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98 MECHANICAL AND ENGINEERING DRAWING 99 A Sphere. Again, if the two lines AC, BC of Fig. 146 be removed and a semi-circle be described on AB as a diameter, as in figure 147, then on revolving this figure AB as an axis, as before, the solid gene- rated will be a " sphere," as in Fig. 147. As the subsidiary solids before referred to are generated by parts of certain sections of the cylinder and cone, their definition is deferred until those sections have been found by projection. Taking the "cylinder" as the first curved-surfaced solid as our object, the prob- lem is Problem 49 (Fig. 148). Given the plan of a cylinder with its axis perpendicular to the UP, to find its elevation when its length is twice its diameter. Let the circle AB, No. 1, be the given plan; then its centre a will be the plan of the axis of the cylinder. Find by projection the eleva- tion of this axis a a. Assume the cylinder to be standing with one end on the HP ; then, as its ends are in the same relative position as the sides of the rectangle which generated them viz., parallel to each other and one of them is on the HP, set off on the axis, from the IL, the length of the cylinder in the point a, and through it draw a line parallel to the IL. Now, in looking at the cylinder in the direc- tion of the arrow in the plan No. 1, the visual rays will impinge upon its surface from A to B ; at A and B the rays will be tangential, and being at the same time perpendicular to the plane of projection, or the YP, they will strike both sides of the cylinder in lines drawn through A and B on its surface, perpendicular to the HP. Therefore through