228 Longitude by Moo its Altitude. 



To reduce the altitude to the point P. — After correcting the 

 observed altitude of the limb for the refraction and applying the 

 apparent semidiameter,* we have the altitude u as affected by- 

 parallax, to correct for which we employ a horizontal parallax n x 

 obtained from the equatorial hor. par. n, by the formula 



sin.^r, =Bsin.7r (3) 



where B = -771 — • — r- -y~c > e e = -0065466 



<S{1 — * * SIB ** 9) 



and log. B is taken from a table with the argument q>. As this 

 table is not given in works on navigation, we may substitute for 



fi 



n x 



■f £f £^sill. 2 qp+ &C. =tt+- /\7t (4) 



The quantity A 7 * is very nearly the same with that which is 

 usually given as the u reduction of parallax/' so that we may 

 with a slight sacrifice of accuracy, obtain it from a table of re- 

 duction, observing however to add it to the hor. par. instead of 

 subtracting, as in the usual methods. The parallax in altitude is 

 then n } cos. « , whence a, = « + ^, cos. a . 



reduce 



We 



relations 



r, cos. S % —rcos.d 



r, sin. 8 1 *=rsin. 8-\-ai 3 



(5) 



where r = moon's distance from the center of the earth. 



r = " « " point P. 



at = distance of the point P from the center of the earth. 

 a = equatorial radius. 



* e sin, y 



• = v/(l^^sin.^) = ^ Bsin ^- 

 From these equations we find directly 



sin. (8 % —8)=z icos.8 1 = tsin.^cos.5, 



r 



or with sufficient accuracy 



8 } - 8 = 66 Trsin.ijpcos.^ (6) 



which is the correction to be added (algebraically, observing the 

 signs of cp and 8) to #, in order to reduce it to the point P. 



These computations are extremely simple and scarcely add to 

 the labor of the ordinary methods. As they are not based upon 

 rigorously exact formulae, however, it is proper to investigate the 

 degree of approximation which they give. 



The 



apparent aemidiameter will be found with sufficient accuracy for the pre 

 ent problem by the usual formula f'Mffl4»dbfc*ap.«j)t where ? and ?' are the 



values of the semidiameter for the center of the earth and the observer respectively; 



but if the smiara of flip nnrnllnv i« rpfniripd in the. series arwl thp rnrrmrpssion of the 



earth also taken into account, a small co 

 to G"-38, when the moon ia in the zenith. 



