THE INVERSE METHOD, OR THE FINDING
OF FLUENTS.

37. IT has been observed, that a Fluent, or Flowing Quantity, is the variable quantity which is considered as increasing or decreasing. Or, the fluent of a given fluxion, is such a quantity, that its fluxion, found according to the foregoing rules, shall be the same as the fluxion given or proposed.

38. It may be further observed, that Contemporary Fluents, or Contemporary Fluxions, are such as flow to. gether, or for the same time-When contemporary fluents are always equal, or in any constant ratio; then also are their fluxions respectively either equal, or in that same constant ratio. That is, if x = y, then is i ý; or if xy::n: 1, then is i ý :: n : 1; or if x = ny, then is i= ný.

39. 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, much 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.

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

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,

41. 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. verse of the 1st rule for finding fluxions. So if the fluxion proposed be

Which is the re

3x5i.

3x5;

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

divide by the index 6, and it is

3° or x,

which is the fluent of the proposed fluxion 3x3i.

FINDING OF FLUENTS.

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

Put a single variable letter for the compound root; and substitute its power 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 by applied.

=

Thus, if the given fluxion be y = (a2 + x2)31⁄23, 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 within the vinculum: therefore, putting ża2 + x2, thence x2=z-a, the fluxion of which is 2xi = ż; hence then x3i = x2ż = ¦ż(z — a2), and the given fluxion ÿ, or ¿1⁄23ż — 1a21⁄23ż; and (a2 + x2)3⁄4x2¿, is = ¿z3¿(z —a2)

hence the fluent y is =

a3) or

=

Or, by substituting the value of z instead of it, the same fluent is 3 (a2+x2)3×(‚'¿x2—‚3‚a2), or ‚¿ (a2+x2) × (x2—fa3).

In like manner for the following examples.

To find the fluent of √a+ cx × x3ï.

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

To find the fluent of (a+cx2)31×dx3¿.

czż

To find the fluent of

or (a + z)*czż.

√a+z

Cz3n-12

To find the fluent of

(a+z”)−‡cx3n~1ż.

√a+z"

43. 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 ży + xy, then the fluent of ży is ry, supposing y constant: and the fluent of ry is also ry, supposing x constant: therefore ry is the required fluent of the given fluxion ży + xỳ.

44. When the given Fluxional Expression is in this Form

iy-xy namely, a fraction, including Two Quantities, being y 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

divided by the latter, by the reverse of rule 4, of finding fluxions. That is,

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

therefore, by that case, is the fluent of the whole

y

45. 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, by rule 6, for finding fluxions.

46. Many fluents may be found by the Direct Method thus:

Take the fluxion again of the given fluxion, or the second i2

fluxion of the fluent sought; into which substitute

y

for x,

for ÿ, &c.; that is, make x, i, ï, as also y, ý, ÿ, &c. to be in continual proportion, or so that x:::: x, and yyyy, &c.; then divide the square of the given