i!+2 



ON QUADRAfUBES AND INTEBPOLATION. 357 



and by giving m any positive or negative value, the series for any diffe- 

 rential-coefficient or integral may be at once found. 



In practice, however, there is' frequently a difficulty in using the 

 series, where mis odd, from the values of V, V,, V2, &c., not being 

 obtainable without a distinct interpolation, where the values of U, U,, 

 U2, U3, &c., only are given. When m is even, the converse may be the 

 case.- "This -may be obviated by using mean difPerences, as follows : 



Makino- Z = :; as before, 



° '■ 1 + A 



2 _ .. , ,._! [ . 1 Z 1 ■ 3 Zy 1 . 3 . 5 Z3 -.. 



2 •+• 2^ ~ ^ ^ ""X 2 4 2 . 4 42 2.4.643 J 



Combining this with the previous expressions, we get 



A2"+4 



VI + A - 



= A2"V,+, + N,A2»+2v„^2 + N2A2"+-'V„+3 + , 



Nj, ^N'jJ N3' being a 'different" s'el "oY coefficients. 

 Now, since (2 + A)V„ = V„ + V„_i, 



j[ log (1 + A) } 2''» = 1 (A2«V„+, + A2«VJ + i Ni (A2"+2V, 



+ A^,.^V„+0 + i- N2(A2-4v„^^ + A2«+4V„+0 + ; 



and similarly, when vi is odd, . 



I log (1 + A)Jf+>= |(A2«+>U„+i + A^-'-iUJ +1 Ni(A2"+3U„+2 



+ A2«+3U„,,) + 1 N2(A2«+5U„+3 + A2"+5[J„+,) + 



The values of the coefficients N are as follows : — 

 N, = - ^ (771 + 3), N2 = ,^, 3, g (577.2 + 52„, + 135), 



N = _ .^ 1 (35m3 + 7777712 + 5749^ + 14175), 



2'° . o* . 5 . 7 



N. = } , „ (175771* + 5720j>i3 + 96794i)i2 + 6197767?! 



2'* . 3* . 52 . 7 



+ 1488375). 



Remember that the sum of two successive differences is the difference 

 of alternate numbers in the preceding column. 



The formulae most frequently occurring in practice are, for integrals,* 



I ^^ulx==-l (« + iH) -i (A^Ui + A%) + 1^ (A^U, + A^UO 



* The higher coefficients cannot be relied upon for accuracy. They -were calcu- 

 lated, with some care, many years ago ; but they are not of much practical use, and 

 have not been satisfactorily verified. 



