and ca the length of the second chord preceding the station considered. When the tangent deflection ƒ is known, the chord deflection do=f(1+) Special Values of Chord and Tangent Deflection. For a chord of 100 ft. preceded by one of the same length the chord deflection for a 1° curve is 1.745; for a 2° curve, it is twice that amount, or 3.49; and so on. The tangent deflection, being half the chord deflection, will be .873 ft. for a 1° curve, 1.745 for a 2° curve, etc. The tangent deflection for a chord of any length equals the tangent deflection for a chord of 100 ft. multiplied by the square of the given chord expressed as the decimal part of a chord of 100 ft. Application of Chord and Tangent Deflection.-Let it be required to restore center stakes on the 4° curve, Fig. 3, at each full station. The points A and B determine the direction of the tangent, the point B being the P. C., which is at Sta. 8+25. For a 4° curve the regular chord deflection for 100 ft. is 4X 1.745 =6.98 ft., and the tangent deflection is 3.49 ft. The distance from P. C. to the next full station is 75 ft.; hence, the tangent deflection CF = .752X3.49=1.96 ft. The point F is found by first measuring 75 ft. from B, thus locating the point C in the line AB prolonged, then from C measuring CF=1.96 ft., at right angles to BC; the point F thus determined will be Sta. 9. Next, the chord BF is prolonged 100 ft. to D; BF is only 75 ft., DG is computed from the preceding formula; thus, do=3.49 (1+7)=6.11. This C distance is measured at right angles to BD; the point G thus determined will be Sta. 10. The point H, which is Sta. 11, and the P. T. of the curve, is determined in the same manner, except that, as the chords FG and GH are each 100 ft. long, the regular A B FIG. 3 G 10 P.T. H 11 chord deflection of 6.98 ft. is used for EH. A stake is driven at each station thus located. Although a chord deflection is not at right angles to the chord theoretically, yet the deflection is so small, as compared with the length of the chord, that for curves of ordinary degree it is usually measured at right angles. Middle Ordinate.-The middle ordinate of a chord is the ordinate to the curve at the middle point of the chord. The following formulas give the relation between the length of the chord c, the radius of the curve R, and the middle ordinate m. To Determine Degree of Curve From Middle Ordinate.-It is sometimes necessary to determine the radius or the degree of a curve in an existing track when no transit is available. By measuring the middle ordinate of any convenient chord, the degree of the curve can be calculated from the relative values of the ordinate and chord. As the track is likely not to be in perfect alinement, it is well to measure the middle ordinate of different chords in different parts of the curve; also, as the middle ordinate of a chord measured to the inner rail will somewhat exceed the middle ordinate of the same chord measured to the outer rail, the ordinate of each chord should be measured to both rails and the average of the two taken as the value of the ordinate. Having measured the middle ordinate of one or more chords, the degree of curve D. can be found by the formula The following rule is sometimes applied in determining the degree of curve: Rule.-Measure the middle ordinate to a chord of 67.71 ft.; express it in feet and decimals of a foot, and multiply by 10; the result will be the degree of the curve. Rules for Measuring the Radius of a Curve.-Stretch a string, say 20 ft. long, or longer if the curve is not a sharp one, across the curve corresponding to the line from A to C, in Fig. 4. Then measure from B the center of the line AC, and at right angles with it, to the rail at D. Multiply the distance A to AC B, or one-half the length of the string, in inches, by B itself; measure the distance D to B in inches, and multiFIG. 4 ply it by itself. Add these two products, and divide the sum by twice the distance from B to D, measured exactly in inches and fractional parts of inches. This will give the radius of the curve in inches. It may be more convenient to use a straightedge instead of a string. Care must be taken to have the ends of the string or straightedge touch the same part of the rail as is taken in measuring the distance from the center. If the string touches the bottom of the rail flange at each end, and the center measurement is made to the rail head, the result will not be correct. In practice, it will be found best to make trials on different parts of the curve, to allow for irregularities. ILLUSTRATION.-Let AC be a 20-ft. string; half the distance, or AB, is then 10 ft., or 120 in. Suppose BD is found on measurement to be 3 in. Then 120 X 120=14,400, and 3X3=9; 14,400+9=14,409, which, divided by 2X3=6, equals 2,401 in., or 200 ft. 1 in., which is the radius of the curve. The AB2+BD2 formula = R, applied to the example is 2 BD = Other Ordinates.