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 BOUGHT WITH THE INCOME OF THE 
 
 SAGE ENDOWMENT FUND 
 
 THE GIFT OF 
 
 HENRY W. SAGE 
 
 1891 
 
Cornell University Library 
 arV18750 
 
 The transition spiral and its introducti 
 
 3 1924 031 215 142 
 olin.anx 
 
Cornell University 
 Library 
 
 The original of tliis book is in 
 tine Cornell University Library. 
 
 There are no known copyright restrictions in 
 the United States on the use of the text. 
 
 http://www.archive.org/details/cu31924031215142 
 
THE TRANSITION SPIRAL 
 
THE 
 TRANSITION SPIRAL 
 
 AND 
 
 ITS INTRODUCTION TO RAILWAY 
 CURVES 
 
 WITH 
 
 FIELD EXERCISES IN CONSTRUCTION AND 
 ALIGNMENT 
 
 BY 
 
 ARTHUR LOVAT HIGGINS 
 
 M.Sc, A.R.C.S., A.M.INSI.C.E. 
 
 Assistant-Lecturer in Engineering in the Victoria University of 
 
 Manchester, and formerly Assistant to the Professor of Civil 
 
 Engineering in the Queen's University of Belfast, Author of 
 
 " The Field Manual " 
 
 NEW YORK 
 
 D. VAN NOSTRAND COMPANY 
 
 EIGHT WARREN STREET 
 
Printed in Gteat Britain 
 
PREFACE 
 
 The aim of this handbook is to afford concise and 
 complete demonstration of the introduction of transition 
 curves, or spirals, in the various aKgnment problems 
 that accrue in connection with the construction or 
 improvement of railway curves. 
 
 Now the extent to which such problems can be 
 satisfactorily essayed depends upon the mathematical 
 development of the form of transition adopted, and 
 the facility with which this development admits 
 utilisation in the field — the effective compromise of 
 mathematical truth with practical precision. 
 
 Hence, besides fulfilling its objective function, the 
 ideal transition curve must be easy and flexible in 
 construction; a curve with simple, definite relations 
 to main curves and straights ; and, above all, one which 
 can be introduced without the aid of special tables, 
 such as hamper the construction of American tabular 
 spirals. 
 
 The curve most readily amenable to these require- 
 ments is to be found in the clothoid X=m\/^> the spiral 
 advanced and investigated in this connection by 
 Mr. James Glover, M.A., A.M.I.C.E., in a paper read 
 to the Institution of Civil Engineers.* 
 
 This spiralt yields an approximation that possesses 
 essentially simple properties, and, without introducing 
 appreciable working error, lends itself to construction 
 
 * Transition Curves for Railways, Proo. Inst. Civ. Engs., 
 vol. 03d. (June, 1900). 
 
 t Ex " Sino Loria." First referred to by Jas. Bemouilli 
 (1664-1705); rediscovered by Comu (1874), and called Oornu's 
 spiral in optics; and fully disouBsed by Cesaro (1890), who named 
 it the " olothoid." 
 
vi PEEFACE 
 
 in so many ways that it may justly be styled the 
 transition spiral. 
 
 1. It is as easily applied to the degree system as to 
 the radius system, and, adapting itself to the curve, 
 is superior to tabular polychord spirals which form 
 the bulk of the multiform compound curves largely 
 employed in America. 
 
 2. It may be set out by linear means solely with the 
 theodolite, or, if occasion arise, with two theodolites, 
 no special tables being required in its various modes 
 of construction. 
 
 3. It is virtually interchangeable with the cubic 
 parabola, and may thus supply dimensions not regu- 
 larly associated with the latter form of transition 
 curve. 
 
 4. Its relations with tangent and circle, or circle 
 and circle, are simple and definite, and thus facilitate 
 its introduction to all cases of alignment. 
 
 The following pages contain an essay on the length 
 of transition curves, a complete demonstration of the 
 various modes of utiUaing the clothoid as a transition 
 curve, a systematic investigation of the introduction 
 of the spiral to the various problems of alignment, 
 and a selection of field exercises, these last being an 
 effort to present the real matter, the actual operations, 
 in practical form. 
 
 The author desires to express his indebtedness to the 
 various sources of information named in the text, and 
 to Messrs. W. F. McMurray, W. E. EngUsh, G. Trotter, 
 and J. D. Gordon, of Queen's University, Belfast, 
 for their assistance in the preparation of the tables and 
 drawings. 
 
 ARTHUR LOVAT HIGGINS. 
 
 Qxtbbn's Uotveesitt, 
 Bblpast. 
 March 1, 1920. 
 
CONTENTS 
 
 INTRODUCTION 
 
 Superelevation and cant — Superelevation and speed — Tan- 
 gent and curve — Transition — Spiral elements — Length of 
 transition: Uniform approach; time approach; rules — 
 Tables: Data of time approaches in the radius and degree 
 systems - ... 
 
 PAET I 
 
 DEVELOPMENT 
 
 Intrinsic equation of clothoid — Derivation of cartesians and 
 deflections — Chord — deflection approximation and its 
 three simple properties — Investigation of modes of con- 
 struction — Construction as polychords ; " spiralling 
 through " and " spiralling in " — Normal deflections and 
 the occupation of spiral points as stations — Interpolating 
 stations — Construction by offsets— Central transition of 
 compound curves — Conclusion — Tables of appendix ex- 
 plained — Spiral notation - - - - 13 
 
 PAET II 
 
 ALIGNMENT 
 
 Classification of problems — The general case of alignment- 
 Transitions replacing the point of compound curvature — 
 The four general problems applied to simple, compound, 
 and reverse curves — Equation of equal routes - - 36 
 
viij CONTENTS 
 
 PAET III 
 FIELD EXERCISES 
 
 PAGIt 
 
 ^Problem] (1) Spiralling through to a simple curve; radius 
 unaltered — (2) Spiralling in to a simple curve; interpolat- 
 ing stations — (3) Offsetting a transition to a simple curve 
 — (4) Replacing a compound by branches of the same 
 degree, but suitably spiralled — (5) Replacmg a compound 
 by branches of the same radii, but with central and end 
 transitions — (6) Superseding a simple curve by one with 
 spirals, the greater portion of the original curve remaining 
 undisturbed — (7) Replacing a compound by one with 
 spirals, the point of compound curvature remaining un- 
 disturbed — (8) 'Replacing a reverse curve by two simple 
 curves with end transitions and spiral reverse — (9) Re- 
 placing a simple curve by a spiralled curve, the latter 
 approximating the original alignment — (10) Replacing a 
 compound by one with spirals that affords the same length 
 of route - - .... - 51 
 
 APPENDIX 
 
 Table I., Spiral elements and data — Radius system - 100 
 
 Table II., Spiral elements and data — Degree system - 104 
 
 Index - 109 
 
THE TRANSITION SPIRAL 
 
 INTRODUCTION 
 
 Superelevation and Cant. — Cant is the transverse 
 slope given to the surface of the rails on a curve in 
 order to counteract, -in particular,* the effect of the 
 centrifugal force of the moving train. • 
 
 Commonly, however, the term is used synonymously 
 with " superelevation," which is the amount by which 
 the outer rail is raised accordingly. 
 
 Thus, if V is the velocity of a train in feet per second, 
 when moving round a curve r feet in radius, the result- 
 ing centrifugal force is to the weight of the train as 
 
 : 1; and this is the ratio that the superelevation 
 
 must bear to the horizontal distance between the centres 
 of the rails, in order that the resultant force shall act 
 perpendicularly to the surface of the rails and so avoid 
 pressure between the wheel flanges and the outer rail. 
 Therefore, if V be the velocity in miles per hour, the 
 superelevation e in feet required on a curve of E chains 
 radius is expressed very nearly by the following relation, 
 
 * The modern bogey obviates the additional cant once neces- 
 sary on account of parallelism of axles and slip of wheels, leaving 
 centrifugal force alone to be considered. 
 
 1 
 
2 THE TEANSITION SPIRAL 
 
 G the distance in feet between the rail centres differing 
 inappreciably from its horizontal equivalent — 
 
 ■ V^ /IS 
 
 '=^9901' ^^^ 
 
 or, in the degree system, 
 
 e-G ^^ (2) 
 
 ^^ 86950' ^ ' 
 
 which for the standard gauge of 4' 8|" and ordinary 
 railheads reduce respectively to 
 
 "200E 
 
 (3) 
 
 ^^<^ '=17500- (^) 
 
 In practice the superelevation is commonly specified 
 in inches. 
 
 Superelevation and Speed. — Since it is impossible to 
 adjust the cant for all speeds, it is best to adapt it to 
 nearly the highest speed occurrent on the line. In 
 practice, however, the superelevation is limited by 
 (a) the comfort of the passengers, (h) the case of stop- 
 pages on the curve, (c) the effect of wind pressure, 
 and (d) the contingency of the outer wheel flange 
 mounting the rail.* 
 
 Six inches appears to be generally favoured as the 
 maximum amount of superelevation permissible; but 
 this limit is often carried to as much as 8 inches. 
 The maximum permissible superelevation may be 
 
 * The limitations of superelevation are thoroughly investi- 
 gated in an excellent paper by Mr. J. W. Spiller, A.M.I.C.E., 
 in the Proceedings of the Institution of Civil Engineers: High 
 Speed on Railway Curves, vol. clxxvi. (1 909). 
 
INTEODUCTION 3 
 
 regarded as fixing the maximum allowable speeds on 
 curves in accordance with the equation Vm= VSOOeE ; 
 but not rigidly, as a somewhat greater velocity, about 
 x^ths, is allowed in practice, causing a lateral pressure 
 on the outer rail, and displacing the centre of pressure 
 slightly laterally.* 
 
 In fact, eleven times the square root of the radius in . 
 chams appears to be the maximum sfeed in miles per 
 hour that is regularly attained on curves. That is: 
 
 V™=11VE, (5) 
 
 which reduces to 102-5 -i-\/'D in the degree system, or, 
 more conveniently. 
 
 Tangent and Curve. — ^It is evident that the tangent 
 point of a circular curve requires two conditions 
 that cannot be realised simultaneously: 
 
 (i.) The rails should be level transversely, the tangent 
 point being on a straight line; 
 
 (ii.) The rails should be canted in accordance with 
 the speed on the curve. 
 
