TF 222 731 UTATiCN RAIL ; OTHER EARTHWORK (AT I liliiil Illliii GIFT OF Charles 1,1. Titus UNIVERSITY FARM 1 i TABLES FOR THE COMPUTATION OF Railway and Other Earthwork. V COMPUTED BY C. L. CRANDALL, C. E., Associate Professor of Civil Engineering, Cornell University Member of American Society of Civil Engineers. SECOND EDITION. NEW YORK: JOHN WII,EY & SONS, 53 EAST TENTH STREET. 1893- UN-IV '* CALIFORNIA LIBRARY BRANCH OF THE COPYRIGHT, 1893, BY CHARTS L. CRANDAU,. ANDRUS & CHURCH, PRINTERS, ITHACA, N. Y. Preface to the First Edition. THE object of these tables is to present a convenient aid in the Com- putation of Railway Earthwork. The volume is first found by the approximate method of averaging end areas, to which can be added, if desired, a correction due to the strict prismoidal formula, or a correc- tion due to a modification of it for irregular ground. This separation reduces the labor, if both parts are computed at once, while it allows of computing only the approximate value at first, which will answer for all purposes except the final estimate, leaving the prismoidal cor- rection to be computed at leisure on those volumes only where it is large enough to be appreciable. The Table for "Volume by Averaging End Areas," is a table for triangular prisms, and it will give the approximate volume of triangu- lar prismoids in cubic yards for a length of one hundred feet, between the parallel triangular ends, by taking the sum of the two numbers found with the base and altitude of each of the two end triangles as arguments. The table is thus independent of side slope, or width of roadbed, and will apply to borrow pits, trenches, etc., by dividing up the cross sections into triangles and trapezoids. The method of dividing up the sections for three-level, five-level and irregular ground, while not essential, is believed to give the least labor in computing, either with or without tables. The method for irregular ground is due to J. W. Davis, and was pub- lished in his Formula for R. R. Earthwork, 1876. The Table for "Prismoidal Correction" is a table for triangular prismoids, and it will give the exact correction to be applied to the ap- proximate volume found above for all solids which can be divided into triangular prismoids, having plane parallel ends, and planes or hyper- bolic paraboloids for side faces. By using A. M. Wellington's method for irregular ground, that for the purposes of the prismoidal correction only, the ground may be called regular, the correction is easily and quickly applied, to three-level, five-level, and irregular ground. It may be noted that when either the bases or altitudes of the end trian- gles are equal, the correction disappears. While quantities are only given for full feet the corrections for tenths are easily taken out from the margin without multiplication and can be added mentally with but slight increase of labor. The Tables have been carefully recomputed in print, and it is confi- dently believed they are free from errors. PREFACE TO THE SECOND EDITION. The first edition was not electrotyped, and advantage has been taken of this to entirely recast the text, adding a proof of the prismoidal for- mula and a comparison with others in use, a formula for correction for curvature, and rules for cross sectioning. ITHACA, N. Y., August, 1893. ZZ 76, Contents. DERIVATION OF PRISMOIDAL FORMULA AND COMPARI- SON WITH OTHERS IN USE. SEC. PAGE. 1. Prisinoidal Formula i 2. Averaging End Areas 2 3. Middle Areas 3 4. Equivalent Mean Heights 3 5. Triangular Prismoid with Curved Center Line 4 VOLUME FOR RAILWAY EARTHWORK IN CUBIC YARDS BY AVERAGING END AREAS. 6. Formula for Three-level Ground 7 7. Five-level Ground 7 8. Irregular Ground 8 9. Tables 10 10. Example, Three level Ground n 11. Example, Five-level Ground n 12. Example, Irregular Ground 12 13. Borrow Pits 13 14. Sidehill Work 13 15. End Sections . . 14 16. Correction for Curvature 14 17. Example, Correction for Curvature 16 PRISMOIDAL CORRECTION IN CUBIC YARDS. 18. Three level Ground 17 19. Five-level Ground 17 20. Irregular Ground, Borrow Pits, etc. . 17 21. Table of Prisinoidal Corrections 18 22. Example, Three- level Ground 18 23. Example, Five-level Ground 19 24. Example, Irregular Ground 19 25. Example, Borrow Pits 20 COMPUTATION, END AREAS WITH PRISMOIDAL CORRECTION. 26. Three-level Ground 21 27. Irregular Ground 21 28. Rules for cross-sectioning 23 Table of Triangular Prismoids for Railway Earthwork, in cubic yards, by Averaging End Areas. Table of Prismoidal Corrections for Triangular Prismoids for Rail- way Earthwork. Derivation of Prismoidal Formula and Comparison with Others in Use. Section x x> First End' cU(d i. Prismoidal Formula. In computing railway earthwork, it is customary to measure sections perpendicular to the center line at sta- tions 100 feet apart, and at as many intermediate points as may be found necessary. To find the volume between two such parallel sections : Take at first the simple case shown in Fig. i, where the section is straight from the center to the slope stake on each side, and where straight lines joining corresponding points of the upper lines of the sections lie in the ground surface. The ground surface thus formed is a hyperbolic parabo- loid, a characteristic property of which is that all planes parallel to the end sections cut the surface in straight lines, whose increments for decrements) in length, starting from either end, are proportional to their distances from that end. With the notation of Fig. i, the area of the right-hand side of the first section (including the grade triangle under the roadbed), A = $(a + c)d. At the distance z, since all the linear dimensions of the section can be interpolated, *. ~1 *1 (a) Plan Section Second Fig .. Elementary volume of length dz = area into length Integrating between the limits z =o to z = /, Volume = C 1 A Z dz J o PRISMOIDAL AND OTHER FORMULAS. [| 2, -c) + (c'-c) (d'-#) Volume = 26 -L <*+0 / 5 , a+c d'l c'c d'l a+cdl^ d >\ 2l \ c + c ' d + d ' 2/ >p .2 6^2 6^2 2 34 2 3 Subtracting Vol. e from the true value, Vol. p , Vol. -Vol =-^di-^d>L + W a l_ 23 2 3*23 * See Roads and Railroads, New York, 1847. Bq-3] EQUIVALENT MEAN HEIGHTS. (2) where Cor. e is the correction to be applied to the volume found by aver- aging end areas in order to obtain the true volume. Usually in passing from lighter to heavier work c' > c and d r > d (or conversely from heavier to lighter), making the above correction usually negative; i. , e 1 ', the eccentricities of the sections, or the distances of their centers of grav- ity from the center of the roadbed ; A, AQ, A / , the areas of the trian- gular sections, with bases b, b, b f , and altitudes a, a, a' ; $ the cen- tral angle from B to any point F; I the length B Cof the solid along the center line of the roadbed, Rty f . At any point D, Elementary volume = A^ D E = A^ (R -\- e^) dq> / R A. (R 4- ed$ o Since all cross section dimensions can be interpolated, Substituting, Volume = -- + b (a'-a) + [ab =\l [ab 4- a(b' b} % + b(a'a) k + ( a +[ ab +a (b'-b) i + d(a'a) i + (a'-a)(b'-b] i ] ^^ = ^\ab + i ab' + i a'b + a'b' +(ab + \ab' + \ a'b + a'V} PRISMOIDAL AND OTHER FORMULAS. \_\ 5, m m - Volume (5) where A m is the area of the mid section, and , etc., adds a term of the form (u { f\\ If //"is omitted, the corresponding term (, r,)^ 1 dis- 2 2 appears and the other term becomes (c rtf-1 , as may be found by leaving off the triangle GH&\ if //"and D are omitted both terms disappear, and (0+^) J becomes (a+c) -i, as may be seen from the ex- pression for the right side. If the next cross section, TOO feet distant, is similar to the first, and the same letters with primes are used for corresponding parts, we shall have, Area second end, = (a + *,') L+K'-^OH- (*'-/ I / ) + etc. 222 22 The volume will be found by multiplying the average area by the length, and dividing by 27 to reduce to cubic yards. Volume in cubic yards = (area 4- area 7 ) 100 2 X 27 Substituting the value of each end area, Volume etc. +(a+c)d r (8) Or, the effect of each cross section on the volume reduces : For the irregular side ; to a constant times the distance to the slope stake times the sum formed by addiug the height next the slope stake to that of the grade triangle ; pins the constant times the distance out to each break, in turn, times the difference formed by subtracting the first height outside from the first height inside that break. For the regular side ; to the constant times the distance out to the slope stake times the sum formed by adding the center height to the height of the grade triangle- 10 VOL UME BY A VER AGING END AREAS. \_\ 9, From the total sum must be subtracted the constant times the base times the altitude of the grade triangle. For the next solid, Volume = (area'4- area") _!2?__ . 2 X 27' the terms comprising area' should therefore be added separately, for convenience in finding the second volume, etc. It may be noted that, if the expressions for the areas of the origi- nal triangles had not been reduced, the constant times the base times the altitude of each, would appear as its effect upon the volume. Hence the computer may divide the sections into any triangles he may wish ; the volume will result by multiplying the constant into the base times the altitude of each triangle, and adding the products. 