Observations on Inclined Planes. 107 
~ 
Arr. XVII.—Observations on Inclined Planes; by J. Tuomson, 
Engineer, and late Professor of Mathematics in the University of 
_ Nashville, Tenn. Giowal Oe ea ye * 
oe 
TO THE EDITOR. gree S i 
Sir—tn the formation of rail roads, the proper location of inclined 
planes forms an important subject of inquiry. The following equa- 
tions have been made out in order to facilitate calculations in refer- 
ence to the ascent and descent of cars on inclined planes, where fric- 
tion forms a considerable part of the resistance to be overcome. 
These equations are given with the expectation that they may be 
found easy of application, in cases where general results, approxi 
mating to the truth, and not strict accuracy, are required. Indeed, 
_it is impossible to give equations that will meet all the causes of ir- 
regularity of motion on inclined planes, and be found accurate in 
practice. 
We will suppose in the first place, that the resistance of friction 
that retards the motion of a car, descending an inclined plane by its 
own gravity, is required. ‘ 
_ Let a=weight of the descending carriage. gencod 
_. g=accelerating force of gravity. 
g’=accelerating force in direction of the plane. 
n=length of the plane divided by the height. 
$=space passed over in time ¢. 
x=force of friction of the wheels, compared with g. 
_ __ f=friction expressed in pounds. 
~ When a body descends a plane without friction, we always have, by 
the laws of descending bodies, 7? =p but when g” is diminished by 
Qs . Qs —. -“£.. £ 
friction, ¢? = 2 whence x=g’ — 725 and since g/=>, a= — 
2s ; Oe 2 
Pro A; we have also the proportion g + @:.a: Piers and by 
2as 
substitution fH, 8 le B. 
The friction f in this equation includes both the friction of attrition 
and of rolling, and also the resistance occasioned by the inertia of 
the wheels in rolling. Mr. Wood, in his “Treatise on Rail roads,” 
makes allowance for this latter resistance, so as to obtain the resis- 
