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APPLICATION OF THE FORMULAS DERIVED IN TRACT 3,FROM THB INQUIRY INTO THE PRINCIPLES OF PEN-DENT BRIDGES, TO DETERMINE THE PROPORTIONSOF THE PARTS OF INSISTENT BRIDGES.

In like manner as the length of a chain of an uni-form thickness hanging vertically, is a measure of thetension at its upper extremity (see Tract 3. Art. 6.page 103), so the length of a column of an uniformarea in section standing vertically will be a measureof the compression at its base.

By a parity of reasoning from the article page 84,Tract 3. and the subsequent theoretical investigationof the pendent Catenary, it may be derived in respectto insistent arches:

That the force of compression at the vertex of aninsistent arch, equally stable in all its parts, and in thesame or horizontal direction every where, is equal tothat of the weight of a prismatick column of the samematerial as the arch is composed of, having its sectionequal in area to a section of the key stone in thedirection of the radius of curvature at the vertex,and its height equal to that radius of curvature ;and supposing the repulsive strength of homogeneousmaterials to vary as the area of the section ; the lengthsof the joints of the voussoirs of such an arch at rightangles thereto must vary as the secant, of the anglethe curve makes with a horizontal line, or of the anglemade by the joint with a vertical one : and it also maybe derived, that such an insistent arch may be pro-duced to any distance from the vertex without alter-ing the compression of any given joint.

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