The expense of horse power upon railroads, at dif ferent rates of speed, will be in the proportion of the useful load which he can drag, at the several rates of speed; and, adopting the conclusions in Art. 5, we have the relative performances, at the several rates of two and a half, four, and ten miles an hour, as per following Table; and the cost likewise apportioned according to the useful effect produced at the different velocities.

TABLE IX.

Table of the Cost of conveying Goods and Passengers on Railroads, at different Rates of Speed, with Horses.

The last column gives the cost of conveyance of railways, according to the average charges of some of the existing railways worked by horses. Upon the Edinburgh and Dalkeith railway, passengers are conveyed at the rate of one penny per mile, the horses travelling at the rate of nine miles an hour.

§ 2. Self-acting Planes.

The impelling force of this kind of motive power, is gravity; it is confined, as previously stated, to descending planes alone; and, when employed in practice, its

object is, to effect, in a given time, the ascent of a train of carriages, by the descent of a similar train, more heavily loaded. The respective weights, and inertia of the descending and ascending train of carriages, being given, and the weights and inertia of the rolls, sheeves, and rope, we shall then have the following known quantities, derived from the preceding experiments; viz., the friction of the loaded and empty carriages, by Chap. VII., and the friction of the ropes denoted by in Chap. VIII.

Art. 1.-Theoretical Conclusions of the Power of Selfacting Planes.

Taking the friction and resistance of the several moving parts, as deduced from the experiments under their respective heads, 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

(w+w+w') xs G-F =

12, 31 T2

(2.)

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

Or, adopting the notation used in calculating the result of the previous experiments, we have

(w+wo+a+a"s +b+c+c) x s

SG

12, 31 T

(w+w)—a+a"

× +9+F+ƒ

Art. 2.-Practical Application of the Power of

self-acting Planes.

It is impossible, in a case of such a compound nature, to give tables of the gradients, or inclinations, required in practice, for all the different lengths of planes; we shall, therefore, give one example, which will render any other a matter of easy calculation, and shall give one upon the Killingworth railway.

Example (1.) Suppose a descending plane, the length of which is equal to 1800 yards, down which it is intended to pass nine loaded carriages at a time, each weighing four tons, and which drag up nine empty carriages, each weighing 24 cwt.: required, the height of plane, or inclination, that will complete the descent, in 400 seconds?

SO

Supposing w = 1312, and wo SG = 2059, we have w+w= 99,171 lbs. ; and, consequently, a' + a" sa = 30,905 lbs.

SO

SG

Let the rope be four inches and a half in circumference, then 1900 yards will be 5562 lbs. = 9. If the sheeves be placed ten yards apart, and weigh 30 lbs. each, then 180 sheeves will be 5400 lbs; whence c=2700 lbs.; c = 227 lbs. The value of

R+R'+R"+P

?=

3.5 r

454+5400+5562+1696
8.5 x 16

=234 lbs. :

F = the 200th part of the weight =403 lbs., and f=

(99171+30905+5562+227 +2700) × 5400

= nearly.

12:31 × 4002

80640-24192

+234 +403+ 120

We may depend upon this result, in practice, where the apparatus is well fitted up, and kept in good order; but,

when this is not the case, we should, perhaps, instead of t, taket, whence, according to the example above, we have

nearly.

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 object 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 as 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.

§3.-Fixed Steam Engine.-Planes.

To elicit the performance of steam engines, fixed, and dragging carriages up planes, inclined or parallel to the horizon, by means of ropes, we have selected the four following experiments, on engines that have been in use for some time, and which, we trust, will be sufficient to furnish data, by which we may calculate the performance of engines upon other planes.

Not to confine the data to one particular kind of engine, we have taken two low-pressure, or condensing, engines, and two high-pressure engines.

Art. 1.-Experiments on the Power of Fixed Engines on Railroads.

EXPERIMENT I.

Boulton and Watt's low-pressure condensing engine, with two thirty-inch cylinders, steam 44 lbs. per square inch above the ordinary pressure of the atmosphere; rope-roll similar to A, Fig. 2, Plate IX.; rope, seven inches and a quarter in circumference, same as employed in Experiment VI., Chap. VIII.; length of plane, 2646 feet; height or ascent, 154 feet, 6 inches, being the same as Experiment VI., Chap. VIII.

Time of drawing up seven loaded carriages, each weighing 9408 lbs., similar to those employed in Experiments II. and III., Chap. VII., 620 seconds, the engine making 374 single strokes, five feet each.

Then (30 x 2) × 7854=1413·72 area of cylinders, and 1413-72 × 19.5 the pressure of steam in the boiler = 27,567 lbs. pressure upon the piston, which, in the experiment, was moved through 374×5=1870 feet; hence 27,567 x 1870-51,550,290 lbs. moved one foot, the power of the engine.