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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, I, y, , &c. So that in the equation y'=ax + bx?, x and y are to be regarded as variable, and a and b as constant.
2. Let us now suppose that any variable quantity x receives a finile augmentation or increment e, so that after having received this increase, its new state may be expressed by «te; 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 3* +2ex te>; and therefore the ratio of the two increments will be
But as e diminishes, this ratio will augment, and 2x te
1 approach nearer and nearer to that of However it will only
2.r 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 x becomes x + e, x will become x3 + 3e x* + 3e* x +e",
1 and the ratio of the increments will be
Zer* + 3er + € 3.r’ +213x +e) now here also, as e diminishes, this ratio approaches towards, and
1 at last coincides with which in this case is the limiting ralio.
3r? And by a similar process we shall find the limiting ratio of the increments of r" and x to be
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 Auxions to find in all cases such limiting ralios 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 *; and if it be a compound quantity as ", the fluxion will be denoted by flux. (x*).
From what we have just shewn, it appears that flux. (x) or x : flux (z") :: 1 : nx); and therefore that the fluxion of x*=nr.. The quantity nz--=Aux
(x"), is called the fluxional coefficient. "
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,
s! which amounts to the same, let it be required to determine the subtangent PT.
T! 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 +Qe? + Re> + &c. (P, Q, R, &c. being I functions of x); therefore we shall have for the expression of rm, the corresponding increment of PM, the quantity Pe +Qe? + Re? + &c. This premised, let there be drawn the secant SMm, and the line
y Mr parallel and equal to Pp; we shall have PS=;
P+Qe + Re? + &c. PS Mr (because
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 + Re: +&c
If the quantity Pp diminishes still farther, and becomes indefinitely small
, the point m will almost coincide with M, and the secant be come almost a tangent. But if e vanishes, the ratio already found is reduced to , PS becomes PT, Sm becomes TM, which will
P 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 equation y=vpx, and we shall have y=p+ (x+e)}=px*+
already found it. See Conic Sections.
Here we must observe that the quantities Pp, or Mr, and rni, 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.
As the finite quantities x and y have their fuxions s' and y, these fluxions have also in turn their flucions.
These are called second flucions, to distinguish them from the first, and are denoted by 2,3. Similarly the third Auxions are marked
RULES FOR FINDING THE FLUXIONS
OF ANY PROPOSED FUNCTIONS.
7. Given the equation y=ar, if we suppose that x 'receives any 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 tar; whence j =ax; and this is the fluxion of the proposed equation.
y If the equation had been b+y=—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—
**+f, we should find by the abovo rule bi+cj= . But if the variable quantities are raised to other powers than the first; if, for example, we have y=xm, then supposing that becomes x+: we shall have y+y=(x+2) =*" +m2*+ &c. Now , 22, &c. vanish in respect to ; consequently there remains j=mrmiz. Hence if m=2, y=2xr; if m=3, y=3r+ 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.
10. When two variables 2 and y are multiplied together, then flur (= y) = (x+)(y+j) — xy=yö +xy+xy=y2 + xy, because my vanishes.
Similarly flux (xyz) =z * finx (xy)+ xyż=ryż+rzy +yzi, and flux (uryz) =yz flux (ux) + ux flur (yz) = uxyż + uxzy + uyzet zyru.
Hence in general, to find the flurion of the product of any number of variable quantities, find the fiuxions of one only at a time, as if
zy", and by taking the Auxion (9 and 10) we have flus (p=r sy-y=
all the rest were constant ; do the same thing with respect to each quan. tity, and then collect together all these fluxions. For example let the quantity be røy. If we consider y only to
у vary, we have woy; and if only I be supposed to vary, we have By s*r. Therefore flux (x’y)=roy+3yrir. Similarly flux (x*y*74) 3y x
** =3x+z+yoy +4x*y*x*2 +Zxyrtë. 11. Let there now be proposed the fraction **; I write it thus,
xy yr-ry Y yy yy Hence to find the fluxion of a fraction containing variable quantities, we must 1°, multiply the denominator by the fluxion of the numeralor ; 2dly, multiply the numeralor by the fuxion of the denominator ; 3dly, subtract the latter product from the former, and divide the remainder by the square of the denominator.
12. Pr=" (az + 3), we may write it thue (98 +2) +, and applying the rule at art. (9) we shall find
ż (a +23)
mm (az+z) Consequently to find the flusion of any radical of the mth degree, we must divide the flucion of the quantity beneath the radical sign by the exponent m, and by the mth root of this quantity raised to the By these few rules, we are enabled to find the fluxion of
ala gebraic quantity whatever.
Examples for Practice.
(liq+y) y Ans.
(2y+yy) of (ax + box + cx').