FLUENT

, or Flowing Quantity, in the Doctrine of Fluxions, is the variable quantity which is considered as increasing or decreasing; or the Fluent of a given fluxion, is that quantity whose fluxion being taken, according to the rules of that doctrine, shall be the same with the given fluxion. See Fluxions.

Contemporary Fluents, are such as flow together or for the same time. And the same is to be understood of Contemporary Fluxions.——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 x. = y.; or if x : y :: n : 1, then is x. : y. :: n : 1; or if x = ny, then is x. = ny..

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 infinite series.

This doctrine, as it was sirst invented by Sir Isaac Newton, so it was carried by him to a considerable degree of perfection, at least as to the most frequent, and most useful forms of fluents; as may be seen in his Fluxions, and in his Quadrature of Curves. Maclaurin, in his Treatise of Fluxions, has made several inquiries into Fluents, reducible to the rectification of the ellipse and hyperbola: and D'Alembert has pusued the same subject, and carried it farther, in the Memoires de l'Acad. de Berlin, tom. 2, p. 200. To the celebrated Mr. Euler this doctrine is greatly indebted, in many parts of his various writings, as well as in the Institutio Calculi Integralis, in 3 vols 4to, Petr. 1768. The ingenious Mr. Cotes contributed very much to this doctrine, in his Harmonia Mensurarum, concerning the measures of ratios and angles, in a large collection of different forms of fluxions, with their corresponding Fluents. And this subject was farther prosecuted in the same way by Walmesley, in his Analyse des Mesures des Rapportes et des Angles, a large vol. in 4to, 1749. Besides many other Authors who, by their ingenious labours, have greatly contributed to facilitate and extend the doctrine of Fluents; as Emerson, Simpson, Landen, Waring, &c, in this country; with l'Hô- | pital, and many other learned foreigners. Lastly, in 1785 was published at Vienna, by M. Paccassi, a German nobleman, Udhandlung uber eine neue Methode zu Integriren, being a method of integrating, or sinding the Fluents of given fluxions, by the rules of the direct method, or by taking again the fluxion of the given fluxion, or the 2d fluxion of the fluent sought; and then making every flowing quantity its fluxion, and 2d fluxion, in geometrical progression; a method however, which, it seems, only holds true in the easiest cases or forms, whose fluents are easily had by the most common methods. See this method farther explained in the rules following.

As 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 do little more than lay down a few rules for such forms of fluxions as are known, from the direct method, to belong to such and such kinds of Fluents or 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 to given flowing quantities. The principal cases of which are as follow:

I. To find the Fluent of a simple fluxion; or 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, and is the result or reverse of the notation only. Thus, The Fluent of ax. is ax. The Fluent of ay. + 2y. is ay + 2y. The Fluent of √(a2 + x2) is √(a2 + x2).

II. When any power of a flowing quantity is multip'ied 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.

So if the fluxion proposed be3x5x.;
Strike out x., then it is3x5;
add 1 to the index, and it is3x6;
divide by the index 6, and it is(3/6)x6 or (1/2)x6;
which is the Fluent of the proposed fluxion 3x5x..

In like manner, the Fluent of 4axx. is 2ax2; of 3x1/2x is 2x3/2; of axnx. is a/(n + 1)xn + 1; of z./z2 or z-2z. is - z-1 or -1/z. of .

III. 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.

So, if the given fluxion be ; where 3, the index of the quantity without the vinculum, increased by 1, makes 4, which is double of 2, the exponent of x2 within the same; therefore putting , thence , and its fluxion is 2xx. = z.; hence then , and the given quantity F. or ; and the Fluent of each term gives ; or, by substituting the value of z instead of it, the same Fluent is

IV. 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 was 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, that quantity will be the Fluent of the whole.

Thus, if the given fluxion be x.y + xy.. Then, the Fluent of x.y is xy, supposing y constant; and the Fluent of xy. is also xy, when x is constant; therefore the common resulting quantity xy is the required Fluent. of the given fluxion x.y + xy..

And, in like manner, the Fluent of x.yz + xy.z + xyz. is xyz.

