OX, meeting QN at P. P is another point on the hyperbola, and by repeating this construction other points may be found as shown. This curve has the property that if from any point P on the curve, perpendiculars PN and PM be drawn to the axes, the product PN × PM is always the same, wherever the point P may be taken. 25. Cycloidal Curves.-If a circle be made to roll along a line and CYCLOID FIG. 30. If remain in the same plane, a point in the circumference of the rolling circle will describe a cycloidal curve. The rolling circle is called a generating circle, and the line along which it rolls is called a director. the director is a straight line, the curve is called a cycloid. If the director is a circle, the curve described is called an epicycloid or a hypo cycloid, according as the generating circle rolls on the outside or inside of the director. The constructions for drawing the cycloid, epicycloid, and hypocy. cloid are shown in Figs. 30 and 31. Make the straight line or arc o'e' equal to half of the circumference of the generating circle. Divide o'e' into any number of equal parts, and divide the semicircle Pco' into the same number of equal parts. Draw the normals a'A, b'B, etc., to meet a line or arc OE parallel to or concentric with o'e' at the points o'e'. A, B, etc. With centres A, B, etc., describe arcs of circles to touch Draw through the points a, b, c, etc., lines parallel to o'e', or arcs concentric with the arc o'e', to meet the arcs whose centres are at A, B, C, etc. The intersections of these determine points on the curve required. The hypocycloid becomes a straight line when the diameter of the generating circle is equal to the radius of the director. The same hypocycloid may be described by either of two generating circles whose diameters are together equal to the diameter of the director. 26. Involute of a Circle.—If a flexible line be wound round a circle, and the part which is off the circle be kept straight, any point in it will describe a curve called the involute of the circle. The involute is a particular case of the epicycloid. If the generating circle of an epicycloid be increased in size until its diameter becomes infinite, the circle becomes FIG. 32. INVOLUTE a straight line, and a point on this line will describe the involute of the director circle if the line is made to roll on that circle. To draw the involute to a given circle OCP (Fig. 32), draw the tangent Om', and make Om' equal in length to half the circumference of the circle. Divide the semicircle OCP into any number of equal parts, and divide Om' into the same number of equal parts. At the points A, B, C, etc., draw tangents to the circle, and make Aa, Bb, Cc, etc., equal to Oa', Ob', Oc', etc., respectively. Pabc m' is a portion of the curve required, which may be extended to any length. 27. True Length of a Line from its Plan and Elevation. It frequently happens that the true length of some line or edge of a solid is not shown by either the plan or elevation, as in Fig. 33, which shows two views of a wedge. In both plan and elevation the sloping edges are shown shorter than their real lengths. To find the true length of a sloping edge, erect a perpendicular at one end of the plan, and make it equal in length to the height of one end of the elevation above the other. The line joining the top of this perpendicular with the other end of the plan is the true length required. ELEVATION TRUE LENGTH PLAN 28. Plane Sections of Solids.-Several examples are here given showing how plane sections of solids may be drawn. Fig. 34 shows how to determine a section of an angle iron. In Fig. 35 is shown a bolt-head which is partly square and partly conical. The curves a' and b'are plane sections of a cone. Fig. 36 shows a hexagonal bolt-head, the end of which is turned to a spherical shape. The curves in this example are FIG. 33. plane sections of a sphere. An example of frequent occurrence in machine drawing is that shown in Fig. 37, where we have plane sections of a surface of revolution—that is, a surface turned in an ordinary lathe. In all these examples the construction lines are fully shown, and further explanation is unnecessary. 1 FIG. 36. FIG. 37. 29. Development of the Surface of a Cylinder.-A surface is said to be developed when it is "folded back on one plane without tearing or creasing at any point." The development of a right circular cylinder is a rectangle whose base is equal in length to the circumference of the base of the cylinder, and whose height is equal to the length of the cylinder. In Fig. 38 the rectangle ABCD is the development of the surface of the cylinder, whose plan and elevation are shown. In order to determine the development of any line on the surface of the cylinder, a number of lines on the surface of the cylinder and parallel to its axis are represented. The points 1, 2, 3, etc., are the plans of these lines, and their positions on the development are shown at 1', 2', 3', etc. Suppose the cylinder to be cut by a plane represented by the line Ae', then the curve AEB would be the top boundary-line of the development of the surface of the portion of the cylinder ADƒ'e. The construction for determining the curve AEB is simple, and will be readily understood from the figure. FIG. 38. 30. The Helix. -If a straight line DH, Fig. 38, be drawn on the development of a cylinder, and then worked backwards on to the cylinder as shown, we get a representation of a curve called the helix. This curve is important in connection with screws. The angle HDC is the inclination of the helix. HC or Dh', the height to which the curve rises in going once round the cylinder, is called the pitch. To draw the elevation of the helix it is not necessary to draw the development. Divide the pitch Dh' into the same number of equal parts that the plan is divided into, and through the divisions draw lines parallel to Dƒ' as shown, to meet the perpendiculars from the plan. 31. Screw Threads and Spiral Springs.-The edges of screw threads are helices, and these are drawn as explained in the preceding article. Examples of these are shown in Figs. 39 and 40. In Fig. 39 the screw is single threaded, while in Fig. 40 it is double threaded. Fig. 41 shows a spiral spring made of square steel. This is drawn in the same way as the square-threaded screw in Fig. 40. To draw the spiral spring shown in Fig. 42, first draw the helix which is the centre line of the spiral. Next draw circles with their centres on the helix, and having a diameter |