CHAP. XXII. OF THE CYCLOID, THE CONCHOID, AND THE CATENARY. (713.) THE diversity of curves which present themselves to the consideration of the geometer is infinite, and consequently the investigation of their individual properties can only be undertaken when those properties are called into play in the sciences or arts. General methods of investigation, by the aid of the language and principles of analysis, may be obtained by those who are willing to prosecute that branch of mathematics; meanwhile there are a few curves which have acquired a peculiar claim upon our attention, from the uses to which they are applied in physical and mechanical science. We shall devote this chapter to a brief exposition of their leading properties. THE CYCLOID. (714.) If a circle whose centre is C (fig.250.), and whose radius is C A, touch a line B B' at A, and while a pencil is attached at V and carried with the circle, the circle itself is rolled along the line A B from A towards B, the pencil V will trace a curve VPB; and if, in like manner, it be rolled in the other direction from A towards B', the pencil will trace an equal and similar curve V B'. (715.) The curve BV B' thus, defined, is called a cycloid, and is the curve which would be traced by a point situated on the edge of a carriage-wheel as that wheel is rolled in a straight direction on a level plane. (716.) While the generating circle rolls from A to B, every point of its semi-circumference A D V applies itself to the line A B, and when the semicircle reaches the point B, the describing point V coincides with B, and the point A takes the position A'. It is evident therefore that A B is equal to half the circumference ADV of the generating circle. And in like manner, when the circle is rolled to B' in the contrary direction, the point A takes the position A', and the describing point V coincides with B'. The distance A B' is therefore rolled over by the semi-circumference A D' V of the generating circle, and is therefore equal to that semi-circumference. (717.) The line B B' is called the base of the cycloid, and is equal to the circumference of the generating circle. (718.) The line AV is called the axis of the cycloid, and is equal to the diameter of the generating circle. (719.) All lines PP' parallel to the base and terminated in the cycloid, are bisected at M by the axis; for the branches of the curve at each side of the axis A V are perfectly equal and symmetrical. (720.) As the generating circle rolls along the base of the cycloid, the describing point P (fig. 251.) has two motions; first; a progressive motion in a direction parallel to the base B B', and secondly, a motion of rotation round the centre of the generating circle. revolution of the generating circle, the describing point P moves by its progressive motion through the space В В ́, while by its motion of rotation it moves through a space equal to the circumference of the circle. Let us suppose the circle to roll from the position A in which the describing point P coincides with the vertex of the cycloid, to the position L in which the describing point has moved to P', and the point which was at A be now at A'. The distance LA will then be equal to the arc LA' of the circle, since that arc has rolled over L A, and since A B is equal to the semicircle A'L P', we have L B equal to the arc of the circle L P'. The point P', in virtue of the two equal motions already explained, one in the horizontal direction P'N, parallel to A B, and the other in the direction of the tangent P'T to the circle at P', will have an actual motion in a direction equally inclined to each of these lines. The direction of the curve at P', or, what is the same, the direction of a tangent to the curve at that point, will therefore be a line bisecting the angle N P' T. But it is easy to show that such a line will be the continuation of the chord of the arc of the circle between P' and the highest point 0; for if LP' be drawn, the angle O P' M will be equal to the angle O L P', because of the similarity of the right angled triangles O QP'and O P'L, and the angle O P' T' will also be equal to the angle O L P'; therefore Ο Ρ' Τ' will be equal to O P' Q, or what is the same, NP'R will be equal to TP'R; the line O P R therefore bisects the angle T P' N, and is therefore a tangent to the cycloid at P'. (721.) Since the arcs Ap and LP' are equal, and also the arcs Pp and O P', the lines A p and L P' are equal and parallel, and the lines Pp and O P' are likewise equal and parallel. (722.) The tangent at P', is therefore parallel to the corresponding chord Pp of the generating circle on the axis. (723.) To draw a tangent therefore to a point P' on a cycloid, draw a line P' M from that point perpendicular to the axis A P, and from the point p, where that line meets the generating circle on the axis, draw a chord p P, and through P' draw a line O P'R parallel to this chord; this line will be a tangent to the cycloid at P'. (724.) The arc Pp of the generating circle on the axis is equal to the parallel p P' to the base, intercepted between that arc and the cycloid. For A L has been already proved equal to A' L; but the latter is equal to O P', and therefore to Pp; but A Lis equal to P'p, being opposite sides of the parallelogram A P'; therefore p P' is equal to the arc Pp. (725.) It is a property of the cycloid, which may be demonstrated by the aid of the higher analysis, that the cycloidal arc P P' is equal to twice the chord Pp, and this will be the case wherever the parallel P'p is drawn. Hence the semi-cycloid P B is equal to twice the diameter of the generating circle, and the entire length of the cycloid B P B' is equal to four times the diameter of the generating circle. (726.) Hence the length of a cycloid is to the length of its base as four times the diameter of a circle is to its circumference. (727.) Since P'O is a tangent to the cycloid at P', and the angle O P' L is a right angle being in a semicircle, the line P' L is the normal to the cycloid at P', and the radius of curvature to the cycloid at P' is twice PL. (728.) One of the most remarkable properties of the cycloid is, that it is its own involute; in other words, the involute of a cycloid is an equal and similar cycloid. Let B B' (fig. 252.) be the base of a cycloid BV B ́, and let B O' and B'O' be two semi-cycloids, each equal and similar to BV, having their vertices at B and B'. Then B O' will be the involute of BV, and B' O' the involute of BV; and in like manner BV will be the evolute of B O', and B' V the evolute of B' O'. If P be any point in the cycloid BV B', and PO be a tangent to the lower cycloid, then O will be the centre and PO the radius of curvature of the point P, and the lines PO will all be bisected by the base B Β ́. (729.) The cycloid, according to the principle by which it is generated, is not supposed to terminate at B and B', the extremities of the base. For the motion of the generating circle may be continued along the line of the base in either direction beyond these points. If it be so continued, the generating point will describe a succession of cycloidal arcs as represented in fig. 253. A cusp being formed at the points B, B', B" &c., where the describing point touches the base. |