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FLUXIONS.

1. In this branch of analysis, quantities are usually distinguished into constant and variable.

A constant quantity, is that which retains always the same value. A variable quantity is that which may increase or decrease; and which, while it changes from one magnitude to another, passes through all the intermediate states.

Thus, in trigonometry, the radius of a circle is a constant quantity; while an arc of the circle, as also its sine, tangent, and secant are variable quantities.

It is usual to denote constant quantities by the first letters of the alphabet, a, b, c, &c., and variable quanties by the last letters, x, y, z, &c. So that in the equation y2=ax+bx2, x and y are to be regarded as variable, and a and b as constant.

2. Let us now suppose that any variable quantity a receives a finite augmentation or increment e, so that after having received this increase, its new state may be expressed by r+e; then our next step must be to ascertain what will be the corresponding increment of any other function of x.

First, it is evident that if x becomes x + e, its square x2 will become x2+2ex+e2; and therefore the ratio of the two increments will be

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But as e diminishes, this ratio will augment, and

approach nearer and nearer to that of

1

2x

However it will only

equal this latter ratio when e vanishes, or at least becomes indefinitely

1

small. Consequently the ratio is the limit of all the ratios which

2x

any simultaneous finite increments of x and xx can bear to one another. Similarly when a becomes a +e, a3 will become a3+3e x2 + 3e2 x+e3, and the ratio of the increments will be

e

3ex+3exe

=

1

3x2+e(3x+e)

now here also, as e diminishes, this ratio approaches towards, and

Gg

1

at last coincides with

which in this case is the limiting ratio.

3x2

And by a similar process we shall find the limiting ratio of the increments of " and x to be

1

Now the indefinitely small increment of any variable quantity is called its fluxion; the variable quantity itself is termed a fluent, and it is the first business of the doctrine of fluxions to find in all cases such limiting ratios between the indefinitely small increments of any function, and of the variable quantity upon which it depends.

If the quantity be represented by a single letter as x, the fluxion is denoted by x; and if it be a compound quantity as x", the fluxion will be denoted by flux. (x").

From what we have just shewn, it appears that flux. (x) or i: flux (x") :: 1 ; nx-1; and therefore that the fluxion of x"=nxTM-1 flux (x), is called the fluxional coefficient. The quantity na”1= Alux (*)

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As an example of the mode of determining these limits, let it be required to draw a tangent to the point M of the curve AMm; or, which amounts to the same, let it be required to determine the subtangent PT.

S

Suppose that the absciss AP=r, is increased A by a finite quantity Pp=e; draw the ordinate PM=y, and determine the ordinate mp, by P substituting x+e, in place of x, in the equation of the curve. Whatever may be the value of this ordinate, we may always represent it by y + Pe+Qe2 + Re3+ &c. (P, Q, R, &c. being P functions of x); therefore we shall have for the

expression of rm, the corresponding increment of PM, the quantity Pe+Qe2+ Re3 +&c.

This premised, let there be drawn the secant SMm, and the line Mr parallel and equal to Pp; we shall have PS=

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y P+Qe+Re+&c.

Now supposing the point p to approach nearer and nearer to the point P, the point m will come nearer to M, and the point S nearer

to the point T; but we shall always have PS=

y P+Qe+ Re2+&c.

If the quantity Pp diminishes still farther, and becomes indefinitely small, the point m will almost coincide with M, and the secant become almost a tangent. But if e vanishes, the ratio already found

is reduced to PS becomes PT, Sm becomes TM, which will

now have only the point M' in common with the curve, and the subtangent is determined by this limit.

Thus if AMm is a parabola, we must substitute x+e to x in the

1

equation y=/pr, and we shall have y=p1 (x+e)2=p2x2+.

x2

&c., which gives P=√, whence PTP=2r, as we have

already found it. See Conic Sections.

P

Here we must observe that the quantities Pp, or Mr, and rm, which gradually diminish as the point p approaches the point P, are the respective elements of the abscissa AP, and of the ordinate MP.

These elements, however small we suppose them, preserve the same ratio to one another as the finite quantities to which they belong, as is visible by the mere inspection of the similar triangles TPM, and Mmr.

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As the finite quantities x and y have their fluxions x and y, these fluxions have also in turn their fluxions.

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These are called second fluxions, to distinguish them from the first, and are denoted by s,y. Similarly the third fluxions are marked 3, y, &c.

RULES FOR FINDING THE FLUXIONS OF ANY PROPOSED FUNCTIONS.

receives any

7. Given the equation y-ax, if we suppose that infinitely small increase, denoted by x, y will also receive an increase, which we shall denote by y, and we shall have y+y= ar+ax; whence yax; and this is the fluxion of the proposed equation.

If the equation had been b+y=a-c, we should have found the same result; for constant quantities have no fluxion.

8. And whenever the variable quantities do not exceed the first degree, we shall obtain the fluxions of the proposed quantities, by erasing the constant terms and substituting in place of the variable quantities their proper fluxions.

If, for example, it were required to find the fluxion of bx+cy—a =mx+f, we should find by the above rule bx+cy=

n

m

n

But if the variable quantities are raised to other powers than the first; if, for example, we have y=xm, then supposing that a becomes x+x we shall have y+y=(x+x)" = x2+mxm-·1 x +

m.m-1
2

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&c. Now, 3, &c. vanish in respect to ; consequently there remains ym. Hence if m=2, y=2xx; if m=3, y=3r1 i, &c.

9. In general, to find the fluxion of a variable quantity raised to any power, diminish its exponent by unity, and multiply the result by its original exponent and by its fluxion.

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10. When two variables and y are multiplied together, then flux (x y) = (x+x) (y+y) — xy=yx+xy+xy=yr+ry, because ry

vanishes.

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Similarly flux (xyz) =z × finx (xy)+ xyż=xyż+xzy+yzx, and flux (uxyz) = yz flux (ux) + ux flux (yz) = uxyz + uxzy + uyzx+ xyzu.

Hence in general, to find the flurion of the product of any number of variable quantities, find the fluxions of one only at a time, as if

all the rest were constant; do the same thing with respect to each quantity, and then collect together all these fluxions.

For example let the quantity be r3y. If we consider y only to vary, we have a3y; and if only x be supposed to vary, we have 3yx. Therefore flux (x3y)=x3y+3yx*x. Similarly flux (x'y3z) =3x2z^y3y+4x2y3z32+2xy3z*x.

11. Let there now be proposed the fraction; I write it thus,

y

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xy1, and by taking the fluxion (9 and 10) we have flux (————y1z— xy yx-xy

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Hence to find the fluxion of a fraction containing variable quantities, we must 1o, multiply the denominator by the fluxion of the numerator; 2dly, multiply the numerator by the fluxion of the denominator; 3dly, subtract the latter product from the former, and divide the remainder by the square of the denominator.

m

12. If x=√(az +2"), we may write it thus (ax+2")",

applying the rule at art. (9) we shall find

m

1.

z (a+2x)
(az+z2)" (ax+2zz) = m√ (az+z3) 1

and

Consequently to find the fluxion of any radical of the mth degree, we must divide the fluxion of the quantity beneath the radical sign by the exponent m, and by the mth root of this quantity raised to the power m--1.

By these few rules, we are enabled to find the fluxion of gebraic quantity whatever.

Examples for Practice.

any

al

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