occur in practice, by irregularities in the tractile power. It is no doubttrue that if we were to make observations under extreme circumstances—as where the engines move on a perfect level, and where they fre-quently mount the steepest grades they can possibly ascend withoutloads—the cost of motive power per mile run, would be found to ex-hibit very considerable variations. The charges for fuel, as well as forrepairs, would be materially increased by the increase of the acclivityof the grades. But we have no such extreme cases in practice; and thefollowing investigation is based on the authorized assumption that thecost of running the locomotive engine and its tender, without other load,is proportional to the distance run, and independent of variations of thetractile force consequent on the ordinary irregularities of grade.

In the preceding remarks, the aggregate annual expenses of a lineof railroad under good management were represented by this formula,

i? + I^ +500/ ‘’ ( A )

In which N is put for the number of miles traveled, during the year,by the locomotive engines, T the number of tons net carried one mile,and h for the length of the line.

This formula, it will be observed, expresses the cost of transporta-tion without any express computation of the effect of the grades overwhich the tonnage is carried. Nevertheless, the result which it yieldsis not independent of the grades; for they enter into the value of N,and control the number of miles traveled by the engines. The vari-ation of the maximum gradient will not cause a sensible variation ofany item of the aggregate cost of transportation excepting that of thelocomotive power.

Now, in deducing the changes in the cost of locomotive power con-sequent on changes of grade, I shall assume that the circumstances ofthe trade are such as will permit that the machine be always startedwith a full train—that is to say, with the largest train which it is certainto control on the limiting grade. In many instances this assumption isnot strictly in accordance with the facts; since it is frequently advisable—particularly where the grades are light, and the trade inconsidera-ble—to effect a portion of the transportation with imperfect loads. Insuch cases greater ascents might obviously be encountered withoutincreasing the cost of carrying every train; and the effect of assumingthat the trains are .all full, when a portion of them is not full, will evi-dently be, to render the estimated increase of cost consequent on thegradient somewhat higher than it really is.

The formula is intended to fix a limit within which the actualvalue of the grade must always be found; and as it should be the ob-ject of every company to do their transportation with the least possi-