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an expression from which we are able to calculate immediately theannual value of any augmentation or reduction of the length of theroad, and consequently the sum which it would be expedient toexpend for the purpose of effecting any given reduction.

But we have already seen that if we diminish the inclination of themaximum gradient any quantity x'—x feet per mile, we simultaneous-ly reduce the aggregate annual expenses the amount represented bythe expression

2 C

x

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t' h.

In order, then, to determine the change in the length of the linewhich will have the same effect on the cost of transportation as anygiven reduction of the limiting gradient, we must equate these twoquantities, and deduce from the equation the value of distance, orh'—h in terms of the change of gradient. This operation produces forthe expression of the quantity which we will be authorized to increasethe length of a road for the purpose of reducing the limiting gradient,of which the ascent is x' feet per mile, down to any other rate x feetper mile,

h t' ( x' — x)

A.( I)

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The augmentation of the length of the road adequate to compensatefor any given reduction of the limiting grade, is therefore directlyproportional to the original length, and reciprocally, or very nearly so,as the power of the engines used in the conveyance of the freight.The greater the tonnage which ascends the limiting gradient, thegreater is the distance that the length of the line may be increased inorder to reduce that gradient any given amount per mile; and thegreater the power of the engine kept in operation at a given expenseper mile run, the less would be the increase of the length of the roadfor the same purpose. The greater the number of passengers that arecarried in light trains, and the greater the proportion of the tonnagewhich is conveyed by engines loaded below the maximum due to thelimiting grade, the shorter must be the line and the steeper the grades.The measure of all these influences is deducible from the formula,which I regard as the true equation of distance. It is expressed in asaccurate terms as it is practicable to obtain with our present expe-rience. In course of time the co-efiicients may, perhaps, be advan-