. The Civil engineer and architect's journal, scientific and railway gazette. Architecture; Civil engineering; Science. That is to say, the forces applied at the several angles of the poly- gon are proportional to the cosines of the halves of those angles. Let us now suppose that the lines A B and B C are equal to each other. Through the points A, B, C, fig. 4, describe the circle A B C D, draw the diameter B D, the arc A E, and E F at right angles to A B. Then li D bisects the /ABC, and because B A D is a right angle (Euc. J). 3L b. 3) :â B A ; B D : : B F : B E : : cos. o : rad. B A .*. cos.
. The Civil engineer and architect's journal, scientific and railway gazette. Architecture; Civil engineering; Science. That is to say, the forces applied at the several angles of the poly- gon are proportional to the cosines of the halves of those angles. Let us now suppose that the lines A B and B C are equal to each other. Through the points A, B, C, fig. 4, describe the circle A B C D, draw the diameter B D, the arc A E, and E F at right angles to A B. Then li D bisects the /ABC, and because B A D is a right angle (Euc. J). 3L b. 3) :â B A ; B D : : B F : B E : : cos. o : rad. B A .*. cos. a = 5-f;. Hence, as th« forces P, Q, R, S, are proportional to B D cos. a, cos. 0, &c., if we suppose all the sides of the polygon to be equal, it is evident they will be inversely, as the radius of the circle passing through the points terminating the two contiguous sides. But if we imagine the sides of the polygon to become indefinitely small, it then assumes the form of a curve, and the circle becomes the osculat- ing circle, or the circle of equal curvature. If, then, a flexible curve, the two extremities of which are immoveably fixed, be acted on at points equidistant from each otherby a number of normal forces, these forces will be inversely as the radii cif curvature of the points of appli- cation, and the forces developed in the direction of the curve will be everywhere the same. If the normal forces be equal, the reciprocals of the radii of curvature w ill be equal, and therefore the radii of curva- ture themselves; consequently, in this case, the curve will be part of a circle. If the normal force vary as the cube of the cosine of the angle formed by the ordinate and tangent at any point, the curve is a para- bola, as is jjroved by Ihe following investigation. Let P A R, fig. 5, be a parabola generated by the action of normal forces, P T the tangent at the point P and N T, the subtangent. Let A N = :r, N P = 7/, and/*, the principal parameter or latus I'ectum
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