UC-NRLF aaa ms THE RAILWAY TRANSITION SPIRAL TAL.BOT cc ^D O THE Railway Transition Spiral ARTHUR N. TALBOT, C. E. Member American Society of Civil Engineers, Professor of Municipal and Sanitary Engineering University of Illinois FIFTH EDITION, REVISED ELEVEXTH THOUSAND NEW YORK: McGRAW-HILL BOOK COMPANY 1915 COPYRIGHT 1901 BY ARTHUR N. TALBOT PREFACE The railway transition spiral here presented is a flex- ible easement curve of general applicability and of com- paratively easy analysis. The conceptions and methods used are similar to those of ordinary circular railroad curves. The definition is based upon degree-of-curve, and degree-of-curve, central angle, and deflection angle may be calculated, and the curve may be located by transit and chain or by co-ordinates from tangent and circular curve. The field work is quite similar to that of circular curves. The spiral is easily applied to a variety of field problems and to a wide range of location and old track conditions. In the principal formulas, angles, co-ordinates, offsets, etc., are expressed in terms of the length or distance along the spiral. The use of series in the development of the prop- erties permits an estimate of the error involved in dis- carding negligible terms The principal easement curves in use give alignments which approach each other very closely, so that, for equal easements, it may be expected that the riding qualities will not differ sensibly. In general, ease of calculation, simplicity of field work, and general applicability and flexibility will determine the form of easement curve to be selected. To establish the underlying principles of an easement curve of any range requires considerable math- ematical analysis. The ordinary treatment of circular rail- way curves assumes previous knowledge of the geometrical properties of the circular curve, but the properties of the railway spiral must be deduced from the beginning. For- tunately the spiral is not complex, and its properties prove to be simple and general. A general treatment of such a curve i.as many advantages over approximate or special treatments. Approximate solutions may overlook important variables, and short methods may be limited to short and inefficient easements. The range of conditions of railway curves is so wide that it is best to develop methods of fairly general applicability, and these may then be simplified in meeting individual conditions. 380514 The treatment herein given has been quite widely used on the railroads of the United States and many engineers have commended its simplicity, convenience, and flexi- bility. The methods and principles are readily taken up by instrument men, and the field work has proved little more difficult than that for circular curves. The use of a regular rate of transition per 100 feet of spiral is advan- tageous, and the tables are in convenient form. The treatment of the railway transition spiral was published in Technograph No. 5, 1890-91, and was pub- lished in field-book size in 1899. Careful attention has been given in this revision to illustrative examples and explanations. The tables have been extended and a treat- ment of the Uniform Chord Length Method and of Street Railway Spirals added. For much of the latter, acknowl- edgment is made to Mr. A. L. Grandy. The writer is indebted to Messrs. J. K. Barker, Alfred L. Kuehn, and many others for valuable assistance in the preparation of tables and text. URBANA, ILLINOIS, A. N. T. November 11, 1901. CONTENTS Nomenclature 1 Use. Definition. Notation. Measurement of length. Theory - - - 6 Intersection angle. Co-ordinate x and y. Spiral deflection angle. Table of corrections. Deflection angle at point of spiral. Ordinates. Offset. Ab- scissa of P. C. Tangent-distance. External-dis- tance. Long chord. Spiral tangent-distances. Middle ordinate. Summary of Principles - 17 Principal formulas. Angles and deflection angles. Angles from tangent. Angles from chord. Diverg- ence from osculating circle. Co-ordinates. Offset. Other distances. Description and Use of the Tables - - 24 Tables for transition spiral. Accuracy. Interpola- tion. General use of Table IV. Corrections for calculations. Other tables. Choice of a and Length of Spiral - - - - - 28 Effect of speed. Attainment of superelevation. Amount of superelevation. Minimum spiral. Se- lection of spiral. Location of P. S., P. C. C., and P. C. - - - - - 33 Laying out the Spiral by Co-ordinates - - - - 35 From tangent, tangent and curve, and spiral tan- gents. Location by Transit and Deflection Angles - - - 38 Transit at P. S. Transit on spiral. Intermediate de- flection angles. Transit at P. C. C. Transit notes. Application to Existing Curves 44 To replace the entire curve Two methods. To re- place part of the curve. To re-align and com- pound. Methods of trackmen. Compound Curves - 54 General method. To insert in old track. Miscellaneous Problems - 60 To change tangent between curves of opposite direction. To change tangent between curves of same direction. Uniform Chord Length Method - - 64 Formulas. Tables. Fractional chord lengths. Use of method. Street Railway Spirals - 72 Theory. Tables. Laying out. Arc excess. Curv- ing rails. Double track. Conclusion 79 Explanation of Tables 84 Tables I-XI. Transition Spirals - - - 86-97 Table XII. Factors for Ordinates 97 Table XIII. Unit Spiral Deflection Angles - 98 Table XIV. Coefficients for Deflection Angles - - 99 Tables XV-XIX. Street Railway Spirals - - 100-101 Table XX. Offsets for Spirals 102 Railway Transition Spiral NOMENCLATURE 1 . A transition curve, or easement curve, as it is some- times called, is a curve of varying radius used to connect circular curves with tangents for the purpose of avoiding the shock and disagreeable lurch of trains due to an in- stant change in the relative position of cars, trucks, and draw-bars and also to a sudden change from level to inclined track. With the spiral the superelevation of the outer rail may be made to correspond to the curvature at all points around the transition curve, and the trucks, springs, draw-bars, and car body will gradually attain their final position for the main curve. The primary object of the transition curve, then, is to effect smooth riding when the train is entering or leaving a curve. The generally accepted requirement for a proper tran- sition curve is that the degree-of-cruve shall increase gradually and uniformly from the point of tangent until the degree of the main curve is reached, allowing the superelevatic' ; to increase uniformly from zero at the tangent to tVie full amount at the connection with the main curve and yet to have at every point the appropriate superelevation for the curvature. In addition to this, an acceptable transition curve must be so simple that the field work may be easily and rapidly done, and should be so flexible that it may be adjusted to meet the varied requirements of problems in location and construction. No attempt will here be made to show the necessity or the utility of transition curves. The principles and some of the applications of one of the best of these curves, the railway transition spiral, will be considered. 2. Definition. The Transition Spiral is a curve whose degree-of-curve increases directly as the distance along the curve from the point of spiral. Thus, if the spiral is to change at the rate of 10 per 100 feet, at 10 feet from the beginning of the spiral the curvature will be the same as that of a 1 curve; at 25 feet, as of a 230' curve; at 60 feet, as of a 6 curve. Likewise, at 60 feet, the spiral may be compounded with a 6 curve; at 80 feet, with an 8 curve, etc. This curve fulfills the requirements for a transition curve. Its curvature increases as the distance measured around the curve. The formulas for its use are compar- atively simple and easy. The field work and the com- putations necessary in laying it out and in connecting it with circular curves are neither long nor complicated, ar?H are similar to those for simple circular curves. The curve is extremely flexible, and may easily be adapted to the requirements of varied problems. The rate of change of degree-of-curve may be made any desirable amount according to the curve used, the maximum speed of trains, or the requirements of the ground. As the derivation of the formulas is somewhat long, their demonstration will be given first. The explanation and application of these formulas to the field work and to the computations will be given separately, a knowledge of the demonstration not being essential to the application. 3. In Fig. 1, DLH is the circular curve and AP the prolongation of the initial tangent wh : ch are to be con- nected by the transition spiral. D is the point where the completed circular curve gives a tangent DN parallel to the tangent AP, and will be called the P. C. of the cir- cular curve. AEL is the transition spiral connecting the NOMENCLATURE 3 initial tangent AP with the main or circular curve LH. A is the beginning of the spiral and will be known as P.S., point of spiral. L is the beginning of the circular curve LH, and will be called P.C.C., point of circular curve. AP will be used as the axis of X, and A as the origin of co-ordinates;. BD is the offset between the tangent AB of a circular curve and spiral, and the parallel tangent DN of an unspiraled curve. The degree-of-curve of the spiral at any point is the same as the degree of a simple curve having the same radius of curvature as the spiral has at that point. The radius of the spiral changes from infinity at the P.S. to that of the main curve at the P.C.C. The spiral and a simple curve of the same degree will be tangent to each other at any given point; i. e., they will have a common tangent. 4. The following notation will be used: P.S. = Point of Spiral. (A, Fig. 1.) P.C.C. = point where spiral compounds with circular, curve. (L, Fig. 1.) P.C.=beginning of offsetted circular curve. (D, Fig. 1.) R = radius of curvature of the spiral at any point. D = Degree-of-curve of the spiral at any point; some- times called D at the end of spiral. Generally D^ is made the same as D Q , the degree of the main curve. a = rate of change of the degree-of-curve of the spiral per 100 ft. of the length. It is equal to the degree-of- curve of the spiral at 100 ft. from the P.S. j = length in feet from the P.S. along the curve to any point on the spiral. L = number of 100- ft. stations from the P.S. along the curve to any point on the spiral; in other words the dis- tance to any point measured in units (or stations) of 100 ft. For the whole spiral (to P.C.C.) it is sometimes called L t . 4- NOMENCLATURE /= Total central angle of the whole curve (intersec- tion angle), or twice BCH of Fig. 1, H being the middle of the circular arc. A = angle showing the change of direction of the spiral at any point, and is the angle between the initial tangent and the tangent to the spiral at the given point. For the whole spiral it is equal to PTL and may be called A x . The lastter is also equal to DCL. = spiral deflection angle at the P.S. from the initial tangent to any point on the 'Spiral. For the point L (Fig. 1) it is BAL. <$ deflection angle at any point on the spiral, between the tangent at that point and a chord to any other point. At L, for the point A, <1> is TLA. x = abscissa of any point en the spiral, referred to the P.S. as the origin and the initial tangent as the axis of X. For the point L, .ar = AM. 3/z^ordinate of the same point, measured at right angles to the above axis. For the point L, y = ML. t = abscissa of the P.C. of the main curve produced backward; i.e., of a simple curve without the spiral. For P.C. at D, t = AB. o = offset between the initial tangent and the parallel tangent from the main curve produced backward, or it is the ordinate of the P.C. of the produced main curve. If D is the P.C., BD is o. It is also the radial distance between the concentric circles LH and BK. T = tangent-distance for spiral and main curve = dis- tance from A to the intersection of tangents. E = external-distance for spiral and main curve. C = long chord AL of the transition spiral. u = distance along initial tangent from P.S. to inter- section with spiral tangent = AT for point L. v = length of spiral tangent to intersection with initial tangent = TL for point L. NOMENCLATURE 5 5. The length of the spiral is to be measured along chords around the curve in the same way that simple curves are usually measured, using any length of chord up to a limit which depends upon the degree-of-curve of the spiral. The best railroad practice, in the writer's opinion, considers circular curves up to a 7 curve as measured with 100-ft. chords, from 7 to 14 as measured with 50-ft. chords, and from 14 upwards as measured with 25-ft. chords; that is to say, a 7 curve is one in which two 50-ft. chords together subtend 7 of central angle, a 14 curve one in which four 25-ft. chords to- gether subtend 14 of central angle. The advantages FIG. 1 of this method are two-fold, the length of the curve as measured along the chords^ more nearly approximates the actual length of the curve, and the radius of the curve is almost exactly inversely proportional to the degree-of-curve. The latter consideration is an important one, simplifying many formulas. With this definition of degree-of-curve, the formula R = 5 ^ will give no error greater than 1 in 2,500. For a 10 curve the error in the radius is .15 feet, and for a 16 curve .06 feet. This approximate value of R will give a resulting error in the length of the spiral, for the ordinary limits of spirals, t> THEORY of less than -^Q-Q of the length, and will not reach 0,1 ft. The resulting error in alignment is -oVo^ an d will not reach 0.01 ft. The difference between the length of the curve and that of these chords is less than 1 in 7000. For spirals measured with lengths of chords as here specified, or shorter, the error either in alignment or dis- tance will be well ( within the limits of accuracy of the field work, and hence the relation R = 5 ^ will be considered true. THEORY 6. Intersection angle A. From the definition of the transition spiral, we have, remembering that the value of a as defined above requires the length of curve to be measured in 100-ft. units (stations) instead of feet, For the P.C.C. this becomes D 1= =aL r From the calculus the radius of curvature R = . Substituting the expression R = ^ and solving, as ds 573000 * as 2 a L 2 Integrating, A = - 1146000 114.6 ' Changing A from circular measure to degrees, A = JaL 2 .................................... (2) which is the intrinsic equation of the Transition Spiral. For the P.C.C. this becomes ^ = J a L*. Since from (i) a = -=, we also have (3) CO-ORDINATES 7 From these equations it will be seen that (a) the change of direction of the spiral varies as the square of the length instead of as the first power of the length as in the simple circular curve, and (b) the transition spiral for any angle A will be twice as long as a simple circular curve. 7. Co-ordinates a? and y. To find the co-ordinates, x and y, of any point on the spiral, we have by the ca'i- culus dy=ds sin A and dx=ds cos A. Expanding the sine and cosine into infinite series, substituting for ds its value in terms of dA, and integrating, we have X = As A here is measured in circular measure and is only J when the angle is 28. 65, these series are rapidly con- verging, especially for smaller angles. Changing the angle A from circular measure to degrees, substituting for A the value given in (2), and dropping the small terms, 3; = .291 a I s -.00000158 a 3 L 7 .. (6) For values of A less than 15 the last term may be dropped, and up to 25 the term will be small. D U may also be written in place of a L 3 . For all except extreme lengths, the last term may be dropped. Using y = 291aL s , it is seen that y varies as the cube of the distance of the point from the P.S. Likewise changing A from circular, measnre to degrees etc., equation (5) becomes x = 100 L .000762 a 2 L 5 + .0000000027 a 4 L 9 (7) Or x = 100 L .000762 D 2 U (8) 8 THEORY The second term in second member of equation (7) or (8) may be used as a correction to be subtracted from the length of the curve in feet. The last term in equation (7) can be omitted, except for extreme lengths. 8. Spiral deflection angle 0. It is desired to find the deflection angle for any point on the spiral, as BAL for the point L (Fig. 1). To show that this is nearly JA, divide equation (4) by equation (5). tan 9 = -J- A + ^ A* + Tir f ^-5- J 5 , etc. But from the tangent series for JA, tan i- A = i A + -J r J 3 + ^3- J 5 , etc. Subtracting one from the other, we get a series which is rapidly decreas- ing when A is less than 40. Investigating this differ- ence, remembering that A is in circular measure, it is found that the error of calling the two equations equal is less than 1' for A = 25 and decrease 1 rapidly below this. As A will rarely reach 25, and aL \he discrepancy is only a small fraction of a minute for any angle ordi- narily used, and as the resultant error of direction will be corrected at the P.C.C. when A is turned off, we may ordinarily disregard this and write * (9) CL where is in degrees. From equation (9) it is seen that the spiral deflection angles to two points on the spiral will be to each other as the square of the distances to the points. 9. The error in equation (9) is dependent upon the value of A or and hence may be expressed independently of the length of spiral and rate of transition. For A between 20 and 40, the number of minutes correction to be subtracted from J A or J a L 2 to give is .000053 A 3 where A is in degrees. The following table gives the deductions for various angles, and for other values inter- polations may be made: SPIRAL TANGENT 9 Correction in minutes to be subtracted from J A or a L 2 to give more precise values of 6. A 12 15 18 Cor. 0.1 0.2 0.3 A 21 24 27 Cor. 0.5 0.7 1.0 A 30 33 36 Cor. 1.4 1.9 2.4 Thus when A is 18, the real value of will be 6 0'.3 = 5 59'.7. For a value of near 6 the same correc- tion may be made. It will be seen that for the spiral de- W A FIG. 2 flection angles ordinarily used the correction may be neglected without material error. For the terminal-point of the spiral, the P.C.C., the value of 0j may be obtained from equation (9). In the extreme cases, where a further term is needed, the cor- rection may easily be made from the above table. 10. Spiral tangent. To find the tangent at any point of the spiral, L, lay off a deflection angle from LA equal to A 0. When A is not over 20, A, or 20 may be used. This since FLT = PTL = A, and FLA = PAL = 0. This is true for any point. 10 THEORY For the terminal point of the spiral, P.C.C., this be* comes A B 1 which is generally expressed with sufficient precision by A r 11. Deflection angle at point on spiral. The de- flection angle from the tangent at any point on the spiral to locate a second point may be found as follows : In Fig. 2, let U be the distance from the P.S. to R, and L the distance from the P.S. to any other point on the spiral, as K. Let FRN be the tangent at R, and RFM = A' its angle with the initial tangent, and ff the corresponding spiral deflection angle, RAF. Let KTM=A be angle of tangent at K with initial tangent, equal to total change of direction of the spiral up to that point, ff and 9 are the deflection angles at the P.S. for R and K respectively. = <= required deflection angle. To show that angle <1>+A' is almost exactly the same 3. Jl as the angle i^4 I - r"0r (J+J*J'*+ A'), the fol- - _ lowing somewhat long and tedious operation may be gone through. It is thought not necessary to give it in detail here. X X\ Substitute for the co-ordinates in the above equation their values from equations (4) and (5), and also develop fo J'"* tan i -^ - into a series, and subtract the latter A^- A'% from the former. An expression for the difference will be found, which amounts to but a small fraction of a minute for any value of A up to 35. Hence we may write DEFLECTION ANGLE 11 By substitutng for A and A' their values in terms of L and L' and reducing, the following value for < is found: (L L')zhja (L L') 2 ........ ...(10) Also, A' P3> = + \D'L ..... ....... . ........ (11) And &'& = ff + 6 + \D'L .............. ,.'. , ...... (12) Even for very large angles these equations are quite accurate if the exact value of is used. In equation (10) the last term ^d(L L') 2 should receive the same cor- rection as an equal value of 0. Of course, for any angle ordinarily used no correction need be made. See cor- rection for 0, page 9. 12. In equation (10) it will be noticed that the first term (J a L' (L L') ) is equal to the deflection angle for a simple circular curve of the same degree as the spiral at the point R (i.e., a I/) and of a length equal to the distance between the two points ; while the second term (%a(L L) 2 ) is equal to the spiral deflection angle at the P.S. from the initial tangent for an equal length of spiral (LU). If the point to be located had been chosen on the side of R nearer to the P.S., the two terms of equation (10) would have opposite algebraic signs, and the difference of the two quantities would be used. To show that the arithmetical difference of the two terms is to be used for a point nearer the P.S. when the distance (L L') is used without regard for the algebraic sign, equation (10) has been written with the plus and minus sign. 13. The spiral then deflects from a circle of the same degree-of-curve at the same rate that the spiral deflects from the initial tangent at the beginning. D'RH, in Fig. 2, represents the circular curve tangent to spiral at R, the two having the same radius at that point and both 12 THEORY being tangent to FRN. The deflection angles between points on the spiral and on the circle RH, and also be- tween the spiral and RD' are the same as the spiral deflection angle for an equal length of spiral from A. In the same way at K, RKT = SKT SKR, the latter angle being equal to the deflection from initial tangent at A for a length of spiral equal to KR. 14. Equation (11) shows that the angle at any point between the chord joining this point with the P.S. and a chord to any other point (the angle, Fig. 2, between AR produced and RK if the point K is to be located from R) is equal to the spiral deflection angle at the P.S., 0, for the point to be located (KAM) plus one third of the deflection angle for a circular curve of the same degree as that of the spiral at the vertex of the angle, R, and of the length of the spiral from P.S. to the point K. This is true whether the point to be located is nearer the P.S., or farther, than the point used as the vertex of the angle. It may also be readily shown from (2) that the dif- ference in direction of the two tangents, A A', is the central angle for this simple curve plus the spiral angle, both for a length equal to the distance between the two points. 15. Equation (12) gives the value of the deflection angle from a line parallel to the initial tangent, the spiral deflection angle ff for the point R being added to the values in equation (11). 16. Ordinates from osculating circle. It may also be shown that the offset distance between a point on the spiral and one on the osculating circle is the same as the ordinate y from the initial tangent at a point the same distance from the P.S. as the former point is from the point of osculation. These ordinates may be measured is, a direction normal to the circular curve. OFFSET 13 17. Offset o. From Fig. 1, BD = BF DF = BF CDversDCL. But o = BD, BF = ;y for the end of spiral, DCL = A foi the whole spiral, and CD = R. Hence, o = y R vers A, Substituting for y, R and A their values in terms of the length of the whole spiral, applying the versed sine series, and reducing we have for o in feet o = .0727 a L* = .0727 >, L*. (13) where I> 1 and L i refer to the whole length of the spiral. The other terms of the series are so small that they may bo dropped when A is less than 30. The next term is .0000002 a 3 .L 7 . It will be seen that o is approximately one fourth of the ordinate of the P.C.C., which, of course, should be true if E, the middle point of the spiral, is opposite D, the P.C. 18. Offset given. From (13) and (3) we have L = 3.709 J-^- (14) A = 1.857 i/ o D (15) y3 - (16) 3f, If and -jV ma 7 be used for these co-efficients with advantage. 19. Abscissa of P.C. , t. From Fig. 1, J = AB = BM = ,r FL = jr R sin A. Expanding and reducing, 1 = 50 L, .000127 a 2 L* ) or [ ................... (17) t = 50 L, .000127 D, 2 L, 3 ) It should be noted that the full length of the spiral is used in the formula. The last term may be used as a correction to be subtracted from the half length of the spiral. It is easily tabulated for the principal spirals, and corrections for other spirals may be found by multi- 14 THEORY plying- the value with a = 1 for the given length of spiral by the square of the a used. 20. A comparison of t with the abscissa found by substituting J L in equation (8) shows that BD cuts the spiral at a point only .0001 a 3 L 8 feet from the middle point of the spiral. This is f of the correction used in equation (1Y) for finding t from J L r For our purpose we may say that BD bisects the spiral. It also follows that the spiral bisects the line BD, since BE = y. This is subject to slight error for large angles. The length of the spiral from the P.S. to BD, therefore, exceeds t by one fifth of the t correction, and the remain- der of the spiral exceeds the length of the circular curve from the P.C. to the P.C.C. by four fifths of the t correc- tion. The entire length of the spiral exceeds the distance measured on t (AB, Fig. 1) plus the distance measured around the circular curve (DL, Fig. 1) by the t correc- tion given in equation (IT). 2 1 . Tangent-distance T. To find T, consider in Fig. 1 that AB intersects CH, H being the middle of the cir- cular curve, at some point P outside the diagram. Then T = AP = AB + BP. BP = BC tan BCH. Hence T = t+ (R + o) tan J / (18) t and o tan -J / may be computed separately and added to the T found from an ordinary table of tangent-dis- tances. 22. Equation (18) gives T for the same transition spiral at each end of the main curve. It may be desirable to make one spiral different from the other. To find an expression for the tangent-distance for this case proceed as follows : In Fig. 3, let RS = HD = 2 , BD == o lt AB = t v RT = f 2 , AE = T 1 , TE = T 2 , R = radius of main curve DLKS, R + 2 = radius of HR, and / = angle PER. TANGENT DISTANCE 15 Then 7\ = f, + HC PE, and T i = t 1 +(R + o s ) tan i/ (o o s ) cot/ (19) Similarly, T s = t a + (R + o s ) tan J/ + (^ 2 ) cosec /. When / is more than 90, the last term of (19) becomes essentially positive. 23. External-distance E In Fig. 1, E = HP. HP = KP + HK. Hence E=(R exsec (20) 24. Long chord C. In Fig. 1, C = AL. ML = AL sin MAL, or C=- . Putting this in terms of the length sin 6' of the curve, C = 100 L (.000338 a 2 L 5 or .000338 > 2 L 3 (21) in which C is in feet and L in stations. It will be seen that the last or correction term is four ninths of the cor- rection for x as given in equation (7). When the cor- 16 THEORY rection term for x is tabulated or otherwise known, the length of the long chord may conveniently be calculated by subtracting four ninths of this x correction term from the whole length of the spiral. The length of a chord which does not go through the P.S. may be calculated from the triangle formed by it and the two long chords drawn from its ends to the P.S. For all except extreme lengths, a chord may be taken as having the same length as the chord of an equal circular arc whose radius is the same as that at the middle point of the given spiral arc. 25. Spiral tangent-distances In. Fig. 1, w=,AT = AM MT. As MT y cot A or v cos A, u = x y cot A. u = x v cos A . }(22) Also v = TL =-^- Expanding sin A into series, sub- sin A. stituting the value of y from equation (6), and reducing, v = ^ L + . 000244 a 2 L 5 .. ..(23) sin A 3 The last term is almost exactly one third the correspond- ing term in equation (7), and hence v may be found by adding one third of the correction term used for deter- mining x to one third of the length of the spiral in feet. 26. Middle ordinate. The middle ordinate for any arc of the spiral is equal to the middle ordinate for an equal length of circular curve of the same degree-of- curve as the spiral at the middle point of the arc consid- ered. This degree-of-curve is the mean of the D's at the end of the given arc. This is an approximate formula which is true whether one end of the chord is at the P.S. or not. SUMMARY OF PRINCIPLES 17 The ordinate from any other point along a chord may be found as follows: Since the spiral diverges from the osculating circle at the middle point of the arc at the same rate as from the initial tangent, the amount of this divergence may be calculated by the method given on page 14 and added to or subtracted from the ordinate for the osculating circular curve. For a point nearer the P.S. than the center of the spiral arc, the divergence will be added to the ordinate of the circular arc, and for one farther away it will be subtracted from the ordinate. As before, the degree of the osculating circular curve is the mean of the D's at the end of the given arc. Other properties may be found by ordinary trigono- metric operations. SUMMARY OF PRINCIPLES 27. For convenience of reference the principal for- mulas will be repeated here. D = aL and L = ....................... I CD D l = aL 1 for whole spiral ................ ... j (2) A = J^/ = J-^^i for whole spiral ........... j y = .291 aL 3 etc ........................... (6) jtr = 100 L .000762 a 2 L 5 + etc ................ (7) " \ (9) = JAj. = J0-^i 2 for whole spiral . 18 SUMMARY OF PRINCIPLES 3> = iaL' (L L')==ja (L L') 2 ........... (10) A' ^=t:^ = r (12) = .0727 aL 3 = .0727 I\ L^ 1 .................. (13) L 1 = 3.709- ........................... (14) = 1.857 j/^ ..... . . . . ................... (15) = .269 .. ..(16) \ o t = 50L X .000127 c?L? ............. . ........ (17) T=t+ (R + o) tan J/ ........ ................ (18) E=(R + o) exsec J/ + ..................... (20) C = 100 L .00034 a 2 L 5 ...................... (21) u = x y cot A ,(22) = ^r v cos A . ........................... j z; = _J^- = ^5 L + .000244 a 2 L 5 . . . '. (23) An inspection of the formulas and demonstrations will show the following properties of the transition spiral: 28. Degree-of-curve The degree-of-curve at any point on the spiral equals the degree at 100 feet from the P.S. multiplied by the distance along the spiral from the P.S. to the point (Eq. 1). This distance must be ex- pressed in units of 100 feet (stations). Thus, if a = 2, at 100 feet from the P.S. the spiral will be a 2 curve; at 35 feet (L = .25) a 030' curve; at 450 feet, (L = 4.5) a 9 curve. ^~ is the number of feet of spiral in which D changes one degree. Thus, for a = 2 the spiral increases SPIRAL DEFLECTION ANGLE 19 its degree of curve one degree for each i-|-^=50 feet; for a = one degree for each -!-. = 150 feet. At the terminal point, the P.C.C., where the spiral con- nects with the main curve, D will sometimes be repre- sented by .Dj, and this should generally equal the degree of the circular curve D . The total length of the spiral will be -. If a = 2, a 6 curve would require a spiral 3 stations (300 feet) long. 29. Angle A. The angle A between the initial tan- gent and the tangent at any point on the spiral (the change of direction corresponding to central angle of circular curves) (LTP, Fig. 1, page 5) in degrees equals (Eq. 2) : (a) One half of a times the square of the distance in 100- ft. stations from the P.S. to the point; thus if a = 2, for 300 ft. from P.S., L = 3, and A = i x 2 x 3 2 = 9. Or (&) One half of the product of this distance L by the degree-of-curve of the spiral at the given point; thus at 300 ft. with a = 2, D = 6, and A = i x 3 x 6 = 9. Or (c) One half of the square of degree-of-curve at the point divided by a ; thus at 300 ft. with a = 2, A == J x | f = 9. For the same angle, then,, the spiral is twice as long is a circular curve, and for the same length the angle is one half that for a circular curve whose D is the same as that at the end o^ the spiral. 30, Spiral deflection angle 0.- The spiral deflection angle at the P.S. from the initial tangent to any point on the spiral, as PAL in Fig. 1, is J A, or J a L 2 . Thus, for a point 300 ft. from the P.S. (L = 3), if a = 2, = 4 x2x 3 2 = 3. If the result is wanted in minutes, since Jx 20 SUMMARY OF PRINCIPLES 60 = 10, use 10 instead of J. For 105.4 ft. with a = 2, = 10 x 2 x (1.054) 2 = 22'. is also one third of the deflec- tion angle for a simple curve of the same degree as the spiral at the given point. Thus, as above, the deflection angle for 300 ft. of 6 curve is 9 and = Jx9 = 3. These values are subject to slight corrections for A larger than 15 or 20 as explained in the derivation of the formula on page 9. 31. Tangent at point on spiral. The deflection angle at any point on the spiral between the tangent at this point and the chord to the P.S. (TLA in Fig. 1) is A 6. This enables the tangent to be found. For A less than 15, the value A or 2 is sufficiently accurate. Thus, for the preceding example, with a = 2, for the point 300 ft. from the P.S., this angle is 20 = 6. 32. Deflection angle at point on spiral. For de- flection angles from a point on the spiral to other points on the spiral, the principle that the spiral diverges from the osculating circle (circular curve of same degree) at the same rate that the spiral deflects . from the initial tangent is of service. The angles may be treated in three ways, as follows: 33. Angles from tangent. By equation (10) the deflection angle between the tangent at a transit point on the spiral and the chord to any other point on the spiral (as CBH, Fig. 4) is the sum or difference of two angles: (1) the deflection angle for a circular curve of the same degree as the spiral at the transit point for a length equal to the distance between the two points, and (2) the spiral deflection angle for a lengh of spiral equal to the dis- tance between the two points. The latter angle. is plus if the desired point is farther from the P.S., and minus if nearer, than the point from which the deflections are made. ANGLES FROM CHORD 21 Thus, if a = 2 and the transit be at B (Fig. 4), 250 ft. from the P.S., the degree-of-curve at the transit point will be 5, and the deflection angle CBH to set a point 150 ft. ahead will be the sum of 3 45', (J of 150 ft. of 5 curve) and 45', (the spiral deflection angle for 150 feet, 10x2x1.5) or 430'. For D, 150 ft. back, it would be 3 45' 45' = 30'. 34. Angles from chord. Likewise by equation (11) the angle CBE, Fig. 4, (deflection angle from chord to P.S.,) may be calculated by adding the spiral deflection angle for the point C (GAC) to J the product of the. degree-of-curve at B by the number of stations from the P.S. to C. For a = 2 and the transit at B, 250 ft. from FIG. 4 the P.S., the degree-of-curve at the transit point is 5, and the angle CBE to locate the point C 150 ft. ahead and 400 ft. from the P.S., will be (J x 2 x4 2 = 520') + (J x 5 x4=r3 20') .-=8 40'. For the point D 100 ft. from the P.S, the angle DBA will be ( J x 2 x I 2 == 20') + ( J x 5 x 1 = 50 / )=110'. This method is applicable whether the point to be located is nearer to, or farther from, the P.S. than the transit point. It permits the calculation of the spiral deflection angles at P.S. for the whole spiral and the determination of the angles between the chords in question by adding ' to these spiral deflection angles 22 SUMMARY OF PRINCIPLES the angles J D' L, where D' is the degree-of curve at the transit point and L is the distance from P.S. to the point to be located. 35. Angles with initial tangent. Equation (12) gives the deflection angles from a line parallel with the initial tangent. The results are the same as if the spiral deflection angle ff for the transit point were added to those from the chord found in the preceding paragraph. 36. Divergence from osculating circle. The spiral diverges from its osculating circle (circular curve of the same degree) at any point at the same rate that the spiral deflects from the initial tangent, and the distance between the circle and spiral is the same as the y for an equal length of spiral. This enables the spiral to be located by offsets measured from the circular curve. By this method half of the spiral may be located from the initial tangent and half from the produced circular curve, the offsets for the two being the same for the same distances from the P.S. and the P.C.C. respectively. See Fig. 5. 37. Abscissa fie. The distance in feet along the ini- tial tangent to the perpendicular to a point on the spiral is equal to the length along the spiral in feet less the quantity .000762 a 2 L 5 , where L is the length along the spiral expressed in units of 100 ft. A convenient way to find x is to have this quantity tabulated for given spirals as a correction, or it may easily be found from tabulated values of such a correction for a = 1 by multiplying by a 2 . For extreme lengths another term may be needed. As an illustration, with a = 1, for 200 ft. the correction to be subtracted from 200 ft. to find x is .000762 x 32 = .02 ft., a small quantity 38. Ordinate y. The ordinate y (perpendicular dis- tance from the initial tangent to a point on the curve) OFFSET 23 in feet equals .291 times the product of a by the cube of the distance along the spiral from the P.S. to the point expressed in units of 100 ft. (stations). It therefore varies as the cube of the distance from P. S. Knowing y for one point, the y for a second point may be computed from it by this relation. D L 2 may be substituted for a L 3 . For extreme lengths, a third term may have to be considered. As an illustration, with a = 1 for 200 ft (L = 2) , y = .291 x 8 = 2.33 ft. For 100 ft, y is one eighth as great; for 400 ft. y may be used as eight times as great, though the use of the next term of the series would change this somewhat. 39. Offset o. The offset o between the initial tan- gent and the parallel tangent from the main curve pro- duced backward, (BD, Fig. 1), in feet equals .0727 times the product of a by the cube of the length of the whole spiral in stations, or .0727 times the square of the length of spiral and the degree of main curve. This ordinate is approximately one fourth of the ordinate y of the end of spiral. The spiral bisects the offset at a point half-way between the P.S. and the P.C.C. (Eq. 11.) BE = ED. AE = EL. (Fig. 1.) The slight error in this is discussed in the derivation of the formulas (page 14). The value of o may best be discussed by means of one of the tables. 40. Calculation from known values. When the length of the spiral is not so great that a second or cor- rection term is needed for the values of 0, y, A, etc., it is seen from equations (9), (6), (2), etc., that these functions vary as the square and cube of the distance L and may be calculated from any known value. Thus r2 if for 400 ft. is 240', for 300 ft, #= 2 #i ={ - \ X 1 [400J (2 400=1 30'. If y for 400 ft. is 18.59, for 300 ft. 24 THE TABLES T 3 fsool 3 y =-7-5 j/i, = \ - - \ x 18.59 = 7.85. The deflection angle i [400J varies as the square of the distance and the ordinate as the cube of the distance from the P.S. 41. t and C. The distance t from the P.S. to this offset (AB ; Fig. 1) is found by subtracting the correction .000127 a 2 L 5 from the half length of the curve in feet. (Eq. 17.) Generally this correction term is quite small. As stated on page 13 this term may be tabulated, and it may also be obtained for a given length of spiral by mul- tiplying tabulated values for a = 1 by the square of the a of the given spiral. For this use see pages 25 and 26. The long chord C is found by subtracting the correc- tion, .000338 a 2 L 5 , from the length of the curve in feet. (Eq. 21.) This correction may be found by multiplying the x correction for the same length of spiral by four ninths 42, u and v. The spiral tangent-distances u and v (AT and TL, Fig. 1) are found by equations (22) and (23). v can be found most easily by taking one third of the tabulated values of the x correction and adding this to one third of the length of the spiral in feet. THE TABLES 43. The computations may be shortened by the use the tables. Tables I-XI gives the values of the principal parts of the transition spiral for the following values of a: J, , f, 1, 1J, If, 2, 2J, 3J, 5, and 10. The column headed "Length" is the distance in feet along the spiral from the P.S. to any point on the spiral, and is equal to 100 times the L of the formulas. The column headed "x COR." INTERPOLATION 25 gives the correction to be subtracted from this distance in feet along the spiral to obtain x, and that headed "t COR." gives the correction to be subtracted from the half length of the spiral in feet to obtain t. Both t COR. and o are to be taken from the line for the full length of the spiral. For example, by Table IV, with a = l, to con- nect with a 5 curve, the length of spiral is 500 ft. and L = 5; the change of direction A ; is 12 30'; the offset o to P.C. of circular curve is 9.07 ft; t is 250 A = 249.6 ft.; x is 500 2.37 = 497.63 ft.; and the values of A A, 6, y y and x COR. for points 200, 210, 220 ft., etc., distant from the P.S. are found in the line with 200, 210, etc. To find the long chord to P.S. C, subtract .445 of x COR. from the length of the curve in feet. To find the spiral tangent-distance, v, add one third of x COR. to one third of the length of the spiral in feet. Tables I-IV have the values of A and calculated to the nearest tenth of a minute, and Tables V-VII to the nearest half minute. While this precision is not usually necessary, it may be of service where the sum of two or more angles is used. 44. Interpolation. To find values intermediate be- tween the distances given in the tables, interpolate by multiplying one tenth of the difference between consec- utive values by the number of additional units. Thus, Table IV gives A for 400 ft. as 8 00'; for 410 ft., 8 24'.3. One tenth of the difference between these is 2'.4. For 406.8 ft, add 6.8 x 2.4 = 16'.3 to 800', giving 816'. For y, add 6.8 x. 143 = .97 to 18.59, giving 19.56 ft For o, add 6.8 times one tenth of .36 to 4.65 giving 4.89. D is 4.068 or 4 4'.08. Interpolation may also be made for other columns. Thus if o is given as 7.0 ft. and a = lj, by Table V the length of spiral will be between 420 and 430 ft. Inter- polating, as o increases 0.5 ft. in 10 ft. of length, the .28 26 THE TABLES will be gained in 5.6 ft. and the length is 425.6 ft. Inter- polation for A, D, etc., may then be made as before. Again, for D = 4 16', still using a = 1 J, the length is be- tween 340 and 350 and is ^ x 10 = 1.33 ft. more than 340, making 341.33. In general this interpolation gives accurate results and no correction need be made. For A the error in inter- polation with values of a greater than 5 may need to be taken into account. To find exact values of A, deduct from the interpolated values a times the following quan- tities: For a length in feet ending with 1, .027'; 2, .048'; 3, .063'; 4, .072'; 5, .075'; 6, .072'; 7, .063'; 8, .048'; 9, .027'. It can easily be determined whether this correc- tion need be considered. The difference arises from the fact that the square of numbers does not increase uni- formly. For the other columns the errors of interpolation are very slight and may be neglected. 45. General use of Table IV. Table IV has been carried to several decimal places to permit its use for values of a other than 1. To calculate values for another a, multiply the tabular value of D, A, 0, o, or y in Table IV for the distance from the P.S. to the point on the spiral by the a of the spiral used, and the x COR. and t COR. by the square of the a of the spiral. Thus if a = 2.2 and L = 3.1, multiply the D, A, 0, o, and y opposite 310 by 2.2, and the x COR. and t COR. by the square of 2.2. The values of y, o, and x COR. obtained in this way are subject to slight errors for large values of a if A is more than 18, but fortunately y for a distance greater than half of the length of the spiral is seldom needed, and as the error of this and the errors in o and x COR. are ordinarily small the correction may generally be neglected. The amount of this error may be found by the method given in a succeeding paragraph. The error in is discussed on page 9. CORRECTIONS 27 To use Table IV for another a, it may be desirable first to determine the length of the spiral by dividing the DI of the required spiral by a or to determine it from o. Thus, for a = 1.5, to connect with a 6 curve, divide 6 by 1.5, which gives L = 4; that is, the whole spiral will be 400 ft. long, and the properties for the spiral may be computed by multiplying those in the line with the re- quired distance by 1.5. In other words it must be borne in mind that the distances in the column of lengths remain unchanged with new values of a, and the quan- tities in all the other columns will be changed for a other than 1 46. Corrections for calculations. For the calcula- tion of tables and other work requiring the recognition of a further term in the equations, the value of the sec- ond term of the o series (.0000002 a 3 L\ eq. (13) ) and of the second term of the y series (.00000158 a 3 U, eq. (6) ) may be obtained by multiplying the quantities in the following table by a 3 ; and the third term of the x series (.00000000268 a 4 L, eq. (7) ) by a 4 . These terms for o and y are negative, and the term fo x is to be subtracted from the x COR. L o y x 2.50 .0010 3.00 .0004 .0035 3.50 .0013 .010 4.00 .0032 .026 .0007 4.50 .0074 .059 .002 5.00 .015 .124 .005 5.50 .030 .241 .012 6.00 .055 .442 .027 6.50 .097 .775 .055 7.00 .163 1.301 1.08 For making corrections on results obtained from Table IV for a other than 1, subtract from the product of 28 CHOICE OF SPIRAL the multiplication used to obtain the desired distance a (a 2 1) times the value from the above table in obtain- ing o and y, and a (a 3 1) times the value from the table in obtaining the x COR. 47. Table of ordinates. By Table XII the ordinate from the tangent or from the circular curve at a decimal part of the half length of the spiral may be obtained by the multiplication of o of the spiral by the factor given in the table. See method by co-ordinates and Fig. 5. It should not be forgotten that Tables I-X give ordinates, and that values for intermediate points may easily be interpolated. 48, Table of offsets. Table XX gives values of o and L for various values of a. Within reasonable limits it will bear interpolation, both for intermediate values of a and D and to determine a for intermediate values of o. It is of service in location problems. Tables XIII and XIV are described under Uniform Chord Length Method. The tables for street railway curves are described under Street Railway Spirals. CHOICE OF a AND LENGTH OF SPIRAL. 49. The selection of a and with it the length of spiral require consideration. The value of a to be used is dependent upon the speed of trains, the maximum degree- of-curve, the length of tangents, the permissible offset of the line for the topographical conditions in question, the distance in which the superelevation of the outer rail may be attained, etc., and hence is subject to a wide range of conditions. It may, however, aid the engineer's judgment to discuss these conditions briefly. I ATTAINMENT OF SUPERELEVATION 29 50. Effect of speed. -For the same rolling stock and for the same comfort in riding, it would seem that a given amount of superelevation must be attained in the same length of time ; and hence it is probable that a should vary nearly inversely as the cube of the speed of the train. This conclusion also emphasizes the desirability of spiral- ing curves used under high speeds. Assuming that a = l is a proper value for speeds of 50 miles per hour, this principle would suggest the follow- ing maximum values of a ; 60 miles per hour, J ; 50 miles per hour, 1; 40 miles per hour, 2; 30 miles per hour, 3J; 25 miles per hour, 5; 20 miles per hour, 10. While for the very high speeds this may seem to require unneces- sarily long spirals and for low speeds short spirals, yet a = 1 has given satisfactory results at speeds of 50 to 60 miles an hour, and a = 2 at 40 to 50 miles an hour, and for 60 miles an hour, a = J is not too small. Of course, in any case, longer spirals and smaller values of a will give smoother riding curves. 51. The speed of trains may be limited by the maxi- mum superelevation allowable on the sharper curves.' Under usual practice the requirement of maximum super- elevation would limit the maximum degree-of-curve for speeds of 60 miles an hour to 3, for 50 miles to 4, for 40 miles to 6, for 30 miles to 12, etc. Where the track is not used for slow trains and a superelevation of more than 7 or 8 inches is allowable, somewhat higher speeds on such curves may be used. The maximum speed of train, however, will be the governing consideration in the choice of a rather than the maximum degree-of-curve. 52. Attainment of superelevation. The rate of attaining the superelevation is sometimes given as the governing consideration, but in reality this rate is gov- 30 CHOICE OF SPIRAL erned by the speed. The distance in which the outer rail should attain an elevation of 1 inch will not be the same for a speed of 60 miles an hour as for one of -iO miles. The schedule of maximum values of a for various speeds as given above involves, approximately, attaining 1 inch of elevation in the following distances : 60 miles, 80 feet; 50 miles, 53 feet; 40 miles, 44 feet; 25 miles, 40 feet. The best rate also depends upon stiffness of car springs, weight and style of cars, and other 'Conditions. Generally speed and amount of superelevation should gov- ern the length of spiral, and rate of attainment is sub- ordinate. 53. It may be convenient for maintenance-of-way work to arrange the spiral so that the superelevation is attained at a definite rate per 100 ft. of length of spiral. Let k be this rate, expressed in inches of superelevation attained in 100 feet. Let h be the superelevation in inches per degree of curve; for a 3 curve, 3h, etc. Then , , D . k k = ah ~ ~h. a = r. L h 54. The following table shows the values of 0, which gives rates of 1, 1J and 2 inches of superelevation attained in 100 ft. for the amount of superelevation per degree of curve given at the head of the columns. VALUES OF a Elevation per degree J J 1 li U 2 2J a for k equal to 1 in. per 100 ft. 2 1% 1 i % % f a for k equal to IK in. per 100 ft. 3 2 \% . f 1 K f a for k equal to 2 in per 100 ft. 4 2% 2 if 1% 1 I Velocity in miles an hour corres- ponding to superelevation 26.9 33.0 38.1 42.6 46.6 53.8 60.2 The length of spiral for a = J is 200 ft. for each degree of curve ; f or a = |, 125 ft. ; for a = , 150 ft., etc. When MINIMUM SPIRALS 31 tables are not given for the a used, the values may be calculated from the tables by multiplication or other process. Thus for a = lj, double the values for a = ; for a = i, multiply those from a = J by 1J or those from a = 1 by f . 55. The amount of superelevation per degree of curve here used is calculated from .00069 F 2 , where V is the velocity in miles per hour. This gives the number of inches per degree to counteract the centrifugal force, and is based on distance from center to center of rail. This amount is used here because it is a value quite commonly quoted; for other superelevations comparisons may read- ily be made with the figure here given. There is a wide divergence of opinion on the proper amount. A super- elevation somewhat less than that required to counter- balance the centrifugal force produces a moderate flange pressure on the outer rail and is held by many to give a smoother riding track. Care must be taken that track so elevated is never used at speeds so far above the assumed speed as to be unsafe. For convenience in main- tenance-of-way work it may be desirable to establish the superelevation at a convenient amount near the value cal- culated for the assumed velocity, an allowable practice since the assumed velocity may not be realized. Thus 2 inches per degree may be used in place of If, etc. 56. Minimum spirals. For a given value of a there may be a question as to how flat a curve may profitably be spiraled. The spiral should certainly vary enough from the position of a simple circular curve that the distinction may not be obliterated by the inaccuracies of track work; otherwise it will be as advantageous to begin the superelevation an equal distance back on the tangent. The limits given in a former edition have been criticised by engineers of maintenance of way, and experience on 32 CHOICE OF SPIRAL prominent roads indicates that the minimum limit of o there set, 0.6 to 1 ft., was too high. It seems that the gradual change of direction between the trucks and the car body, and the fitting of elevation to the curvature by the spiral, are advantageous to -smooth riding even when the change in alignment is slight. Experience seems to indicate that for a = J curves above 30' may be spiraled; for a = l, 1 and above; for a = 2, 2; for a = 3J, 3; for a = 5, 4; for a ^ 10, 6. For curves lighter than these any advantage seemingly found by spiraling would probably be obtained by beginning the supereleva- tion back on the tangent. In any case decreasing the value of a and thus increas- ing the length of the spiral will increase the efficiency of the spiral and better the riding qualities of the curve. This view needs emphasizing, and too much should not be expected of short spirals. 57. Selection of a and length of spiral The selec- tion of a, then, must be a matter to be left to the judg- ment of the engineer. As a guide the following table containing values of a which have given satisfactory results at the speeds noted is given. Lower values of a are of course advantageous; as, for example, at a speed of 40 miles an hour a = lj, or even 1 will make a more efficient easement than the one given. Higher values of a shorter spirals may be necessary in many cases, but it must be understood that they will not be so satisfactory. The column headed "Minimum curve spiraled" is the lightest curve which it is considered desirable to spiral with the value of a given opposite. "Maximum curve" is fixed at the given speed by the limit of superelevation; at lower speeds this a may profitably be used for sharper curves. The speeds are given in miles per hour and the elevations in inches P.S.. P.C.C., AND P.C. 33 MINIMUM SPIRAL FOR MAXIMUM SPEED Maximum s. Maximum Min. Curve Length per Elev. per Speed ^ Curve Spiraled Degree Degree 60 J 3 30' 200 2-1 50 1 4 1 100 15 40 2 7 2 50 1J 30 3J 11 3 30 f 25 5 14 . 4 20 J 20 10 25 5 10 ^ For shorter spirals, the following values of a are con- sistent with each other : 60 miles per hour, 1 ; 50 miles per hour, 1|; 40 miles per hour, 3J; 30 miles per hour, 6| ; 25 miles per hour, 10. LOCATION OF P.S., P.C.C., AND P.C. 58. Location from intersection of tangents. When the tangents have been run to an intersection, the P.S. (A, Fig. 1, page 5) may be located by measuring back on the tangent from the point of intersection a distance equal to the tangential distance T (equation 18). This distance may also be computed by adding t + o tan J/ to the tangential distance of the circular curve as ordinarily calculated. (See section 21.). The P.C. (D, Fig. 1) may be located by calculating o and offsetting this amount at a point on the tangent distant t from the P.S. (See section 19.). t may be found by subtracting the t COR. of the tables from the half length of curve in feet. Thus, for 400 ft. of spiral with a = 2, by Table VII t cor. is .5 ft. and = 200 0.5 = 199.5 ft. The P.C.C. (L, Fig. 1) may then be located by running the spiral from the P.S., or by locating the circular curve from the P.C. for a distance jL . 34 PS V P.C.C., AND P.C. 59. Location from P.C. of a curve without spiral. In case a simple curve has been run without provision for a spiral and without offsets, that is in the usual way, it will be necessary to change the position of the' circular curve. The distance of the P.S. back of the P.C. of the old simple curve will be t + o tan J I, I being the total intersection angle. The new curve will come inside the old but will not be exactly parallel to it. 60. Location from P. C. of offsetted curve. If a simple curve has been run for use with spiral, as DLH in Fig. 1, page 5, o may be computed, the offset measured to B and the distance t (AB) measured to locate the P.S. (A). The length J L measured from the P.C. on the circular curve will locate the P.C.C. (L). Similarly if the tangent is fixed, the curve may be located by first making the offset from the tangent to the P.C. 61. If botk P.C. and tangent are fixed with an offset np o = BD between them, a may be found from a = .269 . or a and L may be found from Table XX. After finding t, the P.S. may be located in the usual manner. For a 5 curve with 0=10 ft., by equations (14) and (16) L = 525.3 ft. and a = .952. Since with a = l and this length of spiral t cor. = .5, the correction to be used here is 0.5 x a 2 , and t = 262.65 .45 = 262.2 ft. This method is a great convenience where it is desired on account of the ground to throw the curve in or out without changing tangent, or where a similar change in the tangent is de- sired without a change in the curve, the connection to be made by means of a suitable spiral. LOCATION BY CO-ORDINATES 35 LAYING OUT THE SPIRAL BY CO-ORDINATES. 62. With the initial tangent as axis of X and the P.S. as the origin of co-ordinates, it is not difficult to locate points on the spiral by means of co-ordinates. These may be calculated from equations (6) and (7), or they may be taken from the tables. Beyond B of Fig 1, page 5, the ordinates become large and the x correction may be considerable. For long spirals, the second term of the 3; series, may need to be considered. The property that the spiral diverges from the circular curve at the same rate as from the initial tangent is of service. Between E and the P.C.C. (L) measure the ordinate or offset from the circular curve, using for this offset at a point a given distance from the P.C.C. the ordinate y of the spiral from the tangent at the same distance from the P.S. Thus for a=l, by Table I, for a point 200 ft. from the P.S., y = 2.33 ft. To locate a point on the spiral 200 ft. from the P.C.C, offset from the cir- cular curve this same distance, 2.33 ft. 63. Knowing the y for any point on the curve, the y for any other point within ordinary limits may be found by multiplying the former number by the cube of the ratio of the distances from the P.S. to the respective points; thus, if for a point 250 ft. from the P.S., y = 4.54 ft., y at 300 ft. equals ( || ) 3 x 4.54 = 7.85 ft. Similarly, points may be located by offsets from the circular curve, the distance being measured from the P.C.C. 64. If the length of the half spiral be divided into an integral number of parts (See Fig. 5), any ordinate from the tangent or from the circular curve may easily be calculated from 0(0 = .0727 DL\ pp. 22 and 25), by mul- 36 LOCATION BY CO-ORDIXATES tiplying o by one half the cube of the ratio of tne number of parts this point is from the P.S. to the whole number of parts. The following table gives the factor by which the o of the spiral may be multiplied to determine the y of the point when the length of the half spiral is divided into 10 parts. RS. 1 RC. FIG. 5 RC.C TABLE OF FACTORS FOR ORDIXATES. To find y, multiply o by the factor. Ratio to half length. 0.1 0.2 0.3 0.4 0.5 0.6 7 0.8 0.9 1.0 Factor. .0005 .004 .014 .032 .063 .108 .172 256 .365 .500 As an example, \vith a = l, o for a 5 curve (spiral 500 feet long) is 9.07 ft. The half length of the spiral is 250 ft. and one tenth of this distance is 25 ft. y at 100 ft. (0.4 of the half length) is .032 x 9.07 = .29 ft. Similarly, .29 ft. will be the offset from the circular curve at a point 100 ft. from the P.C.C. To divide the half-length of spiral into 5 parts and the whole spiral into 10 parts, use the even numbered tenths of the above table. LOCATION FROM SPIRAL TANGENTS 37 For intermediate values, interpolation in the above table will give reasonably accurate results. This enables interpolation for quarter points, and other fractional parts; thus, for .67 of half length the factor is .153. The results by the above table are subject to error in the hundredths place, but for usual cases are within .02 ft. 65. Still another method is to measure ordinates from the initial tangent for about two thirds of the length of the spiral and for the remaining distance to measure ordi- nates or offsets from the terminal spiral tangent (TL, Fig. 1, page 5). The offsets from the terminal spiral tangent will be the difference between the offset for the produced circular curve DL and the y for a spiral, both for a length equal to the distance from P.C.C. to the point to be located. As this distance will be less than one third the spiral length, the approximate formula for tangent offset, .873 Dj L 2 , may ordinarily be used. D^ is the degree of the spiral at the P.C.C. and L here will be used as the distance from the P.C.C. to the desired point. The offset will then be .873 D^ L 2 .291 a L 3 . While the off- sets are longer than those from the circular curve, the measurements are made from the tangent and the circular curve need not be run. As an example take a = 1, and ^ = 4. Length of spiral is then 400 ft. For a point 50 ft. back of P.C.C. (L = .5) the offset is .87 .04 .83 ft. For 100 ft. from P.C.C. the offset is 3.49 .29 = 3.20 ft. This is a very convenient method. 66. Many engineers prefer the co-ordinate method. The circular curve is run from the P.C. established by making the offset from the initial tangent, and the spiral is then located by setting off ordinates from the simple curve between P.C. and P.C.C. and by ordinates from the initial tangent back to the P.S., or for the latter portion 38 LOCATION BY TRANSIT ANGLES by laying off o y from a tangent at the P.C. parallel to the initial tangent, the ordinates being calculated by one of the preceding methods; or offsets from the terminal spiral tangent may be made for the last third of the spiral length. This method is particularlly applicable to location work and to short spirals, though under many conditions it may readily be applied to setting track centers. LAYING OUT THE SPIRAL BY TRANSIT AND DEFLECTION ANGLES. 67. The spiral may be run in with the transit by turn- ing off deflection angles and making measurements along chords in much the same manner as circular curves. The deflection angles are easily calculated, and the field work is not more difficult than for circular curves. The ordi- nary transit-man will find no difficulty in understanding the work. Since it is not necessary to keep succeeding chords the same length as the first, the stationing may be kept up, and the even stations, -|-50's, and other points put in as usual. Herein is an advantage over methods requiring a regular length of chord to be used. 68. Transit at P. S. With the transit at the P.S., which has been located by one of the methods previously described, the deflection angle (BAL, Fig. 1, page 5) will locate points on the spiral. may be taken from the tables, or it may be calculated from equation (9), = 4- A = i a L~. For this calculation, if desired, the square of L may be taken from a table of squares, the lower deci- mals dropped, and the multiplication by the simple factors remaining may be made easily and rapidly. Thus, when a 2, to determine for a point 234 ft. (2.34 stations) from the P.S., find the square of 234 (54756), change the TRANSIT ON SPIRAL 39 decimal point so that it will become the square of 2.34 (5.48), and = I a U = j x 2 x 5.48 = 1 49'. If the re- sult is wanted in minutes, since J x 60 = 10, use 10 instead of J. The slide rule may be used with advantage. For a tabulated spiral, the spiral deflection angles may be taken from the table. Thus, for a = 1 J, by Table V for 110 ft. is 15'. For 114 ft. interpolate proportionally between the tabulated value of for 110 and that for 120 ft. giving 14J'. If it is not desired that the even stations be located, the spiral may be located by 50-ft. chords, or chords of other length, directly from the P.S. and the labor of cal- culation will be reduced. 69. Transit on spiral. With the transit on an in- termediate point on the spiral, the tangent to the spiral at this point may be obtained by turning off from the chord to the P.S. as a back-sight the angle A (ARF, Fig. 2, page 9), where A is the spiral intersection angle and is the spiral deflection angle at the P.S. for the given transit point (R). Except for extreme lengths this is equal to 20. Thus, by Table I (a = J) for a transit point 400 ft. from the P.S., the required angle is 4 1 20'= 2 40' (or 2 (1 20') ). 70. For deflection angles from an intermediate transit point on ordinary circular curves, three methods are in use among engineers : (a) The measurement and record of the angle between the tangent to the curve at the transit point and the chord to the point to be located. (b) The use of the angle between the chord connect- ing the transit point to the P.C., and the chord to the point to be located. 40 LOCATION BY TRANSIT ANGLES (c) The use of the angle between a line through the transit point parallel to the initial tangent and the chord to the point to be located. Three corresponding methods may be used with the spiral and will be treated separately. 71. Intermediate deflection angles, (a) From tan- gent. By equation (10), the angle between the tangent at the transit point and any chord (as CBH, Fig. 4, page 21) is ' the degree-of- curve at B is 3. By Table IV, ff (I- = 3) is 1 30', and ABG is 3, giving the position of the tangent at B. For C 100 ft. from B,. add Jx3xl = l30' and Jxlxl 2 ^ 10', giving 1 40' for CBH. For D 100 ft. from B, DBG 72. (b) From chord to the P. S. This is the meth- od generally to be recommended. By equation (11) the angle between the chord from transit point to P.S. and any chord (as CBE, Fig. 4, page 21) is + JZ7L, being the spiral deflection angle from the initial tangent for the point to be located, D' the degree-of-curve at the transit point, and L the distance in stations from P.S. to the point to be located. (See also pages 10 and 21.) This method involves the following steps: With transit at B and vernier reading zero, back-sight on the P.S. TRANSIT AT P.C.C. 41 To locate C turn off an angle equal to the sum of (1) the spiral deflection angle for a distance equal to the distance from C to the P.S. and (2) one sixth of the product of the degree-of-curve at the transit point and L for the point C. Thus for a = 1, with the transit 300 ft. from the P.S., D' at B is 3. For C 100 ft. from B and 400 ft. from the P.S., add J x 1 x 4 2 =:2 40' (which is the spiral deflection angle for 400 ft.) and J x 3 x 4 = 2 (which is J D" L) giving 4 40' for CBE. For D 100 ft. from B, DBA = 40' + 1 = 1 40'. To facilitate the calculation the transit point may be chosen at a point where the spiral has an even degree-of- curve, as in the above example, but this is not essential. It may be seen that T V D' gives the minutes per foot in J-2XL. 73. (c) Angles with initial tangent. The use of angles with the line parallel to the initial tangent (BK, Fig. 4) is the same as (b) except that ff, the spiral deflec- ion angle to the transit point, must be added to all angles. Otherwise the method is the same as (b). Use equation (12). 74. Transit at P. C. C. With the transit at the P. C.C., the tangent to the curve may be found by turning off from the chord to the P.S. an angle A O v these being the angles for the full spiral. Within ordinary limits this angle equals 20 A . The main circular curve may be run as usual. In case the P.S. can not be seen from the P.C.C., the chord to the P.S. may be located by turning off from the chord to an intermediate point on the spiral an angle A! #! (ACB, Fig. 4, page 21) where < is the angle between the chord and the tangent at P.C.C. (BCF). (See page 10.) To locate the chord from P.C.C. to P.C. (not shown in 42 LOCATION BY TRANSIT ANGLES any diagram), deflect from the chord to the P.S. the angle J A t O r To locate chord to P.C. from a chord to an intermediate point on spiral, deflect from chord to the intermediate point the angle J A <>. With the data already at hand, it may be easier to calculate this angle as + J D* L i A X + V remembering that and L refer to the intermediate point and D', A 1? and to the P.C.C. 75. For the circular curve some engineers prefer to measure the deflection angles from the tangent at the P.C.C., and others prefer to measure from the chord from P.C.C. to P.C. and thus maintain the same notes as though the spiral had not been used. By the use of the angles discussed in preceding paragraphs, either method may be used. 76. To run from the P.C.C. toward the P.S. Two methods may be used, (a) using L as the distance from P.C.C. and deflecting from the tangent, and (b) using L as measured from the P.S. and deflecting from chord to P.S. 77. (a) Angles from tangent. -Using the distance L as measured from the P.C.C., deflect from the tangent to the curve an angle equal to the difference of (1) one half of the product of D t (degree-of-curve at P.C.C.) and distance L to point (which is the same as the deflection angle for D circular curve) and (2) spiral deflection angle for distance L, (J a L 2 ). This is the same as method (a) of "Transit on Spiral." The method depends upon the principle that the spiral deflects from the oscu- lating curve at the P.C.C. at the same rate that it deflects from the initial tangent at P.S. As an example take 400 ft. of spiral connecting with a 4 curve (0 = 1). Measure L from P.C.C. For a point 150 ft. from P.C.C. (L = 1.5) 3 take the difference between ANGLES FROM CHORD Jx4xl.5 = 3 and 22J' (spiral deflection angle for 150 ft.) which is 2 37 J'. This angle is to be deflected from the tangent at P.C.C. By the same method the angle to locate the P.S. is x 4 x 4 = 8 minus 2 40', or 5 20', the result found by the usual method. 78. (b) Angles from chord to P. S. Using the dis- tance L as measured from the P.S., deflect from the chord to the P.S. an angle equal to the sum of (1) the spiral deflection angle for distance L from the P.S. (J a U) and (2) one sixth of the product of D 1 (degree-of- curve at the P.C.C.) and L, (j D^ L). This is the same as (b) of "Transit on Sprial." Using the example cited in the preceding method, 150 ft., from P.C.C. will be 250 ft. from P.S., and L=r2.5. = jxlx (2.5) a =l2i'. j J D 1 L = jx4x2.5 = l40' Th e sum of these is 2 42 J', the angle to be deflected from the chord to P.S. By the same method, for P.S. L is 0, and the deflection angle proves to be 0, as it should be. 5TA. POINTS iD'L +29 o PCC. 6 5-20' J-37' f L*57l \ 2ef Vernier at

ack-5/g/)t on PS.. and furn off angte +50 /'37' in brackets /8 O*5 d' /7 O/O Curve to ft/aftf /6 -r29<* S*S 0*0' a=2, *-,= *, A=/6. FIELD NOTES 79. Transit notes. For a spiral with a = 2 connect- ing with an 8 curve, L = 4, and if the P.S. has been found to be at 16 + 29, the notes may be made as follows, using method (b) for the transit on the spiral. At Sta. 19 the 44 APPLICATION TO EXISTING CURVES deflection angle from the chord to the P.S., as a back- sight, is the sum of those given in third and fourth col- umns, and it is here inclosed in brackets. APPLICATION TO EXISTING CURVES 80. When a road has been constructed without transi- tion curves, the ordinary application of the preceding principles will require a new line to be built inside the old curve, and the cost of construction may be consider- able. To retain as far as possible the old roadbed, three methods are applicable: (0) To replace the old curve with a new and sharper curve located so as not to vary far from the old align- ment. (0) To replace a part of the existing curve with a curve of slightly smaller radius, compounding with the old curve. (c) To make a new alignment for the main part of the curve close to the old and replace a part of this with a curve of smaller radius. 81. To replace the entire curve. First method. In Fig. 6, the dotted line TNH is the old curve, T being its P.C. It is desired to throw the line out at H, the middle point of the curve, a distance of HK = />, and replace the curve by a sharper curve whose P.C. will be at D, thus permitting the spiral AEL to be inserted. P is the inter- section of tangents, which comes outside the diagram. Let R^ be the radius of the eld curve and R of the new. HP KP = p, or R! exsec J / (R + o) exsec J / o = p. TO REPLACE THE ENTIRE CURVE Hence R R = o + O -t-/> O + p exsec J7 ~~ vers 4 7 vers i 7 o (24) cos 4 / = (^ 7? o) exsec 4 7 o..(25) Also AT = AP TP=t(R l ^R o) tan 4 7 = ;_(0 + />) C ot i 7 (26) by which the P.S. (A) may be located; or if T is not known, the tangent distance AP may be calculated and A located. T B M OM i 'H\\ FIG. 6. 82. Values of p from zero to Jo may be used. If the new curve comes inside the old at the center, p must be used as negative and its sign in the formula must be changed. It must be borne in mind that the o used in the above formula must be the o of the new curve. As this will not be known, first use the value of o for the old 46 APPLICATION TO EXISTING CURVES curve in (22), select a radius and degree of new curve near the resulting value, and then determine p and AT with the o for the new curve. 83. As an example take 7 = 60, D = 6, a = 2. Then o for a 6 curve is 3.93. Take 1.0 ft. as a trial value of p. By equation (24) the radius of the new 'curve will be approximately 35.8 ft. shorter than the old and by consult- ing a table of radii of curves it will be seen that a 6 14' curve may be used. - - =3.117; there will be 311.7 ft. a of spiral at the end.. The o for a 6 14' curve will be 4.4 ft. and the resulting p is found by (25) to be 0.5 ft. There will be 9 43' in each of the spirals and 40 34' in the remaining circular curve. The P.S. may be located by measuring the tangent-distance T, or the middle point K of the curve may be located by means of the external distance, E. 84. Second method. The method just described may be modified to use measurements along the external secant as follows, using Fig. 6 as before : Intersect tan- gents at P. (Intersection outside of diagram.) Measure PK along external secant to the point K where it is de- sired to have middle point of new curve come. By equa- tion (20) (page 15) calculate the radius and the degree- of-curve which will give PK as the external-distance E of a spiraled curve. It will be necessary to use the value of o for a degree-of-curve equal to that of the original curve, since the degree of the new curve is not yet known. Next select a curve whose degree will give a radius close to that found by the above calculation. For this D, com- pute 0, and also PK. As the real o was not know r n in the first calculation and the new curve will not have exactly the R found, the point K as now located may not coincide with that first chosen. Having located K anew, the curve TO REPLACE A PART OF THE CURVE 47 may be run in from K with back-sight on P, or the tan- gent-distances* may be measured to locate P.S. Instead of using equation (20), PK may be found by adding o sec J / to the external-distance for 7 of circu- lar curve without spiral. Likewise in finding the desired D, subtract o sec J / from the measured distance PK, and use the remainder as the external-distance for an un- spiraled circular curve. By this means a table of external- distances for a 1 curve may be utilized and the calcula- tions shortened. 85. This method is applicable on short curves and \vhere the ground will permit of easy and accurate meas- urement of the external-distance. Take the same example as before. Consider that the measured PK is 146.7 ft. Using = 3.93, o sec J 7 = 4.5 ft. The circular curve whose external-distance is 146.7 4.5 = 142.2 ft. lies between 6 14' and 6 13'. Choosing a 6 14' curve and recalculating, the external-distance for a simple curve is found to be 142.2 and o sec J / 5.1, mak- ing PK 147.3 ft. After K is located the curve may be run in. 86. (b) To replace a part of the curve. -In Fig. 7, B is the P.C. of the old curve whose degree is D . It is desired to go back on this curve a distance BD and there compound with a curve of somewhat sharper curvature, D lf which if run to a point E where its tangent is parallel to the original tangent shall be at a distance EF = o from it. The tangent and D curve may then be connected by a spiral having this o. It is required to locate D and the P.S. and P.C.C. so that a selected curve, D 19 will give a calculated or assumed distance EF as o. Let R be the radius of the D curve and R^ that of the D^ curve, and / the angle to be replaced. o = EF == FH EH == (# J?J vers /,. 48 APPLICATION TO EXISTING CURVES vers 7, - . (27) Having I lt back up on the curve to D, run the D^ curve to G, the P.C.C. of spiral, and locate the spiral. The P.S. may' be located from B by AB = t(R 9 R 1 o) tan/, = t(R R 1 ) sin 7, ................... (28) -Or /ft I? a/ Curve FIG. 7. 87. Thus, consider that a part of a 4 curve is to be replaced with a 430' curve, and that a = 1. o = 6.62. By equation (27), 7 1 =16 35'. Take out BD = 414.6 ft. of 4 curve and locate D. 16 35' of 4 30' curve requires 368.5 ft. The half length of the spiral is 225 ft. The P.C.C. is then found by running from D 368.5 225 = 143.5 ft. of 4 30' curve to the P.P.C., G. Likewise by (28) A.B. = 224.8 45.4 = 179.4 ft. TO RE-ALIGN AND COMPOUND 49 88. The limiting values of D i will be on the one hand | D and on the other a value which will make BD one half of the length of the original curve. Ordinarily, D t should not be one fifth more than D ; better less than one tenth more on sharp curves. 89. It may be convenient to calculate a standard set of values for the curves on a road, The following gives a few such values. D Q a A ^o^i /, AB GD 2 i 215' 318.3 3.31 816' 179.2 142.4 2 i' 230 / 572.9 4.53 712' 178.1 38.0 3 1 3 30' 272.8 3.12 840' 133.8 72.6 4 1 5 286.4 9.07 1427' 178.1 39.0 5 1 6 190.9 15.65 2321' 223.4 89.2 5 2 6 ' 190.9 3.91 ll38 r 111.4 43.9 90. (c) To re-align and compound. When the middle portion of the curve is in fair alignment and it is desired not to disturb it, or when it seems best to re-locate the central part of the curve, a method by taking up points on the old track and not running the principal tan- gents to an intersection, may be used. See Fig. 8. Select M, N, and O on the curve on the portion not to be dis- turbed. Set transit at M, measure the distances MN and NO, and by the usual methods for circular curves deter- mine the degree of curve, D , which will fit this middle portion. Or select points that will locate the curve in a desirable position, and determine D Q . The selection of points in this way will probably not give a curve whose tangent coincides with the track tangent. When this curve is run back until its tangent is parallel to AH at B, the distance from the track tangent will be called m. From M, intersect with tangent at H and measure /. Determine m by running out MDB and measuring the APPLICATION TO EXISTING CURVES offset or by calculation from m=HG sin / and HG=HM GM, remembering that GM is the tangent-distance for / of D curve. Let ED be the new D t curve which must be run in from D so that EF shall be the o for the D t curve. Call the radius of the D curve R and that of the D curve R r The D^ curve will have / of central angle. Then EJH=o m=(R R 1 ) vers / t . o m vers /. = - .(29) If o is less than m, then R t must be greater than R a . FIG. 8. If K comes outside of AH, m must be added to o. Care must be taken that M is far enough back on the curve. 91. To locate the curve, run / /.of D curve from M to D. Run DE to locate the P.C., or run such part of this D 1 curve as will give the P.C.C. for the spiral. METHODS OF TRACK MEN 51 The P.S. (A) may be located as follows: AH = * + BG BK HG cos / (30) BG = GM, the tangent-distance for 7 of D curve, BK = (^ RJ sin 7,, and HG cos 7 is also m cot 7. The P.S. may also be located by offsetting o from E to F and measuring t to A. 92. For example, if D has been found to be 4 and 7 at H 20 and m 1.2 ft., select D^ = 4 30' and a = I. Then c = 6.62. From equation (29) 7,= 15 Then calculating the length of the curves from the angles 7 and I lf as MB is 500 ft. and DB 375 ft, MD = 125 ft. DE = 333.3 ft. since there is to be 15 of 4 30' curve. The P.C.C. for spiral is 333.3 225 = 108.3 ft. from D toward E, since the half length of the spiral is 225 ft. AH is 224.8 + 253.6 41.1 3.3 = 434.0 ft. The spiral may be run in by usual methods. 93. The limiting values of D 1 are similar to those given in the preceding method. Generally D 1 may be from one tenth to one fourth more than D , depending upon the amount of the curve and its degree. 94. When the new 7? curve is so much sharper that it is desired to connect it with the old by a spiral, the following method is applicable. Call o t the offset to tan- gent, and o the offset between the two curves, the latter to be found as for compound curves. Then by a method similar to the foregoing, (31) 95. Methods of track men. When curves are left without transition curves, many track men "ease" the curve by throwing the P.C. inward a short distance and gradually approaching the tangent a few rail lengths 52 APPLICATION TO EXISTING CURVES away, while the main curve is reached finally by sharpen- ing the curve for a short distance. 96. Another simple method for track which is aligned to a circular curve, consists in utilizing one of the proper- ties of the transition spiral. In Fig. 1, page 5, let ABK be the original track line, B being the P.C. Select a length of spiral and calculate o, or select o and calculate the length, by a preceding method. At a distance from B equal to half the length of spiral (point of the curve opposite L) throw the track inward to L a distance equal to o. At B, the old P.C., throw the track to E, a distance half as great. Measure back from B half the length of the spiral to A for the beginning of the easement. Be- tween A and L, line the track by eye, or calculate offsets from Table IX. The remainder of the main curve must then be thrown in the same distance as at L. 97. On long curves the latter work would be objec- tionable. It may be avoided by using a spiral running up to a curve whose degree-of-curve is one third greater than that of the main curve and compounding directly with the main curve. To do this, first select length of spiral for a curve one third sharper than the circular curve which call L. See Fig. 9. Call the circular curve D and the curvature of the end of the spiral D r Measure back from the old P.C. on tangent a distance J L, which will locate the P.S. Measure forward on the curve from the P.C. a distance J L to locate the middle of the spiral, and offset from prolongation of tangent a distance equal to J o, or J o from the circular curve. Measure also along the curve from the P.C. a distance | L to the P.C.C. where the track will not be changed. The spiral will pass the old P.C. at -^0 from it, and at a point J L from the P.C.C. will be J0 distant from the circular curve. METHODS OF TRACK MEN 53 The spiral is one and one half times as long as the cir- cular curve replaced. The o used must be that for the full spiral and for the sharper curve, |r D Q , and the true position of the cir- cular curve should be known. As the last fourth of this spiral is sharper than the main curve, the elevation of the other rail up to the P.C.C. must be greater than that on the main curve, gradually reducing beyond to the regular amount. J_ ,0/JP.c. ,J. P. 5. 2. o U"' FIG. 9. 98. Thus, for a 3 curve using a = l, the degree at the end of the spiral will be 3 x |^ =4, and the length of spiral required is 400 ft., = 4.65. The P.S. will be 133.3 ft. back of the P.C., the middle of spiral 66.7 ft. ahead of P.C. and the P.C.C. 266.T ft. ahead of P.C. At the P.C. the track must be thrown in 0.69 ft., at the middle point 2.32 ft. from tangent (1.16 ft. from curve), and at the third quarter point .58 ft. from curve, while at the 54 COMPOUND CURVES P.C.C. there will be no change. Between these points the track may be aligned by eye, or ordinates may be calcu- lated by Table XII. However, while such methods are easements, they are at best makeshifts and should give place to better methods. COMPOUND CURVES. 99. The spiral may be used to connect curves of dif- ferent radii, choosing that part of the spiral having curv- ature intermediate between the degrees of the two curves, thus, connect a 3 and an 8 curve by omitting the spiral up to D = 3 and continuing until D = 8. In Fig. 10, DKM is a D i curve, and LNP a D 2 curve, the two curves having parallel tangents at M and N. D 2 is greater than D r Call the distance MN o. It is desired to connect the two curves by a spiral shown by the full line KP. The degree of curve of the spiral at K must be DI and at P ? D 2 . Consider the spiral to be run backward from K to a tangent at A. Then the spiral from K to P is the portion of the regular spiral from where its degree is D^ to the point where it is D 2 . Since the spiral diverges from the osculating circle at the same rate as from the tangent at the P.S., PN = MK and the spiral bisects MN. MN, or o, is the offset for a spiral for a curve whose degree is D 2 D r Hence, find o for a D z D 1 curve, and make the offset at MN. Measure MK and NP each equal to J 2 ~ * thus locating the P.C.C. of each curve K and P. Run in the spiral from K or P by the method for point on spiral heretofore described, AK being omitted. The angle between tangents at K and TO INSERT IN OLD TRACK 55 P is A for a D 2 spiral minus A for D 1 spiral, and may also be expressed as (D 1 + D 2 ) limes KP in stations. Thus, with a = 2, to connect a 3 and an 8 curve o =2.27, the value for a 5 spiral. The portion of the spiral used will be 250 ft. long. K is 125 ft. from M. and N is 125 ft. from P. If greater accuracy is required, the t COR. for FIG. 10. this length should be subtracted from 125. The angle between tangents at K and P is x 2.50 (3 + 8) = 13 45'. The spiral may also be used to connect two curves having a given offset between them. 100. To insert in old track. It may be desired to insert a spiral between the two curves of an existing com- pound curve by first replacing a part of the sharper curve with a curve of slightly smaller radius. In Fig. 11, let AB be a > curve and BG a D 3 curve, B being the P.C.C. and the D 3 -curve having the smaller tadius. Q is the center of the D 1 curve, not on the cut, 56 COMPOUND CURVES It is desired to go back on the D 3 curve to a point D and there compound with a D z curve which shall be run to a point E where its tangent shall have the same direction as the tangent to the DI curve produced backward to F has at F. The radial distance EF corresponds to the offset of the usual spiral and will be called o. It is de- sired to locate D and F L'O that a selected cuve, D 2 , will give a calculated or assumed distance EF as o. The distance EF is made up of FK and KE, the first being the divergence of the D curve from the D 3 curve in the distance BF and the second the divergence of the D 2 curve from the D s in the distance DE. Call the distance BF L v and DE L 2 .. For the small angles used these divergences may be calculated accurately enough by the approximate formula for tangent offset, y==.87 DU y and we shall have EF = .87 (D 2 D 3 ) L/+.87 (> 3 >,) L? = o,or (D 2 D Z ) L 2 2 +(D 3 D l ) L^ 1.150 (32) Since the amount of D^ curve in BF plus the amount TO INSERT IN OLD TRACK 57 of D 2 curve in DE (total angle) must be equal to the amount of D s curve taken out, we have I>,L, + I> 1 L 1 = .y,(L 1 + L 1 ) or (^.-^.) L 2 =(^-^) /-,... ... ..... ..(33) Combining (32) and (33) and solving, V 3 I/ /OK\ - ^ ........... (35) 101. Having L and L 2 , the points D, E and F may be located, and the D 2 curve may be run in from D as far as necessary. The problem is then identical with that of putting a spiral between two curves having an offset o (EF) between their parallel tangents. By the principles governing the placing of a spiral be- tween two curves, it is seen that the length of the con- necting spiral L' is that of a spiral for a curve of degree equal to the difference of degree of the two connected; that is a The offset is equal to that for a (D 2 D^) degree curve from a tangent or o = .0727 (D 2 DJ L'' 2 = .0727 aL" ............ (36) Half of this spiral will lie on one side of the offset and half on the other, hence in Fig. 12 JL to the right of F will give the beginning of the spiral, H, and JL to the left of E will give the end of spiral, I. 102. The method of field work will then be as follows : Measure from B, the P.C.C., (Fig. 12) back on the D t curve a distance BH = J L' L to locate the point of spiral H. Measure from B on the D z curve the distance BD = L t + L 2 to D, the new P.C.C., run in the D 2 curve to I, DI being L 2 i I/ . The spiral is then to be run in 58 COMPOUND CURVES from H to I. The dotted line in Fig. 12 shows the spiral. The field work for the spiral is simple. The spiral may be run in by offsetting from the D^ curve HF (Fig. 12) knowing that the offset from the curve to the spiral is the same as that of a spiral from the tangent using the distance from H as the distance on the spiral. Likewise the remainder of the spiral may be ofTsetted from the D., curve IE using distances from I in the calculations. If the field work on the spiral is to be done by deflec- r FIG. 12. tion angles, the spiral may be run in from H by using as deflection angles the sum of the deflection angle for the circular curve HF and the spiral deflection angle from a tangent for the same distance ; or the transition spiral may be run backward from I in a similar manner. In either case the work will be no more difficult than for spirals for simple curves. TO INSERT IN OLD TRACK 59 103. As an example let us consider that a 2 and an 8 curve are compounded at B. Consider that the degree of the new curve to be run in is 8 30', and that the value of a to be used is 2. Then D = 2, D 3 = 8, D 2 = 8J. For a spiral from 2 to 8 30', the value of the offset o (EF) is the same as the o for a 6 30' curve from a tan- gent. Hence = 4.99. By formula (34), L l= =.271, and by formula (35), L 2 = 3.255. Hence the point D will be back on the > 3 curve 325.5 + 27.1 or 352.6 ft. from B. The length of the spiral to be used will be L'= 8 ^~ 2 = 3.25. Of this 162.5 ft. will be to the left of E and 162.5 ft. will be to the right of F. Hence H and I, the ends of the spiral, may readily be located and the spiral may be run in. 104. By this method the value of a may be chosen beforehand, the value of o may be easily calculated, and the preliminary field work is small. It may be stated that the limiting values of D 2 will be, on the. one hand, a value so near D 3 that the resulting L 2 will carry the new point of compound curve back to the end of the old curve, and on the other hand such that the length of the D 2 curve shall be at least equal to half the length of the transition spiral, a value which may be shown to be D 2 = J (4 *>,- D,). For large angles the above method is subject to slight error. 60 MISCELLANEOUS PROBLEMS MISCELLANEOUS PROBLEMS 105. To change tangent between curves of oppo- site direction. Having given two curves of opposite direction connected by a short tangent, it is required to find the position of a line to which both curves may be connected by spirals. This involves determining the angle which must be added to each curve to get the position of the new P.C. of each curve for spiraling. 106. In Fig. 13 let AB be the original tangent con- necting the two curves and I its length in feet. It is required to run the curve KA to E, which will be the Or/ginal Tangent ^ [^L^^ C N ^^ r 4=^ ,,- ' L-- P \^ AI L r ^r __ ) .i^^^--' >i A^*v Tangent - D FIG. 13 P.C. for the spiraled curve, and MB to F for its spiraled P.C., and also to find the position of the line CD, which will be the common tangent for the two spirals. Call the CURVES OF OPPOSITE DIRECTION 61 angle APC a. It is the same as that of the additional amount of curve AE and BF. Let R be the radius of the curve KE and R 2 that of ME, and o l and o 2 be the respective spiral offsets NE and OF for the spirals chosen lor the two curves. AC + BD = NE + EG + OF + FH. Then, since AC + BD = / sin a and EG = ^ vers a, etc., Z sin a =o l + o 2 + (R^ + # 2 )vers a, (37) 107. Since a will be small, we may substitute, using a in degrees, sin a .01745 a and vers <* = . 000152 2 which are close approximations below 8. Transforming, o i + o 2 .000152 (flj + R.,) a 2 a ~ .01745 I "*" .01745 / This quadratic may be solved, but usually the follow- ing approximate root gives sufficiently close results : o, + o t .000152 (R,+R 9 ) (i+o m - (.01745 /)* Having a the lengths AE and BF may be found, the position of the P.C.C. of each curve found, and the new tangent located by offsetting EN and FO, or by offsetting AC (equal to o 1 + R 1 versa) and BD. For very short tangents, spirals must be chosen short enough not to over- lap on the tangent. 108. As an example take a 3 curve and a 4 curve connected by 600 ft. of tangent. Use a = 1. Then o i = 1.96 and 2 = 4.65. By equation (38) a = .63 + .02 = .65 = 39'. This result checks equation (37) very closely. 039' gives 21.7 ft. of 3 curve (AE) and 16.2 ft. of 4 curve (BF). There will be 300 ft. of spiral for the 3 curve and 400 ft. of spiral for the 4 curve. The P.S. and P.C.C. of each spiral will be half of the spiral length from the points E and F. The P.C.C. of one will be (150 21.7=128.3) ft. back of A, and of the other (200- 16.2 62 MISCELLANEOUS PROBLEMS =183.8) ft. back of B. AC will be (1.96 + .11 = 2.07) ft. and BD (4.65 +.09 =4.74) ft. The distance from C to P.S. will be 150 + 21.7, and from D to the other P.S, 200 + 16.2, neglecting the t correction. The spirals may then be run in as usual. 109. This solution may also be applied to the case where a tangent thrown off from the curve KA does not strike the curve MB but is parallel to this curve at a point opposite B distant m from it. Since cos a is nearly 1, equations (37) and (38) may be modified by subtract- ing m from (o^ + oj wherever it occurs. This modifica- tion is of convenience in revising old lines. The engineer should make his own diagram. 110. To change tangent between curves of same direction. Having given two curves of same direction connected by a tangent it is desired to find the position of a line to which the two curves may be connected by spirals. As in the preceding problem this involves deter- mining the change in the angle of the two curves and the position of the P.C. of each curve for spiraling. 111. In Fig. 14 let AB be the original tangent con- necting the two curves and I its length in feet. It is required to back up on the curve AK to E for the P.C. for spiraled curve and to run the curve MB to F for its spiraled P.C., and to find the position of the line CO which will be the common tangent for the two spirals. Call the angle BAD' a. It is the same as that in AE and BF. Let R^ be the radius of the curve KE and R 2 that of MF, and o^ and o 2 be the respective spiral offsets NE and OF for the spirals chosen for the two curves. BD AC = OF+ FH NE EG. Then, since BD AC or BD' equals / sin a and EG equals R l vers a, etc. CURVES OF SAME DIRECTION sin a = o 2 o l (R t R.J) vers a. A/ew Tangent -^ N C \ O 63 .(39) FIG. 14. 112. Since a will be even smaller than in the preced- ing problem, we may substitute sin a = .01745 a and versa = .000152 a 2 , using a in degrees. Transforming, Q.-OI .000152 (R-R 2 ) 2 .01745 I .01745 I As in the preceding problem, the approximate solution of this quadratic may be used. , A .000152 (RR 9 ) (o , (in degrees) =-~ ^-^ ~ .01745 I (.01745 O 3 " The last term here is very small. Having a the lengths AE and BF may be found, the position of the P.C.C. of each curve found, and the new tangent located by offsetting EN and FO, or bv offsetting AC (equal to 0^ + ^ vers a ) and BD (equal to o. 2 + R : vers a. The length of spiral must not be so great that the spirals will overlap on short tangents. 64 UNIFORM CHORD LENGTH METHOD 113. As an example take a 3 curve and a 4 curve connected by 600 ft. of tangent. Use a = 1. Then O L = 1.96 and 2 = 4.65. By equation (40) a =.257 .006 = .251 = 015i'. 15-J' gives 8.6 ft. of 3 curve ( AE) and 6.4 ft. of 4 curve (BF). There will be 300 ft. of spiral for the 3 curve and 400 ft. for the 4 curve. The P.C.C. of spiral will then be (150 + 8.6 = 158.6) ft. back of A and (200 6.4 = 193.6) ft. back of B. AC will be 1.98 ft. and BD 4.67 ft. The distance from C to P.S will be 150 8.6 and from D to the other P.S. 200 + 6.4, neglecting the t correction. The spirals may then be run in. 114. This solution may also be applied to the case where a tangent thrown off from the curve KA misses the curve MB by a distance m from B, the point of par- allelism. In this case equations (39) and (40) may be modified by subtracting m from (o 2 oj wherever it occurs. If the second curve is one of larger radius, it will be necessary to construct a new diagram and deter- mine the signs of the terms. UNIFORM CHORD LENGTH METHOD 115. The treatment of the spiral heretofore given is based upon principles which permit the use of any chord length, either uniform or variable, throughout the length of the spiral. Regular chord lengths, like 20 or 25 feet, may be used, if desired and the excess if any used as a fractional chord at the beginning or the end of the spiral. If it is desired to use chords of common length, another method known as uniform chord length method, may be derived by modifying the preceding formulas. A further modification of this method may be made to allow the UNIFORM CHORD LENGTH METHOD 65 use of fractional chord lengths at the beginning or the end of the spiral, so that it will not be necessary to make the uniform chord length an aliquot part of the length of the spiral. Thus, if the spiral is to be 203.2 ft. long, ten 20-ft. chords, or eight 25-ft. chords, or thirteen 15-ft. chords, etc., may be used the first or last chord or both being fractional. 116. The notation used will be the same as heretofore except as noted, and the equations will be numbered the same, using the prime mark to distinguish them. Let c be the chord length used. This will be expressed in hun- dreds of feet, that is in the number of 100-ft. stations. FIG. 15. For a chord length of 20 ft., c =.2 ; for one of 15 ft., c = .15, etc. Let n (an integer) represent the number of full chords from the P.S. to a desired point. In Fig. 15, A is the P.S. and its n is 0. The n of B is 1, of C 2, etc. Let O i = spiral deflection angle at P.S. from initial tangent for a single full chord length (BAE) (called unit spiral deflection angle), and O n for n chord lengths. For Dn = 66 UNIFORM CHORD LENGTH METHOD 3 and n = DAE. Similarly, A n , D n and L n are for a point n chord lengths from the P.S. For the instrument at other points than P.S., let n' be the number of chord lengths from the P.S. to the chord point at which the instrument is located, reserving n still as the number of chord lengths from the P.S. to the point to be located, and let 3> n represent the deflection angle from tangent at the instrument point to this desired point. Thus, if the instrument is at C two chord lengths from the P.S. n f = 2, and to locate D three chord lengths from the P.S., n = 3 and < w will represent the deflection angle DCK to locate the third chord point D. 117. The length L = nc. By substitution in equations (1), (2), and (9) of the spiral, we have for any point on the spiral distant L = nc from the P.S. D = a.L = a n c... (V) A = i a L 2 = J a n 2 c 2 (2') For end of first full chord and for end of n full chords, respectively, (9') Also 3> n = 1 an' c (n ') c =b J a (n H') = JW (n n'):=(n ')'] #1 In formula (10'), the arithmetical difference of the numbers of the chord points is taken, rather than the algebraic difference. If the latter is used, the signs of. operation should all be plus. 118. The first step is to calculate the value of the unit spiral deflection angle X by means of equation (9'), using a and c or other terms. For a chord length of 20 ft, and a value of a = 2, c = 3 and ^=J x 2 x (2) 3 = DEFLECTION ANGLES 67 0.8'. If a spiral 250 ft. long is to connect with a 4 curve using 25-ft. chords, D n = 4 and L n = 2.5, O l = J + 4X T 1 6X^=-01 / . If A = 9, L = 3andc = .2 (20ft), ft = i (I) 2 x 9 = 0.8'. 119. The value of gives a basis for computing the deflection angle for other points; thus for a point 5 chord lengths from the P.S., n = 5 and the deflection angle by equation (9') is 25 times the value of . For the instru- ment at the fifth chord point (w' = 5), the deflection angle from the tangent at the instrument point to a point 8 chord lengths from the P.S. (n = S) is by equation (10') : $ n z=:[3x5 (8 5) == (8 5) 2 ] ^ = 54^. To locate from the same instrument point a point 3 chord lengths from the P.S. (2 from the instrument point), the deflec- tion angle is 26 O r 120. Table of unit spiral deflection angles. A table giving 0! for various chord lengths for many of the values of a used in field work may be of service. Table XIII gives spiral deflection angles for first chord length (unit spiral deflection angles). The angle is given in minutes. , It is well in the calculations to express decimals as com- mon fractions; thus, for 20- ft. chords with a = l use 0, = .5J'; for 16-ft. chords with a = l use ^ = .421'. 121. Table of coefficients for deflection angles. The values obtained from (9') and (10') may be considered as coefficients of O lt and a general table prepared. Table XIV is table for coefficients for a spiral up to 15 chord lengths for use with the instrument at any -chord point. 122. For the instrument at the P.S., multiply the co- efficient in the columns headed opposite the chord point to be located by the value of the spiral deflection angle for a single chord length (unit spiral deflection angle 0J. 68 UNIFORM CHORD LENGTH METHOD 123. To find the deflection angle from the tangent at any chord point, enter the column whose heading gives the number of the chord point at which the instrument is placed and take the coefficient opposite the number of the chord point to be located; then multiply the spiral deflec- tion angle for a single chord length (0J by this coefficient. Thus, as in example cited above, for the instrument at 5, the deflection angle from the tangent at this point to locate a point 8 chord lengths from the P.S. is found to be 540,, and to locate a point 3 chord lengths from the P.S. is 260,. This table may easily be extended. The variation in the tabular differences in horizontal, vertical, and diagonal directions is readily discerned, and if pre- ferred the method of differences may be used for calcu- lating deflection angles for a particular case in place of a multiplication of these coefficients. 124. To end the spiral with a fractional chord. If the number of chord lengths is not integral, the first and succeeding chords may be made of uniform length until the last, one is reached and the full deflection angle may be turned off for the P.C.C. Thus, for 218.4 ft. of spiral, ten 20-ft. chords may be used and the full deflec- tion angle turned off for the remaining 18.4 ft. 125. To begin the spiral with a fractional chord length. In case it is desired to begin the spiral with a fractional chord length, the following modification may be made. Let m be the ratio of this fractional chord length to a full chord length, and O m be the spiral deflection angle from the initial tangent for this fractional chord length, which from the general formula for 9 may be seen to be m 2 r Let Q n+m be the spiral deflection angle from initial tangent to locate a point (n + m) chord lengths away (n an inte- ger and m fractional), and $ w+m the deflection angle from tangent at instrument point (w' + ;w) chord lengths from DEFLECTION ANGLES 69 P.S. to locate a point (n + m)chord lengths from P.S. Substituting in formula (9) page 8, = O n + n (2m 0J + O m (9") 126. In the last member of equation (9"), the first term is the spiral deflection angle for n full chords, the second term is n times a constant, and the third term is the spiral deflection angle for the fractional chord. The calculations may be simplified by the method of differ- ences. For example, for a chord length of 20 ft. let the be- ginning chord be 8.4 ft. Then w = -^r- = .42. If 1.2' be zo the unit spiral deflection angle lf OT =.21'. To locate a point 88.4 ft. from P.S., the spiral deflection angle at 1 .S. will be 6 n+m = 16 x 1.2 + 4 x 2 x .42 x .21 + .2 = 23'.4. Use Table XIV in calculating n . 127. For the instrument at a chord point (n' chord lengths from the P.S., the deflection angle from the tangent at this chord point to locate a chord point (w+ *w) chord lengths from the P.S. is found from equation (10) page 11 to be $ n+w =C3(n / + w) (n n')(n n') 2 ] ^ = $ + ( ') (3w ^) (10") In the last member of equation (10") the first term is the deflection angle for full chords, and the second term is (n n') times a constant. If n' is greater than n, the second term is still added numerically to the first. Tabular differences may also be used. 128. To use equation (10") first calculate & n using Table XIV; then add the last term. Thus, for a chord lenecth of 20 ft. an,d a beginning: chord of 8.4 ft. and a YO UNIFORM CHORD LENGTH METHOD unit spiral deflection angle of 1.2', with the instrument 88.4 ft. from P.S., n' + m = 4.42. To locate a chord point 148.4 ft. from P.S. ( = 7), the deflection angle from the tangent is found to be, taking the coefficient for n from the column headed 4, 3> n+m = 45 x 1.2' + 3 x 3 x .42 x .21' = 54.8'. To locate a chord point 48.8 ft. from P.S., using the same instrument point, (n = %, n'=4:, w .42), we have for the deflection angle 3> n+m = 20 x 1.2 + 2 x 3 x .42 x .21'= 24.5'. 129. To illustrate the use of these methods, take the following examples. It is desired to spiral a 6 40' curve with 200 ft. of spiral, using 20-ft. chords, c = .20. n = 10. D n = 6%. By equation (9') the spiral deflection angle X 9O 2 for a 20-ft. chord is J x^- x 6 40' = lj' = r The multi- Li plication of lj' by the coefficients in Table XIV for instrument at and for instrument at 10, gives the de- sired deflection angles. If it is desired to connect a tangent with a 4 curve so that the offset o shall be 5.0 ft., proceed as follows. By equation (14) the length of spiral is 415.2 ft. Using 25-ft. chords, the deflection angle in minutes is by equa- 1 x 4 x f i V tion (9') * ^=.602 = 0,. Table XIV will give the TT.-LO^J coefficients for multiplication, and the fractional chord may be left for the last measurement. 130. To show the use of fractional beginning and ending chords, consider that a spiral 138.4 ft. long is to connect with a 10 curve and that it is desired to use 15-ft. chords but that the first chord shall be 9.3 ft. O c=.15. w--^.==.62. By equation (9'), the spiral deflec- -LO tion angle for a 15-ft. chord is found to be If', and for DEFLECTION ANGLES 71 the point 9.3 ft. from P.S. O m is .62 2 x 1| = .62'. The table below gives values for field work, considering that P.S. is at Sta. 322 + 13.7. In the column headed "number of Survey Station No. of Chord Point Central Angle INSTRUMENT POINT P.S. 0.62 1.62 4.62 5.62 322+13.7 P.S. 1.2' 109.4' +23 0.62 1.8' 0.6' 104.1' +38 1.62 12.8' 4.3' 4 6' 53.0' +53 2.62 33.5' 11.2' 12.5' 38.6' +68 3.62 103.9' 21.3' 23.7' 20.9' +83 4.62 144.1' 34.7' +98 5.62 234.0' 51.3' 24.1' 323+13 6.62 333.6' 111.2' +28 7.62 134.3' +43 8.62 200.7' +52.1 End 655.3' 218.4' chord point" the integer of the number is n and the frac- tional part is m. In the succeeding column headings for the instrument points, the integer is n'. Thus, to deter- mine the deflection angle with the instrument at to locate 323 + 43, by equation (9") and Table XIV, O n+m = (64 x If) + (8 x 2.01) + .6 =-- 2 00.7'. To determine the deflection with instrument at 322 + 83, ..(' = 4), to locate 322 + 68, (n = 3), by equation (10"), 3> n+m = (llxlf) + (1x3.02) =20.9'. 131. To run in the spiral from the P.C.C., this method may likewise be used if the P.P.C. is at n or at n + m chord lengths from the P.S. If it is not, that is, if both the first . and last chord lengths are to be fractional, the points on the spiral as far as the next instrument point may be set by the principle that deflection angles will equal the difference between the deflection angle for a circular curve and the deflection angle for a spiral from 72 STREET RAILWAY SPIRALS initial tangent, both for a distance equal to the distance to the desired point. After the next instrument point is reached, calculate deflection angles as though working from the P.S. 132. The method of uniform chord lengths is subject to the same correction for as is given on page 9. As it is not likely that this method will be used for large deflection angles, the error will usually be negligible. For < the correction needed is almost exactly that for the which enters into it; thus in equation (10') make a correction which would be necessary for a equal to (n n') z 0^ This is not often necessary. 133. It will be seen that the method of uniform chord lengths may have advantages where a chord of full length may be used at the beginning of the spiral, especially where the rate a is fractional, and that it is also appli- cable when the beginning chord is fractional. It is more especially applicable where evenly spaced points are wanted. Table XIV is a convenient table. STREET RAILWAY SPIRALS 134. For use in connection -with curves of short radii, as street railway curves, the formulas for the transition spiral may be modified with advantage. The variable radius R of the spiral may replace the degree-of-curve D. The product of the radius at any point by its distance from the P.S. will be shown to be constant for a given spiral, and this product may be used as the characteristic constant, taking the place of a. The offset o may be used as one fourth of the ordinate y of the terminal point of the spiral except for extreme lengths. Certain other THEORY approximations may be made which are not always allow- able with curves of large radius. 135, Theory. The general notation will not be changed. Fig. 16 shows the co-ordinates x and y, spiral intersection angle A, spiral deflection angle 0, and spiral tangent-distances u and v for a point on the spiral, and X j zj-u^^ - -^rjjl 1 PCC, FIG, 16. also o and t for the full spiral, together with the produced circular curve. As before, R = radius of curvature at any point and s = length of the spiral arc in feet from P.S. to any point of the spiral. From equation (1) page 6, D _ . 100D 573000 Hence L s , 573000 s R ", a constant. Represent this constant product of ^ and R by k. Then d7?= _ a s The property of the transition spiral that R varies in- versely as the distance along the spiral is satisfied by this equation. 74 STREET RAILWAY SPIRALS 136, Modification of the formulas already derived may be made as follows. The angle subtended by a cir- cular arc in degrees is equal to = ' ~ m Since A v R R is one half as much as the angle of the same length of circular curve having a radius equal to the terminal radius of the spiral, 28.65 j_ 28.65 / A = ~^~ ~~k~ ( * This may also be derived directly from equation (2) page 6 by substitution for the values of a and L. Also 9.55 s 9.55 r * = * A= -/r : (53) For large angles, if the precise values of are desired, the corrections given in the table on page 9 may be made. However, for the short distances involved this correction may generally be neglected. 137. Consider that = J y v using the subscript 1 to designate the y, etc, of the terminal point of the spiral. By trigonometry y 1 o = R^ vers A. Then y l =^R i vers A and = 1 R vers A ...(54) Also by substitution for a and L in eauations (6), (7), (13), (17), (21) and (23), pages 17 and 18, the follow- ing formulas are obtained: V _ s (55) J -v- k / 9 y 2 (56) f~l 40 2C 10 5 (57) 24 k -1- Ci r 24 & " v u * ) . . (58) TABLES 75 , 5 v =i j+ _ ........................ . . ____ (60) 120 k* In the last three equations, note that the t correction is J of the x correction (last term in equation (56) used in finding x from s) ; the C correction is ^ of the x correction; and the v correction is J of the x correction. For extreme cases, the values of y and o given by equations (55) and (57) will be slightly too large. For y, subtract .003 g- from the results of equation (55). k For o subtract one eighth as much. For x, t, C, and v, the terms given in the equations above will generally be sufficiently accurate. 138. The following equations may be repeated here: T = t+(R 1 + o) fan J / ...................... (18) E=(R i + o) exsec 1 I + o ......... ...... ..... (20) u = x v cot A .... . (22) = x v cos A. 139. The Tables Tables' \v ill .facilitate the applica- tion of these equations. Tables XV-XIX give properties of five spirals. The spiral is to be used up to that length which gives the required radius. The x correction is the amount to be subtracted from the length of the spiral to give the abscissa x. The long chord C may be found by subtracting four ninths of the x correction from the length of spiral. Similarly the spiral tangent-distance v is found by adding one third of the x correction to one third of the length of spiral. Interpolations for distances between those given in the tables may be made, but it is best to compute R and the angles. 76 STREET RAILWAY SPIRALS 140. Laying out spiral. The same methods may be used in laying out spirals of short radii as have been described for curves of large radii. The location of the P.S., P.C.C., and P.C. is generally not difficult. If the lines have been run to an intersection, as is generally desirable, the tangent-distance T may be measured to locate the P.S. The P.C.C. may be located by turning off the full spiral deflection angle 0,, at the P.S. and measuring the long chord C; or the spiral tangent dis- tances u and v may be calculated and the angle A turned at their intersection point. In either case the tangent at the P.C.C. may readily be found. Another method is to locate the P.C. by offsetting the distance o from the ini- tial tangent and then running in the circular curve to the P.C.C. 141. Centers for the spiral may be set (a) by meas- uring ordinates from the initial tangent for the full length of spiral; (b) by ordinates from the initial tangent as far as the middle of the spiral and from the produced circular curve for the remaining half length; (c) by ordi- nates from the initial tangent for about two thirds of the spiral and from the terminal spiral tangent for the remainder of the spiral. The offsets from the circular curve will be the same for given distances from the P.C.C. as the y for an equal length of spiral. The offsets from the terminal spiral tangent will be the difference between the offset for the circular curve and the y for a spiral, both for a length equal to the distance from 2 2 P.C.C. to the point located. This will be y where s is the distance of the point from the P.C.C. and R is the radius of the circular curve. 142. Location by means of deflection angles from initial tangent at P.S. is so similar to that for railway ARC EXCESS 77 spirals already described that it need not be further dis- cussed. If the rails have previously been bent to their proper curvature very few centers need be set. 143. Arc excess. It must be borne in mind that the actual length of arc is considered in the formulas here given, and care must be taken to provide for the difference between arc and chord measurement. The long chord is easily found by the x correction of the tables as already indicated. For other chords of spiral arcs (not from P.S.), it will be sufficiently accurate to use for the excess of arc over chord the excess of the same length of circular arc having a radius equal to that of the middle point of the spiral arc under consideration. The excess length of a circular arc over its chord may be calculated from the approximate formula where c may be used as the length of either chord or arc. It may also be noted here that the number of degrees of angle in a circular arc is =~ and the deflection J\. angle from tangent is of course half of this. - 144. Curving rails. The principles here outlined are for the center line of track, but it may be desirable to have measurements for the curves formed by the rails for use in curving rails, etc. Although the outer and inner rails will be parallel to the center line, their lines will not be true spirals, and allowance should be made for this. For data in bending rails, it will be well first to get the variation in length of rail from the length of the center line. For a given point on the center line, first find the A of the spiral. The outer rail will be : 180 TT x %G longer than the center line, and the inner rail will be as much shorter, G being the gauge of track. 78 STREET RAILWAY SPIRALS Thus for = 1500 and % a spiral distance of 30 ft. A = 17 17 18 11' and the excess length in outer rail is - x 3.14 x 180 2.35 = .70 ft. The outer rail distance will be 30.70 ft. and the inner rail distance 29.30 ft. The ordinate of the rail from its own initial tangent will be the 3; for the center line spiral d= J G vers A ; plus to be used for the outer rail and minus for the inner one. Thus for the example above cited, the ordinates for the rails opposite a center distant 30 ft. from the P.S. (30.70 along outer rail and 29.30 ft. along the inner rail) will be 3.00 .11 = 3.11 and 2.89. In locating points on the rails, allow- ance should be made for the difference between the cen- ter line distance and the rail distance. The x for the Doint on the rail will be the x for the corresponding point on the center spiral d= \ G sin A. These principles will apply to any point on the spiral. It will be well to tabu- late values for the sets of curves most used. 145. If it is desired to locate the last third of the spiral from the terminal spiral tangent for the two rails, the length and position of these tangents may be calculated from tneir ordinates and the A already used, and points on the rails located by offsets from these tangents. These offsets may readily be calculated by the principles already outlined. 146. Double track. Double track curves will gen- erally need radii of different lengths. The spirals for both curves may be taken from the same table, or the one for the outside curve may be taken from the table hav- ing a value of k next higher than that used with the inside curve. In the latter case the two spirals will be of nearly the same length and their ends will be nearly opposite. If the . distance between center lines on the curve be made equal to the distance between center lines DOUBLE TRACK 79 on tangent plus the difference in the o for the inside and outside spiral, the circular parts of the two curves will be parallel and have the same center, and the radius of the outside curve will be equal to the radius of the inside curve plus the distance between tracks on the curve. 147. In case consideration of clearance requires greater distance between the tracks on curves than on tangents, care must be exercised in the selection of the spiral. The calculation of the external-distance E will best enable the distance between center lines at their middle points to be determined. If the inside radius is assumed, first find its external distance E, to this add the distance from one P. I. to the other, and subtract the required distance between the curves. The remainder will be the external-distance E for the .outside curve from which the desired radius may be found. As by this ar- rangement the two curves will be closer at their ends than at their middle, care must be taken to secure suffi- cient clearance. The selection of curves and spirals for double track is more complex than for single track. CONCLUSION 148. Besides the problems and methods here pre- sented, many other applications may be made. For par- ticular conditions the engineer may develop speci-l methods. The preceding methods generally have been based upon the principle that the spiral is to have the same degree- of-curve at the end as the main curve, and slight modi- fications may be necessary when not so. The value of o and of the angle in the circular curve omitted must be that for the spiral used. Thus with a = 2 the spiral at the 80 CONCLUSION end of 300 ft. will be a 6 curve. It may, however, be there compounded with a curve of different radius, as a 6 30' curve, provided the offset is 3.93 and the central angle between the P.C. and the P.C.C. is 9. Generally, in order to utilize the problems given for old track, etc., the formulas will need to be modified if D does not agree with D r 149. In field work most of the usual formulas of the various location problems, like ''Required to change the P.C. so that the curve may end in a parallel tangent," may be used without modification with curves having transition endings, by simply considering the whole inter- section angle including the angle in the spirals. This is true whenever the same amount of spiral is used with the new curve. If the degree-of-curve changes and with it the length of the spiral, the difference between the o's in the two cases must be allowed for. With a little practice in using such formulas with spirals, the engineer will find no difficulty. 150. The transition spiral has the merit of compara- tive simplicity and extreme flexibility. It is a natural method, since it is so similar to the methods used in lay- ing out circular curves. Like circular curves, the length along the curve is the principal term, and the degree-of- curve, central angle, deflection angles and ordinates are obtainable from this variable. It may be used with any main curve, even of fractional degree; any length of chord may be used in measurement under the same re- strictions as circular curves, and as it is not necessary to restrict the measurements to a common chord length, intermediate points may be readily located. The calcula- tions for angles and distances are easily made. If de- sired, the tangent and the circular curve may be run out CONCLUSION 81 and the spiral put in by co-ordinates, one half from the tangent and one half from the circular curve. This is especially applicable to location work and to short spirals. The engineer should not be frightened by the mathe- matics in the demonstration of the formulas; the prin- ciples and methods may be understood without mastering the demonstrations. Experience has shown that the ordi- nary transit man, with a little thought and study, can understand and use the transition spiral as easily as cir- cular curves, and that young assistants without previous training readily take up the work. 151. With reference to the use of the cubic parabola as an easement it may be said that, except for the rela- tion between x and y, it has no properties of value for a transition curve which are not merely approximations of the transition spiral. Within small limits, the radius of curvature and the angles to be used approach some- what closely to those for the transition spiral. As soon as x differs materially from the length of curve, a cor- rection has to be made. The radius of the curvature finally begins to increase. The investigation of the cubic parabola in reference to its radius of curvature, its angle turned, the angular deflection to points on it, and the length of the curve, require as long mathematical equa- tions as those governing the transition spiral. Many at- tempts have been made to utilize this curve, but both field work and computations are too intricate and inconvenient if the curve has any considerable length, and it has no advantage over the transition spiral. 152. The question of the efficiency of easement curves is of considerable importance. The objection is some- times raised that even if track is laid out with a carefully fitted spiral there would be no possibility of keeping it in place by the methods of the ordinary trackman. This 82 CONCLUSION identical objection could be made with the same force against carefully laid out circular curves, yet no engineer would recommend abolishing that practice. Even if, in re-lining, the transition curve is considerably distorted, it remains an easement, and will be in far better riding condition than a distorted circular curve. By marking the P.S. and the P.C.C with a stake or post, with inter- mediate points on long spirals, the trackman will be able to keep the spiral in as good condition as though it were of uniform curvature. The short spirals advocated by some engineers have proved to be insufficient. For effi- cient service, a length of spiral which will give an o of considerable amount must be used, even if this necessi- tates widening the roadbed. 153. Properly constructed spirals would frequently allow the use of sharper curvature since the riding qual- ity of curves may be the governing consideration in the selection of a maximum and thus make a saving in construction. By fitting curves with proper transition spirals, roads using sharp curves may partially relieve the objection of the public to traveling by their routes. The in- troduction of fast trains has made it necessary to take every precaution to secure an easy-riding track. The disagree- able lurch and necessary "slow order" for fast trains at certain curves on many roads has been entirely eliminated by the construction of proper spirals, and passengers do not now know when such curves are reached. The transition curve has, then, a financial value largely over- balancing its cost. The adoption of such curves by many of our principal railways proves their efficiency, and the future will see a much more general adoption. EXPLANATION. OF TABLES EXPLANATION OF TABLES. In Tables I-XI, the columns give the following prop- erties : 1. The distance in feet from the P.S. along the spiral to a point on the spiral; i. e., 100 L. The full length of spiral will give values for the terminal point, the P.C.C. of main curve. 2. D, the degree-of-curve of the spiral at any point. It becomes D at the P.C.C. ^ -R\C. * ^ >\PC.C. FIG. 17. 3. A, the spiral angle or change of direction of the spiral to the point. 4. 0, the spiral deflection angle at the P.S. from the initial tangent to locate the point. 5. o, the offset from the initial tangent to the P.C. of main curve produced backward. Enter the table with the full length of the spiral used. 6. y, the ordinate from the initial tangent as the axis of X." EXPLANATION OF TABLES 85 7. x COR., an amount to be subtracted from the dis- tance in feet from the P.S. along the spiral to find the abscissa, x, of the point, x = 100 L x COR. 8. t COR., an amount to be subtracted from half the full length of the spiral in feet to find t, the abscissa of the P.C. Enter the table with full length of the spiral used. t=^L t COR. To find the long chord to P.S., subtract four ninths (.444) of x COR. from the length of the spiral in feet. C = 100 L f x COR. For chords not ending at P.S., see pages 16 and 77. To find the terminal spiral tangent-distance, add one third x COR. to one third the spiral distance to the point. z ,= i|o.L + J x COR. Intermediate values may be found by interpolation. With transit at intermediate point on spiral, for deflec- tion angle <, see pages 10, 20, and 40. To use Table IV for other values of a, multiply the tabulated values of D 9 A, 9, o, and y in Table IV opposite the given distance from the P.S. by the a of the desired spiral, and x COR. and t COR. by the square of a. For inaccuracies of this method see page 27. If a and D or o are given, first find L. Table XII permits ordinates to be calculated from o. See Fig. 5. For the use of Tables XIII and XIV see page 67, and for Tables XV, XIX, see page 75. Table XX gives values of o and L for values of a and D. For full nomenclature, see page 3. For equations and summary of principles, see page 17. For fuller explanation of tables and errors of interpo- lation, see page 24. For choice of a, see page 28. 1 in 200 ft. TABLE I. TRANSITION SPIRAL. Length D A 9 y x COR. /COR. 25 QQ7}4' ooo:9 ooo:3 .00 .00 50 15 03.8 01.3 .00 .02 75 22^ 08.4 02.8 .02 .06 100 30 15. 05. .04 .15 125 37X 23.4 07 8 .07 .29 150 45 33.8 11.3 .12 .49 175 52/2 45.9 15.3 .20 .78 .00 200 1 00 1 00. 20. .29 1.16 .01 225 1 07^ 1 15.9 25.3 .41 1.66 .01 250 1 15 1 33.8 31.3 .57 2.27 .02 .00 275 1 22X 1 53.4 37.8 .76 3.03 .03 .01 300 1 30 2 15. 45. .98 3.93 .05 .01 325 1 37^ 2 38.4 52.8 1.25 5.00 .07 .01 350 1 45 3 03.8 1 01.3 1.56 6.23 .10 .02 375 1 52^ 3 30.9 1 10.3 1.92 7.67 .14 .02 400 2 00 4 00. 1 20. 2.33 9.31 .19 .03 425 2 07^ 4 30.9 1 30.3 2.79 11.16 .26 .04 450 2 15 5 03.8 1 41.3 3.31 13.25 .35 .06 475 2 22/ 2 5 38.4 1 52.8 3.89 15.58 .46 .08 500 2 30 6 15. 2 05. 4.54 18.16 .59 .10 525 2 37^ 6 53.4 2 17.8 5.26 21.03 .75 .13 550 2 45 7 33.8 2 31.3 6.04 24.17 .95 .16 575 2 52/2 8 15.9 2 45.3 6.91 27.62 1.20 .20 600 3 00 9 00. 3 00. 7.84 31.36 1.48 .24 '625 3 07^ 9 45.9 3 15.3 8.87 35.45 1.81 .30 650 3 15 10 33.8 3 31.3 9.97 39.85 2.21 .37 675 3 22X 11 23.4 3 47.8 11.16 44.63 2.66 .44 700 3 30 12 15. 4 04.9 12.45 49.73 3.20 .53 725 3 37^ 13 08.4 4 22.7 13.83 55.22 3.81 .64 750 3 45 14 03.8 4 41.2 15.30 61.09 4.51 .75 775 3 52K 15 00.9 5 00.1 16.88 67.37 5.31 .89 800 4 00 16 00. 5 19.8 18.56 74.05 6 22 1.04 1 in 150 ft. TABLK II. TRANSITION SPIRAL. Length D A 9 y x COR. /COR. 25 010' 001'.3 000'4 .00 .00 50 20 05. 01.7 .01 .02 75 30 11.3 03.8 .02 .08 100 40 20. 06.7 .05 .19 125 50 31.3 10.4 .10 .38 150 1 00 45. 15. .16 .65 0.00 175 1 10 ^ 1 01.3 20.4 .26 1.04 0.01 200 1 20 1 20 26.7 .39 1 55 01 TABLE II. Continued. 1 in 150 ft. Length D A 8 y x COR. /COR. 225 130' i4i:3 033'.8 .55 2.21 .02 .00 250 1 40 2 05. 41.7 .76 3.03 .03 .01 275 1 50 2 31.3 50.4 1.01 4.04 .05 .01 300 2 00 3 00. 1 00. 1.31 5.23 .08 .01 325 2 10 3 31.3 1 10.4 1.66 6.66 .12 .02 350 2 20 4 05. 1 21.7 2.08 8.31 .18 .03 375 2 30 4 41.3 1 33.8 2.56 10.23 .25 .04 400 2 40 5 20. 1 46.7 3.10 12.40 .35 .06 425 '2 50 6 01.3 2 00.4 3.72 14.88 .47 .08 450 3 00 6 45. 2 15. 4.41 17.66 .62 .10 475 3 10 7 31.3 2 30.4 5.19 20.76 .82 .14 500 3 20 8 20. 2 46.7 6.05 24.20 1.06 .18 525 3 30 9 11.3 3 03.8 7.01 28.02 1.35 .22 550 3 40 10 05. 3 21.7 8.05 32.19 1.70 .28 575 3 50 11 01.3 3 40.4 9.20 36.78 2.12 .36 600 4 00 12 00. 3 59.9 10.45 41 76 2.63 .44 TABLK III. TRANSITION SPIRAL. 1 in 125 ft. Length D 4 y x COR. t COR. 25 012' 001^' 000^' .00 .00 50 24 06 02 .01 .03 75 36 13^ 04)4 .02 .10 100 48 24 08 .06 .23 125 1 00 37^ 12^ .11 .46 150 1 12 54 18 .20 .79 .00 175 1 24 1 13^ 24^ .31 1.25 .01 200 1 36 1 36 32 .47 1.86 .02 225 1 48 2 01^ 40^ .66 2.65 .03 .00 250 2 00 2 30 50 .91 3.64 .05 .01 275 2 12 3 01/2 1 00^ 1.21 4.84 .08 .01 300 2 24 3 36 1 12 1.57 6.28 .12 .02 325 2 36 4 13^ 1 24^ 2.00 7.99 .18 .03 350 2 48 4 54 1 38 2.49 9.97 .26 .04 375 3 00 5 37^ 1 52^ 3.07 12.27 .36 .06 400 3 12 6 24 2 08 3.72 14.88 .50 .08 425 3 24 7 13K 2 24^ 4.47 17.85 .68 .11 450 3 36 8 06 .2 42 5.31 21.18 .90 .15 475 3 48 9 01X 3 00^ 6.23 24.90 1.18 .20 500 4 00 10 00 3 20 7.26 29.02 1.52 .25 TABLE IV. TRANSITION SPIRAL in 100 ft. Length D A e y x COR. /COR. 10 0.1 ooo:3 ooo:i .000 .000 .000 .000 20 0.2 01.2 00.4 .001 .002 30 0.3 02.7 00.9 .002 .008 40 0.4 04.8 01.6 .005 .019 50 0.5 07.5 02.5 .009 .036 60 0.6 10.8 03.6 .016 .063 70 0.7 14.7 04.9 .025 .100 80 0.8 19.2 06.4 .037 .149 90 0.9 24.3 08.1 .053 .212 100 1.0 30. 10. .073 .291 .001 110 1.1 36.3 12.1 .097 .387 .001 120 1.2 43.2 14.4 .126 .503 .002 130 1.3 50.7 16.9 .160 .639 .003 140 1.4 58.8 19.6 .199 .798 .004 150 1.5 1 07.5 22.5 .245 .982 .006 .001 160 1.6 1 16.8 25.6 .298 1.191 .008 .001 170 1.7 1 26.7 28.9 .357 1.429 .011 .002 180 1.8 1 37.2 32.4 .424 1.696 .014 .002 190 1.9 1 48.3 36.1 .499 1.995 .019 .003 200 2.0 2 00. 40. .582 2.327 .024 .004 210 2.1 2 12.3 44.1 .673 2.690 ,031 .005 220 2 2 2 25.2 48 4 .774 3.097 .039 .006 230 2.3 2 38.7 52.9 .885 3.538 .049 .008 240 2 4 2 52.8 57.6 1.005 4.020 .061 .010 250 2.5 3 07.5 1 02.5 1.136 4.544 .074 .012 260 2.6 3 22.8 1 07.6 1.278 5.111 .090 .015 270 2.7 3 38.7 1 12.9 1.431 5.724 .109 .018 280 2.8 3 55.2 1 18.4 1.596 6.383 .131 .022 290 2.9 4 12.3 1 24.1 1.773 7.091 .156 .027 300 3.0 4 30. 1 30. 1.963 7.850 .185 .031 310 3.1 4 48.3 1 36.1 2.166 8.66 .218 .036 320 3.2 5 07.2 1 42.4 2.382 9.53 .255 .043 330 3.3 5 26.7 1 48.9 2.612 10.45 .298 .050 340 3.4 5 46.8 1 55.6 2.857 11.42 .346 .058 350 3.5 6 07.5 2 02.5 3.116 12.46 .400 .067 360 3.6 6 28.8 2 09.6 3.391 13.56 .460 .077 370 3.7 6 50.7 2 16.9 3.681 14.72 .528 .088 380 3.8 7 13.2 2 24.4 3.988 15.94 .603 .100 390 3.9 7 36.3 2 32.1 4.311 17.23. .686 .114 400 4.0 8 00. 2 40. 4.651 18.59 .779 .130 410 4.1 8 24.3 2 48.1 5.01 20.02 .SSI .147 420 4.2 8 49.2 2 56.4 5.38 21.51 .994 .166 430 4 3 9 14.7 3 04.9 5.78 23.08 1.118 .186 440 4.4 9 40.8 3 13.6 6.19 24.73 1.254 *'.2Q? 450 4.5 10 07.5 3 22.5 6.62 26.45 1.403 .234 1 in 100 ft. TABLE TV. Continued. f Length D J (-) y ^COR. /CoR. 460 4.6 1034'.8 331'6 7 07 28 24 1.57 .26 470 4.7 11 02.7 3 40.9 7.54 30.12 -1.74 .29 480 4.8 11 31.2 3 50.4 8 03 32.07 1.94 .32 490 4.9 12 00.3 4 00.1 8.54 34.11 2.15 .36 500 5.0 12 30. 4 10. 9 07 36.23 2.37 .40 516 5.1 13 00.3 4 20.1 9.63 38.44 2.62 .44 520 5.2 13 31.2 4 30.4 10.20 40.73 2.89 .48 530 5 3 14 02.7 4 40.9 10.80 43.12 3.17 .53 540 5.4 14 34.8 4 51.4 11.42 45.59 3.49 .58 550 5.5 15 07.5 5 02.3 12.07 48.15 3.82 .64 560 5.6 15 40.8 5 13.4 12.74 50.83 4.18 .70 570 5.7 16 14.7 5 24.7 13.43 53.56 4.56 .76 580 5.8 16 49.2 5 36.2 14.14 56.40 4.98 .83 590 5.9 17 24.3 5 47.8 14.89 59.34 5.42 .90 600 6.0 18 00. 5 59.7 15.65 62.39 5.89 .98 610 6.1 18 36.3 6 11.8 16.44 65 52 6.40 1.07 620 6.2 19 13.2 6 24 1 17.26 68.77 6.94 1.16 630 6.3 19 50.7 6 36 5 18.10 72.11 - 7.51 1 25 640 64 20 28.8 6 49.1 18.97 75.56 8.13 1.36 650 6.5 21 07.5 7 02.0 19.87 79.11 8.78 1.47 660 6 6 21 47. 7 15.1 20.79 82.79 9.48 1.57 670 6.7 22 27. 7 28.5 21.74 86.56 10.22 1.69 680 6.8 23 07. 7 41.8 22.73 90.43 11.00 1.82 690 6.9 23 48. 7 55.3 23.73 94.42 11.82 1.96 700 7.0 24 30. 8 09.3 24.79 98.50 12 70 2.10 TABLE V. TRANSITION SPIRAL. 1 in 80 ft. Length D J 9 o y x COR. /COR. ~~T(T 007!' 000!' 000' .00 .00 .00 .00 20 15 01! 00! .00 .00 30 22! 03! 01 .00 .01 40 30 06 02 .00 .02 50 371 09! 03 .01 .04 60 45 13! 04J .02 .08 70 52! 18! 06 .03 .12 80 1 00 24 08 .05 .19 90 1 07! 30! 10 .07 .26 100 1 15 37! 12! .09 .36 110 1 22J 45! 15 .12 .48 120 1 30 54 18 .16 .63 130 1 37! 1 03! 21 .20 .80 140 1 45 1 13! 24! .25 1.00 150 1 52! 1 24! 28 .31 1.23 .00 .00 1 in 80 ft. TABLE V. Continued. Length D A B y x COR. i /COR 160 200' 136' 032' .37 1.49 .0 .0 170 2 07* 1 48* 36 .45 1.77 180 2 15 2 01* 40* .53 2.12 190 2 22* 2 15* 45 .62 2.50 200 2 30 2 30 50 .73 2.90 210 2 37* 2 45* 55 .84 3.36 220 2 45 3 01* 1 00* .97 3.87 230 2 521 3 18* 1 06 1.10 4.42 240 3 00 3 36 1 12 1.25 5.02 250 3 07* 3 54* 1 18 1.42 5.67 .1 260 3 15 4 13* 1 24* 1.59 6.38 .1 270 3 221 4 33* 1 31 1.79 7.15 .2 280 3 30 4 54 1 38 1.99 7.98 .2 290 3 371 5 15* 1 45 2.21 8.86 .2 300 3 45 5 37* 1 52* 2.45 9.81 .3 310 3 52* 6 00* 2 00 2.70 10.74 .3 320 4 00 6 24 2 08 2.98 11.91 .4 330 4 07* 6 48* 2 16 3.26 13.06 .4 340 4 15 7 13i 2 24* 3.57 14.28 .5 350 4 22* 7 39* 2 33 3.89 15.57 .6 360 4 30 8 06 2 42 4.23 16.95 .7 .1 370 4 37* 8 33* 2 51 4.59 18.40 .8 .1 380 4 45 9 01* 3 00* 4.97 19.92 .9 .2 390 4 52* 9 30* 3 10 5.38 21.54 1.0 .2 400 5 00 10 00 3 20 5.80 23.23 1.2 .2 410 5 07* 10 30* 3 30 6.26 25.00 1.4 .2 420 5 15 11 01* 3 40* 6.72 26.86 1.6 .3 430 5 22J 11 33* 3 51 7.22 28.82 1.7 .3 440 5 30 12 06 4 02 7.74 30.87 2.0 .3 450 5 37* 12 39* 4 13 8.28 33.02 2.2 ' .4 460 5 45 13 13* 4 24* 8.84 35.25 2.4 .4 470 5 52* 13 48* 4 36 9.41 37.59 2.7 .5 480 6 00 14 24 4 48 10.03 40.02 3.0 .5 490 6 07i 15 00* 5 00 10.67 42.56 3.4 .6 500 6 15 15 37* 5 12* 11.33 45.20 3.7 .6 510 6 22* 16 15J 5 25 12.03 47.95 4.1 .7 520 6 30 16 54 5 38 12.74 50.79 4.5 .8 530 6 37* 17 33* 5 51 13.48 53.76 5.0 .8 540 6 45 18 13* 6 04 14.26 56.84 5.4 .9 550 6 52* 18 54i 6 18 15.07 60.02 6.0 1.0 560 7 00 19 36 6 32 15.90 63 34 6.5 1.1 570 7 07* 20 18* 6 46 16.76 66.72 7.1 1.2 580 7 15 21 01* 7 00 17.65 70.26 7.8 1.3 590 7 22* 21 45* 7 14* 18.57 73.90 8.4 1.4 600 7 30 22 30 7 29 19.52 77 68 9.2 1 5 TABLE VI. TRANSITION SPIRAL. 1 in 60 ft. Length D A 8 y x COR. / COR. 10 010' 000l' 000' .00 .00 .0 20 20 02 00* 30 30 041 01* 40 40 08 03 .03 50 50 121 04 .06 60 1 0.0 IS 06 .03 .10 70 1 10 241 08 .04 .17 80 1 20 32 101 .06 .25 90 1 30 401 13* .09 .35 100 1 40 50 161 .12 .48 110 1 50 1 00* 20 .16 .64 120 2 00 1 12 24 .21 .84 130 2 10 1 24* 28 .26 1.06 140 2 20 1 38 321 .33 1.33 150 2 30 1 52* 371 .41 1.63 160 2 40 2 08 421 .50 1.98 170 2 50 2 24* 48 .59 2.38 180 3 00 2 42 54 .70 2.82 190 3 10 3 00* 1 00 .83 3 32 200 3 20 3 20 1 061 .97 3.88 210 3 30 3 40 1 131 1.12 4.48 .1 220 3 40 4 02 1 201 1.29 5.15 .1 230 3 50 4 24* 1 28 1.47 5.90 .1 240 4 00 4 48 1 36 1.67 6.69 .2 250 4 10 5 12* 1 44 1.89 7.58 2 260 4 20 5 38 1 521 2.13 8.52 .2 270 4 30 6 041 2 Oil 2.38 9.54 .3 280 4 40 6 32 2 101 2.65 10.64 .4 290 4 50 7 001 2 20 2.94 11.82 .4 300 5 00 7 30 2 30 3.26 13.07 .5 310 5 10 8 00* 2 40 3.60 14.43 .6 .1 320 5 20 8 32 2 501 3.96 15.87 .7 .1 330 5 30 9 041 3 Oil 4.34 17.40 .8 .1 340 5 40 9 38 3 121 4.75 19.02 .9 .2 350 5 50 10 12* 3 24 5.18 20.74 1.1 .2 360 6 00 10 48 3 36 5.64 22.56 1.3 -.2 370 6 10 11 24* 3 48 6.12 24.50 1.4 .2 380 6 20 12 02 4 001 6.63 26.53 1.7 .3 390 6 30 12 40* 4 131 7.16 28.67 1.9 .3 400 6 40 13 20 4 261 7 73 30.92 2.2 .4 410 6 50 14 00* 4 40 8 34 33.27 2.4 .4 420 7 00 14 42 4 54 8.96 35.73 2.8 .5 430 7 10 15 241 5 08 9.6i 38.32 3.1 .5 440 7 20 16 08 5 22J 10.30 41.07 3.5 6 450 7 30 16 52* 5 371 11.01 43.90 3.9 .6 1 in 60 ft. TABLE VI. Continued. Length D J y x COR. /COR. 460 740' 1738' 552' 11.75 46.86 4.3 .7 470 7 50 18 24* 6 08 12.50 49.94 4.8 .8 480 8 00 19 12 6 24 13.35 53.16 5 4 .9 490 8 10 20 00* 6 40 14.19 56.52 5.9 1.0 500 8 20 20 50 6 56 15.07 60.01 6.6 1.1 510 8 30 21 401 7 13 16.00 63.64 7.2 1.2 520 8 40 22 32 7 30 16.94 67.36 8.0 1.3 530 8 50 23 24* 7 471 17.93 71.25 8.8 1.5 540 9 00 24 18 8 05 18.95 75.31 9 6 1.6 550 9 10 25 12i 8 23 20.03 79.53 10.5 1.8 560 9 20 26 08 8 42 21.13 83.88 11.5 1.9 570 9 30 27 041 9 00* 22.26 88.31 12.6 2.1 580 9 40 28 02 9 19* 23.42 92.92 13.7 2.3 590 9 50 29 00* 9 39 24.67 97.70 14.9 2.5 600 10 00 30 00 9 59 25.91 102.66 16.2 2.7 TABLE VII. TRANSITION SPIRAL. 1 in 50 ft. Length D J e o y x COR. /COR. 10 012' 000*' 000' .00 .00 .0 .0 20 24 02* 01 30 36 05i 02 .02 40 48 09i 03 .01 .04 50 1 00 15 05 .02 .07 60 1 12 211 07 .03 .13 70 1 24 29* 10 .05 .20 80 1 36 38* 13 .07 .30 90 1 48 48*- 16 .10 .42 100 2 00 1 00 20 .15 .58 110 2 12 1 121 24 .19 .77 120 2 24 1 26 J 29 .25 1.00 130 2 36 1 411 34 .32 1.28 140 2 48 1 57* 39 .40 1 60 150 3 00 2 15 45 .49 1.96 160 3 12 2 33i 51 .59 2.38 170 3 24 2 53* 58 .71 2.86 180 3 36 3 141 1 05 .85 3.39 .1 190 3 48 3 36* 1 12 1.00 3 99 .1 200 4 00 4 00 1 20 1.16 4 65 .1 .0 1 in 50 ft. TABLE VII. Continued. a=2. Length D A . y ;rCOR. 2fCOR. 210 4 12' 424F 128' 1.35 5.38 .1 220 4 24 4 50i 1 37 1.54 6.19 .2 230 4 36 5 17* 1 46 1.76 7.07 .2 240 4 48 5 45i 1 55 2.00 8.04 ' .2 250 5 00 6 15 2 05 2.27 9.09 .3 260 5 12 6 45|- 2 15 2.55 10.22 .4 270 5 24 7 17i 2 26 2.85 11.45 .4 280 5 36 7 50| 2 37 3.18 12.75 .5 290 5 48 8 24i 2 48 3.54 14.18 .6 .1 300 6 00 9 00 3 00 3.91 15.68 .7 .1 310 6 12 9 36-1- 3 12 4.32 17.31 .9 .1 320 6 24 10 14-|- 3 25 4.75 19.03 1.0 .2 330 6 36 10 53i 3 38 5 21 20.87 1.2 .2 340 6 48 11 33i 3 51 5.70 22 81 1.4 .2 350 7 00 12 15 4 05 6.22 24.87 1.6 .3 360 7 12 12 57i 4 19 6.77 27.05 1.8 .3 370 7 24 13 41i 4 34 7.34 29.35 2.1 .3 380 7 36 14 26i 4 49 7.95 31.79 2.4 .4 390 7 48 15 12* 5 04 8.60 34.35 2.7 .4 400 8 00 16 00 5 20 9.28 37.04 3.1 .5 410 8 12 16 48 5 36 10 00 39.85 3 5 .6 420 8 24 17 38i 5 53 10.73 42.79 4 .7 430 8 36 18 29i 6 10 11.53 45.88 4 4 .7 440 8 48 19 211 6 27 12.34 49.14 5.0 .8 450 9 00 20 15 6 45 13.20 52.55 5.6 .9 460 9 12 21 09J 7 03 14.09 56.05 6.3 1.0 470 9 24 22 05i- 7 21 15.02 59.73 6.9 1.2 480 9 36 23 02i 7 40 15.99 63.55 7.7 . 1.3 490 9 48 24 00* 8 00 17.00 67.55 8.5 1 4 500 10 00 25 00 8 19 18.05 71.72 9.4 1.6 510 10 12 26 OQi 8 39 19.15 76 00 10.4 1.7 520 10 24 27 02| 9 00 20.27 80.04 11.4 1.9 530 10 36 28 05i 9 21 21.45 85.08 12.6 2.1 540 10 48 29 09* 9 42 22.68 89.88 13.8 2.3 550 11 00 30 15 10 03* 23.96 94.85 15.1 2.5 560 11 12 31 21* 10 26 25.27 99.97 16.5 2.8 570 11 24 32 29* 10 48 26.62 105 19 18.0 3.0 580 11 36 33 38J 11 10* 28.01 110.62 19.6 3.3 590 11 48 34 48* 11 34 29.48 116.27 21.3 3.6 600 12 00 36 00 11 58 30.97 122.13 23.2 3 9 1 in 40 ft. TABLB Vlli. TRANSITION SPIRAL. Length D A o y x COR. t COR. 10 015 001' 000' "Too" "Too"" .0 .0 20 30 03 01 30 45 07 02 .02 40 1 00 12 04 .01 .05 50 1 15 19 06 .02 .09 60 1 30 27 09 .04 .16 70 1 45 37 12 .06 .25 80 2 00 48 16 .09 .37 90 2 15 1 01 20 .13 .53 100 2 30 1 15 25 .18 .73 110 2 45 1 31 30 .24 .97 120 3 00 1 48 36 .31 1.25 130 3 15 2 07 42 .40 1.60 140 3 30 2 27 49 .50 2.00 150 3 45 2 49 56 .61 2.45 160 4 00 3 12 1 04 .74 2.97 170 4 15 3 37 1 12 .89 3.57 180 4 30 4 03 1 21 1.06 4.24 .1 190 4 45 4 31 1 30 1.25 4.99 .1 200 5 00 5 00 1 40 1.45 5,81 .2 210 5 15 5 31 1 50 1 68 6.72 2 220 5 30 6 03 2 01 1.93 7.74 .2 230 5 45 6 37 2 12 2.20 8.85 .3 240 6 00 7 12 2 24 2.51 10.05 .4 250 6 15 7 49 2 36 2.84 11.37 .5 .1 260 6 30 8 27 2 49 3.19 12.77 .6 .1 270 6 45 9 07 3 02 3.57 14.29 .7 .1 280 7 00 9 48 3 16 3.98 15.94 .8 .1 290 7 15 10 31 3 30 4.42 17-70 1.0 .2 300 7 30 11 15 3 45 4.89 19.59 1.2 .2 310 *7 45 12 01 4 00 5.40 21.61 1.4 2 320 8 00 12 48 4 16 5.94 23.76 1.6 .'3 330 8 15 13 37 4 32 6.51 26.05 1.9 3 340 8 30 14 27 4 49 7.12 28.46 2.2 .4 350 8 45 15 19 5 06 7.77 31.03 2.5 .4 360 9 00 16 12 5 A 8.46 33.74 2.9 .5 370 9 15 17 07 5 42 9.18 36.62 3/3 .5 380 9 30 18 03 6 01 9.95 39.64 3.7 .6 390 9 45 19 01 6 20 10.75 42.82 4.3 .7 400 10 00 20 00 6 40 11.60 46.16 4.9 .8 410 10 15 21 01 7 00 12 47 49.65 5 5 .9 420 10 30 22 03 7 21 13.39 53.28 6.2 1.0 430 10 45 23 07 7 42 14.38 57.10 6 9 1.2 440 H 00 24 12 8 04 15.39 61.12 7 8 1.3 450 11 15 25 19 8 26 16.45 65.32 8.7 1.5 ( 1 in 30 ft. TABLE IX. TRANSITION SPIRAL. Length ' D J 8 y x COR. /COR. 10 020' oor 000' .00 .00 .0 .0 20 40 04 01 .00 .01 30 1 00 09 03 .01 .03 40 1 20 16 05 .02 .06 50 1 40 25 08 .03 .12 60 2 00 36 12 .05 .21 70 2 20 49 16 .08 .53 80 2 40 1 04 21 .12 .50 90 3 00 1 21 27 .18 .71 100. 3 20' 1 40 33 .24 .97 110 3 40 2 01 40 .32 1.29 120 4 00 2 24 48 .42 1.68 130 4 20 2 49 56 .53 2.13 140 4 40 3 16 1 05 .67 2.66 150 5 00 3 45 1 15 .82 3.27 .1 160 5 20 4 16 1 25 .99 3.97 .1 170 5 40 4 49 1 36 1.19 4.76 .1 180 6 00 5 24 1 48 1.41 5.65 .2 190 6 20 6 01 2 00 1.66 6.65 .2 200 6 40 6 40 2 13 1.94 7.75 .3 210 7 00 7 21 2 27 2.24 8.97 .3 .1 220 7 20 8 04 2 41 2.58 10.31 .4 .1 230 7 40 8 49 2 56 2.95 11.77 .5 .1 240 8 00 9 36 3 12 3.35 13.38 .7 .1 250 8 20 10 25 3 28 3.78 15.11 .8 .1 260 8 40 11 16 3 45 4.25 17.00 1.0 ,2 ? 270 9 00 12 09 4 03 4.76 19.02 1.2 .2 280 9 20 13 04 4 21 5.31 21.20 1.4 .2 290 9 40 14 01 4 40 5.90 23.55 1.7 .3 300 10 00 15 00 5 00 6.53 26.05 2.0 .3 310 10 20 16 01 5 20 7.20 28.72 2.4 .4 320 10 40 17 04 5 41 7.92 31.57 2.8 .5 330 11 00 18 09 6 03 8.69 34.59 3.3 .5 340 11 20 19 16 6 25 9.49 37.80 3.8 .6 350 11 40 20 25 6 48 10.35 41.19 4.4 .7 360 12 00 21 36 7 11 11.25 44.78 5.1 .8 370 12 20 22 49 7 36 12.21 48.56 5.8 1.0 380 12 40 24 04 8 00 13.22 52.53 6.6 1.1 390 13 00 25 21 8 26 14.28 56.71 7.6 1.3 400 13 20 26 40 8 52 15.39 61.10 8.6 1.4 410 13 40 28 01 9 19 16.56 65.69 9.7 1.6 420 14 00 29 24 9 47 17.79 70.49 10.9 1.8 430 14 20 30 49 10 15 19.07 75.51 12.3 2.1 440 14 40 32 16 10 43 20.41 80.74 13.7 2.3 450 15 00 33 45 11 13 21.81 86.19 15.4 2.6 TABLE X. TRANSITION SPIRAL. . 1 in 20 ft. Length D A o y x COR. /COR. 10 030' 001' 000' .00 .00 .0 .0 20 1 00 06 02 .01 30 1 30 13 04 .01 .04 40 2 00 24 08 .02 .09 50 2 30 37 12 .05 .18 60 3 00 54 18 .08 .31 70 3 30 1 13 24 .12 .50 80 4 00 1 36 32 .19 .74 90 4 30 2 01 40 .26 1.06 100 5 00 2 30 50 .36 1.45 110 5 30 3 01 1 00 .48 1.94 120 6 00 3 36 1 12 .62 2.51 130 6 30 4 13 1 24 .79 3.20 140 7 00 4 54 1 38 .99 3.99 .1 150 7 30 5 37 1 52 1.22 4.90 .1 160 8 00 6 24 2 08 1.48 5.96 .2 170 8 30 7 13 2 24 1.78 7.15 .3 180 9 00 8 06 2 42 2.11 8.49 .4 190 9 30 9 01 3 00 2.49 9.98 .5 200 10 00 10 00 3 20 2.90 11.62 .6 .1 210 10 30 11 01 3 40 3.36 13.45 .8 .1 220 11 00 12 06 4 02 3.86 15.44 1.0 .2 230 11 30 13 13 4 24 4.41 17.63 1.2 .2 240 12 00 14 24 4 48 5.01 20.01 1.5 .3 250 12 30 15 37 5 12 5.66 22.60 1.8 .3 260 13 00 16 54 5 38 6.37 25.38 2.2 .4 270 13 30 18 13 6 04 7 12 28 39 2.7 .5 280 14 00 19 36 6 32 7.94 31.62 3.3 .6 290 14 30 21 02 7 00 8.82 35.10 3.9 .7 300 15 00 22 30 7 29 9.76 38.83 4.6 .8 310 15 30 24 02 8 00 10.76 42.73 5.4 .9 320 16 00 25 36 8 31 11.82 46.92 6.3 1.1 330 16 30 27 13 9 04 12.95 51.36 7 4 1.2 340 17 00 28 54 9 37 14.15 56.05 8.6 1.4 350 17 30 30 37 10 11 15.43 61.09 9.9 1.7 360 18 00 32 24 10 46 16.75 66.31 11.3 1.9 370 18 30 34 14 11 19 18.16 71.63 13.0 2.2 380 19 00 36 06 12 00 19.65 77.35 14.8 2.5 390 19 30 38 02 12 38 21.21 83.41 16 8 2.8 400 20 00 40 00 13 17 22.87 89.83 19.0 3.2 TABLE XI. TRANSITION SPIRAL. 1 in 10 ft. Length D A B y ^COR. /COR 10 100' 003' 001' .00 .00 .0 .0 20 2 12 04 .01 .02 30 3 27 09 .02 .08 40 4 48 16 .05 .19 50 5 1 15 25 .09 .36 60 6 00 1 48 36 .16 .63 70 7 2 27 49 .25 1.00 80 8 3 12 1 04 .37 1.49 90 9 4 03 1 21 .53 2.12 100 10 5 00 1 40 .73 2.91 . 1 110 11 00 6 03 2 01 .97 3 87 .1 120 12 7 12 2 24 1.26 5.02 .2 130 13 8 27 2 49 1.60 6.38 .3 140 14 9 48 3 16 1.99 7.97 .4 .1 150 15 11 15 3 45 2.45 9.79 .6 .1 160 16 00 12 48 4 16 2.97 11.87 .8 .1 170 17 14 27 4 49 3.56 14.23 1.1 .2 180 18 16 12 5 24 4.23 16.87 1.4 .2 190 19 18 03 6 01 4 97 19.81 1 9 .3 200 20 20 00 6 39 5.79 23.07 2.4 .4 210 21 00 22 03 7 20 6.70 26.65 3.1 .5 220 22 24 12 8 03 7.69 30.58 3.9 .6 230 23 26 27 8 48 8.78 34.86 4.8 .8 240 24 28 48 9 35 9.96 39.49 6 1.0 250 25 31 15 10 23 11.24 44.49 7.3 1.2 TABLE XII. FACTORS FOR ORDINATBS. To find jv, multiply o by the factor for the ratio found by dividing the distance of the point from the P. S. or P. C. C. by the half-length of spiral. Ratio to Yz length 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 C' Factor... .0005 .004 .014 032 063 108 172 .256 .365 .500 TABLE XIII. SPIRAL DEFLECTION ANGLES FOR FIRST CHORD LENGTH. Values of 6 l in minutes for use with Table XIV. The first line of captions gives the distance in feet in which 1 of degree of-curve is attained. The second line of captions gives the value of a Angles are given in minutes Chord 200 150 125 100 SO 75 Chord Length X 2 A % - 1 IX . IK Length 10 .050 .067 .080 .100 .125 .133 10 11 .060 .081 .097 .121 .151 .161 11 12 .072 .096 .115 .144 .180 .192 12 13 .084 .113 .135 .169 211 .225 13 14 .088 .131 .157 196 .245 .261 14 15 .112 .150 .180 .225 .281 .300 15 16 .128 .171 .205 .256 .320 .341 16 17 .144 .193 .231 .289 .361 385 17 18 .162 .216 .259 .324 .405 .432 18 19 .180 .241 .289 .361 .451 .481 19 20 .200 .267 .320 .400 .500 533 20 25 .312 .417 .500 .625 .781 .833 25 30 .450 .600 .720 .900 1 125 1.200 30 35 .612 .817 .980 1.225 1.531 1.633 35 40 .800 1.067 1.280 1.600 2.000 2.133 40 45 1.012 1.350 1.620 2.025 2.531 2.700 45 50 1.250 1.667 2.000 2.500 3.125 3.333 50 Cho rH rHrHrHrHpH INSTRUMENT AT CHORD POINT NUMBER 10 r I iO co rH ON I^N. 10 ft ft ft CO CO CO ft VO VO ft O cxi ON VO co O CO CM CM CM CM ft vO VO ft vo CM oo ftr-s rH rH V* rH CM t^ O rH O 1^ ON t^ VO ft Cxi ON CO co co co co CM CM iO VO tO CM t^ ft rH CO tO CM CM CM rH rH gSSoS? CO rH CO ft CO O O CO co CM O ON !>. ft CO CO CO CM CM CM ft CO O O CO CM ON t^ ft O CM -H rH r-i rH SS ?S3 rH OO iO O co ft O CO ON 00 CO to CM ON VO rH rH rH CO / s co t>. i ' Q - p rH rH rH CM O VO O CM CM ft CO rH O CO VO CM CM CM CM rH rH O VO O CM CM ft rH ON VO CO rH rH ft O 00 CO Q co 1>. O ft O iH ON VO ON O co 00 IO -H- 10 CO CO rH iH rH CM CM LO O ft VO VO ft IO T CO CM HQ VC ft ft VO O rH CO to J> O rH VO ft ft VO O CM IO GO rH IO rH rH rH CM CM ftf CM t^ O rH CO CO CM CM rH O 1 """ 1 OO IO ft IO 00 CM ft VO OO O CO O ON O co CO VO CO CM iO rH rH rH CM CM OO OO ft CO O CM i-H rH Q rH CM VO CM O O CM CO IO t^ ON rH rH VO CM O O M CO VO ON CM to iH rH rH CM CM oq O """^ O 10 CM rH CM ft 10 t^-ONrH to o t>i vo t^ CO VO CO rH ft rH rH rH CM CM rH CM ft O 00 00 O rH rHCM O ft O 00 00 ft IO t^ OO O rH O ft O CO CO CO iO CO O CO rH rH r-l CM CM O rH ft ON VO O o IHCSI VO ON ft rH o CO ft VO 00 O rH rH ft ON VO IO CM ft VO ON CM rH rH rH rH CM pioj pjoq<) OrHOqOXtflO COt-CQOSO rH rH rH rH rH rH TABLE XV. STREET RAILWAY SPIRAL. =2000. =2000. Length Radios A y .rCoR. /COR. 5 400 021' 007' .01 .00 .00 10 200 1 26 29 .08 .00 15 133.33 3 13 1 04 .28 .00 16 125 3 40 1 13 .34 .01 20 100 5 44 1 55 .17 .67 .02 21.06 95 6 21 2 07 .19 .78 .03 22.22 90 7 04 2 21 .23 .91 .03 .00 23.53 85 7 56 2 39 .27 1.09 .04 -.01 25 80 8 57 2 59 .32 1.30 .06 .01 26.67 75 10 11 3 24 .39 1.58 .08 .01 28.57 70 11 41 3 54 .48 1.94 .12 .02 30 66.67 12 54 4 18 .56 2.24 .15 .02 30.77 65 13 34 4 31 .61 2.42 .17 .03 33 33 60 15 55 5 18 .77 3.07 .26 .04 36.36 55 18 56 6 19 1.00 3.98 .40 .07 40 50 22 55 7 38 1.32 5 27 .64 .11 44.44 45 28 17 9 25 1.81 7.19 1.08 .18 50 40 35 49 11 54 2.56 10.13 1.95 .32 55 36.36 43 20 14 23 3.39 13.29 3.15 .52 56.05 35.68 45 00 14 55 3.59 14-02 3 46 .58 TABLE XVI. =1500. STREET RAILWAY SPIRAL. = 1500. Length Radius A y .r COR. /COR. 5 300 029' 010' .01 .00 .00 10 150 1 55 38 .11 .00 12 125 2 45 55 .19 .CO t f \j 100 4 18 1 26 .38 .01 15.79 95 4 46 1 35 .43 .01 16.67 90 5 19 1 46 .51 .01 17.65 85 5 58 1 59 .61 .02 18.75 80 6 43 2 14 .18 .73 .03 20 75 7 38 2 33 .22 .89 .03 .00 21.43 70 8 46 2 55 .27 1.09 .05 .01 23.08 65 10 10 3 23 .34 1.36 .07 .01 25 60 11 56 3 59 .43 1.74 .11 .02 27.27 55 14 13 4 44 .56 2.24 .17 .03 30 50 17 11 5 44 .75 2.98 .27 .04 33.33 45 21 13 "7 04 1.02 4.07 .45 .07 35 42.86 23 24 7 47 1.18 4.71 .57 .09 37.50 40 26 51 8 56 1.45 5.77 .82 .14 40 37.50 30 34 10 10 1.76 6.96 1.14 .19 42.86 35 35 05 11 40 2.16 8.51 1.61 .27 45 33.33 38 41 12 51 2.49 9.80 2.05 .34 i TABLE XVII. STREET RAILWAY SPIRAL. -^1250 = 1250 Length Radius 4 y x COR. /COR. 5 250 034' oii' .02 .00 .00 10 125 2 17 46 .13 15 83.33 5 11 1 44 .45 .01 15.62 80 5 36 1 52 .51 .01 16.67 75 6 22 2 07 .15 .62 .02 17.86 70 7 19 2 26 .19 .76 .03 .00 19.23 65 8 29 2 50 .24 .95 .04 .01 20 62,50 9 10 3 03 .26 1.04 .05 .01 20.83 60 9 56 3 19 .30 1.20 .06 .01 22.73 55 11 50 3 57 .39 1.56 .10 .02 25 50 14 19 4 46 .52 2 07 .15 .02 27.78 45 17 41 5 54 .71 2.84 26 .04 30 41 66 20 38 6 52 .89 3.56 38 .06 31.25 40 22 24 7 28 1.01 5.04 .47 .08 35 35.71 28 05 9 20 1.40 5.62 .83 .14 TABLE XVIII. -1000 STREET RAILWAY SPIRAL. = 1000 Length Radius A 8 o y x COR. /COR. 5 200 043' 014' .02 .00 .00 10 100 2 52 57 .17 .01 15 66.67 6 27 2 09 .56 .02 15.39 65 6 47 2 16 .15 .61 .02 16.67 60 7 57 2 39 .19 .77 .03 .00 58.18 55 9 28 3 09 .25 1.00 .05 .01 20 50 11 27 3 49 .33 1 33 .08 .01 22.22 45 14 09 4 44 .46 1.83 .13 .02 25 40 17 54 5 57 .64 2.58 .24 .04 28.57 35 23 23 7 47 .97 3.84 .47 .08 30 33.33 25 47 8 34 1.12 4.45 .60 .10 33.33 30 31 50 10 35 1.53 6.05 1.03 .17 TABLE XIX. STREET RAILWAY SPIRAL, = 750 = 750 Length Radius A 8 y x COR. /COR. 5 150 057' 019' .03 .00 .00 i r\ \J 75 3 49 1 16 .22 .01 15 50 8 54 2 52 .19 .75 .03 .00 16.67 45 10 37 3 32 .26 1.03 .06 .01 18.75 40 13 24 4 28 .36 1.46 .10 .02 20 37.5 15 17 5 05 .44 1.75 .14 .02 21.43 35 17 32 5 50 .57 2.27 .20 .03 25 30 23 52 7 -S7 .86 3.43 .43 .07 TABLE XX. OFFSETS FOR SPIRALS. I) 3 4 5 6 7 a L L L L L 0.5 7 83 6.000 0.6 5 44 5.000 12.86 6.667 0.7 4 00 4 286 9.47 5.714 18.47 7-143 0.8 3.06 3-750 7-24 5-000 14-06 6.250 24-34 7.500 38-46 8.750 0.9 2-41 3-333 5.71 4-444 11.14 5-555 19-22 6-667 30.41 7.778 1.0 1.96 3.000 4.64 4-000 9.04 5-000 15-60 6.000 24.71 7. 000 1.1 1.63 2.727 3.84 3-636 7.48 4.546 12.88 5.455 20-42 6-364 12 1.36 2.500 b.c2 3-333 6.30 4.167 10.84 5-000 17-16 5-833 1.3 1.16 2.308 2.75 3-077 5-36 3 846 9.26 4.615 14-63 5-384 1.4 1 00 2.143 2.37 2-857 4.61 3-572 7.99 4-286 12 65 5-000 1.5 .87 2.000 2.07 2.667 4 01 3 334 6.95 4.000 11-04 4.667 1.6 1.81 2.500 3-51 3.125 6.11 3-750 9-71 4.375 1.7 1.60 2 353 3.13 2.941 5-41 3-529 8-59 4.117 1.8 1.43 2.222 2-80 2.778 4.81 3-334 7-62 3-889 1.9 1.30 2.105 2.50 2.632 4.35 3.158 6-90 3-684 2.0 1.16 2.000 2.27 2-500 3.91 3.000 6.22 3.500 2.2 .96 1.818 1.88 2.272 3.22 2-727 5-13 3.182 2-4 .80 1.667 1.57 2.084 2-72 2-500 4-33 2.917 2.6 1.20 1.923 2.32 2-308 3.67 2.692 2.8 1.14 1-785 1.99 2.143 3.17 2.500 3.0 1.00 1.667 1.74 2.000 2.78 2.333 D 8 9 1O 11 12 a o L L o L L o L 1.0 36.70 8-000 1.2 25-53 6.667 1.4 18.80 5-714 26.72 6.429 1.6 14.45 5-000 20.47 5-625 1-8 11.40 4.445 16.23 5-000 22.15 5.556 2.0 9.28 4.000 13 20 4.500 18.05 5.000 2-2 7.67 3.636 10.90 4.090 14.94 4 545 19-78 5.000 2.4 6.43 3.337 9-17 3-750 12.50 4-167 16.63 4.584 21 54 5.000 2.6 5.47 3.077 7-81 3.461 10.64 3-846 14-19 4.231 18 43 4.615 2.8 4.74 2.857 6.77 3.214 9.21 3-571 12.19 3.927 15.83 4. 285 3-0 4.14 2.667 5.88 3.000 8-02 3.333 10.70 3.667 13-84 4 000 3-2 3.62 2-500 5.15 2.812 7-04 3-125 9.39 3.438 12.15 3-750 3-4 3.20 2.353 4-58 2-647 6.25 2-941 8.34 3 234 10.79 3-528 3-6 2.86 2.222 4.08 2 500 5.59 2-778 7.44 3.056 9.61 3-334 3-8 2.56 2-106 3.67 2.369 5-01 2.632 6.64 2.895 8-67 3.158 4.0 2.32 2.000 3-30 2.250 4-52 2.500 6.02 2.750 7.81 3 000 4.5 1.84 1.778 2.61 2.000 3-57 2-222 4.74 2.444 6.20 2.667 5.0 1.48 1.600 2 11 1-800 2.90 2.000 3.86 2.200 5.01 2.400 5-5 1.23 1.454 1-75 1 636 2.40 1-818 3 19 2.000 4.14 2.182 6-0 1.03 1.3J4 1.47 1.500 2 02 1 667 2.63 1.834 3.48 2.000 X >ATE LIBRARY USE TURN TO DESK FROM WHICH BORROWED f TS LOAN DEPT. HIS BOOK IS DUE BEFORE CLOSING TIME ON LAST DATE STAMPED BELOW TURN ALTY JRTH DAY JN25 1972 LD62A-30m-2,'71 (P2003slO)9412A-A-32 General Library University of California Berkeley Y A 380514 OF CALIFORNIA LIBRARY