282 Mr. Adams on [Oct. 



f, = {X + ?<;)"'= (l + ")" . x'" ; f = (x + 2 «;)" = 



(1 + ^-^y .x-;<p^ = (x + n wT = (l + ^j". x-, &c. 



Then by substituting in problem 2, we have 

 A-(x-) = (l + "-^y.x- - ;. (l + (Jl^y . ^« + !L(!L_:_L) 



(l + ^1^) . ^™ . 8cc. ± X- 



Or, A" ix-) = X" \{l + "-^y -n{l + (jL^y + »l!^ 



Example 2. — To find the nth increment of — . 



^ X"' 



Put f = — and A a; = w, a constant quantity, then will <pi 



; f, = 



I 



a. ss — = , &c. By substituting in pro- 



^" (x + n t£))'» / nieV" J Or 



blem 2, we have A" (1,) = ^ W^^,. - ^TEZllEf "^ 

 "^"-'^ -&c + 1? 



Example 3. — To find the nth increment of a'. 



Put (p = a' and A x = w, a constant quantity, then will 



f, = «(^+"', ^^ = a<" + "'', (p„= «<=' + ""•', &c. By substituting in 



problem 2, A" (a') = a' ja""' — na^"-'^'' + " ^" ~ '^ a("-»"" — 

 &c + Ij 



Example 4. — To find the wth increment of—. 



Put ip = — and A X = w, a constant quantity, then will 



f. =-^)' "P^ = „7^' ^» = ■^^T^'^''' By substitution 

 in problem 2, A» (-L) = - ^-^ r^ + " ^"^ 1^ " &c. 



.... + l| 



Example 5. — To find the r<th increment of the log. x. 



Put f z= I X and A j; s= w, a constant quantity, then will 



