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OF SECOND, THIRD, &c. FLUXIONS.

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13. The second Auxion of a quantity is the fluxion of the first fluxion. The third fluxion is the fluxion of the second, and so on: thus a signifies the fuxion of x ; C, that of ö. The

square

of the fuxion ë is written **; its mth power is written ti", &c.

From what we have just said respecting first fluxions, it will be easy to find the second fluxions, &c. Thus if we require the second fluxions of xa; we shall have for the first fluxion 203; consequently the second fluxion will be 2.cx+2x8=2c* +2rë. Similarly, since flux (3)=mxmc, we shall 2d-flux (3*)=mxml ö+m.m-l:**** again flux (xy)=xy+yč; therefore 2d flut (xy)=rý+yö +2yc. Since flur yx - XY xy

#y, we easily infer that 2d flux (5) y yy y yy xy. ay, 2xya _2.x

ya

my _2xy y yy yyyy

yo

Y

yy 2xy*+yöx— ryy2yxy, and in the same manner with other func

y3 tions.

By the same principles we are enabled to find the third, fourth, &c. Auxions, and in general the fluxions of any degree, of all sorts of quantities affected with x and y. For example the fluxion of ye is yö +oye.... that of ✓ (**+y*)

yz yx is *x+yy

that of ye is a +

yx уху

; and that of the (2a +ya)

y

y

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yy

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14. In order to shorten the calculation of the second fluxions of several variable quantities, we generally suppose one of the first fluxions to be constant; that is, we refer the other fluxions to that one, as to a fixed term of comparison. We shall soon see examples of this process.

This kind of supposition simplifies the labour by causing all those terms to disappear which are affected with the fluxion of the quantity assumed as constant.

Forexample, if we required the fluxion of Yx; on the supposition

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that i is constant, we should find the required fluxion to be i-_yty

ух.

and if we considered y as constant, we should obtain : +

y 15. Hitherto we have supposed that the variable quantities of which the fuxions were required, increased all at the same time. If some increased, while others diminished, this would cause no difficulty, for I and y may be positive or negative, as well as any other algebraic quantities. OF LOGARITHMIC AND EXPONENTIAL

FLUXIONS.

+

C

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16. Let it be required to find the fluxion of the natural logarithm of the variable t. We shall denote this logarithm by lx, and making lx=x, we have z+=1(+i); this gives z or flux (lx)=1

23 (2+1) –lx=/(1+y=

&=

la = neglecting 2.x2

' 3x3 the higher powers of x. Hence the fluxion of the logarithm of any quantity is equal to the flurion of that quantity divided by itself. Consequently in a system in which the modulus=m, we shall have flux (lx),= But in the subsequent articles we shall treat only of the natural or hyperbolic logarithms whose modulus=1. By the above rule, we easily find flux (l) =

&c.

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17. If we had the powers of logarithms, or even logarithms of logarithms, it would be easy to find their fluxions. For example, let y=(lt)", we shall have y=m (l.s)If we had y=s" (13)",

m we should obtain y=mars (Lx)" + nam); (lx)*1 = 7*!(l:)*? (n + mlx), &c. Again let y=lr; make l=, and we shall have j= =

-1

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alx

18. The equation fuz (1x) = gives : = flux (la). Conse

quently the fluxion of any quantity is equal to the product of that quantity by the the flusion of its logarithm. This rule may be applied to facilitate finding the fluxions of quantities of any sort, even

mr x if algebraic. {For example flux (**) = flux 2x1 =

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mamla.

xy)

Flux (ry)=xy(+=yx+ xy.

== yi($) =

+ y)=ya+aj. Flux *_y)

y) = y*+

Flux ý

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The same principle may be applied with

ya success in finding the fluxions of exponential quantities; by which name is to be understood, such quantities as have variable exponents.

Such are a®, x, &c. which are of the first order; m, which is of the second, &c.

