AG 2

FG= ; and from the similar triangles A EG,

2AE

DCB; AG: AE::DB: BC; from which AG=

AEX DB

BC

MN

; but t; r representing the velocity with

и

which the space MN is traversed in the time t. By making these substitutions, the former equation becomes DB2

K".

Q v2 4g MN

From this expression the following conclusions may be drawn:

1. That the resistance arising from a surface of this description is proportional to the load.

2. That the draught or force of traction is proportional to the square of the velocity; and consequently pebble or rough pavements are more adapted for heavy loads, with a slow velocity, than for light carriages with quick velocities.

3. That the draught increases in the inverse ratio of MN; that is, as the distance between the paving-stones diminishes, or as the stones are narrow, the cavities remaining the same.

4. That the draught increases in the ratio of the width of the cavity to the radius of the wheel.

When the stones made use of for paving are of a good shape, well dressed, and sufficiently large, and laid on a firm and substantial foundation, they form the most perfect road surface for general purposes. The cavity between the stones should not exceed half an inch in width, by which means carriage wheels would pass over them without the least shock or resistance, and consequently without producing the noise often complained of in towns, at the same time that the surface would be sufficiently rough to prevent the horses from slipping.

NOTE C. Page 70.

THE next resistance, friction, which we shall consider, is that which arises from the wheels being forced over obstacles which break down under their weight, or when they are drawn through mud or other soft substances, or when the material of which the road is composed (such as gravel or small stones) is put on a soft or yielding substratum in layers so thin that the weight of the wheel can make an impression on it, and force it down so as to form a rut.

Let A B C represent a carriage wheel resting on the

horizontal road BD, the surface of which is hard and solid, but covered with mud, sand, or gravel, to the height of the line A E: if it be very soft, the wheel, as it rolls along, will press through it as if it was water, and rest on the hard and firm surface B D. If it be of a more tenacious nature, as some clays, or composed of sand or gravel which the wheels will only compress, without displacing it, the wheel will not go to the hard

surface, but approach it in proportion to the weight on the axle or wheel and the compressibility of the material over which it passes. A heavy wheel will sink deeper than a light one into a soft road, if both wheels be of the same dimensions. At the point A, where there is no weight, the surface is undisturbed; and at the point B, the material composing the road is compressed and sunk as much as it can be by such a weight: all the intermediate part between A and B is pressed by a less weight, decreasing from B towards A, and is proportionally compressed or lowered. The resistance which is opposed to the wheel evidently arises from its action. upon that portion of sand or mud contained between A and B; and the power necessary to overcome this will depend upon the length of lever at which it acts, or the depth to which the wheel sinks, and the stiffness or incompressibility of the substance which covers the road. Hence it is impossible to say or calculate the power or draught necessary to draw a carriage over a road so circumstanced, without experiments being made to ascertain the compressibility of each substance, and the consequent effect on the draught of carriages with wheels of different construction, and different loads. It is, however, within the power of mathematical investigation to furnish formulæ by which the law of increase in the power necessary to overcome such resistances is known, and by combining these with experiments the power necessary to draw a carriage over any line of road may be determined.

If the resistance arises from the wheel sinking into a soft stratum, instead of through an accumulation of mud or dust, until it rests on a firm surface, the investigation will be similar: the only difference is, that in one case the wheel can only sink a limited depth; for, arriving at the

hard surface of the road, it can penetrate no farther, the leverage at which the power acts will remain constant, if the weight be sufficient to press the wheel through the soft covering to the solid surface. The resistance will depend upon the nature of the material through which it rolls; but, if there be no solid or hard substratum under the outer crust, there will be no limit to the depth to which the wheel will sink. Thus, when a cart is drawn through a ploughed field, it is well known that the wheels will penetrate to a depth proportionate to the load, and the labour of the horses will be increased accordingly.

This effect is nearly the same as that which takes place when a carriage is drawn over a weak gravelly road, and is evidently more injurious to the horses employed in draught than when they work on a solid and firm road, although it be covered with an inch or two of mud.

M. Gerstner has investigated this subject, and given formulæ for the resistance arising from a wheel passing over a soft stratum of different degrees of compressibility. These formulæ are

the resistance;

half the chord of the segment of the wheel in the

ground;

A radius of the wheel;

q=weight of load;

W resistance of the soil when compressed by the wheel to the depth of an inch or any other unit; m=an indeterminate number, expressing a power of the depth to which the wheel penetrates, proportional to the resistance of the soil at that depth, and which is to be determined by experiment.

From these formulæ it is evident that the resistance is caused by the wheels sinking into the ground; and therefore it will be better, under such circumstances, to divide a heavy load between two or more carriages than to carry a heavy load on one carriage; and also that the resistance will be diminished by increasing the width of the wheels.