The friction of carriages, moved along Rail-roads, will be afterwards shewn not to differ materially from that of uniform resistance; we may, therefore, express the resistance opposed by friction to the body moving freely down the plane by F, and consider the diminution of the gravitating force of the body, by this cause, equal to the amount of the friction; hence, retaining the former symbols,

We can, therefore, determine the friction F of any carriage or waggon by the latter formula, in causing them to descend a certain known declivity; and, ascertaining the space passed over in a given time, the difference between the space actually passed over, and that which the body ought to have described in descending freely, will be the diminution by the effect of friction, and will be a correct estimate of its amount.

Thus, find the gravitating force of the body down the plane, by multiplying the weight of the body by the height of the plane, and dividing the sum by the length; then multiply the weight of the body by the space passed over, and divide this sum by the square of the time in seconds, multiplied by 16 feet, and subtract this quotient from the gravitating force of the body, and it will give the friction.

This comprehends a body, or system of

the ascent of a train of carriages by the descent of a similar train more heavily loaded in a given time. The respective weights W and w of the descending and ascending train of carriages being given, we shall then have the following known quantities derived from the preceding experiments, viz. F and f by Table I. p. 194, by Table II. p.214.

Then, taking the friction and resistance of the several moving parts as deduced, by the foregoing experiments, and making T = the required time of descent of the carriages down the plane = t, we have

whence the preponderance of gravity necessary to effect the descent, in all states of the weather, in the time T, will be

and, having the weight of the descending and ascending trains of carriages, the inclination of the plane will be

In the caes of a single train of carriages dragging a rope after them, we have

This expression of G, which represents the gravitating force of the carriages, will, also, represent the action of any other power; and will shew the force required to urge the body, or train of carriages forward, whether they have to descend or ascend planes inclined to the horizon, or upon level planes.

In practice, therefore, we must either elevate the plane, or increase the number of carriages,

until we obtain the requisite preponderance; but, in every case, it will be necessary, in order to secure the constant action in winter and summer, that the excess amount to that given by the above formula.

Before dismissing the subject of self-acting planes, it may be necessary to state, that considerable regard should be observed in forming the line into a proper descent, or into that in which the velocity of the carriages, on all parts of it, shall be nearly equable as possible. The action of gravity causing bodies to descend with velocities uniformly accelerated, the motion of the carriages upon a plane with a uniform descent will be very variable; being accelerated, as the square of the times employed in traversing the plane. The plane should not, therefore, be made with a regular and uniform descent; but such as will give a greater preponderance of gravity at the commencement, and then diminish in such a ratio that the diminution of preponderance will abstract as much gravitating force as will compensate for the increasing velocity of the carriages, so that the two will counteract each other, and thus produce a comparatively uniform velocity in the carriages on the plane. The line of descent to perform these conditions is rather difficult to determine, but perhaps will approach near to the curve called a cycloid.

investment of capital in a Rail-road, and the greater certainty of transit, may make it superior to a canal; but, unless the disparity of cost is great between a Rail-road entering into competition with an existing canal, or unless some extraordinary circumstances in the nature of the traffic occur, it may be difficult to say, when horses are the motive power on each, which is superior.

There is one very important property in a Rail-way, which gives it great advantage over a canal, viz. the range of undulation which its nature permits; a straighter and shorter line can mostly be made between one place and another, which, from the necessity of having canals always perfectly level, or at least that level only broken at certain intervals by the occurrence of locks, occasions frequently a difference in distance of considerable magnitude, and this may diminish the comparative cost of transporting goods, and give a superiority to Rail-roads.

And again, in many cases, where the principal part of the goods are to be conveyed in one direction by a proper inclination of the Railway, the weight of goods may, in some instances, be considerably augmented without presenting a greater average-resistance than previously stated, when the relative perform

ance upon Rail-roads will be proportionably

increased.

Having thus given a few hasty remarks on the comparison of Rail-roads with Canals in the use of animal power, I shall also give a brief comparison between the use of mechanical power on Rail-roads, and animal power on canals; and here, as in every other case, where the two species of action come into competition, we shall find the mechanical power outstrip the animal in general economy.

Table of the relative performances of horses dragging boats on canals; and loco-motive engines, dragging carriages upon Rail-roads. The former supposed to be without locks, and