Thus, the expression or sum a+br, evidently increases as x, or the term br, increases; therefore the given expression will be the greatest, or a maximum, when a is the greatest, or infinite: and the same expression will be a minimum, or the least, when x is the least, or nothing.

Again, in the algebraic expression a2 bx, where a and b denote constant or invariable quantities, and r a flowing or variable one. Now, it is evident that the value of this remainder or difference, a2 bx, will increase, as the term bx, or as x, decreases; therefore the former will be the greatest, when the latter is the smallest; that is a2 b.r is a maximum, when x is the least, or nothing at all; and the difference is the least, when x is the greatest.

50. Some variable quantities increase continually; and so have no maximum, but what is infinite. Others again decrease continually; and so have no minimum, but what is of no magnitude, or nothing. But, on the other hand, some variable quantities increase only to a certain finite magnitude, called their Maximum, or greatest state, and after that they decrease again. While others decrease to a certain finite magnitude, called their Minimum, or least state, and afterwards increase again. And lastly, some quantities have several maxima and minima.

Thus, for example, the ordinate BC of the parabola, or such-like curve, flowing along the axis AB from the vertex A, continually increases, and has no limit or maximum. And the ordinate GF of the curve EFH, flowing from E towards H, continually decreases to nothing when it arrives at the point H. But in the circle ILM, the ordinate only increases to a certain magnitude, namely, the radius, when it arrives at the middle as at KL, which is its maximum; and after that it decreases again to nothing, at the point M. And in the curve NOQ, the ordinate decreases only to the position OP, where it is least, or a minimum; and after that it continually increases towards q. But in the curve RSU &c, the ordinates have several maxima, as ST, wx, and several minima, aş vʊ, yz, &c.

51. Now,

34. It is easy to find the fluxions to all the given forms of fluents; but, on the contrary, it is difficult to find the fluents of many given fluxions; and indeed there are numberless cases in which this cannot at all be done, excepting by the quadrature and rectification of curve lines, or by logarithms, or by infinite series. For, it is only in certain particular forms and cases that the fluents of given fluxions can be found; there being no method of performing this universally, a priori, by a direct investigation, like finding the fluxion of a given fluent quantity. We can only therefore lay down a few rules for such forms of fluxions as we know, from the direct method, belong to such and such kinds of flowing quantities : and these rules, it is evident, must chiefly consist in performing such operations as are the reverse of those by which the fluxions are found of given fluent quantities. The principal cases of which are as follow.

35. To find the Fluent of a Simple Fluxion; or of that in which there is no variable quantity, and only one fluxional quantity.

This is done by barely substituting the variable or flowing quantity instead of its fluxion; being the result or reverse of the notation only. Thus,

The fluent of ax is ar.

The fluent of aỷ + 2ỷ is ay + 2y.
The fluent of √ a2 + x2 is √ a2 + x2.

36. When any Power of a flowing quantity is Multiplied by the Fluxion of the Root:

Then, having substituted, as before, the flowing quantity, for its fluxion, divide the result by the new index of the power. Or, which is the same thing, take out, or divide by, the fluxion of the root; add 1 to the index of the power; and divide by the index so increased. Which is the reverse of the 1st rule for finding fluxions.

So, if the fluxion proposed be

Leave out, or divide by, x, then it is add 1 to the index, and it is

divide by the index 6, and it is

3.xx.

which is the fluent of the proposed fluxion 3x3x.

In like manner,

The fluent of 2arx is ax2.

The fluent of 3xx is x3,

The

The fluent of r*x + 3y3j is ‡x3 + {y}•

The fluent of x is ".

The fluent of ny"-1y is

37. When the Root under a Vinculum is a Compound Quantity ; and the Index of the part or factor Without the Vinculum, increased by 1, is some Multiple of that Under the Vinculum :

Put a single variable letter for the compound root; and substitute its powers and fluxion instead of those of the same value, in the given quantity; so will it be reduced to a simpler form, to which the preceding rule can then be applied.

2

Thus, if the given fluxion be ǹ = (a2 + a2)3×3, where 3, the index of the quantity without the vinculum, increased by 1, making 4, which is just the double of 2, the exponent of a within the vinculum: therefore, putting z = a2 + x2, thence 2=% a2, the fluxion of which is 2xx = 2; hence

-

then 3==(za2), and the given fluxion F, or

5

=3z3

Or, by substituting the value of z instead of it, the same fluent is 3(a2+x2)† × († ̧22—z3å3)‚or·‚3⁄4(a2+x2) × (x2 — ja2),

In like manner for the following examples.

To find the fluent of Na+ cx × x3¿

To find the fluent of (a + cx)3x2x.

To find the fluent of (a + cx2)} × dx31⁄2.

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 + xỷ, then the fluent of xy is xy, supposing y constant: and the fluent of ry is also ry, supposing r constant: therefore xy is the required fluent of the given fluxion xy + xÿ.

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:

Though, indeed, the examples of this case may be per formed by the foregoing one. Thus, the given fluxion -

*y

ży - xÿ

reduces to

x xy x

or

y2

xjy2; of which,

y

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.