point in reference to it to which his attention is to be directed. The pencil being made of two kinds of material, this is indicated in the projection by drawing the sectional lines referred to in the last prob- lem in opposite directions across the materials, as shown. This point will, however, be more fully enlarged upon later on. From the projection of the pencil we pass on to that of the hollow tube, whose elevation is given in Sheet 2 in the YP at C, the section plane being shown by the line x y. .Now in the plan of this object, given in Fig. 97, Sheet 1, it will be noted that as there are only three side edges, and the two ends, seen from above, its "plan" is obtained by the projection of these edges into the lower plane or HP, and joining them up, as shown. But in the problem before us the upper part of the tube is assumed to be cut away, leaving the section of two of its sides exposed, together with part of its interior. To find its plan under these conditions we proceed as follows : From a and b in the elevation C, let fall projectors perpendicular to the IL into the HP ; and as the tube is assumed to be of the same length as the beam and pencil, and its back end the same distance from the YP as their ends are, at this distance from the IL, and parallel to it, draw the line a" b", cutting the projectors from a and b in those points ; and parallel to this line and at a distance a" a equal to the length of the tube from it draw the line a 1 b'. We have so far obtained the bounding lines of the plan sought. For the plan of the parts of the tube cut by the section plane x y, let fall projectors from points 1, 2, 3, 4 in the eleva- tion, cutting the lines a" b", a b', in the plan in points 1', 2', 3', 4', 1", 2", 3", 4" ; join these points by lines as shown, and the surfaces between each of these pairs of lines will be the plan of the parts of the