 It is thus essential that the full superelevation of the 
 outer rail should be attained by the time the train 
 enters the circular portion. Before the introduction 
 of transition curves, it was customary to effect this — 
 in opposition to (i.) — by commencing the elevation 
 of the outer rail on the tangent itself, with the result 
 that a sharp jerk occurred when the train suddenly 
 entered the curve from the tangent. Besides, the 
 
 J^* N.B. — 11 ;^R corresponds to a superelavation of 0-6 foot for 
 centrifugal force. 
 
4 THE TEANSITION SPIRAL 
 
 widening of gauge for curves was gradually applied 
 on the straight, so that the requisite width was at- 
 tained at the tangent point. 
 
 Transition. — The objects of a transition curve are 
 therefore: 
 
 1. To afford a gradual increase of curvature from 
 zero at the tangent to the specified radius or degree of 
 the circular portion, or main curve; and 
 
 2. To afford the elevation of the outer rail in accord- 
 ance with such curvature, so that the full supereleva- 
 tion is attained simultaneously with the curvature 
 of the circular portion. 
 
 The effect of introducing transition curves, or spirals, 
 to a circular curve of given radius is to displace that 
 curve, or to change its radius, or to produce both dis- 
 placement and change. 
 
 Fig. 1 illustrates the fundamental case, a simple 
 curve being replaced by one of the same radius pro- 
 .vided with suitable spirals. Here the apex of the curve 
 is shifted from v to V, moving the tangent point from 
 a on the main tangent to a point A on the shift tangent, 
 the central angle A being reduced by 2$, where <I> 
 is the total angle of the spiral. 
 
 The perpendicular distance between the main tangent 
 and the shift tangent is styled the shift. This quantity 
 s, with the radius R of the main curve, constitute what 
 may be termed the general specification of a transition 
 curve. It is the characteristic element of such curves. 
 Whether the main curves be simple, or compound, or 
 reversed, and whatever be the form of transition em- 
 ployed, the shift bisects the transition curve, and the 
 transition curve bisects the shift. 
 
INTEODUOTION 5 
 
 Thus, in Fig. 1 the transition curve PQ passes 
 through the point E, bisecting AS, while the length 
 PEQ is itself bisected by AS. 
 
 Spiral Elements. — The elements of a spiral are those 
 dimensions which determine the limits and ruling points 
 
 MAia'&ngeaC 
 
 of spiral approaches without regard to the data by 
 which the spiral itself will be constructed. These are 
 in general the following: (Fig. 1) 
 
 1. The Length L. 
 
 2. The Shift s=Y -E vers $. 
 
 3. The Shift Increment, aS=s tan |A, A being the 
 angle of intersection. 
 
 4. The Spiral Extension, 8P=X -E sin <J>, * being 
 the total angle of the spiral. 
 
 5. The Tangent Excess, aP=X -E sin *+s tan JA, 
 being the sum of the increment and extension. 
 
6 THE TEANSITION SPIEAL 
 
 6. The Total Tangent Length, T,=(E+s) tan |A + 
 X - E sin ^, which is the sum of the tangent excess 
 and the normal length for a circular curve of radius E. 
 
 7. X, the terrnvnal x abscissa. 
 
 8. Y, the termmal y ordinate. 
 
 9. The Long Chord PQ and its Total Deflection, the 
 terminal deflection angle XI. PQ =X sec O. 
 
 It will be seen that the tangent distance IS is equal 
 to (E+s) tan |A, and, this corresponding to a curve 
 of radius (E+s), affords another aspect of regarding 
 the case in question. (Fig. 1) 
 
 Length of Transition Curves. — Considerable diver- 
 gence of opinion exists as to what constitutes the proper 
 length of a transition curve, numerous rules being 
 advanced, and these not infrequently being conflicting. 
 It is, therefore, advisable at this juncture to reduce 
 the subject to a tangible basis, bearing in mind that 
 local conditions may render an otherwise suitable length 
 impracticable. 
 
 Existing rules may be grouped with reserve into what 
 may be styled (1) Uniform Af^roaches, and (2) Time 
 Approaches. 
 
 1. Unifoem Approaches. — Some rules assume a 
 uniform rate of elevation gradient, running up the 
 required superelevation in the length of the transition 
 curve — 1 in 300; 1 in 500; 1 in 1,000, as the case may 
 be. Here it is assumed that the superelevation is 
 given in accordance with the allowable speed, e being 
 
 equal to m -p- or wV*D, where m and n are respectively 
 equal to 1^ and g|g^. (See p. 2.) 
 
INTEODUOTION 7 
 
 Hence if L and I are lengths of transition correspond- 
 ing to radii E and r and speeds V and v, then for the 
 
 same gauge, :^ and - equal a constant fc, E and e being 
 ill e 
 
 the cants accordingly. 
 
 That is— 
 
 LE Ir 
 
 and 
 
 or 
 
 mV" 
 
 'mv^' 
 
 L 
 
 r 
 
 
 L 
 
 r 
 
 -v^d' 
 
 D and d being the degrees taken as 5730 -> radius. 
 
 Thus, for the same radius the lengths of transition 
 would vary as the squares of the speeds employed. 
 
 2. Time Appkoachbs. — The rational rule for varying 
 speeds is that the same amount of superelevation should 
 be attained in the same time. 
 
 That is— 
 
 mV^ ■ • L 
 
 ^p- is the superelevation attained in time :;^> 
 
 and — is the superelevation attained in time -• 
 
 Hence the superelevation attained in unit time is 
 Y^ and ^— ; and, this being the same in each case, 
 
 T=^^ and L=a^g- 
 
 or T=-^ and L=a'V''D 
 
 I v^d ) 
 
 Thus, for the same radii the lengths of transition 
 
 would vary as the cubes of the speeds employed. 
 
8 THE TEANSITION SPIEAL 
 
 Hence it follows if a maximum superelevation, 6", 
 say, be given consistently, the maximum speeds 11 -\/Er 
 
 or — ^ being permissible, 
 
 L 
 l~ 
 
 
 L 
 
 Vd 
 
 r 
 
 WD 
 
 (8) 
 
 and L=a-\/E 
 
 or T= ^7^ and L=—^^ 
 
 And, conversely, under these conditions the shift s 
 and the superelevation E will be constant. 
 
 Shortt's Rule. — Mr. W. H. Shortt, A.M.I.C.E., 
 considers the length essential for a transition curve 
 from an entirely new standpoint, and advances that 
 this should be such as will eliminate the sense of dis- 
 comfort inherent in passing from straight to circle, the 
 rate of gain of acceleration serving as a measure of the 
 amount of discomfort experienced in passing the tangent 
 point.* 
 
 Thus, if a car c feet in length moves at a velocity of 
 
 V feet per second on a circle of r feet radius, the radial 
 
 v^ c 
 
 acceleration is — , this being gained in - seconds, the 
 
 time the car takes to pass the tangent point; and, assum- 
 ing the rate of gain of acceleration to be uniform during 
 the time the ear is passing the tangent point, its value 
 
 is - divided by -; that is — , which also may be re- 
 r '' V cr •' 
 
 garded as expressing the degree of sensational dis- 
 comfort. 
 Therefore, when there is a transition curve of length 
 
 * A Practiced Method for the Improvement of Existing Bailtoay 
 Curves, Proc. Inst. Civ. Engs., vol. clxxvi. (1909). 
 
INTEODUGTION 9 
 
 I feet, it follows that the acceleration will have to be 
 gained at the rate of ^ feet per second^ in a second, 
 
 assuming it to be gained uniformly. Now, if - be the 
 
 CL 
 
 maximum rate of change of acceleration that will pass 
 
 1 v^ 
 
 unnoticed, - should then equal ^; and, therefore, 
 
 the length I of the transition curve should equal — . 
 
 r/iiis, for all transition curves to have an equal effect, 
 their lengths must be proportional to the cube of the 
 velocity and inversely as the radius of the circular 
 curve. 
 
 Hence, Mr. Shortt's rule is rational in the sense of 
 time approach, while his observations from the stand- 
 point assumed afford a value for the constant a in ex- 
 pressions (7) and (8) of pp. 7 and 8, which may be written 
 
 accordingly 1= — and a-\/r respectively. 
 
 However, in Mr. Shortt's experience, a rate of change 
 
 of acceleration of 1 foot per second^ in a second is the 
 
 maximum that will pass unnoticed; and, in consequence, 
 
 a may be taken as unity, the length of the transition 
 
 v^ 
 curve becommg — 
 
 Or, changing the units, V being the speed in miles per 
 hour, E the radius in chains, and L the length of transi- 
 tion in chains, yj 
 
 ^=iMiE- («) 
 
 Since, however, the speed may vary, the fastest 
 trains must be considered when calculating the length 
 
10 THE TEANSITION SPIEAL 
 
 of the transition curve. But IIVE represents the 
 maximum speeds in miles per hour that are regularly 
 attained on curves in practice, and in consequence 
 V may be eliminated from Eq. 9, leaving L equal to 
 •96-s/B:; that is, for practical purposes — 
 
 L=VR. (10) 
 
 which expresses the length in feet as six times the 
 maximum speed, the time approach length advanced by 
 Mr. J. E. Stephens, an American authority.* 
 
 Accordingly the corresponding dimension in the 
 degree system is conveniently 
 
 L'=^ feet, (11) 
 
 Now the relation V=ll-13-\/E obtains only up to 
 speeds of about 82 miles per hour, and Eq. 10 there- 
 fore applies only to curves under 54-3 chains radius, 
 lengths exceeding the square root of this limit being 
 wasteful, as also are those calculated accordingly for 
 speeds considerably under ll-y/E. 
 
 However, the length of transition suitable for any 
 other velocity, V miles per hour, can be obtained from 
 Eq. 9; and in the case of curves over 54-3 chains radius 
 it is merely necessary to substitute V=82, such speed 
 being possible, reducing the expression to 
 
 T 400 , . 1 
 
 L=^ chams, I ^j^) 
 
 or L'=300D feetneaHj/.J 
 
 Mr. Shortt measures the efficiency of a transition 
 
 * The Six Ghord Spiral, by J. R. Stephens, G.E. 
 
INTEODUCTION 
 
 11 
 
 curve by the reciprocal of the rate of change of accelera- 
 tion, which, written as — :^ — , gives values greater 
 
 or less than unity according as lengths L longer or 
 shorter than ^/'B, are employed. Although somewhat 
 anomalous, such usage of the term " efficiency " is 
 defensible, being sufficiently incongruous to emphasise 
 that a transition is uselessly or unnecessarily long. 
 
