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line, of which Gp and Ct are its projections. This is self-evident, for if the right-angled triangle Gth, which may be assumed to be lying on the HP, with its base line coinciding w^ith Ct, be raised to a vertical position, moving on Ct as a hinge, its base and hypothenuse will then be coincident with Ct, and its third side lit is a vertical line perpendicular to the HP represented by the point t. Fig. 72. The other position a line may have, with respect to the planes of its projection, is that of being parallel to the HP, but making an angle with the YP. Putting this in the form of a problem, we will say Qtk. Let a given line be parallel to the HP, but inclined to the VP, and its projections required. In this case, let the given line at first be perpendicular to the VP ; its elevation when in that position in the VP will be a point as e, and its plan a line EF at right angles to the IL. But as EF is perpendicular to the VP it is parallel to the HP. While keeping it so, let it be con- ceived to swing on its end F as a joint in its direction of the arrow, until it makes any desired angle with the VP or IL, or until, say, E MECHANICAL AND ENGINEERING DRAWING 41 has moved into the position f ; its elevation in that position is found by drawing a projector through f, perpendicular to the IL, and a line through e parallel to it to cut the projector from/" in g, then the line eg is the vertical projection of EF when making the angle LF/" with the VP. Here it is again seen that the projected length of a line, although parallel to one of its planes of projection, is determined on the other plane by the amount of its inclination to that plane ; for had the given line EF in this case been moved through any greater or less angle than