1820.] On the Direct Method of Finite Diferences. 281 



Spring. — September, October, and November. 

 Simmer. — December, January, and Februar)\ 

 Autumn.— Udirch, April, and ]May. 

 iy/„^e7-.— June, July, and August. 



Article VII. 



On the Direct Method of Finite Differences. 

 By Mr. James Adams. 



(To Dr. Thomson.) 



SIR Stonehouse, ntar Plyviouth, June 30, 1820. 



If, in your opinion, the following problems and examples rela- 

 tive to the direct method (f Jinite ^differences, are likely to be of 

 service to the young analyst, your inserting them in the Annals 

 of Philosophy will much oblige, 



Your obedient servant, 



James Adams. 



Problem 1 .—To find the «th order of the function <(>. 



n(n - 1) . ., , n (n - I") (n - 2) . , , o 



f . = ^ + H A ^ + — -^— A'^ ? + ^75 A' ^ + fc^c. 



= (1 + A)" If!.— (See Annals of Philosophy for Feb. 1820.) 

 Corollary \. — By omitting the terras affected with A^ (p, A' f, 

 &c. or supposing Af constant, we have ^, = 9 + » A ^. 



Corollary 2. -^'mcQ <p, = (1 + A) ^, ?, = (!+ A)- ip, 

 ^. = (1 + A)" ^. it plainly appears that the result in the pro- 

 blem is obtained by operating on the symbol only. 



Problem'2. — To'find the n\h increment of the function ^. 



. n(n- 1) n(n - li (" - 2) 

 A"?i = f. - «?._, + 5 ?._a ^TS ^"-^ 



+ ip = ((p. — 1)", writing ip for unity in the expansion of 

 (^, _ l)-.-(Ibid.) 

 Corollary .—'Ry Dr. Taylor's theorem, and detachmg the 

 symbol of operation, we have ip„ = 9 e""*, <p,_^ = (pe'"""''', ?,_, 

 = ^e'"""'', &.C. then by substitution we get A"<p = ip 



^e--* - He'—"" + " ^'\~ '^ e'"-^'"- Sec.) =«(«"- l)"- 



Or, A- p = e-"^ - n eC-""^ + "^" ~'^ e'"-''"* - &c. e = 



2-718, 8cc. and r/ denoting differential. 



Example 1. — To find the /(th increment of x". 



Put f = x" and A x = w a constant quantity ; then will 