 Tables. — The following tables exhibit in the radius 
 and degree systems the relations between the various 
 data of time approaches, the lengths of transition being 
 taken respectively as ■\/B, chains and 6Vto feet, in 
 accordance with maximum speeds V™ of ll*13\/Ii 
 and IOOh- y'D miles per hour. Under these conditions 
 the shift s is constant, being 33 inches and Mtw inches 
 
 respectively, as in the relation s=orR" 
 
 TABLE I. 
 Radius System. 
 
 Radius 
 
 Max. Speed 
 
 Length of 
 
 Superelevation 
 
 Normal 
 
 Gradient 
 
 R 
 
 V 
 
 Transition 
 
 E 
 
 Shift 
 
 of 
 
 Chains. 
 
 M.P.H. 
 
 L Chains. 
 
 Inches. 
 
 a. 
 
 Adjustment. 
 
 20 
 
 49-8 
 
 4•472^ 
 
 
 
 /'I in 580 
 
 25 
 
 55-6 
 
 5-000 
 
 
 
 
 1 in 660 
 
 30 
 
 6I-0 
 
 5-477 
 
 
 
 2 feet 
 
 1 in 723 
 
 35 
 
 65-8 
 
 5-916 
 
 
 6 inches 
 
 9mohes, 
 
 1 in 781 
 
 40 
 
 70-4 
 
 6-325 
 
 
 or 
 
 1 in 835 
 
 45 
 
 74-7 
 
 6-708 
 
 
 
 ■^^ chain 
 
 1 in 885 
 
 50 
 
 ■ 78-7 
 
 7-071 
 
 
 
 
 1 in 933 
 
 55 
 
 82-5 
 
 7-416J 
 
 
 
 
 U in 979 
 
 60 
 
 86-2 
 
 7-746 
 
 
 
 1 in ion 
 
 80 
 
 100-0 
 
 8-9U 
 
 
 
 1 m, 1181 
 
12 
 
 THE TEANSITION SPIEAL 
 
 TABLE II. 
 Dbgebe System. 
 
 Degree 
 
 Max. Speed 
 V 
 
 Length oj 
 Spiral 
 
 f 'Sv/perdevation 
 E 
 
 Normal 
 Shift 
 
 Gradient 
 of 
 
 M.P.H. 
 
 L Feet. 
 
 Inches. 
 
 s. 
 
 Adjustment. 
 
 1 
 
 lOO-O 
 
 eoo-oy 
 
 
 
 (1 in 1200 
 
 2 
 
 70-7 
 
 424-2 
 
 
 
 1 in 848 
 
 3 
 
 67-7 
 
 346-2 
 
 
 31-416 
 
 1 in 692 
 
 4 
 
 50-0 
 
 300-0 
 
 
 inches, 
 
 1 in 600 
 
 5 
 
 44-7 
 
 268-2 
 
 6 inches 
 
 or 
 
 1 in 536 
 
 6 
 
 40-8 
 
 244-8 ■ 
 
 2 feet 
 
 1 in 490 
 
 7 
 
 37-7 
 
 226-5 
 
 
 7A 
 
 1 in 452 
 
 8 
 
 35-4 
 
 212-4 
 
 
 inches 
 
 1 in 425 
 
 9 
 
 33-3 
 
 199-8 
 
 
 
 1 in 400 
 
 10 
 
 31-6 
 
 189-6J 
 
 
 
 1 in 379 
 
 N.B. — It is evident tkat the length of transition in feet 
 in either table is equal to the maximum speed multiplied 
 by the full superelevation in inches. Hence, if a curve is 
 elevated by an amount e less than 6 inches for a speed V 
 less than Vm, the necessary length of transition can be found 
 by multiplying the speed V by the superelevation e in 
 inches. 
 
PART I 
 DEVELOPMENT 
 
 The present section of this work deals with the 
 development of the clothoid as a railway transition 
 curve. 
 
 The intrinsic equation of this spiral is — 
 
 \=m-\/4> ; (i.) 
 
 where X is the 4istance of any point on the curve as 
 measured along the latter from its origin, m a constant 
 for any particular curve, and the angle between the 
 tangent at any point and the tangent at the origin — 
 an angle which in the present connection may be styled 
 the spiral angle. 
 
 Now the radius of curvature p of any point on the 
 spiral is 
 
 dX , dA. m ,.. , 
 
 whence on substituting for 0, 
 
 which indicates that the radius of curvature varies 
 inversely as the distance measured along the curve 
 from the origin, being infinite when X is zero, and r 
 
 13 
 
14 THE TEANSITION SPIRAL 
 
 when \ becomes I, the spiral merging into a circle of 
 radius r. (Pig- 2) 
 
 Hence, since /3=r when X=i, 
 
 m—VM, (iv.) 
 
 thus leading to an equation in terms of the data, the 
 
 following — 
 
 X=V2rZ^. (1) 
 
 It now becomes necessary to reduce Eq. 1 to some 
 tangible form in order that the curve may be readily 
 set out in the field. Now, by form, this spiral lends 
 itself the more readily to construction in a manner 
 akin to that of setting out circular curves with the 
 theodohte, a series of equal chords c being laid down 
 with deflection angles w, the latter being measured from 
 the tangent at the origin P (Pig. 2). This necessitates 
 
 the determining of the cartesians x and y, - being equal 
 
 to tan CD. 
 
DEVELOPMENT 15 
 
 Now dx=dX cos <f>= — - /T ' hence by expand- 
 ing cos and integrating, 
 
 ^^\ 5x2!^9x4! 13x6! / ^ ' 
 
 Similarly, dy=dX sin <f)= — ^ >? ; and by expand- 
 ing sin ^ and integrating, 
 
 Therefore, 
 
 tan „=-^=|+^g+^+jgL+ . . . (vu.) 
 
 Now on expanding tan <u, it can be shown that 
 0)=^ for all practical purposes, seldom exceeding 6°, 
 
 or, roughly, a tenth of a radian. 
 
 1. Thus the total deflection angle is one-third of the 
 total angle of the spiral. 
 
 Hence on substituting ^=1 and 0=<1> in Eq. 1, 
 
 Z=2r* (2) 
 
 and fi=573 -, (3) 
 
 r 
 
 fl being the total deflection angle expressed in minutes. 
 Further, Eq. 1 may be written as follows, nc approxi- 
 mating the distance \mn chords of length c — 
 
 (u=573 — =- minutes; (4) 
 
 Or, generally, co=n^k minutes, (5) 
 
 where fc is a constant depending upon the method by 
 which the curve is constructed. 
 
16 THE TEANSITION SPIEAL 
 
 2. Thus the deflections from the tangent at the origin 
 vary as the squares of the nurnber of chords or units of 
 distance along the spiral. 
 
 Also, if Y be the terminal ordinate in Eq. vi. — 
 
 Y-.<*-*i+...). 
 
 But^=7r; hence — 
 
 ^=6rV "14+ ■ • -h&r ""^""^ '^^^''^^' ^^^ 
 the ratio 1 - jr: 1 being -99922 when $=6°. 
 
 Further, if ^ is small, 0" is neghgible, as also are 
 higher powers. 
 
 Then a;— m-v/0 and z/=-n02; (viii.) 
 
 whence 2/=s — ^, a cubic parabola. 
 
 But m^=2ri; 
 
 therefore y~Rl' ('^) 
 
 Now the length of circular curve displaced by the 
 transition curve is exactly one-half the length of the 
 latter, these values being respectively r(p and 2r^ by 
 the spiral and circle accordingly. 
 
 Hence QN=Y=AS+^^ (Pig. 1) 
 
 =Shift s+o" by tbe circle. 
 But ^~a' by the spiral. 
 
 "6r 
 
 I _ 
 
 ^&r 8r 24r' 
 
 Therefore s=^'-^'=^. (8) 
 
DEVELOPMENT 17 
 
 In fact, the last relation applies more closely to the 
 clothoid than to the cubic, the curve with which it is 
 usually associated. 
 
 3. Thus the elements of the clothoid are for the lengths of 
 transition occurrent i/n -practice i/nterchangeahle with those 
 of the cubic paraiola, which, not without reason, remains 
 the engineer's conception of the ideal transition curve. 
 
 Construction. — The methods by which this curve 
 can be constructed in the field are substantially two — 
 
 I. By deflection angles from the main tangent, using 
 the theodolite and chain. 
 
 II. By rectangular co-ordinates, offsetting the curve 
 from the main tangent as though it were Proude's 
 parabola. 
 
 I. By Deflections. — ^The clothoid lends itself 
 readily to the three characteristic modes utilised 
 in constructing American Polychord Spirals — ^viz., 
 (1) Constant chord length, (2) Number of chords constant, 
 and (3) Constant deflection angles. 
 
 In all these an integral number of chords is under- 
 stood, and the points located are termed " spiral 
 points," as distinct from stations, which latter are to 
 be interpolated, often by an approximate process, with 
 the result that spiral points alone can be occupied in 
 the case of obstructions. 
 
 1. Constant Chord Length. — Using consistently as 
 
 the chord c an even fraction of a chain or 100 feet, 
 
 573c^ 
 and calculating the deflection in minutes from ^ps- 
 
 lOc^D 
 or -p — , L and c being in chains and hundreds of 
 
 feet accordingly. 
 
18 THE TEANSITION SPIEAL 
 
 In general, a chord length of about 30 feet is desir- 
 able; hence 83 feet in the radius system and 83^ feet 
 in the degree system. Here, however, the length of 
 transition may be 15 feet longer or shorter than that 
 given tentatively by known rules. 
 
 2. Chord Number Constant. — Using n equal chords, 
 10 preferably, and calculating deflections in minutes 
 
 from — r- :f^ and — LB, L being in chains and hundreds 
 
 n" n n^ 
 
 of feet respectively. 
 
 Here, as above, the deflections will run in geometrical 
 progression, but will normally be fractional values. 
 
 3. Deflections Constant. — Using a common initial 
 deflection a, one minute preferably, and calculating 
 
 the chord length c in chains or feet from V -fsq- or 
 
 V 
 
 573 
 
 lOmL i. , 
 
 jr— respectively. 
 
 Here, with « one minute, the deflections will run as 
 the square of the chord number n: V, 4', 9', 16', etc.; 
 but the length of transition may be as much as nearly 
 30 feet too long or short of the value given tentatively 
 by known rules. 
 
