. Railroad construction : theory and practice : a textbook for the use of students in colleges and technical schools . m-parison, as well as in other similar comparisons elsewhere made,the volume of the grade prism invariably cancels out, and so forthe sake of simplicity it will be disregarded. This expressionfor volume may be transposed to IsVxix/ /\/i7^ \/^?%^\ 2 , Xl^Xr I 6L s^ ^^\~2s~^^—2s—) +^^J- 88 RAILROAD CONSTRUCTION. § 79. The true volume of the prismoid with sloping ends is (see § 77) The difference of the two volumes OS ^^WxLXr-\/xix/y. -2\/xi/Xrx/W-Xrx/) ^ (58) This shows that ^ e
. Railroad construction : theory and practice : a textbook for the use of students in colleges and technical schools . m-parison, as well as in other similar comparisons elsewhere made,the volume of the grade prism invariably cancels out, and so forthe sake of simplicity it will be disregarded. This expressionfor volume may be transposed to IsVxix/ /\/i7^ \/^?%^\ 2 , Xl^Xr I 6L s^ ^^\~2s~^^—2s—) +^^J- 88 RAILROAD CONSTRUCTION. § 79. The true volume of the prismoid with sloping ends is (see § 77) The difference of the two volumes OS ^^WxLXr-\/xix/y. -2\/xi/Xrx/W-Xrx/) ^ (58) This shows that ^ equivalent level sections do not ingeneral give the true volume, there being an exception whenxixr =^xi^x/. This condition is fulfilled when the slope isuniform, , when a = a, When this is nearly so the erroris evidently not large. On the other hand, if the slopes are in-clined in opposite directions the error may be very considerable,particularly if the angles of slope are also large. 79. Three-level sections. The next method of cross-section-ing in the order of complexity, and therefore in the order of. Fig. 51. accuracy, is of three-level sections. The area ofthe section is \{a-\-d){vcT-^wi)—^, which may be written § 79, EARTHWORK. 8& ^(a-{-d)w — ~, in which w=Wr + wi If the volume is computed by averaging end areas, it will equal ~[{a + d)w-ah + {^a-{-d)w---ahl . . (59) If we divide by 27 to reduce to cubic yards, we have, whenZ = 100, Vol (,,,,. )=ff(a + c^0^^-||a&4-|f(a + rf V-|f«?>For the next section Vol 0,.. ^) = ^(a + d)w-f^ah + ma + d)w-i^abFor a partial station length compute as usual and multiply result by —^ — The prismoidal correction may be obtained by applying Eq 46 to each side in turn For the leftside we have —[(« + ^0-(« + O](^/-i^/), which equals For the right side we have, similarly,~L(d-d)(Wr-Wr). The total correction therefore equals -l^{d-d)[(wi + wr)-iwi + wr)}= ~{d-d){w-w). Reduced to cubic yards, and with Z = 100,
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