different degrees of perfection, according to the sum of money expended on the works; but it is not evident which of these plans is the best, nor does it follow that the same sum of money would produce an equally beneficial improvement, if laid out in raising the valley without lowering the summits, or in lowering the summits without raising the valley. In order to solve this important problem, and to arrive at an accurate result in this and similar investigations, it is necessary to know correctly the expense of horse labour under the varying circumstances of velocity and force of traction on different inclined planes, and also the draught of carriages, and the ratio of the increase of draught in consequence of increase of velocity.

By the experiments lately made on the Holyhead Road by order of the Parliamentary Commissioners, these circumstances have been accurately ascertained from practical experience, which has enabled me to deduce the necessary formulæ from actual practice, without having recourse to theoretical investigations or abstruse calculations.

To go into the detail by which these formulæ were deduced would be in this place unnecessary; it is sufficient to state that correct tables have been calculated from these formula, which show the expense of drawing a given weight with a given velocity over every rate of acclivity and declivity, and length of inclined plane.

By means of these tables the expense of drawing a ton weight over any line of road may be determined with great accuracy. Hence all that is necessary in the present investigation is, to calculate by the tables the expense of transporting a ton weight over the existing line of road, and also over the proposed improvements.

The difference will be the saving in expense of drawing one ton with the given velocity over the proposed improvement. This, multiplied by the number of tons that pass over the road each day, and by the number of days in the year, will give the annual saving, which, compared with the interest of the money necessary to be expended in making the improvement, will clearly show whether the saving in horse labour is commensurate with the proposed expense. By applying the same criterion to each of the proposed plans, it will at once be made evident which of them should be adopted, as that which would produce the most beneficial result at the smallest expense. By this method, which is new, and founded on correct principles, I have endeavoured to determine the most advantageous method of improving the present line of road across the Stowe Hill Valley from the sixty-fifth milestone to the Crown publichouse at Foster's Booth, a distance of two miles.

PLAN, NO. I.

By this plan it is proposed to leave the present road near the sixty-fifth milestone, and to pass at an elevation of twenty-seven feet lower than the present road: from thence it would descend through a natural valley at a rate of inclination of one in thirty to an angle in the Lichborough and Northampton road: from this point it would pass in a straight and horizontal line, at an elevation of fifty feet over the brook, to the junction of the present road; here it would cross the road, and, skirting along the side of the hills at an inclination of one in thirty, running nearly parallel to the present road, and at about fifty yards distance from it, would pass the summit at twenty-seven feet lower level, and join the present road, near the Crown public-house.

For the purpose of ascertaining the comparative merit of this plan, the following calculation, as above described, has been made:

Pages 411, and 412. contain the calculation of the expense of drawing one ton over the present line of road between the given points, in both directions; which amounts to 82.0647 pence.

Page 413. contains the calculation of expense of drawing one ton over the proposed improved road, as above described, between the same points; which amounts to 76 1724 pence. By this it appears that the saving in horse labour on each ton will be 5.8923 pence; and for 170 tons the daily saving will be 47. 3s. 6d., which, at 5 per cent., is interest for 30,310l. 10s. The estimate for making this improvement is 23,7571. (See page 418.)

The difference between the amount of the estimate and the saving to the public by the proposed improvement is, therefore, 65531., which is the actual sum the public would gain by this improvement, supposing the present traffic to continue; if the traffic increased, the saving would be still more.

PLAN, No. II.

By this plan an embankment is proposed to be raised across the valley, 70 feet high, and 1313 yards long, the embankment to be formed with earth taken from the most convenient place, without having recourse to cutting from the summits: by this plan part of the present road would be retained.

By a calculation similar to that in the first investigation, and given in pages 413. and 414., it appears that the mean expense of drawing a ton between the sixty-fifth milestone and the Crown Inn, at Foster's Booth, would be 79.4584 pence; and as the expense of drawing a ton over the

present road between the same points is 82.0647 pence, the saving in expense by this improvement would be 2.6053 pence, and for 170 tons it would be 17. 16s. 11d., which is interest for 16,3947. The estimate for this improvement, as detailed in p. 419., is 28,8907. The difference between this sum, and the saving, is 12,4967. which is the loss the public would sustain by making this alteration, calculated on the present state of the traffic over the road.

PLAN, No. III.

By this plan it is proposed to raise the valley fifty feet, to cut twenty-seven feet from the summit at Foster's Booth, and reduce the inclination from 1 in 16, 17, and 18, to 1 in 30, on the east side of the valley; and from 1 in 14 and 15 to 1 in 30 on the west side of the hill.

The embankment would commence near the sixtyfourth milestone, and terminate at the turn of the road leading to Northampton. The mean expense of drawing a ton weight over the road, when improved in this manner, between the sixty-fifth milestone and the Crown Inn at Foster's Booth, as given at p. 415., would be 78.4715 pence: here the saving would be 3.5942 pence on each ton, and for 170 tons it would be 21. 10s. 11d., which is interest for 18,4837.

The estimate for this work, as detailed in p. 420., is 20,1447.; the loss would therefore be 16617. in making this alteration.

PLAN, No. IV.

By this plan it is proposed to raise the valley forty feet, to cut twenty-seven feet from the summit at Foster's Booth, and ten feet from the summit at the Angel. The embankment would extend from near the turn of

the road to Lichborough to the upper angle of Mr. Drayson's osier plantation, and the lowering of the summit at Foster's Booth would extend into Cold Higham Fields, nearly similar to that in the first plan. All the rest of the road would remain as at present.

If this improvement were made, the mean expense of drawing one ton over it between the sixty-fifth milestone and the Crown Inn at Foster's Booth, would be 79.8450 pence, as shown in p. 416.: the saving, therefore, per ton, would be 2.2207 pence, and for 170 tons it would be 377.5190 pence; which would be interest for 11,4207.

The estimated expense of this improvement, as detailed in p. 421., is 14,1717. The difference is 2,7517.; which shows the amount of loss the public would sustain by completing the work here described.

PLAN, NO. V.

By this plan it is proposed to raise the valley forty feet, to lower the summit at Foster's Booth twentyseven feet, and the summit at the Angel eighteen feet. If this improvement was adopted, the new line would run along the south side of the present road to near the sand pits, where it would cross it obliquely, and entering. a small ravine on the opposite side, it would cross the valley in a straight line for the upper angle of the osier plantation. At this point it would again cross to the south side of the present line, and follow the direction described in the first plan.

The saving in expense of drawing a ton over this line of road, when made as above described, would be 50200 pence (p. 417.), and for 170 tons it would be 475.39 pence, which would be interest for 25,8157.

The estimated expense of this improvement, as