 Now, a transition curve is subsidiary to a main curve, 
 and the latter is never constructed through " curve 
 points," the points actually located being stations of 
 the curve. Further, an integral number of chords is 
 desirable, though such is only of accidental occurrence 
 in main curves. With the clothoid, however, neither 
 condition is imposed, and the spiral may therefore be 
 run in in a manner akin to the ranging of circular 
 curves. Thus, the first mode may be modified as follows : 
 
DEVELOPMENT 19 
 
 4. Spiralling Through. — " Chaining through " with 
 38 feet or 38^ feet Chords consistently, calculating ihe 
 deflections in minutes per ohain^ or per 100 feet^ from 
 
 ■pp and 10 Y> L being in chains or hundreds of feet 
 
 respectively. 
 
 Spiralling through is the most feasible mode of con- 
 structing transition curves, the procedure being con- 
 sistent with that of setting out the main curves. The 
 fractional deflections are certainly inconvenient, but, 
 after all, such are common to all circular curves. 
 Besides, latitude in selecting the value of L in either 
 
 573 D 
 
 Pp or 10 Y allows simplification in many cases, 
 
 analogous modification in the circular curve being 
 impossible. 
 
 5. Spiralling In. — But in regard to simplicity and 
 accuracy the third mode commends itgelf, particularly 
 in tracing spirals before the final centring. Again, 
 an integral number of chords with inordinate length of 
 transition is not obligatory, a subchord removing the 
 latter objection, while, on the other hand, both sub- 
 chord and inordinate length are avoided by adopting f ' 
 or J' as the initial deflections, construction thus with V 
 reading theodolites being as accurate as in other modes. 
 
 Spiralling Through. — In this mode the expression 
 
 — J— reduces to ^-, c being unity; the result — 
 
 fc=;-S; - (9) 
 
 "LE' 
 
 101 
 
 L 
 
 or fe'=^, (10) 
 
20 
 
 THE TEANSITION SPIEAL 
 
 being the deflection angle in minutes per chain, 66 feet 
 or 100 feet respectively. 
 
 Normally 83 feet or 33J feet chords will be em- 
 ployed, giving alternate points as stations in the 
 radius system, and third points as such in the degree 
 system. 
 
 Eadius System. — ^Thus, if the chainage of P, the 
 point of spiral, is 26' 82, the first point on the spiral will 
 be Sta. 27 at the end of a subchord of -18 chain; 
 the next point will be at -68 chain from P; the next, 
 Sta. 28, at M8 chain from P; and so on. All that is 
 necessary, therefore, is to tabulate the distances of the 
 spiral points for the whole transition, and to calculate 
 the corresponding deflections by multiplying the squares 
 of these distances by the deflection angle per chain, 
 
 which is — i, the length L being equal to -\/E chains. 
 
 Hence, if the spiral in question is to approach a 
 86 chain main curve, the deflections will be as follows, 
 
 573 
 
 the deflection angle in minutes per chain^ being 
 
 or 2-653: 
 
 86x6 
 
 Spiral point 
 
 P 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 Chainage . . . . 
 
 26-82 
 
 27 
 
 27-5 
 
 28 
 
 28-5 
 
 29-0 
 
 29-5 
 
 Distance . . . . 
 
 
 
 0-18 
 
 0-08 
 
 M8 
 
 1-68 
 
 2-18 
 
 2-68 
 
 
 
 9 
 
 o 
 
 K. 
 
 H' 
 
 "S 
 
 "S 
 
 
 
 1— ' 
 
 Oi 
 
 (— < 
 
 05 
 
 Ai 
 
 Ci 
 
 
 
 00 
 
 b3 
 
 lO 
 
 00 
 
 00 
 
 
 00 
 
 
 
 X 
 
 X 
 
 X 
 
 X 
 
 X 
 
 X 
 
 
 
 K) 
 
 CO 
 
 to 
 
 u> 
 
 to 
 
 » 
 
 
 
 03 
 
 OS 
 
 (35 
 
 & 
 
 Oi 
 
 
 Defleotion angle 
 
 
 w 
 
 s 
 
 8 
 
 S 
 
 a 
 
 w 
 
 in minutes . . 
 
 
 
 0-09 
 
 1-23 
 
 3-69 
 
 7-49 
 
 12-Cl 
 
 19-05 
 
DEVELOPMENT 
 
 21 
 
 673 
 The total deflection angle il is also equal to -7=5= 
 
 — -=95-50 minutes. 
 
 " Shifting Forward." — When, as in the foregoing 
 example, the theodolite is stationed at the beginning 
 of the spiral, the deflections from the tangent thereat 
 to any points on the spiral may be said to be the normal 
 deflections for those points. However, it is sometimes 
 expedient or necessary to occupy some point on the 
 spiral other than its beginning P, and in such cases a 
 tangent at the point occupied should be determined, 
 the deflections being set out from this tangent. 
 
 It follows from Eq. vii. that the tangent at any point 
 q of the spiral can be determined by backsighting on the 
 origin P, and laying off an angle equal to twice the normal 
 deflection from P to q, or, expressed generally, 2q^k, 
 where fc is the normal initial deflection per chord or 
 chain. But no such simple rule obtains in deter- 
 mining the deflections from the tangent at q to 
 other points p of the spiral, or, conversely, the 
 angles between the points p and the tangent at q. 
 
 ^ 
 
 
 
 
 
 
 
 
 Spiral point 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 Q 
 
 Chainage . . . . 
 
 30 
 
 30-5 
 
 31-0 
 
 31-5 
 
 32-0 
 
 32-5 
 
 32-82 
 
 Distance . . 
 
 3-18 
 
 3-68 
 
 4-18 
 
 4-68 
 
 5-18 
 
 S-68 
 
 6-00 
 
 
 u 
 
 « 
 
 *. 
 
 If' 
 
 =? 
 
 V 
 
 3 
 
 
 1— i 
 00 
 
 X 
 
 X 
 
 to 
 
 t— 1 
 B 
 
 X 
 
 to 
 
 05 
 00 
 
 X 
 
 to 
 
 .25 
 X 
 
 J 
 
 'To 
 
 X 
 
 -Si 
 
 X 
 
 Deflection angle 
 in minutes . ., 
 
 26-83 
 
 CO 
 
 35-93 
 
 OS 
 
 46-35 
 
 58-10 
 
 OJ 
 
 s 
 
 71-18 
 
 85-59 
 
 95-61 
 
22 THE TRANSITION SPIRAL 
 
 Both are comprehended in the general case, which in- 
 volves finding the differences of the ordinates y^, and y^ 
 and those of the corresponding ahscissse aij, and x^, the 
 ratio of these differences being approximately the 
 deflections from a line through q parallel to the main 
 tangent, the tangent at q being incHned to this parallel 
 at an angle of 3g^/e, or ^^. 
 Thus: 
 
 Xp-x^ mV'f)p-mV(}>a mV^p^k -mV^q'li 
 
 Subtracting the inclination Qq% of the tangent at q, 
 the deflections from qto p are expressed by (j>^-q^)Jc 
 +q{p-q)ls or {q^-p^)k+q{q-p)'k, according as p, the 
 point sighted, is forward or rear of q, the point occupied. 
 
 The foregoing expressions reduce respectively to 
 N(3g+N)fc and N(3g-N)fe when the points p are 
 counted 1, 2, 3, etc., or (j3-g)=N chords forward or 
 rear of q, which retains its numerical value in chords or 
 in chainage from the origin P. Spiralling through will 
 not admit the use of the multipliers tabulated on p. 24. 
 
 Degree System. — ^The calculations in the degree 
 system may be made in the manner described for the 
 radius system, the 100 feet unit merely superseding 
 the chain. Accordingly spiral points would occur in 
 the main at intervals of 50 feet, which is somewhat 
 high for precise spiralling. But the reductions for a 
 33 J feet chord merely require that the station distances 
 shall be expressed in chordages, the initial deflection 
 per chord being one-nrnth that for a 100 feet length. 
 
 Thus, if P is at 72+39, stations will occur at distances 
 
DEVELOPMENT 23 
 
 therefrom of 61', 161', and 261', the ohordages of 
 which will be 1-83 for 78, 4-83 for 74:, and 7-83 
 for 75, while intermediate spiral points will inter- 
 sperse these at chordages -83 . . . 2-83, 3-83 . . . 5-83, 
 6-83 . . . 8-83. Hence, if the /spiral in view is^ to 
 approach a 4° main curve, thennitial deflection per 
 
 33^ feet chord will be - 10 =^, or - — - minutes; and 
 
 the deflections will run as the squares of the chordages , 
 viz., (■83)^xl-481, {1-83YX1-481, (2-83) ^x 1-481, 
 (3-83)^ X 1-481, {4-83yxl-481, (5-83) ''x 1-481, (6-83)^x 
 1-481, {7-83yxl'481, and (8-83)2x1-481, the last giving 
 the subchord which closes on Q. 
 
 Observation. — Siace stations are of less frequent occur- 
 rence in tte degree system than in the radius system, it is 
 doubtful if the foregoing procedure is to be preferred to 
 that of using the same chord length and length of spiral, 
 but commencing a full chord at P and closing with a sub- 
 chord on Q. Here the iuitial deflection for a 33 J feet chord 
 would be merely multiplied by 1^, 2^, 3^, 4^, etc., stations 
 being interpolated during progress to deflections given by 
 the squares of the chordages of the station distances multi- 
 plied by the initial deflection. 
 
 Nevertheless, such procedure cannot economically super- 
 sede " chaiaing through," as given for the radius system. 
 
 Spiralling In. — In this method the chord length c 
 
 573c^ 
 is calculated such that k is unity — i.e., —, — =1. 
 
 Hence — 
 1 
 
 23-94 
 
 VEL chains, L and E being in chains, (11) 
 or g_/y'12i^ feet, L also being in feet. (12) 
 
24 
 
 THE TEANSITION SPIEAL 
 
 Here stations are subsidiary to the construction, and 
 the curve is set out from its commencement with chords 
 of length c, the only subchord occurring at the end of 
 the spiral. Accordingly the deflections run as the 
 squares of the chord number, 1', 4', 9', 16', etc., the 
 
 last being ( - j or 573 ^ or lOLD, which may be made 
 
 integral in the selection of the length L. 
 
