290 Astronomical and Nautical Collections. 



site to the third and fourth terms will hed^+^d^ + —d* =D2 



2 2 



the first difference opposite to the same interval will be = d* + 2d^ 

 + d^ = D', and the third term of the series will be = a + 2d' + 

 d* = c. 

 Hence 



d* = D^-JLd' 

 2 



d» a: D3 - d* = D3 - D* + JL d5 



2 



d2=D2— i_d3 - i.d*=: D2- — D3 H- D*- i-d5 

 2 2 2 2 



d» = D' - 2d2-d3 = D' — 2D2 + 2D3~D4 + i-d^ 



2 



«=:c — 2d» - d2 == c - 2D» + 3D2 - Ad^ + D* - J-d^ 



2 2 



And substituting these values of d*, &c. in formula (B) it becomes 

 c + 2/ D> + {y^-.y)^ + (22/3 ^ 32,^+ 2/) -^ + &* ~ 



2y3 - 3,2 + 22,) ^ + (21/5 - 5y* + 5y^^ - 22/) "^^ 



24 " " " "' 240 

 N0W3/D1 is the simple proportional part of the middle first dif- 

 ference; {y^ —y) " D2 X 3' X ^"" is the equation of 



J)3 



mean second difference; (22/3 — Si/ 2 +2/) is the equation of 



1 z 



third difference in the annexed Table 1; {y^^^y^ — y^ + ^2/) 



24 

 is the equation of mean fourth difference in Table II ; and the re- 



maining term (2t/^ — by^ + by^ — 2?/) is neglected, as at 



its maximum it can only amount to -j-f^ of the fifth difference, and 

 is therefore insensible. Hence applying all these equations, we 

 have the value of the term required, the same as if the common 

 formula of interpolation had been employed. 



