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aš Exam. 27. To find the fluent of

x ax2 Exam. 28. To find the fluent of 23 v 2.x ma. EXÁM. 29. To find the fluent of a š. Exam. 30. To find the fluent of 3a2% . EXAM. 31. To find the fluent of 3z*x log.z + 3.x3. Exam. 32. To find the fluent of (1 + x3).xx. Exam. 33. To find the fluent of (2 + **).x**. EXAM. 34. To find the fluent of riva? + 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 in 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 all together by their proper signs, and the series will be the fluent sought, after it is multiplied by any constant factor or co-efficient 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 inay be a converging one, rather than diverging: and this is effected h placing the greater terms foremost in the given i When these are known or constant quantities, the series will be an ascending one; that is, the power variable quantity will ascend or increase ; but if the quantity be set foremost, the infinite series pred a descending one, or the powers of that crease always more and more in the succe crease in the denominators of them, whic

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For example, to find the fluent of

Here, by dividing the numerator by the denominator, the

1 + x

' 22.

1

proposed fluxion becomesi - 2x:+3x; — 5x3; +8x4— &c;
then the fluents of all the terms being taken, give
x - 2? + 33 - 4x4 + 2.25 - &c, for the fluent sought.

Again, to find the fluent of iN x?.
Here, by extracting the root, or expanding the radical
quantity v1 – x?, the given fluxion becomes
3 - 4x** - - * *

&c. Then the fluents of all the terms, being taken, give x - 3x3

3x3 – 4625 Thir? - &c, for the fluent sought.

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OTHER EXAMPLES.

/

bxx EXAM. 1. To And the fluent of both in an ascend

a - X
ing and descending series.

bi
EXAM. 2. To find the fluent of in both series.

atx

31 EXAM. 3. To find the fluent of

(a + x)2".

1 x2 + 2x4 EXAM. 4. To find the fluent of

-a, 1 + x x2

bi EXAM. 5. Given ż=

to find z. a + 729

a2 + x2 EXAM. 6. Given z =

jc to find %.

a + x
EXAM.-7. Given z = 3iva + x, to find z.
EXAM. 8. Given z = 2in ax + xż, to find %.
EXAM. 9. Given z = 4* a? x?, to find

5ai
EXAM. 10. Given z =

to find z.

V x2
Exam. 11. Given ż= 28a3 F, to find z.

Зах
EXAM. 12. Given z =

to find z.

var
Exam. 13. Given z = 233,23 + x4 + xs, to find %.
EXAM. 14. Given z = 54 N 0.38

TO

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%.

XX

XX, to find ;.

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, as i may be either x, 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+c have 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 fore. going 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.

EXAMPLES.

47. EXAM. 1. To find the correct fluent of 2 = 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 mis 0, when x = : 0; then writing o for both x and z, the general equation becomes 0 = 0 +c, or = c; so' that, the value of c being 0, the correct fluents are 2 = ax4.

But if z be = 0, when x is = b, any known quantity; then substituting 0 for %, and b for X, in the general equation, it becomes 0 = abt + c, 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,

ab4;

Or, if it be known that z is = some quantity d, when : is = some other quantity as b; then substituting d for z, and b for x, in the general fluential equation z = ax4 + c, it becomes d = abt tc; and hence is deduced the value of the correction, namely, crd - abt ; consequently, writing this value for c in the general equation, it becomes Zar4

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 % = axt; write d for z, and b for x, then d = abt; hence, by subtraction,

d = axt aba, or 2 = ax4 ab4 + d, the correct fluents as before. EXAM. 2. To find the correct fluents of 2 = 5.x*; z being

= 0 when x is = a.

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Exam. 3. To find the correct fluents of ź = 3x va tri z and .r being = 0 at the same time.

2ax EXAM. 4. To find the correct fluent of ź = ; sup

att posing z and x to begin to flow together, or to be each = 0 at the same time.

23 ExAM. 5. To find the correct fluents of 3 = ;

posing z and 2' to begin together.

a' + i sup

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,

Thus, the expression or sum a* + br, 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?. bır 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.

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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 now, the ordinate decreases only to the position OP, where it is least, or a minimum; and after that it continually increases towards e. But in the curve Rsu &c,' the ordinates have several maxima, as st, wx, and several minima, as Vu, yz, &c.

51. Now,

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