NATURAL riiiLOsornY. 107 



by Mr. Warner. The communication presents a theoretical discussion, with 

 practical illustrations of the method known to engineers, as the method of 

 transverse r/round slopes, which treats the surface of the ground as a plane, 

 the position of which is given, in respect to the roadbed or formation level, 

 by the centre height at each end of the work, and the transverse slope of the 

 ground. The tables may also be used in calculating the solidity under a 

 warped surface. 



Mr. Warner's formula? admit of various transformations and applications. 

 We shall notice briefly some which pertain to those cases of the method of 

 transverse slopes, wherein the cross section of the work is a quadrilateral 

 figure, and the side slopes alike. To find the solidity, it is necessary to rind 

 an expression for the area of the cross section at any point, which is bounded 

 by the traces of the two side planes, and the traces of the roadbed and 

 ground surface. If the traces of the side planes be prolonged to meet, and 

 the point of their intersection assumed as the pole to which the polar equa- 

 tions of the roadbed and ground-plane traces are referred, then the area 

 sought may be derived from a known formula for the area included between 

 two radii vectores and two lines given by their polar equations. This being 

 done, the solidity between two given cross sections may be found. For this 

 solidity Mr. Warner gives the following formula: 



Let L denote the length of the work perpendicular to -the parallel end sec- 

 tions, B the width of base, a- and j the inclinations of the side and transverse 

 slopes respectively to the base or horizon. Let also S denote the sum of the 

 end heights measured from the intersection of the side slopes, and 1) the dif- 

 ference of these heights; then the solidity will be 



T/9 9 9 9 >> , D~ \ tan CT 



lL(SB 2 tan ff +B 2 tany+- I- - - - - 1 



' 'tan- a tairy 



If B and D be both = 0, the solid in question becomes a triangular prism 

 whose end height is 4 S, and whose solidity is 



L S 2 tan a- 

 J- -- ............ ... 2 



tanrff tmry 



If, iii formula 1, the square root of the quantity within the parenthesis be 



i 

 denoted by S, the solidity expressed by that formula may be put under the 



l o 



, L S- tanff 



form - r, --- 5 -- 



- 



tun" V tan- J 



1 



which, by its similarity with 2, shows that 2 S is the end height of a triangu- 

 lar prism, whose solidity is equal to that of the work. We may also consider 

 the solidity of the work as made up of the solidities of two or more prisms, 

 with similar bases, and of equal length. 



Hence it is evident that tables containing the solidities of such prisms may 

 be employed in the computation of earthwork. Of this description were 

 some of Mr. Warner's tables, adapted to several of the most usual side 

 slopes. They may also be used by finding the whole content included 

 between the side slopes and the surface, and deducting therefrom the redun- 

 dant prism lying above or below the roadbed. This process has been par- 

 tially developed by previous writers. 



Other tables exhibited by Mr. Warner, based on the same general forn:;ihu 

 and adapted to logarithmic computation, were directly applicable to any one 

 of thirteen different side slopes, combined with any one of foi-tv uliTc-rcut 



17* 