9. Tables. The table gives the value of (a-\-c)dk& (or/rf|f), for a -f- c (height) and d (width) as arguments ; or it gives the effect which any triangle in the cross section has upon the volume, for a length be- tween stations of 100 feet, by using the altitude and base of the trian- gle as arguments. After looking ont the partial volume for each tri- angle for both ends and adding, if the length be not TOO, multiply the sum by T ^ of the length. To look out the partial volume for a triangle : Find the integral part of the height factor in the line of height headings at the top of the page ; then move down this column until opposite the integral part of the width factor in the width column and note the number ; then move to the left on this line (or the next below if the tenths of width be greater than 5) and take out the " Correction for tenths of height" ; them move to the bottom of the page and under the column first used (or the next to the right if the tenths of height be greater than 5) take out the " Correction for tenths of width." Take the sum of the three mentally. Look out the quantity for each of the other terms ; the algebraic sum will be the total volume. The terms coming from "area second end," should be kept separate as they will be used in finding the volume of the next solid. If one of the arguments cannot be found in the table divide it by two, look out the number and multiply by two ; if both cannot, divide each by two and multiply by four. In order to impress upon the mind the fact that the correction for tenths of width is placed under the height column, it should be remem- bered that the differences down the column are constant, so that cor- rections or proportional parts of these differences can be placed under it, but the widths are increased as we move down the column, or these corrections are for tenths of width. Similarly for the lines horizon- tally, or the corrections for tenths of height. After becoming familiar with the table, it will often save time to call height "width," and width "height," and to take the correction for tenths from the side or bottom of the page as most convenient, pro- vided always that it is opposite (or under) the heading number to which it belongs. Eq. 8] EXAMPLE, FIVE-LEVEL GROUND. n 10. Example, Three-level Ground. Sta. Center. Left. Right. 20 . . 10.8 . 21 24.2, .... 32-6 19-6 ' 24.1 Hi! 32.6 23-7' ' ' 55-9 21+20 . . 46.7, . .,. 5Q.6 37-0 82.9 a =4.7, 6 = 14. f Substituting in formula (6), or following the rule $ 6, we shall have, in connection with \ 9 : Sta. 20, a + c = 4.7 + 10.8 = 15.5, w = 19.6 + 24. i = 43.7. From the Table of Triangular Prismoids, Under height 15, right of width 43, 597 Left of width 44, correction for 0.5, .......... 20 Under height 15, correction for 0.7, 10 627 Arguments, = 4.7 and b =- 14 (grade prism), 61 566 Sta. 21, a + ^'=4.7 + 24. 2 = 28.9, ^' = 23.7 + 55.9 = 79.6, . . . 2131 = 4.7, = 14, 6 1 2070 Total in yards, ... 2636 From above, for sta. 21, 2070 Sta. 21 + 20, a + '=4. 7 + 46. 7 = 51.4, ze/'=37-o + 82.9 =119.9, 5712 a = 4. 7, = 14 (grade prism), 61 5651 7721 Total in yards 7721 X .20= 1544 Grand total, 4180 ii. Example, Five-level Ground. Sta. Center. Left. Right. 46 . . 10.8, . . IL4 . . !_', . . . 9^ . . A_ 6 7 25.1 7 19.9 47 - - 6.4, . . M . . 1. . . At > .' K 7 17-8 7 14-5 * The numerator is the height above or below grade at the point and the denominator is its distance from the center. t The side slope s' is here assumed to be one and one-half horizontal to one vertical, so that 12 VOLUME BY A VE RAGING END AREAS. [g 12, Substituting in formula (7), using the Table for the multiplications, Sta. 46, arguments, 25.1 and 11.4, 265 19.9 " 9.9, 183 14.0 " 10.8, 140 588 Similarly for Sta. 47, 272 Total in yards, 860 12. Example, Irregular Ground. Sta. Center. Left. Right. 20 . . 10.8, . M, , . IL . . JH 19.6 18 24.1 21 24.2 IM . . ^ . 20 4 32.6 16 23.7' 42 55.9 21+20 . 46.7, . 4 iZ . . 2ao , . . 4*L . i _5 . . 50- 6 20 37.0 20 58 82.9 b = 14, 5 = 1.5 to i, a = 4.7. Substituting in formula (8), using the Table for the multiplications, Volume from sta. 20 to sta. 2 1 . Sta. 20, a-\-t= 4.74-11.2=15.9, o? r =24.i ; c r r =io.8 11.4= 0.6, - r =i8 ; #+=15.5, ^,=19.6; #=4.7, =14. Under height 15, right of width 24, .. . . 333 I/eft of width 24, cor. for 0.9, 20 Under height 16, cor. for o. i, i 354 Arguments, 0.6 and 18, 10 19.6 and 15.5, 281 14 and 4.7 (grade prism), .... 61 564 Sta. 21, Arguments, 20.4+4.7=25.1, and 55.9, .... 1299 24.2 32.6= 8.4, and 42, . . . . 327 18.6+4.7=23.3, and 23.7, .... 512 24.2 11.1=13.1, and 16, 194 ' 14 and 4.7 (grade prism), .... 61 1617 Total in yards, 2181 Volume from 21 to 21+20. From above for sta. 21, 1617 Sta. 21+20, . 50+4.7=54.7, and 82.9, 4200 48.1 50.6= 2.5, and 58, . . . . 134 46.7 50= 3.3, and 20, 62 40.7+4.7=45.4, and 37, 1556 46.7 20=26.7, and 20, 494 14 and 4,7 (grade prism), .... - 61 5993 7610 76lO X. 20= 1522 3703 Eq. 8] SIDE HILL WORK. 13 Any other method of dividing the sections into triangles can be used if preferred, although it will result in increased labor. 13. Borrow Pits. In borrow pits, ditches or other work, where the cross sections are made up of triangles or trapezoids, take the altitude and base of each triangle, or the sum of the parallel sides, and the per- pendicular distance between them for each trapezoid, as arguments, as with the two sections below. /^i"> - '" iX" ^^^~, M5/^ ""vi ;' 0-< a- a ! so I/in s "-'- Sta. I.50. Sta. i, 27 and 14 2, . 355 50 " 28.0, . 1296 50 " 26.4, 1223 20 " 12.6, . 233 - 3107 Similarly for sta. 1+50, 2672 5779 5779 X -5<=> = 2890 yards. 14. Sidehill Work. When a "grade point" occurs in a cross sec- tion, one side being in cut, the other in fill, to compute the cutting, put in a break at the grade point and call the cuts on the fill side zero, and the distance to the slope stake b -=- 2. Then proceed as for irregular sections. In computing fills the process would be reversed. EXAMPLE. Sta. Center. Left. Right. 46 . . -2.4, . . .. ^ - *4 4.3 12.3 16.6 A*J J^ T *7 _*2." Roadbed 18 feet in cut, 14 feet in fill, slopes \\ to i. Altitude of ade triangle in cut = 9 -r- 6, For cutting, the notes would be grade triangle in cut = 9 -v- ii 6, in fill = 7 -s-ll = 4.7. 46 . . . O.O, . 0.0 2.2 0.0 An I 7 4-3 12.3 3-8 o.o 9.0 0.0 ' 14.7' ' 2.6 9.0 Giving, Sta. 46, 0.0+6.0= 6, and 12.3, . . 69 0.02.2==: 2.2, and 4.3, . 9 0.0+6.0= 6, and 9, .... 50 Grade triangle, 6 and 18, 100 10 Sta. 47, similarly, . . 60 Total in cu. yards, ... 7 VOLUME BY AVERAGING END AREAS. 16 Or it may be more convenient to leave out the grade prism, thus : Sta. 46, height=2.2, width= 9 4.3= 4-7, giving -.59 " 47, " =i-7, " =14-7+2.6=17.3, " . . 28 " =3-8, " 9.0. " . . 32 Total in yards, . For filling, the notes would be Sta. Center. Left. o.o o.o 4-3 7-o' ' 69 Right. 46 . . 2.4, 47 - Giving, Sta. 46, o.o o.o 7.0' 2.6 0.0+4.7=4.7, and 7, . . 31 2.40.0=2.4, " 4.3, . . 9 2.4+4.7=7.1, " 16.6, . . 109 Grade triangle, 4.7 (< 14, . . 61 Sta. 47, similarly, -6.4 16.6 - 2-9 11.4 88 n Total fill in cu. yards, ... 99 15. End Sections. A cross section should be taken wherever either edge of the roadbed in cutting reaches grade. This will require two sections in passing from a cut to a fill, or vice versa, while a third is sometimes added where the center line reaches grade. The solid, in cut, between the two sections, is a pyramid with volume equal to area of base into one- third the altitude, or it may be taken under the rule, base into one-half the altitude, and afterwards corrected by the prismoidal formula. The solid in fill, between the two sections, can usually be taken as having the same altitude, the volume not be- ing required as accurately as for cut ; the exact values can be interpolated when necessary. 16. Correction for Curvature. For a solid corresponding to the triangular prismoid, as described in 5, we have from (5), 8. Eq. 13] CORRECTION FOR CUR VA TURE. 1 5 Correction = ^ \_(A -f 2 AJ e + (2 ^ m -f ^4/) */] ( 9 ) ''] (approx.) (10) The latter equation is the one in common use, giving, with less labor, sufficient accuracy for earthwork solids. Three-level Ground. To find e, the eccentricity of the center of gravity, we have, d, d T Fig- 9 v ^' For the right side, 2 Center of gravity from center line = -i 3 (Product), = (a + c) r 6 Similarly, for the left side, (Product)! = Hence f (product), + (product), (a-\-c) 6 total area ~ (a-f-^r) d r -\- d\ Or, ,-ftz& (ii) . 