-Any ordinate y to the curve at a distance a from the middle point of a chord may be determined by means of the formula: y= √R2-a2-R+m By using long chords, a curve may be laid out or obstacles passed by means of ordinates. Suppose that it is required to run out the curve AEH, Fig. 5, with several obstacles in the direct line of the curve, as shown, Sta. 3 being the P. C., and the regular stations on the curve being in the positions indicated by the numbers 4, 7, 8, etc. The positions of Stas. 5 and 6 are indicated by the letters C and D. The stations are to be located in their proper positions on the curve, between the obstructions, wherever it is possible to do so. In addition to this, it is customary to mark with a tack or otherwise the point where the line of the L curve intersects each obstruction. Beginning at the point of curve A, which is at Sta. 3, the curve can be run in as far as the first obstruction, which is the building P, setting the stakes on the curve at Stas. 4 and 5, and a tack in the side of the building P at the point where the line of curve intersects it, according to the deflection angle as determined by its distance from Sta. 5. It is not possible to proceed further in the regular manner, however, because Sta. 6 cannot be seen from the P. C. Therefore, it is necessary to locate Sta. 7 by deflection angle V'BE, from B or PC. Sta. 4, to determine the chord 4-7, which, in this case, is a long FIG. 5 H chord of three stations, and to calculate the ordinates D'D and C'C by substituting for a in the preceding formula the value of MC' = MD' = half a station or 50 ft. Fig. 5 shows also another method of passing a building S, namely, by running an equilateral triangle FLG. In this method, the instrument is set up at Sta. 8 and sighted back to the P. C. Then, the telescope is reversed and the deflection angle for Sta. 9 is turned off the same as if no obstruction existed. The telescope will then be sighted on the line FG, although the point G will not be visible. The angle GFL, equal to 60°, is then turned, and the point L is located so that FL=FG= 100 ft. The instrument is next moved to L, and the line LG is run, making 60° with FL. On this line the distance LG-100 ft. is measured, giving the point G, which is Sta. 9. The transit is then set up at this point and sighted to L, and an angle of 60° is turned off to the right, giving the direction of the line 9-8, the intersection of which with S is marked. The remainder of the curve may be run in the following manner: Set the vernier at an angle equal to the deflection angle of the chord 9-8 to the left from the 0; clamp the upper plate, sight at the point set in the line 9-8; then clamp the lower plate and set vernier at 0. The line of sight will then be in the tangent at point 9, and by plunging the telescope the remainder of the curve can be run as if the point 9 were the P. C. FIELD NOTES FOR CURVES Various styles of field notebooks are published, in which the pages are ruled to suit the different kinds and methods of field work. The accompanying, which are the field notes of a portion of a line containing a curve, represent a good form for recording the field notes of a curve that is run in by the method of zero tangent. In the first column are recorded the station numbers; in the second column, the deflections with the abbreviations P. C. and P. T., together with the degree of curve and the abbreviation R or L, according as the line curves to the right or left. At each transit point on the curve, the total or central angle from the P. C. to that point is calculated and recorded in the third column. This total angle is double the deflection angle between the P. C. and the transit point. In these notes, there is but one intermediate transit point between the P. C. and the P. T. The deflection from the P. C. at Sta. 3+20 to the intermediate transit point at Sta. 4+50 is 2° 36'. The total angle is double this deflection, or 5° 12', which is recorded on the same line in the third column. The record of total angles at once indicates the stations at which transit points are placed. The total angle at the P. T. will be the same as the angle of intersection, provided the work is correct. When the curve is finished, the transit is set up at the P. T., and the bearing of the forward tangent taken, which affords an additional check upon the previous calculations. The magnetic bearing is recorded in the fourth column, and the deduced, or calculated, bearing is recorded in the fifth column. EARTHWORK Cuts and Fills.-When building a railroad, cuts and fills are introduced to equalize the irregularities of the natural soil. Figs. 1 and 2 show a typical fill and cut in ordinary firm earth or gravel. Slope Ratio. In cuts in the hardest rock, the average slope is made usually 1:1; that is, horizontal to 1 vertical. As the soil becomes less firm, the slope must be flattened until, for a soil of firm earth or gravel, a slope of 1 to 1 may be permissible, although a slope of 1:1 is commonly adopted. In very soft soil, the slope ratio is sometimes cut down even as far as 4 horizontal to 1 vertical. The standard practice in a fill is 1 horizontal to 1 vertical. When a fill is made of the material from a rock cut, it is possible to make a stable embankment with a slope ratio of 1:1. On side-hill work, where a slope ratio of 14:1 or even 1:1 might require a very long slope, it is often advisable to make a rough dry wall of the stones from a rock cut that will have a slope ratio of 2:1, or it may even be steeper. Width of Excavations and Embankments.