. Graphical and mechanical computation . xes AX andBY and a curved axis CZ (Fig. 57). Draw any index line cutting theseixes in u, v, and w respectively. Draw wD parallel to AX, cuttinsr ABin D, and draw wE and vF parallel to AB. The triangles uEw and wFv Art. 58 STORM WATER RUN-OFF FORMULA 107 are similar, hence Eu : Fw = Ew: Fv = AD : DB. Therefore, if Au = x,Bv = y, AD = 21, Dw = z and AB = k, we have x — z : z — y = Zi : k — Zi Z\ or and if x = Wi/i(w), x + ;y = m2f2(v), or (fc — Zi) * + Ziy = kz,k k — Z\ y = — Zi k — Z\ 7U2 RZ k — Zi = Wi/4(w), this relation becomes /i(w) + j2{v) • /3(w) =
. Graphical and mechanical computation . xes AX andBY and a curved axis CZ (Fig. 57). Draw any index line cutting theseixes in u, v, and w respectively. Draw wD parallel to AX, cuttinsr ABin D, and draw wE and vF parallel to AB. The triangles uEw and wFv Art. 58 STORM WATER RUN-OFF FORMULA 107 are similar, hence Eu : Fw = Ew: Fv = AD : DB. Therefore, if Au = x,Bv = y, AD = 21, Dw = z and AB = k, we have x — z : z — y = Zi : k — Zi Z\ or and if x = Wi/i(w), x + ;y = m2f2(v), or (fc — Zi) * + Ziy = kz,k k — Z\ y = — Zi k — Z\ 7U2 RZ k — Zi = Wi/4(w), this relation becomes /i(w) + j2{v) • /3(w) = /4(w). Solving for Zi and z we get Zi = z = Wife • /s(w), W1/3O) + W2 mim2 _fi{w).Wi/3(w) + ra2 Hence to chart equation (X) pro-ceed as follows: Construct the scalesx = mlfi(u), y = m2f2(v) on two par-allel axes AX and BY extendingin the same direction. If AB — kinches, construct the points of thecurved scale CZ by assigning valuesto w, and laying off along AB, Zi = AD = allel to AX, z = Dw = f3(w), and par- 7WiW2. ~-^W .J*B Fig. 57-fi(w), and marking the point thus mj3(w) -f m2 found with the corresponding value of w. Then any index line will cutthe three scales in values of u, v, and w satisfying equation (X). To chart the equation fx{u) — f2(v) «/3(w) = fi(w), we construct thescales x = Wi/i(w) and y = — m2f2(v) in opposite directions. 58. Storm water run-off formula, q + Nq* = P. — This equationarises in the storm water run-off formula given by C. B. Buerger, in theTrans. Am. Soc. C. Vol. LXXVIII, p. 1139, where iVand P are quan-tities which depend upon the sewer run, the area, and the rainfall, and qis the run-off in cu. ft. per sec. per acre. If we write the equation P — Nq* = q, we have an equation of theform (X), with the scales m^k 4 raim2 x = m\P, y = — m2N, Z\ = ,1 z = miqz + w2 m\q^ -f- mt ~ Let P, N, and q vary from o to 10, and take Wi = m2 = 1 and k = 14. io8 NOMOGRAPHIC OR ALIGNMENT CHARTS Chap. V
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