3 Correction = J\_A (d r d l ) -f ^/ 7 (rf/ rf/)] (12) 6 /c where ^4 and ^ x must include the grade triangle. Or, Correction y = -^LQ K ~ ^i) + Q f W ~ ^/)] ( 13) where Q and Q' are the tabular quantities or volumes corresponding to A and A' (A = f ^ Q) and Correction y is in cubic yards. Five-level and irregular ground can usually be taken as three-level, with the same distances to slope stakes, for the purposes of this cor- rection, unless the intermediate heights show a marked tendency to lie above, or below, the lines joining the side heights with the center, when the section can be plotted and an equalizing line drawn, or in extreme cases each triangular prismoid can be taken separately by (10). Sidehill Ground, neglecting all breaks between slope stakes. The center of gravity will be on the line joining the center of the base with the vertex, at one-third its length from the base. 16 VOLUME BY AVERAGING END AREAS. [ 17, The distance from the lower end of this line to the center, Fig. 10, 2 Distance from the upper end = d r . Hence = 119,9, 79-6 = + 40.3; 22.5 and 40.3 as arguments = 272 -\- 6 -f 2 = 280 280 X .20= 56 Total correction to the volume found by " End areas " = 204 23. Example, Five-level Ground. Sta. Center. Left. Right. 46 . 10.8, . IM . . "^ 9^9 . M 8 20.1 8 16.6 47 . 6.4, . M . 7^ . ^ . . 1 8 15.2' 8 13.0 fy= 16, side slope ratio 5= i to i. See Fig. 5 and (18). /, // = n.4 6.8 = + 4.6, 34 an d IO -4> 50 and 21.4, 50 and 24.6, 17 and 13.6. The corrections will all disappear except for the first and last pro- duct at each station. Arguments, 27 34 = 7, 10.4 14.2 = 3.8, corr. = + 8 " 20 17 = + 3, 13.6 12.6 = + 1.0, " = + i Correction in yards = 9 + .50 = 5. Computation End Areas with Prismoidal Correction. 26. Three-level Ground. St o .. Center. 7 I Left. 6-4 Right. 8.0 . 9 8 14.4 7-6 16.0 12.4 . 12.4 . ' 15-6 12.4 20.4 10.3 8 7 20.4 9-8 18.3 8.1 . . IO.2 ' 17.8 ' u. 6 19.6 16.1 9.8 ' 17.8 4 Roadbed 16 ft., side slopes i to I. COMPUTATION. Sta. Width. Height, orh.+a. Yds. by Knd areas w w'. ht' ht. Yards, pris'dl. o i 2 3 4 30-4 36.0 38.7 33-9 37-4 I5-I 17.8 20.4 16.7 18.2 426 594 M 73 I 525 631 782 1087 1018 918 -5.6 2.7 +4-8 3-5 +2.7 +2.6 3-7 + 1-5 5 2 ' 6 2 Approx. volume = 3805 Prismo'l cor. = 15 True volume = 3790 27. Irregular Ground. 5-2 7-4 3-4 V 13.2 8 11.4 IO . . 4.9 9-6 4.1 17.6 9 ' 3-6 ii. 6 7-2 5.0 8.4 5-2 8.9 10,14 1 * . 9.0 . 8 12 16.4 7 16.9 16 | 12 12 6.8 11.4 12.8 7-1 16 20.8 15.1 T1 8 6 7-7 7-4 5-2 M * 12 15-4 13.2 *The breaks at the left of n run out 10 ft. and 14 ft. respectively to the left of 10, etc. 22 END AREAS WITH PRISMDL. CORRECTION. 27, +47 - 2.o . 4 5-4 0.0 10 13-4 8.0 0.0 4-2 2.0 1 4.o 12.2 IO.O 8| +2O . 6.O . . . O.O 4.4 8.0 ' 13-6 Roadbed 16 ft, slopes i to I in cut. Roadbed 14 ft., slopes i^ to i in fill. Computation on the assumption that, for the purpose of moidal correction only, the ground is regular. the pris- Sta. Width. Height, or h.+a. Yds. by Bnd areas w w'. ht' ht. Yards, pris'dl. 9 13.2 148 182 11.4 15-4 163 8 3-4 25 10 17.6 12.9 211 1 1.6 12. 1 130 9 i-3 II 484 -4.6 1.9 + 2 ii 16.4 13.0 I 9 8 12. 1.2 13 8 +4 30 16.9 13.2 ON 207 7 +0.1 M I 537 4.1 +4-1 5 12 20.8 19.4 1 374 16.0 6.0 -89 15-1 14.8 206 676 2.6 2.2 + i 13 15.4 15-7 224 12. 1.2 13 13.2 16.6 203 693 +7.3 + 1.8 +4 +47 13.4 12.0 148 10. 3-4 32 8 IO.O _ 74 184 +7-2 6.6 7 Pris'l for 14 r 5-4 o 14 4.0 4.2 1 16 46 + 12.0 1.2 3 +20 O.O 0.0 oo 3 + 4.0 4.2 i Approx. volume = 2623 Prismo'l cor. 9 Corrected volume = 2614 16 9 Eq. 18] RULES FOR CROSS SECTIONING. Computation With the true prismoidal correction. Sta. Width. Height, or h.+a. Yds. by End areas w w'. ht' ht. Yards, pris'dl. 9 13.2 14.8 182 11.4 15.4 163 8 3-4 25 10 17.6 12.9 211 4.4 1.9 +2 Pris'l* Ji7.6 16.6 for ii 1 J ^" 1.9 1 10 3-7 n.6 I2.I I 3 0.2 3-3 9 1-3 II 484 1.0 2.1 +i ii 16.4 13.0 I 9 8 + 1.2 -3-6 i 12 1.2 13 +2.0 +0.7 8 +4.0 30 +2.0 +7-7 +4 16.9 13.2 2O7 5-3 +1.1 2 7 +0.1 I 537 +2.0 1.2 I 12 20.8 19.4 374 4-4 +6-4 8 16 6.0 oT~ 89 Pris'l* / J 6 1.4 4.0 O.2 for ii Ii6 -4.6 H I 8.0 8.6 +21 15.1 14.8 1 206 Pris'l* /i5-i 15.0 +1.8 +1.8 + 1 for ii 1 12 o-3 676 5-0 0.4 + 1 13 15-4 15.7 224 +5-4 3-7 6 12 1.2 13 +4-o +7-2 +9 13.2 16.6 203 693 +1-9 +1.8 +47 13-4 12.0 148 +2.0 3-7 i IO 3-4 32 + 2.0 4.6 i 8 IO.O 74 184 +5-2 6.6 5 14 12.2 8.0 90 + 1.2 4.0 i 4 42 16 +6.0 0.8 i 8 8.0 59 45 0.0 2.O +20 Pris'l* ( 8 1 8 8.0 0.0 +4.2 4.0 O.O +4.2 i for 14 I 8 8.0 3 0.0 0.0 Approx. volume + 2622 + 40 Prism'l. Cor. =+12 28 True volume = 2634 28. Rules for Cross Sectioning. In deriving the prismoidal for- mula, the ground surface of each prismoid was assumed to be a hyper- bolic paraboloid. Hence if the prismoidal formula, or its equivalent, is to be used : * These can usually be avoided in practice by placing the same num- ber of breaks in adjacent sections as already indicated in $ 20. 24 RULES FOR CROSS SECTIONING. [ 28, Place cross sections at the regular stations (usually 100 feet apart), treating the ground as three-level if the surface is very regular ; five- level if at all broken, as the increased labor in field and office is slight ; and irregular if more broken. The object being to obtain points enough in each cross section so that straight lines joining them will lie in the surface, or will equalize the surface, and to adopt the meth- od which will secure this at most of the sections without extra breaks, even if it gives more points than are necessary at some of the more regular stations. An occasional extra break can often best be treated by itself, as add- ing or subtracting a computed amount to or from the regular quantity which would be obtained without it. It should also be noted that the formula requires the same number of sides or points in adjacent cross sections (dividing the solid into tri- angular prismoids), requiring that when a ridge or hollow runs out its vanishing point in the more regular section be taken and the section be made up with the break for the solid on one side and without it for the solid on the other side of the section. The height of this vanish- ing point can be computed, but it is usually better to measure it in the field. Intermediate cross sections should be added whenever straight lines joining corresponding points of the regular sections will not lie in the surface or equalize the surface, and enough should be added to fulfill this condition. When the center line is curved, the lines joining cor- responding points of adjacent cross sections should be estimated at the same curvature in judging of their coincidence with the ground sur- face. A cross section should also be placed wherever a grade point occurs on either edge of the roadbed (usually limited to the edges in cutting) and one is often placed where a grade point occurs on the center line. In light work and where the transverse slope is small, a section is sometimes taken where the center line comes to grade and the other two sections omitted. See $ 15. The plus stations are preferably placed at 50 feet, at multiples of 10, or at whole feet (except grade points) from the last full station, for convenience of computing grades in the field and quantities in the office. Breaks on a cross section should be put at multiples of 10, or at whole feet, from the center, when possible without a sacrifice of ac- curacy. When the method by averaging end areas is used, without the pris- moidal correction, additional sections should be taken whenever the change in transverse slope or in area between adjacent sections is great. The reason for this can be seen by referring to (2) ; halving the distance between sections will halve both factors (c c f } and (w' w], or quarter the product, so that when the two half length solids are added together the correction or error will be reduced one- half. The error due to averaging end areas can thus be made as small as desirable, but the method if extended far would soon involve more extra labor than to apply the prismoidal correction. The same remarks will apply to the method by middle areas. TABLE OF Triangular Prismoids for Railway Earthwork IN CUBIC YARDS, BY AVERAGING END AREAS. (See Prismoidal Correction.) [Width 1-50: Ht. i-io.J VOL UME BY A VERA GING .7 2 3 4 5 6 7 8 9 WIDTH. 1 2 345 678 9 10 i i i i I i 2 3 4 5 6 6 7 8| 9 I i i i i i 2 2 2 4 6 7 9 ii 13 15 17 19 i i i 2 2 2 3 3 3 6 8 ii 14 17 19 22 25 28 i i 2 2 3 3 3 4 4 7 n 15 19 22 26 30 33 37 i 2 2 3 3 4 4 5 5 9 H 19 23 28 32 37 42 46 2 2 3 3 4 4 5 6 6 n 17 22 28 33 39 44 '50 56 2 3 3 4 5 5 6 7 6 13 19 26 32 39 45 52 58 65 2 3 4 4 5 6 7 8 7 15 22 30 37 44 52 59 67 74 2 3 3 4 5 6 7 8 9 8 17 25 33 42 50 58 67 75 83 2 3 4 5 6 6 7 8 IO 9 28 37 46 56 65 74 83 93 2 3 4 5 6 7 8 9 ii IO 20 31 41 5I 61 7i 8 1 9 2 IO2 2 3 4 6 7 8 9 IO 12 ii 22 33 44 56 67 78 89 IOO III 2 4 5 6 7 8 10 n 13 12 24 36 48 60 72 84 96 108 120 3 4 5 6 8 9 10 12 14 13 26 39 52 65 78 91 104 117 130 3 4 6 7 8 10 II 12 15 H 28 42 56 6 9 83 97 in 125 139 I 3 4 6 7 9 10 12 13 16 15 30 44 59 74 89 104 119 133 148 2 3 5 6 8 9 ii 13 17 16 31 47 63 79 94 no 126 142 157 2 3 5 7 8 10 12 13 15 18 17 33 50 67 83 IOO 117 133 150 I6 7 2 4 5 7 9 ii 12 H 16 19 18 35 53 70 88 1 06 123 141 158 I 7 6 2 4 6 7 9 n 13 15 17 20 19 37 56 74 93 in 130 148 167 2 4 6 8 10 12 M 16 18 21 Ig 39 58 78 97 117 136 156 175 194 2 4 6 8 10 12 14 16 18 22 20 61 81 102 122 '43 163 183 2O4 2 4 6 9 ii 13 15 17 19 23 21 43 64 85 1 06 128 149 170 192 213 2 4 7 9 n 13 16 18 20 24 22 44 67 89 III 133 156 178 200 222 2 5 7 9 12 16 19 21 25 2 3 46 69 93 116 139 162 185 208 231 2 5 7 10 12 I 4 17 !9 22 26 24 48 72 96 120 144 169 193 217 241 3 5 8 10 12 15 18 20 2 3 27 2 