356 • EEPOET — 1880. 



It is not, however, commonly used in this form for mere quadrature ; bat 

 the method is used in the calculation of tables in the form 



A'"/^" dx" = A"' I log (1 + A) j~" 



and in most cases in practice, m is taken equal to n. The algebraical 

 process consists in the development of the right-hand side of the last 

 equation in terms of 



A2 



1 + A 



Z = ^orA = |z+^(z>|z^) 



log (1 + A) = logjl + Iz +^Z + \ Z^} •-• ^■ 



J ^VZ^ 4 7 



• ^^ I . 2 3.22^2.4 5. 2^ .2.4.67.26^ , / 



Repfesenting the mth power of the series in < > by 



1 + MiZ + M2Z2 + M3Z3 + M4Z'' + . . . , 



and restoring A-(l + A)-iin place of ZJ 



_^!5^ + M. ^^^^'^^ + M, - ^'""^ + ! ., • 



"' ' 711,1 ' ^ m.o 



(1 + A)^ (H-A)^+' (1 + A)^^' 



which has to be interpreted. 



Denote the successive values of « by . . . . U„ . . . . U2, Ui,TT or u, iiy, 

 U2, M3, .... M,„ in which « = ^ (x), u„ = ^ (x + nh), U„ = ^ (a; — iih). 

 Then the scale of relation is 



U„ = ■ ^\ , u„ = (1 + A)"u. If u, OT(j>(x + -- v^) be also denoted 

 (i + A; 2 V ■^ / 



by V or v, it gives rise to a parallel scale, V„ Vj, V or u, v,, i^2 • • • • *-'«) 



with the same relation between its successive terms, and for its connect- 



i 

 ing relation with the other scale, V,. = (1 + A ) 2 U,.. 



If lit be even (= 2n), direct substitution gives 



I log (1 + A) j 2"w = A2«U„ + M, A2''+2U„+, + M2 A2"+^U„+2 + 



If m be odd (= 2n + 1), 



{ log (1 + A) j 2n+iu = A2«+iV„+i + M,A2"+3V„+2 + MaA^-'+sV^+a + -. 

 The values of the coefficients are as follows : — 



