262 ANNUAL OF SCIENTIFIC DISCOVERY. 



ON THE WAEPED SURFACES OF GROUND OCCURRING IN ROAD 

 EXCAVATIONS AND EMBANKMENTS. 



The following is an abstract of a paper read before the American Associa- 

 tion, at Montreal, by Professor TV. M. Gillespie, of Schenectady, N. Y. 



To determine the amount of earth necessary to be moved in making 

 the " cuts " and " fills " of roads, engineers take " cross-sections/' or " pro- 

 files," of the ground at right angles to the line of road, at convenient in- 

 tervals, and then calculate by various methods, usually near approxima- 

 tions, the volume included between each pair of these cross-sections. The 

 distances apart at which these cross-sections are taken, are determined by 

 the engineer according to the nature of the ground ; his aim being that 

 there shall not merely be no abrupt change of height between each pair 

 of these cross-sections, but that the surface from one to the other shall vary 

 uniformly ; gradually passing, for example, from a small to a great degree 

 of slope, or from a slope to the right into a slope to the left, without any 

 sudden variation at any one place. 



The surface fulfilling this condition, since it is everywhere straight in 

 some direction, is evidently a ruled surface ; and since the extreme profiles 

 are seldom parallel, it will be a warped or twisted surface. 



Our engineers have been accustomed to consider these surfaces as not 

 admitting of precise calculation, but only of a degree of approximation 

 varying with the nearness of the cross-sections. The object of this paper 

 is to examine the correctness of this assumption. It will therefore have 

 two parts : firstly, a discussion of the precise nature of the surface ; and 

 secondly, an investigation of a formula applying to it. 



I. What sort of a warped surface is the one in question ; that is, what is its 

 mode of generation ? 



To determine this, Ave must examine what the engineer means when he 

 says that the ground " varies uniformly " from the place at which he stands 

 to the place at which he decides it will be proper to take the next cross- 

 section ; whether he means that the ground is straight lengthwise, or straight 

 crosswise straight in the direction the road runs, or straight at right angles 

 to that direction. 



Probably few engineers ask themselves this question in so many words ; 

 but it would seem that the latter, or straightness crosswise, is the more likely 

 to be what is meant, for the reason that any deviation from straightness in 

 that direction, at right angles to the line along which we look, is much 

 more easily seen than in the other direction. We can therefore much more 

 readily determine whether the surface of the road is straight or curved from 

 side to side than from end to end. In geometrical language the former 

 surface (which we will call No. 1) is generated by a straight line resting 

 on the two straight lines which join the extremities of the two profiles, 

 and moving parallel to their planes, or perpendicular to the axis of the 

 road. 



This Surface is a " Hyperbolic Paraboloid." 