5 50 75 IOO 125 150 175 200 225 250 3 5 8 10 13 16 18 21 2 3 28 26 52 78 104 130 156 181 207 233 259 3 5 8 II 13 16 '9 21 24 29 27 54 81 107 134 161 1 88 215 242 269 3 6 8 II 17 '9 22 2 5 30 28 56 83 in 167 194 222 250 2 7 8 3 6 9 11 14 17 20 23 26 31 2 9 57 86 115 144 172 201 2 3 258 287 3 6 9 12 15 18 21 24 27 32 30 59 89 119 I 4 8 178 207 237 267 296 3 6 9 12 15 18 21 24 28 33 3 1 61 92 122 153 183 214 244 275 306 3 6 9 13 16 19 22 25 28 34 63 94 126 157 189 22t 252 283 3J5 3 6 IO 13 16 19 23 26 29 35 32 65 97 130 162 194 227 259 292 324 3 7 IO 13 17 20 2 3 27 30 36 33 67 IOO 133 I6 7 200 233 267 300 333 3 7 IO 14 17 21 24 27 3 1 37 34 69 103 137 171 206 240 274 308 343 4 7 II 14 18 21 25 28 32 38 35 70 106 I 7 6 211 246 28l 317 352 4 7 II 14 18 22 2 5 29 33 39 36 72 1 08 144 181 2I 7 253 289 325 361 4 7 II 15 19 22 26 30 33 40 37 74 in I 4 8 185 222 259 2 9 6 333 370 4 8 II 15 !9 2 3 27 3 34 4i 38 76 114 152 190 228 266 304 342 380 4 8 12 16 19 2 3 27 3' 35 42 39 78 117 156 194 233 272 311 350 389 4 8 12 16 20 24 28 32 36 43 40 80 119 I 5 C 199 239 279 319 358 398 4 8 12 16 20 24 29 33 37 44 81 122 163 204 244 285 326 367 407 4 8 13 17 21 2 5 2 9 33 45 42 83 125 167 208 250 292 333 375 4'7 4 9 13 17 21 26 3 34 & 46 43 85 128 170 213 256 2 9 8 34i' 383 426 4 4 9 9 13 13 17 1 8 22 22 26 27 30 3 1 35 36 39 40 47 48 44 44 87 89 133 174 178 218 222 26l 267 305 3 11 348 392 356 400 435 444 5 9 14 18 23 27 3 2 36 49 45 91 136 181 227 272 318 363 4081 454 5 9 14 19 23 28 32 37 42 50 46 93 139 185 231 2 7 8 324 370 417 463 .7 .2 3 4 5 .6 7 .8 9 1 2 3 4 5 6 7 8 9 10 O.I i i i i| i O.2 i i i I i i 2 2 0-3 i i i i 2 2 2 3 3 Correction for tenths 0.4 i i i 2 2 3 3 3 4 0-5 i i 2 2 3 3 4 4 5 of width. 0.6 i i 2 2 3 3 4 4 5 6 0.7 i i 2 3 3 4 5 5 6 6 0.8 i i 2 3 4 4 5 6 7 7 0.9 i 2 3 3 4 5 6 7 8 8 (2) END AREAS. [Width 51-100: Ht. i-io.] ./ .2 3 -4 5 .6 7 .8 9 WIDTH. 1 2 3 4 5 6 7 8 9 10 5 5 5 5 9 10 IO IO 14 H 15 15 19 20 20 24 24 25 25 28 29 29 30 33 34 34 35 38 39 39 40 43 43 44 45 51 52 53 54 47 48 49 50 94 96 98 100 142 144 147 150 189 '93 196 200 236 241 245 250 283 289 294 300 331 337 344 350 378 385 393 400 425 433 442 450 472 481 491 500 5 10 *5 2O 25 3i 3 6 4i 46 55 51 102 153 204 255 .306 356 407 458 509 5 5 5 5 6 IO II II II II 16 16 16 16 17 21 21 21 22 22 26 26 27 27 28 3i 32 32 33 33 36 37 38 38 39 4i 42 43 44 44 47 48 48 49 50 56 57 58 59 60 52 53 54 55 56 104 106 107 109 III 156 158 161 164 167 207 211 215 2I 9 222 259 264 269 273 278 317 322 328 333 363 369 376 382 389 415 422 430 437 444 467 475 483 492 500 519 528 537 546 556 6 6 II II 17 17 23 23 28 2 9 34 34 40 40 45 46 52 61 62 56 57 113 115 169 172 226 230 282 287 339 344 395 402 452 459 508 565 574 6 6 6 12 12 12 18 18 18 23 24 24 29 30 30 35 36 36 4i 42 47 48 53 53 54 63 64 65 58 59 60 117 119 120 175 178 181 233 237 241 292 296 301 350 356 408 415 421 s- 467 474 481 525 533 542 ji t 583 593 602 6 6 6 6 12 12 13 13 18 19 *9 19 24 25 25 26 3 1 32 37 37 38 38 43 43 44 45 49 50 50 55 56 57 58 66 67 68 69 61 62 63 64 122 124 126 128 183 1 86 189 192 244 248 252 306 310 319 367 3/2 378 383 428 434 441 447 489 496 504 550 558 567 575 611 620 630 639 6 13 19 26 32 39 45 52 58 70 65 I 3 194 259 324 389 454 519 583 648 7 13 20 26 33 39 46 53 59 71 66 I 3 I 197 263 329 394 460 526 592 657 7 13 20 27 33 40 47 53 60 72 67 133 200 267 333 400 467 533 660 667 7 H 20 27 34 4i 47 54 61 73 68 135 203 270 338 406 473 608 676 7 7 14 14 21 21 27 28 34 35 4i 42 48 49 55 56 62 63 74 75 69 69 137 206 208 274 2 7 8 343 347 411 417 480 486 548 556 617 625 685 694 7 I 4 21 28 35 42 49 56 63 76 70 141 211 28l 352 422 493 563 633 704 7 14 21 2 9 36 43 50 57 64 77 71 143 214 285 356 428 499 570 642 713 7 H 22 29 36 43 51 58 65 78 72 144 217 289 433 ^ 506 578 650 722 7 15 22 2 9 37 44 51 59 66 79 73 146 2I 9 366 439 512 585 658 7 15 22 30 37 44 52 59 67 80 74 148 222 296 370 444 519 593 667 74i 8 8 15 15 23 23 30 30 11 45 46 53 53 60 61 68 68 81 82 75 76 150 152 225 228 300 304 375 380 450 456 525 600 607 31 750 759 8 15 23 3 1 38 46 54 61 69 83 77 154 231 307 384 461 538 615 692 769 8 16 23 3i 39 47 54 62 70 84 78 156 233 389 467 544 622 700 778 8 16 24 3i 39 47 55 63 7i 85 79 157 236 315 394 472 630 708 787 8 16 24 32 40 48 56 64 72 86 80 159 239 319 398 478 557 637 717 796 8 16 24 32 40 48 56 64 73 87 81 161 2 4 2 322 403 483 564 644 725 806 8 16 24 33 4i 49 57 65 73 88 81 163 244 326 407 489 570 652 733 815 8 16 25 33 4i 49 58 66 74 89 82 165 247 330 412 494 577 659 742 824 8 17 25 33 42 50 58 67 75 90 83 167 250 333 417 500 583 667 750 . 833 8 17 25 34 42 5i 59 67 76 91 84 169 253 337 421 506 590 674 758 843 9 17 26 34 43 5i 60 68 77 92 85 170 34i 426 596 681 767 852 9 17 26 34 43 52 60 69 78 93 86 172 258 344 43 1 517 603 689 775 861 9 17 26 35 44 52 61 70 78 94 87 174 261 348 435 522 609 696 783 870 9 18 26 35 44 53 62 70 79 95 88 176 264 352 44 528 616 704 792 880 9 18 27 36 44 53 62 7i 80 96 89 178 267 356 444 533 622 711 800 889 9 9 18 18 27 27 36 36 45 45* 54 54 8 72 73 81 82 97 98 90 9 1 180 181 26 9 272 359 363 449 454 539 544 629 635 719 726 808 817 898 907 9 18 28 37 46 55 64 73 83 99 9 2 183 275 367 458 550 642 733 825 917 9 19 28 37 46 56 65 74 83 100 93 185 2 7 8 370 463 556 648 741 833 926 .7 .2 3 4 5 .6 7 .8 9 1 2 3 4 5 6 7 8 9 10 O.I i i i i I O.2 i i i i i i 2 2 0-3 i i i i 2 2 2 3 3 Correction for tenths 0.4 i i i 2 2 3 3 3 4 0.5 i i 2 2 3 3 4 4 5 of width. 0.6 i i 2 2 3 3 4 4 5 6 0.7 i i 2 3 3 4 5 5 6 6 0.8 i i 2 3 4 4 5 6 7 7 0.9 i 2 3 3 4 5 6 7 8 8 (3) [Width 1-50: Ht. 21-30.] VOL UME BY A VER AGING .7 .2 -3 4 5 .6 7 .5 9 WIDTH. 21 22 23 24 25 26 27 28 29 30 i i i i I 19 20 21 22 23 24 25 26 27 28 i i i i i i 2 2 39 41 43 44 46 48 50 52 54 56 i i i i 2 2 2 3 3 58 61 64 67 69 72 75 78 81 83 i i i 2 2 3 3 3 4 78 81 85 89 93 96 100 104 107 in 1 i 2 2 3 3 4 4 5 97 102 106 in 116 120 125 130 134 139 I 2 2 3 3 4 4 5 6 117 122 128 133 139 144 150 156 161 167 I 2 3 3 4 5 5 6 7 136 143 149 156 162 I6 9 175 181 188 194 i 2 3 4 4 5 6 7 8 156 I6 3 170 178 185 !93 200 207 215 222 2 3 3 4 5 6 7 8 9 175 I8 3 192 200 208 217 225 233 242 250 2 3 4 5 6 6 7 8 IO 194 204 213 222 231 241 250 259 269 2 7 8 2 3 4 5 6 7 8 9 ii 214 224 234 244 255 265 275 285 295 306 2 3 4 6 7 8 9 10 12 233 244 256 26 7 278 289 300 3 J i 322 333 2 4 5 6 7 8 10 ii 13 253 265 277 289 301 3!3 325 337 349 36i 3 4 5 6 8 9 10 12 M 272 285 298 3" 324 337 350 363 376 389 3 4 6 7 8 IO ii 12 15 292 306 319 333 347 36i 375 389 403 417 I 3 4 6 7 9 IO 12 13 16 3n 326 34i 356 370 385 400 4i5 430 444 2 3 5 6 8 9 ii 13 '4 17 33i 346 362 378 394 409 425 441 456 472 2 3 5 7 8 10 12 13 15 18 350 367 383 400 417 433 450 467 483 500 2 4 5 7 9 n 12 14 16 19 369 387 405 422 440 457 475 493 5io 528 2 4 6 7 9 ii 13 15 17 20 389 407 426 444 463 481 500 519 537 556 2 4 6 8 IO 12 H 16 18 21 408 428 447 467 486 506 525 544 564 583 2 4 6 8 IO 12 M 16 18 22 428 448 469 489 509 530 550 570 59i 611 2 4 6 9 ii 13 15 *7 19 23 447 469 490 5ii 532 554 575 596 618 639 2 4 7 9 ii 13 16 18 20 24 467 489 5ii 533 556 578 600 622 644 667 2 5 7 9 12 H 16 *9 21 25 486 509 532 556 579 602 625 648 671 694 2 5 7 10 12 14 17 19 22 26 506 530 554 578 602 626 650 674 698 722 3 5 8 10 12 15 18 20 23 27 525 550 575 600 625 650 675 700 725 750 3 5 8 IO 13 16 18 21 23 28 544 570 596 622 648 674 700 726 752 778 3 5 8 II 13 16 19 21 24 29 564 591 618 644 671 698 725 752 779 806 3 6 8 II 14 17 19 22 25 30 583 611 639 667 694 722 750 778 806 833 3 6 9 II 14 17 20 23 26 31 603 631 660 689 718 746 775 804 832 86 1 3 6 9 12 15 18 21 24 27 32 622 652 681 711 741 770 800 830 859 889 3 6 9 12 15 18 21 24 28 33 642 672 703 733 764 794 825 856 886 917 3 6 9 13 16 r 9 22 25 28 34 661 693 724 756 787 819 850 88 1 9 T 3 944 3 6 10 13 16 19 23 26 2 9 35 681 7i3 745 778 810 843 875 907 940 972 3 7 10 13 17 20 23 27 30 36 700 733 767 800 833 867 900 933 967 IOOO 3 7 10 H 17 21 24 27 31 37 719 754 788 822 856 891 925 959 994 1028 4 7 II H 18 21 25 28 32 38 739 774 809 844 880 915 950 985 1020 1056 4 7 II 14 18 22 25 2 9 33 39 758 794 831 867 903 939 975 ion 1047 1083 4 7 II 15 19 22 26 30 33 40 778 815 852 889 926 963 IOOO 1037 1074 IIII 4 8 II 15 19 23 2 7 30 34 4i 797 835 873 911 949 987 1025 1063 IIOI 1139 4 8 12 16 J 9 23 27 31 35 42 817 856 894 933 972 ion 1050 1089 1128 1167 4 8 12 16 20 2 4 28 32 36 43 836 876 916 956 995 1035 1075 IH5 1155 1194 4 8 12 16 2O 24 29 33 37 44 856 896 937 978 1019 1059 IIOO 1141 1181 1222 4 8 13 17 21 25 29 33 38 45 875 917 958 IOOO 1042 1083 1125 1167 1208 1250 4 9 13 17 21 26 3 34 38 46 894 937 980 1022 1065 1107 1150 H93 1235 1278 4 9 13 17 22 26 3 35 39 47 914 957 1001 1044 1088 1131 1175 1219 1262 1306 4 9 13 18 22 27 3 1 36 40 48 933 978 IO22 1067 mi 1156 I2OO 1244 1289 1333 5 9 14 18 23 27 32 36 4i 49 953 998 1044 1089 H34 1 1 80 1225 1270 1316 I 3 6l 5 9 14 19 23 28 32 37 42 50 972 1019 1065 IIII H57 1204 1250 1296 1343 1389 .7 .^ 3 4 5 .6 7 .8 9 21 22 23 24 25 26 27 28 29 30 O.I 2 2 2 2 2 2 3 3 3 3 O.2 4 4 4 4 c 5 5 6 -3 6 6 6 7 7 7 8 8 8 8 Correction for tenths 0.4 8 8 9 9 9 10 10 10 ii ii 0.5 10 10 ii ii 12 12 12 13 13 U of width. 