Page images
PDF
EPUB
[ocr errors][merged small][ocr errors][merged small]

y

[ocr errors][ocr errors][merged small][merged small][merged small][merged small]

a- y

or 3x2 = a2, and xa, the absciss. Hence also, from the original equation,

y=

a3

a.x2
a2+x2 + a2

=

flexion sought.

= a, the ordinate to the point of in

EXAM. 2. To find the point of inflexion in a curve de

fined by the equation ay aar + xx.

EXAM. 3. To find the point of inflexion in a curve defined by the equation ay2 ax + x3.

ABGE

FGH

EXAM. 4. To find the point of inflexion in the Conchoid of Nicomedes, which is generated or constructed in this manner: From a fixed point P, which is called the pole of the conchoid, draw any number of right lines PA, PB, PC, PE, &c, cutting the given line FD in the points F, G, H, I, &c: then make the distances FA, GB, HC, IE, &c, equal to each other, and equal to a given line; then the curve line ABCE, &c, will be the conchoid; a curve so called by its inventor Nicomedes.

TO FIND THE RADIUS OF CURVATURE OF CURVES.

73. THE Curvature of a Circle is constant, or the same in every point of it, and its radius is the radius of curvature. But the case is different in other curves, every one of which has its curvature continually varying, either increasing or decreasing, and every point having a degree of curvature peculiar to itself; and the radius of a circle which has the same curvature with the curve at any given point, is the radius of curvature at that point; which radius it is the business of this chapter to find.

74. Let AEe be any curve, concave towards its axis AD; draw an ordinate DE to the point E, where the curvature is to be found; and suppose EC perpendicular to the curve, and equal to the radius of curvature sought, or equal to theradius of a circle having the same curvature there,and with that radiusde-. scribe the said equally-curved circle

B

A

BEe;

BEe; lastly, draw Ed parallel to AD, and de parallel and indefinitely near to DE: thereby making Ed the fluxion or increment of the absciss AD, also de the fluxion of the ordinate DE, and Ee that of the curve AE. Then put

= AD, y = DE, z AE, and r = CE the radius of curvature; then is Edx, dey, and Ee = 2.

Now, by sim. triangles, the three lines Ed, de, Ee,

[blocks in formation]

and the flux. of this eq. is GC.+ GC.

[ocr errors]

or i, j, ż,

GE, GC, CE;

Gc. x = GE.j;

[ocr errors]

GE.ÿ + GE.ÿ, or, because GC = — BG, it is GC. - BG . * = GE . j + GE .j.

But since the two curves AE and BE have the same curvature at the point E, their abscisses and ordinates have the same fluxions at that point, that is, Ed or x is the fluxion both of AD and BG, and de or j is the fluxion both of DE and GE. In the equation above therefore substitute for BG, and j for GE, and it becomes

[merged small][merged small][merged small][merged small][ocr errors][merged small][ocr errors][merged small][ocr errors][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

radius of curvature, for all curves whatever, in terms of the fluxions of the absciss and ordinate.

75. Further, as in any case either r or y may be supposed to flow equably, that is, either or j constant quantities, or * or j equal to nothing, it follows that, by this supposition, either of the terms in the denominator, of the value of r, may be made to vanish. Thus, when x is supposed constant, * being then 0, the value or r is barely - 3 orris =

xy

23

[ocr errors]

when y is constant.

EXAMPLES.

EXAM. 1. To find the radius of curvature to any point

[blocks in formation]

of

of a parabola, whose equation is ar = y2, its vertex being A, and axis AD.

Now, the equation to the curve being ax = y, the fluxion of it is ax = 2yy; and the fluxion of this again is a = 2jo, supposing y constant; hence then r or

[merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small]

for the general value of the radius of curvature at any point E, the ordinate to which cuts off the absciss AD = x.

Hence, when the abciss x is nothing, the last expression becomes barely la: =r, for the radius of curvature at the vertex of the parabola; that is, the diameter of the circle of curvature at the vertex of a parabola, is equal to a, the parameter of the axis.

EXAM. 2. To find the radius of curvature of an ellipse, whose equation is a2y2 = c2. ax - x2.

Ans. r =

(a2c2 + 4 (a2 − c2) × (ax − x2)ž

2a4c

EXAM. 3. To find the radius of curvature of an hyperbola, whose equation is a22 = c2 . ax + x2.

EXAM. 4. To find the radius of curvature of the cycloid.

Ans. r = 2aa

ax, where x is the absciss, and a the diameter of the generating circle.

OF INVOLUTE AND EVOLUTE CURVES.

76. AN Evolute is any curve supposed to be evolved or opened, by having a thread wrapped close about it, fastened at one end, and beginning to evolve or unwind the thread from the other end, keeping always tight stretched the part which is evolved or wound off: then this end of the thread will describe another curve, called the Involute. Or, the same involute is described in the contrary way, by wrapping the thread about the curve of the evolute, keeping it at the same time always stretched.

77. Thus,

B

A E

D

77. Thus, if EFGH be any curve, and AE be either a part of the curve, or a right line: then if a thread be fixed to the curve at H, and be wound or plied close to the curve, &c, from H to A, keeping the thread always stretched tight; the other end of the thread will describe a certain curve ABCD, called an Involute; the first curve EFGH being its evolute. Or, if the thread, fixed at H, be unwound from the curve, beginning at A, and keeping it always tight, it will describe the same involute ABCD.

H

78. If AE, BF, CG, DH, &c, be any positions of the thread, in evolving or unwinding; it follows, that these parts of the thread are always the radii of curvature, at the corresponding points, A, B, C, D; and also equal to the corresponding lengths AE, ALF, AEFG, AEFGH, of the evolute; that is,

AE AE is the radius of curvature to the point A,
BF = AF is the radius of curvature to the point B,
CG = AG is the radius of curvature to the point c,
DHAH is the radius of curvature to the point D.

79. It also follows, from the premises, that any radius of curvature, BF, is perpendicular to the involute at the point E, and is a tangent to the evolute curve at the point F. Also, that the evolute is the locus of the centre of curvature of the involute curve.

[merged small][ocr errors][merged small][merged small][ocr errors][merged small][ocr errors][merged small][merged small]

Then, by the nature of the radius of curvature, it is

[merged small][merged small][ocr errors]

= BC= AE + Ec; also, by sim. triangles,

[blocks in formation]

××

yx xy

[blocks in formation]
[merged small][ocr errors][merged small][ocr errors][ocr errors][ocr errors][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

which are the values of the absciss and ordinate of the evolute curve EC; from which therefore these may be found, when the involute is given.

On the contrary, if v and u, or the evolute, be given: then, putting the given curve EC=S; since CB = AE + EC, or r = a+s, this gives r the radius of curvature. Also, by similar triangles, there arise these proportions, viz.

[merged small][ocr errors][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][ocr errors][merged small][ocr errors][merged small][ocr errors][merged small][ocr errors][merged small][ocr errors][merged small][merged small][merged small]

which are the absciss and ordinate of the involute curve, and which may therefore be found, when the evolute is given. Where it may be noted, that 22+u, and ż2 = *2 + j2. Also, either of the quantities x, y, may be supposed to flow equably, in which case the respective second fluxion, * or ÿ, will be nothing, and the corresponding term in the denominator jÿ will vanish, leaving only the other term in it; which will have the effect of rendering the whole operation simpler.

--

81. EXAMPLES.

EXAM. 1. To determine the nature of the curve by whose evolution the common parabola AB is described.

Here

« PreviousContinue »