avoided the issue because of the immense amount of work involved in the solution of the various problems. The author has tried of late years on several occasions to persuade some of his brother engineers to undertake the task; but not meeting with any success, he finally decided to do the work himself. The results of his efforts are here-with presented to the members of the Society with a most earnest request for a thorough discussion, particularly by those who specialize in bridgework.

The principal economic problems concerning the design of steel arch-bridges which have been solved for this paper are the following:

First. The economic ratio of rise to span-length.

Second. The economic depths for the ribs.

Third. The economic location for the crown-hinge in three-hinged, spandrel-braced arches.

Fourth. The ratios of weights of metal required for the solid-rib, the braced-rib, and the spandrel-braced types.

Fifth. The ratios of weights of metal required for the hingeless, the two-hinged, and the three-hinged types.

Sixth. The economics involved by making arches three-hinged for the dead load and two-hinged for the live load.

Seventh. The economy of the cantilever-arch with suspended end-spans, as compared with an ordinary arch and two flanking simple-truss-spans.

Eighth. The ratios of weights of metal required for certain portions of arch bridges as compared with the corresponding portions of simple-truss bridges of the same span and same live-load-carrying-capacity.

In answer to this request, a number of, engineers, both at home and abroad, discussed the paper; and their discussions have been printed in several copies of the Society's "Proceedings." These discussions have been thoroughly analyzed and replied to in the closure of the memoir.

With one exception, the discussions did not result in the modification of any of the author's findings-and in that one the change was not serious. It was pointed out by him that the first two problems were solved by the use of certain semi-rational, semi-empirical formulae for weights of metal in arches established in Chapter XXVI of "Bridge Engineering," but that the other six were settled by means of actual computations of weights of metal determined from specially-prepared diagrams of stresses and sectional areas of main members.

Mr. Charles Evan Fowler in his thorough and elaborate discussion, including a tabulation of the salient features and dimensions of one hundred of the world's largest steel-arch-bridges, showed that the author's finding for economic ratios of rise to span-length agreed. fairly well with the averages computed from existing structures, but that his deduced economic ratio of depth of arch-ring to span-length did not; whereupon the author agreed to settle this question beyond a doubt by submitting it to the incontrovertible test of making actual designs and estimates of weights for arches of 500-feet span, of economic ratio of rise to span-length, and having regularly-varying depths of arch ribs, so as to determine the depth producing the least weight of metal. This was done; and the results of the special calculations were given in the résumé, which showed that, for short-span railroad arch-bridges, the previous finding was correct, but that,