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mologous sides. Through KK'G draw the curve, Fig. 12.which is a parabola.

Through any point f in the intrados of the arch,draw fg parallel and equal to AV, and gi horizontal,cutting^'//, drawn at right angles to the tangent at fin h ; make ih equal gh, and from hi let fall the tri-angle hih similar to fgh, fg and hi homologous sides.Through Yk draw the extrados to the arch, whichwill touch the parabola GK'K in some point K'. FromF at right angles, to the tangent at F, draw FK",cutting the extrados in K". FK is the abutting jointbetween the arch and pier, vk K'K is the extradosof the arch, and its abutment pier, and is a curve,limiting the greatest range of any body projectedfrom VgG at the height VA, above any point A,jlF, O, of the intrados of the arch and pier, with thehorizontal velocity VA.

From any point K in the extrados, let fall KWparallel and equal to VA, and draw the horizontalline WY, cutting KY, drawn at right angles to thetangent at the point K in Y. KY is the secant ofthe angle YKW to AV radius. Through YY' y drawthe curve A y Y'Y. Then the arch and its abutmentalways contains in its thickness an arch VAy Y'YKK'A V,whose thickness, in a direction at right angles to theextrados =. AV x secant of the angle the extradosmakes with its ordinate, and whose strength in everypart is proportional to the compression on that part.