230 Longitude by Moon's Altitude. 



whence (substituting also the value of d8 = i* cos. 8) 



i 7i cos. « cos. M 



d'k = - = 7- 



cos. (p sin. h 



Now 



k' = h+dh t h K =h-dh+d'h 

 h'-h x =2dh-d'k 



l n COS. « 



cos. « sin. gp - cos. S cos. M 

 cos. 8 cos. <p sin. h 



and if A = moon's azimuth, we have the relation 



whence 



M = sin. « cos. <p cos. A, 



sin. a COS. a COS. A 



1 cos. d sin. A 



which by means of the relation 



cos. « sin. A —cos. 8 sin. h 



is finally reduced to 



sin. a 



h! - h t =in . ; r - 



1 tan. A 



The maximum value of in = ee n sin. <jp=25" sin. <j>, so that 

 the error of the common method may be expressed by 



sin. « sin. qd 

 A 7 - A. =25".— A — 



1 tan. A 



This formula shows clearly enough that the common method 

 is too inaccurate to be generally employed, for the error in the 

 hour angle may even exceed 25", when the moon is within 45° 

 of the meridian or tan. A<1. In the problem here considered, 

 however, the error will always be much- less, since the observa- 

 tions will always be taken within 45° of the prime vertical, and 

 in many cases upon the prime vertical, when the error will be 

 nothing. This method then might be used in certain cases, but 

 it will be preferable to proceed by the method above given, which 

 is always precise, and requires very little additional labor. 



The method of determining the longitude by the moon's alti- 

 tude has not the advantage which distinguishes moon culmina- 

 tions, occultations, etc., of a direct comparison with similar ob- 

 servations in other places, or the same or nearly the same time, 

 but like the lunar distance, depends principally upon the accuracy 

 of the ephemeris. It must therefore rank as a subordinate 

 method, valuable chiefly to the traveller on account of its prac- 

 ticability with that " portable observatory," the sextant and arti- 

 ficial horizon* 



Annapolis, Bid, June, 1849. 





