In the very same manner it is found, that the fluxion of vuxyz is vux y z + vuxyz+vuxyz + vuxyz + vuxyz; 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 x2, or x3, or x2, &c. For, in the

3

product or rectangle xy, if x = =y, then is xy =xx or x2, and

also its fluxion xy+xy=xx+xx 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 + xyz+xyz=xxx+xxx+xxx or 3x2x, the fluxion of x3.

Ꮖ ;

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 (x), multiplied by the exponent of the power (n), and by the power of the same root whose index is less by 1, (xn−1).

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

that of (a+cx) is 2cx X(a+cx) or 2acx+2c3xx,
that of (a+cx2) is 4cxxx(a+cx2) or 4acxx+4c 2x23 x,
that of (x2+ y2)2 is (4xx+4yy) ×(x2+y2),

that of (x +cy2)3 is (3x+6cyÿ) ×(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

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

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

(by substituting the value of q, or),

gy

Y

8ig|ཨཱུ

**

That

1

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.

y

xy-xy axy-axy

yz

or

y2

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

1

xn

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 1 nxnxx fluxion of x-, or is xn

or

x2n

nx xn+1

or-nx -N1 x; 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 pro

1

xn

posed fraction, or quotient q; then is qx= 1; and taking the fluxions, we have

gxn+qnxn-1x=0, hence qx2-qnxn-1; divide by 2", then

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

4аx

5x or

any

ԴՈՆ :

any fractional power of a fluent quantity, as of x, orx

m
n

For, put the proposed quantity x"=g; then, raising each side to the n power, gives x?

taking the fluxions, gives mom-x-ng-1g; then

mxm-1 m

'm

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 ax is 3 ax2x

that of ax2 is ax x = 1 αx

ax

ax

2x 2/x; and that of

√✓ (α2 —x2) or (a2 —x2)3 is 1 (a2 — x2)3× — 2xx =

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.

23. Now, to determine this from the nature of the hyperbolic spaces. Let a be the principle 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 CBXBA; 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 XPQ is always equal to DB, that is, xy=m: hence y= and the fluxion of the

mx

m

space, xy is raqp the fluxion of the log. of x, to the mo

dulus m. And, in the hyperbolic logarithms, the modulus m being 1, therefore, is the fluxion of the hyp. log. of x;

which is therefore equal to the fluxion of the quantity, divided

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 z, namely, z=ex; then take the logarithm of each, so shall log. z=xXlog. e;

take the fluxions of these, so shall xX log. e, by the last

2

article: hence z zx Xlog. e=ex Xlog. e, which is the fluxion of the proposed quantity e orz ; 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)nx × nx × log, (a+c).

26. In like manner, in the second case, put the given quantity yz; then the logarithms give log. zx Xlog. y, and the xxlog.y+x×2; hence z=zx X log. y+

fluxions give

Y

(by substituting y* for z) y*x Xlog. y+xy-ly, which is the fluxion of the proposed quantity y; and which therefore consists of two terms, of which 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.

VOL. II.

41

OF

t

OF SECOND, THIRD, &c. FLUXIONS.

HAVING explained the manner of considering and determin ing 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 x, x, x ɔ x, x, &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 306 where, if x denote the absciss AP, and y the ordinate re: and if the ordinate re or y flow along the absciss AP or x, with a uniform motion; then the fluxion of x, namely, x=pp or qr, is a constant quantity, or x= O, in all the figures. Also, in fig. 1, in which AQ is a right line, y=rg, or the fluxion of rq, is a constant quantity, or y 0; for, the angle e, the an gle A, being constant, er is to rq, or x to y, in a constant ratio. But in the 2d fig. rq, or the fluxion of PQ, continually increases more and more; and in fig. 3 it continually decreases more and more, and therefore in both these cases y has a second fluxion, being positive in fig 2, but negative in fig. 3. And so on, for the other orders of fluxions.

Thus if, for instance, the nature of the curve be such, thạt a3 is every where equal to asy; then, taking the fluxions it is a3y3xx; and, considering always as a constant quantity, and taking always the fluxions, the equations of the several orders of fluxions will be as below, viz.

the 1st fluxions a2y = 3x3x,
the 2d fluxions a2y=6xx2,

the 3d fluxions a2 =6x3,

the 4th fluxions a2 ÿ=0,

and all the higher fluxions also ≈ 0, or nothing.