0.6 12 12 13 X 3 14 14 15 16 16 17 0.7 14 14 15 16 16 17 18 18 19 19 0.8 16 16 17 18 *9 19 20 21 21 22 0.9 18 18 19 20 21 22 23 23 24 25 (6) END AREAS. [Width 51-100: Ht. 21-30.] .1 2 3 ^ 5 6 7 .8 9 WIDTH. 21 22 23 24 25 26 27 28 29 30 5 9 14 19 24 28 33 3843 51 992 1039 1086 U33 1181 1228 1275 1322 1369 1417 5 10 H 1924293413943 52 IOII 1059 1107 1156 1204 1252 1300 1348 1396 1444 5 10 15 20252913439 44 53 1031 1080 1129 1178 1227 1276 1325 1374! 1423 1472 5 10 15 20 25 30 35 40 45 54 1050 I IOO 1150 1 200 1250 1300 1350 1400 1450 1500 5 10 15 2O 25 31 3b 4i 46 55 1069 1120 1171 1222 1273 1324 1375 1426 1477 1528 5 10 16 21 26 3 1 36 4i 47 56 1089 II4I 1193 1244 1296 1348 1400 H52 J 504 1556 5 II 16 21 26 32 37 42 48 57 1108 1161 1214 1267 1319 1372 1425 1478 i53i 1583 5 II 16 21 27 3 2 38 43 48 58 1128 1181 1235 1289 1343 1396 1450 1504 1557 1611 5 II 16 22 27 33 38 44 49 59 1147 1202 1256 I3II 1366 1420 M75 1530 1584 1639 6 II 17 22 28 33 39 44 50 60 1167 1222 1278 1333 1389 1444 1500 1556 1611 1667 6 II 17 23 28 34 40 45 51 61 1186 T 243 1299 1356 1412 1469 1525 1581 1638 1694 6 II 17 23 29 34 40 46 52 62 1206 1263 1320 1378 M35 1493 1550 1607 1665 1722 6 12 18 23 29 35 4i 47 53 63 1225 1283 1342 1400 1458 1517 1575 !633 1692 1750 6 2 18 24 3 36 4i 47 53 64 1244 1304 1363 1422 1481 1541 1600 1659 1719 1778 6. 2 18 24 3 36 42 48 54 65 1264 1324 1384 1444 1505 T 565 1625 1685 1745 1806 6 2 18 24 3' 37 43 49 55 66 1283 1344 1406 1467 1528 1589 1650 1711 1772 1833 6 2 19 25 3i 37 43 50 56 67 1303 1365 1427 1489 i55i 1613 J675 1737 1799 1861 6 ^ 19 25 3i 3 44 50 57 68 1322 1385 1448 I5U 1574 1637 1700 1763 1826 1889 6 3 19 26 32 38 45 5i 58 6g 1342 1406 1469 1533 1597 1661 1725 1789 1853 1917 6 3 T 9 26 32 39 45 52 58 70 1361 1426 1491 1556 1620 1685 1750 1815 1880 1944 7 3 20 26 33 39 46 53 59 7i 1381 1446 1512 1578 1644 1709 1775 1841 1906 1972 7 3 20 27 33 40 47 53 60 72 1400 1467 J533 1600 1667 1733 1800 1867 J 933 2000 7 M 20 27 34 4i 47 54 61 73 1419 1487 1555 1622 !6 9 1757 1825 1893 1960 2028 7 14 21 27 34 4i 48 55 62 74 M39 1507 1576 1644 1713 1781 1850 1919 1987 2056 7 M 21 28 35 42 49 56 63 75 H58 1528 1597 1667 1736 1806 1875 1944 2014 2083 7 14 21 28 35 42 49 56 63 76 1478 1548 1619 1689 1759 1830 1900 1970 2041 2III 7 M 21 29 36 43 5l57 64 77 H97 1569 1640 I7II 1782 1854 1925 1996 2068 2139 7 14 22 2 9 36 43 5158 65 78 1517 1589 1661 1733 1806 1878 1950 2022 2094 2167 7 15 22 2 9 37 44 5i 59 66 79 1536 I60 9 1682 1756 1829 1902 1975 2048 2121 2194 *j 15 22 30 37 44 52 59 67 80 1556 1630 1704 1778 1852 1926 2000 2074 2148 2222 8 15 23 30 37 45 53 60 68 81 1575 1650 1725 I800 1875 1950 2O25 2IOO 2175 2250 8 15 23 30 3 46 53 61 68 82 1594 1670 1746 1822 1898 1974 2050 2126 2202 2278 .8 15 23 31 38 46 54 61 69 83 1614 1691 1768 1844 1921 1998 2O75 2152 2229 2306 8 16 23 3 1 39 47 54 62 70 84 1633 I7II 1789 1867 1944 2022 2100 2I 7 8 2256 2333 8 16 24 3i 39 47 55 63 7i 85 1653 I73 1 1810 l88 9 1968 2046 2125 22O4 2282 2361 8 16 24 32 40 48 56 64 72 86 1672 1752 1831 igil 1991 2070 2150 2230 2309 2389 8 16 24 3 2 40 48 56 64 73 87 1692 1772 1853 1933 2OI/ 2094 2175 22 5 6 2336 2417 8 16 24 33 4i 49 57 65 73 88 1711 1793 1874 1956 2037 2119 2200 228l 2363 2444 8 16 25 33 4i 49 58 66 74 89 I73i 1813 1895 1978 2060 2143 2225 2307 2390 2472 8 17 2 5 33 42 50 58 67 75 go 1750 1833 1917 2000 2083 2167 2250 2333 2417 2500 8 17 2 5 34 42 5i 59 67 76 9i 1769 1854 1938 2022 2106 2I 9 I 2275 2359 2444 2528 9 17 26 34 43 516068 77 92 1789 1874 1959 2044 2130 2215 2300 2385 2470 2556 9 17 26 3443 5260 69 78 93 1808 1894 1981 2067 2153 2239 2325 2411 2497 2583 9 9 17 18 2635 26J35 44 44 52 53 61 62 70 70 78 79 94 95 1828 1847 1915 1935 2002 2023 2089 2III 2176 2199 2263 2287 2350 2375 2437 2463 2524 2551 26ll 2639 9 18 27 36 44 53 62 7i 80 96 1867 1956 2O4/ 2133 2222 2 3 II 2400 2489 2578 2667 9 18 27 36 45 54 63 72 Si 97 1886 1976 2066 2156 2245 2335 2425 2515 2605 269^ 9 18 27 3 6 45 54 64i73 82 98 1906 1996 2087 2178 2269 2359 2450 2541 2631 f r> 2722 9 9 18 19 28 28 37 37 46 46 55 56 64 65 73|83 7483 99 IOO 1925 1944 2017 2037 2IO8 2130 22OO 2222 2292 2315 2383 2407 2475 2500 2567 2593 2658 2685 2750 2778 ./ .2 .J -4 5 .6 7 .8 9 21 22 23 24 25 26 27 28 29 30 O.I 2 2 2 2 2 2 3 3 3 \ 0.2 4 A A 4 5 5 5 j 5 6 -3 6 6 6 7 7 7 8 8 Correction for tenths 0.4 8 8 9 9 9 10 10 1C ii ii o-5 10 10 ii ii 12 12 12 13 *3 ii of width. 0.6 12 12 J 3 i' 14 14 15 16 16 17 0.7 M 14 15 it 16 17 IS 18 19 19 0.8 16 16 17 18 19 *9 2C 21 21 22 0.9 iS 18 *9 20 21 22 23 23 24 25 (7) [Width 1-50 : Ht. 31-40.] VOL UME BY A VERA GING ./ .2 3 4 5 .6 7 .8 9 WIDTH. 31 32 33 34 35 36 37 38 39 40 I i i i I 29 30 31 (\i 31 f\f 3 2 33 A7 35 36 37 i i i 2 2 2 2 3 2 3 57 86 59 89 OI 9 2 b 3 94 65 97 07 IOO 60 69 103 70 1 06 72 1 08 74 in i i 2 2 3 3 3 4 H5 119 122 126 130 133 137 141 144 148 i 2 2 3 3 4 4 5 144 148 J 53 157 162 167 171 176 181 185 I 2 2 3 3 4 4 5 6 172 178 183 189 194 2OO 206 211 217 222 I 2 3 3 4 5 5 6 7 201 207 214 220 227 233 240 246 253 259 I 2 3 4 4 5 6 7 8 230 237 244 252 259 267 274 28l 289 2 9 6 I 2 3 3 4 5 6 7 8 9 258 367 275 283 292 300 308 317 325 333 I 2 3 4 5 6 6 7 8 10 287 296 306 315 324 333 343 352 36i 370 I 2 3 4 5 6 7 8 9 ii 316 326 336 346 356 367 377 387 397 407 I 2 3 4 6 7 8 9 10 12 344 356 367 378 389 400 411 422 433; 444 I 2 4 5 6 7 8 10 ii 13 373 385 397 409 421 433 445 457 469 481 I 3 4 5 6 8 9 10 12 14 402 415 428 441 454 467 480 493 5o6 519 I 3 4 6 7 8 10 1 1 12 15 43i 444 458 472 486 500 5i4 528 542 556 I 3 4 6 7 9 10 12 13 16 459 474 489 504 5i9 533 548 563 578 593 2 3 5 6 8 9 ii !3 14 17 488 504 519 535 55i 567 582 598 614 630 2 3 5 7 8 10 12 13 15 18 5T7 533 550 567 583 600 617 633 650 667 2 4 5 7 9 i i 12 14 16 19 545 563 58i 598 616 633 651 669 686 704 2 4 6 7 9 ii U '5 '7 20 574 593 611 630 648 667 685 704 722 741 2 4 6 8 10 12 14 16 18 21 603 622 642 66 1 68 1 700 719 739 758 778 2 4 6 8 10 12 '4 16 1 8 22 631 652 672 693 7i3 733 754 774 794 815 2 4 6 9 ri T 3 '5 17 19 23 660 681 703 724 745 767 788 809 831 852 2 4 7 9 1 1 !3 16 18 20 24 689 711 733 756 778 800 822 844 867 889 2 5 7 9 12 14 16 19 21 25 718 741 764 787 810 833 856 880 903 926 2 5 7 10 12 H 17 J 9 22 26 746 770 794 819 843 867 891 9i5 939 963 3 5 8 10 12 15 1 8 20 23 27 775 800 825 850 875 900 925 950 975 IOOO 3 5 8 10 13 16 18 21 23 28 804 830 856 881 907 933 959 985 ion 1037 3 5 8 ii 13 16 19 21 24 29 832 859 886 9*3 940 967 994 IO2O 1047 1074 3 6 8 ii H 17 19 22 25 30 861 889 917 944 972 IOOO 1028 1056 1083 IIII 3 6 9 ii M 17 20 23 26 31 890 919 947 976 1005 1033 1062 1091 1119 1148 3 6 9 12 15 18 21 24 27 32 919 948 978 1007 1037 1067 1096 1126 1156 1185 3 6 9 12 15 18 21 24 28 33 947 978 1008 1039 1069 IIOO 1131 1161 1192 1222 3 6 9 13 16 19 22 25 28 34 976 1007 1039 1070 1 102 1133 1165 1196 1228 1259 3 6 10 ^3 16 *9 23 26 29 35 1005 1037 1069 1 102 H34 1167 1199 1231. 1264 1296 3 7 10 13 17 20 2 3 27 30 36 1033 1067 IIOO H33 Il67 I2OO 1233 1267 1300 1333 3 7 10 H 17 21 2 4 27 31 37 1062 1096 1131 Il65 1199 1233 1268 1302 1336 1370 4 7 ii 14 18 21 25 28 32 38 1091 1126 1161 1196 1231 1267 1302 1337 1372 1407 4 7 ii 14 18 22 25 29 33 39 1119 1156 1192 1228 1264 1300 1336 1372 1408 1444 4 7 ii '5 r 9 22 26 3 33 40 1148 1185 1222 1259 1296 1333 1370 1407 1444 I48l 4 8 ii 15 J 9 23 27 30 34 4i 1177 1215 1253 1291 1329 1367 1405 H43 1481 1519 4 8 12 .6 19 2 3 27 3i 35 42 1206 1244 1283 I 3 22 1361 1400 1439 1478 1517 1556 4 8 12 16 20 24 28 32 36 43 1234 1274 I3H 1354 1394 1433 1473 1513 1553 1593 4 8 12 16 20 24 2 9 33 37 44 1263 1304 1344 1385 1426 1467 1507 1548 1589 1630 4 8 13 17 21 25 29 33 3 45 1292 1333 1375 1417 1458 1500 1542 1583 1625 1667 4 9 13 17 21 26 30 34 38 46 1320 1363 1406 1448 1491 1533 1576 1619 1661 1704 4 9 *3 17 22 26 3 35 39 47 1349 1393 M3 6 1480 1523 1567 1610 1654 1697 1741 4 9 13 18 22 27 3 1 36 40 48 1378 1422 "1467 IS" 1556 1600 1644 1689 1733 1778 5 9 M 18 2 3 27 32 36 4i 49 1406 1452 1497 1543 1588 1633 1679 1724; 1769! 1815 5 9 H J 9 23 28 32 37 42 50 1435 1481 1528 1574 l62O 1667 J713 1759 1806 1852 ./ .2 J 4 5 .6 7 >8 9 31 32 33 34 35 36 37 38 39 40 O.I 3 3 3 3 3 3 3 4 4 4 0.2 6 6 6 6 6 7 7 7 7 7 -3 9 9 9 9 10 IO IO ii ii ii Correction for tenths 0.4 ii 12 12 13 !3 J 3 14 14 14 15 0-5 14 15 15 16 16 17 17 18 18 J 9 of width. 0.6 17 18 18 19 19 20 21 21 22 22 0.7 20 21 21 22 23 23 24 25 25 26 0.8 23 24 24 25 26 27 27 28 2 9 30 0.9 26 27 28 28 29 30 31 32 33 33 (8) END AREAS. [Width 51-100: Ht. 31-40.] -/ .2 3 4 5 .6 -7 .8 -9 WIDTH. 