« 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
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
; or ris
when y is constant.
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
for the general value of the radius of curvature at any point
Hence, when the absciss x is nothing, the last expression
EXAM. 2. To find the radius of curvature of an ellipse,
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.
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,
113. If AE, DF, CG, DH, &c. be any positions of the thread,
AE AE is the radius of curvature to the point A,
114. It also follows, from the premises, that any radius of
Then by the nature of the radius of curvature, it is
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
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=
by making x
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
y = 4x
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.
TO FIND THE CENTRE OF GRAVITY.
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 sum of all the ds ; whether ABD be a b
universally denoted by
line, or a plane surface, or a curve superficies, or a solid. But the sum of all the ds, is the same as the fluent of db, and b is the same as the fluent of b; therefore the general expression for the distance of the centre of gravity, is Ac= fluent of b fluent xb
fluent of b
; putting ad the variable distance
118. CASE 1. When AE is some line, as a curve suppose.
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 generated by the rotation of a line AEB, about the axis AH. Then, putting c=3.14159 3.14159 &c. 2cy will denote the circumference of the generating circle, and 2cyz the fluxion of the fluent of 2cyxz__ fluent of yxz surface; therefore ac = fluent of yz
fluent of 2cyz
be the distance of the centre of gravity for a surface generat
ed by the rotation of a curve line z.
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 yxx is the dis
fluent of y2x
tance of the centre of gravity below the vertex in a solid.
EXAM 1. Let the figure proposed be the isosceles triangle
It is evident that the centre of gravity c, will be somewhere in the