-The width required for a standard-gauge single-track roadbed may be estimated as follows (see Figs. 1 and 2): The tie will be between 8 and 9 ft. long, usually 8 ft. 6 in., and at the ends the ballast will slope down to subgrade. The extra width required for this will be about 1 or 2 ft. at each end of the tie. Usually, the embankment is widened for about 2 ft. beyond the ballast on each side. The absolute minimum for the width of subgrade for a fill is, therefore, 8 ft.+2× (1+2) ft. = 14 ft. This width would be used only for light-traffic, cheaply constructed roads; 16 to 18 ft. is far more common, while 20 ft. and even more is frequently used, as the danger of accident due to a washing out of the embankment is materially reduced by widening the roadbed. = In cuts, the proper width for two ditches should be added. Unless the soil is especially firm, the ditches should have a side slope of 1:1. If the ditch is 12 in. wide at the base and 12 in. deep, with side slopes of 14:1, each ditch will require a total width of 4 ft. This will add 8 ft. to the width of the cut at the elevation of subgrade. The usual distance between track centers for double track is 13 ft. Therefore, whatever rate of side slopes and width of ditches is required for single-track work,. the width for double-track work must be 13 ft. greater. When excavation is made through rock, the side slopes of the ditches may properly be made much steeper; the danger of scouring during heavy rain storms being eliminated, the total required width may be very materially reduced from the figures just given. The heavy expense of excavating through solid rock requires that such economy shall be used if possible. Grade Profile. For the purpose of constructing a road as well as for calculating the earthwork, a grade profile is prepared by setting stakes on the center line at every full station and also at all intermediate points at which the inclination of the natural surface of the ground changes abruptly; then, by leveling, the elevation of the natural surface at each stake is determined and plotted, as explained under Leveling. The established grade is then drawn in. It consists of a series of straight lines, the elevations of the ends of which are clearly indicated. These elevations are those of the subgrade ac, Figs. 1 and 2. A short portion of a profile is shown in Fig. 3. The horizontal line XX' represents a reference plane, and the broken line AGH shows the position of the established grade. The station numbers are written along the line XX', and the elevations of the corresponding points of the established grade are written along the grade line. Thus, in Fig. 3, the elevation of subgrade at Sta. 90, or A, is 100 ft.; at Sta. 93, or G, it is 102.28 ft.; and at Sta. 94, or H, it is 101.78 ft. The gradient of the established grade the per cent. of rise or fall of grade; that is, the number of feet by which the elevation increases or decreases in 100 ft. It is usually marked on the grade line in the manner shown in Fig. 3. The depth of center stake is the difference between the elevation of the natural surface at any stake and the elevation of the subgrade. The elevation of the natural surface is found in the level notes, while the elevations of the subgrade are computed from the gradients and also entered in the level notes. The difference for each stake is then figured and entered in a column headed Depth of Center Stake, being preceded by the letter C or F to indicate cut or fill. EXAMPLE. Stakes are set at the stations indicated in the first column of the accompanying field notes. The gradient is +.76% from Sta. 90 to Sta. 93, and -50% beyond Sta. 93. The elevation of the established grade at Sta. 90 is 100 ft.; the elevation of the natural surface at each stake is given in the third column. Find the center depth at each stake. (See Fig. 3.) SOLUTION.-The elevations of the subgrade at the station stakes are determined as follows: The center depth is the difference between the corresponding numbers in the second and third columns. This is a fill if the subgrade is higher than the natural surface; otherwise, it is a cut. Slope Stakes. In addition to center stakes, slope stakes are used to mark the points where the side slopes of a cut or a fill intersect the natural surface of the ground. In Fig. 4, c is the center stake and m and m' are the slope stakes. The method of locating slope stakes is as follows, all letters referring to Fig. 4: Let b be the width Il' of the roadbed; d, the depth ce of the center stake; and s the slope ratio=lk÷mk=l'k'÷m'k'. For the upper stake at m, let x be the distance mq from the slope stake to the center line; y+d, the elevation of m above the subgrade = gc+ce=mk. Similarly for the lower stake at m', |