31 32 33 34 35 36 37 38 39 40 9 *4 19 24 28 33 38 43 51 1464 i 5 n 1558 1606 1653 1700 1747 1794 1842 1880 10 10 It 19 20 24 2 5 29 29 34 34 39 39 43 44 52 53 H93 1521 1541 1570 1589 1619 1637 1669 1685 1718 1733 1767 1781 1816 1830 1865 1878 1926 10 15 20 2 5 3 35 40 45 54 1550 1600 1650 1700 1750 1800 1850 1900 IQSO 2OOO 1.0 15 2O 2 5 3i 36 4i 46 55 1579 1630 1681 1731 1782 1833 1884 1935 7O 1986 2037 10 II II II VO vo vo vo 21 21 21 22 26 26 27 27 32 32 33 36 I 42 43 44 47 48 48 49 56 57 58 59 1607 1636 1665 169^ l6 59 1689 1719 1748 1711 1742 1772 1803 1763 1794 1826 1857 1815 1847 1880 1912 1867 1900 ^933 1967 1919 1953 1987 2021 1970 2006 2041 2076 2022 2058 2094 2131 2074 2III 2148 2185 6 II 17 22 28 33 39 44 50 60 1722 1778 1833 1889 1944 2000 2056 2III 2167 2222 6 6 II II 17 17 2 3 2 3 28 2 9 34 34 40 40 45 46 52 61 62 ITSO 1807 1837 1864 189^ 1920 1952 1977 2009 2033 2067 2090 2124 2146 2181 2203 223C 2259 22Q6 6 2 18 2 3 2 9 35 4i 47 53 63 1808 1867 192' 1983 2042 2100 2158 2217 2275 mm-jM 6 6 ^ 0000 24 24 30 30 36 36 4i 42 47 48 53 54 64 65 1837 1866 1896 1926 1956 1986 2015 2046 2074 2106 2133 2167 2193 2227 2252 2287 / ^ 2 3 II 2347 2370 2407 6 2 18 2 4 3i 37 43 49 55 66 1894 1956 2017 2078 2139 2200 226l 2322 2383 2444 6 2 19 2 5 31 37 43 50 56 67 1923 1985 2047 2109 2171 2233 2295 2357 2419 2481 6 6 \ 19 19 25 26 32 38 38 44 45 50 1 68 69 1952 1981 2015 204/1 2078 2108 2141 2172 220^ 2236 2267 2 3 00 2330 2364 2393 2428 2456 2492 2519 2556 6 *3 19 26 32 39 45 52 58 70 2009 2074 2139 2204 2269 2333 2398 2463 2528 2593 7 13 20 26 33 39 46 53 59 71 2038 2104 2169 2235 2301 2367 2432 2498 2564 2630 7 13 20 27 33 40 47 53 60 72 2067 2133 2 2OO 2267 2333 2400 2467 2533 2600 2667 7 14 20 27 34 4i 47 54 61 73 2095 2163 22 3 I 2298 2366 2433 2501 2569 2636 2704 7 H 21 27 34 4i 48 55 62 74 2124 2193 2261 2330 2398 2467 2535 2604 2672 2741 7 14 21 28 35 42 49 56 63 75 2153 2222 2292 2361 2431 2500 2569 2639 2708 2778 7 14 21 28 35 42 49 56 63 76 2181 2252 2322 2393 2463 2533 2604 2674 2744 2815 7 14 21 29 36 43 50 57 6j 77 2210 228l 2353 2424 2495 2567 2638 2709 2781 2852 7 14 22 29 36 43 5i 58 65 78 2239 2311 2383 2456 2 5 2 2600 2672 2744 2817 2889 7 15 22 29 37 44 5i 59 66 79 2268 2341 2414 2487 2633 2706 2 7 80 2853 2926 7 15 22 30 37 44 52 59 67 80 22 9 6 2370 2444 2519 2593 2667 2741 2815 2889 2963 8 15 23 30 37 45 53 60 68 81 2325 2400 2475 2550 2625 2700 2775 2850 2925 3000 8 15 23 30 38 46 53 61 68 82 2354 2430 2506 2581 2657 2733 280 9 2885 2961 3037 8 15 23 31 38 46 54 61 69 83 2382 2459 2536 2613 2690 2767 2844 2920 2997 3074 8 16 23 31 39 47 54 62 70 84 2411 2489 2567 2644 2722 2800 2878 2956 333 3111 8 16 24 31 39 47 55 63 7i 85 2440 2519 2597 2676 2755 2833 2912 2991 3069 8 16 24 32 40 48 56 64 72 86 2469 2548 2628 2707 2787 2867 2946 3026 3106 3185 8 16 24 3 2 40 48 56 64 73 87 2497 2578 2658 2739 2819 2900 2981 3061 3142 3222 8 16 24 33 4i 49 57 65 73 88 2526 2607 2689 2770 2852 2933 3015 3096 3178 3 2 59 8 16 25 33 4i 49 58 66 74 89 2555 2637 2719 2802 2884 2967 3049 3131 3214 3296 8 17 25 33 42 50 58 67 75 90 2583 2667 2750 2833 2917 3000 3083 3167 3250 3333 8 17 25 34 42 5I 59 67 76 91 26l2 2696 2781 2865 2949 333 3118 3202 3286 3370 9 17 26 34 43 51 60 68 77 92 2641 2726 28ll 2896 2981 3067 3152 3237 3322 3407 9 17 26 34 43 52 60 69 78 93 2669 2756 2842 2928 3 OI 4 3100 3186 3272 3358 3444 9 17 26 35 44 52 61 70 78 94 2698 2785 2872 2959 3046 3133 3220 3307 3394 3481 9 18 26 35 44 53 62 70 79 95 2727 2815 2903 2991 3079 3167 3255 3343 3431 3519 9 18 27 36 44 53 62 71 80 96 2756 2844 2933 3022 3111 3200 3289 3378 3467 3556 9 18 27 36 45 54 63 72 81 97 2784 2874 2964 3054 3144 3233 3323 3413 3503 3593 9 18 27 36 45 54 64 73 82 98 2813 2904 2994 3085 3176 3267 3357 3448 3539 3630 9 18 28 37 46 55 64 73 83 99 2842 2933 3025 3117 3208 3300 3392 3483 3575 3667 9 r 9 28 37 46 56 65 74 83 100 2870 2963 3056 3241 3333 3426 3519 3611 3704 .7 2 3 4 5 6 7 8 9 31 32 33 34 35 36 37 38 39 40 O.I 3 3 3 3 3 3 3 4 4 4 0.2 6 6 6 6 6 7 7 7 7 7 0.3 9 9 9 9 10 10 10 ii ii ii Correction for tenths 0.4 ii 12 12 13 13 13 H 14 14 15 0-5 14 15 15 16 16 17 17 18 18 19 of width. 0.6 17 18 18 19 19 20 21 21 22 22 0.7 20 21 21 22 23 23 24 2 5 25 26 0.8 23 24 24 25 26 27 27 28 29 30 0.9 26 27 28 28 29 3 31 32 33 33 (9) [Width 1-50: Ht. 41-50.] VOL UME BY A VER AGING ./ .2 3 4 5 .6 7 .8 9 WIDTH 41 42 43 44 45 46 47 48 49 50 i i i I 3* 39 4c > 41 42 43 44 44 45 46 i i i i i 2 2 76 7 8c 81 83 8 5 87 89 9 1 93 i i 2 2 2 3 H4 117 "9 122 125) 128 131 133 136 139 i i 2 2 3 3 4 152 156 J 59 163 167 170 174 17* 181 185 i 2 3 3 4 5 I 9 C 194 195 204 208 213 218 222 227 231 2 2 3 4 4 6 22S 233 239 244 250 256 261 267 272 278 2 3 3 4 5 5 6 7 266 272 279 285 292 298 305 324 2 J L 4 5 6 7 8 304 326 333 341 348 356 363 370 2 3 J L 5 6 7 8 ~*9 342 350 358 367 375 383 392 4oc 408 417 2 3 i : 6 6 7 8 IO 380 389 398 407 417 426 435 444 454 463 2 3 t c 6 7 8 9 ii 418 428 438 448 458 469 479 489 499 509 2 3 i_ 6 7 8 9 10 12 456 467 478 489 500 5n 522 533 544 556 2 A c 6 7 8 10 ii 13 494 506 530 542 554 566 578 590 602 j 4 c *- 6 8 9 10 12 53i 544 557 570 583 596 609 622 635 648 3 4 6 7 8 IO n 12 15 569 583 597 611 625 639 653 667 681 694 I ^ 4 6 7 9 10 12 13 16 607 622 637 652 667 681 696 711 726 74i 2 J 5 6 8 9 ii 13 I/, 17 645 661 677 693 708 724 740 756 771 787 2 3 ^ 7 8 10 12 15 18 683 700 717 733 750 767 783 800 817 833 2 4 5 7 9 ii 12 H 16 19 721 739 756 774 792 809 827 844 862 880 2 4 6 7 9 ii 13 15 17 20 759 778 796 815 833 852 870 889 907 926 2 4 6 8 10 12 I 4 16 18 21 797 817 836 856 875 894 914 933 953 972 2 2 4 4 6 6 8 9 10 ii 12 '3 14 15 16 17 18 19 22 23 835 873 856 894 876 916 896 937 917 958 $ 957 1001 978 1022 998 1044 1019 1065 2 4 7 9 ii 13 16 1820 24 911 933 956 978 1000 IO22 1044 1067 1089 mi 2 5 7 9 12 H 16 19 21 25 949 972 995 1019 1042 1065 1088 IIII ii34 H57 2 5 7 10 12 14 17 19 22 26 987 IOII 1035 1059 1083 1107 1131 H56 1180 1204 3 5 8 IO 12 15 18 20 2 3 27 1025 1050 1 100 1125 H50 H75 1200 1225 1250 3 5 8 IO 13 16 18 21 23 28 1063 1089 1115 1141 1167 1193 1219 1244 1270 1296 3 5 8 II 13 16 19 21 24 29 IIOI 1128 "55 1181 1208 1235 1262 1289 1316 1343 3 6 8 II 14 T 7 19 22 25 30 "39 1167 1194 1222 1250 1278 1306 1333 1361 1389 3 6 9 II 14 17 20 23 26 31 1177 1206 1234 1263 1292 1320 1349 1378 1406 1435 3 6 9 12 15 18 21 24 27 32 1215 1244 1274 1304 1333 1363 1393 1422 1452 1481; 3 6 9 12 15 18 21 2 4 28 33 1253 1283 1344 1375 1406 1467 1497 1528 3 6 9 13 16 19 22 25 28 34 1291 1322 T 354 1385 1417 1448 1480 1543 1574 3 6 IO 13 16 r 9 23 26 29 35 1329 1361 1426 1458 1491 1523 1556 1588 1620 3 7 o 13 17 20 23 27 30 36 1367 1400 1433 1467 1500 1533 1567 1600 1633 1667 3 7 IO 14 17 21 24 27 31 37 1405 H39 1473 1507 1542 1576 1610 1644 1679 1713 4 7 II 14 18 21 25 28 32 38 H43 1478 1548 1583 1619 1654 1689 1724 1759 4 7 II 14 1 8 22 25 29 33 39 1481 1517 J553 1589 1625 166! 1697 1733 1769 1806 4 7 II 15 J 9 22 26 30 33 40 1519 1556 1593 1630 1667 1704 1741 1778 1815 1852 4 8 I 15 I9 23 27 3 34 4i 1556 I594 1632 1670 1708 1746 1784 1822 1860 1898 4 8 2 16 19 23 27 3 1 35 42 1594 1633 1672 I7II 1750 1789 1828 1867 1906 1944 4 8 2 16 2O 24 28 3 2 36 43 1632 1672 1712 1752 1792 1831 1871 igil 1951 1991 4 8 2 16 2O 24 29 33 37 44 1670 1711 1752 1833 1874 1915 1956 1996 2037 4 8 3 17 21 25 2 9 33 38 45 1708 1750 1792 1833 1875 1917 1958 2000 2042 2083 4 9 3 17 21 26 30 34 38 46 1746 1789 1831 1874 1917 1959 2002 2044 2087 2130 4 9 3 17 22 26 3 35 39 47 1784 1828 1871 1915 1958 2002 2045 2089 2132 2176 4 9 1318 22 27 3 1 36 40 48 1822 1867 1911 1956 2000 2044 2089 2133 2178 2222 5 9 14 18 2 3 27 32 36 4i 49 1860 1906 I 95 I 1996 2042 2087 2132 2178 2223 2269 5 9 14 19 23 28 32 37 42 50 1898 1944 1991 2037 2083 2130 2176 2222 2269 2315 .7 * 3 4 5 6 7 8\,9 41 42 43 44 45 46 47 48 49 50 O.I 4 4 4 4 4 4 4 4 5 5 O.2 8 8 8 8 8 9 9 9 9 9 0.3 ii 12 12 12 13 13 13 13 14 14 Correction for tenths 0.4 15 16 16 16 17 i7 17 18 18 19 0-5 *9 *9 20 20 21 21 22 22 23 23 of width. 0.6 23 23 24 24 25 26 26 27 27 28 0.7 27 27 28 29 29 30 30 31 32 32 0.8 30 3 1 S 2 33 33 34 35 36 36 37 - . - . . . 0.9 34 35 36 37 38 38 39 40 4i 42 (10) END AREAS. [Width 51-100: Ht. 41-50.] .1 .2 3 4 5 .6 7 .8 9 WIDTH. 41 42 43 44 45 46 47 48 49 50 5 9 14 !9 24 28 33 38 43 51 1936 1983 2031 2078 2125 2172 2219 2267 2314 2361 5 10 14 J 9 24 29 34 39 43 52 1974 2022 2070 2119 2167 2215 2263 2311 2359 2407 5 IO 15 20 25 29 34 39 44 53 2OI2 2061 2IIO 2159 2208 2257 2306 2356 2405 2454 5 10 15 2O 25 30 35 40 45 54 2050 2 IOO 2150 2200 2250 2300 2350 2400 2450 2500 5 10 15 20 25 3i 36 4i 46 55 2088 2I 39 2190 2241 2292 2343 2394 2444 2495 2546 5 IO 16 21 26 3i 36 4i 47 56 2126 2178 2230 228l 2333 2385 2437 2489 2541 2593 5 5 II II 16 16 21 21 26 27 32 32 37 38 42 43 48 48 57 58 2l64 2202 2217 2256 2269 2309 2322 2363 2375 2417 2428 2470 2481 2524 2533 2578 2586 2631 2639 2685 5 II 16 22 27 33 38 44 49 59 2240 2294 2349 2404 2458 2513 2568 2622 2677 2731 6 II 17 22 28 33 39 44 50 60 2278 2333 2389 2444 2500 2556 2611 2667 2722 / \j 2778 6 II 17 23 28 34 40 45 5i 61 2316 2372 2429 2485 2542 2598 2655 2711 2768 2824 6 II 17 23 29 34 40 46 52 62 2354 2411 2469 2526 2583 2641 2698 2756 2813 2870 6 12 1 8 23 29 35 4i 47 53 63 2392 2450 2508 2567 2625 2683 2742 2800 2858 2917 6 12 18 24 3 36 4i 47 53 64 2430 2489 2548 2607 2667 2726 2785 2844 2904 2963 6 12 18 24 30 36 42 48 54 65 2468 2528 2588 2648 2708 2769 2829 2889 2949 3009 6 12 18 24 3i 37 43 49 55 66 2506 2567 2628 2689 2750 2811 2872 2933 2994 3056 6 12 19 25 31 37 43 5o 56 67 2544 2606 2668 2730 2792 2854 2916 2978 3040 3102 6 13 J 9 25 3 1 38 44 50 57 68 2581 2644 2707 2770 2833 2896 2959 3022 3085 3148 6 13 T 9 26 32 3 45 5i 58 69 2619 2683 2747 28ll 2875 2939 3o3 3067 3*3! 