116 
FORMULAE. 
and even of 250 feet, may be advantageously overcome by locomotive power; it being clearly 
understood that such grades are to be resorted to only in cases of absolute necessity—economy 
in working the road rendering low gradients very desirable. 
The accompanying formulae and their applications show what work is to be expected from any 
given engine over given grades, and make the loss of economy in any particular case a question 
of easy solution. 
It is evidently the fact that there is at present a strong tendency to use much higher grades than 
were formerly considered practicable or advisable. Even in England and on the “Continent,” 
the American system of cheap roads, with high grades, to avoid the great expense of long tunnels, 
deep cuts, and high embankments, appears to be, to a certain extent, rapidly rising in repute. 
The use of inclined planes with stationary power, (within the limits before mentioned,) may, as 
a general rule, be considered obsolete, except in cases similar to that of the Pennsylvania Central 
road, where the amount of traffic is becoming so great as to require more than a double or even 
triple track; in this case it has been proposed to pass the surplus freight over the mountains by 
means of stationary power, reserving the locomotive power for passengers and freight requiring 
rapid transportation. 
Planes for stationary power should not exceed one mile in length. The number required to 
overcome any given ascent will depend more upon the elevation to be surmounted than the length 
of the ascent. 
The opinion has been expressed by one of the most reliable railway engineers in the country, 
that where the gradient does not exceed 132 feet per mile, locomotive is cheaper than stationary 
power, without reference to the element of the first cost of grading for the two plans of operating 
the road ; also, that the difficulty and danger in descending high grades is more important in 
determining their inclination than the resistance in their ascent. 
In estimating the loss of economy of power in overcoming high gradients, the comparison 
should be made between the loads habitually drawn over the more favorable portions of the road, 
and the maximum load that can be drawn over the gradient in question. 
FORMULAE. 
To obtain the maximum load due any engine of given weight, upon a given grade, and to 
obtain the maximum grade up which an engine of given weight can draw a given load: 
0.2A 
(1) X 
0.4242!/+ 8 
(2)/ 
0.2A—8;r 
0.4242 x 
(3) x 
0.143A 
0.4242/+ 8 
0.143A-8z 
(4)/ 
0.4242 x 
The engine and rail being in good order. 
The rail being in bad order, slippery, greasy, &c. 
In these formulae, A represents the adhesive weight of the engine ; that is to say, the portion of 
the weight of the engine actually supported by the drivers ; it is expressed in pounds. In 
engines with four drivers about 0.6 of the whole weight of the engine rests upon the drivers, 
sometimes as high as 0.67 ; 0.64 may be taken as the average. With six drivers the whole 
weight of the engine will rest upon them, and, consequently, be the value of “A.” 
