INCREMENTS, 



O' 



y \x — x A y _ 



^ + -^^' y 



And therefore from the expanfion of {y' + y ■^y)~ ^ into 

 a feries, we have 



^ (:!) = iy^I:z^)^(.-^+^- +-"-£+ Sec.) 

 \y / r \ y y- y' ' 



And if the propofed fraflion be effefted with any conftant 

 faftor a, the whole of the above increment muft be multi- 

 plied by a. 



Ex. 6. — To find the firft increment of the quantity 

 ^{a' + x^. 



Here it io obvious, on the fame principles, that 

 A ( ,/(a-- + :r)'-) = ../{a- + (x + Ax)) - ^'{a' + :t-) 



^/( {a + x) + (2 .V A .r + A .V ) ) - ^'{d' + .r^) 

 And if now we confider thefe two exprefHons as two bino- 

 mials, to be raifed to the power }, it is obvious that from 

 the developement of the firft, there will be cancelled the firit 

 term, and the other terms will reprefent the increment re- 

 quired ; and thus we have 



2.vAi- + Ajr' (2xA.r + A.vT 



A v(a- + x) = 



(2.vA.r+ A.V-)-' 



2(0' 



- Sec. 



^)- 



(a + >:■)-' 



i6 («' + .v) 



Et. 7. — Having given the equation y' — a x + x' = o 

 which expreffes the relation between the conftant quantity a, 

 and the two variable quantities x and y ; to find the equa- 

 tion whith ought to exprefs the relation between a, and the 

 firft increments of .v and y. 



H^re we muft fubftitute x + A x for .r ; and y + A^ 

 for y ; which gives 



(.V + A j)^ -aix + A.v) + (.V + A .r)'- = o ; 

 from which, fubtraifting the original equation, there re- 

 mains 



2y 



+ 2 xAx -^- A.V- + A/- 



which.isthe equation required. 



And in a fimilar manner the firft increments of any quan- 

 tities whatever may be afcertained, as alfo of any algebraical 

 equation. 



When it is required to find the fecond, third, &c. incre- 

 ments of any propofed fnnftion, it is only neceffary to conli- 

 der the preceding order of increments as variable quantities ; 

 and we ftiall thus pafs from the firft increments lo the 

 fecond, from the fecond to the third, from the third to the 

 fourth, &c. ; in the fame manner as we pafs from the ori- 

 ginal function to the firft increment. 



Ex. I. — To find the fecond increment of .r'. 



Here we have, in the firft place, 



A (.v^) = 2 X A .r -f A .v' the firft increment. 



And if now, in this expreflion, we fubftitute .r + A.v for 

 X, and Ax -t- A' j- for -i x, we have 



A' i^) = A (2 A- A .r + A .x^) = 2 (.V -f- A x) 



(A.V -I- A't) + (A;,- -f- A'x)-- - (2.V Aat + A.v') 

 = 2 A x' -^ 4 A jr A- X + 2 .V A= .V + A' .vS 

 which is the fecond increment required. 



And in the fame way we find the third, fourth, &c. in- 

 crements of a qujntity or funftion, by fubftituting A' .v 

 -r A'jrinftead of .J.' x j and A'x + A-' x inftead of A' .v ; 

 &iid fo or:. 



Remark. — If we confider the fecond increments as being 

 conftant, it adds very much to the limplicity of the operation ; 

 for, after having found the firft increments, viz. 

 A (.v') = 2.V A.I- -f- A-r', 

 A (.v') = 3 .V- A X -}- 3 .-c A .V- 4- A .v', 

 A (v') — 4 x^ A .V -j- 6 JT^" A .v^ -f- 4 .V A a'' -I- A *■', 

 &c. &c. &c. &c. 



we fliall have for the higher order of increments, 

 A- (x) = 2 Ax% A' x' = o, A' (x^) = o, &c. 

 A^ (.1- ) = 6 X A .v' + 6 A x\ A'- (.v) = 6 A x\ 

 A* {x) = o. A* (<;•) = o, &c. 

 AM-v ) = 12 .v" A .V- + 24 X A.v' + 14 A x\ 

 A^ (.V ) = 24 .r A .v' + 36 A X-, A' (.r') = 24 A*-*, 

 A' (.1 ) = o, A'' {x') = o, &c. 

 &c. &c. &c. 



Scholium It \j eafv to find, on the fame principles, the 



fecond increments of all forts of fun(riions. For example, to 

 find the fecond increment of the prcduft x )■, without fup- 

 pofing any increment as conftant, we muit firft find the 

 firft increment, which is, 



A {xy) = _j. A :v + ,r A J. + A .V A J. 

 And fubftituting now in this expreflion, 

 X + A .r for x, and y + A ji for ^ ; 

 A .V + A' X fcr A X, and Ay + A' y for A v, 

 it becomes 



0' +Ay) (A.V + A-x) + (.v + Ax) 

 {Ay -\- A\y) -f (A x + A' x) {Ay + A» ; 

 from which, fubtrafting the firft increment, there remains 

 A' {xy) =yA'xJrxA^y-\-zAxAy 

 + 2 _\ _y A' X + 2 A .V A' J)' + A- X A' y. 

 And if we fuppofe A x conftant, this expreflion reduces to 

 A' (x^) = xA'_y+ 2 Ax Ay -\- 2 Ax A' y. 

 And fimiiar metliods apply in all cafes ; it will, therefore, 

 be unnecelfary to give any farther examples, except in the 

 csfe of an exponential expreflion, which is foraewhat dif- 

 ferent. 



Let it be propofed, for example, to find the increment of 

 the hyperboHc logarithm of x. 



Let y = h .1 .x; then, as x becomes .v + A .r ; fo will / 

 become _}i -\- Ay, that is, 



y + Ay - hi {x + Ax) ; 

 and hence, Cncej = h . I . x, we have 



y + ^y - 



^y = hl(> 



= h.l{x + Ax) 



h. l.x, or 

 A, 



Ax)-U.x=hl{ I -t- 



Now, by the well-known logarithmic feries, we have 



, ,/ A.v\ Ax A.v' Ax' A.' 



V X / X 2.V 3x' 4. 



'(-^-> 



— i- &c. 



A.1-^ 



A.v^ 



y.xnce Ay {j x) ^ ^ ^^ + ^ ^^ ^ ^^ 



+ &c. as required. And the higher order of increments 

 of /' /.v will be found as above, by taking the increments of 

 the terms of this feries. 



Ex. 2. — Required the firft increment of the exponential 

 expreflion a'. 