3 J 94 6 13 19 26 3 2 39 45 52 58 70 2657 2722 2787 2852 2917 2981 3046 3111 3176 3241 7 !3 20 26 33 39 46 53 59 7i 2695 2761 2827 2893 2958 3024 3090 3156 3221 3287 7 J 3 20 27 33 40 47 53 60 72 2733 2800 2867 2933 3000 3067 3i33 3200 3267 3333 7 14 20 27 34 4r 47 54 61 73 2771 2839 2906 2974 3042 3109 3i77 3244 33 12 338o 7 14 21 27 34 4i 48 55 62 74 2809 2878 2946 3015 3083 3152 3220 3289 3357 3426 7 14 21 28 35 42 49 56 63 75 2847 2917 2 9 86 3056 3125 3i94 3264 3333 3403 3472 7 14 21 28 35 42 49 56 63 76 2885 2956 3026 3096 3167 3237 3307 3378 3448 3519 7 14 21 2 9 36 43 50 57 64 77 2923 2994 3066 3137 3208 3280 335i 3422 3494 3565 7 14 22 29 36 43 5i 58 65 78 2961 3033 3106 3178 3250 3322 3394 3467 3539 3611 7 15 22 2 9 37 44 5i 59 66 79 2999 3072 3145 3219 3292 3365 3438 35ii 3584 3657 7 15 22 30 37 44 52 59 67 80 3037 3 III 3185 3259 3333 3407 348i 3556 3630 3704 8 15 23 30 37 45 53 60 68 81 3075 3150 3225 3300 3375 3450 3525 3600 3675 3750 8 15 23 30 38 46 53 6r 68 82 3 J i3 3^9 3265 3341 3417 3493 3569 3644 3720 379 6 8 15 23 31 38 46 54 61 69 83 3 J 5i 3228 3305 3381 3458 3535 3612 3689 3766 3843 8 16 23 31 39 47 54 62 70 84 3189 3267 3344 3422 35oo 3578 3656 3733 3811 3889 8 16 24 31 39 47 55 63 7i 85 3227 336 3384 3463 3542 3620 3699 3778 3856 3935 8 16 24 32 40 48 56 64 72 86 3265 3344 3424 3504 3583 3663 3743 3822 3902 398i 8 16 24 32 40 48 56 64 73 87 3303 3383 3464 3544 3625 3706 3786 3867 3947 4028 8 16 24 33 4i 49 57 65 73 88 334i 3422 3504 3585 3667 3748 3830 39 11 3993 4074 8 16 25 33 4i 49 58 66 74 89 3379 346i 3544 3626 3708 379 1 3873 3956 4038 4120 8 17 25 33 42 50 58 67 75 90 3417 35oo 3583 3667 3750 3833 3917 4000 4083 4167 8 17 25 34 42 5i 59 67 76 91 3455 3539 3623 3707 3792 3876 3960 4044 4129 4213 9 17 26 34 43 5i 60 68 77 92 3493 3578 3663 3748 3833 3919 4004 4089 4174 4259 9 17 26 34 43 52 60 69 78 93 353 i 3617 3703 3789 3875 3961 4047 4133 4219 4306 9 17 26 35 44 52 61 70 78 94 3569 3656 3743 3830 3917 4004 4091 4178 4265 4352 9 18 26 35 44 53 62 70 79 95 3606 3694 3782 3870 3958 4046 4134 4222 43 10 4398 9 18 27 36 44 53 62 7i 80 96 3644 3733 3822 39" 4000 4089 4178 4267 4356 4444 9 18 27 36 45 54 63 72 81 97 3682 3772 3862 3952 4042 4I3 1 4221 43 11 4401 449 i 9 9 CO CO 27 28 36 37 45 46 54 55 64 64 73 73 82 83 98 99 3720 3758 3811 3850 3902 3942 3993 4033 4083 4125 4174 4217 4265 4308 4356 4400 4446 4492 4537 4583 9 r 9 28 37 46 56 65 74 83 IOO 3796 3889 398i 4074 4167 4259 4352 4444 4537 4630 .7 .2 -3 -4 -5 .6 7 .8 9 41 42 43 44 45 46 47 48 49 50 O.I 0.2 8 8 8 8 8 4 9 4 9 4 9 5 9 5 9 -3 ii 12 12 12 13 13 13 13 14 14 Correction for tenths 0.4 15 16 16 16 17 17 17 18 18 19 0.5 19 19 20 20 21 21 22 22 23 2 3 of width. 0.6 23 23 24 24 25 26 26 27 27 28 0.7 27 27 28 29 29 30 30 31 32 32 0.8 3 3 1 3 2 33 33 34 35 36 36 37 0.9 34 35 36 37 38 38 39 40 41 42 (II) TABLE OF Prismoidal Corrections for Triangular Prismoids for Railway Earthwork, IN CUBIC YARDS, To be applied to the approximate volume found by averaging end areas. [w x w 1^50 : h h' i-io.] PRISMOIDAL .1 2 j # 5 <5 7 8 9 w' w. 1 2 3 4 5 6 7 8 9 10, I i i i 2 2 2 2 3 3 i 2 i i 2 2 3 4 4 5 6 6 i i i i 3 i 2 3 4 5 6 6 7 8 9 i i i i 4 i 2 4 5 6 7 9 IO ii 12 I i i i i 5 2 3 5 6 8 9 ii 12 14 15 i i i i 2 6 2 4 6 7 9 ii 13 15 17 19 i i 2 2 2 7 2 4 6 9 ii 13 15 17 19 22 i i 2 2 2 8 2 5 7 10 12 15 17 2O 22 25 i i i 2 2 2 2 9 3 6 8 ii 14 17 19 22 25 28 I i 2 2 2 2 3 10 3 6 9 12 15 19 22 25 28 31 i 2 2 2 3 3 ii 3 7 10 14 17 20 24 27 31 34 i 2 2 3 3 3 12 4 7 ii 15 19 22 26 30 33 37 i 2 2 2 3 3 4 13 4 8 12 16 20 24 28 32 36 40 i 2 2 3 3 3 4 14 4 9 J 3 17 22 26 3 35 39 43 i 2 2 3 3 4- 4 15 5 9 14 19 23 28 32 37 42 46 i 2 2 3 3 4 4 16 5 10 15 20 25 30 35 40 44 49 I 2 2 3 3 4 4 5 17 5 10 16 21 26 31 37 42 47 52 I 2 2 3 3 4 4 5 18 6 ii 17 22 28 33 39 44 50 56 I 2 2 3 4 4 5 19 6 12 18 2 3 29 35 41 47 53 59 I 2 2 3 4 4 5 6 20 6 12 19 25 3 1 37 43 49 56 62 I 2 3 3 4 5 5 6 21 6 13 19 26 32 39 45 52 58 65 I 2 3 3 4 5 5 6 22 7 14 20 27 34 4i 48 54 61 68 I i 2 ^ 4 4 5 6 6 23 7 u 21 28 35 43 50 57 64 7i I i 2 3 4 4 5 6 7 24 15 22 30 37 44 52 59 67 74 I 2 2 3 4 5 5 6 7 25 8 15 23 31 39 46 54 62 69 77 I T 2 2 2 3 ^ 4 5 5" 6 6 6 7 26 27 8 8 16 24 32 40 48 CQ 56 ;8 64 6? 72 7c 80 83 I 2 3 2 4 c 6 7 8 W 28 9 ^7 26 35 43 J^ 52 D u 60 w 69 /. 78 86 I 2 3 A 4 6 7 8 29 9 18 27 36 45 54 63 72 81 90 I 2 4 5 6 6 7 8 30 9 19 28 37 46 56 6 5 74 83 93 I 2 4 5 6 7 8 9 3i 10 19 29 38 48 57 67 77 86 96 I 2 4 5 6 7 8 9 32 10 20 30 40 49 59 6 9 79 89 99 I 2 4 5 6 7 8 9 33 10 20 31 4i 5i 61 71 81 9 2 102 I 2 4 5 6 7 8 9 34 10 21 31 42 52 63 73 84 94 105 I 2 4 c 6 8 9 10 35 ii 22 32 43 54 65 76 86 97 108 I 2 4 6 7 8 9 10 36 ii 22 33 44 56 67 78 89 100 in I 2 6 7 8 9 10 37 ii 23 34 46 57 69 80 9i 103 114 I 2 3 6 7 8 9 ii 38 12 23 35 47 59 70 82 94 106 117 I 2 4 6 7 8 10 ii 39 12 1L 36 48 60 72 84 96 108 1 20 I 4 6 7 9 10 ii 40 12 25 37 49 62 74 86 99 in 123 I 4 6 8 9 10 ii 4i 13 25 38 5i 63 76 89 IOI 114 127 I 4 6 8 9 10 2 42 J 26 39 52 65 78 9 1 IO/ 117 130 I 4 7 8 9 ii 2 43 fl 27 40 53 66 80 93 106 119 133 I 4 7 8 10 n 2 44 L 27 41 54 68 81 95 109 122 136 I 4 6 7 8 10 ii ' 45 L 28 42 56 69 83 97 in !25 139 I 4 6 7 9 10 ii , 46 i 28 43 57 7J 85 99 ii/ 128 142 I 4 6 7 9 10 12 \ 47 I 29 44 58 73 87 102 116 131 H5 I 4 6 7 9 10 12 \ 48 : 3 44 59 74 89 IO/ 119 133 148 r- 6 8 9 n 12 Iz 49 ( 3 45 60 76 9i 1 06 121 136 151 2 c w 6 8 9 ii 12 \i 50 I 3i 46 62 77 93 108 123 139 154 .7 .^ J 4 5 .6 7 .8 9 1 2 3 4 5 6 7 8 9 10 O.I 0.2 i ] 0.3 i I i I Correction for tenths 0.4 i i I i I 0-5 i i I i 2 of width 7 width. 0.6 i i i I 2 2 0.7 i i 2 2 2 2 0.8 ] i 2 2 2 2 0.9 i i i 2 2 2 2 (14) CORRECTIONS. [ w /_ w j_ 50: h h x 11-20.] .t .2 3 -4 5 .6 7 .8 9 w' w. 11 12 13 14 15 16 17 18 19 20 I 3 4 4 4 5 5 5 6 6 6 i 2 7 7 8 9 9 10 10 ii 12 12 i i i 3 10 ii 12 13 14 15 16 18 19 i i i i 4 14 15 16 17 19 20 21 22 23 -7 2 5 i 1 i i i 5 17 19 20 22 23 25 26 28 29 31 i i i i i 2 2 2 2 6 7 20 24 22 26 24 28 26 30 28 32 30 35 31 37 33 39 35 37 43 i 2 2 2 8 27 30 32 35 37 40 42 44 47 i i 2 2 2 2 9 31 33 36 39 42 44 47 50 53 56 i 2 2 2 2 3 10 34 37 40 43 46 49 52 56 59 62 2 2 2 3 3 ii 37 41 44 48 5I 54 58 61 65 68 2 2 3 3 3 12 41 44 48 52 56 59 63 67 70 74 2 2 2 3 3 4 13 44 48 52 56 60 64 68 72 80 2 2 3 3 3 4 14 48 52 56 60 65 69 73 78 82 86 2 2 3 3 4 4 15 5 1 56 60 65 6 9 74 79 83 88 93 i 2 3 3 3 4 4 16 54 59 64 69 74 79 84 89 94 99 2 2 3 3 4 4 5 17 58 63 68 73 79 84 89 94 100 105 2 2 3 3 4 4 5 18 61 67 72 78 83 89 94 100 106 in 2 2 3 4 4 5 5 19 65 7o 76 82 88 94 100 106 in 117 2 2 3 4 4 5 6 20 68 74 80 86 93 99 105 in 117 123 2 3 3 4 5 5 6 21 71 78 84 9I 97 104 no 117 123 130 2 3 3 4 5 5 6 22 75 81 88 95 102 109 115 122 129 136 2 3 4 4 5 6 6 23 78 85 92 99 106 114 121 128 142 2 3 4 4 5 6 7 24 81 89 96 104 in 119 126 T 33 141 148 2 2 3 4 5 5 6 7 25 85 93 loo 108 116 123 J3 1 139 147 154 I 2 2 3 4 5 6 6 7 26 88 96 104 112 120 128 136 144 152 160 I 2 3 3 4 5 6 7 8 27 92 TOO 108 117 125 133 142 150 158 167 I 2 3 3 4 5 6 7 8 28 95 104 112 121 130 138 147 156 164 173 I 2 3 4 4 5 6 7 8 29 98 107 116 125 134 143 152 161 170 179 I 2 3 4 5 6 6 7 8 30 102 III 1 20 130 139 148 157 167 176 185 I 2 3 4 5 6 7 8 9 31 105 U5 124 134 144 153 163 172 182 191 I 2 3 4 5 6 7 8 9 32 I0 9 119 128 138 148 158 168 178 188 198 I 2 3 4 5 6 7 8 9 33 112 122 132 143 153 163 173 183 194 204 I 2 3 4 5 6 7 8 9 34 115 126 136 147 157 168 178 189 199 210 I 2 3 4 5 6 8 9 10 35 II 9 130 140 162 173 184 194 205 216 I 2 3 4 6 7 8 9 10 36 122 133 144 156 I6 7 178 189 200 211 222 I 2 3 5 6 7 8 9 IO 37 126 137 148 160 171 183 194 206 217 228 I 2 4 5 6 7 8 9 II 38 I2 9 141 152 164 176 188 199 211 223 235 I 2 4 5 6 7 8 10 II 39 I 3 2 144 156 169 181 193 205 217 229 241 I 2 4 5 6 7 9 10 II 40 136 148 160 173 185 198 2IO 222 235 247 I 3 4 5 6 8 9 10 II 41 139 152 165 177 190 202 215 228 24O 253 I 3 4 5 6 8 9 10 12 42 143 156 169 181 194 207 22O 233 246 259 I '3 4 5 7 8 9 ii 12 43 146 159 173 186 199 212 226 239 252 26 5 I 3 4 5 7 8 10 ii 12 44 149 163 177 190 204 217 231 244 258 272 I 3 4 6 7 8 10 ii 13 45 153 I6 7 181 194 208 222 2 3 6 250 264 2 7 8 I 3 4 6 7 9 10 ii 13 46 156 170 185 199 213 227 241 256 270 284 I 3 4 6 7 9 10 12 13 47 160 174 189 203 218 232 247 26l 2 7 6 290 I 3 4 6 7 9 10 12 13 48 163 I 7 8 193 207 222 237 252 267 28l 2 9 6 2 3 5 6 8 9 II 12 49 166 181 197 212 227 242 257 272 287 302 2 3 5 6 8 9 II 12 14 50 170 185 201 216 231 247 262 2 7 8 293 309 .7 .2 3 .* 5 .(5 7 .8 9 11 12 13 14 15 16 17 18 19 20 O.I i i i i 0.2 i i i i i i i i i i 0.3 i i i i i i 2 2 2 2 Correction for tenths 0.4 i i 2 2 2 2 2 2 2 2 o-5 2 2 2 2 2 2 3 3 3 3 of width ' width. 0.6 Qfj 2 2 2 3 3 3 3 3 4 4 i 0.8 3 3 3 3 4 4 4 4 5 5 0.9 3 3 4 4 4 4 5 5 5 6 (15) [ w / w 1-50: h h' 21-30.] PRISMOIDAL .1 .2 3 4 5 .6 7 .8 9 w 7 w. 21 22 23 24 25 26 27 28 29 30 I 6 7 7 7 8 8 8 9 9 9 i 2 13 14 14 15 15 16 17 17 18 19 i i i i 3 19 20 21 22 23 24 25 26 27 28 i i i i 4 26 27 28 30 3i 32 33 35 36 37 i i 1 i i 5 32 34 35 37 39 . 40 42 43 45 46 i i i i 2 6 39 4i 43 44 46 48 50 52 54 56 i i 2 2 2 7 45 48 50 52 54 56 58 60 63 65 i i 2 2 2 8 52 54 57 59 62 64 67 69 72 74 i 2 2 2 2 9 58 61 64 67 69 72 75 78 81 83 I 2 2 2 2 3 10 65 68 7i 74 77 80 83 86 90 93 i 2 2 2 3 3 ii 7i 75 78 81 85 88 92 95 98 102 i 2 2 3 3 3 12 78 81 85 89 93 96 IOO 104 107 III 2 2 2 3 3 4 13 84 88 92 96 IOO 104 108 112 116 120 2 2 3 3 3 4 14 9 1 95 99 104 1 08 112 117 121 125 130 2 2 3 3 4 4 15 97 102 106 in ii6 120 125 I 3 134 139 i 2 3 3 3 4 4 16 104 109 114 119 123 128 133 138 143 148 I 2 2 3 3 4 4 5 17 no "5 121 126 J 3! 