. Railroad construction, theory and practice; a text-book for the use of students in colleges and technical schools . be sufficiently accurateto serve as final. In any case the approximate result is firstcomputed and then the prismoidal correction is computedif necessary. The mathematical necessity for a correction maybe at once appreciated from the consideration that the volumeof a prismoid having dissimilar and unequal ends is NOT equalto the length times the average of the end areas but is usuallysomewhat less. In an extreme case the correction is one-thirdof the approximate volume, or one-
. Railroad construction, theory and practice; a text-book for the use of students in colleges and technical schools . be sufficiently accurateto serve as final. In any case the approximate result is firstcomputed and then the prismoidal correction is computedif necessary. The mathematical necessity for a correction maybe at once appreciated from the consideration that the volumeof a prismoid having dissimilar and unequal ends is NOT equalto the length times the average of the end areas but is usuallysomewhat less. In an extreme case the correction is one-thirdof the approximate volume, or one-half of the true volume. Theamount of the prismoidal correction for a triangular prism willbe first determined and from that the correction for any kind ofprism may be deduced. Let Fig. 54 represent a triangular prismoid. The two tri-angles forming the ends lie in parallel planes, but since the anglesof one triangle are not equal to the corresponding angles of the §110. EARTHWORK. 129 other triangle, at least two of the surfaces must be warped. Ifa section, parallel to the bases, is made at any point at a dis-. •——67 -* Fig. 54. tance x from one end, the area of the section will evidently be The volume of a section of infinitesimal length will be Axdx,and the total volume of the prismoid will be * < bihix+{lh-bi)hi^j^+bi{lh-hi)^ + (62-61) (A; -^4][ 1. — 2l &iW+[(62-6i)/ii+&i(/i2-W]|+(62-6i)(fe-w|} =^[ibihi+ibih2+ih2hi+ib2h2] = -]Ai+iAm+A,],D (52) ^ * Students unfamiliar with the Integral Calculus may take for granted the fundamental formulae that | dx=x, that | xdx = ^x\ and that | x^dx=^x^; also that in integrating between the limits of I and 0 (zero), the valueof the integral may be found by simply substituting I for x afterintegration. 130 RAILROAD CONSTRUCTION. §111. in which Ai, A2, and A^i are the areas respectively of the twobases and of the middle section. Note that A,» is not the meanof Ai and A2, although it does not necessarily differ very greatlyfrom i
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