288 On the Direct Method of Finite Differences. [Oct. 



A (x y) = X y { A + A' (1 + A)} 

 A« {xy) = xy {A + A' (1 + A)}^- 

 A' {xy) = xy {A -r A' (1 + A)}' 



A-'ixy) = xy{A + A'{i + A)}"- 

 Therefore, 

 A- {X y) = xy J A" + « A"- A' (1 + A) + "_i!^) a-' A'* 



(1 + A)^ + &C.J 



= 7/ A" a; + « A y A"-' (1 + A) r + !!i^l 

 A^yA— ^(1 + A)* + &.C. 

 Or by substituting for (1 + A) x, (1 + A)- .r, Sec. as in Prob. 1, 



A" {xy) =y A"x +n Ay A""' a-, + "-i^i^ A« y A"-'x, + &c. 



Example 16. — To find the wth increment of aye. 



By separating the symbols from their corresponding quantities, 

 as in the last example, and denoting the symbol of z by A'\ we 

 have 



A{xy z) = (x + A x) (y + A' y) {z + A" z) - xy z 

 = (1 + A) (1 + A') ( I + A") x y z - xyz 

 = {(1 + A) (1 + A') (1 + A") - 1} ,r y 'z 

 = (A + A' + A" + AA' + AA" + A'A" + AA'A")a:j/S 



Then by operating on the si/mbols o)ih/, we have 

 A" {xy z) = xy z (A + A' + A" + A A' + A A" + A'A" + A A' A'0% 

 which being expanded by the muliinomial theorem, and the 

 symbols placed before their corresponding quantities, the incre 

 ment required will be shown. 



Example 17. — To find the «th increment of xy z, &c. 

 A {xyz,^c.)= {x + Ax){y + A'y){z+ A" zj&ic.—x y z, &c 

 = {(1 + A)(l + A') (1 4- A") &c.}x3/2&c.-a-3/2, &c 

 = {(1 + A) (1 + A') (1 + A") &.C.-1} xy 2, &c. 



Then by operating on the symbols onlt/, we have 

 A" {xyz,kc.) = {( 1 + A) (1 + A') (1 '+ A")&c. - 1}" xyz, &c. 

 which being expanded and so arranged that each symbol be 

 placed before its respective quantity, the required increment will 

 be exhibited. 



See Appendix to the Translation of Lacroix's Differential and 

 Integral Calculus, by Messrs. Babbage, Peacock, and Herschel, 

 for the three last examples. 



I am not aware that the foregoing examples have before been 

 applied to the two general problems herein given. Since the 

 publication of the Annals of Philosophy for May, 1820, I have 

 seen demonstrations of the series therein printed at p. 69, vol. iii. 

 Dr. Button's Course. After I had finished my solutions, and 

 before 1 sent them to be printed, I saw demonstrations of the 

 same series in Cagnoli's Trigonometry : the three solutions are 

 very different from each other. 



