Richmond Paper - Problems in Hauling and Hoisting

Of the following problems, some have been suggested by recent articles in technical journals, which have led me to believe that the mathematical discussion here submitted may be of use to mining engineers. Problem I. To Find the Grade or Angle of Inclination upon which the Power Required to Move a Loaded Car Down-hill Will Haul an Empty One Up-hill. Let a be the desired angle of inclination; C, the coefficient of' friction; W and w, the weight in pounds of the loaded and of the empty car, respectively.