 ® 
 
 Sighting @ 
 
 Em 
 
 
 
 1 
 
 2 
 
 3 
 
 i 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 
 
 
 
 1 
 
 4 
 
 9 
 
 16 
 
 25 
 
 36 
 
 49 
 
 64 
 
 81 
 
 100 
 
 121 
 
 144 
 
 1 
 
 2 
 
 
 
 4 
 
 10 
 
 18 
 
 28 
 
 40 
 
 54 
 
 70 
 
 88 
 
 108 
 
 130 
 
 154 
 
 2 
 
 8 
 
 5 
 
 
 
 7 
 
 16 
 
 27 
 
 40 
 
 55 
 
 72 
 
 91 
 
 112 
 
 135 
 
 160 
 
 3 
 
 18 
 
 14 
 
 8 
 
 
 
 10 
 
 22 
 
 36 
 
 52 
 
 70 
 
 90 
 
 112 
 
 136 
 
 162 
 
 4 
 
 32 
 
 27 
 
 20 
 
 11 
 
 
 
 13 
 
 28 
 
 45 
 
 64 
 
 85 
 
 108 
 
 133 
 
 160 
 
 5 
 
 50 
 
 44 
 
 36 
 
 26 
 
 14 
 
 
 
 16 
 
 34 
 
 54 
 
 76 
 
 100 
 
 126 
 
 154 
 
 6 
 
 72 
 
 65 
 
 56 
 
 45 
 
 32 
 
 17 
 
 
 
 19 
 
 40 
 
 63 
 
 88 
 
 115 
 
 144 
 
 7 
 
 98 
 
 90 
 
 80 
 
 66 
 
 54 
 
 38 
 
 20 
 
 
 
 22 
 
 46 
 
 72 
 
 100 
 
 132 
 
 8 
 
 126 
 
 119 
 
 108 
 
 95 
 
 80 
 
 63 
 
 44 
 
 23 
 
 
 
 25 
 
 52 
 
 81 
 
 112 
 
 9 
 
 162 
 
 152 
 
 140 
 
 126 
 
 110 
 
 92 
 
 72 
 
 50 
 
 26 
 
 
 
 28 
 
 58 
 
 90 
 
 10 
 
 200 
 
 189 
 
 176 
 
 161 
 
 144 
 
 125 
 
 104 
 
 81 
 
 56 
 
 29 
 
 
 
 31 
 
 64, 
 
 11 
 
 242 
 
 230 
 
 216 
 
 200 
 
 182 
 
 162 
 
 140 
 
 116 
 
 90 
 
 62 
 
 32 
 
 
 
 34 
 
 12 
 
 288 
 
 275 
 
 260 
 
 243 
 
 224 
 
 203 
 
 180 
 
 115 
 
 128 
 
 99 
 
 68 
 
 35 
 
 
 
 The foregoing table affords multipUers for the initial 
 normal deflection k when a point other than the origin 
 
DEVELOPMENT 25 
 
 P is occupied, Q being comprehended only in spirals 
 consisting of an integral number of chords. It applies 
 only to spirals commencing with a full chord at P, the 
 case of "spiralling through" requiring individual 
 calculations, q not being integral in N(3g±N), where N 
 increases by increments of | or ^. 
 
 It is evident that the tabular values are the summa- 
 tions forward and rear of tangent points q of arithmeti- 
 cal series with a common difference of 2, the first 
 forward and rear values being respectively 3g+l and 
 3g -1. Thus, if the instrument is at Point 4, the first 
 forward value is 13, and the first rear value 11; and 
 if, as indicated, sigma denotes the instrument at Point 
 4, the summation of the following series gives the re- 
 quired multipliers, which themselves are deflections 
 in minutes in spiralling in, k being unity. 
 
 Spiral Points : P1234 56789 10 
 Differences: 5 + 7 + 9 +11 2 13 + 15 + 17 + 1Q + 21 +23 
 
 Deflections: 32 27 20 U 13 28 45 6i 85 108 
 
 Note. — Since the long ciiord and the tangents at P and Q 
 enclose angles of fl and 2i2 respectively, it follows that two 
 theodolites may be used, one at P, the other at Q, the deflec- 
 tions from the tangents thereat being respectively N^A 
 and N(3L - N)^, where N is counted in chords or chains 
 from P and Q, and L is the length of transition in chords 
 or chains. This fact suggests the occupation of Q and 
 reversed construction: a process that may be expediently 
 considered in spiralling through and in spiralling in with 
 an integral number of chords. 
 
 Interpolating Stations. — Stations at even chain- 
 ages around the spiral may be interpolated in a number of 
 ways, the following being preferable with the theodolite. 
 
26 THE TEANSITION SPIEAL 
 
 (i.) By calculating the chordage of the station points and 
 setting out the deflections as the squares of these values. 
 
 (ii.) By adding a deflection correction to the pre- 
 ceding deflection, the differences in deflection being 
 regarded proportional to the length of the chord. 
 
 Consider these with reference to the case cited on 
 p. 20, a 36 chain curve with spiral approach, the latter 
 leaving the main tangent at 26*82 chains. 
 
 Here stations will occur on the curve at the following 
 distances from the point of spiral P — 
 
 •18 M8 2-18 3-18 4-18 5-18, 
 
 and if the length of transition is taken as V36 chains, 
 
 the chord length c will be ^^VMx6, or -614 chain. 
 
 The chordages of the stations will then be in order — 
 
 •18 1-18 2-18 3-18 4-18 5-18 
 ■614 -614 •eU -614 -614 ^614 
 
 or ^29 1^92 3^55 5-02 6-81 8-44, 
 
 the third decimal place being negligible. 
 
 (i.) Then the deflections for the instrument at the 
 point of spiral P will be as the squares of these values,viz. : 
 
 •08 3^69 12-60 25-20 46-37 71-23. 
 
 This process is tantamount to calculating the deflec- 
 tions for the given distances in chains from P, by 
 multiplying the squares of those distances by the deflec- 
 tion angle per chain. 
 
 (ii.) Now the differences in deflection per chord run 
 in the series 1, 3, 5, 7, etc., and the corrections are 
 accordingly — 
 
 •29x1' -92x3' ^55X7' ^02x11' -81x13' ^44x17', 
 
DEVELOPMENT 27 
 
 the multipliers being one more than double the integers 
 in the chordages. 
 
 Adding these to the squares of the integers in the 
 chordages, the following deflections are obtained: 
 
 •29' 3-76' 12-85' 25-22' 46-63' 71-48'. 
 
 It will be seen that these are approximate values, 
 being too great by the product of the fraction into its 
 difference from unity. In no case, however, will the 
 error exceed 0-25'. But the exact value can be obtained 
 by multiplying the fraction into the sum of twice the 
 integer plus the fraction. 
 
 Thus, for chordage 3-55, the exact deflection will be 
 32+-55(2x3+-55)=9+3-60, or 12-60 minutes. It is 
 thus more accurate to take the difference into twice the 
 integer when the fraction is under 0-5. 
 
 Stations inserted by approximate means should 
 never be occupied in the construction. Besides, when 
 fractional angles occur, it is doubtful whether with 
 coarse instruments stations are located any more 
 accurately than if they are placed on the chord, or 
 offset by some simple rule. 
 
 This subject introduces the matter of offsets from 
 the chord. 
 
 Offsets from the Chord. — While the angles between 
 successive tangents are in the series 3, 9, 15, etc., the 
 angles between successive chords and their tangents 
 are in the series 1, 4, 7, etc., back values being greater 
 by unity in the series 2, 5, 8, etc. It is therefore 
 evident that the farther the chord is from the origin, 
 the more nearly its arc approaches that of a circle, 
 and, in practice, no appreciable error will be intro- 
 
28 THE TEANSITION SPIEAL 
 
 duced by regarding each arc as circular in ascertaining 
 the offsets from the chord in interpolating stations. 
 Hence, if c be the chord length in chains or 100 feet 
 units, / the fraction of a chord at which a station falls, 
 and" a the deflection angle from the tangent at the chord 
 considered, then the offset z will be given by /(I -f)ca. 
 But a is initially one minute, or -00029 radian, its values 
 increasing in the series (3N -2), which may be written 
 (31+1), where I is the integer in the chordage. Whence 
 the rule, z=aj{l -/)(3I+l)c, where a is I" or f " accord- 
 ing as c is in chains or hundreds of feet. 
 
 Thus, in apphcation to the 36 chain curve of pp. 20 
 and 26, where the chordages of the stations are -29, 
 1-92, 3-63, 5-02, 6-81, and 8-44, the offsets will be in 
 order i"x-29(l -•29)(0+l)-614=0, J" X -92(1 --92) x 
 (3+l)x-614=0, i"x-53(l--53)(9+l)-614=|", i"x 
 •02(l--02)(15+l)-614=Ji,", i"x-81(l -•81)(18+1)-614 
 =A", and i"x-44(l -■44)(24+l)-614=ir. 
 
 II. By Offsets. — ^In this method the spiral is con- 
 structed in the same manner as Froude's cubic para- 
 bola. In fact, the data of the clothoid and cubic 
 become identical when abscissae x along the main tangent 
 are regarded equal to spiral distances, which is a safe as- 
 sumption for most of the radii ocourrent on main lines. 
 
 Here x=1n^/(j) and J/== q" ^^ (Bq. viii., p. 16). 
 Whence y^^j^ (Eq. 7, p. 16). 
 
 But s=2^ (Bq. 8, p. 16). 
 
 Hence 2/=4s-|j, (13) 
 
 where 4s is Y, the terminal y ordinate. 
 
DEVELOPMENT 29 
 
 The method may be effected in various ways, the 
 following being preferable : 
 
 1. Measuring the length of transition L along the 
 main tangent from the point of spiral P, and taking 
 the abscissse x as successive tenths of that length, 
 the y offsets from the main tangent running as the 
 cubes of the intervals, being in order: 
 
 k 8k 21k Mk 125fc 216fe didk 512k 
 729k lOOOfc 
 
 where k is equal to 4s -4- 1000. 
 
 Thus if s be 2' 6", k will be -01 foot, and the offsets 
 in inches will be as follows: 
 
 I" A" 31" IW I'S" 2'\W 3'5tV' 5'1t^" 
 7'3|" 10' 0". 
 
 When, however, it is desired to introduce the angu- 
 lar dimensions of the clothoid, fixing only P and Q 
 in the original location, the spiral itself may be subse- 
 quently inserted by measuring the intervals as chords, 
 and tying the ends of these with the corresponding 
 tangent offsets. Here the shift will bisect the actual 
 length of transition, not the terminal abscissa; but, on 
 the other hand, slight lateral deviations will be intro- 
 duced, yet never to such an extent as would displace 
 the alignment by an eighth of an inch. 
 
 For similar reasons, resort may be made to the 
 following process which is involved in the case of com- 
 pound curves (and reverse curves byFroude's method).* 
 
 2. Measuring on one side of the shift centre R, half 
 the length of transition L along the main tangent, 
 
 * See p. 43. 
 
