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base and vertical height, has its axis inclined to its base at 45 ; required its elevation and the development of its curved surface, together with that of a frustum of the same solid of a given height. Fig. 201 Assuming the cone to be resting with its base on the HP, to draw its elevation, take any convenient point, a in the IL as a centre, and with half the intended diameter of its base as radius, describe a semi- circle cutting the IL in B and C. At a draw a line, making with BC an angle of 45, and another perpendicular to it. On the perpen- dicular line set off from a in a the intended vertical height of the cone, and through a' parallel to BC draw a projector to cut the inclined line from a in A, which will be the apex of the intended cone. Join B and C with A by right lines, and the triangle ABC will be the elevation. To find the development of its curved surface, divide the semi- circle drawn on BC into any number say eight of equal parts, and from A let fall a vertical projector to the IL or base BC of the cone produced to cut it in x. Now it is evident that the longest and shortest meridians that can be drawn on the surface of the cone, will be its bounding lines AB and AC ; then to determine the lengths of any intermediate ones for that is what is required to be known proceed as follows : From x in BC produced, draw right lines to the several points of division in the semi-circle, and take each of these lines as a radius, 206 FIRST PRINCIPLES OF and x as centre, and describe arcs to cut BC in points 1, 2, 4, etc. Join each of these by right lines with A, the apex, then will they be