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the flu. of the next in the series, or

will be found,

by assuming / (x2 + 2ax) = y; and so on for any other ¤

of the same form. As, if the fluent of


be given

c; then, by assuming x-1 (x2+2ax)=y, the fluent

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2n — 1




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found; and thus the series may be continued exactly as in the 3d ex. only taking- 2ax for + 2ax.

80. Again, having given the fluent of





✓ (2αx-x2 ')'

X circular arc to radius a and versed sine x, the fluents

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assigned by the same method of continuation. Thus,

Ex. 1. For the fluent of




y; the required fluent will be found(2ax-x)+4 or arc to radius a and vers. x.

Ex. 2. In like manner the fluent of

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3 α xx √(2ax--x2) where a denotes the arc mentioned in the last example. xxx Ex. 3. And in general the fluent of √(2αx-x2) 2n-1 1



·ac· x2-1 ✓ (2ax-x), where c is the fluent of



✓ (2ax-x3)

the next preceding term in the series.

81. Thus also, the fluent of x(x-a) being given, =

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X a), by the 2d form, the fluents of xx(x — α); x 3 & √
x—α), &c.
x^x √ (x—a), may be found. And in gene-
ral, if the fluent of x-(xa) c be given; then by

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assuming x”(x —a)¤=y, the fluent of xx(x—a) is found=

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2na 2n+3


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82. Also, given the fluent of (xa)m; which is m+1 (x-a)m+1 by the 2d form, the fluents of the series (x—a)mxx, a)mx3x&c... (a)m can be found. And in general, the fluent of (x-a)-1 being given c; then by assuming (x — a)mtiny, the fluent of (x-a)" is found x(x-a)m+1+nac


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Also, by the same way of continuation, the fluents of (ax) and of xx (ax) may be found.

83. When the fluxional expression contains a trinomial quantity, as (b + cx + x2), this may be reduced to a binomial, by substituting another letter for the unknown one x, connected with half the coefficient of the middle term with its sign. Thus, put z=x+c: then 22=x+x+1c; theref z2 — 1c2 = x2+cx, and z+b1c2 = x2+cx+b the given 22 trinomial which is z2 + a2, by putting a2 = b

Ex. 1. To find the fluent of

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Here z = x + 2; then z2 = x2 + 4x + 4, and 2a + 1 5 + 4x + x2, also * z; theref. the proposed fluxion re

duces to


the fluent of which, by the 12th form in

this vol. is 3 hyp. log. of 2 + √ (1 + z) =3 hyp. log. z+2 + √ (5+ 4x + x2).

Ex. 2. To find the fluent of (b+cx+dx2)=x √✓ à × √


(2 + 2 x + x3).

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Here assuming x+ =2; then x=2, and the proposed

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flux. reduces to ż✔✅d×√(z2+ d 4da) = ż√/ d× √ ( z2+a2),

b putting a2 for d 4d2

; and the fluent will be found by a simi lar process to that employed in ex. 1 art. 75.

Ex. 3. In like manner, for the flu. of x-(b + cx2 +

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dr2"), assuming x2+~=z, nx^~1x=z, and x^~1=−z; hence 2d

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; hence the given fluxion becomes - ž✔d × √✓

(23±a3), and its fluent as in the last example.

Ex. 4. Also, for the fluent of


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xn-1x ; assume an. b+cx+dx2 =z, then the fluxion may be reduced to the form X

and the fluent found as before.


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So far on this subject may suffice on the present occasion. But the student who may wish to see more on this branch, may profitably consult Mr. Dealtry's very methodical and ingenious treatise on Fluxions, lately published, from which several of the foregoing cases and examples have been taken or imitated.


84. 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 a+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 a2 bx, where a and b denote constant or invariable quantities, and a flowing or variable one. Now, it is evident that the value of this remainder or difference, a2-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 a2-bx is a maximum, when x is the least, or nothing at all; and the difference is the least, when x is the greatest.


85. 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

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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 continually increases, and has no limit or maximum. And the ordinate GF of the curve EFH, flowing from E towards ¤, 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 or, where it is least, or a minimum ; and after that it continually increases towards & But in the curve RSU, &c. the ordinates have several maxima, as st, wx, and several minima, as vu, vz, &c.

86. Now, because the fluxion of a variable quantity, is the rate of its increase or decrease: and because the maximum or minimum of a quantity neither increases nor decreases, at those points or states; therefore such maximum or minimum has no fluxion, or the fluxion is then equal to nothing. From which we have the following rule.

To find the Maximum or Minimum.

87. From the nature of the question or problem, find an algebraical expression for the value, or general state of the quantity whose maximum or minimum is required; then take the fluxion of that expression, and put it equal to nothing; from which equation, by dividing by, or leaving out, the fluxional letter and other c quantities, and performing other proper reductions, as in common algebra, the value of the unknown quantity will be obtained, determining the point of the maximum or minimum.

So, if it be required to find the maximum state of the compound expression 100x 5x2±c, or the value of x when 100% 5x3 c is a c is a maximum. The fluxion of this expression is 100x10x=0; which being made = 0, and divided by 10%, the equation is 10 — x 0; and hence = 10. That is, the value of x is 10, when the expression VOL. II. 46




100x - 5x is the greatest. As is easily tried: for if 10 be substituted for x, in that expression, it becomes +c+500: · but if, for x there be substituted any other number, whether great or lesser than 10, that will always be found to be less than + c + 500, which is therefore its greatest possible value, as its maximum.


88. It is evident, that if a maximum or minimum be any way compounded with, or operated on, by a given constant quantity, the result will still be a maximum or minimum. That is, if a maximum or minimum be increased, or decreased, or multiplied, or divided, by a given quantity, or any given power or root of it be taken; the result will still be a maximum or minimum. Thus, if a be a maximum or minimum, then also is xa, or x- a, or ax, or

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or 2a, or Xar


x, still a maximum or minimum. Also, the logarithm, of the same will be a maximum or a minimum. And therefore, if any proposed maximum or minimum can be made simpler by performing any of these operations, it is better to do so, before the expression is put into fluxions.

89. When the expression for a maximum or minimum contains several variable letters or quantities; take the fluxion of it as often as there are variable letters; supposing first one of them only to flow, and the rest to be constant ; then an other only to flow, and the rest constant; and so on for all of them; then putting each of these fluxions = 0, there will be as many equations as unknown letters, from which these may be all determined. For the fluxion of the expression must be equal to nothing in each of these cases; otherwise the expression might become greater or less, without altering the values of the other letters, which are considered as constant. So, if it be required to find the values of x and y when 4x2-xy+2y is a minimum. Then we have,


First, 8xx ay=0,
xy = 0, and 8x - y = 0, or y = 8x.
Secondly, 2y — Xy 0, and 2

And hence y or 8x


x = 0, or x = 2.

* The numbers here found, viz. 2 and 16 for s and y, do not render the propos

ed formula 4x2 —xy+2y either a maximum or minimum.

If x = 1, y=10, then 4x3-xy+2y=14,

x=2, y=16, then 4x3-xy+2y-16,
x=3, y=10, then 4xxy+2y=26.

In general put x=2+e, y=16+ƒ, and by substitution we have

4x2xy+2y= 16+4e2 —ef.

It is evident that if 4e-ef were always positive, whatever values positive or

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