The teeth are completed as in previous examples. If the point circle for the tops of the teeth of wheel B cuts the line of contact D F in D, and the point circle of the teeth of wheel A cuts it in F, then D F is the length of the line of contact. EXAMPLES. EX. 1.-Draw an arc of a circle of 8′′ radius and consider it as part of the pitch circle of a toothed wheel. Then draw completely four teeth as follows:-a, b, and c to have epicycloidal faces and hypocycloidal flanks, rolling circles, (a) 4" diameter, (b) 2" diameter, (c) 5" diameter, (d) involute teeth radius of base circle 7.7". To get sizes of teeth assume them as of 2" pitch. (The object of this example is to show the effect of using rolling circles of different sizes, hence the teeth had better be drawn separate from each other, say about 3′′ centres.) * EX. 2.-Draw a rack and pinion showing four teeth on each as follows:-Pitch of teeth 2", number of teeth in pinion twenty; pinion teeth to have involute faces and radial flanks, rack teeth to have cycloidal faces and radial or straight flanks. * EX. 3.-Draw two equal spur wheels in gear showing five teeth in each. Pitch of teeth 24", number of teeth ten. Faces of teeth epicycloids, flanks hypocycloids. Rolling circles 13′′ radius. * EX. 4.-Draw two spur wheels A and B in gear, showing five teeth in each. Involute teeth 2" pitch, twelve teeth in wheel A, seventeen in wheel B, angle of obliquity 15°. EX. 5.-The diameters of two spur wheels are 24′′ and 36′′, the pitch 21", and the path of contact a straight line at 75° to the line of centres. Draw a pair of teeth in contact of such length that two pairs of teeth may always be in contact. (Vict. Hon., 1891). (Length of path of contact must oe twice normal pitch, the normal pitch is distance from face of one tooth to face of next along line of contact; make a right angled triangle with hypotenuse equal to pitch, and base angle equal angle of obliquity-then base is normal pitch.) * After drawing accurately three or four teeth on each wheel of Exs. 2, 3, 4, the student would do well to work as follows :-Cut the paper carefully round the teeth, leaving enough for the whole wheel, thus making a pattern of the wheels; and fix the wheels the right distance apart by sticking pins through their centres. Then place a sheet of paper beneath the teeth, and move the wheels as in actual working; prick through at the point of contact of the teeth upon the paper below, thus obtaining the path of the points of contact, compare the results with Figs. 57 and 59. Also draw the normals to the teeth at one or two different points of contact, and see if the normal at the point of contact of any two teeth is common to the curves of both teeth. SECTION IX. SOLID GEOMETRY. Projection of Points, Lines, Surfaces, and Simple Solids.— Solid geometry or orthographical projection, as its name implies, deals with the drawing of solids, and enables the three dimensions of a solid length, breadth, and thickness to be shown upon a flat surface, such as a sheet of drawing paper. It must not be confounded with perspective, with which it has no connection, beyond the fact that both use måny similar methods and terms, as perspective geometry depicts a solid as it appears to the eye to be, and shows its three dimensions in one drawing, whereas solid geometry depicts a solid as it really is, and requires at least two separate drawings to show its three dimensions. If, for example, we look down upon the top of a table, we see a view of the table which gives no idea of its height from the floor, but only shows its width and length, while if we look on one end of the table with our eyes on a level with the table top, we then see the height of the table, but cannot form any idea of its length. The principles of solid geometry recognise these facts, and suppose a solid to be looked at from different positions, and views to be drawn of its appearance from each position. These views have distinct names and are drawn in accordance with certain laws of projection, which it is very important should be clearly understood. From the illustration of the table, we see that by looking at it from two different positions, we are able to show its three dimensions, length, width (or breadth), and height. The way in which solid geometry enables us to draw these two views is by supposing that the view of the table looking from above is drawn upon the floor underneath it, and that the view looking on the table end is drawn upon the wall at the further side of it. The floor and the wall are flat surfaces, called planes, mutually at right angles, and the way in which the views are depicted upon them, is to suppose that lines are drawn from each corner of the table perpendicular to the floor and wall, meeting these surfaces in points, which, if joined by lines, will form the two views of the table. Now, if we suppose the floor and wall to have been covered with a sheet of paper, and the drawings made upon it, and then the paper spread out flat, we should possess what is recognised as a drawing of the table, showing practically all its dimensions. We Plan, Elevation, Projections, Planes of Projection, Projectors.