136 142 H7 152 157 I 2 2 3 3 4 4 5 18 117 122 128 133 139 144 150 156 161 I6 7 I 2 2 3 4 4 5 5 19 123 129 135 141 M7 152 158 164 170 I 7 6 I 2 2 3 4 4 5 6 20 130 136 142 148 154 1 60 167 173 179 185 I 2 3 3 4 5 5 6 21 136 143 149 156 162 I6 9 175 181 1 88 194 I 2 3 3 4 5 5 6 22 H3 149 156 163 170 177 183 190 197 204 I 2 3 4 4 5 6 6 23 149 156 163 170 177 I8 5 192 199 206 213 I 2 3 4 4 5 6 7 24 156 I6 3 170 178 185 193 200 207 215 222 I 2 2 3 4 5 5 6 7 25 162 170 177 185 193 201 208 216 224 2 3 I I 2 2 3 4 5 6 6 7 26 169 177 185 193 20 1 209 217 225 233 241 I 2 3 3 4 5 6 7 8 27 175 183 192 200 208 217 225 233 242 250 I 2 3 3 4 5 6 7 8 28 181 190 I 99 207 216 225 233 242 251 259 I 2 3 4 4 5 6 7 8 29 188 I 9 7 206 215 224 233 242 251 260 269 I 2 3 4 5 6 6 7 8 30 194 204 2I 3 222 231 241 250 259 269 2 7 8 I 2 3 4 5 6 7 8 9 3i 20 1 2IO 220 230 239 249 258 268 277 287 I 2 3 4 5 6 7 8 9 32 207 217 227 237 247 257 26 7 277 286 2 9 6 I 2 3 4 5 6 7 8 9 33 214 224 234 244 255 265 275 285 295 306 I 2 3 4 5 6 7 8 9 34 220 231 241 252 262 273 283 294 34 315 1 2 3 4 5 6 8 9 10 35 227 238 248 259 270 28l 2 9 2 302 3i3 324 I 2 3 4 6 7 8 9 10 36 233 244 256 267 278 289 300 3H 322 333 I 2 3 5 6 7 8 9 10 37 240 251 263 274 285 297 308 320 33i 343 I 2 4 5 6 7 8 9 II 38 246 258 270 28l 293 305 317 328 340 352 I 2 4 5 6 7 8 10 II 39 253 265 277 28 9 301 313 325 337 349 36i I 2 4 5 6 7 9 10 II 40 259 272 284 296 309 321 333 346 358 370 I 3 4 5 6 8 9 10 II 4i 266 2 7 8 2 9 I 3^4 3i6 329 342 354 367 380 I 3 4 5 6 8 9 10 12 42 272 285 298 311 324 337 350 363 376 389 I 3 4 5 7 8 9 II 12 43 279 2 9 2 305 319 332 345 358 372 385 398 I 3 4 5 7 8 10 II 12 44 285 299 312 326 340 353 367 380 394 407 I 3 4 6 7 8 10 II *3 45 29? 306 319 333 347 36i 375 389 403 417 I 3 4 6 7 9 10 II 13 46 298 312 327 34i 355 369 383 398 412 426 I 3 4 6 7 9 10 12 13 47 305 319 334 348 363 377 392 406 421 435 I 3 4 6 7 9 10 12 13 48 3ii 326 34i 356 370 385 400 415 430 444 2 3 5 6 8 9 II 12 H 49 3i8 333 348 363 378 393 408 423 439 454 2 3 5 6 8 9 II 12 14 50 324 340 355 370 386 401 417 432 448 463 .7 .2 3 * 5 .6 7 .8 9 21 22 23 24 25 26 27 28 29 30 O.I i i i i i i i i i| i 0.2 i i i i 2 2 2 2 2 2 0-3 r 2 2 2 2 2 3 3 3 3 Correction for tenths 0.4 c\ c \ 3 3 3 3 3 3 3 4 i i of width / width. Ut o 0.6 i. 4 4 A 5 ^ i 5 5 6 0.7 ; 5 5 5 5 6 ( 6 6 6 0.8 c 5 6 6 e 6 7 7 7 7 0.9 6 6 6 7 7 7 g 8 8 i (16) CORRECTIONS. [ W /_ W 1-50: h h' 31-40.] .? 4 -5 .5 w' w 31 32 33 34 35 36 37 38 39 40 i 1C ) 10 I 1C i i i i i 12 2 H ) 2 2 2 2 2 2 2 2; 25 3 2C ! 3 3 3 3 3 3' 3 3 37 4 3* * 4 4 43 4 4 4 4 4 49 5 4* * 4 5 5' 5 5 5 5 6c 6 57 5 6 6; 6 6 6 7 7 7 7 67 6 7 7: 7 7 8 8 / * 86 8 11 I 8 s^ 8 8 9 94 9 99 9 86 8 9 9' 9 IOC 10 106 ii 10 9 9 10 ii 1 1/ ii J2C 12 ii 10 IOC ii HP ii 12 12 I2C I3 136 12 13 ii ii 12 12 13 I2( I4C 13 144 13 14 14 15 I* 14 13- 13 14 147 15 15 160 164 169 173 15 144 14 15 157 16 16 17 176 18 185 i 16 i 53 15* 163 i6 173 17 18 188 I93 198 2 ; t 17 163 1 6 173 IT? 184 18 19; 199 20 210 2 ^ i 18 172 178 183 189 194 200 206 21 217 222 2 ^ t 19 182 188 I 9 4 199 205 21 217 223 229 2 35 2 3 ' ' 5 1 20 J 9 198 204 2IC 211 22 22 235 241 247 ^J 2 ^ ^ ; 5 6 21 20 207 214 220 227 233 240 246 253 259 b/ i 2 '. i j 6 1 1 22 23 2IO 22O 217 227 22; 234 231 241 238 248 244 256 25 263 258 270 265 277 III * 2 ; ; ; 6 7 24 230 237 244 2 5 2 259 267 274 28l 289 296 1 I 2 c 6 i 2 5 2*Z< 247 ore 262 O*7 1 285 ^ 5/-|i 2 3 4 1 6 6 7 26 249 *T/ 257 265 273 28l 289 297 305 o*-* 1 313 321 5 I 2 i 3 ^ c 6 * 1 27 258 267 275 283 292 300 308 317 325 333 4-i I 2 J 3 ^ c 6 7 c 28 268 277 285 294 302 3 11 32O 328 337 346 01 I 2 ; 4 ^ c 6 * 8 29 277 286 295 304 313 322 331 340 358 5 I 2 4 6 6 f 8 30 287 2 9 6 306 315 3 2 4 333 343 352 361 370 ! I 2 4 6 7 8 9 3i 297 306 316 3 2 5 335 344 354 364 373 383 1 2 4 4 6 6 7 8 8 9 9 32 33 306 316 326 336 336 346 346 356 356 367 365 377 ill 385 397 395 407 I 2 4 6 7 8 9 34 325 336 346 357 367 378 388 399 409 420 2 4 , 6 8 9 35 335 346 356 367 378 389 400 410 421 432 y 2 2 * 6 8 9 36 344 356 367 378 389 400 411 422 433 444 o 2 5 6 8 9 o 37 354 365 377 388 400 411 4 2 3 434 445 457 / 4 5 6 8 9 i 38 3 6 4 375 387 399 410 422 434 446 457 469 2 4 , 5 6 8 i 39 373 385 397 409 421 433 445 457 469 481 2 4 , 5 6 9 i 40 383 395 407 420 432 444 457 469 481 494 3 4 . 5 6 8 9 o i 41 392 405 418 430 443 456 468 481 494 506 i 4 , > 6 8 9 o 2 42 402 4 I 5 428 441 454 467 480 493 506 519 3 4 . > 7 8 9 i 2 43 411 425 438 465 478 491 504 3 4 . 5 7 8 o i 2 44 421 435 448 462 475 489 502 530 543 3 4 < > 7 8 o i 3 45 43i 444 458 472 486 500 528 542 556 3 4 < > 7 9 i 3 46 440 454 469 483 497 5" 525 540 554 568 3 4 < > 7 9 2 3 47 450 464 479 493 508 522 537 566 580 3 4 < > 7 9 2 3 48 459 474 489 504 519 533 548 563 578 593 3 5 < i 8 9 I 2 4 49 469 484 499 5 r 4 529 544 560 575 590 605 5 3 5 * > 8 9 I 2 4 50 478 494 509 525 540 556 586 602 617 .2 3 -4 .j <5 7 ^ p 31 32 33 34 35 36 37 38 39 40 O.I 0.2 2 2 2 2 2 2 2 2 2 2 Correction for tenths o 3 3 3 3 3 3 3 3 c 4 c 4 r 4 K 0-5 5 5 5 5 5 6 6 6 6 6 of width'' width. 0.6 6 6 6 6 6 7 7 7 7 7 0.7 7 7 7 7 8 8 8 8 8 9 0.8 8 8 8 8 9 9 9 9 10 10 0.9 9 9 9 9 10 10 10 ii ii ii (17) [ w / _ w ^50 : h h' 41-50.] PRISMOIDAL CORRECTIONS. ./ .2 3 -4 5 .6 7 .8 9 w' w. 41 42 43 44 45 46 47 48 49 50 I 13 13 13 14 14 14 15 15 15 15 i 2 25 26 27 27 28 28 29 30 30 31 i i i i 3 38 39 40 41 42 43 44 44 45 46 i i i i 4 51 52 53 54 56 57 58 59 60 62 i i 1 i i 5 63 65 66 68 69 7i 73 74 76 77 i i i 2 6 76 78 80 81 83 85 87 89 9 1 93 i 2 2 2 7 89 9i 93 95 97 99 102 104 1 06 1 08 i 2 2 2 8 101 104 106 109 in 114 116 119 121 123 2 2 2 2 9 114 117 119 122 125 128 131 133 136 139 2 2 2 2 3 10 127 130 133 136 139 142 145 148 151 154 2 2 2 3 3 ii 139 H3 146 149 153 156 1 60 163 1 66 170 2 2 3 3 3 12 152 156 159 I6 3 167 170 174 178 181 185 2 2 2 3 3 4 13 165 169 173 177 181 185 189 193 197 20 1 2 2 3 3 3 4 14 177 181 186 190 194 199 203 207 212 216 2 2 3 3 4 4 15 190 194 199 2O4 208 213 218 222 227 231 i 2 3 3 3 4 4 16 202 207 212 217 222 227 232 237 242 247 2 2 3 3 4 4 5 17 2I 5 220 226 2 3 I 2 3 6 241 247 252 257 262 2 2 3 3 4 4 5 18 228 233 239 244 250 256 261 267 272 278 2 2 3 4 4 5 5 19 240 246 252 258 264 270 276 28l 287 293 2 2 3 4 4 5 6 20 253 259 265 272 2 7 8 284 290 2 9 6 302 309 2 3 3 4 5 5 6 21 266 272 279 285 2 9 2 298 305 311 318 324 2 3 3 4 5 5 6 22 2 7 8 285 292 299 306 312 319 326 333 340 2 3 4 4 5 6 6 23 291 298 35 312 319 327 334 341 348 355 I 2 3 4 4 5 6 7 24 304 311 3!9 326 333 34i 348 356 363 37o 2 2 3 4 5 5 6 7 25 316 324 332 340 347 355 363 370 378 386 2 2 3 4 5 6 6 7 26 329 337 345 353 36i 369 377 385 393 401 2 3 3 4 5 6 7 8 27 342 350 358 367 375 383 392 400 408 417 2 3 3 4 5 6 7 8 28 354 363 372 380 389 398 406 415 423 432 2 3 4 4 5 6 7 8 29 367 376 385 394 403 412 421 430 439 448 2 3 4 5 6 6 7 8 30 380 389 398 407 417 426 435 444 454 463 2 3 4 5 6 7 8 9 31 392 402 411 421 43i 440 450 459 469 478 2 3 4 5 6 7 8 9 32 405 415 425 435 444 454 464 474 484 494 2 3 4 5 6 7 8 9 33 418 428 438 448 458 469 479 489 499 509 2 3 4 5 6 7 8 9 34 430 441 45i 462 472 483 493 504 5H 525 2 3 4 5 6 8 9 10 35 443 454 465 475 486 497 508 519 529 540 2 3 4 6 7 8 9 10 36 456 467 478 489 500 5ii 522 533 544 556 2 3 5 6 7 8 9 10 37 468 480 491 502 5H 525 537 548 560 57i I 2 4 5 6 7 8 9 II 38 481 493 504 5i6 528 540 55i 563 575 586 I 2 4 5 6 7 8 10 II 39 494 506 5i8 530 542 554 566 578 590 602 I 2 4 5 6 7 9 10 II 40 506 5i9 53i 543 556 568 580 593 605 617 3 4 5 6 8 9 10 II 4i 519 531 544 557 569 582 595 607 620 633 3 4 5 6 8 9 10 12 42 531 544 557 570 583 596 609 622 635 648 3 4 5 7 8 9 II 12 43 544 557 57i 584 597 610 624 637 650 664 3 4 5 7 8 10 II 12 44 557 570 584 598 611 625 638 652 665 679 3 4 6 7 8 10 II 13 45 569 583 597 611 625 639 653 667 681 694 I 3 4 6 7 9 10 II T 3 46 582 596 610 625 639 653 667 68 1 696 710 I 3 4 6 7 9 10 12 13 47 595 609 624 638 653 667 682 696 711 725 I 3 4 6 9 10 12 13 48 607 622 637 652 667 681 696 711 726 741 2 3 5 6 8 9 II 12 14 49 620 635 650 665 68 1 696 711 726 74i 756 2 3 5 6 8 9 II 12 14 50 633 648 664 679 694 710 725 741 756 772 ./ .2 v? -4 5 .6 7 .8 -9 41 42 43 44 45 46 47 48 49 50 O.I i i i i i i i i 2 2 0.2 3 3 3 3 3 3 3 3 3 3 0.3 4 4 4 4 4 4 4 4 5 5 Correction for tenths 0.4 5 5 5 5 6 6 6 6 6 6 -5 6 6 7 7 7 7 7 7 8 8 of width' width. 0.6 8 8 8 8 8 9 9 9 9 9 0.7 9 9 9 10 10 10 10 10 ii ii 0.8 10 10 ii ii ii ii 12 12 12 12 0.9 ii 12 12 12 T 3 13 13 13 14 14 (18) THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW AN INITIAL FINE OF 25 CENTS WILL BE ASSESSED FOR FAILURE TO RETURN THIS BOOK ON THE DATE DUE. THE PENALTY WILL INCREASE TO SO CENTS ON THE FOURTH DAY AND TO $1.OO ON THE SEVENTH DAY OVERDUE. DEC 1 4 HI SEP 1 o ' LIBRARY, BRANCH OP THE COLLEGE OP AGRICULTURE 5m-9,'35(s) 22769 Crandall Tables fcr and other e Q 1 4 '41 SEP 1 Q '56 L-. the OT222 12. 1893 TFZZZ C7 1833 LIBRARY, BRANCH OF THE COLLEGE OF AGRICULTURE, DAVIS UNIVERSITY OF CALIFORNIA