134 G. H. KNIBBS. 



CE = c, is ordinarily determined by measurement, 1 so that if the 

 'augmented centre height' DE be denoted by C, we shall have 



C = c + f^ (12) 



If the surface, f g Fig. 3, is level across, in which case all the sides 

 are planes, and the area of any triangle f Dg is rC 2 , the volume 

 of m longitudinally contiguous solids will be 



V^llr | C 2 + C,? l+ 2l = ct + 2 = C;C k+1 -|mi^ ] (13) 



{ k=i k=o r £ ) 



the negative term k«) 2 /4r, being the volume of the prism, whose 



constant section is ABD, beneath or above the road-bed. 



7. Solids of quadrilateral section with one warped surface. — If 

 in Fig. 3, the upper line fg instead of being horizontal, i.e., 

 parallel to AB, makes an angle therewith whose cotangent is s; 

 that is takes up some such position as FEG the slope of which is 

 s horizontal units to 1 perpendicular; then the projections of EG, 

 EF, on f g produced if necessary, are 



D=rC (— + 4)=rcJ^ (14) 



\s-r s + r' S* -r 2 



r and s being regarded as always positive : consequently the area 



of the triangle FDG is 



A = rC2 ^ < 15 >; 



that is to say s 2 /(s 2 - r 2 ) = q, is the factor, 2 which multiplied into 

 the area f Dg, gives the area FDG. Since s is necessarily greater 

 than r, this factor q is generally greater, and can never be less 

 than unity, that being the limit for s = oc . 



In a figure like DFEGD, Fig. 3, the area will always be \CD 

 whether FE and EG be in the same straight line or not; or the 



1 For example by * levelling ' the profile of the ' centre-line ' of the road, 

 and determining the levels for the formed road by a general consideration 

 of the best grade, having regard to all relevant circumstances. 



2 It is sometimes convenient to express this factor as a series 



e being the angle which the sloping line makes with the horizontal. 



