m

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

taking the fluxions, gives mxml=nq-q; then dividing by

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 ar1 is fax1i;

that of axt is ax-1i=}ax−1¿=

at

=

αν

; and that of

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

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.

1

C

B PP

Now, put

CB or BA the side of the square or rhombus DB; m = the modulus, or CB X BA X sin. c; or area of DB, or sine of the angle c to the radius 1; also the absciss cp = x, and the ordinate rq = y. Then, by the nature of the hyperbola, CP X PQ is always equal to DB, that is, xy=m; hence mi

m

And, in

the fluxion of the log. of x, to the modulus m.

the hyperbolic logarithms, the modulus m being 1, there

i

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

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 ea, 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 = e*; then take the logarithm of each, so shall log. z = x×

log. e; take the fluxions of these, so shall =iX log. e,

by the last article; henceż zi × log. e = e2i X 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)nxx ni x log. (a+c).

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

and the fluxions give = i log. y + x

zxy y

= zx. log. y + yayy, 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.

(by substituting y for 2)yi. log.

27. The fluxions of the usual trigonometrical quantities, sin. z, cos. z, &c. are easily found by blending these principles with the analytical formulæ at pa. 395, vol. i.

We assume the proportionality of the increments, and of their contemporaneous fluxions, and proceed thus:

To find o sin. z, we suppose that by a motion of one of the legs including the angle, it becomes z+x' or z + ż. Then o sin. zsin. (z + ż) — sin. z. But by equa. 9. p. 395, vol. i. we have

sin. (z + ż) = sin. z. cos. ż + sin. ż cos. z. But the sine of an arc indefinitely small does not differ sensi. bly from that arc itself, nor its cosine differ perceptibly from radius; hence we have sin. żż, and cos. = 1; and therefore sin. (≈ + ż) = sin. z +ż cos. z; whence sin. (z+ż)—sin. z, or q(sin. z) : = =ż cos. z, viz. the fluxion of the sine of an arc whose radius is unity, is equal to the product of the fluxion of the arc into the cosine of the same arc.

-

28. In like manner, the fluxion of cos. z, or cos. (z + ż) -cos. zcos. z cos. ż- sin. z sin. ż cos. z, or since cos. (z+ż) =cos. z cos ż — sin. z sin. z; therefore, because sin. żż, and cos. ż = 1, we have o cos. z = cos. z -ż sin. zcos. z=— ż sin. z, that is, the fluxion of the cosine of an arc, radius being 1, is found by multiplying the fluxion of the arc (taken with a contrary sign) by the sine of the same arc.

29. By means of these two formulæ, many other fluxional expressions may be found, viz.

sin."z =m sin.m-1z ż cos. z.

❤ cos."z= m cos."-lz ż sin. z.

30. Hence, by the way, will flow this useful practical conclusion, that if x be any arc, then

OF SECOND, THIRD, &c. FLUXIONS.

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

31. If the rate or celerity with which any flowing quan. tity changes its magnituse 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 i ï, , &c.; where each is the fluxion of the one next before it. 32. This description of the higher orders of fluxions may be illustrated by the figures exhibited in art. 8, where, if x denote the absciss AP, and y the ordinate pq; and if the or. dinate PQ or y flow along the absciss AP or x, with a uniform motion; then the fluxion of x, namely, i = rp or qr, is a constant quantity, or 0, in all the figures. Also, in fig. 1, in which AQ is a right line, ý rq, or the fluxion of PQ, is a constant quantity, or y 0; for the angle q, = the angle A, being constant, or is to rq, or i to y, in a constant ratio. But in the 2d fig. rq, or the fluxion of ro, continually increases more and more; and in fig. 3 it continually de. creases 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, that is every where equal to a'y; then, taking the fluxions, it is a'y 3x2; 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 4th fluxions a3y = 0,

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

Also the higher orders of fluxions are found in the same

manner as the lower ones.

Thus,

the first fluxion of y3 is

3y'y;

is 2d flux, or the flux. of 3y2ỷ, con.

sidered as the rectangle of 3y', { 3y2ÿ + 6yÿ' ;

and y, is

and the flux. of this again, or the 3d3y2 + 18yÿÿ + 6ÿ3. flux. of y3, is

}

33. If the function proposed were ax", we should find ¤ ax" = nax”—1; the factors n, a, and i being regarded as constant in the first fluxion naz^-1 i, to obtain the second fluxion it will suffice to make "- flow, and to multiply the result by nai; but o̟ x”−1 = (n − 1) xn−2i; we have, there

fore,

1) ax”—2 ¿a.

I) (n − 2) an−31.

− 1) (n − 2) (n − 3) ax”—1 ¿a.

&c.

mth o ax" = n (n − 1) (n − 2) . . . . (n − m + 1)

ɑn-m ¿m,

m being supposed not to exceed n, for it is manifest that in the case of n being integral, the function ax" has only a limited number of fluxions, of which the most elevated is the nth, and which of course is expressed by the formula,

nth q ax” = n (n − 1) (n − 2) ....3.2.1. ain in which state it admits no longer of being put into fluxions, as it contains no variable quantity, or, in other words, its fluxion is equal to zero.

34. In the foregoing articles, it has been supposed that the fluents increase, or that their fluxions are positive; but it often happens that some fluents decrease, and that therefore their fluxions are negative and whenever this is the case, the sign of the fluxion must be changed, or made contrary to that of the fluent. So, of the rectangle xy, when both x and y increase together, the fluxion is ży + xy: but if one of them, as y, decrease, while the other, x, increases; then, the fluxion of y being y, the fluxion of ry will