The fluxion of a“, will be according to this rule as flux la'=a* ffux (xla) = a'ila. Therefore if e be the number 2.7182818, of which the logarithm is 1, we shall have flux (e") =ér. Similarly flux (zy) = x* Aux ( ylx) = z; (jlx + y),

flux

zy &c. 19. We might also have found these fluxions in the following We have seen that n=1+ In +

(In)? _ (in3
+

2.3 Suppose now that n=a", and substitute this value in place of n, and

, we shall find a*=1+la" +

(las)? _ (lar)

+ &c. Now la*=xla, and

2 2.3 (la") = (xla)' = xola; therefore a*=1+ala+

xla, r3 13 a

+

2.3

**x 1a and consequently flux (ax) = xla + xx l'a +

+ &c. = rila x?la, xyla 1 + xla +

+ &c.}=cika 2.3

manner.

+ &c.

2

+

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With respect to such exponentials as s', their fluxion is easily found : for we have fur (")=x** flux (y*lu) = "{w+y'ls (žly + * + ay la + žlaly). If x=y=e, we shall have o ti for the *

é y fluxion of er.

In a similar manner we may find the second, third, &c. fluxions of logarithmic and exponential quantities, but it is needless to dwell longer on this head. OF THE FLUXIONS OF SINES, COSINES, &c.

AND OTHER CIRCULAR FUNCTIONS.

=

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sin a

COS I

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20. Let sin x=y, we shall have y+y=sin (x+c) = sin x cos z + sin r cos x. Now representing an infinitely small arc, we shall have 1st, cos c=1; 2dly, sin a=s. Therefore y+y=sin x ICOS 3, or y=flux sin x=ä сos r. Consequently the fluxion of the sine of any arc is equal to the fluxion of that arc multiplied by its cosine.

21. Since flux sin x=x cos x, if we make x = 90° —y we shall have a = -ý, and flux cos y=-ý sin y, a formula which we might have obtained in either of the two following manners.

ist. Sin? t + cos x = 1. Therefore sin x (flux sin x) + cos x (Aus cos x) =0, and flux cos x = (flux sin x) = - sin x .... or 2dly, flux cos « =cos (x+3) cos a = cos x cose sin

cos x = - I sin x. Hence the fluxion of the cosine of any arc is equal to the negative flucion of that arc multiplied by its sine. 22. Let lang x=2

we shall have cos (flux sin x) flux cos x x sin x cos ar + sin ?x cos ?

cos ? = flux tang r.

Hence the flusion of the tangent of any cos 2x arc is equal to the fluxion of that arc divided by the square of its cosine.

If instead of supposing the radius = 1, we had supposed it = a, we should have had flux tang x =

cos ** 23. Let x=90°-y, we shall have flux cot y =. ; similarly

sin 'y

sin

COS I

aar

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sin y

sin' y

sin 'y

sin y

E

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G

or,

1 fhuiz sec y = flux

fless cos y_ y sin y_ tang y; and

-
cos y
cos'y

сов у
1
flux cosec. y=flux

flux sin y = y cos y y cot y. These same formula may also be found in the following manner. Let the arc AB be denoted by z; its cosine CD by x; its sine BD by y; and suppose the radius to be unity.

Conceive now AB to be lengthened by the indefinitely small arc Bm, and draw mr perpendicularly on BG; we

DA" shall have Bm = 2, Br = -2, and mr =ý. The similar triangles CBD, Bmr, give

Bm : BC :: Br : BD :: mr : CD,

3:1::--:(1-):: yiv (1-y') conseyuently = =

v (1-x)(1-4) Now y=v (1-), or flux sin 2 =

; but r=

(1-7) Co$ %, and we have just shewn that

=;; therefore

V(1-x) flux sin z=z cos z.

Similarly con AB or x = (1-y), and therefore i or flur cos =-yx

=-ż sin .

(1-y') Let AT or tan z=t, we shall have t:1::y: V (1–y'), and cony

y -; therefore i=

y sequently i=

х ✓(1-y")

(1–y")
1-y

✓ (1-ye 1 (1-y) cos z

or flux cot = =

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Hence it follows that flux (), or

17)

Х

as before.

1

y yV(1—y) ya ✓(1-y) sino

These rules enable us to find the first, second, &c. fluxions of any quantity in which there enter sines, cosines, &c.

Thus, for Example.
Flux (sin x)" = m (sin x) *-1 * cos x=m sin xm x cot x.
Second flux (sin x) = x cos x xsin x.
Second flux (cos x)

- flux (-sin is) * sin r-** cosa. Flux (sin mx) = mx cos mi. Flux (cos mx) =

më sin mz.

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