-The view of the table, and therefore of any solid, as seen from above is called its plan, and the view as seen from the end its elevation. The views of objects drawn on the principles of solid geometry are called its projections, the imaginary planes on which they are drawn are called the planes of projection, and the lines from the object to the planes are called projectors. have seen that only two projections are necessary-one on the floor, a horizontal plane, the other on the wall, a vertical plane, and as these two planes are always required for the plan and elevation of a solid, they are termed the horizontal plane of projection, usually denoted by the capital letters HP, and the vertical plane of projection, usually abbreviated to V P. Evidently the two planes of projection intersect in a line, which, owing to the horizontal plane being supposed the plane of the ground, is called the ground line, and is generally denoted by the capital letters XY. All this will be clearly understood by reference to the following example.-In Fig. 60a is shown a representation of a simple solid, an equal armed cross made of square wood, with the horizontal and vertical planes of projection, having its plan and elevation drawn upon them Lines perpendicular to the HP are drawn through each corner of the solid, meeting the HP in the points marked a, b, c, d . . ., and similarly lines perpendicular to the V P are drawn through each corner, meeting the VP in the points marked a', b', c', d'... To join these points in the right order we look at the solid, and see that A joins B, and that B joins C, and C joins D, and so on, and, therefore, by joining a to b, and a to b', &c., we obtain on the HP and V P a plan and an elevation of the solid. Notice that the plan really represents two faces of the cross, the upper and lower, which are similar, and, therefore, that each point in the plan shows at least two corners of the cross, and similarly with the elevation. On the right hand of the cross is shown a point P with its plan p, and its elevation p', the line Pp being its projector to the HP, and the line Pp' its projector to the V P. The plan and elevation of these projectors are drawn at po and p'o, and it should be specially noticed that they make two lines, meeting on the ground line, each being perpendicular to it. The student should suppose that the plan and elevation of the projectors of the cross are drawn in this way, although in the figure they are omitted for the sake of clearness. Now, imagine the VP to be turned upon the ground line as a hinge, away from the solid, as shown by the arrow, until it becomes horizontal and forms a continuation of the HP. We shall then have the representation of Fig. 606, which is the usual solid geometry projection or drawing, showing a plan and elevation upon a single flat sheet of paper. With the aid of these two figures the student should now verify the following X statements, all of which are important and should be remembered : (a) The plan is below the ground line, and the elevation above it. (b) The plan and elevation of the same point are exactly one under the other, in a line perpen dicular to the ground line, therefore the plan of a solid should be directly under the elevation. (c) Heights above the HP are shown in the elevation. (e) The projectors are shown by lines, joining points in the plan and elevation, and perpendicular to the ground line. (f) The elevation of a point in the H P, and the plan of a point in the V P, are shown by a point on the ground line, for if p and p' be these points, their elevation and plan are both shown by the point o (Fig. 606). From this it follows that when a solid has one face or edge in the ground plane or H P, its elevation will begin from the ground line, and similarly if it has a face or edge in the VP its plan will also begin from the ground line. We see from Fig. 606 that the plan and elevation are separated from one another, and that the distance between them depends only on the height of the solid above the HP and its distance in front of the V P. In examples of solid geometry these distances may be given of any desired length, or may be left to the will of the student, in which case it is convenient to assume the solid, as standing on the HP and in front of the V P, as this gives an elevation starting from the ground line and a plan removed from it, thus separating the two drawings and adding to their clearness. Marking Plans and Elevations. It will have been noticed in Fig. 60a that each point in the solid is denoted by a capital letter, as A, B, C, while its plan is marked by the same letter in small type, as a, b, c, and its elevation by a similar letter with the addition of a dash, as a', b', c'. This is a convenient notation, usually adopted in solid geometry, and will be adhered to in all following examples. A solid is bounded by surfaces, a surface by lines, and a line by points, and we shall, therefore, lead up to the projection of solids by examples dealing with points, lines, and surfaces. In commencing solid geometry it will be found very helpful to make up a rough model of the planes of projection, and of the objects to be drawn. A book or instrument box opened at right angles very well represents the HP and V P, a drawing pin may represent a point, a pencil a line, and a set square or piece of card a surface or plane, while models of simple solids can be easily made. It is only in this way that the beginner can hope to gain an intelligent and useful knowledge of the subject, and be able to proceed with confidence to advanced problems and to machine drawing, where the objects to be drawn exist only as a mental picture, and where their positions relative to, and their projections upon, the planes of projection have to be vividly imagined before they can be represented upon the paper. All engineering draughtsmen use the results of the principles of solid geometry, although, as the student will see in |