30 THE TEAN8ITI0N SPIEAL 
 
 and on the other side half along the shifted curve; 
 then at intervals of one-tenth the length L, offsetting 
 the spiral with ordinates which vary as the cubes of the 
 distances from the beginning and end, P and Q respec- 
 tively. 
 
 Thus, if the rule L^V^^ be embodied, s being con- 
 sistently 33 inches, the offsets for any spiral will be as 
 follows: 
 
 5^ 
 10 
 
 t.- 
 
 LfromPorQ, 16-5" (|y=16J" @E 
 , „ 16-5"(^)'= 8t^"@4.6 
 
 1l „ „ 16-5"(|) = 3tV"@3,7 
 
 ^L „ „ 16-5"(|)= 1A"@2,8 
 
 ^L „ „ 16-5" Q)'= V @1,9. 
 
 @ P and Q 
 
 It will be seen that both offsets at the shift centre E 
 are equal to half the shift s. 
 
 Although the preceding method is theoretically the 
 most exact, the results compare unfavourably with 
 those derived entirely by tangent offsets when the 
 points located on the shifted curve are not abreast of 
 the spiral points. (See p. 64.) 
 
 Conclusion. — Normally, four elements are necessary 
 in fixing the limits of the spiral, these being the same 
 whatever data be utilised in the construction: (1) The 
 
 La 
 
 length L; (2) the shift s^^m'' (^) *^^ ^^^^^ increment, 
 
DEVELOPMENT 81 
 
 s tan J A (aS, Pig. 1); (4) the spiral extension 
 X -E sin $ (SP, Eig. 1), the last two together forming 
 the tangent excess aP. 
 
 It is desirable, however, that tangent distances should 
 be expressed in terms of the radius, the length, and the 
 shift. Now, it follows that — (Eq. v., p. 15.) 
 
 X=L(l-g+...);and*=2^ 
 
 Further, the spiral extension SP=X-E sin <l>; and 
 on expanding sin ^ and substituting ™ for $, the 
 third and higher terms being neglected — 
 
 Hence, on substituting for X— 
 
 or jj L when E is over 25 chains. 
 
 (a) Thus the tangent excess iu the fundamental case— 
 
 aP = s tan jA+g (l~5 gj' 
 (6) And the total tangent distance IP, 
 
 T.= (R + s)tanjA + ^(l-g^)- 
 
82 ^ THE TEANSITION SPIEAL 
 
 The remaining elements of pp. 5 and 6 may be added, 
 being often desirable either as checks or as means of 
 modification. 
 
 (7) The terminal x abscissa, X=Lf l=p ipV 
 
 (8) The terminal y ordmate, Y=4 s=«w* 
 
 (9) The whole chord PQ and its deflection 12, or 
 
 5 ^, where $ is the total angle of the spiral. 
 
 PQ=X sec n. 
 
 Tables. — Tables I. and II. of the Appendix give the 
 elements of spiral approaches to main curves of radii 
 20 to 80 chains and degrees 1° to 6"". In both, time approach 
 lengths are assigned, those of the former table being calcu- 
 lated from the rule L= -y/R chains, and those of the latter 
 from L'= 600-f-'v/D° feet. The data both for spiralling 
 through and spiralling in are added, these being respectively 
 the deflections Jo per chain^ or per 100 feet^ and the chord 
 lengths c per minute^ of deflection. Incidentally it may be 
 noted that the deflections for a spiral of ten integral chords 
 may be found by taking one-hundredth part of the total 
 deflection angle fl, the corresponding chord lengths being 
 one-tenth of the tabular values of the length L or L', as 
 the case may be. 
 
 After 55 chains radius in Table I., the product LR is 
 constantly equated to 414; in accordance with the rule 
 
 L= ^„Q^T> chains, V being limited to about 83 miles per 
 
 hour; and, in consequence, the chord c and the deflection 
 k per chain^ become constant, whUe the shift s decreases 
 in accordance with the rule 7141-f-R' chains. Similarly 
 above 1° 30' in Table II., L'= 330D°, in accordance with 
 
 the rule L'= •^\Tq) D° feet, but is taken as 333-33D° feet 
 
DEVELOPMENT 33 
 
 for simplicity, ¥ and c' becoming constant, and s accordingly 
 reducing to -SOSD' feet. 
 
 Finally, it slio;iId be remembered tkat there is consider'- 
 able latitude in selecting tbe length of transition; and, in 
 consequence, the tabular values may be regarded as being 
 tentative, a little modification often leading to considerable 
 simplification in the data. For instance, in " spiralling 
 through " to a 27 chain curve, the deflection k per chain^ 
 could be taken as 4 minutes, giving the length L as 
 5-315 chains, the total deflection fi as 112-80 minutes, 
 and the shift s as -0436 chain, or 2' lOJ". Similarly, 
 in " spiralling in " to a 36 chain curve, the chord c 
 could be taken as -628 chain, giving the length L as 10c, 
 £2 as 100 minutes, and s accordingly -0457 chain, or 
 3' OJ". In fact, a system of integral-chorded spirals 
 may be developed with the basis of 20", 40", and 60" as 
 the initial deflections in spiralling in, the whole embracing 
 all practiea' requirements without introducing lengths of 
 transition that difier appreciably from the lengths that 
 would otherwise be employed. Such a system would 
 not preclude the use of instruments reading to one 
 minute, division to 20" being desirable; for in all tabular 
 spirals there are fractions over \', which are necessarily 
 dropped in construction with minute reading theodo- 
 lites. 
 
 Notation. — ^Before passing to the subject of Align- 
 ment, it will be advisable to adopt some explicit 
 notation identifying the limits of spirals, the points 
 P and Q of the first spiral becoming P' and Q' 
 in the second spiral of the same curve (Pig. 1). But 
 in progress from P to P', the end Q' of the second 
 spiral becomes the constructional iegmnmg, while P' 
 becomes likewise the constructional end. Hence the 
 
 3 
 
34 THE TEANSITION SPIEAL. 
 
 exclusive adoption of the following terms and ab- 
 breviations: 
 
 V-.Pomt-Spvral, P.S. 
 Q: Povnt-Spvral-Oune, P.S.G. 
 Q': Pomt-Curve-Spvral, P.G.S. 
 P': Pomt-Spiral-Tangent, P.S.T. 
 
 Similarly, in compound curves, the fomt of com- 
 pound curvature, P.G.O., may be superseded thus — 
 
 C : Poimt of Spi/ral Compound, P.S.S. ; 
 
 and, in reverse curves, the pomt of reversed curvature, 
 P. B.C., thus— 
 
 E: Point of Spiral Beverse, P.S.B., 
 
 the tangent points of circular arcs being designated 
 as follows : a or A, Point of Curve, P.C. ; a' or A', Point 
 of Tangent, P.T. 
 
PAET II 
 ALIQNMENT 
 
 This section deals with the various problems of align- 
 ment that arise when transition curves are introduced, 
 either in the original location or in easing existing lines, 
 or spiralling old track, as the Americans style it. 
 
 The insertion of transition curves, or spirals, may be 
 said to introduce four general problems of alignment, 
 the following : 
 
 1. Shiftmg the mam curves so as to admit the transition 
 curves without change of radii. 
 
 2. Assuming the original radius to embody the shift and 
 so preserve the conditions of tangency.* 
 
 3. Sharpening the radii of the maim curves so as to 
 preserve the original location as nearly as possible. 
 
 4. Changi/ng the -radii of the main curves so as to 
 preserve as nearly as may he the original length of the 
 line. 
 
 As a rule, straights are not shifted and, in conse- 
 quence, the following is not considered — 
 
 5. Shifting the straights so as to admit the transition 
 curves without disturbing the main curves. 
 
 Nevertheless, it must be noted that the improvement 
 of existing lines may allow such latitude as will relax 
 
 * The second of these may be regarded as affording another 
 
 aspect of the first, and, in consequence, will be con6id«red with 
 
 that case. 
 
 35 
 
36 
 
 THE TEANSITION SPIEAL 
 
 the conditions imposed in the foregoing problems; 
 and, in this connection generally, it may be added that 
 only the constructional lines are regarded in the cases 
 treated, the artifices with which the engineer would 
 transfer to or from such alignment being a matter for 
 his ingenuity. 
 
 General Problems. — A fitting introduction to the 
 present subject is afforded by the following construc- 
 tion, which may be regarded as the general case of 
 alignment. (Fig. 3) 
 
 P' 
 
 ^^ Main Tangent 
 
 ria. 3. 
 
 Here the curve AV of radius E^ replaces the original 
 curve of radius Eo, the vertex being displaced by an 
 amount h from v to V. 
 
 Then ao=NQ+MH+Ho. 
 
 But ao=Eo; NQ=Y, MH=QO cos <I'=E„ cos 0); 
 
 oH=oO cos ^A=Qi+'Ro -E„) cos 
 
 2 
 
 Hence E, 
 
 Eo vers |A 
 
 h cos -j+Y 
 
 ^ ^ ^ A' 
 
 cos * - cos -5- COS <t> - COS -rr 
 
 (1) 
 
ALIGNMENT 37 
 
 And h cos ^=(E„ - E„) vers y+Ea vers * - Y. (2) 
 
 But s, the shift=QN -LA=Y -E„ vers *. 
 
 Hence h cos ^=(^0 - Ra) vers -k--s- (3) 
 
 When h=0, s=(Eo - E,,) vers | A . (See p. 44.) 
 
 When Eo=E<„ h=s sec -^, being plus since it is 
 
 measured in the opposite direction. 
 Also, the tangent excess, 
 
 Pa=X - E„ sin * - (/i+E„ - E J sin ^ A . (4) 
 Or, substituting for h in terms of the shift, 
 Pa=X - Ea sin 0) - (E„ - E„) tan | A +s tan | A . (5) 
 When ft=0, Pa=X -E„ sin * -(E, -E„) sin |A. 
 
 (See p. 44.) 
 When E„=Ea, Po=X -E„ sin *+s tan JA. 
 
 (See p. 5.) 
 The spiral extension X - Eg sin <J> has been shown 
 
 to be equal to „ L(i -^ =r~)' o^ 9" "when Eo is over 
 25 chains. (See p. 31.) 
 
 Compound Curves. — At this juncture it will be con- 
 venient to note that two cases arise in connection with 
 compound curves, viz. : 
 
 (a) When spirals are to be inserted between the 
 branches of the compound, and also at the beginning 
 and the end of the composite curve. 
 
