and the Methods of calculating their Residts. 345 



quantities referring to the equator become more troublesome 

 than the calculation of the longitude and latitude of the zenith: 

 in that case the ecliptic deserves the preference. The tables 

 give the longitude, latitude, and parallax of the moon for the 

 time T, and likewise their variations for the preceding and fol- 

 lowing-hours. These three longitudes and latitudes are to be 

 converted into right ascension and declination, if the equator 

 is to be used. This trouble is saved if we calculate with longi- 

 tude and latitude ; but, on the contrary, in using right ascen- 

 sion and declination we dispense with the calculation of the 

 longitude and latitude of the star and the zenith. If, there- 

 fore, in the case that the places of the moon in relation to the 

 equator are unknown, only two or three observations are to 

 be calculated, the ecliptic appears to deserve the preference. 

 If a single observation is to be compared with the epheme- 

 rides or the tables, it is more advantageous to calculate p' and 

 q' from the hourly variations of a, 8, 71-, than to derive them 

 from the three values of P and Q. In this case it will be 

 most convenient to assume for T, the time of the observation 

 reduced to the first meridian, by applying an approximate 

 value of the difference of meridians, and to calculate for this 

 moment P and Q, as also their differential quotients p' and q\ 

 by the formulae / 



'cosScos(a— A) da. sin § sin (a— A) dl p d <x 





dt at sin it dt u tana- n dt 



/'inV f /cos S sin D sin (a— A) da cos5cosD-|-sin2sinDcos(fls— A) d 

 I * \ &isinw dt w sin «• d 



d; 

 q d T \ 



a tang •*• dt / 



in which -r-, ■— -, -7— signify the hourly motions, and 00 the 



dt dt dt ° f J 



radius of the circle expressed in seconds. 



[9.] It now remains further to develope that part of formula 

 (6) which is dependent on the corrections of the elements of 

 the calculation. The result obtained by the method here ex- 

 plained, is not so much the difference of the meridian of the 

 place of observation, as the relation between it and the ele- 

 ments used in the calculation, and by the combination of se- 

 veral observations one 'or more of the elements of the calcula- 

 tion are eliminated, and the result is thus made partly or en- 

 tirely independent of the tables. 



In section [5] the quantities i and i' have been so assumed 

 as to give these equations : 



p' i — q'.i' = #A« + &A&-f-cA 7r +^A^ 3 

 q< i + p' i' - a' A « + #A8 + c'A tt + d'A e* 

 N. S. Vol. 8. No. 47. Nov. 1830. a Y A «, 



