time, the aim's true alt;- 



