242 Mr. Baily on the Computation of 



series, for every hour of the day. The table, however, which 

 he has computed, being frequently found too limited for ge- 

 neral use, I have here enlarged it, by calculating the terms of 

 the series for every ten minutes during the clay; whereby 

 the requisite quantities for any intermediate time may be taken 

 out almost on inspection ; or at least, in the most essential 

 points, with very little trouble: since it will seldom be found 

 necessary to interpolate, except in the values given in the co- 

 lumn B. 



M. Bessel's formula is as follows ; viz. make <7, b, c 9 d, e 

 respectively equal to the first difference, the mean of the two 

 middle seconcl differences, the third difference, the mean of 

 the two fourth differences, and the fifth difference of the moon's 

 right ascension, as taken from the Nautical Almanac. Then 

 will 



2 





, 



n\ 

 "ISO" 



denote the moon's semi-diurnal motion in right ascension 

 corresponding to that fractional part of the twelve hours, from 

 the preceding noon or midnight, indicated by n : and which 

 must always be assumed equal to the middle point of time 

 between the two observations. Or, preserving the same value 

 of c and x as are adopted in my Memoir above mentioned, 

 the value of n must be assumed equal to J (c + x). If there- 

 fore we express the co-efficients of b, c, d, e by the letters 

 B, C, D, E, respectively, the semi-diurnal motion (M) of the 

 moon in right ascension will be denoted by 



M = a + Rb + Cc + Dd + Ee. 



The following table contains the logarithms of B, C, D, E, 

 for every ten minutes of the twelve hours from the preceding 

 noon or midnight, as above mentioned; to which must be 

 added respectively the logarithms of b, c, d, e; the natural 

 numbers thence resulting, being added to the first difference, 

 will give M, or the semi-diurnal motion required. And, re- 

 taining the value of the symbols x> s and A> as adopted in 

 the Memoir above quoted, the true difference of longitude 

 will be 



which is (I believe) the most simple form in which the general 

 solution of the problem can be at present expressed : but, 

 when the new Nautical Almanac appears, 7%s will be a con- 



stant 



