INCREMENTS. 



finite number of quelHons, in which, from their nature, it 

 is not I! -ceflary that any order of their increments (hould be 

 conftant : yet as we may attribute to a certain quantity 

 whatever variation we pleafe, providing that the variations 

 of the other q'lantitics depending upon the firft be fuch as 

 to accord with the variation wc have attributed to it ; it 

 follows, that in any problem, we may at pleafure make any 

 order of increments of a quantity chofen at will be con- 

 Uant ; obferving only, that the other quantities ought to 

 vary in confequencc ; and, therefore,' we cannot make an- 

 other order of increments alfo conftant, unlefs, from the na- 

 ture of the problem, fome other increment has a certain 

 ratio to that which is fo alTumed. 



6. The whole of the method of increments confifts of 

 two problems ; viz. l ft. Finding the increments of all or- 

 ders of any variable quantity, raifed to any power ; the 

 prodnft of different variable quantities ; and generally of 

 any funftion of variable quantities ; which problem is al- 

 ways folvible, and prefcnts but little difficulty in any cafe ; 

 and this is called the DireH Method of Increments. The 

 other problem, which is the reverfe ot the preceding, is 

 that of finding the integral of any given increment, which 

 is frequently infolvible ; at leall, without infinite feries, or 

 fome other mode of approximation ; and this is termed the 

 Jnverfe Method of Increments : which two problems we will 

 confider under their diftindt heads. 



Of the Direa Method of Increments. 



7. Since the increment of a variable quantity is the dif- 

 ference between the funis in any two conlecutive ftates, it is 

 obvious in general, that in order to find the increment of 

 any fundlion of variable quantities, we muft fuppofe, that 

 each of thofe quantities is increafed or diminifhed by their 

 refpeftive increments ; and fubftitute thefe quantities, thus 

 changed, into the propofed funtlion ; and from this refult, 

 if there be fubtrafted the original expreffion, the remainder 

 will be the increment fought. 



Ex. I. — Find the firil increment of the fum x -\-y + s. 

 Thefe quantities, augmented by their refpeftive increments, 

 become 



(.v-f-AA-) + + Aj-)+ (si-H Ak) 

 from which fubtrafting the original expreffion, there re- 

 mains A X + Aji + A s, as is e\'ident ; fince the whole 

 increment muft neceffarily be equal to the fum of each par- 

 ticular one. 



In the fame manner wc find the increment of .v -\-y — z, 

 or A (.V + y — z) =1 A .r + A;i — A z. 



And if we had to find the increment of a + x + y — z, 

 we fnould have, confidering a as a conftant quantity, 

 A (<J + .V + > — z) = A A- + A jf — A a, 

 the fame refult as before, becaufe the conftant quantity 

 a has no increment, or its increment is equal to zero. 



Hence it appears, that if to the fum of any variable 

 quantity we add or fubtruft any conftant quantity what- 

 ever, the increment of the whole function will ftill be the 

 fame. 



£x. 2. Find the firft increment of «•'". 



This is, from what is obferved above, the difference be- 

 tween (*• + A x)"" and x"" ; which by the binomial theo- 

 rem becomes m x"'-' A a- + ^"""^ A .-c' + 



the increment of .v', or 



A (x') =2xAx + Ax'' 

 aIx') = Sx"-Ax + 3xAx' + Ax' 

 A (jt') = 4.r' A.V + 6x'Ax'- + ^xAx' + x* 

 &c. &C. &C. &.C. 



And if the propofed quantity, of which the increment ii 

 required, be a x", a being a conftant multiplier, then it is 

 obvious that 



r a(x + Ax)'^-ax- = 

 A {ax") = i c(x ^A .vj" - a (.v-) = 



I .Z(x + Axr-x-:i^aA{x") 

 whence the increment a x" is equal to a times, the incre- 

 ment of x"'. 



Ex. 3. To find the firft increment of the produA xy. 

 Here x becomes x 4 A x 

 and _y becomes _j' + A_y 



whence the produft = .v y -Jr y .^-v + x Ay -\- A .r A v : 

 from which, fubtrafting the original quantity xy, we have 

 A (.r ^) — y Ax -^r x Ay ir Ax Ay. 

 And in the fame way we find 



^K^y^) - I +yAxAz->rzAxAyJf.AxAyAz.. 

 And in like manner may the increment of any other pro- 

 duft be readily afcertained. 



If the produCl was axy, a x y z. Sec, a being a con- 

 ftant quantity, we ftiould have 



A {a xy) = aA (xy), A (a xy z) - n A (xy z), &c. 

 that is, we muft find the increment as above, and multiply 

 the refult by the conftant faftor ci. 



Ex. 4 — To find the firft increment of any quantity of 



the form .v (x + a) {x ~ 2 a) {x -t- 3 a) (.r -f ra). 



It is obvious that this may be referred to the preceding ex- 

 ample, by making x + a ~ u; x -f- 2 a - y ; .r -j- 3 a = s; 

 &c. under which fubftitution, the funflion, of which the 

 increment is required, reduces to x u y z, &c. and,confe- 

 quently, A (x u y z) may be found as above. But if, 

 without this fubftitution, we find the aftual produft, it is 

 obvious that it will take the following form (where A, B, 

 C, D, &c. reprefent conftant quantities) ; viz. 

 x" + A.v"-' + B:r"--4- C .x"- ' + D.r-" + &C. 

 and hence by finding the increments of each of thofe terms 

 by example 2, the fum of them will be the increment of 

 the fundlion propofed. Thus, 



A (^x (.r + «) j := A {x' + <7 .v). 



Now A (.r)- = 2 .V A .r + A .v , 



A (ax) =<7A(.x) = a Ax; 

 whence A ^.v (x + a)) = (2x + a) Ax + A x\ 

 And in the fame manner 



a(^x(x +a) (x + 2a)) = A (a' + 3ax' + 2a .x). 



Now A (.v) = 3 .r' A.V 4- 3 .V A x^ + A.i-' 

 A ( 5 a .V*) = 6 <j X A .r 4- 3 j A x^ 

 A (2 a x) = 2a Ax. 

 The fum of which particular increments will be the incre- 

 ment of the original fundion propofed: and in exatlly 

 the fame way, the increment of any fimilar funftion may 

 be afcertained. 



>_-2)^ 



•3 



•' A.v^ 4- &C. A: 



Tin 



find 



-To find the firft increment of the fraftion ■ 



Here 



