APPLICATION AND DEVEOPMENT OF PRISMOIDAL FORMULA. 133 



and the other two surfaces will be planes. Then, I denoting, as 

 in § 4, the perpendicular distance between the parallel terminal 

 planes, the volume will be 



V=-h-l $c[2(a + b) + (a' + b')] + c'[2(a' + b') + (a + b)]\ (10) 



c and c being the rectangular distances between the parallel sides 

 of the trapezoids. If a and a become zero, we have a solid 

 whose end planes are triangles, with the bases b and b', parallel, 

 and altitudes c and c', the mantle of the solid consisting therefore 

 of one plane and two warped surfaces ; its volume will be 



V=- 1 \-l{c(2b + b') + c'(2b' + b)} (II). 1 



6. Solids of quadrilateral section with plane surfaces. — In earth- 

 work computations, the 

 data are to hand in a 

 form which requires 

 special consideration. In 

 a roadway it is usual to 

 make the bed, or formed 

 surface of the road, a 

 constant width, AB = w 

 say, and to keep the sides 

 AF, BG to a constant 

 slope • see Fig. 3. This 

 slope is defined by the 

 cotangent (r) of the angle 

 which the sides make with the horizontal bed; that is to say its 

 grade is expressed as r horizontal units to 1 perpendicular. The 

 side planes will consequently intersect one another at the distance 

 CT> = w/2r, beneath in the case of ' cuttings,' or above in that of 

 * embankments,' the centre of the road-bed AB; and the area of 



; centre height,' 



Fig. 3. 



the triangle ABD thus formed is w" 2, jir. The 



1 If the factor be made \ I, the expression will give the volume of a 

 solid whose parallel ends are rectangles with the sides b, c and b'c 1 ; as 

 was shown by Hutton in 1770. 



