« PreviousContinue »
fluxions at that point, that is, ed, or x is the fluxion both of AD
or GC-GFyx2 + y2 = z2.
Now multiply the three terms of this equation respectively,
by these three quantities,
it becomes yx-xy
for the general value of the radius of curvature, for all curves whatever, in terms of the fluxions of the absciss and ordinate.
; or ris
GC GE CE
110. Further, as in any case either x or y may be supposed to flow equably, that is, either a or y constant quantities, or x or y equal to nothing, it follows that, by this supposition, either of the terms in the denominator, or the value of r may be made to vanish. Thus, when z is supposed constant, x being then =0, the value of r is barely
when y is constant.
or ; and hence is found r =
EXAM. 1. To find the radius of curvature to any point of a parabola, whose equation is axy2, its vertex being a, and
Now, the equation to the curve being ary, the fluxion of it is ax= 2yy; and the fluxion of this again is ax = 2y2, supposing y constant; hence then or
Hence, when the absciss x is nothing, the last expression
EXAM. 2. To find the radius of curvature of an ellipse,
a2 c2 + 4 (α2 — c2) × (ax
EXAM. 3. To find the radius of curvature of an hyperbola,
whose equation is a2y2=c2. ax+xa ̧
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.
111. AN Evolute is any curve supposed to be evolved or opened, which 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.
112. 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 Invo lute; the first curve EFGH being its evolute. Or, if the thread, fixed at н, be unwound from the curve, beginning at a, and keeping it always tight, it will describe the same involute ABCD.
AE AE is the radius of curvature to the point A,
113. If AE, DF, 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, aef, aefĠ, AEFGH, of the evolute: that is,
114. It also follows, from the premises, that any radius of curvature, BF, is perpendicular to the involute at the point B, 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.
AE +ɛc; also, by sim. triangles,
YX XY and FC AD AE+GC=x-αt
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 ECs, since CB AE+EC, or r=a+s, this gives r the radius of curvature. Also, by similar triangles, there arise these proportions, viz.
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 ¿2+u2, and z2=x2+ y2. Also, either of the quantities x, y, may be supposed to flow equably, in which case the respective second fluxion, x or ÿ will be nothing, and the corresponding term in the denominator yx-xÿ will vanish, leaving only the other term in it.
which will have the effect of rendering the whole operation simpler.
EXAM. 1. To determine the nature of the curve by whose evolution the common parabola AB is described.
Here the equation of the given involute AB, is cx = y 2 where c is the parameter of the axis AD. Hence then
y — √cx, and y = ‡x √, also ÿ=
constant. Consequently the general values of v and u, or of the absciss and ordinate, EF and Fc, above given, become, in that case,
x2 + y2
ป by making x
y = 4x
s denote any section EF of the figure, =AG, its distance below PQ, and b= the whole body or figure ABD ;
FC U =x at
But the value of the quantity a or AE, by exam. 1 to art. 75, was found to be c; consequently the last quantity, re or u is barely 3x.
Hence then, comparing the values of v and u, there is found 3v/c = 3u/x, or 27cv2 16u3; which is the equation between the absciss and ordinate of the evolute curve Ec, showing it to be the semicubical parabola.
EXAM. 2. To determine the evolute of the common cycloid. Ans. another cycloid, equal to the former.
117. By referring to prop. 42, &c. in Mechanics, it is seen what are the principles and nature of the Centre of Gravity in any figure, and how it is generally expressed. It there appears, that if PAQ be a line, or plane, drawn through any point, as suppose the vertex of any body, or figure, ABD, and if
CENTRE OF GRAVITY.
then the distance ac, of the centre of gravity below re, is
; putting ad the variable distance
fluent of b
118. CASE 1. When AE is some line, as a curve suppose. In this case ¿ is zor y2, the fluxion of the curve fluent of xz. fluent of x2+ y2
and bz: theref. Ac
is the distance of the centre of gravity in a curve.
119. CASE 2. When the figure ABD is a plane; then byx; therefore the general expression becomes AC = fluent of yxx for the distance of the centre of gravity in a plane. Auent of yo
120. CASE 3. When the figure is the superficies of a body
surface; therefore ac =
121. Case 4. When the figure is a solid generated by the rotation of a plane ABH, about the axis AH.
Then, putting c=314159 &c. it is cy2 the area of the circle whose radius is y, and cy2=b, the fluxion of the solid; therefore
fluent of x¿ fluent of cy2xx fluent of b fluent of cy2x tance of the centre of gravity below the vertex in a solid.
fluent of yxx is the disfluent of y2x
EXAM 1. Let the figure proposed be the isosceles triangle
It is evident that the centre of gravity c, will be somewhere in the