Longitude by Moon's Altitude. 229 



The expression for the normal (a,) or the distance of the ob- 

 server from the point P is 



a 



1 \/{\ -ee sin. 2 g>) 

 a » a 



whence — = Bx-> ot nearly sin. *, =B sin. n as in formula (3). 



Strictly sin. 7T is not equal to — > but to — • To estimate the 



error we hare from the equations (5), 



r f =r+ai sin, <$ 

 whence 



a. a. a. a 



8111.*,= — = — - -+-- = — = (l--i sin. <J-&c.) 



1 r, r-fatsm. £ r v r y 



or with extreme precision, 

 | n x ~{n) -yr 2 isin. <?sin. I 77 , 



so that (tt) found by (3) is in error by the quantity 7r 2 isin. 8 

 sin. 1", the value of which for *«61! s <? = 29° is only 0''-21 sin. <*>. 



To find the error of formula (6) in which the factor B is as- 

 sumed equal to unity, we easily find that the maximum value of 

 3 , - 8 is about 25", and the value of B lies between 1 and 1*0033, 

 so that the maximum error in (J, - S is 25" x '0033 or 0"08. 



>/ 



In this 



method the hor. par. is reduced by the quantity A* of formula 

 (6) and the geocentric declination and geographical latitude are 

 employed in the formula (1). Let « be the altitude that would 

 be obtained by neglecting the compression of the earth, that is, 

 employing the equatorial hor. par. * without reduction ; «' that 

 which is obtained by employing ji-A 71 , and «, that which is 

 obtained as above, by employing 4A". Let h, h' and A, be 

 the corresponding hour angles. Differentiating (1) taking k and 

 « as variables, we find 



d« COS . « 



dh 



cos. d cos. <p sin. h 



But «'=«- A^cos. «; therefore substituting da= -A* cos. « 



\ 



i n cos. 2 « tan, y 

 rfA = T^^mTX' 



6* tUOi v Qiii* '* 



Again, differentiating (1) taking h and $ as variables 



cos. 8 sin. <p -sin. <* cos. <r cos. A 



d7t = dS \ bos.*cc».f8in.i; 



If M = the angle at the moon included by the vertical and de- 

 clination circles, we have the relation 



cos. 8 sin. <jp — sin. <* cos. <p sin. h =cos. « cos. M 



