Astronomical and Nautical Collections, 



289 



Moon's A. R. July 4th, Midnight 

 Proportional part of first diff. 

 Equation of mean second diflf. 

 Equation of third difif. 

 Equation of fourtli diflf. 



Moon's A. R. July 5th, at 18 hours 



The formulse from which the Tables were constructed are thus 

 investigated. 



Let a be the first of the series of six terms given in the Ephe- 

 meris, and rf', d^^ d^, d'* and d^, be the first terms of the several 



orders of differences. 



(A.) 

 Then by the usual formula of interpolation, the value of any 

 term of the series at the distance x from the first term a is = 



a + a:d' + X x ini cZ^ + x x - 



— 1 ^ x^2 ,, , ^^ x—l 



— X d^ -{- X y. 



2 3 2 



xiz5x^:i2d4+^^fzlxfi:2xi=:5x ^^d 



3 4 2 3 4 5 



= a + xd^ + (x2 — a-) ^ -I- (^3 - 3x2 + 2x) — + (x^-ex* 

 2 6 



+ 11^2 - 6x) — + (a'5 - lOx* + 35x3 _ 50a; + 24*2) Jt^, 

 "^ 24 ^ M20 



(B.) 

 Now the term to be interpolated being between the third and 

 fourth terms of the series, let j/ be its distance from the third term, 

 then X =: y + 2, which being substituted in the above formula it 



becomes a + (j/ + 2) d' + (ys + sy + 2) f^ + (ys + 3^2 +2y) 



A) 



^ + (y« + V-y" -2y)|J + (y* - 5y» + 4y) ^. 



Tlie fifth difference being d^, the mean of the two fourth diflfer- 



d^ 

 ences will be = fi^ + — = D* the third difference standing oppo- 



site to the interval between the third and fourth terms will be 

 =: rf3 ^ d* ^1)3 ^Q mean of the two second differences oppo- 

 VoL. XIX. U 



