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In the very same manner it is found, that the fluxion of vuxyz is vuxyz + vuxyz + vuxyz + vuxyz + vuxyż; and so on, for any number of quantities whatever; in which it is always found, that there are as many terms as there are variable quantities in the proposed fluent; and that these terms consist of the fluxion of each variable quantity, multiplied by the product of all the rest of the quantities.

18. Hence is easily derived the fluxion of any power of a variable quantity, as of x*, br .r'?, or x4, &c. For, in the product or rectangle xy, if x=y, then is xy = xx or r?,

Y and also its fluxion xy + xy = ** + xi or 2 , the fluxion of r.

Again, if all the three x, y, z be equal; then is the product of the three xyz = *}; and consequently its fluxion xyz + xyz + xyż=ixx+xäx+xxå or 3x:, the fluxion of x3. In the same manner, it will

appear

that
the fluxion of x* is = 4x3+, and
the fluxion of xs is = 5.x*; and, in general,

the fluxion of his = nx-'*; where n is any positive whole number whatever. That is, the Auxion of any positive integral power, is equal to the fluxion of the root (*), multiplied by the exponent of the power (n), and by the power of the same root whose index is less by 1, (*-).

And thus, the fluxion of a + cx being cé, that of (a + cx) is 2c* X (a + cx ) or 2aci + 2c*xi, that of a +cx+)2 is 4cxx x (a + c**) or 4ačxi + 4c***t, that of (x2 + y2)? is (4x3 + 4yý) (+ yą); that of (r + cy?)} is (36 * 6cyj) x (x + cy?).

19. From the conclusions in the same article, we may also derive the fluxion of any fraction, or the quotient of one variable quantity divided by another, as of For, put the quotient or fraction q; then, multiply.

y ing by the denominator, * = qy; and, taking the fluxions, ů =qy + qj, orqy=i-qj; and, by division,

qj
9
(by substituting the value of q, or

у
--
у

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y

#ys, the Auxion of , as required.

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That

OX

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That is, the fluxion of any fraction, is equal to the fluxion of the numerator drawn into the denominator, minus the Auxion of the denominator drawn into the numerator, and the remainder divided by the square of the denoininator.

my – xj aty — axj So that the fluxion of

is a X

or 9

ge 20. Hence too is easily derived the fluxion of any negative

1 integer power of a variable quantity, as of x-", or which is the same thing. For here the numerator of the fraction is 1, whose fluxion is nothing; and therefore, by the last article, the fluxion of such a fraction, or negative power, is barely equal to minus the fluxion of the denominator, divided by the square of the said denominator. That is, the

naxhm

ni fluxion of x", or is

nx-1-'*; rn

x2 or the fluxion of any negative integer power of a variable quantity, as x-*, is equal to the fluxion of the root, multiplied by the exponent of the power, and by the next power less by 1; the same rule as for positive powers.

The same thing is otherwise obtained thus: Put the proposed fraction, or quotient = 9; then is q.r” = 1; and, taking the fluxions, we have gro + qn.xn-*=0; hence qr" =-qn.xn- ;; divide by x'", then

1

nă = (by substituting for. 9), **** or = --x--*; the same as before.

1 Hence the fluxion of x-? or is * 2*, or

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c

2

1
that of (a+*)' or is - (a + *)-or
ata

(a + x)}, that of c(a + 3.x2) · or is – 12cx* * (a + 3.x*),

,
(a + 9.72)2

12c.x*
(m + 3.72)3•

21 Mush

or

m

21. Much in the same manner is obtained the fluxion of any fractional power of a fluent quantity, as of x", or ".

For, put the proposed quantity = 9; then, raising each side to the n power, gives xim = 9";

q taking the fluxions, gives mxml=

ng"-fq; then

mrmdividing by nqn-, gives q =

-I

n

mom-do
nghai

m

nam

a

Which is still the same rule, as before, for finding the flusion of any power of a fluent quantity, and which therefore is general, whether the exponent be positive or negative, integral or fractional. And hence the fluxion of art is fafti;

ai that of axă is _ac3-=ļax

and that of

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-xx' V(4222) or (a* – x2) is (a*

1 vlaz

{

2.a)
X - 2x*

(Va-x) 22. Having now found out the fluxions of all the ordinary forms of algebraical quantities; it remains to deter: mine those of logarithmic expressions; and also of exponential ones, that is, such powers as have their exponents variable or flowing quantities. And first, for the fluxion of Napier's, or the hyperbolic logarithin.

