Ignoring the latter contingency, the two errors indicated, notwithstanding the fact that their effects are additive, are so small as not to affect materially the correctness of the results of this investigation concerning economic span lengths.

This demonstration proves that, in any layout of spans, with the conditions assumed, the greatest economy will be attained when the cost of the substructure per lineal foot of bridge is equal to the cost per lineal foot of the trusses and lateral systems. Of course no such condition as a bridge of indefinite extent ever exists, nor is the bed-rock often level over the whole crossing; nevertheless the principle can be applied to each pier and the spans that it helps to support by making the cost of each pier equal to one half of the total cost of the trusses and laterals of both spans. Since working out this demonstration some ten years ago, the author has made a practice of checking the correctness of the principle thereby established, by comparing the cost of substructure and superstructure in a number of bridges which he has designed and built, with the result that he finds it to be exact.

The principle will apply also to trestles and elevated roads, for in the latter, if we make the cost of the stringers or longitudinal girders of one span equal to the cost of the bent at one end of same, including its pedestals, we shall obtain the most economic layout. In an ordinary railroad trestle, consisting of alternate spans and towers, it will be necessary for greatest economy to have the cost of all the girders in two spans (one span being over the tower) plus the cost of the longitudinal bracing of one tower, equal to the cost of the two bents of said tower, including their pedestals.

On page 235 of the first edition of Prof. J. B. Johnson's "Theory and Practice of Modern Framed Structures," Mr. Bryan uses this method of the author's in a slightly different form for determining the most economic number of spans to adopt at any crossing, establishing the equation,