tageously modified, to suit the more precise determination of the valueof railway constants, which we may anticipate from the increasingfacilities which are offered, and the more accurate observations whichsuch facilities will permit.

In order to apply the formulas, we must determine the values ofC, C' and c. Of the two first, C may be estimated, in ordinary cases,at three-tenths of a dollar, and C' at a quarter of a dollar; and experi-ment will justify the assumption of a half mill per mile for the averageincrease of the cost of motive power, due to each gross ton added tothe load of the engine—a fact which is expressed by the equationc=IS Wvs °f a dollar. In the formula t" is put for the gross tonnageof the line, consisting of passengers and freight and the cars whichcontain both, whether attached to engines conveying full or partialloads.

Let us take the following case for an example. The length of theroad, by the preliminary survey, is 30 miles; the limit grade whichthe eye and judgment have established on the profile, is 40 feet permile; the anticipated freight is 60,000 tons net, and is likely to be sodivided in direction that there will be found an annual gross weightof 40,000 tons ascending this limit grade. The tonnage trains aresupposed to be full, and the gross weight of cars, freight and passen-gers is 130,000 tons. The number of through passengers is 20,000,and the power of the engines intended to be used is adequate to thecommand of 300 tons gross on a level road. Now,

I. How much would it be good policy to expend in the construc-tion of this road in order to reduce the grade which limits the loadfrom 40 feet down to 30 feet per mile?

II. How much could we afford to pay for the reduction of the lengthof the road one mile ?

III. How much would we be authorized to increase the length ofthe road in order to reduce the rate of ascent of the limit grade tenfeet per mile?

IV. How much would we be authorized to increase the length ofthe same road in order to reduce the limit grade ten feet per mile, inthe supposition that no passengers arc to be conveyed, and that thegross tonnage is all conveyed in full trams, and amounts, exclusive ofpassengers and their baggage, to 120,000 tons?

Now wo have C’ = C = T 3 T ; c = t = 60,000; t' =* 40 , 000 ;t" = 130,000, and for the fourth problem 120 , 000;77 = 20 , 000 ; h = 30;x' =40; x — 30, and W= 300.

The appropriate quantities being substituted in equation (F) willgive, in answer to the first problem,

40,000 X 30 X (40—30)