In the very same manner it is found, that the fluxion of vuxyz is vuxyz + vuxyz + vuxyz + vuxýz + 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 r2, or r3, or 4, &c. For, in the product or rectangle xy, if y, then is xy = xx or x2, and also its fluxion xy + xý = *x + x* or 2xx, the fluxion of x2.

Again, if all the three x, y, z be equal; then is the product of the three xyz=x3; and consequently its fluxion xyz+ xjz+xyż=xxx+xxx+xxx or 3r2x, the fluxion of x3.

In the same manner, it will appear that

the fluxion of x4 is = 4x3x, and

the fluxion of x is

the fluxion of r" is =

5xx and, in general,
nx3¬x;

where n is any positive whole number whatever.

That is, the fluxion 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, (x-1).

And thus, the fluxion of a + cx being cx,

that of (a+cx) is 2cx × (a+cx) or 2acx + 2c2xx, that of (a+cx2)2 is 4cxx × (a + cx2) or 4ačxx + 4c2x3×, that of (x2+2)2 is (4xx+4y) × (22 + y2),

that of (r+cy2)3 is (3x + 6cyỳ) × (x + cy2)2.

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

x

x For, put the quotient or fraction = q; then, multiply

و

ing by the denominator, x = qy; and, taking the fluxions, x = qy + q3, or qy = -qj; and, by division,

That is, the fluxion of any fraction, is equal to the fluxion of the numerator drawn into the denominator, minus the fluxion of the denominator drawn into the numerator, and the remainder divided by the square of the denominator. xy xj axy-axy So that the fluxion of

ax

is @ x y

or

20. Hence too is easily derived the fluxion of any negative integer power of a variable quantity, as of x ̄”, or which

1

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

nx-1x

or the fluxion of any negative integer power of a variable quantity, as a ̄”, 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

1

proposed fraction, or quotient =q; qan then is q.

and, taking the fluxions, we have

qx"+qnx"1=0; hence qr"-qna; divide by x", then

qnx

-n; the same as before.

nx

Hence the fluxion of xTM2 or

2x

that of

3x

.2.49

21. Much in the same manner is obtained the fluxion of

m

any fractional power of a fluent quantity, as of x", oryxTM.

For, put the proposed quantity "q; then, raising each

side to the n power, gives x = : q" ;

taking the fluxions, gives m=nq-q; then

dividing by nq"—1, gives q

mxm-x
ngh

mxm-1x

nxm

n

m

n

Which is still the same rule, as before, for finding the fluxion 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 ar* is jaxx;

that of axis ar11x=1ax→± *=

√ (a2 — x2) or (a2 — x2)1⁄2 is ž(a2 — x2)

ax

2x

I

=

ax

24x3

and that of

-xx

(√ a2 - x2)

22. Having now found out the fluxions of all the ordinary forms of algebraical quantities; it remains to determine 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 logarithm.

D

PP

23. Now, to determine this from the nature of the hyperbolic spaces. Let A be the principal vertex of an hyperbola, having its asymptotes CD), CP, with the ordinates DA, BA, PQ, &c, parallel to them. Then, from the nature of the hyperbola and of logarithms, it is known, that any space ABPQ is the log. of the ratio of CB to CP, to the modulus ABCD. Now, put 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 cr = X, and the ordinate PQ= y. Then, by the nature of the hyperbola, CP XPQ is always equal to DB, that is, xy = m; hence

the fluxion of the log. of x, to the modulus m. the hyperbolic logarithms, the modulus m being

fore,

x

fore is the fluxion of the hyp. log. of x; which is there

20

fore equal to the fluxion of the quantity, divided by the quantity itself.

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, ze; then take the logarithm of each, so shall log. zxx log.e; take the fluxions of these, so shall =* × log. e,

by the last article; hence ż z x log. eexxx log. e, which is the fluxion of the proposed quantity e* or z; and which therefore is equal to the said given quantity drawn into the fluxion of the exponent, and into the log. of the root. Hence also, the fluxion of (a + c) is (a + c) x nx x log. (a + c).

26. In like manner, in the second case, put the given quantity; then the logarithms give log. z=xx log. y,

and the fluxions give

ช

zxy

= log. y + xx

-; hence

zx X log. y + = (by substituting y for z) yx x

log.ay, which is the fluxion of the proposed quantity ; and which therefore consist of two terms, of which

th

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

OF SECOND, THIRD, &c, FLUXIONS.

HAVING explained the manner of considering and determining the first fluxions of flowing or variable quantities; it remains now to consider those of the higher orders, as second, third, fourth, &c, fluxions.

27. If the rate or celerity with which any flowing quantity changes its magnitude, be constant, or the same at every position; then is the fluxion of it also constantly the same. But if the variation of magnitude be continually changing, either increasing or decreasing; then will there be a certain degree of fluxion peculiar to every point or position; and the rate of variation or change in the fluxion, is called the Fluxion of the Fluxion, or the Second Fluxion of the given fluent quantity. In like manner, the variation or fluxion of this second fluxion, is called the Third Fluxion of the first proposed fluent quantity; and so on.

These orders of fluxions are denoted by the same fluent letter with the corresponding number of points over it: namely, two points for the second fluxion, three points for the third fluxion, four points for the fourth fluxion, and so on. So, the different orders of the fluxion of x, are *, *, *, *, &c; where each is the fluxion of the one next before it.

28. This description of the higher orders of fluxions may be illustrated by the figures exhibited in art. 8, page 288; where, if r denote the absciss AP, and y the ordinate PQ; and if the ordinate PQ or y flow along the absciss Ap or x, with a uniform motion; then the fluxion of x, namely, * = Pp or or, is a constant quantity, or = 0, in all the figures. Also, in fig. 1, in which AQ is a right line, j = rq, or the fluxion of PQ, is a constant quantity, or y = 0; for, the angle a, the angle A, being constant, or is to rq, or to j, in a constant ratio. But in the 2d fig. rq, or the fluxion of Po, continually increases more and more; and