218 COALFIELDS AND COLLIERIES OF AUSTRALIA. 



the force with which the waggon tends to move down hill i? 

 exactly held in equilibrium by the amount of friction. 

 The force with which a loaded waggon tends to move down th& 

 plane when the angle of inclination exceeds the angle of 

 friction is 



W sin a WC cos a 



and under the same conditions, the force with which the empty 

 waggon resists motion up the hill is 



W sin a + WC cos a. 



The smaller the difference between (W) and (w) the greater 

 the angle of slope required to make a self-acting plane. 



One must also consider the weight of the rope and its- 

 friction on the rollers of the incline, and the friction on the 

 periphery and axle of the drum round which the rope passes. 



The principal factors in determining the coefficient of 

 friction for wheeled carriages moving on rails are the ratio 

 of the diameter of the wheel to that of the axle, the quality 

 of the lubricant, and the smoothness of the contact surfaces. 

 Take the coefficient of friction for the rope on the rollers a& 

 being the same as that of the waggon, though it should really 

 be a little greater on account of the sag of the rope and the 

 roughness of its surface, the resistance offered by the rope 

 will be continually decreasing as the empty car ascends the 

 plane. The required angle of inclination will increase with 

 the length of the incline. 



So long as W sin a > (w + r) sin a + C (W + w + r) 

 cos a, the conditions permit a self-acting plane, but when 

 W sin a is equal to or smaller than the second member of this 

 formula, no motion can be produced by gravity alone. 



As the weights for steel ropes are nearly in proportion to 

 their respective safe working strengths, if the load is increased, 

 the weight of the rope in the same ratio must also be increased. 

 Therefore, the angle sought would be the same for any 

 number of waggons per trip as for one waggon. But if the 

 rope used for a one-waggon trip is stronger than necessary, so 

 that additional waggons can be put on without using a heavier 

 rope, then it may be possible to make the plane self-acting 

 by simply increasing the number of waggons in a set. 



As the resistance of the empty waggon and rope to the 

 motion up the plane is 



(w + r) sin a + C (w + r) cos a 



the strain executed by the loaded car to move down must be at 

 least equal to this ; hence the strain on the drum round which 

 the connecting rope passes must be at least 



2[(w + r) sin a + C (w + r) cos a]. 