 (&) When spirals are to be inserted only at the 
 beginning and the end of the compound curve. 
 
 This distinction introduces that important matter, 
 the difference in the radii E and r that requires a transi- 
 tion between the branches of the compound. 
 
38 THE TEANSITION SPIEAL 
 
 Now it may be shown that, as in Proude's parabola, 
 the shift s,. for the compound is the difference of the 
 shifts 8 and s of the branches, the relation being ex- 
 pressed algebraically as 
 
 where E and r are the radii and l^ the length of transi- 
 tion connecting the shifted curves. 
 
 It is evident that such transition will be necessary 
 only when the radii E and r differ materially. 
 
 Assume the curves to be placed so that they yield 
 the requisite shift along a common radial, E being 
 greater than r. 
 
 Now, if e be the superelevation required by the curve, 
 of radius r, and E that required by the curve of radius 
 E, the latter being the lesser value under the controlUng 
 speed of the sharper curve; then, since the super- 
 elevation of the sharper curve when run out on to the 
 straight should be run out in the same time as into the 
 flatter curve, the velocity v being constant — 
 
 Elev. e lost in - secs.=Elev. (e -E) lost in - sees., 
 
 where l^ and l^ are the respective lengths of transition 
 to straight and to greater circle. 
 
 iVr E. 
 
 Hence 'f='^=f-^=^\ 
 
 li e M. 1 E ' 
 
 r 
 M being equal to ^^. (Page 2) 
 
 Whence l,=li(l-^. (7) 
 
ALIGNMENT 39 
 
 Mr. Shortt arrives at the same result by equating the 
 rate of change in acceleration in entering the smaller 
 curve from the straight to the rate of change in accelera- 
 tion in entering from the larger radius, the lengths of 
 transition remaining Zi and l^ respectively. That is — 
 
 r V \r E/ u 
 Further, assuming ■\/r to be the length of the transi- 
 tion li in chains, 2(,=f l-^pj-y/r; and, on substituting 
 
 for Z,, in (6), 
 
 1 / r\' 
 ^"=24 V~Ti.) ^"^^^^^ (^) 
 
 Since a track cannot be lined to closer than |-th inch, 
 this value of s^ may be said to determine the limit 
 beneath which it is impossible to insert a transition 
 curve between the branches. 
 
 Hence, if s„ be put equal to ^-^rr chain, f 1 - =^ j = V 
 
 TBT) 
 
 and ^=85 per cent, very nearly. But with con- 
 siderably larger shifts than this the curves would be 
 practically identical with the transition; so if Sc=3" 
 be fixed as the limit at which the latter is intro- 
 duced, the difference in the radii E and r may be as 
 much as 40 per cent. However, a lower difference of 
 radii may call for a transition curve, and its inefficient 
 length may be by no means derogatory to the alignment. 
 
 1. Shiflmg the mam curves so as to admit the transi- 
 tion curves without change of radii. 
 
 (i.) Simple Curves. — This case was considered in 
 determining the elements of transition curves, the 
 
40 THE TEANSITION SPIEAL 
 
 La 
 shift s being generally Y - E sin *, or ^jp in the case 
 
 of the clothoid and cubic. Also, in general, the tan- 
 gent excess Pa=X -E sin$+s tan |A, which reduces 
 
 to 2^ fl-i pj+s tan |A, or ^L+s tan ^ A very nearly. 
 
 Here it should be noted that if the original radius 
 Eo is taken such that Eo=Ea+s, the shift term 
 s tan I A disappears, and the remainder, the spiral ex- 
 tension ^ (l-R ^d)) is measured from the original point 
 
 of tangency {a, Pig. 1). 
 
 (ii.) Compound Curves. — (a) When the spirals are 
 to be inserted between the branches of the compound, 
 and also at the begiiming and the end of the composite 
 curve. 
 
 When the transitions are inserted in the original 
 location it may be possible to retain the conditions 
 of tangency by assuming radii E+S and r+s, where 
 E and r are the given radii, and S and s the end shifts 
 accordingly, the difference of which is the central 
 shift, or gap d. (Pig. 4) 
 
 Otherwise, as is the more usual, d will be equal to 
 s sec /8 - S sec a, the curves moving along a common 
 radial to a and ^ as central angles. 
 
 Now in either case, it seldom happens that d is 
 
 equal to s„, the value required by ^ (^^-^). and the 
 
 question arises as to whether the central transition be 
 omitted or not, and in what manner must the entire 
 construction be modified. This introduces the follow- 
 ing artifice, which especially apphes when the com- 
 
ALIGNMENT 
 
 41 
 
 pound curves have been run in without provision for 
 transition curves. 
 
 Consider the case when the shifts have been calcu- 
 lated for transition to the straights from curves of radii 
 E and r: 
 
 Gap C2Ci=d=C2C -CiC=s sec ;8 -S sec a. (i.) 
 
 ^0. 
 
 Fig. 4. 
 
 Caution. — Now if time approaches be adopted, it is 
 not to be inferred ttat S= s, and d= s (sec j8— sec a) 
 consequently; for the speed of the sharper radius will 
 obtain, and wiU thus reduce the length L and the shift S 
 accordingly. 
 
 As a rule the gap d will not accord with the correct 
 shift s„, but will be much greater. However, consider- 
 able latitude is to be countenanced rather than resort 
 
42 THE TEANSITION SPIEAL 
 
 to modification of data or, particularly, shifting the 
 end transitions in the manner hereafter described. 
 Often, however, a central transition corresponding to 
 the gap d would be incongruous, and the latter operation 
 becomes inevitable. 
 
 Therefore, let a suitable length l,. be assumed and 
 the corresponding shift s,. be calculated, the sharper 
 curve he alone being shifted. 
 
 Then cCi=cC2 - Sc=s sec /3 - s^. (ii.) 
 
 Now imagine the curve ac shifted inwards until c 
 comes to Ci and a moves to A. 
 Then 
 
 AA'=aA cos a=CiC cos a=(s sec /3 -s„) cos a. (iii.) 
 
 It is unlikely that AA' will be equal to the end shift 
 S. Therefore if AA' is greater than S, the point B 
 must be shd along the tangent to b' by an amount Bb% 
 which is equal to Aa', the latter distance being along 
 a tangent parallel to H. 
 
 Now the angle a'AA'=90° -(a+/8); and Aa'= 
 (AA'-S)seca'AA'. 
 
 Hence ^>a J^ sec ^-.,) cos a-S_ 
 
 sm (a+p) ^ ' 
 
 Had it been necessary to pull back B, AA' being less 
 than S, the foregoing distance would have been 
 
 S-(ssec^+s,) cosa ,. , 
 
 sin {a+^) • ^^^' 
 
 (b) When spirals are to be inserted only at the begin- 
 ning and the end of the compound curve. 
 
 Here the gap C2C1 becomes zero, and S sec a= 
 s sec jS. But, as is so often the case, S and s do not 
 
ALIGNMENT 48 
 
 together accord with suitable lengths of transition; and 
 unless modification of length is permissible, the com- 
 pound must be shifted by the foregoing process, s or S 
 being taken at its correct value. The shift along the 
 tangent in this case will be — 
 
 S(sec /3 cos g - 1) .^^. 
 
 sin (a+jS) ' ^"^ 
 
 S(l-sec/3cos«) 
 
 sm (a+yS) ^ ' 
 
 according as AA' is less or greater than S. 
 
 (iii.) Reverse Curves. — ^The transition between the 
 branches of a true reverse should consist of two separate 
 spirals, the superelevation changing from zero to the 
 opposite rails at the junctions of these.* Hence the 
 first and second cases should be reduced to those of 
 simple curves, the radii being sharpened so as to admit 
 suitable lengths of transition through the medium of a 
 separating piece of straight. Here the simplest con- 
 struction is that in which the centres of curvature 
 are moved inwards along the radials at the original 
 tangent points, the latter points remaining unchanged. 
 As a rule the central angles A and S will be equal, or 
 the reverse will have a common radius E,,. Even then 
 the process is somewhat complex, since it involves 
 finding an auxiliary angle x, an angle 0, and thence the 
 
 * Froude's cubic reverse is certainly an exception, not being 
 a true transition reverse, but a curve in which the point of 
 spiral reverse R is virtually a point of inflexion, outside which 
 the radius of curvature is double that of similar reversed circular 
 curves. Such is constructed in the manner outlined for simple 
 curves, cubic offsets being erected from the redundant portion 
 of each circular curve. (See p. 29.) 
 
44 THE TEANSITION SPIEAL 
 
 reduced central angles a or y8, or both, the trigonometri- 
 cal relations giving the inner tangent points of the 
 reduced circular curves. The calculations are simplified 
 by embodying the shift in the reduced radii, thus elimin- 
 ating the shift term s tan \a or s tan |/3, or both. 
 
 3. Qharpenmg the radii of the mam curves so as to 
 preserve the original location as nearly as possible. 
 
 (a) Consider first the case of the vertex remaining 
 unaltered, the entire curve being superseded by one of 
 reduced radius. 
 
 (i.) Simple Curves. — Let E^ and Ea denote the 
 original and amended radii respectively, a shift tan- 
 gent being drawn parallel to the main tangent at A. 
 
 (Fig. 5) 
 
 Then 'Ra+s=B,„ ~ on. 
 
 But on=(E<, -Ea) cos -^-j where A is the intersection 
 or central angle. 
 Hence s=(E„-Ea) vers ^. (13) 
 
 Also the distance Pa= 
 
 X-EaSin<D-(E<,-Ea)siniA. (14) 
 
 The foregoing result may be derived by substituting 
 h=0 in Eq. 5 of the General Construction (p. 37). 
 
 (6) As a rule, however, the problem will consist 
 in compounding the portions of the main curve 
 adjacent to the transition curves, thus leaving the 
 central portion of the main curves undisturbed. 
 Here V becomes the point of compound curvature, and 
 
 g- the angle 7 cut of the curve of radius E,, and replaced 
 
 by the Ea curve. 
 
ALIGNMENT 
 
 Hence s=(Eo -Ea) vers 7. 
 
 Also, in general, the distance Pa= 
 
 X -Ea sin O - (K,, -R„) sin 7; 
 or, in terms of the shift and length, 
 
 45 
 
 (15) 
 (16) 
 
 (17) 
 
 Fia. 5. 
 
 The angle 7 must be less than -«-; in fact, it should 
 
 not exceed -r- in practice. 
 
