Now since 7-1/2 pounds is very nearly the 300th part of a ton, it follows that if an inclination upon a railroad rises at the rate of 1 foot in 300, or, what is the same, 17-1/2 feet in a mile, such an acclivity will add 7-1/2 pounds per ton to the force of traction. This acclivity therefore would require a force of traction twice as great as a level. In like manner a rise of 35 feet in a mile would require three times the force of traction of a level, 52-1/2 feet in a mile four times that force, and so on. In fact, for every 7 feet in a mile which an acclivity rises, 3 pounds per ton will be added to the force of traction. If we would then ascertain the power necessary to pull a load up any given acclivity upon a railroad, we must first take 7-1/2 pounds as the force necessary to overcome the common resistance of the road, and then add 3 pounds for every 7 feet which the acclivity rises per mile. For example, suppose an acclivity to rise at the rate of 70 feet in a mile, the force of traction necessary to draw a ton up it would be thus calculated:--
Friction 7-1/2 lbs. 70 feet = 10 times 3 lbs. 30 ------ Total force 37-1/2
It will be apparent, therefore, that if a railroad undulates by inclined planes, even of the most moderate inclinations, the propelling power to be used upon it must be of such a nature as to be capable of increasing its intensity in a great degree, according to the elevation of the planes which it has to encounter. A plane which rises 52-1/2 feet per mile presents to the eye scarcely the appearance of an ascent, and yet requires the power of traction to be increased in a fourfold proportion.
It is the property of animal power, that within certain limits its energy can be put forth at will, according to the exigency of the occasion; but the intensity of mechanical power, in the instance now considered, cannot so conveniently be varied, except indeed within narrow limits.
In the application of locomotive engines upon railways the difficulty arising from inclined planes has been attempted to be surmounted by several methods, which we shall now explain.
1. Upon arriving at the foot of the plane the load is divided, and the engine carries it up in several successive trips, descending the plane unloaded after each trip. The objection to this method is the delay which it occasions,--a circumstance which is incompatible with a large transport of passengers. From what has been stated, it would be necessary, when the engine is fully loaded on a level, to divide its load into four parts, to be successively carried up when the incline rises 52 feet per mile. This method has been practised in the transport of merchandise occasionally, when heavy loads were carried on the Liverpool and Manchester line, upon the Rainhill incline.