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In both cases, the extrados with its intrados willrepresent a beam*, bressummer, lintel, or girder,whose fibres or laminae are horizontal, equally strongin every part; in the first case, a weight would affectthe rigidity of the material; in the second case, thecohesiveness of the material.

When a vertical line.

From any point U in this catenary draw the verticalline UK cutting AR in K, and through any point Habove U in the catenary draw the horizontal line YWcutting UK in Y. Make YW equal to HF, the thick-ness of the catenary at right angles to it at H. Inlike manner, find other points O, and X. ThroughXWO draw the curve which is a logarithmic curve,whose subtangent equals the radius of curvature ofthe catenary at its vertex A, from which it is derived,and KYUOWX is a section of a wall equally strongin every part.

TO DETERMINE THE BATTERING OF A POST, PIER, ORWALL, SO THAT ANY HORIZONTAL SECTION OF IT MAYBE PROPORTIONAL TO THE VERTICAL COMPRESSIONAT THAT SECTION, AND TO DETERMINE THE WEIGHTIT WILL SUSTAIN.

In the case of a Pier.

Fig. 4. Let CY be the height of a prismatick pier or cylin-der, which by its own weight would begin to crush at

* Through the cusp O where the intrados VFO of the archcuts the absciss, draw the horizontal line OQ, cutting the loga-rithmic curve VSEQ in Q. It happens in this case that VO is toOQ as 4 to 5, and that the logarithmic curve VSEQ is, or a veryclose approximation to, the semi-curve of the arch of equilibration,which Dr. Hutton, vol. i., Tracts, page 62., terms, “ an arch ofequilibration, whose extrados shall be a horizontal line.”