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1-X For example, to find the fluent of
1 + x Here, by dividing the numerator by the denominator, the proposed fluxion becomesi – 2x:+ 3x?; – 5.x38 +8x^3 - &c; then the fluents of all the terms being taken, give x - 212 + x3 – 4x4 + 8.25 - &c, for the fluent sought.
Again, to find the fluent of iN l - x2 Here, by extracting the root, or expanding the radical quantity v 1 - x?, the given fluxion becomes á - 1x?* - fx43
Taxi - &c. Then the fluents of all the terms, being taken, give r – fx3 – 4925 – TEX' - &c,
x tri for the fluent sought.
bxă EXAM. 1. To find the fluent of both in an ascend
a ing and descending series.
bä EXAM. 2. To find the fluent of in both series.
3 EXAM. 3. To find the fluent of
(a + x)2".
1 x2 + 2x4 EXAM. 4. To find the fluent of
-6, 1 + x x2
bic EXAM. 5. Given 2 =
to find z. a2 + x2
a2 + x2 Exam. 6. Given z =
å to find z.
a + x
z 431 '?
5 ai EXAM. 10. Given z =
to find z. V x2 a?
ExAM. 11. Given z = 2837a3 X?, to find z.
to find z.
XX, to find :
TO as i
To Correct the Fluent of any Given Fluxion. 46. The fluxion found from a given fluent, is always perfect and complete; but the fluent found from a given Auxion is not always so; as it often wants a correction, to make it contemporaneous with that required by the problem under consideration, &c: for, the fluent of any given fluxion, may
be either which is found by the rule, or it may be x + c, or X c, that is x plus or minus some constant quantity c; because both x and x Echave the same fluxion x, and the finding of the constant quantity c, to be added or subtracted with the fluent as found by the foregoing rules, 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, supposed to be found by the foregoing rules, to that point or time; then if the equation be true, it is correct; but if not, it wants a correction, and the quantity of the correction, is the difference between the two general sides of the equation when reduced to that particular point. 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.
47. EXAM. 1. To find the correct fluent of ż = axis.
The general fluent is z = ax4, or z = ax+ + c, taking in the correction c.
Now, if it be known that z and x begin together, or that is 0, when x = 0; then writing 0 for both x and %, the general equation becomes 0 = 0 +c, or = c; so that, the value of c being 0, the correct fluents are x = ax4.
But if z be = 0, when x is = b, any known quantity; then substituting o for %, and b for x, in the general equation, it becomes 0 = abt to, and hence we find c=which being written for c in the general fluential equation, it becomes z = ax4 ab, for the correct fluents.
Or, if it be known that z is = some quantity d, when z is = some other quantity as b; then substituting d for z, and 6 for x, in the general fluential equation z = ax4 + c, it becomes d ab* tc; and hence is deduced the value of the correction, namely, c = d - abt ; consequently, writing this value for c in the general equation, it becomes z = ax* - abt + d, for the correct equation of the fluents in this case.
48. 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 or position ; 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, the general equation being % = a.x4 i write d for 2, and b for x, then d=
abt ; hence, by subtraction, % - d = ax4 - aba,
or z = ax4 ab4 + d, the correct fluents as before. Exam. 2. To find the correct fluents of ź = 5x*; z being
= 0 when x is = a.
EXAM. 3. To find the correct fluents of į = 3iva + xi z and x being = 0 at the same time.
2a: EXAM. 4. To find the correct fluent of į =
at X posing z and x to begin to flow together, or to be each = 0 at the same time.
2 EXAM. 5. To find the correct fluents of z =
a? + posing z and l' to begin together.
OF MAXIMA AND MINIMA; OR, THE GREATEST AND
LEAST MAGNITUDE OF VARIABLE OR FLOWING QUANTITIES.
49. MAXIMUM, denotes the greatest state or quantity attainable in any given case, or the greatest value of a variable quantity: by which it stands opposed to Minimum, which is the least possible quantity in any case.
Thus, the expression or sum ax + bx, evidently increases as X, or the term bx, increases; therefore the given expression will be the greatest, or a maximum, when x 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 a? – bx, where a and b denote constant or invariable quantities, and x a flowing or variable one.
Now, it is evident that the value of this remainder or difference, a' - 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 a bir 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 g. But in the curve Rsu &c, the ordinates have several maxima, as st, wx, and several minima, as Vu, yz, &c.
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 ax.
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 ist rule for finding fluxions. So, if the fluxion proposed be
3.75i. Leave out, or divide by, x, then it is add 1 to the index, and it is
3.2; divide by the index 6, and it is
2.2.6 or 2.216, which is the fluent of the proposed fluxion 3x**.
In like manner,