Ohservaiions on Inclined Planes. 107 



Art. XVII. — Observations on Inclined Planes ; by J. Thomson, 

 Engineer, and late Professor of Mathematics in the University of 

 Nashville, Tenn. 



TO THE EDITOR. 



Sir — In 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 refci- 

 ence to the ascent and descent of cars on'inclined planes, where Mic- 

 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. 



*Y/e 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. 

 g*= accelerating force of gravity. 

 ^'=accelerating force in direction of the plane. 

 w=length of the plane divided by the height. 

 s= space passed over in time t. 

 a;=force of friction of the wheels, compared with g. 

 /= friction expressed in pounds. 

 When a body descends a plane without friction, we always have, by 



2s 

 tlie laws of descending bodies, t^ =^, but when g' is diminished by 



t> 

 2s 2s S S 



friction, ^^= ,_^ , whence a; =^'-^, and since ^'=-, a:=~ - 



o 



2s . o,^ 



— A ; we have also the proportion g \ x'.'.a : ~=/j and by 



* o 



a 2as 

 substitution f'^n~^ ^' 



The friction / in this equation includes both the friction of attrition 

 and of rolling, and also the resistance occasioned by the inertia of 

 tlie wheels in rolling. Mr. Wood, in his " Treatise on Rail roads," 

 makes allowance for this latter resistance, so as to obtain the resis- 



