1820.] the Direct Method of Finite Differences. 283 



tp^ = I (x + iv) = I (^l -r "j X, 9, = I (x + 2w) = 



/ (l + ^) X, <p^:=.l(x+n ^v) = l(l + ^) r, &c. By sub- 

 stituting in problem 2, A" {I x) = I {x + n w) — tc I 

 (x +(« - 1) x) +'^^^^ .l[x + (« -2) lo) - &c 



. / (1 + (lZ_£lJf) _ &c. 



The latter expression may appear plainer by considering that 



ih + ^^ X = I (\ + —) + I X, and that in the expansion 



of (x — yY, the sum of the affirmative is equal to the sum of the 

 negative coefficients. 



In like manner we find A" (-— ) = — ; ; rr— r 



Example 6. — To find the «th increment of x log. x. 

 Put ip = X I X and A a; = Wj a constant quantity, then will 

 <()„ = {X + nw) I {x + n w)5 <p„_^ = {x + (n - 1) w] I 

 {x + (h — 1) w} &c. 

 By substituting in problem 2, 

 A" (x I x) = {x + 71 w) l{x + n w) 



— n {x + {n — 1) lo} I \x + (rt — 1) zf} 



+ " V'^ ^-^ + ("- 2)«;}/ir + («-2)«;} .... 

 + a: / T. 

 By multiplying and dividing by x I x, and putting 1 H = f, 



1 + ~ w = <?;' 1 4- " ~ . ^ w = v" &c. we shall have 



X X 



A''{xlx)=^xlx {ylv - nv'lv' + ""-^^^ .v" Iv'' .••.±l) 



Example 7. — To find the «th increment of (log. ar)". 



Put (p — {I x)'" and A x = «;, a constant quantity, then will 



^. = {lix + nw)}-= J/(l + "-^)xf 



<f>,-, = \i(x+ («- \)w)f= \i (i +^«')'^r 



&c 



