109 



TffE Mathematics of Haul. 



H. O. Gab MAN. 



Haul or the average rlistance earth is moved when taken from exca- 

 vation and placed in embankment has been the source of much discussion 

 at different times for many years. For a review of the literature on the 

 subject, "Overhaul," the writer will call attention to the "Proceedings of 

 the American Eailwa.v Engineering and Maintenance of Way Associa- 

 tion," for 1906, vol. 7, pages 357 to 428. Among the contributors to the 

 subject will be found Italians, French and Germans, but it seemed to 

 excite more interest among our American engineers. 



The methematics of haul deals, of course, with the methods of com- 

 puting haul and overhaul, but it is the purix)se here to discuss more par- 

 ticularly the means for locating the center of mass. These centers of mass 

 may be located by any one of four methods, two algebraic and two graph- 

 ical. All four methods for locating the center of mass fail completely 

 for the volumes adjoining the grade point unless several extra interme- 

 diate sections are taken. 



In all tlie calculations a close rapid approximation was used at these 

 points 



"""^V 



When the ground was a plane surface, the v(jlume next the grade point 

 was assumed at least a wedge, and the center of gravity then taken i 

 the length of tlie wedge from its base. When the ground was a parabola 

 in longitudinal section, the center of gravity was taken § of the length 

 between section and grade point from the section. 



The four methods of computation were carried on under the condi- 

 tions of the three general types of profile, i. e. : 



