2

COS &

Flux (sin x cos x) = * cos fei sin I = cos 2c. Since (1 +cos x) =cos 1 , we have fluxul

1+cos

*)= flux cos 2 x=-1: sin

Similarly we shall find that flux (cos log x)=- flux log I sin log * = sin log w; and that flux (sin )=rsin x +zz cos z. 24. If x be any arc, its Auxion x =

flux sin s

flux cos a

sin * flux tans = cos ** ftux tan x=

flux tan x

- flux cot x sin 's sec **

1+ tang ** - flux cota - flux cot

&c. &c. coseca

1+cot ** APPLICATION OF FLUXIONS TO THE

THEORY OF CURVES. 25. Of all the problems that can be proposed respecting a curve, the most simple is that which requires us to draw a tangent to any point of the curve.

Suppose the curve to be AM, its axis, AP; its coordinates AP and PM; it is evident that to draw a

M

В. tangent to the point M, we have only to determine the subtangent PT.

Let us imagine the arc Mm to be infinitely small, draw the ordinate np, infinitely near to AP, and

T
I

A
sup-

N

PP pose Mr parallel to Pp. Let, as usual, AP=*, PM=y, and we shall have Pp, or Mrzi, mr=y; and by similar triangles Mrm, TPM we have mr : rM::MP: PT ory::::9: PT=Y.

- yx Consequently we

y have only to throw the equation of the curve into fluxions, in order to obtain the value of *, and then to substitute this value in the

formula for the subtangent just found, and PT will be determined. 26. The expression for the tangent MT is ✓ (yo +

y ya

ข Comet j'); that of the subnormal PN is y* yy; the normal

PT'

و = (+)

or

MN = V(+9);

; and if through the point A we draw the line

AQ parallel to MP, we shall have PT : AT::MP : AQ,
that is yx : yo

ty
-::: AQ=y- These values

of AQ and y y AT enable us to find the asymptotes of the curve AM if it have any ; for if, after having substituted in these two values that of ý obtained from the equation of the curve, we suppose x infinite, there will be as many asymptotes as there are different values of the lines AQ and AT. The position of the asymptotes will always be determined by the points T and Q. We shall now apply these formula to a few examples.

The equation of the circle is yʻ=a*—*°; therefore yy=-xi, and yg - " or the subtangent = - y_ (a*—2)

The sign – indicates

y that the subtangent must be taken in the same direction as the abscissa, because in the construction of the formula it was taken in a contrary direction. If we had taken the vertex of the curve for the origin of the abscissæ, the equation would have been y? = 2ar --XX, and this would have given a positive result as in the formula.

T'he equation y'=a? — gives yy or the subnormal =

m; and

the normal = v( 99 +

=N (*+y)=a=the radius, as it evidently should be.

In the parabola yo=pr; therefore yy = \p, the subnormal ; and

= 2.x.

2

ya, or the subtangent=

2y2
P
62

h?
In the hyperbola y:=
(20x+xx); therefore YY,

x (a +x);

s and yx q? 12 2ax tix

62 (a +x) y

a + x We have also AT (see preceding figure) = - -, an expression

atx which is reduced the quantity a, if we suppose e infinite. On the same supposition we find that AQ=y- TY=y

622

(a +x)=

a®y a*yo—b*: (a+r) = b^x

bNT -), becomes simply=b. Thest ay

ar

two values of AT and AQ give the position of the asymptotes the same as we found in article (49) conic sections.

Ay. In the logarithmic curve, we have r = A log y, and i =

ข Therefore y&_A. Consequently its subtangent is always equal to

y the modulus (art 69, page 429)

27. Let there be any curve BOC with another curve BMA, such, that if we prolong the ordinates OP of the first till they meet with

a á the second curve, the line MO) may be some function of arc BO;

91

M it is proposed to draw through the

Ľ given point M the tangent MT.

Conceive the ordinate mp.indefinitely near to MP, and Mr parallel to the tangent at the point 0; if we make the arc BO MO = u, we shall have mr = u, „M=00=Ż, and : 2::u: 0T Now u being a function of

D

P

=%;

%, we shall have

by taking the fluxions of the equation which

a

expresses the given relation; and consequently TO, or the point T will be determined, whence it will be easy to draw the tangent MT.

6

bz Suppose, for example, that u= 2, we shall have i = and ОТ:

=z=arc BO. If BOC is a circular arc, then AMB, is a cycloid, and this construction is the same as we have already given.

In the quadratrix, if we compute the abscissæ from the centre, we have (art 76 page 432) y= w.cot; thereforey =

cx

cx cot

a

a

a

sin 2 CX,

a

ved by the two similar triangles MOT, and the ffuxional triangle formed by drawing an ordinate indefinitely near to MP. (We employ — y, because y diminishes, as x augments.) Therefore OT=

. ; and adding to each side sin

crt

CX

cot

=

aa

When CM=CT, or at the point D, we have, as before explained,

СМ: the base CD=4a, and consequently CT = We must there

CD fore take CT a third proportional to the base CD and the radius CM; this will give the point T, through which and through the point M if we draw the line MT, it will be the tangent required.

28. To draw tangents to spirals, we must resolve the following problem. Let there be described a circle with any radius CA, and

let there be a curve CKM such, that drawing the radius CMN, the line CM may be any function of the arc ABŇ, it is required to draw through the given point M the tangent MT.

Conceive the two radii CMN, Cmn indefinitely near to each other, and the little arc Mr described from the centre C, and with the radius CM; and then draw CT perpendicular to CM.

This done, let CM=y, ABN = x, CA=a; we shall then have

=

CA: CM::Nn: Mr

or

ye

a

a : y :: * : Mr =
and again rm : Mr::CM: CT

ý :*
:: y : CT=

ау For example, let y then CKM will be the spiral of Archi

or

a

ar

and CT = y*r—axy_xy_

a

aa

aa

a

a

M.19

MQO.

In the hyperbolic spiral whose equation is xy=ab, we shall have ay+yi=0, y:=-xy, and CT =

'xyy
xy

- b; as we

ay before found it (art 85 page 435).

29. In the logarithmic spiral in which the angle CMT is constant, conceive the radii CM, Cm indefinitely near together, and from C as a centre and with any radius CN, describe a circle; make CM=%, CN= a; and marking on the cir

B

D cumference of the circle a fixed

K point A, suppose the abscissa

A AN=r, which will give this T proportion, CN: Nn::CM : Mr

a : m :: z : Mr

72 N

therefore log x= ==

+ a constant quantity C; because the

a ;

at fluxion of the equation log x=

= the same as that of log

Now the equation log == +C, shews I”, that this spiral makes

++

an infinite number of revolutions about its centre, as well in apa proaching towards it, as in receding from it; for in place of we may successively substitute : +6, 7 + 29, x + 37, &c. + 2, - 2x +*, &c. # being the circumference ANB.