Required the fluxion of the logarithm of

ion of the number is x (a−x)+x (a+x)

(a-x)2

=

and,

ding this by irithm.

a + x

we obtain

2ax a2

for the fluxion of the

The most useful forms of the fluxions of logarithms are the

T

PROB. LXXV.-To express the fluxions of circular cs in terms of the sine, tangent, secant, &c. Let the radius AC ber, the versed sine B=x, the sine BD=y, the tangent AT t, the secant CTs, and the arc AD = v. raw the tangent Ds, and the line sm parallel BD, and Dn parallel to AC, and let sin meet e arc in v, then ns nv. Therefore the rao of Dn to nv is always greater than that of n to ns, but by diminishing Dn it continu

Cm B

ally approaches to that ratio, and at length comes nearer to than any given ratio greater than that of Dn to ns; therefor the ratio of Dn to ns is the limit or fluxion of the ratio of Di to nv, and of course Dn: Ds is the fluxion of the ratio d Dn: Dv. But the triangles nDs, CDB are similar, for CD being a right angle, nDs = BDC; therefore BD: DC::n): Ds, and nD = x, and Ds=v; therefore y:r::x:v= I:

y

like manner, BC: CD: : ns: sD, and ns =y; therefore r—:

ry

:r::y: v=; now r — x — √r2 — yo, and y=√2rx—z

T

therefore v=

2

ry √ra-y2

ry

: AT, or r-x: y: r: t=,, whence i=

v; also CB: CD:: CA: CT, or r—x:r::r:s=

; therefore s::: y:r::x:v, whence again =

r2

and s=

v=

These are the most useful forms of fluxions of circular ares

TO FIND THE SINE AND COSINE OF AN ARC v.

Assume sin. v=av+bv2+cv3, &c. and cos. v=1+mv‡ nv2 + pv3, &c. then sin. v] = av+2bvv+3cv2v, &c. an cos. v) = mv+2nvv+3pv2v, &c. but sin. v=v cos. v, and cos. v=v sin. v, whence we have two equations a+b +3cv2, &c. =1+mv+nv2+pv3, &c. and av+bv2+cr3, &c.m ·2nv · 3pv2, &c. and equating the coefficients we have a = 1,—mo, b=o, n =—12, c=

-

-1

2.3, P=0, d=

&c.; therefore, substituting these value

TO FIND THE LENGTH OF THE ARC OF WHICH THE

TANGENT IS t.

Assume v=at+bt2+cts, &c. then v=ai+2bit +3ct2i, &c.

ut v=m=i—p2i+1a i—toi, and, equating the coeffi

1+12

ents, we have a=1, b=o, c==1, d=o, e=+1, &c.;

=꿍,

OF FLUENTS OR INTEGRALS.

PROP. LXXVI. The limit or fluxion of any quanity may be found by the preceding rules; but it is ften difficult to find the quantity which will produce a iven fluxional expression. This quantity is called the luent or integral: The following are the most general nd simple rules for finding fluents.

RULE 1. If the quantity be simple, and have one variable, dd unity to the index, and divide by the increased index, and y the fluxion of the root.

Thus, because the limit of an+1 is (n+1)" x, therefore the

quantity of which 2 x is the limit will be

xn+1x

(n+1)x

RULE 2. If the quantity be a compound power or radical, and the quantity without the vinculum have a given ratio to the fluxion of the quantity under the vinculum, the fluent may be found by the preceding rule.

Because the limit of (2ax—x 2)3 is § x (2a—2x) (2ax—x2)−1, therefore the integral of å (a—x) (2ax—x2)—§ is

(2 +1

x

■ 2ax — 1 2) −4+1 (a — x); — (2ax — x2)3.

(a-x) 2x

RULE 3. If the quantity consists of as many terms as there

are variable quantities, and each term be the product of the fluxion of one of the variables by all the rest. Take the fluen: of any term, upon the supposition of all the quantities being constant, except that which has its fluxion in it, and it will be the fluent of the whole.

Because the fluxion of xvz is vzx+xzv+xvż, the whole integral of vzx+xzv+xvz will be xvz.

And because the limit of the logarithm of

2:

therefore if a limit occur of the form, we know that it

the fluxion, is the arc of which the tangent is : Thus we may determine fluents when the limit is of any form mentioned in logarithms or arcs of the circle.

RULE 4. If the quantity be a compound power, or radical. and the index of the variable without the vinculum increased by one be a multiple of that within it, the power, or radical, may be expanded into a series, and multiplied by the quantity without the vinculum, and then the fluent of each term may be found separately. Or a letter may be taken for the quantity under the vinculum, and the whole expressed in terms of that letter and expanded, which will be often more simple than the other, and the fluent of each term is to be taken as before. When the exponent of x without the power, or radical, increased by 1, is a multiplier of that within it, the expansion will consist of a finite number of terms, and some of these may be limits of logarithmic or circular functions.

and, taking the fluent of each term, we have y=x+

NOTE. Constant quantities connected with the variable ones, by addition or subtraction, disappear in taking th

xion; it is necessary to restore these when the fluent is en. Consider whether the fluent becomeso, or to some ɔwn quantity at the time it ought; if not, annex to it such onstant quantity as will make it its proper value. We nmonly annex C for this constant, the value of which may determined afterwards.

L means the hyperbolic logarithm, or the common logarithm aultiplied by 2.302585.

SECTION V.

OF THE LENGTHS AND AREAS OF CURVES.

PROP. LXXVII. Problem. To determine the length of any curve ABC.

C G

H
L

B

Let AE=x, EB=y, and the curve AB=2. Draw GF parallel to BE and BG to touch the curve at B, and BL parallel to AD. Then, while AE has increased to AF, BE has increased to FH, and the tangent is BG, and GL is always greater than LH. But as BL decreases, GL becomes more nearly equal to HL, and at length

DF E A