an economic investigation, to be of any value, must include both substructure and superstructure; and the costs of the former are likely to be very different in arch designs and simple-truss designs for any crossing.

Some of the solutions of the "side issues" referred to in the preceding "Synopsis" are the following:

First. The percentages to apply to weights of metal in simple-truss spans, in order to find the weights for arch ribs and the superimposed columns with their bracing to carry the same live loads, are given by the following equations:

For steam-railway structures:

For electric-railway structures:

For highway structures:

In these equations S is the span-length in feet, and P is the percentage to apply to the weight of metal in the trusses of any simple-truss bridge, in order to ascertain the weight of metal in the corresponding arches and the superimposed columns with their bracing. It must not be forgotten that the superior limit of S in these equations is about 1000-ft., which is as far as the recorded weights of simple-truss spans are carried, and that the inferior limit is 100-ft. In Fig. 26d are plotted curves giving the same information as that presented in the three preceding equations.

Second. It is evident from the computations and the resultant diagrams that the arch is more economical for highway bridges and for combined highway and electric-railway bridges than for steam-railway structures. This is because of the smaller ratio of live load plus impact to total load in the former. The larger the dead load of the flooring and floor-system, the more advantageous is it for the arch structure in comparison with the truss bridge.

Third. Based on the numerous weight computations made specially for the preparation of the paper, the following formulae for weights of metal in the arches alone, per lineal foot of span, in arch bridges of the several types, have been established:

For Three-Hinged Arches:

For Two-Hinged Arches:

For Combined Two-Hinged and Three-Hinged Arches: