m - - BCAL 'DEPIiEKS Jj'j! l|i;i' ; ; ! ' : fifflffliiiiiir" THE SIX-CHORD SPIRAL BY J. R. STEPHENS, C.E. NEW YORK THE ENGINEERING NEWS PUBLISHING CO. 1907 -< < COPYRIGHT, 1907 BY THE ENGINEERING NEWS PUBLISHING COMPANY PREFACE THE Six-Chord Spiral is an ordinary multiform compound curve of six arcs of equal length, whose degrees of curvature increase in the order of the natural numbers, and so arranged that the seventh arc always exactly coincides with the main circu- lar curve. As herein outlined it has several valuable features. 1st. It is perfectly, flexible and always fits. 2d. No special tables whatever are required for general use. Hence such tables cannot be lost or mislaid. If desired, special tables of the usual form may be quickly computed from Table IV and formulas (1) and (8). 3d. The spiral is adapted to the curve, and not the curve to the spiral of fixed offset or length, as is the case with table spirals. 4th. Odd curves are as readily fitted as even ones, which saves time and trouble in spiraling old track. 5th. Intermediate transit points may be set at any plus, and do not lead to complex deflec- tion calculations. 6th. The method is quickly grasped, memorized, iii 382062 iv PREFACE and applied by transitmen with no previous knowl- edge of spirals, being based on what they already know; and the mathematical treatment being elementary throughout. On location it is not even necessary to run in the six-chord, a terminal curve of half the degree of the main curve and giving the same length as the spiral line being substituted. In this connection note that curves are usually traced a number of times and by different men before the final centering. 7th. It is perfectly interchangeable with the cubic parabola, the two being, within the common limits of spiraling, practically identical. It should be noted that no spiral changes its degree of curvature directly with the elevation of the outer rail, when the elevation approach has vertical curves at the beginning and end. In this respect all spirals are misfits. The importance of a proper length of spiral is dwelt upon, and methods are given to insure con- sistency in this respect with varying conditions of speed and curve. Comparisons are made between spirals commonly used, which, with the same conditions, define their relations, not only in length and total angle, but also laterally. The second part deals with methods for shifting old tracks to make room for spirals, pointing out PREFACE v that this question is entirely independent of the kind of spiral used. Acknowledgment is due to Professor Talbot for the method of swinging tangents to make room for spirals, and also the method of formulas (27) and (28) for inserting a spiral between the two arcs of a compound curve (see Talbot's "Transi- tion Spiral"). J. R. STEPHENS. DENVER, COLORADO, November 5, 1906. NOTE. Natural versed sines are much used in this book. When not given in the ordinary field tables, they may be found by mentally subtracting the natural cosine of the given angle from .9999 (10), working from left to right, and calling the last decimal used 10. To find the angle corresponding to a given natural versed sine, subtract the latter from .9999 (10), as above. The remainder will be the natural cosine of the required angle. CONTENTS PART I LOCATION AND CONSTRUCTION OF SPIRALS PAGE General Forms of Spirals 1 The Six-Chord Spiral 4 The Six-Chord Spiral and Terminal Curve Having a Radius Twice that of Main Curve .... 7 Formula for Substituting Spirals between Two Curves by Shifting Original Tangent Main Curves Undisturbed 12 Compound Curves 14 To Shift the Two Members of a Compound Curve so that Suitable Spirals may be Inserted . . .17 The Length of Spirals 22 The Length of Spirals Joining Compound Curves . 25 To Run in the Six-Chord Spiral by Deflections . . 26 The Track Parabola 29 Relative Lengths and Total Angles of Spirals . . 31 Demonstration of the Six-Chord Spiral . . . . 32 Comparative Tabulations : Relations between the Six- Chord Spiral and Terminal Curve, Each Being Exactly and Independently Calculated ... 35 Comparison of Spirals and Summary 40 PART II SPIRALING OLD TRACK Methods 46 Compound Curves 53 Space-Shifts Preserving Original Length of Line , 59 vii THE SIX-CHORD SPIRAL. PART I. LOCATION AND CONSTRUCTION OF SPIRALS. There are two general forms of spirals in com- mon use. 1st. The Track Parabola, in which the deflec- tions from the point of spiral vary as the squares of the distances measured from the same point along the curve. With the track parabola, any given values of EM and p, Fig. 1, are fitted exactly. Further, any intermediate point can be set exactly, and, the instrument being moved up, work continued in a manner similar to that used in laying out circular curves. This, however, sometimes results in trouble for inexperienced men. 2d. The Polychord Spiral, in which the degree of curve increases with each chord, in arithmetical progression. The polychord spiral with an infinite number of chords is the track parabola. Reduced to its simplest form, the polychord be- comes what might be called a One-Chord Spiral. The latter is a terminal circular curve having a radius 2 R M (see dotted curve, Fig. 1). 1 2 THE SIX-CHORD SPIRAL The values of p and R M being fixed, all poly- chord spirals will fall between the one-chord spiral and the track parabola, and the greater the number of chords, the nearer the approach to the track parabola. THE SIX-CHORD SPIRAL 3 For fixed values of p and R M , each form of spiral has its own appropriate length, the one- chord being the shortest and the track parabola the longest, all the poly chords falling in between; the greater the number of chords the longer the spiral. In practice, the maximum lateral variation of a six-chord from a parabola will not exceed 0.02 feet. The usual variation is negligible in this class of work. Hence the principal easement curves in use yield alinements which approach each other so closely that their riding qualities are the same. The total length of track, between common points on the main tangent and main curve, is also the same, no matter what spiral be used, so that, after track is laid to a one-chord, it may be thrown into a track parabola without altering the expansion. The three principal classes of polychords are: 1st. With deflections constant, while chord length and number of chords vary (such as the Searles form). 2d. With chord length constant, while deflec- tions and number of chords vary. 3d. With number of chords constant, while de- flections and chord lengths vary. Most of these spirals depend for their usefulness on specially prepared tables, which must be con- 4 THE SIX-CHORD SPIRAL suited in the field, and their efficiency for varying values of p and R M increases with the number of tables. Thus, Searles has provided 500 tabulated spirals from which to "select the one coming nearest to given values of p and R M - The spiral used in the following discussion is of the third type and has invariably six chords. The Six-Chord Spiral is chosen: 1st. On account of its extremely simple rela- tion to the one-chord spiral or terminal arc of half the degree of the main curve (see Fig. 2). 2d. On account of its close approximation to the track parabola, and all polychords commonly used. It will first be considered as a curve to be offset from the one-chord spiral. The offsets are small, and may usually be esti- mated in a manner analogous to the use of the self-reading rod in leveling. The instrument is to be kept on the one-chord spiral, and all calculations, shifts, etc., are made by the ordinary rules and tables for circular curves. Notes are kept and plats made precisely as for compound curves. The one-chord is sufficiently exact for right-of- way descriptions. Since the one-chord and the six-chord have the same length between common points, no equation THE SIX-CHORD SPIRAL 5 of distance is introduced in passing from one to the other. To aid the eye in offsetting in the field of view of the instrument, a 2J-inch wrought-iron washer FIG. 2. may be put on the transit rod. This will give a 0.1 it. offset on each side of the center of rod, which is usually a sufficient help for setting stakes. A more exact makeshift may be obtained as follows : 6 THE SIX-CHORD SPIRAL Take a two-foot rule, cut off the two outside hinged legs, thus leaving the pivot joint with a six-inch leg on each side. Screw one of these legs along a face of an ordinary wooden octagon rod. The other leg will make a folding offset sight. This movable leg should have fastened to its face a strip of sheet-iron, say 6 in. long and 1 in. wide, in which F-shaped notches are cut, deep ones for the full tenths from rod center, and shallow for the half tenths. When the vertical hair cuts the scale at the proper offset, set tack at point of rod. In case the spiral is so long that a division into six parts gives too great a distance between track centers, it may be divided into twelve equal parts by taking every fifth point in Table I. This will not constitute the regular twelve-chord spiral, which would be longer and include a greater total angle than the six-chord. As a guide to section foremen in determining track elevation it is preferable to divide the spiral into some fixed number of equal parts, regardless of the full stationing. FORMULAS (see Fig. 2). p = R M (l- cos 7\) (1) cos Ti - u - (3) THE SIX-CHORD SPIRAL (5) (6) The inferiors "M," "1," and "6" indicate re- spectively "main curve," "one-chord," and "six- chord." L and R are lengths of arc and radius in feet, and D x = degree of one-chord. THE SIX-CHORD SPIRAL AND TERMINAL CURVE HAVING A RADIUS TWICE THAT OF MAIN CURVE. This spiral (Fig. 2) has six chords, each one- fourth length of terminal curve, hence spiral is 1J times length of terminal curve, and the quarter points H^H 2 H^H. > of the terminal curve, are abreast the one-sixth points S t S 2 $ 4 S 5 of the spiral. $ 3 and H 3 coincide. S 3 A = S S B. One-half the terminal curve is inside the spiral, the other half outside; and the offsets between them, at equal distances from H 3 or S 3 , are equal. H 1 S t = H 5 S 5 = .036 p, and H 2 S 2 = H 4 S 4 = .054 p. The offset p - EM (1 - cos 7\) = R M X versed sine T 19 where R M = radius of main curve, and T l = the terminal angle. (1) R M = ^ 8 THE SIX-CHORD SPIRAL To locate the spiral, take the distance for gain- ing the required elevation = L 6 = 6 C (at the nearest multiple of six feet, to avoid fractional chaining) . Here C = chord, and L 6 = 6 C = length of spiral. Then 2 C X DM ' where 100 DM = degree of main curve. 7\ = terminal angle in degrees. C = length of chord in feet. Next calculate p from equation (1) above - run in the terminal curve and offset to spiral. Locate P. S. and $ 6 , on outer tangent and main curve, one chord-length from H^ and H & respect- ively. NOTE. TV the total angle of six-chord = 1 TV Note particularly that the length of six-chord = L 6 is 1.5 times the length of the one-chord = L x ; also, as an aid to the memory, that the offset .054 = 1.5 times .036. In practice, taking p at 4 feet, the offsets would be 4 times .054 = 0.216 ft., and 4 times .036 = 0.144 ft. Example. Take a 14 curve having a spiral approach of six chords, each 25 ft. long or 150 ft. in all, to connect with a 7 approach, and calcu- late the offsets to spiral. R M = 5730 * 14 = 409.3. L 6 = 25 X 6 = 150. THE SIX-CHORD SPIRAL 9 7\ (the terminal angle) = J L 6 X Z> = 150 X 14 -r- 3 = 7, L 6 being expressed in one hundred- foot units. The main offset p = R M (1 - cos 7\) = 409.3 X .00745 = 3.05 ft. The offsets HA = H 5 S 5 = 3.05 X .036 = 0.11 ft. H 2 S 2 = HjS 4 = 3.05 X .054 = 0.16 ft. H 3 S 3 = Zero. . The P. S. and S 6 are set as shown in Fig. 2. The 7 approach from H l to H 5 , or the one- chord spiral, will be four 25-ft. chords. Whenever intermediate offsets are required, as in centering trestle bents, etc., the following table is used: 10 THE SIX-CHORD SPIRAL TABLE L TABLE FOR INTERMEDIATE OFFSETS TO SIX-CHORD SPIRAL FROM MAIN TANGENT AND MAIN CURVE WITH ONE-CHORD APPROACH. (To BE MEASURED INWARD FROM THE MAIN TANGENT HALF OF SPIRAL AND OUTWARD FROM THE MAIN CURVE HALF). "So * to 1 * bO 5 * H a r sl a 1 1 |l a o 1 1 I 1 TJ 2 a "S t-<*H "g ^ fl ? Mt "S l3 "S t,4H o '^' o o o .a o ^"^ o o o ^a o p o C , t - 5 ji o S^l J3 o 5 "8 1 -5g "3 Sic *o *s Ste o"g_ O to "o o'B m 02 S o J3 43 C n so 5fi 03 III ,) 50 5fi? 03 ||| ^ S >. M a S^ J 1 1^1 1 J.S M 1 P.S. .000 S 6 Si .036 S 5 S 2 .054 S 4 .0000 .0006 .0004 1 .000 9 1 .042 9 1 .050 9 .0001 .0006 .0004 2 .001 8 2 .048 8 2 .046 8 .0002 .0004 .0005 3 .003 7 3 .052 7 3 .041 7 .0003 .0004 .0005 4 .006 6 4 .056 6 4 .036 6 .0003 .0002 .0005 5 .009 5 5 .058 5 5 .031 5 .0004 .0001 .0005 6 .013 4 6 .059 4 6 .026 4 .0005 .0000 .0006 7 .018 3 7 .059 3 7 .020 3 .0005 .0000 .0006 8 .023 2 8 .059 2 8 .014 2 .0006 .0002 .0007 9 .029 1 9 .057 1 9 .007 1 .0007 .0003 .0007 Si .036 S 5 S 2 .054 S 4 S 3 .000 S 3 Example. In the preceding example let the P. S. be at station 7 + 07, chords 25 feet; required the offset at the even station 8. The curve may be tabulated thus: THE SIX-CHORD SPIRAL 11 p. S. = 7 + 07 5 1 = 7 + 32 5 2 = 7 + 57 5 3 = 7 + 82 4 = 8 + 07 J Hence 8 = S 3 + if = S 3 + 0.72 toward S 4 , which, by interpolation in Table I, equals .042; and .042 X p or 3.05 = .128 ft. If the numbering ran in the opposite direction, the offset at 6 + 40 being required, then : p q _ n _(_ C\<-7 S = 6 + 82 ' _ __ Here 6 + 40 = S 3 + A = S 3 + 0.32 towardS 2 ,which,byTableI, equals -021 X 3.05 = .064 ft. In case a simple curve has been run in connect- ing the main tangents, as in Fig. 1, no provision being made for spiraling, the circular curve is moved inward, without altering the original radius, along the line BC, for the distance EF = p -=- cos J 7, where p is the principal offset and / the total angle turned between tangents, EF being parallel to BC. Also EG = p tan \ I. The distance back to the P. C. from G of the one-chord spiral approach at H is (see Fig. 1) and GH EH AH R M sin 7\ R M sin T 1 + p tan (R M + p) tan J / - (8) I (9) fl M sin 7\ (10) In order to avoid small equations and to fit the ground from the start, the one-chord spiral should be run in on the first located line that is likely to become final. 12 THE SIX-CHORD SPIRAL FORMULA FOR SUBSTITUTING SPIRALS BETWEEN Two CURVES, BY SHIFTING THE POSITION OF THE ORIGINAL TANGPJNT TO MAKE ROOM FOR THE SPIRALS, LEAVING MAIN CURVES UNDIS- TURBED. Let A be the angle between the old and new tangents ; L = length of original tangent; p l and p 2 = values of principal offsets selected for the two curves respectively; Ri and R 2 = radii of the two curves respectively. Then, when the curves are in opposite directions, A (in minutes) = 344 ( ?' + ft) + /3440 (p + P2 )V x OOQ145 (B. + B.). \ L/ / L and when the same curves are in the same direction, 3440 (p, - TO A (in minutes) = L X .000145 L Example. Given alinement as follows : Zero = P. C. 9 R for 36. 4 = P. T. 7 = P. C. 6 L for 30. 12 = P. T. To insert spirals between the curves: THE SIX-CHORD SPIRAL 13 By Rule 1, page 26, for length of spiral, with speed at 33J miles per hour, the six-chord for 9 = 331 x 6 in. elevation = 200 ft.; and six- chord for 6 = 33 X 4 in. elevation = 133.3 ft. The lengths of terminal curves are: 133.3 of 4 30' for 9 = 6, total angle. 88.9 of 3 for 6 = 2 40', total angle. R, + R 2 = 955 + 637 = 1592. 637 X vers 6 = 3.49 - ft. 955 X vers 2 40' = 1.03 = ft, and ft + ft = 4.52. Then, by above formula: A = - ^^ 2 + f 344 * 4 ' 52 Y X .000145 X 1592 300 ' V 300 J ' 300 (The original tangent being 300 feet long), A = 51.83' + 2.07' = 53.9' = 54', approx. This is 10 feet on 9 curve, and 15 feet on 6, and the corrected alinement without terminal curves would read: Zero = P. C. 9 R = 36 54'. 4 + 10 = P. T. 6 + 85 = P. C. 6 L = 30 54'. 12 + 00 = P. T. Then, as one-half of each terminal curve lies either way from the P. T. of 9 and the P. C. of 6, the new alinement (ignoring the small equation which should be made to fall on the new tangent between the spirals) will be : 14 THE SIX-CHORD SPIRAL Zero = P.O. 9 R for 30 54', total angle. 3 + 43.3 = P.C.C. 4 30' R for 6, total angle. 4 + 76.7 = P.T. 6 + 40.6 = P.C. 3 L for 2 40', total angle. 7 + 29.5 = P.C.C. 6 L for 28 14', total angle. 12 + 00 = P.T. COMPOUND CURVES. Whenever the degrees of curvature of the two members of a compound curve differ materially, they should be connected by a spiral. This spiral should be run in on the original loca- tion, to save the trouble of subsequent shifts, equations, etc. The general method before described, of offsets from a one-chord to a six-chord spiral, may be applied equally well in this case. The one-chord connection averages the degrees of the adjacent main curves. Thus, a 4 compounding into an 8 will have a one-chord connection of J (8 + 4) = 6. To make room for this intermediate 6, a suffi- cient offset between the two main curves must be allowed, and the sharper curve must lie inside the lighter one. The length of the one-chord spiral, the principal offset or gap p, and the intermediate offsets, are determined as follows: Take the 4, 6, and 8 combination and assume THE SIX-CHORD SPIRAL 15 that the whole curvature is uniformly "bent" outward until the 4 becomes a tangent, the 6 a 2, and the 8 a 4. We then have the conditions of a 4 curve from tangent, and the necessary calculations are made, as before shown, to fit these conditions. P. S. to S, = S 5 to S G = i SA - i H,H 5 AS 3 = AH, = BS 3 - BH 3 . H,H 3 = H 3 H 5 NOTE. All "H" points are on one-chord spiral; all "S" points are on six-chord spiral. Example. (See Searles' " R. R. Spiral," page 63, Art. 55.) Given a compound curve in which d f = 6 and d" = 10 40', to replace the P. C. C. by a spiral having six chords of 25 ft. each (P. S. to 6 , Fig. 3). 16 THE SIX-CHORD SPIRAL First determine the data for the one-chord, #A, Fig. 3. Its degree d t = J (10 40' + 6) - 8 20'. Its length I, = 4 X 25 = 100 ft. Its total angle / t = 8J X 100 = 8 20', of which d' X i /, = 6 X .50 = 3 is deducted from the 6, and d" X K = 10 40' X .50 = 5 20' is deducted from the 10 40'. The total angle of the six-chord spiral will be 8 20' X 1.5 = 12 30'; of this, d' X j 1, = 6 X .75 - 4 30' is deducted from the 6, and d" X J I, = 10 40' X .75 = 8 is deducted from the 10 40'. Note that in this case the choice of a six-chord spiral in Searles is accidental. The above rea- soning would not obtain had any other chord number been chosen. Now, assuming as before that the 6 curve (the lightest of the three) be bent straight, the 8 20' curve becomes a 2 20', and the 10 40' becomes a 4 40'. Hence the conditions are a 2 20' one-chord approach from tangent to a 4 40' main curve. The terminal angle for 100 feet of 2 20' curve = 2 20', and p, = .436 X 2.33 X 1 = 1.02 [see (4), page 9]: or p 1 = 1228 X .00083 = 1.02 [see (1), page 8], which is the value given by Searles, page 65. Then with the instrument at H^ or H 5 (each THE SIX-CHORD SPIRAL 17 being two chord lengths or 50 feet from the middle point S 3 or H 3 ), run in the 8 20' one-chord spiral and offset. HA = H 5 S 5 = 1.02 X .036 = .037 ft. HA = H,S, = 1.02 X .054 - .055 ft. Intermediate offsets are interpolated from Table I, as before shown. Since in this particular case the maximum differ- ence between the one-chord and six-chord is but f in., the six-chord might well be omitted until it comes to the final adjustment of the track. Note the direction of the offsets, outward from the one-chord line on sharper curve half, and in- ward on lighter curve half. Similarly to the above, the length of the one- chord when P! is given may be determined from formulas 3, 5 and 6, page 9, taking 4 40' as the main curve. To SHIFT THE Two MEMBERS OF A COMPOUND CURVE so THAT SUITABLE SPIRALS MAY BE IN- SERTED. Let LEF, Fig. 4, be a compound curve, with B and C as centers (b and c being the total angles), which has been run in without provision for spirals. Required to insert spirals without changing the degree of either branch of the original compound. The required offsets p and P, Fig. 4, are to be 18 THE SIX-CHORD SPIRAL taken for spirals having a length suitable for the speed and elevation proposed. Assume that the curve EF is slid inward, along the radial line EB common to both curves, until F falls on G, E on D, and C on C7 THE SIX-CHORD SPIRAL 19 p Then FG, parallel and equal to ED = - , where cose P = GN and angle FGN = c; also FN = P tan c. Next determine the proper offset p l for a one- chord JK uniting the two members of the com- pound (see Fig. 3 and following). Then EH = ED - Pl = - - p,. (11) Assume that the curve EL is moved inward until E falls on H and L on M, EH being equal and parallel to ML. Since angle TML = 6, M7 7 = AfL cos 6; hence cos c - Pl }cosb. (12) ^ ) If the curve had been thus run in, the P. T. at M would be a distance, M U, too far out to fit the spiral selected, whose principal offset is p. To make this fit, the provisional P. C. at G must be pushed ahead along YG produced, for a dis- tance (- - p,J cos b p GV = VCQSC . ,/ - - - - WN. (13) sin (b + c) If (b + c) exceeds 90, its sine will be sin [180 - 9 + en FW = P tan c - WN. (14) From W add the distance back to S, making 20 THE SIX-CHORD SPIRAL W S = CF X sin 7\, where R^ = 2 CF (see also equation (8) page 13). The whole curve, with one-chord spirals, may now be run in, remembering to deduct from the total angle c the terminal angle of its spiral 'to tangent plus the angle KC f D of the one-chord KJ. Similarly, the total angle b is reduced by its ter- minal spiral angle plus the angle JB'H. Angle KC'D = i JK X degree of curve EF. Angle JB'H = \ JK X degree of curve EL. In the case of a long compound, minor differ- ences in running may be adjusted by shifting J, the end of the one-chord (see rule, page 56). This should be done by first running out the full curve JM, and before attempting to put in the final spiral. In some cases it will be necessary to shift the original P. C. C. before room can be made for end spirals. In making any or all of these shifts, the nature of the ground should be kept in mind, in order to gain the advantages of a general revision of the line. For this purpose, a large-scale special plat is often of use. Fig. 5 indicates the process when the curve is to be run in from the lighter end. Here angle FGN = b. THE SIX-CHORD SPIRAL 21 Then FG, parallel and equal to ED, = --. (15) cos b (16) FN = p tan &, EH = ED + Pl = LM Angle TML = c. Then cos 6 22 THE SIX-CHORD SPIRAL TM = LM X cos c = -- + p cos c. (17) \cos 6 r V TU = P, the required offset = TM + MU. Hence the shift required is and the necessary pull back is P - ^ + Pi) COS C sin (6 + c) (19) FN + TFAT = p tan 6 + TFAT (20) The rest of the process is the same as in the pre- ceding case after GV has been obtained. THE LENGTH OF SPIRALS. There is no definite rule for determining the length of spirals. This depends on both speed and elevation. The rate on which the given elevation is to be obtained is also important. Some rules for spiral length are based on a uni- form rate of elevation grade, such as 1 in 300, 1 in 400, etc. The rational rule for varying speeds is that the same amount of super-elevation should be attained in the same time. This may be called the "time approach." It follows that curves of the same degree, oper- THE SIX-CHORD SPIRAL 23 ated under different speed conditions, should have spiral lengths proportional to the cubes of the speeds used. The following tables indicate the relations be- tween spirals, and the data used to determine their lengths. Speeds are in miles per hour. Distances and elevations are in feet. TABLE II. fl J *c3 sq i .i-a f Jo "3 I CO O_t- CO.tj J ii it 1 *- | a .2 *, < o. oco vii a oco > || 1 0> 5*8 Mo 5"H |j-i-^ |l | |1 5 w j J6 'cd O rt X 1 0-30 100. .5 3.5 400.0 600.0 1 in 1200 2 1- 70.7 .5 3.5 282.8 424.2 1 in 848 3 1-30 57.7 .5 3.5 230.8 346.2 1 in 692 4 2- 50.0 .5 3.5 200.0 300.0 1 in 600 5 2-30 44.7 .5 3.5 178.8 268.2 in 536 6 3- 40.8 .5 3.5 163.2 244.8 in 490 7 3-30 37.7 .5 3.5 151.0 226.5 in 452 8 4- 35.4 .5 3.5 141.6 212.4 in 425 9 4-30 33.3 .5 3.5 133.2 199.8 in 400 10 5- 31.6 .5 3.5 126.4 189.6 in 379 In the above table, the maximum safe speeds are, for convenience, taken as the reciprocals of the square roots of the degrees of main curve X 100; also, Length of one-chord spiral = max. speed X 4; Length of six-chord spiral = max. speed X 6. Note from Table II that for curves operated under the same conditions of safe speed and with 24 THE SIX-CHORD SPIRAL time approaches, the offsets p and the elevation are constant. The following convenient rules for lengths of spirals are also indicated by the table: Rule 1. Length of six-chord equals speed in miles per hour multiplied by elevation in inches. Here maximum p = 3.5 ft. Rule 2. If somewhat longer spirals be desired, then length of one-chord spiral equals speed in miles per hour multiplied by elevation in tenths of feet. Here maximum p = 5.45 ft. Other rules, yielding longer or shorter spirals as desired, may be formed on the same plan. The following table of elevations explains itself. The elevations are in decimals of a foot, and the speeds in miles per hour are given at the heads of the columns. TABLE III. Degree of Main 31.6 33.3 35.4 37.7 40.8 44.7 50.0 57.7 70.7 100.0 Curve 1 .05 .06 .06 .07 .08 .10 .13 .17 .25 .50 2 .10 .11 .13 .14 .17 .20 .25 .34 .50 3 .15? '.17 .19 .21 .25 .30 .38 .50 4 .20 .22 .25 .29 .33 .40 .50 5 .25 .28 .31 .36 .42 .50 6 .30 .33 .38 .43 .50 7 .35 .39 .44 .50 8 .40 .44 .50 9 .45 .50 10 .50 THE SIX-CHORD SPIRAL 25 Now, finding a 5 curve which is elevated .36 ft. and giving satisfaction as to rail wear, comfort, etc., a glance at Table III shows that it belongs to the 7 maximum series, having a speed of 37.7 miles per hour. The length of spiral required would be (adopting Rule 1 under Table II), .36 X 12 = 4.32 ins., and 4.32 X 37.7 == 162.9 ft. for the length of a six- chord spiral; and 162.9 X f = 108.6 ft., the cor- responding one-chord spiral. If Rule 2 be adopted, then 10 X .36 X 37.7 = 135.72 = length of one-chord, and 135.72 X 1.5 = 203.58 = length of six-chord. It may sometimes be advisable to use longer easements on certain curves, so that, if the speed limit be increased, the elevation only need be changed, the alinement remaining fixed. For construction purposes it is necessary to divide the line into speed sections of suitable length, treating each section by itself. A speed section may sometimes be as short as a single sharp curve, or even the sharp member of a compound curve. THE LENGTH OF SPIRALS JOINING COMPOUND CURVES. This should obviously be sufficient to gain the proper difference of elevation between the two curves, or what is the same thing, the length for a spiral from tangent to a curve whose degree is the 26 THE SIX-CHORD SPIRAL difference between the two members of the com- pound; for example: A 5 curve compounds with a 3; required the length of one-chord connection, using Rule 2. 5 - 3 = 2. Then, assuming speed at 40.8 miles, column 6 ; Table III, gives elevation for a 2 = .17 ft. Then 1.7 X 40.8 = 69.4 = length of one-chord spiral. Length of six-chord = 69.4 X 1.5 = 104.1 ft. To RUN IN THE SIX-CHORD SPIRAL BY DEFLECTIONS. The degrees of curvature of the six arcs of the spiral are: D 2D 3D 4D 5D . 6Z>. 7D 7' T'T' 7-> T and y ; y ==A being the degree of main curve (see Fig. 2). The angle of crossing of the six-chord and one- D X C chord at S 3} or H 3 , = ? when both D and the crossing angle are expressed in degrees and decimals, and C equals the length of the single chords in feet. D X L The total angle of the six-chord = .200 TABLE IV. DEFLECTION COEFFICIENTS AND THEIR LOGARITHMS FOR SIX-CHORD SPIRAL. These coefficients multiplied by (C X -D) , where C equals chord length in feet, and D equals degree of main curve in THE SIX-CHORD SPIRAL 27 degrees, give deflections from tangent at transit in minutes and decimals. Add the logarithms to log (C X -D). S 7 is on main curve, one chord length beyond S 6 , and is given to provide an alternative set-up when S 6 falls on bad ground. The transit being over any point in the first vertical column, the deflection coefficients are read from this transit point horizontally. TABLE IV. Transit over P.S. s, S 2 S 3 S 4 S 5 S 6 87 P.S. coef. . . log .0429 8.63202 .1071 9.02996 .2000 9.30103 .3214 9.50708 .4714 9.67342 .6500 9.81291 .8571 9.93305 Si coef. . . log .0429 8.63202 .0857 8.93305 .1929 9.28524 .3286 9.51663 .4929 9.69272 .6857 9.83614 .9071 9.95768 S 2 coef. . . log .1500 9 . 17609 .0857 8.93305 .1286 9.10914 .2786 9.44494 .4571 9 . 66005 .6643 9.82236 .9000 9.95424 S 3 coef. . . log .3143 9.49733 .2357 9.37239 .1286 9.10914 .1714 9.23408 .3643 9.56144 .5857 9.76769 .8357 9.92206 84 coef. . . log .5357 9.72893 .4429 9.64626 .3214 9.50708 .1714 9.23408 .2143 9.33099 .4500 9.65321 .7143 9.85387 S 5 coef. . . log .8143 9.91078 .7071 9.84951 .5714 9.75696 .4071 9.60975 .2143 9 . 33099 .2571 9.41017 .5357 9.72893 S 6 coef. . . log 1 . 1500 0.06070 1.0286 0.01223 .8786 9.94378 .7000 9.84510 .4929 9.69272 .2571 9.41017 .3000 9.47712 S 7 coef. . . log 1.5429 0.18833 1.4071 0.14834 1.2429 0.09442 1.0500 0.02119 .8286 9.91833 .5786 9.76236 .3000 9.47712 Total 1 coef. angle i log P.S. to 0.0857 8.93305 0.2571 9.41017 0.5143 9.71120 0.8571 9.93305 1.2857 0.10914 1.8000 0.25527 2.4000 0.38021 The total angle of the six-chord spiral in min- utes = C X D X 1.8. The degrees of curvature of the six-chord spiral 6D . p = length of spiral X sine of deflection angle P. S. to S 3 . See also formula (4), page 9. , arcs are to 28 THE SIX-CHORD SPIRAL Example. Take a 14 curve having a spiral approach of six chords, each 25 ft. long or 150 ft. in all, to calculate the deflections. Here C X D = 25 X 14 = 350. (log = 2.54407). Then from Table IV, instrument on P. S., 8.63202 9.02996 2.54407 2.54407 1.17609 1.57403 15' 015' 37.5' 37J' 9.30103 2.54407 1.84510 70' 9.50708 2.54407 112.5' 1 52 J' 9.67342 2.54407 165' 2 45' 9.81291 2.54407 2.05115 2.21749 2.35698 227.5' 3 47 J' With instrument at $ 6 , to turn tangent to the six-chord and main curve at S 9 : Sight on P. S. with vernier set at (see Table IV) 1.15 X C X D = 1.15 X 350 = 402J' - 6 42J', and then turn vernier to zero. Or, sighting on S 3 , 0.7 X 350 = 245' = 405', which is to be turned off at S 9 to obtain tangent. These computations may be made by logarithms, as before. For instrument at S 3 the crossing angle be- tween the spiral and the 7 curve (one-chord THE SIX-CHORD SPIRAL 29 spiral) will be C X D + 700 = 350 -*- 700 = 0.5 = 30', and from this one may pass from one curve to the other. The total angle of the six-chord is D X L -i- 200 = 14 X 150 4- 200 = 10 3t)'. To calculate the deflections for a six-chord spiral joining two members of a compound curve (see ex- ample under Fig. 3) : First calculate the deflections by Table IV for 150 ft. of six-chord spiral joining a tangent with a 10 40' - 6 *= 4 40' main curve. Then to each deflection thus found add that of a 6 curve for the length of sight taken. Thus, from P. S. to S, add 45' ; from S 3 to S Q add 2 15'. If so desired, necessary tabulations may be pre- pared in advance, giving once for all the deflections required for the general run of curves in use, pre- cisely as is customary with all table spirals. THE TRACK PARABOLA. Table V may be used in offsetting from the one- chord spiral to the track parabola. Tables I and V are on the same six-chord base and may be similarly used. It will be noticed that the differences between the corresponding offsets in Tables I and V are, for any usual value of p, too small to be note- worthy. 30 THE SIX-CHORD SPIRAL In actual service, the parabola has no advantage whatever over the polychord spiral, and a choice between them should be governed by their rela- tive adaptability to field and office use. The offsets in Table V are to be measured in- ward from the main tangent half of spiral, and outward from the main curve half. Note that the offsets at P. S. are insignificant. For p = 10 ft. they are 0.01 ft. TABLE v. TABLE OF INTERMEDIATE OFFSETS TO TRACK PARABOLA FROM MAIN TANGENT AND MAIN CURVE WITH ONE- CHORD APPROACH. ! X^ js 8 o< 1 JS ! ,c "So 1 ii 1 J3 * ^ , X X o> O **-! 1 jj> 1 Tenths of chord Coefficients whi give offsets in Tenths of chord Differences for < hundredth of chord length I Tenths of chord Coefficients whi give offsets in Tenths of chord Differences for i hundredth of chord length Tenths of chord Coefficients whii give offsets in Tenths of chord Differences for < hundredth of chord length P.S. .001 S 6 Si .038 S 5 S 2 .055 S 4 .0001 .0007 .0003 1 .002 9 1 .045 9 1 .052 9 .0001 .0006 .0004 2 .003 8 2 .051 8 2 .048 8 .0002 .0004 .0005 3 .005 7 3 .055 7 3 .043 7 .0003 .0003 .0005 4 .008 6 4 .058 6 4 .038 6 .0003 .0002 .0006 5 .011 5 5 .060 5 5 .032 5 .0004 .0001 .0006 6 .015 4 6 .061 4 6 .026 4 .0004 .0000 .0006 7 .019 3 7 .061 3 7 .020 3 .0005 .0001 .0006 8 .024 2 8 .060 2 8 .014 2 .0007 .0002 .0007 9 .031 1 9 .058 1 9 .007 1 .0007 .0003 .0007 s, .038 S 5 S 2 .055 S 4 s a .000 S 3 THE SIX-CHORD SPIRAL 31 RELATIVE LENGTHS AND TOTAL ANGLES OF SPIRALS, p AND R M CONSTANT (see Fig. 1) : Let L! = length or total angle of one-chord spiral. L 6 = length or total angle of six-chord spiral. L P = length or total angle of track parabola. Then L 6 = 1.5 L, L, = f L 6 L, = .577 L P L P = 1.733 L l L P =1.155 L 6 L 6 = .866 L P Example. Given R M = 1432.5 = 4 curve, p - 4.65; The total angle of a one-chord will be [ (3), page 8] 4 37' - 4.617 The total angle of a six-chord = 4.617 X 1.5 = 6.926 = 6 55 \' The total angle of track parabola = 4.617 X 1.733 = 8 00' Length one-chord = 4.617 -f- 2 = 230.85 ft. Length six-chord = 230.85 X 1.5 = 346.28 ft. Length parabola = 230.85 X 1.733 = 400.00 ft. These lengths are bisected at S 3 , which is the middle point of all spirals. In the foregoing example, 400 - 230.85 = 169.15 ft. is the difference, L P - L 1 = .733 L t . Hence, 169.15 H- 2 = 84.58 ft., is the distance to be laid off along main tangent or main curve from the beginning or ending of the one-chord, in order to obtain the beginning or ending of the track parabola. This may be used in connection 32 THE SIX-CHORD SPIRAL with Table V, when it is desired to lay off the track parabola. The total angles of the spirals will be divided at the middle point S 3 as follows: One-chord 4.617, J - 2.31 on tangent half. One-chord 4.617, J - 2.31 on main curve half. Six-chord 6.926, f = 1.98 on tangent half. Six-chord 6.926, f = 4.95 on main curve half. Track parabola 8, J = 2 on tangent half. Track parabola 8, f = 6 on main curve half. In all spiral running it is important to keep a watch on the total angles of the various parts, so that the grand total, from tangent to tangent, will check with the intersection angle of the whole curve. DEMONSTRATION OF THE SIX-CHORD SPIRAL. In this spiral (Fig. 6), if the total angle of the first arc, P. S. to S ly be taken as 2, that of the second, Sfiz, will be 4, the third, S 2 S 3 , 6, and so on, S 6 S 7 being 14. $ 6 *S 7 coincides with the main curve, the end of spiral being at $ 6 , all chords being of the same length. Hence the angles which the spiral makes with the outer tangent will be at S lt S 2 , etc., 2, 6, 12, 20, 30, 42, and 56, the angle 42, at S 9 , being the total angle of the spiral. The angle which each chord of the spiral, P, S. Si, S 1 S 2 , etc., makes with the outer tangent THE SIX-CHORD SPIRAL 33 34 THE SIX-CHORD SPIRAL will be the total angle to the end of that chord less the deflection angle of the last arc. From P. S. to S l it equals 2-1 = 1; S&, 6 2=4, and so on, or as the squares of the natural numbers. Since the sines of small angles are proportional to the angles, the ordinates from S 1} S 2 , etc., will be as the sums of these squares, or as 1, 5, 14, 30, 55, 91, and 140, as marked on the figure. AB = 140 - 91 = 49. Since the total angle of the spiral to $ 6 is repre- sented by 42, and to S 7 by 56, the angle S 7 OS & equals 56 - 42 = 14, both on main curve and spiral. Now, as 14 is one-fourth of 56, continuing the main curve back to D through S Q and H 5 will make the tangent at D parallel to the outer tan- gent. The angles K, L, and M each being equal to S Q OS 7 , will be at right angles to DF at D. Assuming that the versed sines of small angles are proportional to the squares of those angles, we have AD: BD::4?:& = 16:9. Hence, AD - BD:BD::IQ - 9:9. But AD - BD = 49, consequently, 49:D::7:9. :.BD = 63, and^D = 112. Take H 5 on the main curve, so that S 6 H 5 sub- tends the angle M and equals S Q S 7 ', then AD : CD : : 4 2 : 2 2 , and CD = 28 = } AD. Also ED = 140 - 112 = 28, and DS 3 = 14 = S 3 E. Now a circle of twice the radius OS 7 , tangent at #, will in 4 chord-lengths have a versed sine = THE SIX-CHORD SPIRAL 35 CE or 28 X 2 = 56, and be tangent to the outer tangent at H^ Taking the ordinates to this circle proportional to the square of the number of chords, it will pass through S 3 , and the ordinates to it will be at H 1 = zero, at H 2 = f| = 3}, # 4 = T 9 e X 56 = 31}. Hence H& = 1, H 2 S 2 = 5 - 3} = 1}, H,S 4 = 31} - 30 = 1}, and H 5 S 5 = 56 - 55 = 1, or, in terms of the main offset, p = 28, H^ = H 5 S 5 = .036 p, and H 2 S 2 = H,S 4 = .054 p. COMPARATIVE TABULATIONS SHOWING THE RELA- TION BETWEEN THE SIX-CHORD SPIRAL AND TERMINAL CURVE WHEN EACH is EXACTLY AND INDEPENDENTLY CALCULATED. The following tables give the coordinates of the H points and the S points, by corresponding pairs, on three typical spirals. In each case the spiral and terminal curve are taken to run in a north- westerly direction from a main tangent running due north. For convenience in taking out sines and cosines from table direct, each chord is 100 feet long. Other spirals having the same total angle may be formed by multiplying the tabular quantities by the selected chord length -=- 100. In this case the degrees of the main and terminal curves will equal the degrees given in the tables chord luU curve at each point is also given. 36 THE SIX-CHORD SPIRAL The differences between corresponding pairs of points $! and H lf S 2 and H 2 , etc., are taken from each H point as an origin or zero; thus the differ- ence between H 2 and S 2 (in Table VI) of W. .437 and S. .002 means that S 2 lies west and south of H 2 , .437 and .002 feet respectively. TABLE VI. COORDINATES FOR 600 FEET OF SIX-CHORD-SPIRAL APPROACH TO 2 20' MAIN CURVE, AND ALSO FOR 400 FEET OF 1 10' TERMINAL CURVE JOINING SAME TANGENT AND CURVE. R M = 2455.7 ft., p = 8.15 ft., spiral angle = 7, terminal angle = 4 40'. Spiral angle of corresponding track para- bola = 8 05'. p X .036 = .293 ft.;" p X -054 = .44 ft. 0^ ~ "84. d c 3 "S "c 1 o .-. W) d ** 'S . W) rf *-> PH JJ a 3 IP 1 a H, N 0.000 100.000 H 4 N 30 30'W 9.160 399.818 s, N 20'W .291 100.000 N 3 20'W 8.726 399.851 W.291 0.000 E.434 N.033 H, N 10 10'W 1.018 199.995 H 5 N 40 40'W 16.281 499.564 So N 1 W 1.455 199.993 S. N 50 00' W 15.992 499.587 W.437 S.002 E.289 N.023 HS N 20 20'W 4.072 299.948 Ho N 7o W 26.445 599.046 S 3 N 20 W 4.073 299.959 Se N 7 W 26.445 599.039 W.001 N.011 0.000 S.007 TABLE VII. COORDINATES FOR 600 FEET OF SIX-CHORD SPIRAL APPROACH TO 4 40' MAIN CURVE, AND ALSO FOR 400 FEET OF 2 20' TERMINAL CURVE JOINING SAME TANGENT AND CURVE. R M = 1228.1 ft., p = 16.26 ft., spiral angle = 14, terminal angle = 9 20'. Spiral angle of corresponding track para- bola = 16 10'. p X .036 = .585 ft.; p X .054 = .878 ft. THE SIX-CHORD SPIRAL 37 O +* "S-a a 3 ** 3 a M 2 | 2 || a 5 11 1 1 rH I H a 3 H & 5 Hi N OOW 0.000 100.000 H 4 N 70W 18.305 399.274 Si N 40'W .582 99.998 S 4 N 60 40'W 17.438 399.401 W.582 S.002 E.867 N.127 H 2 N 20 20'W 2.036 199.979 H 5 N 90 20'W 32.510 498.260 S 2 N 20W 2.909 199.971 S 5 N 10 OO'W 31.931 498.345 W.873 S.008 E.579 N.085 H 3 N 40 40'W 8.141 299.792 Ho N 140 OO'W 52.732 596.194 S 3 N 40W 8 143 299.834 N 140 OO'W 52.722 596.160 W.002 N042 E.010 S.034 TABLE VIII. COORDINATES FOR 600 FEET OF SIX-CHORD SPIRAL APPROACH TO 7 MAIN CURVE, AND ALSO FOR 400 FEET OF 3 30' TERMINAL CURVE JOINING SAME TANGENT AND CURVE. R M = 819.9, p = 24.35 ft., spiral angle = 21, terminal angle = 14. Spiral angle of corresponding track para- bola = 24 15'. p X .036 = .877 ft.; p X .054 = 1.315 ft. a P | g "c 1 1 1 9 I 1 1 V I 3 HI N W 0.000 100.000 H 4 N 10^30'W 27.416 398.369 Si N P W .873 99.996 S 4 N 10W 26.126 398.654 W.873 S.004 El. 290 N.285 H 2 N 3030' W 3.054 199.953 H 5 N 140W 48.634 496.092 S 2 N 30W 4.363 199.935 S 5 N 150W 47.770 496.284 W 1.309 S.018 E.864 N.192 H 3 N 7W 12.204 299.533 HO N 2FW 78.705 591.464 S 3 N 60W 12.209 299.627 SK N 2PW 78.672 591.390 W.005 N.094 E.033 S.074 An inspection of these tables shows : 1st. That in all three cases the six-chord spiral practically passes through H 3 . In Table VIII (an extreme case of high values for p and spiral angle) 38 THE SIX-CHORD SPIRAL S 3 is W. .005 and N. .094 of H 3 , and the tangent to the curve at S 3 bears N. 6 W. Tracing the six- chord south for .094 of latitude would reduce its departure .094 X tangent 6 (.105) - .010, which would cause the six-chord to pass .010 .005 = .005 feet due east of H 3 . In Table VII S 3 would fall .001 feet due east of H 3 . 2d. That in all three cases, the six-chord spiral (continued) practically passes through H 6 , which is on the main curve one chord length beyond H 5 . Thus, in Table VIII, S 6 lies E. .033 and S. .074 feet of H 6 , and the tangent to the curve bears N. 21 W. A continuation along this tangent for N. .074 feet would make a westing of .074 X tan- gent 21 (.38) = .028 feet, and the six-chord would pass .033 - .028 = .005 feet due east of# a . It is to be noted that continuing the six-chord .074 north would lengthen it along the 7 curve .074 *- cos 21 (.93) = .08 feet, thus increasing the total angle to the point abreast of H Q by 7 X .6 X .08 = J minute,, which, in this extreme case, would be the error in total angle. Note also that the coordinates of S 6 divided one by the other give 78.672 -j- 591.39 = .13303 = tangent 7 34f. Now the table of deflections for a six-chord previously given shows a deflection from P. S. to & of C X D X 0.65 - 100 X 7 X THE SIX-CHORD SPIRAL 39 0.65 = 455' = 7 35'. Here also is an error of J minute. A tabulation similar to VIII but reversed, i.e., starting from S Q and running back to the P. S., gives for the quotient of the coordinates of P. S., 138.492 - 580.303 = .23865 - tangent 13 25}'. By table of deflections this angle is C X D X 1.15 = 100 X 7 X 1.15 = 13 25', or again an error of } minute. Similar computations will show that the errors for all intermediate deflections are insignificant. The same treatment of Tables VI and VII will show no material error whatever, that in Table VII from P. S. to S 6 , or S 6 to P. S., being only i 1 -^ of one minute. 3d. A comparison of the actual offsets between the two curves at H lt H 2 , H 4 , and H 5 is best made by platting the coordinates of the S points with reference to their corresponding H points, on a scale of ten inches to the foot, and drawing the tangents through each pair of points from the bearings given in the tables. By this it will be found that in every case (measuring at right angles to the H line) the coefficients .036 and 0.54 mul- tiplied by p will give the correct distance between the two curves, almost exactly. From the foregoing the conclusion is drawn that, even for unusually large values of p and the spiral angle, the method of offsets from the ter- 40 THE SIX-CHORD SPIRAL niinal curve to the six-chord spiral is practically exact, and that the methods of offsets and deflec- tions are interchangeable, i.e., one method will duplicate the other theoretically much closer than either can be made to duplicate itself on the ground, with the customary appliances and methods. COMPARISON OF SPIRALS AND SUMMARY. The railroad spiral provides for a gradual change from the position of car and trucks on a tangent to that assumed by them on a curve. This change is effected by an intermediate curve having an average curvature usually one-half that of the main curve. Figure 1 shows the general problem. Here FJN is the main curve with center at C", and HGE the main tangent. The main curve has been moved inward a distance BN from its original position. This shift is necessary to allow room for the insertion of the lighter intermediate curve. The new main curve merges into the shift tangent (which is parallel to the main tangent) at F. The simplest form of spiral is that shown by the dotted curve HKJ, which has twice the radius or half the degree of the main curve. This is called the terminal curve or one-chord spiral. The point K, which practically bisects the principal offset, FG = p, is the middle point of the length of the spiral. THE SIX-CHORD SPIRAL 41 H is the P. C. and J the P. C. C. of the one-chord. If two curves be used in passing from tangent to curve, the degree of the first will be from tangent to K = J degree of main curve, and of the second from K to main curve = f of the same. As before, K is the half-way point of the spiral. This con- stitutes a two-chord spiral. Calling the total length of the one-chord unity, that of the two-chord will be 1.225. Hence the latter starts from the tangent to the left of H, and passing through K merges into the main curve between J and N. It thus lies inside the one- chord from tangent to K, and outside from K to main curve, the two spirals crossing each other at K. This condition is indicated by the dotted spiral in Fig. 2, where H lf H 3 , and H 5 are points on the one-chord and respectively correspond with H, K, and J of Fig. 1. Spirals having any number of chords (N) are so taken that the degree of curve of the first arc = degree of main curve H- (N + 1), that of the second twice, of the third thrice that of the first arc, and so on, the (N + 1) arc coinciding with the main curve. The lengths of spirals for fixed values of p and main curve increase with the number of chords or arcs used; that is, they start further back on the tangent, and, passing through the common point 42 THE SIX-CHORD SPIRAL K (where they are bisected), reach further around the main curve toward N, Fig. 1, before merging into it. The greater the number of arcs in a spiral, the greater the lateral deviation from the one-chord on the inside of KH and the outside of KJ. The limit is reached when the number of arcs becomes infinite. The spiral then increases uni- formly in curvature from start to finish and with- out pause. This constitutes the usual track parabola whose length is 1.733 that of the one- chord. The curvature of all spirals increases at the same rate from K toward the main curve as it decreases from K toward the main tangent. Hence, with degree of main curve and p fixed, the total angle of a spiral is proportional to its length. The total angles and lengths of various spirals are given in the following table, those of the one- chord being unity: 1-chord = 1.000 10-chord = 1.581 2-chord = 1.225 11-chord = 1.593 3-chord = 1.342 12-chord = 1.604 4-chord - 1.414 13-chord - 1.613 5-chord = 1.464 14-chord = 1.621 6-chord = 1.500 15-chord = 1.628 7-chord - 1.528 16-chord = 1.634 8-chord = 1.549 17-chord - 1.640 9-chord = 1.567 parabola = 1.733 THE SIX-CHORD SPIRAL 43 Example. Take R M = 286.5 ft. (20 curve) and p = 4.35 ft., the total angle of the one-chord being 10. These conditions will be fitted by: 100 feet of one-chord, total angle 10. 150 feet of six-chord, total angle 15. 156.7 feet of nine-chord, total angle 15 40'. 173.3 feet of parabola, total angle 17 20'. Each chord of the six-chord will be 150 -=- 6 = 25 ft., and of the nine-chord, 156.7 -j- 9 = 17.4 ft. The degrees of curve of the six arcs of the six- chord will be -V-, -VH - 6 T-> > --?-> and 1 f-- Th e seventh or (N + 1) arc is J-f- 2 -, which is the 20 main curve. In this example, the difference (173.3 - 150) divided by 2 ( = 11.65 ft.) is the amount the parabola overlaps the six-chord at each end. The lateral variation of any of these spirals from the one-chord or from each other is the same at equal distances from K measured along the spiral, but these offsets are to be made inward from the one-chord on the main tangent side of K, and outward on the main curve side. From this it follows that the total length of track between common points on the main tan- gent and main curve is the same for fixed values of RM and p, no matter what spiral be used, so that after track has been laid to a one-chord it may be shifted to a track parabola or any inter- mediate spiral without altering the expansion. 44 THE SIX-CHORD SPIRAL For any one form of spiral with a fixed value of R M , the principal offset p varies as the square of the length of the spiral; that is, doubling the length of spiral increases p four times. If the distances along the one-chord from the middle point K or from either end be expressed in fractions of the length of the one-chord, then the offsets from the one-chord to any fixed form of spiral at any given point will equal p X constant coefficient for that point, regardless of the degree of main curve or length of spiral. Thus, in Fig. 2, the offsets S 2 H 2 or H 4 S 4 , which are at the quarter points of the one-chord, will always for a six-chord spiral, equal p X .054. From the same quarter points of the one-chord to the parabola the offsets are always p X .055. The complete coefficients for the six-chord and track parabola are given in Tables I and V; see also Fig. 2. Since the length of the six-chord is always 1.5 times that of the one-chord, the quarter points of the one-chord lie abreast of the sixth points of the six-chord. Both Tables I and V give the offsets at the various points along the six-chord, from its beginning at P. S. through S lt S 2 , etc., to its end at S 6 . This is solely for convenience in setting off and in making comparisons. These coefficients are the offsets in feet when p = 1 ft. For any other value of p, multiply by p. THE SIX-CHORD SPIRAL 45 Thus, when p = 10 ft. (an unusually large value), Table I shows that the maximum distance from the six-chord to the one-chord is .059 X 10 = .59 ft. at 1.7 chord lengths from either the beginning or end of the six-chord. Table V shows that the maximum distance of the parabola from the one-chord is at 1.65 chord lengths from the beginning or end of the six-chord, and equals (with p = 10 ft.) 0.061 X 10 = .61 ft. Comparing I and V shows that the greatest divergence of the parabola from the six-chord occurs at 1.2 chord lengths from the beginning or ending of the six-chord and equals (.051 .048) X 10 = 0.03 ft. Table V also shows that the offset from main tangent and main curve at the beginning and ending of the six-chord (at P. S. and $ 6 ) equals p X 0.001, which, when p = 10 ft., becomes .01 ft. Hence, for easement purposes, the excess of length of the parabola over the six-chord is negligible. PART II. SPIRALING OLD TRACK. Spiraling old track consists mainly in com- pounding to make room for the spirals. The methods used for the shifts are entirely in- dependent of the form of spiral, for, with fixed values of p and R M , any spiral from the one-chord to the track parabola may be inserted, differing from each other, of course, in length and total angle, according to Table IX, but all giving prac- tically the same length of line between common points. In making room for a six-chord spiral, the obvious method is to first provide for a one-chord, remembering that the one-chord radius must be double that of the revised curve into which it com- pounds, and not double that of the existing curve, unless the latter be unchanged. With this condition imposed, any of the formulas for three-center compound curves may be used direct. Space-shifts for inserting spirals are usually made according to one or the other of the following assumptions : 1st. To leave as much of the original line un- disturbed as circumstances permit. 46 THE SIX-CHORD SPIRAL 47 2d. To preserve the original length of line, thus avoiding numerous equations of distance. In either case the tangents are usually undis- turbed, the necessary changes being confined to the curves. When working on the first assumption, the following compound-curve formulas are most useful (see following example for application): R = R - vers/ (21) P p =(R - RN) vers /. (22) Vers/--^-^-. (23) where R N = radius of new main curve, R = radius of original main curve, p = principal offset. / equals the angle cut out of the R -curve and replaced by the R N -curve. The degree of the R N -curve is usually taken from one-tenth to one-fifth greater than the degree of RO- The one-chord terminal angle T l is determined from vers 7\ =^-, and either the one-chord or six-chord run in. The P. C. of the one-chord, 2R N} will be back along the main tangent a distance from the original P. C. =R N sin T l - pcoti / (24). 48 THE SIX-CHORD SPIRAL Example. To replace one end of a 4 curve with enough 4 30' to give an offset p = 6.62 ft. Here vers / = ' = -^ = .0416. HO HN 159 Hence, / = 16 35'; and 16 35' of 4 = 414.6 ft., also, 16 35' of 4 30' = 368.5 ft. Hence, the last 414.6 feet of the 4 is to be re- placed by 368.5 feet of 4 30' curve. Again, vers T l = ^- = 7 9 ~^ = .0052 T, = 5 50.6' 5 50.6' of 4 30' = ^|^ = 129.87 ft., which in its turn is replaced by 129.87 X 2 = 259.74 ft. of 2 15' one-chord approach. From the preceding formula this 2 15' one- chord will begin on main tangent back from origi- nal P. C. a distance = (1273.6 X .1018) - (6.62 X 6.862) = 84.2 ft. It is clear that in the preceding formula / may be made as large as half the intersection angle of the original curve. If it be desired to throw the middle of the original simple curve out along a radial offset for a distance THE SIX-CHORD SPIRAL 49 h y then, 7 2 being half the intersection angle of the original curve, (25) Example : Take / = 60, hence / 2 = 30 R = 955.4 (6). p = 4.4 ft. h = 0.5 ft. Then R N = 955.4 + 0.5 - 4 ' 4 ^' 5 - 919.3. Hence R N = 6 14' curve. Also, vers^l--^-. 00479. 7\ = 5 36.6'. Hence 5 36.6' of 6 14' curve are to be replaced by 5 36.6' of 3 07' one-chord approach. The P. C. of this 3 07' one-chord will be back along the main tangent a distance = R N sin 7\ - (p + h) cot J 7 2 (26) from the original P. C., or 919.3 X sin 5 36.6' - (4.4 + 0.5) cot .15 = 71.60 ft. It is usually best to run such curves as the above from both ends, making the junction at the middle of the curve or on the radial line through the vertex. If, in recentering old track, the best-fitting curve should merge into a tangent parallel to, and either inside (i) or outside (o) of the existing 50 THE SIX-CHORD SPIRAL tangent (which is to be maintained), then the amount o must be added to, and the amount i subtracted from, p in formulas (23) to (26). Example. To replace part of a 4 curve that merges into a parallel tangent 2 ft. outside the existing tangent, by enough 4 30' to make p, with relation to the existing tangent, = 6.62 ft. As the 2-ft. offset o is outside, equation (23) becomes : , 6.62 + 2 8.62 ._.. vers/ = _ -- _ = = .0542 KO KN 159 / == 18 57' 18 57' of 4 = 473.75 ft. to be replaced by 18 57' of 4 30' == 421.11ft. Again, vers r, = - jg ~ . 