for the corresponding long-span structures, it was somewhat too great—also that the conclusion to the effect that the economic depth for arch ribs is the same for both railway and highway bridges is wrong. The reason for the variation in the two findings is that the weight formulae, upon which the first investigation was made, were of too empirical a nature to warrant their use for an exact determination of economic depth of arch rib.

From the original paper, the discussions, and the résumé, the following answers to the set questions have been reached:

First. For three-hinged arches with the grade line approximately tangent to the top chord of the arch at the crown, the average economic ratios of rise to span-length are as follows:

These values may be either increased or decreased by 0.025 without making any material difference in the economics.

For three-hinged, half-through arch-bridges, the average economic ratios for rise to span-length are as follows:

For three-hinged, high-deck arch-bridges, they are as follows:

For two-hinged arches and combined two-hinged and three-hinged arches, the economic ratios of rise to span will be practically the same as for three-hinged structures.

For the hingeless arch, a somewhat greater ratio of rise to span than that for the three-hinged arch is economical. The single test of this made for the 500-ft. span indicates that the best ratio is about 0.28 with low-grade deck, 0.33 for half-through arches, and 0.38 with high-grade deck.

Second. In respect to solid-rib arches, the question of economic rib-depth does not arise; for the depth should always be made as great as a proper consideration of the section for resisting compression will permit, and with due regard to shipping restrictions concerning limiting sizes of single pieces. For braced-rib, three-hinged arches in steam-railroad bridges, the economic rib-depth varies from 7.8% of the span-length for 100-ft. spans to 5.3% thereof for 1000-ft. spans; and for highway bridges the corresponding variation is from 5.8% to 4.2%, as shown in Fig. 26a.

The effects on rib weights from using uneconomic rib-depths in braced-rib arches for both railway and highway bridges are given in Fig. 26b.

Third. It was found for all cases that the most economic location for the crown-hinge in a spandrel-braced arch is in the top chord. In braced-