INCREMENTS. 



Now, expanding (a* 



I ) into a fcries, 



) _^ A:r--(/.^r 



("') 



t 1 1.3 1-2.3 J 



as required. And the higher 

 be found bv the ufual method. 



rder of its incremenis may 



Of the Inverfe Method of Increments. 



P. In the inverfe method of increments, the queftion 

 find the integral, 



f Ax = 



5 -^ A- 2 



&C, &C. 





3 



&c. &c 



where it is only necefTary to obferve, that if the propofed 

 funftion, from its increment being given, increment have - any eonftant mulliplier, the integral above 

 We muft, therefore, examine with attention the fteps by found mutl have the fame. 

 ■which we defcend from a variable quantity to its increment ; Cor. i — Hence we may find the integral of any funftion 

 and then, by the reverfe operation, we may afcend to the compofed of the powers of x, affefted with any eonftant co- 

 integral, when the increment is known. But this reverfe efficients a, 6, c, &c. For in order to find the integral of 

 operation is attended with the very fame difficulties as the in- fuch an increment, it is only necefTary to find thofe of the 

 Terfe method of fluxions, for, as in that, every fluent may be different powers of x, and their fum will be the integral re- 

 readily put into fluxions, fo may the increment of any fiinc- quired. 



tion be readily obtained ; but it is frequently difficult, and £^_ , — Required the integral of the increment a + iiff 



fometimes impoffible to find the fluent of a given fluxion ; ^ ^ ^^ . confidering Ax as eonftant. 

 and fo in the method of increments, there are many cafes 



that will not admit of integration ; we fliall, however, give /"^ _ ^ C l =■ " '' 



fome of t!ie mod ufual and obvious rules, and which will -^ .' Ax 



apply to tho generality of examples. ,-. , ^ __ o -f' ix 



Let us firft attend to the powers of a variable quan 

 tity X. 



fi 



a + i X 



2 A.V 



1. Since Ax = A{x); therefore, reciprocally, j Ax J J' ^Ax 

 = X. And if we fuppofe A jr as eonftant (a fuppolition that And hence by addition, 

 has place in all that followr.), we fliall have /"a .r x i — x, 



erAxfi = X ; therefore /"i = — • 



2. Since A (x'') = 2 .v A .v + A x', therefore, recipro- 

 cally, /'(2a' A.v + A .v") = .v-; or, which is the fame, 

 f 2 X A X + f A x^ = A-' ; whence again alfo f x + 



= ; and hence, by tranfpofition, / x — — — 



2 2 Ax' ' '^ ;/ 2A.V 



A J." 



..-/.. 



) = ^ + :^., ^ 



5A» 



Ax 



6~' 



Ex. 2. — Required the integral of ax*— I x^, confidering 



.V as conllant. 

 Here we have 



d r' A jf a X A x' 



r^x ^ _£ Ax r^ ^ x2_ 



J Z 1 Ax 2 ^J ' 2 A A- 



3. Again, fince A {x) = 3 

 therefore, reciprocally, 

 \\lx^Ax^lxAx 

 /3x'A.vM-/3,tA. 



C A X 



I.-+Axjx + -^^ 



■Ax + 



+ A*^ 



+ A .v') = .m" ; or, which is the fame, 

 -f /A .V' = .V ' ; or dividing by 3 A .» 



r 1 = — -r— ; whence, again. 



./'' 



3 Ax 



A x fx — 



.h: 



I'hich is the fame. 



r /t .r' ax' d r' .i 



ax' = — H 



5^-^ 2 3 i^ 



3 A x 2 6 



the fum of which exprcffions will be the whole increment 

 fouglit. 



Cor. 2. — When it is required to find the integral of a 

 quantity of any of the following forms, (Aa- being fiip- 

 pofed eonftant,) viz. 



(.V + a) 



(x + a) (.V 4- 2«) 



{x -r a) (.r -;- 2a) (x + ^a) 



{x i-a) {x + 2 a) (x + 3 fl) {x + j^a) 

 we arrive at them by taking the aAual produft of thofe 

 quantities, and finding fucceffively the increments of each of 

 the terms : thus. 



We find in a fimi!;!r manner, by continuing to fuppofe Ax 

 as eonftant, and fubftitutiiig always for tiie quantities con- 

 tained under the particular fum their refpeftive values ; the /.,>_,-/• _ x'' ^ , "^ 

 following refults for the integrals of the fucceffive powers " ./ ^* "^ "' ~J '*-'''~2Ax 3 A.v 

 of x; iit which we have repeated the two preceding ones, ^ % r 

 f«r the fake of uniformity. J. / (.v -t- fl) (.v + 3 a) = y (*' -f 3 a at -j- 2 0") 