0052 T l = 5 50.6' of 4 30' = 129.87 ft. to be replaced by 259.74 ft. of 2 15 r one-chord. The P. C. of this one-chord will be back on main tangent from the original P. C. 4 a distance of R N sin T t - (p + o) cot i / = (1273.6 X .1018) - 5.992 (6.62 + 2) = 78 ft. If the tangent falls inside of the existing tangent an amount i (less than p) of 2 ft., then Hence / = 13 51', and T, = 5 50.6' as before. THE SIX-CHORD SPIRAL 51 The distance of the new P. C. 2 15' one-chord back from the original P. C. 4 will be (1273.6 X .1018) -8.233 (6.62 - 2) = 91.61 ft. If i be made larger, / will become smaller and the 4 30' curve will soon be too short to serve as the base for a 2 15' one-chord with the given value of p. A lighter curve must then be taken, say 4 20', 4 10', etc., until, when i becomes equal to p, the 4 curve is connected directly with the tangent by means of the proper length of 2 one- chord. When i exceeds p, a curve lighter than 4 must be taken. In all cases the total angle / of the terminal branch must be at least 1J times 7\ in order to make room for the six-chord, and at least 1.733 T 1 for the track parabola. In addition to the formulas above given, the following rule for shifting the P. C. C. of the last arc of any compound curve (without changing the degrees of curve) in order to offset the last tangent parallel to itself, is of constant use. Rule (Modified from Shunk's " Field Book/' page 101) : Divide the required offset by the difference of the radii, and call the quotient Q; call the nat. cosine of the total angle of the located last arc C. Then either Q + C or Q C will be the nat. cosine of the new last arc, and the difference between the angle whose cos = C and the angle whose cos = Q C gives the required angular shift of the P. C. C. 52 THE SIX-CHORD SPIRAL This angular shift is reduced to feet according to the degree of curve of the next-to-the-last branch, and on which it must be used. It is evident that 1st. To offset the last tangent out requires more of a lighter or kss of a sharper final arc. 2d. To offset the last tangent in requires more of a sharper or less of a lighter final arc. 3d. Less final arc requires more cosine , hence use Q + C. 4th. Afore final arc requires less cosine, hence use Q C. If C be greater than Q, use C - Q. Example. A3 compounds into a 5, which latter has a total angle of 30 22'. It is desired to throw the final tangent inward 34 ft. Here R-r= 1,910 - 1,146 = 764, and nat. cos 30 22' = .8628 = C. nat, cos 35 05' = .8183 = C - Q. In this case the tangent is to be thrown in, hence more of the sharper last arc (5) is required. Therefore use C - Q = nat. cos 35 05'. 35 05' - 30 22' = 4 43' Since more sharper last arc is required, the P. C. C. must be moved back along the 3 curve 4 43' = 157.22 ft. To throw the tangent out 34 ft., proceed as follows : THE SIX-CHORD SPIRAL 53 .0445 + .8628 = .9073 = nat. cos 24 52' . 30 22' - 24 52' = 5 30'. Here the P. C. C. must be advanced along the 3 produced 5.5 - 3 = 183.33 ft. This rule may be used to shift the P. C. C. of a one-chord spiral. In this case the difference of the radii = 2R M R M = the degree of the main curve, and the final arc of one-chord spiral is always the lighter. Hence, to offset the last tangent out requires more one-chord, and for this use Q C. To offset the last tangent in requires less one-chord, and for this use Q + C. Since, in this case, the original value of p is always known, it is preferable to add to or sub- tract from p (as the case may be) the required offset, thus forming a new p which is then used to determine the new T l by formula (1). COMPOUND CURVES. Space may be made at the P. C. C. for a spiral between the two members of a compound curve by employing one of the following methods: (1) By replacing part of the sharper curve with a still sharper one. (2) By replacing part of the lighter curve with a still lighter one. (3) By a combination of (1) and (2), preferably by adding as many minutes to the degree of the sharper curve as are subtracted from the degree of the lighter one. 54 THE SIX-CHORD SPIRAL By the third method, the center of the spiral practically falls on the original P. C. C., and the length of line is unchanged. First Method. When sharpening the sharper curve (of degree = D s ) to D N for a length L^, the original lighter curve D L must be produced for a distance L L , so that tangents at the end of L N of D s and the end of L L of D L are parallel to each other and p feet apart. (Inferiors: S Sharper; L = Lighter; N = New). Example. A 2 (D L ) and 8 (D s ) compound, and it is required to insert a spiral, p being taken at 3 ft., the 8 to be changed to an 8 30' (D N ) ; here L L = .2103 Stations = 21.03 ft. L N = 2.5237 Stations = 252.37 ft. Hence, the beginning of the new 8 30' (D N ) curve will be back along the 8 curve a distance of 252.37 + 21.03 = 273.4 ft. from the original P. C. C., and the resulting condition is 8 30' and 2 main curves parallel to each other at a point THE SIX-CHORD SPIRAL 55 0$,) 21.03 ft. from the original P. C. C. along the original 2 produced, and distant apart 3 ft.; re- quired to connect them with a oo q/y _i_ oo ^- = 5 15' one-chord spiral. This 5 15' curve starts from the 2 and ends on the 8 30' (or vice versa) at a distance from S 3 (Fig. 3) of dfa (29) where d = the difference between the degrees of the two final curves (8 30' - 2), and l t = length of one-chord in stations of 100 ft. Hence, J / 1= J- - = .7284 Stations. ' .o7 X o.o The total length of the 5 15' = 72.84 X 2 = 145.7 ft., and its total angle = 7 39'. (See also example under Fig. 3.) Second Method. When lightening the lighter curve D L for a length L N , the original sharper curve D s must be produced a distance L s so that tangents at the end of L N of D L and the end of L s of D s are parallel to each other and p feet apart. Example. A 2 (D z ) and an 8 (D s ) com- 56 THE SIX-CHORD SPIRAL pound, and it is required to insert a spiral, p being taken at 3 ft. and the 2 to be changed to a 1 30' (D N ). Here V = L15 (8 i 2 - ( ^ 3 15)= . 04423. L=21.03 ft. L N = 252.37 ft. Hence the beginning of the new 1 30' curve will be back along the 2 curve a distance of 252.37 + 21.03 = 273.4 ft. from the original P. C. C., and the resulting condition is 1 30' and 8 main curves parallel to each other at a point (S a ) 21.03 ft. from the original P. C. C. along the original 8 produced, and distant apart 3 ft. oo I -I o o/y Required to connect them with a - = 4 45' one-chord spiral. This 4 45' curve starts from the 8 and ends on the 1 30' (or vice versa) at a distance from S 3 (Fig. 3) = where d is the difference between the degrees of the two final curves (8 - 1 30'), and Z t = length of one-chord. Hence } I, = \ - - = .7284 Stations, or ,o7 X o.o 72.84 ft., as before. THE SIX-CHORD SPIRAL 57 The total length of the 4 45' one-chord = 72.84 X 2 = 145.7, and its total angle = 6 55'. (See also Example under Fig. 3.) Third Method. When both the sharper curve is sharpened and the lighter curve lightened by equal amounts, the method is as follows: Example. A 2 compounds with a 10. It is desired to replace 150 ft. of the 2 by a 1 30', and 150 ft. of the 10 by a 10 30', the increase and decrease being each 30'. Here the line will be thrown both in and out at the P. C. C. for a dis- tance of J p = .87 KL N \ (30) where K is the increase or decrease expressed in degrees and decimals, and L N the length of change of each curve in stations of 100 feet. In this case i p = .87 X .5 X 2.25 = .98 ft., hence p = .98 x 2 = 1.96ft. The resulting condition is 1 30' and 10 30' curves parallel to each other at the original P. C. C. and 1.96 ft. apart. Required to connect them .,, 10 30' + 1 30' u j i with a - = 6 one-chord spiral. t As before, this 6 curve starts from the 1 30' and ends on the 10 30' (or vice versa), at a distance from S 3 (Fig. 3) of P 58 THE SIX-CHORD SPIRAL where d is the difference between the degrees of the two final curves and / x = length of one-chord, or v/ ' 25 = " 5 stations ' Hence the 6 will have a total length of .5 X 2 = 100 ft., or 50 ft. each way from the original P. C. C. When the offset p is given and also the increase and decrease of the degree of curve, proceed as follows : Example. A 2 compounds with a 10, p is to be taken at 1.96 ft., and 1 30' and 10 30' curves used. Here where L N = length of 10 30' or 1 30' (to be used measured from original P. C. C.), p = principal off- set, and K = increase or decrease of degree of curve expressed in degrees and decimals. In this case L N = .758 \ ~- = 1.5 Stations = .5 150 ft., and l is found as above. These methods for spiraling