In like manner for the following examples.

To find the fluent of Va + cx xxxi

To find the fluent of (a + cx)*x*x.

To find the fluent of (a + cx2) x dxx.

38. When there are several Terms, involving Two or more Variable Quantities, having the Fluxion of each Multiplied by the other Quantity or Quantities :

Take the fluent of each term, as if there were only one variable quantity in it, namely, that whose fluxion is contained in it, supposing all the others to be constant in that term; then, if the fluents of all the terms, so found, be the very same quantity in all of them, that quantity will be the fluent of the whole. Which is the reverse of the 5th rule. for finding fluxions: Thus, if the given fluxion be xy + xy, then the fluent of xy is xy, supposing y constant: and the fluent of ry is also ry, supposing & constant: therefore ay is the required fluent of the given fluxion xy + xj.

√y 2yVy

1

,

xy-xj 39. When the given Fluxional Expression is in this Form y2 namely, a Fraction, including Two Quantities, being the Fluxion of the former of them drawn into the latter, minus the Fluxion of the latter drawn into the former, and divided by the Square of the latter:

Then, the fluent is the fraction -, -, or the former quantity

y

is And, in like manner,

y

•

2x2yj

Though, indeed, the examples of this case may be per

formed by the foregoing one. xy

y2

reduces to

the fluent of is supposing y constant; and YY the fluent of - xjy is also ry or -, when r is constant; y therefore, by that case, is the fluent of the wholey

40. When the Fluxion of a Quantity is Divided by the Quantity

itself:

Then the fluent is equal to the hyperbolic logarithm of that quantity; or, which is the same thing, the fluent is equal to 2.30258509 multiplied by the common logarithm of the same quantity.

or xx, is the hyp. log. of x.

is 2 x hyp. log. of x, or = hyp. log. r.

*, is a x hyp. log. x, or hyp. log. of r.

The fluent of

In like manner for the following examples.

To find the fluent of a + c x x x

3

To find the fluent of (a + cx)*x*x.

To find the fluent of (a + cx2) x dx**.

38. When there are several Terms, involving riable Quantities, having the Fluxion of eac other Quantity or Quantities :

Take the fluent of each term, as if t variable quantity in it, namely, that whe tained in it, supposing all the others to term; then, if the fluents of all the term very same quantity in all of them, that fluent of the whole. Which is the re for finding fluxions: Thus, if the give then the fluent of xy is xy, supposinc fluent of ry is also ry, supposing x co the required fluent of the given fluxio

ad call

G; then onal F be

taking the e proposed herwise, it is

r=G;

F::G: F, irant sought.

-. and y for j =20+2; then

2xyxy = F,

s of Fluxions and

the most usual forms

en of problems, sate to them; by my proposed fluxion se duent of it will,

Forms.

ogarithms, in the above forms, are the hyperwhich are found by multiplying the common 2-302585092994. And the arcs, whose sine, Xc, are mentioned, have the radius 1, and are e common tables of sines, tangents and secants. numbers m, n, &c, are to be some real quantities, ans fail when m = 0, or n = 0, &c.

the foregoing Table of Forms of Fluxions and Fluents. using the foregoing table, it is to be observed, that lumn serves only to show the number of the form; nd column are the several forms of fluxions, which Saerent kinds or classes; and in the third or last ..e the corresponding fluents.

method of using the table, is this. Having any given, to find its fluent: First, Compare the given with the several forms of fluxions in the second cothe table, till one of the forms be found that agrees done by comparing the terms of the given parts of the tabular fluxion, namely, he one, with that of the other; and the