Ex. 4. The intrinsic equation of the cycloid.

If s begins at the highest point (the vertex) of the cycloid,

Ex. 5. The intrinsic equation of the catenary whose equation is

and if s begins at the lowest point of the curve, s = a coty.

In this case

P = —a (cosec↓)2.

Ex. 6. In the curve whose equation is x+y3

8 =

(1-cos 24)

=

4.

(88)

Ex. 8. If ra°; then in the formula (82), dp = 0; and dy do; so that 04-Vo, where yo is the value of

=

when

ifs 0, when 0 = 0; that is, when y = √。.

=

Ex. 9. Find the intrinsic equation to the Epicycloid whose

From the preceding equations and (79) we have

(92)

Now let us suppose

to begin at the cusp, where 0 = 0, and

decreases as s increases; then we have

which is the required intrinsic equation.

If b=a, the epicycloid becomes the cardioide, and (94) becomes

Ex. 10. The intrinsic equation to the involute of the circle is

SECTION 5.-Involutes of Plane Curves.

169.] Another geometrical problem of considerable interest, which requires single integration in its solution, is that of the

discovery of an involute of a curve, when the curve is given. Our investigation will comprise the determination of certain general formulæ which express reciprocal properties of evolutes and involutes, and the application of these formula to certain examples.

Let An, fig. 11, be a part of the curve whose involute is to be found. Let ON = έ, Nпn, and let the equation to an be

n=f($);

(97) let the length-element of this curve be do, and let PП be the tangent at п, whose extremity P is the generating point of the involute; so that Pn is the radius of curvature of the involute. Let PПp, then, by Vol. I, Art. 292, equation (40),

c being a constant, the value of which depends on the position of the point on an, at which σ begins. Thus in fig. 11, if ▲П = σ, and пР Aп, then σ = p, and c = 0; and if Рn be longer than An, then c is the excess of length; that is, if a string of the length PП is wrapped round An, and ultimately becomes a tangent at A, c is the length of the string that remains when the wrapping is complete.

Let OM = x, MP = y; then, since

dn

αξ

in which equations p must be expressed in terms of έ and ŋ; and έ and ʼn having been eliminated from them and (98), the resulting equation will contain x and y only, and be that to the required involute.

170.] From (100) by differentiation we have

But by Art. 285, Vol. I, (21) and (22), the numerator and denominator of the right-hand side of (102) are proportional to the direction-cosines of the tangent of the evolute; and therefore we conclude that it is perpendicular to the tangent of the involute. Again, squaring and adding the two equations (101), we have

But if p' is the radius of curvature of the evolute at the point (s, n), then, by Vol. I, Art. 285, (19),

and therefore by means of (34), Art. 311, Vol. I,

Ρ

= ±

3 ds d2s (dx d'y―dy d2x) — ds2 (dx d3y — dy d3x)
(dx dy-dy d2x)2

For a problem in

p' kp, so that

(103)

illustration of these equations, I will suppose

ds=

but if s and are intrinsic coordinates, ds = pdy;

Ex. 1. To find the involute to the catenary, the generating point being in contact with it at its lowest point.

the equation to the tractory, the form of which is evident from fig. 7.

Ex. 2. To find the equation to the involute of the cycloid, the generating point being in contact with it at its vertex.

Let the cycloid be placed as in fig. 12; and let ON=έ, Nп=n; OM = x, MP = y; then the equation to the cycloid is

the equation to a cycloid in an inverted position, as OPD in the figure, and lying below the axis of x.

Ex. 3. To find the involute of a point.

Let the coordinates of the point be
PRICE, VOL. II.

=

a, n = b; and let

G g