fluxions at that point, that is, ed, or x is the fluxion both of AD
and BG, and de or y is the fluxion both of DE and GE. In the
equation above therefore, substitute for BG, and y for GE,
and it becomes

or GC-GFyx2 + y2 = z2.

Now multiply the three terms of this equation respectively,

by these three quantities,

it becomes yx-xy

Yxxy

CE

for the general value of the radius of curvature, for

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

X

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

xy

23 YX

when y is constant.

EXAMPLES.

:

EXAM. 1. To find the radius of curvature to any point of a parabola, whose equation is axy2, its vertex being a, and

axis AD.

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
E, the ordinate to which cuts off the absciss ad=x.

Hence, when the absciss x is nothing, the last expression
becomes barely ar, for the radius of curvature at the ver-
tex of the parabola; that is, the diameter of the circle of
curvature at the vertex of a parabola, is equal to ́a, the
rameter of the axis.

the pa

EXAM. 2. To find the radius of curvature of an ellipse,
whose equation is a2y2c2. ax-x2
x2.

1

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.

A E

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.

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,

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

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 in-
volute curve.

115. Hence,

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

yx xy

Hence EFGB-DB

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.

116. EXAMPLES.

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

C

y — √cx, and y = ‡x √, also ÿ=

X

-x2

4x

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

and

y

xy

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

B

then

374

<

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

b

fluent of b
AG. Which will divide into the following four cases.

; 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

will

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

AC

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.

122. EXAMPLES.

EXAM 1. Let the figure proposed be the isosceles triangle

ABD.

It is evident that the centre of gravity c, will be somewhere in the