compound curves, though approximate, give excellent results in practice. It is to be remembered that in such cases the spiral notes are not used to replace original records; hence there is no real need of absolute accuracy. THE SIX-CHORD SPIRAL 59 SPACE-SHIFTS PRESERVING THE ORIGINAL LENGTH OF LINE. As previously indicated, when p and R M are fixed, the one-chord, six-chord and track parabola all give the same length of line between common points on main tangent and main curve. Hence, for convenience and simplicity, the one- chord will be considered in the following computa- tions : Given two tangents intersecting at a fixed angle and joined by simple curves of various radii. Call the distance from P. C. to P. T. along the tangents, via the vertex, the tangent route, and the distance P. C. to P. T., via the curve, the curve route. Then the difference between the tangent and curve routes varies in direct proportion with the radius used. Further, any two curve routes are of equal length when they are equally less than the tangent route common to both. On these principles ^the following solutions are based : Example. Given a 4 curve for 70 30', to substitute a curve with spirals, retaining the same length of line. Assume a terminal angle (7^) of 3 24'. First, compute the elements of 3 24' of 2 one- 60 THE SIX-CHORD SPIRAL chord on each end of (70 30' - 6 48' = ) 63 42' of 4 main curve. By formula (1) p = vers 7\ X R M = .00176 X 1432.7 - 2.52 ft. By formula (10) the distance from the apex to P. C. of one-chord is (R M + p) tan J / + RM sin T l (see Fig. 1). R M + p = 1435.22 log = 3.156918 i / =35 15' log tan = 9.849254 GA = 1014.31 log = 3.006172 R M = 1432.7 log = 3.156151 T t = 3 24' log sin = 8.773101 GH = 84.97 log = 1.929252 AH = GA + GH = 1099.28 Hence tangent route = 2 (1014.31 + 84.97) = 2198.56 By the curve route there is 6 48' of 2 = 340 ft., and 63 42' of 4 = 1592.5 ft. : total = 1932.50 Tangent route less curve route = 266.06 Now, to preserve the original length of line, this difference must be reduced by shortening the radii until it equals the original difference between the tangent and curve routes. The latter is calculated thus: R M = 1432.7 log = 3.156151 i 7 = 35 15' log tan = 9.849254 EA = 1012.52 log = 3.005405 THE SIX-CHORD SPIRAL 61 1012.52 X 2 = 2025.04 Length 4 for 70 30' = 1762.50 Tangent route less orig- inal curve route = 262.54 Then 262.54 : 266.06 : :4: 4.054 (= 4 03.2'). Hence each terminal curve will consist of 3 24' of 2 01.6' curve. The new apex distance A H will be 266.06 : 262.54 :: 1099.28 : 1084.74. Similarly, the new p = 2.487. In working out these proportions use logarithms. For running in such a curve as a 4.054, the decimal vernier (formerly supplied on transits by Young & Sons, of Philadelphia) is a great con- venience. These instruments had one decimal vernier, the opposite one being of the usual sex- agesimal form. If, instead of the terminal angle 3 24', the final value of p = 2.487 be given, proceed as follows : 1st. Calculate the difference between the tan- gent and original curve routes; call this A. 2d. Multiply twice p by the tangent of half the whole intersection angle; call this product B. Then A :A + B ::D :D N where D = degree of original main curve, D N = degree of new main curve, \ D N = degree of new one-chord. 62 THE SIX-CHORD SPIRAL Example. Given a 4 curve for 70 30', to substitute a curve with spirals, retaining the same length of line and assuming p = 2.487. From the preceding example, tangent route less the original curve route = 262.54 = A. Also, 2p = 4.974 log = 0.696706 i / - 35 15' log tan = 9.849254 B = 3.52 log = 0.545960 A + B = 266.06 Then 262.54 : 266.06 :: 4 : 4.054 (= 4 03.2'). Other elements of the curves are found from formulas (1) and (10), as before. Similarly, spirals may be inserted at the ends and between the members of a compound curve, while preserving the original length of line. This is shown by the following example, which also serves as a general review. Given a compound curve, as follows (see Fig. 4) : Sta. 10 P. C. 4 R for 32 of angle = b. Sta. 18 P. C. C. 10 R for 50 of angle = c. Sta. 23 P. T. Required to insert spirals at P. C., P. C. C., and P. T. without changing length of line. Maximum speed on 10 curve = 31.6 miles per hour, which will also be taken on the 4. By Rule 2, length of one-chord spiral equals elevation in tenths of feet multiplied by speed. Hence, from Table III: THE SIX-CHORD SPIRAL 63 Length of one-chord for 10 curve = 31.6 X 5 = 158.0 ft. Length of one-chord for 4 curve = 31.6 X 2 = 63.2 ft. Length of one-chord between 4 and 10 = that for 10 - 4 = 6. Length of one-chord for 6 curve = 31.6 X 3 = 94.8 ft. In this example radius = 5730 -r- degree of curve, The values of p are as follows: Since p = R M X vers T lf Terminal ang. 7\for 10 = 7 54', p = 5.44 = P. Terminal ang. 7\ for 4 = 1 16', p = .34 = p. Terminal ang. 7\for 6 = 2 51', p = 1.18 = p 4 By formula (13): / 44 \ - -1.18 cos 32 -0.34 = 5.893. sm 82 By formula (14) : FW = (5.44 X tan 50) - 5.893 - .590. TL = ML X sin b = EH X sin b. EH =-^-- Pl = 7.283. cos C Hence 7.283 X sin 32 = 3.860 = TL. Next calculate the effect of the shift GV meas- ured along LT produced. 64 THE SIX-CHORD SPIRAL This will equal WN X cos (b + c) = 5.893 X cos 82 = 0.820 = TZ. To this add TL = 3.860. Hence shift LZ = 4.680. The notes of a new curve with one-chord spirals inserted, but retaining the original degrees of curve, would be as follows : P. C. 4 = point L (Fig. 4) = Station 10 +00. Less LZ 04.68 Point Z (Fig. 4) = 9 + 95.32 Deduct GH (Fig. 1) = 1432.5 X sin 1 16', [from (8)] 31.60 9 + 63.72 Then 9 + 63.72 P. C. 2 one-chord for 1 16' + 63.20 10 + 26.92 P. C. C. 4 main curve for 28 50' 7 + 20.83 17 + 47.75 P. C. C. 7 one-chord for 6 38' 94.80 18 + 42.55 P. C. C. 10 main curve for 37 22' 3 + 73.67 22 + 16.22 P. C. C. 5 one-chord for _ 7 54' .1 + 58 23 + 74.22 P. T. 82 00' Thus far the procedure has been the same as though the change was to be made in the first line THE SIX-CHORD SPIRAL 65 prior to construction. It remains to reduce the above to a similar figure, having the same length between common points as the original line. For this purpose first calculate the original apex dis- tances, A to L and F (Fig. 4), from the following formulas, A being the intersection of the main tan- gents through L and F produced (not shown in figure) : AF = fl. tan \I + , sin/ ~ fi ' ) f vers sin/ where R 2 is the larger radius = 1432.5 R l is the smaller radius = 573.0 / = the grand total angle = 82 / t = the R t total angle c 50 / 2 = the R 2 total angle b 32 AL is on the side of the lighter curve and A F on that of the sharper. Hence AL = 935.21 AF = 629.96 Original tangent route = 1565.17 Original curve route (2300 - 1000) = 1300.00 Difference = 265.17 The new curve route = 23 + 74.22 less 9 + 63.72 or 14 + 10.50 The new tangent route is obtained thus: 66 THE SIX-CHORD SPIRAL 1st. Distance from apex to P. C. 2 one-chord. Original tangent = AL = 935.21 LZ = 4.68 R 4 X sin 1 16' 31.60 971.49 2d. Distance from apex to P. T. 5. Original tangent AF = 629.96 FW .59 R 10 X sin 7 54' 78.75 709.30 Total via tangent route 1680.79 Total via new curve route 1410.50 Difference 270.29 Hence required ratio = 270.29 = 1.0193, or, inversely, 265.17 265.17 270.29 = .9810. Hence 4 becomes 4 X 1.0193 = 4 04.6', and 10 becomes 10.1930 = 10 11.6'. The new tangents will be 971.49 X .981 = 953.03 for AL. 709.30 X .981 = 695.82 for AF. Final tangent route = 1648.85 971.49 - 953.03 = 18.46 And the final alinement notes will read : 9 + 63.72 trial P. C. 2. 18.46 9 + 82.18 P. C. 2 02.3' for 1 16' 62. THE SIX-CHORD SPIRAL 67 10 + 44.18 P. C. C. 4 04.6' for 28 50' 7 + 07.13. 17 + 51.31 P. C. C. 7 08.1' for 6 38' 93. 18 + 44.31 P. C. C. 10 11.6' for 37 22' 3 + 66.57. 22 + 10.88 P. C. C. 5 05.8' for 7 54' 1 + 55. 82 00' 23 + 65.88 P. T. - 9 + 82.18 P.O. 13 + 83.70 = final curve route. 16 + 48.85 = final tangent route. 265.15 = final difference. 265.17 = original difference. The quantities added in the above tabulation are those in the preceding alinement table X .981. Having thus computed the required one-chords, the corresponding six-chord spirals or track para- bolas may be traced, as previously shown. If no spiral be required between the two mem- bers of the compound curve, make p = zero in formulas (11) to (19). The spiral at the P. C. C. may be subsequently run in by the methods of formulas (30) and (31). For the case of a simple curve terminating in unequal spirals use formulas (11) to (19), making 68 THE SIX-CHORD SPIRAL b + c = p t = zero. P and p = their assumed values. Calculations such as the preceding should be made in the office after a careful survey of the existing track has been made. On the plat of this survey the most suitable points for widening cuts and fills to make room for spirals, must be noted. Each division, at least, of the road should be treated by one experienced man. This will insure uniform and consistent spiraling. The whole should be formally approved by the highest available operating officer before being traced on the ground. UNIVERSITY OF CALIFORNIA LIBRARY