14

or, it would be worth while to expend 20,000 dollars in grading theroad to reduce the ascent of this limiting slope ten feet per mile, pro-viding we do not thereby incur any increase of the cost of main-tenance, or any augmentation of distance.

Again, by inserting the values provided by our data in equation(H) we obtain, for the solution of the second problem, in the suppo-sition that the passenger trains make two trips a day, or that N' =» 730

730 3 20 + 40 130,000 14x 60,000 20,000

4 + 5 20 X 300 * ’ + 2,000 1,000 + 100

600 =

2,027 dollars, for the annual charge incident to each mile of the

length of this particular road.

We multiply this quantity by

100

to obtain the equivalent capital, which we find to be $ 33,783.

It would be worth about $ 34,000 to reduce the length of this roadone mile.

Again, by inserting these values in equation (I) we will obtain insolution of the third problem

3_

10

X 30 X 40,000 (40 — 30)

3000

(

730 3

4+5

^±ii X 40,000-

20X300*

130,000 14x60,000

2,000

1,000

20,000

100

1200

202T

or about four-sevenths of a mile. On such a road, the value often feet in the limiting grade is only equal to four-sevenths of a milein distance, although that grade is not opposed to the heavy trade—afact which should prompt us to be on our guard when we attempt toreduce the grades of a road by an increase of distance before we haveexamined well into the character and amount of the trade which is tobe accommodated.

Finally, we shall have, for the solution of the fourth problem,

3

T<T

X 30 X 40,000 (40 — 30)

nnnn .~ 20+40 120,000

3000#- -—Z.— x 40,000 + -

14 X 60,000

5 20 X 300 '

2,000

1,000

+ 50o)

40

‘55

or about five-sevenths of a mile :—showing that we can increase thelength of this road 25 per cent, more, for the purpose of reducing thegrade any given amount when we carry no passengers, although theamount of freight is the same, than when the road is destined, in part,for the accommodation of travel. This is simply to say that the valueof distance increases with the increase of every part of the businessoftheline; while the value of a gradient increases only with that por-tion ol the business which ascends it in full trains.

It may be objected to the method which is here pursued that the