INCREMENTS. 



Here, by t-^king the differences as before, we (hall have This quantity admits of the following Jccompofition : 



Ir,^} 



(x + ^x) {.v + 2Sx) .v(^+ A.v) .V {x + A.x) {x +2^x) A .v 



2AXX X. -- . - r-^' .-^-. = ^ X i + 



r(.v + A x) (x + 2 A .v) • A.v (.V + 2 A a:) A .v -v A ..• x + A .■< 



And in the fame manner we find the increment, or, ^ ^^ I . (..^^i, ^f „ Inch parts is evidently of 



. ^ -J _ ^ .1- X + 2 A .f 



' 1 V (x -t- A v) (v 4- 2 A x) i ~ ~ the form that has bc-en invelligated in the preceding para- 



graph. By maliing firil n — o, >i = i, and « = 2 ; thus 

 liave 



.r(.v + Ax) [x + 2 A.r) (.r + s^x) 

 l^x + Ax) (x + Z A.r) {x + 3A.I-) J 



4A.r 



by which means the integral of the propofed quantity take 

 the following form : vi-z. 



X (x + Ax) (x -h 2 A.r) (x + 3 A.v) (.r + 4 A.f) 

 znA fo on of others. Whence it follows, that in order to 



d^nermine the increment of any expreffions of the above But the above formula gives alfo 

 form, we muft mcreafe the denommator by one fadtor, and ° 



mnltiplv the ne>v fraction by the conllant increment taken 



negatively and affefted by fuch a co-efficient, as is equal to 

 the number of faftors in the denominator of the refulting 



means the integral 

 ing form : vi-z. 



It the above formula gives al 



fraftion. 



the whole integral is 



<! .r - A .r 



And hence, again, converfely, in order to find the integral _}_ x { ~ ^ — ^ 



correfponding to any increment of the form -^ x V-i-t--i-i- x / x A .v (.v 4- A a) 



a A X as required. 

 X {x + A x) (x + 2 A J-) (x + 3 A .v) . . (.I- + n A .r) £x. 2. — Required the integral of the quantity 

 we muft fupprefs the laft faflor in the denominator, and S -^ ^' 



afterwards divide the refulting fraftion, taken negatively, by x Ix + 3~aT) 



the produft of A x, into the number of factors comprifed in 



the denominator of the faid fraiftion ; thus, pj^ ^p „.g ],.,yg 3 ^ 



.A 



(x+ Ax) (.V + 2 A^x) (.1- -^ 3 A .v) . . (.V + n A X ) a"J, confequently, the integral of the quantity will be 



«.r (.v4-Ax)(.v-f 2A.f)(^-+3-^«)--- (^• + «-l^-') -^ ""' J «+3---' 



12. On the fame principles we find Now the above formula gives 



and, therefore, reciprocally, p j _ i /• 



J X + (« + 1 ) A jr J x + nAx s + nSx whence the whole integral is exprefled by 



The integration of thefe two latter expreflions cannot be /• 1 / \ 1 



cffefted feparately, but the difference of them is evidently / ^^r~^. ~ / .v + 2 -^ .v ~ ~x ~ 



eoual to the algebraical fraftion -n /^ i /^ • 



, ^+"^\ . , . ^"7 rv-:^^ -J .^^-^- '- 



Whence it appears, that quantities may iometimes admit 



of integration, by being decompofed into many parts ; which, by the fame formula, and, conlequently, we have at length 



though they will not admit of it in their (late of feparalio:i, /•-Ax i I I 



may, notwithftanduig, be fo combined with each other, that / — — — - , = — ^ r— "" "• 



the final refuh (hall be algebraical ; a circumftance that com- J ■»'(•'•' i" 3 ■^•''; •"* + ■■* ■^' "^ •■* •*" 



monly happens in praflical operation*. Such are the elements of the direft and inverf : method of 



increments, and which will be found to embrace a very great 



Examples of Integration. number of cafes ; but thofe who wifli for a more complete de- 



n. Ex. i.-Find the integral of the increment velopement of the principles, cannot con (ult a work better 



*J' ■^■' ° calculated to convey the ncccffary information, than the 



3 a- -I- 2 A .V third volume of the " Traitc du Calcul Diffcrentiel, 



,r (.1- + A x) (x -f 2 A x) &c.'' par La Croix. We (hall now conclude this article, 



by 



t- A 



