Longitude by Moon's Altitude. • 227 



horizon, it admits of considerable accuracy, and properly compu- 

 ted may be rendered quite as correct as lunar distances. Indeed 



can 



more correctly than they can measure the distance of the moon 

 from a star, it may even prove superior to the " lunar." With a 

 view to this application, I propose to point out a slight inaccuracy 

 in the common method of computing the hour angle of the moon 

 from the altitude, to investigate the amount of this error, and to 

 give a more correct process. 



The outline of this method is as follows. With the true alti- 

 tude («), the declination (8) and the geographical latitude (<?), the 



(A) 



sin. « -sin. 8 sin. $> 

 cos. h= * 



COS. COS. q> 



(1) 



/cos. £>-f-<r + «)sm.£(p + T-") 



or sin.4/i=\/ : 



^ v sin. jt? cos. q> 



where ^> =, 90° - 5. 



Then from the known local sidereal time (T) we have the moon's 



right ascension by the formula R=T±A; and corresponding to 



this right ascension, the ephemeris gives the Greenwich time of 



the observation, and consequently the longitude. 



Now in order to compute the formula (1) correctly, it is neces- 

 sary to allow for the compression of the earth, and in every work 

 upon practical astronomy that I have seen, we are directed sim- 

 ply to reduce the horizontal parallax by S quantity that is propor- 

 tional to the diminution of the earth's radius at the latitude of the 

 observer, and with this reduced parallax to correct the altitude by 

 the usual methods. If the vertical line of the observer passed 

 through the center of the earth, this process would obviously be 

 correct, but the earth's radius and the vertical, making in general 

 an angle, it involves an error in all cases except when the ob- 

 server is on the equator, (or at the pole,) or when the moon is on 

 the prime vertical. The amount of this error will be estimated 

 by means of the accurate process, which follows. The principle 

 of this process is familiar to astronomers, from the use made of it 

 by Bessel, but it does not appear to have been applied to this 

 problem. 



Let the altitude be reduced to the point in which the vertical 

 intersects the axis of the earth, which for brevity we may desig- 

 nate as the point P ; and let the moon's declination be reduced to 

 the same point. Since this point is in the axis, these reductions 

 do not affect the hour angle ; and since the zenith is not changed, 

 w e still employ the geographical latitude, so that we shall have 



sin.«,-sm.<J, sm.f 



cos./i, = 7~TXTZ, 



' COS. d , COS. if 



(2) . 



in which « , and <5, are the altitude and declination reduced to the 



point 



