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Note. The logarithms, in the above forms, are the hyperbolic ones, which are found by multiplying the common logarithms by 2.302585092994. And the arcs, whose sine, or tangent, &c. are mentioned, have the radius 1, and are those in the common tables of sines, tangents, and secants. Also, the numbers m, n, &c. and to be some real quantities, as the forms fail when m―0, or n=0, &c.
The Use of the foregoing Table of Forms of Fluxions and Fluents.
43. In using the foregoing table, it is to be observed, that the first column serves only to show the number of the forms; in the second column are the several forms of fluxions, which are of different kinds or classes; and in the third or last column, are the corresponding fluents.
The method of using the table, is this. Having any fluxion given, to find its fluent: First, Compare the given fluxion with the several forms of fluxions in the second 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, namely, the radical quantity 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 in the general or tabular form of the fluent, and the result will be the particular fluent of the given fluxion: after it is multiplied by any coefficient the proposed fluxion may have.
EXAM. 1. To find the fluent of the fluxion 3x3 This is found to agree with the first form. And, by comparing the fluxions, it appears that x=x, x, and n 1 , or n; which being substituted in the tabular fluent, or gives, after multiplying by 3, the coefficient, 3×x3, or fx, for the fluent sought.
EXAM. 2. To find the fluent of 5x2x3-x3 ̧or 5xa x (c3—x3).
This fluxion, it appears, belongs to the 2d tabular form : for a c3, and X1---- x*, and n = 3 under the vinculum, also m— 1=1, or m=3, and the exponent 1 of x-1 without the vinculum, by using 3 for n, is n 1=2, which agrees with x2 in the given fluxion: so that all the parts of the form are found to correspond. Then, substituting these values into the general fluent, ——— (α-xn)m.
it becomes - §X (c3 — x3) *
EXAM. 3. To find the fluent of
This is found to agree with the 8th form; where
++ in the denominator, or n=3; and the numerator - then becomes x2, which agrees with the numerator in the given fluxion; also a=1. Hence then, by substituting in the general or tabular fluent, log. of a +x", it becomes log. 1+x3.
EXAM. 4. To find the fluent of ax*x.
EXAM. 5. To find the fluent of 2 (10+x2)*xx.
EXAM. 28. To find the fluent of 2x √√ 2x − x2
EXAM. 29. To find the fluent of axx.
EXAM. 30. To find the fluent of 3a2x.
EXAM. 31, To find the fluent of 3zx log. z + 3xzx-1ż.
EXAM. 33. To find the fluent of (2 + x1) x2 x
To find Fluents by Infinite Series.
44. When a given fluxion, whose fluent is required, is so complex, that it cannot be made to agree with any of the forms of the foregoing table of cases, nor made out from the general rules before given; recourse may then be had to the method of infinite series; which is thus performed:
Expand the radical or fraction, in the given fluxion, into an infinite series of simple terms, by the methods given for that purpose in books of algebra; viz. either by division or extraction of roots, or by the binomial theorem, &c.; and multiply every term by the fluxional letter, and by such simple variable factor as the given fluxional expression may contain. Then take the fluent of each term separately, by the foregoing rules, connecting them altogether by their proper signs; and the series will be the fluent sought, after it is multiplied by any constant factor or coefficient which may be contained in the given fluxional expression.
45. It is to be noted however, that the quantities must be so arranged, as that the series produced may be a converging one, rather than diverging: and this is effected by placing the greater terms foremost in the given fluxion. When these are known or constant quantities, the infinite series will be an ascending one; that is, the powers of the variable quantity will ascend or increase; but if the variable quantity be set foremost, the infinite series produced will be a descending one, or the powers of that quantity will decrease always more and more in the succeeding terms, or increase in the denominators of them, which is the same thing.