V. When the given fluxional expression is in this form (x.y - xy.)/y2, viz, a sraction including two quantities, being the fluxion of the former 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 x/y, or of the former quantity divided by the latter. That is, The Fluent of (x.y - xy.)/y2 is x/y; and the Fluent of (2xx.y2 - 2x2yy.)/y4 is x2/y2.

Though the examples of this case may be performed by the foregoing one. Thus the given fluxion (x.y - xy.)/y2 reduces to x./y - xy./y2 or x./y - xy.y-2; of which the Fluent of x./y is x/y when y is constant; and the Fluent of xy.y-2 is + xy-1 or x/y when x is constant; and therefore, by that case, x/y is the Fluent on the whole (x.y - xy.)/y2.

VI. 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 &c, multiplied by the common log. of the same quantity. |

So, the Fluent of x./x or x-1 x. is the hyp. log. of x; of 2x./x is 2 X hyp. log. of x, or = h. l. of x2; of ax./x is a X h. l. of x, or h. l. of xa; of x./(a + x) is the h. l. of a + x; of 3x2x./(a + x3) is the h. l. of a + x3.

VII. Many Fluents may be found by the direct method of fluxions, thus: Take the fluxion again of the given fluxional expression, or the 2d fluxion of the Fluent sought; into which substitute x.2/x for x.., and y.2/y for y.., &c, that is, make x, x., x.., as also y, y., y.., &c, in continual proportion, or x : x. :: x. : x.., and y : y. :: y. : y.., &c; then divide the square of the given fluxional expression by the 2d fluxion, just found, and the quotient will be the Fluent sought in many cases.

Or the same rule may be otherwise delivered thus: In the given fluxion F., write x for x., y for y., &c, and call the result G, taking also the fluxion of this quantity, G.; then make G. : F. :: G : F, so shall the 4th proportional F be the Fluent as before. And this is the rule of M. Paccassi.

It may be proved if this be the true Fluent, by taking the fluxion of it again, which, if it agree with the proposed fluxion, will shew that the Fluent is right; otherwise, it is wrong.

Thus, if it be proposed to find the Fluent of nxn-1x.. Here F. = nxn-1x.; write first x for x., and it is nxn-1x or nxn=G; the fluxion of this is G. = n2xn-1x.; therefore G. : F. :: G : F becomes n2xn-1x. : nxn-1x. :: nxn : xn = F, the Fluent sought.

For a 2d ex. suppose it be proposed to find the Fluent of x.y + xy.. Here ; then, writing x for x., and y for y., it is xy + xy or 2xy = G; the fluxion of which is ; then G. : F. :: G : F becomes , the Fluent sought.

VIII. To find Fluents by means of a table of forms of Fluxions and Fluents.

In the following table are contained the most usual forms of fluxions that occur in the practical solution of problems, with their corresponding Fluents set opposite to them; by means of which, viz, comparing any proposed fluxion with the corresponding form here, the Fluent of it will be found.

Where it is to be noted, that the logarithms in the said forms, are the hyperbolic ones, which are found by multiplying the common logs. by 2.3025850929940 &c. Also the arcs whose sine, or tangent, &c, are mentioned, have the radius 1, and are those in the common tables of sines, tangents, &c.—And the numbers m, n, &c. are to be some quantities, as the forms fail when n = o, or m = o, &c. |

The Use of the foregoing Table of Forms of Fluxions and Fluents.—In the use of this table, it is to be observed, that the first column serves only to shew the number of the form, as a mark of reference; in the 2d column are the several forms of fluxions, which are of different kinds or classes; and in the 3d or last column are the corresponding Fluents.

The method of using the table is this. Having any fluxion given, whose Fluent it is proposed to find: First, compare the given fluxion with the several forms of fluxions in the 2d column of the table, till one of the forms be found that agrees with it; which is done by comparing the terms of the given fluxion with the like parts of the tabular fluxion, viz, the radical quan<*> tity of the one, with that of the other; and the exponents of the variable quantities of each, both within and without the vinculum; all which, being found to agree or correspond, will give the particular values of the general quantities in the tabular form. Then substitute these particular values, for the same quantities | in the general or tabular form of the Fluent, and the result will be the particular Fluent sought; after it is multiplied by any coefficient the proposed fluxion may have.