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23. Now, to determine this from the nature of the hyperbolic spaces. Let A be the principal vertex of an hyperbola, having its asymptotes Cl), CP, with the ordinates DA, BA, PQ, &c, parallel to them. Then, from

PP the nature of the hyperbola and of logarithms, it is known, that any space ABPO is the log. of the ratio of CB to CP, to the modulus ABCD. 1 = CB or BA the side of the square or rhombus DB; m = the modulus, or CB X BA; or area of DB, or sine of the angle c to the radius 1 ; also the absciss cp = x, and the ordinate PQ = y. Then, by the nature of the hyperbola, CP X PQ is always equal to do, that is, ay = m; hence

mi y

and the fluxion of the space, sy is = POP the fluxion of the log. of x, to the modulus m.

And, in the hyperbolic logarithms, the modulus m being 1, there- .

fore,

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fore

is the fluxion of the hyp. log. of x; which is therefore equal to the fluxion of the quantity, divided by the quantity itself.

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of or + z is

X +
of a +.?
r * (a – x) + *(a + x)

A - X
is

Х
(a - x)2

atx
naram
of a.x" is

a

x

ar"

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24. By means of the fluxions of logarithms, are usually determined those of exponential quantities, that is, quantities which have their exponent a flowing or variable letter, These exponentials are of two kinds, namely, when the root is a constant quantity, as e*, and when the root is variable as well as the exponent, as y.

25. In the first case, put the exponential, whose fluxion is to be found, equal to a single variable quantity , namely, %=e; then take the logarithm of each, so shall log.z = x x

ż log.“e; take the fluxions of these, so shall

= * x log. lg by the last article; hence z = zi x log. e=e* x log. e, which is the fluxion of the proposed quantity ox or z; and which therefore is equal to the said given quantity drawn into the ffuxion of the exponent, and into the log. of the root.

Hence also, the fluxion of (a + c)ax is (a + c) x nx x log. (a + c).

26. In like manner, in the second case, put the given quantity y =; then the logarithms give log. Z=x x log. Y, and the fluxions give * = * * log. y + x.X

-; hence

y xxy = zi X log. y +

= (by substituting gue for z) y,* * * log. 9 + 2yo?-?;, which is the fluxion of the proposed quantity yt; and which therefore consist of two terms, of which

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X

th

the one is the fluxion of the given quantity consi exponent as constant, and the other the fluxion of : quantity considering the root as constant.

OF SECOND, THIRD, &c, FLUXIC

HAYING explained the manner of considerin mining the first fluxions of flowing or variable remains now to consider those of the high second, third, fourth, &c, fluxions.

27. If the rate or celerity with which any tity changes its magnitude, be constant, or the position; then is the fluxion of it also consta But if the variation of magnitude be contini. either increasing or decreasing ; then will the degree of fluxion peculiar to every point or the rate of variation or change in the fluxio Fluxion of the Fluxion, or the Second Fluxis fluent quantity. In like manner, the variati this second fluxion, is called the Third Flu. proposed fluent quantity; and so on.

These orders of fluxions are denoted by letter with the corresponding number of namely, two points for the second fluxion, the third fluxion, four points for the fourti on. So, the different orders of the fluxic 3, 3, &c; where each is the fluxion of the

28. This description of the higher e. may be illustrated by the figures exhibited i where, if x denote the absciss AP, and ' and if the ordinate pe or y flow along t1 with a uniform motion; then the fluxi * = PP or er, is a constant quantity, o figures. Also, in fig. 1, in which AQ is a or the fluxion of pg, is a constant quant the angle Qs = the angle A, being consi á to j, in a constant ratio, But in the Auxion of pg, continually increases mo

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