INCREMENTS. 



the feries foiight. Now the integral of {x + i) (•» + -) 

 ^ X(:.+ !)(>:+ 2) . 



3 



and fir 



the prefcnt ex- tioii 



this form, that tlie theory of ir.cremenls becomes fo uiiivcrfally 

 applicable to alr.ioll every fpecies of mathematical inveiliga- 



ample = loo, we have — 



lOI X 102 



= 343-1 



for 



the fum required. 



£x. 2. — Required the fura of n terms of the feries 

 I . 2 . 3 + 2 . 3 . 4 + 3 . 4. 5 + &c. n (» + I ) (n -f s). 



Here, by writing x inftead of n + i, we (hall have for 

 the fucceeding term x {x -\- i) {.v +2), which is the in- 

 crement of the feries, and therefore the integral of 



*(:.• 



(■>•■ 



_ (x-l)x{. 



.)(.< 



2) 



vill 



be the fum required ; which, by re-eftabli(hing the value of 



= n + I, becomes 



r,{n+ I) (n + 2) (n +3) 



the 



fum of n terms. 



Ex. 3. — Required the fum of n terms of the natural 

 feries of fquares i' + 2' + 3' + 4^ -j- . . . . . n. 



Here writing .v for «, the fucceeding term is (v + l)* 

 =: x' + 2 X + I = X {x + 1) 4- (j.- + i), which is the 

 increment ; and it confifts of two parts. Now the 



integral ofx(. + x) = ^"+'^;("+' ^ 



integral of w + i = ~ ' 



And therefore fince x = n, we have (- 



2 



{n — 1) n (n + 1) _ ^j^^ j-^^ required. 

 3 

 Ex. — Let it now be propofed to find the fum of the infinite 



feries 1 1 1 1- &c. 



1.2 2.3 3.4 4.5 



Here it will be neceffary for us to confider the feries as 

 generated from the extreme term, which is o, and therefore 



as its laft term, which will therefore be the increment 



of the feries -f + + &c. ; and, confe- 



2-3 3-4 4 • > 

 quently, the integral of this will be the fum of the feries, 



wanting only the term . Make, therefore, x^= 2, or 



J . 3 X {^ 



in which cafe the increment ■ 



the increment is — 1) ; therefore the fum of the feries, be- 

 ginning at the term ,is- , to which adding the firft 



2-3 2 



term = — , we have the fum of the whole feries = i. 



1.2 2 



This will ferve to cxpkin the method purfued by M. Ni- 

 chole in his iirft paper, and will bo ufcful as an introduftion 

 to what follows : in which we (hall not limit ourfeivcs Icj con- 

 iideriiig the continual increafe of x as conllant as is done 

 above; but as being variable like jr iifclf; fo» it is under 



Notation and Definiiioni. 



... "? 



more than the difference between that quantity in its firlt 

 ftate, and what it becomes after a certain increafe, this dif- 

 ference may be properly reprefentcd by V) x, or :i x ; and 

 in the fame manner, if j' be any funftion of a variable quan- 

 tity, D J', or i\y, will reprefent the increment oi y. And 

 as in the fiuxional or differential calculus, yis made the cha- 

 rafter of integration, fo in the prefent inftance, we (hall em- 

 ploy it to reprefent the integral of any increment. 



2. The increment of a variable quantity being, as we have 

 obferved above, only the excefs of this quantity in one ilatc, 

 over the fame quantity in the confecutive flr.te, it follows 

 that if a variable magnitude x become fucccflively x, x, x'', 

 x'", &.C. we (hall have A x = :r' — .v ; A .v' = .v" — .v' ; 

 A A-" = .r'" — *" ; A .r'" = .-c"' — .t'", S:c. 



It may happen that an increment may be pnfitive, or ne- 

 gative, according as the variable quantity of wliich it is the 

 increment is augmented or dinuinlhed, with regard to feme 

 other magnitude or magnitudes which we fuppofe to in- 

 cre.ife, and of which the increments are therefore neceflarily 

 poiitive. 



3. The increments of quantities being themfelves quan- 

 tities, if they be variable, we may take the increments of 

 them, thefe are called fecond increments ; and if tliefe fecond 

 increments be alfo variable, we may in like manner take the 

 increments of thefe alfo; which are called third increments, and 

 fo on as long as the differences or increments are variable. 



In all thefe cafes, the condition of the increments being 

 variable, is neceiTary ; becauie if, in any cafe, they become 

 coniiant, then it is obvious that the increments are o, 

 whether it be the tirll, fecond, third, &c. increment that 

 thus become conllant or invariable. 



Thus the feries of fquares i, 4, 9, 16, 25, &c. is an ex- 

 ample of a cafe in which th.e fecond differences or increments 

 are conflant : for this feries may be confidered as generated by 

 a variable quar.tity x' ; which is fuch, that if the difference 

 between the fuccelTive terms be taken, they will form a feries 

 of quantities in arithmetical progreffion, and confequently 

 the differences of thefe differences, or the fecond increment 

 of .-«- will be conflant. In the fame manner we find the 

 third differences of the feries of cubes, 1,8, 27, 64, I2J, or 

 the third increment of x , is conftant ; and therefore the 

 fourth increment — o ; for after any order of increments 

 becomes conflant, ail the ulterior orders muft, neccfTarily, 

 become zero. 



4. As A.v denstes the firft increment of any variable 

 quantity x ; fo A'.v, A' a-, i^* x. Sec. will reprefent the fe- 

 cond, tliird, fourth, &c. increments of the fame quantity .v ; 

 which c.^ipreffions are fufficlcnlly di.linft from Ajr% Ax\ 

 Aa- , &c. which reprefent the powers of tliofc increments; 

 and if it be required to exprefs the power of any increment 

 pafl the firfl, as lor inflance the nl\\ pswer of the fecond, 

 third, &c. increment of .r, that will be done thus, A'*', 

 AV, &c. 



5. In fome problems it is neceflary to confider a certain 

 order of the increments as conllant ; thi.s, in any arith- 

 metical progrefiion ttie firft incrtments are conllant. \n 

 the feries cF nr.lural fquares, the fecond increments are 

 neceffarily conllant ; as are alfo the third increments of 

 cubes, &c. as we have fcen above. But there is an in- 



B 2 Suite 