For Ex. To find the Fluent of the given fluxional expression 3x(5/3)x.. This agrees with the first form; and by comparing the fluxions, it appears that x = x, and , or n = <*>/3; which being substituted in the tabular Fluent, or (1/n)xn, gives, after multiplying by 3 the coefficient, 3 X (3/8)x8/3 or (9/8)x8/3 for the Fluent sought.

Again, To find the Fluent of 5x2x.√(c3 - x3), or 5x2x..―(c3 - x3)1/2. This belongs to the 2d form; for under the vinculum, , and the exponentn - 1 of xn - 1 without the vinculum, by using 3 for n, is n - 1 =2, which agrees with x2 in the fluxion given; and therefore all the parts of the form are found to answer. Then, substituting these values into the general Fluent, , it becomes .

Thirdly, To find the Fluent of x2x./(1 + x3). This agrees with the 8th form; where in the denominator, or n = 3; and the numerator xn - 1 then becomes x2, which agrees with the numerator in the given fluxion; also a = 1. Hence then, by substituting in the general form of the Fluent 1/n logarithm of a + xn, it becomes 1/3 logarithm of 1 + x3.

IX. To find Fluents by means of Infinite Series.— When a finite form cannot be found to agree with a proposed fluxion, it is then usual to throw it into an infinite series, either by division, or extraction of roots, or by the binomial theorem, &c; after which, the Fluents of all the terms are taken separately.

For Ex. To find the Fluent of (1 - x)/(1 + x - x2)x.. Here, by dividing the numerator by the denominator, this becomes x. - 2xx. + 3x2x. - 5x3x. + 8x4x. &c; and, the Fluents of all the terms being taken, give x - x2 + x3 - (5/4)x4 + (8/5)x5 &c, for the Fluent sought.

To Correct a Fluent.—The Fluent of a given fluxion, found as above, sometimes wants a correction, to make it contemporary with that required by the problem under consideration, &c: for the Fluent of any given fluxion, as x., may be either x (which is found by the rule) or it may be x ± c, that is x plus or minus some constant quantity c; because both x and x ± c have the same fluxion x.: and the finding of this constant quantity, is called correcting the Fluent. Now this correction is to be determined from the nature of the problem in hand, by which we come to know the relation which the Fluent quantities have to each other at some certain point or time. Reduce therefore the general Fluential equation, found by the rules above, to that point or time; then if the equation be true at that point, it is correct; but if not, it wants a correction, and the quantity of that correction is the dif- ference between the two general sides of the equation when reduced to that particular state. Hence the general rule for the correction is this:

Connect the constant, but indeterminate, quantity c with one side of the Fluential equation, as determined by the foregoing rules; then, in this equation, substitute for the variable quantities such values as they are known to have at any particular state, place, or time; and then from that particular state of the equation find the value of c, the constant quantity of the correction.

Ex. To find the Correct Fluent of z. = ax3x.. First the general Fluent of this is z = ax4, or , taking in the correction c.

Now if it be known that z and x begin together, or that z = 0, when x = 0; then writing 0 both for x and z, the general equation becomes , or c = 0; so that, the value of c being 0, the Correct Fluents are z = ax4.

But if z be = 0, when x is = b, any known quantity; then substituting 0 for z, and b for x, in the general equation, it becomes , from which is found c = - ab4; and this being written for it in the general equation, this becomes , for the correct, or contemporary Fluents.

Or lastly, if it be known that z is = some quantity d, when x is equal some other quantity, as b; then substituting d for z, and b for x, in the general Fluential equation , it becomes ; and hence is deduced the value of the correction, viz, ; consequently, writing this value for c in the general equation, it becomes , for the Correct equation of the Fluents in this case.

And hence arises another easy and general way of correcting the Fluents, which is this: In the general equation of the Fluents, write the particular values of the quantities which they are known to have at any certain time; then subtract the sides of the resulting particular equation, from the corresponding sides of the general one, and the remainders will give the Correct equation of the Fluents sought. So, as above, the general equation being ; write d for z, and b for x, then ; hence by subtraction , or , the Correct Fluents as before.

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Entry taken from A Mathematical and Philosophical Dictionary, by Charles Hutton, 1796.

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FLEXION
FLEXURE
FLIE
FLOOD
FLOORING
* FLUENT
FLUID
FLUTES
FLUIDITY
FLUX
FLUXION