201.] The Cycloid. Figs. 39 and 40.

DEF.-A cycloid is the curve traced out by a point in the circumference of a circle, as the circle rolls along a fixed straight line.

(a) Let the given straight line, fig. 39, be taken as the axis of x, and the radius of the rolling circle be a, and the origin be at the point o, where the generating point P is in contact with the fixed line; and let RPQ be a position of the generating circle, such that oq is equal to the arc PQ. Let E be the point in the line OAE, at which the generating point is again in contact with it, so that OAE the circumference of the circle = 2 ′′α. Bisect OE in A, and at a draw the ordinate AB = 2CP; then by the geometry of the figure, в is the highest point.

which two equations are those to the cycloid, when the startingpoint is the origin; and if 0 is eliminated, we have

x = a versin-12 — {2 ay — y2} 4.

a

(29)

Since sin and versin 0 have the same values whenever is increased by 2, or by 4, ...... it appears from (28) that the values of y recur whenever a is increased by 2ña, oг bу 4ñа.....; hence there is a series of curves similar and equal to OBE placed along the straight line oa E, parts of which at o and E are drawn in the figure; this is evident from the mode of generation of the curve.

The line OAE is called the base, and AB the axis, and в the highest point of the cycloid.

(B) It is also frequently convenient to refer the cycloid to the highest point as origin, and to its axis as the axis of æ, in which case its equation may be found as follows.

Fig. 40. Let RPT be the circle in its generating position, P being the generating point, the arc PR being equal to the line

BR; from

let MP be drawn at right angles to oa, and let oqa

be a semicircle described on the axis oA.

=

Let OM = x, MP=y, Oc=CQ = ca= a, Qco = 0. Then since AB the semi-circumference RPT', of which the parts RB and arc RP are equal, therefore AR = the arc PT' the arc oq, on account of the similar positions and equality of the two semicircles; whence

which two equations, taken simultaneously, are those to the cycloid; and by the elimination of we have

202.] The Companion to the Cycloid. Fig. 41.

(31)

It appears from the first of equations (30) of the last Article, that the ordinate to the cycloid is equal to the sum of the ordinate of the circle, viz. мQ of fig. 40, and of a part produced, viz. PQ, which is equal to the intercepted arc oq; but if the ordinate to a circle is produced until the whole is equal to the intercepted arc of the circle, the locus of the extremity is called the Companion to the Cycloid.

Let OM, MP = y, OC = C A = a, qco = 0; then, since MP is equal to the arc oq,

y = a0

x = a (1- cos 0)

;

(32)

which two equations are those to the companion to the cycloid; and, if is eliminated, we have

203.] Epitrochoidal and Hypotrochoidal Curves. Figs. 42, 46. DEF.-An epitrochoid is the curve generated by a point within

or without the circumference of a circle, which rolls on, that is, outside, another circle of given radius.

If the generating circle rolls inside the given circle, the generated curve is called the Hypotrochoid.

And if the generating point is on the circumference of the rolling circle, it is called an Epicycloid or Hypocycloid, according as it rolls without or within the fixed circle.

We shall consider the epitrochoid to be the normal case, and deduce the equations to the other curves from its equations by changing the signs and values of the constants.

Let o, the centre of the fixed circle, be the origin, and a be the centre of the generating circle, and p the generating point; and suppose в, in the line QBP, to have been originally in contact with the fixed circle at A, and let oa be the axis of a; see fig. 42.

Let oм= x, MP=y, QOA = 0, OR=0A=a, QR=QB=b, QP = mb. Then, since one circle rolls on the other, the arc AR the arc BR;

and (34) and (35), taken simultaneously, are the equations to

the epitrochoid.

If the generating circle rolls inside instead of outside the fixed circle the sign of b must be changed, and the curve is an hypotrochoid; the equations to which are, fig. 43,

204.] And if m = 1, the generating point is on the circumference of the rolling circle, and the curves become respectively the epicycloid and the hypocycloid; and the equations are

the curves expressed by which are those dotted respectively in figs. 42 and 43.

When a and b are commensurable numbers, the branches of the curve re-enter after a certain number of revolutions of the generating circle: in which cases the subsidiary angle @ may be eliminated, and the equation expressed in an algebraical form; but when a and b are incommensurable, the branches never reenter, and the equation can only be expressed in a transcendental form equivalent to the above equations.

Some varieties of the above curves, in which the equations assume particular forms, are subjoined.

205.] Suppose that the generating circle of the epicycloid is equal to the fixed circle, then a = b; and equations (37) become y = 2a sin 0 - a sin 20;

x = 2 a cos 0

a cos 20,

whence, squaring and adding,

Again,

x2 + y2 = 5a2-4a2 cos 0,

x2 + y2—a2 = 4a2 (1-cos 0).

x = 2a cos 0-a {2 (cos 0)2-1};

which is the equation to the curve expressed in rectangular coordinates; o, the centre of the fixed circle, being the origin. Fig. 44.

Let us change the origin to A, and transform the equation to polar coordinates, by putting a = a +r cos, y = r sin ; whence

r = 2a (1—cos 4).

The curve is called the Cardioid, from its heart-like shape.

(40)

206.] In the equations to the hypocycloid, let b = which case equations (38) become

a

see fig. 45.

X =

4

Again, in the equations to the hypotrochoid, let b = which case equations (36) become

which equation represents an ellipse, the axes of which are

(42)

(43)

a (1+m) and a (1-m);

see fig. 46.

Again, in equations (38) of the hypocycloid, let b =

(44)

which equations express a straight line on the axis of x, of length 2a, which is coincident with the diameter of the circle.

SECTION 4.-On certain general properties of curves as expressed by algebraical equations of the nth degree.

207.] In the following Chapter I shall enter on a formal inquiry into the properties of plane curves referred to rectangular coordinates, as exhibited by means of differentiation; but preliminary to that investigation it is necessary to explain certain general properties of a particular class of such curves, those, viz., which are expressed by algebraical equations; because these will receive further illustration hereafter; and it cannot be that such illustration will be adequately compre