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119. That if we make C=log C, we shall have logo log e, or = că, and ==C'eá ; therefore at the point A where

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C =0, we have CD=C.

IIIdly. That if the abscissæ are taken in arithmetical progression as 8, 26, 3x, &c. the ordinates will form the geometrical progression C'cas, C'ea, cei &c.

IVthly. That if t=0, we have 2=d', a property of the circle which, as we already know, cuts all its radii at right angles.

These examples will enable the student to draw tangents to all sorts of curves whether geometrical or mechanical.

Examples for Practice. 1. Let it be required to draw a tangent to the ellipse whose

67 equation is ge= (a —s").

ye: '— 2. Required the expression for the subtangent to the cissoid of which the equation is y

3. Required the subtangent of the conchoid whose equation is : b+y

(aa-yy). 4. Required the subtangent of the parabolic spiral of which the equation is y=a-w px, (See art. 82 page 484) OF INVOLUTE AND EVOLUTE CURVES.

a

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30. Suppose a thread ABC to be lapped closely upon any curve BC whose origin is at B, and to which AB is a

M

M tangent at that point; if we gradually unlap this thread, keeping it alwaye equally stretched, its extremity Aiwill

B describe a curve AM, which A PIP will have the following properties.

Ist. The tangent MC of the curve BC will always be perpendiular to the curve AM;

Ildly. The length of MC will be equal to the line AB + the arc BC;

IIIdly. The indefinitely little arc Mm may be considered as a cir. cular arc described from the centre C with the radius CM;

IVly. The point C will be the point of concidence, or re-union of the two normals MN, mn, which are supposed to be indefinitely close to each other.

31. The curve BC is called the cvolute of the curve AM: and reciprocally the curve AM, the involule of the curve BC; the line MC is the radius of the evolute, it is also called the radius of curvature, or of osculation, &c. This premised, we shall proceed to determine for any point M the radius MC of the evolute BC which we suppose known,

Let MP, mp, be two perpendiculars to the axis AQ, indefinitely near to each other, and CO, rM two parallels to the same axis ; if we call MO=u, AP=r, PM=y, Mm, or (+:+ya)=s, we shall have

Mr : Mm :: MO: MC that is

it :

: MC=

US

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u

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But while AP, PM, and MO vary, MC becoming mc, does not change ; therefore the fluxion of the equation MC="" being taken, we shall have (us+su) =us š’and since i=mr=y, we shall find

and consequently that MC =

that u =

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szy
3

3
– ( x+y =) 9 3–4 9

(3 For the sake of simplifying this expression, let us suppose one of these Auxions to be constant, the element s of the curve, for example, and we shall have. MC = sy _ ý v(**+y)

j if we had supposed y constant, we should have found š š = * **: whence's =

_(+*+y+) which gives MC=

Y 2

y x But if, as is usually done, we suppose x to be constant, then

23 _(+y); MC=

.

X y 32. As the curvature of circles varies in the inverse ratio of their radii, it follows that in two different points of any curve, the degrees of curvature are inversely as the radii of the evolute. Therefore in order to find in what points the curve has the greatest

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curvature, we must determine the minimum or least value of the rac dius of curvature.

If the tangent at A is perpendicular to the axis, then to determine the right line BA, or the distance from the vertex A to the origin of the evolute, we must make s=o in the expression of the radius MC, and we shall have the value of BA. Lastly to find the equation of the evolute, draw CQ perpendicular to the axis, and call AB, a; BQ, t; CQ, 2; in the first place supposing & constant,

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and x =

+ya

y. And then

we shall have MOS

- y

- y mr : M :: MO : CO

*

: - y

that is,

ä

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:: 3 ::*+y":Co=PQ=j (*+yi)

() Therefore AP +PQ-AB=i=r_a+ ý (x+y); values which

4

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;

together with the equation of the curve, enable us to find the equation of the evolute.

33. Hitherto we have supposed the ordinates to be parallel to one another. If they proceed from a fixed point or pole B, we may determine the radius MC in the following manner.

Conceive two ordinates BM, Bm indefinitely near each other, and CO, Co two perpendiculars to these ordinates; then from B as a centre describe the arc Mr. This done, let BM=y, Mrzi, mr=ý, Mm=s =V (**+y), MO=4. Because B of the similar triangles Mrm, CMO, we have Mr : MO :: mr : CO :: Mm: MC

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Taking the fluxion of this last equation (on the supposition that that : is constant,) we have us, and the luxion of Co

CO

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con

This expression becomes merely when y=00, or when

X y the ordinates are parallel, as we have already found. We shall now proceed to give a few examples.

The equation to the ellipse and hyperbola, when we place the origin of the abscissæ at the vertex, is expressed generally by yy= pr

prx + and it is evident that when a=00, we have yy=px; an

2a equation of the parabola, which consequently is merely an ellipse or hyperbola whose transverse axis is infinite. Hence the equation wy = px + pet is general for all the conic sections. It

2a will therefore serve to find their radius of curvature.

34. Observing first that y. v(x2+”), being equal to the normal (26), if we call it n, the radius of curvature, supposing

123 ze stant, will be expressed by; and since in this example yy=

--yoy pxx, we have 2yj=pr+px; and again taking the Auxion of

pirs this last equation we have aj j+2j*=+ , and therefore yöj= + – y*yo =* {= a (px + port) + y y = { P

pr

= - ==2a

m3 Hence

or the radius of curvature= -y5 y conic sections the radius of curvature is equal to the cube of the normal divided by one-fourth of the parameter. Hence in the circle where n = $p, the radius of curvature is always equal to the nora

;

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;

2a

a

a

2a

2a

2

2a

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Ipp; that is, in all the

+PP (1+

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ca

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O

=

mal, as is evident. As for the evolute of the circle, it is obviously the point which serves as the centre of the circle. 35. We haven="V (+) = v {px=

px.x

23

2a and at the vertex where x=0, n=1 p, and the radius of the evolute or the

M rightline AB=\ p. In the ellipse the evolute has four branches BD, Db, bd, Bd,

A equal and making with

B one another four cusps or points of reflexion. The distance CH = Cb = a

м
p, and ED=ed = one-
half the parameter of the
conjugate axis.
In the parabola, the radius
MN3

MN
Х

PN'
consequently CO or PQ-NT-
2x+p; therefore AQ=3x+\p
=3* + AB, and consequently N
BQ=3 x. This gives a very
simple construction to determine

N 12 the point C, or the centre of the

TAI PB osculating or equi-curve circle

; take BQ=3 AP, and draw CQ

C perpendicular to AQ, then the

M point of concourse C of the two lines MC, CQ, will be the centre of the required circle.

To find the equation of the evolute, let BQ=2, CQ=u; we shall have x=} , and pry:: QN: CQ :: 2x:u=

4 ay _ 4 X px P

P 1 Consequently 16

16 the evolute of the common parabola is the second cubical parabola", whose parameter is

27

of that of the given parabola. 16

MC = 1 pp = NT *

; and

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pu?

X=

272, and z'=ipu; this shews that

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* All the curves represented by the general equation ymern amh are called parabolas, while n and m are positive; thus if yz-a? r the curve is called the first cúbical parabola, because n=i; if y=a x?, the curve is the second cubical parabola, because n=2, &c.

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