 (ii.) Compound Curves. — (a) In this connection the 
 radius of each arc may be shortened, giving two new 
 arcs compounded at the same point V (Fig. 5), the 
 central angles becoming a and /3 corresponding to 
 radii E^ and Tg, which reduce to E^ and Ta respectively. 
 
(19) 
 
 46 THE TEANSITION SPIEAL 
 
 Hence by analogy with Eq. 13, 
 
 S=(E„-E„)ver3al , 
 
 s=(r„-ra) vers yS J 
 And Po = X -Eo sin <E> - (E^ -Eo) sin a\ 
 Ya'=x - r„ sin ^ - (r,, - r^ sin )S J 
 
 (&) But either arc may be compounded again, leaving 
 the central portion undisturbed. This is merely an 
 extension of the case of a simple curve. (See p. 44.) 
 
 (iii.) Beverse Curves. — Here the reverse branches 
 will be sharpened so that the point of spiral reverse E 
 as a point of spiral P coincides with the original point 
 of reverse curvature. That is, by Eq. 5 of the General 
 Cons'truction, h, being 0, and Pa also 0, 
 
 X - E„ sin <I>=(E„ - EJ sin ^.- (20) 
 
 Since X-E^ sin ^=|L=|vll<, when E„ is over 
 25 chains and D is under 4°, 
 
 ^«+ o''>^°A -Bp=0, (21) 
 
 2 sm JA ^ ' 
 
 where A is the central angle of the branch considered, 
 a being unity or eight according as the radii are ex- 
 pressed in chains or feet.* 
 
 The respective solutions of the foregoing quadratic 
 are — 
 
 ^»= - 4 sin JA - '^^»+16lES|A '^'^'- (21") 
 
 * This asaumption increases the shift in the degree system 
 from 2-618 to 2-667 feet, or 2 feet 8 inches. 
 
ALIGNMENT 47 
 
 Also from Fig. 5, aS=E<, tan ^A -(Ea+s) tan |A 
 =(E„-E<,) sin|A. (22) 
 
 It is evident that the foregoing equations give the 
 necessary length of transition for each portion of the 
 reverse, and the same lengths occur where the ends of 
 the reverse join the straights. 
 
 4. Changing the radii so as to preserve as nearly as 
 may he the original length of the line. 
 
 (i.) Simple Curves. — The expression leading to this 
 result in the case of a simple curve reduces to — 
 
 h cos JA==(E(, -Ea) vers J A -s, 
 
 (Eq. 3, p. 37) 
 
 where s is the shift and h the displacement of the 
 vertex along the centre line. 
 Accordingly, the expression for the tangent excess is 
 
 Pa=X -Ea sin $ -(E„ -E,,) tan JA+s tan fA. 
 
 (Eq. 5, p. 37) 
 
 It is often possible to insert a number of transition 
 curves so that the total distance between common 
 points is unaltered, the relation between the original 
 and amended half-lengths being generally — 
 
 X - Ea sin * - (E„ - E„) tan J A +s tan J A +E<,^ 
 =LH-E„(|^-*), (28) 
 
 where L is the length of transition. 
 
 But it is only by introducing the displacement h 
 that the same length of line can be obtained in any 
 individual curve when inserting transition curves. 
 
 Let the original main curve of radius Eo be super- 
 
48 THE TEANSITION SPIEAL 
 
 seded by one of radius Eo, the latter's vertex v being 
 displaced centrally. 
 
 Then, since the original and amended half-lengths 
 must be equal, 
 
 Original h alf-length 
 
 X -E„ sin * -(E„ -E„) tan^A+s tan JA+E,,^ 
 
 E, 
 
 ih-) 
 
 Amended half-length, 
 
 the length of the clothoid being 2Ea3>. 
 
 Now, X - Ea sin <S> is |L very nearly-^.e., Ea<I>. 
 Hence 
 
 E„(|^+^)=E„* -(E„ -E„) taniA+stan JA+E„y 
 
 which leads to the Equation of Equal Routes — viz., 
 
 s tan 4 A =(E„ -Ea) (tan JA - 1 A). (24) 
 
 Therefore, if s be assumed, Eo is determinate. But s 
 is constant in time approach, being normally 33 inches 
 or 31-42 inches by the rules given on pp. 11 and 12. 
 
 (ii.) Compound Curves. — Compound curves without 
 central transitions may be dealt with by an analogous 
 process. Here E q. 3 becomes h cos a=(E(, - Eo) vers a - S, 
 for the larger radius, and h cos /3=(ro-ro) vers j8-s, 
 for the shorter radius, where S and s are the shifts, 
 E„ and r^ the original radii, and a and /8 the central 
 angles. 
 
 But since the central displacement must have the 
 same value h in each case — 
 
 (?»:i.^?) 7.^^^ a^S_(r„ -rg) vers 8-s . 
 
 cos a cos ;8 ^ ' 
 
ALIGNMENT 49 
 
 If time approaches are considered, the shift s for 
 the sharper radius will be known, leaving as unknowns 
 Ea, ra, and S. It will therefore be necessary to deter- 
 mine the correct values of these by trial, assuming 
 Efl and r,,, and determining S for the radius Ea under 
 the controlling speed of the radius Va- 
 
 Another apphcation of the general equation may be 
 to determine dimensions when the displacement h is 
 limited, the equalising of routes being a secondary 
 consideration. In this, Eq. 3 will contain two un- 
 knowns, h and r^, s being constant in time approaches 
 (pp. 37 and 48). Hence, if the vertex displacement h 
 is known, then is the reduced radius r,, determinate. 
 But this will be evident in the succeeding case, where 
 the problem is extended to reverse curves. 
 
 However, if the Equation of Equal Eoutes be em- 
 bodied, the assuming of h, r^, or Eo will be precluded; 
 
 2 2 
 
 for then will s=g (r^ -r^) vers /3, S=g (E^ -Ea) vers a, 
 
 o o 
 
 and s cos a=S cos ^, s being known by the controlling 
 
 speed of the sharper branch. 
 
 (iii.) Reverse Curves. — In the case of the true reverse, 
 the point of reversed curvature becomes the common 
 point of spiral of two separate amended curves which 
 overlap the original ones. The tangent excess, there- 
 fore, will be the same at opposite ends of each curve, 
 being zero in Bq. 4. (See p. 37.) 
 
 Thus: 
 
 X - E„ sin $ - (/i+E„+E<,) sin | A =0, (26) 
 
 where A is the central angle and E^ the original radius 
 of either branch. 
 
 4 
 
50 THE TEAN8ITI0N SPIEAL 
 
 Therefore, since X-Eo sin *=|L very nearly, and 
 L=avTlo in time approach, 
 
 a being unity or eight according as Ea is expressed in 
 chains or feet. 
 
 Whence, if h be assumed, Ea is determinate. 
 
 Further, from Eq. 5, h being unlimited and time 
 approaches being assumed — 
 
 a being unity or eight according as Va is expressed in 
 chains or feet. 
 
 Here s has its normal value, and the consequent 
 displacement of the vertex is expressed by the relation 
 
 A cos I A =(ro - Ta) vers -g- - s. (Eq. 2, p. 37) 
 
 If the radii differ, E„ being that of the flatter branch, 
 the latter's amended value Ea will have to be deter- 
 mined, leading to a length L and a shift S to the speed 
 of the sharper curve. But in Sharpening generally 
 from equal original radii, it seldom happens that Ea 
 will be equal to r^; and, in practice, it would be advis- 
 able to adopt throughout the length of transition I of the 
 sharper amended branch. In fact, existing condi- 
 tions would seldom permit transitions in accordance 
 with the rule aVo) and, in consequence, reduction to 
 
 g a-y/fa may be necessary. 
 
PAET III 
 FIELD EXERCISES 
 
 Problem 1. — The pickets indicate two intersecting 
 straights which are to be connected by a 30 chain 
 circular curve with transition curves, the highest 
 working speed being the maximum for the radius. 
 
 Continue the pegging out of the centre line from 
 84-00 chains, the peg last driven " chaining through," 
 and inserting additional pegs at half-chain intervals 
 on the spirals. 
 
 Procedure. — ^1. Set up the theodolite over the 
 point of intersection I, and measure the angle between 
 the straights supplementary to the intersection angle A . 
 
 For purposes of calculation, let the angle A be 56° 20'. 
 
 2. Calculate IP, IP', the total tangent distances 
 Ts=(E+s) tan I A +JL, where L is the length of transi- 
 tion in chains and s the shift. But, since the maximum 
 speed 11 VE is permissible, L will be ^/B. chains in 
 time approach, and s 33 inches accordingly. 
 
 Thus : T, will be (30 + Jj) tan 28° 10' + ^ VSO = 
 (30 +^) X •53545+2-739= 18-829 chains. 
 
 3. Measure the total tangent distances IP, IP', and 
 drive three pegs, two guarding the centre peg at the 
 points of spiral P, P'. Continue the chaining forward, 
 and record the chainage of the peg P. 
 
 Let chainage 84-00 be -72 chain short of P; then the 
 51 
 
52 THE TEANSITION SPIEAL 
 
 chainage of P will be 84-72, and the chainage of Q 
 90197. 
 
 4. Calculate the remaining elements of the spirals — 
 viz., (a) the Long Chords PQ and P'Q', and the Total 
 Deflection Angle 12, also {h) the Deflection per chain^ 
 of the spirals, and (c) the Tangential Angle per chain 
 of the circular portion. Complete the data of the 
 composite curve, finding the chainages of Q' and P', 
 and tabulating the deflection and tangential angles for 
 the pegs on the curves. 
 
 (a) Total deflection angle 12= 573 g= 1° 44-61', and *, 
 the total spiral angle= 5° 13-83'. 
 Long chords PQ, P'Q'= X sec fi= L^ 1 -| ^) sec 1° 45'= 
 
 5-473 chains. 
 
 573 
 (&) Deflection k per chain^= ¥-5= 3-507 minutes. 
 
 1 1719 
 (c) Tangential angle per chain sin" ^ se or -w- minutes 
 
 (nearly) = 57-3 minutes. 
 
 Length of circular portion= - — »? — -= i ° 54 .«' ~ 24-017 
 chains. 
 
 Hence chainage of Q'= 114-214; of P'= 119-691. 
 
 Deflection angles from tangent at P in minutes : 
 {•28Yx 8-507 ; (-78)2x3-507; {1-38^ x 3-507 ; (1-78)8x3-507; 
 (3-28)^ X 8-507 ; (2-78)" x 3-597 ; {3-88)^ x 3-507 ; (3-78)" x 
 3-507; (4-28)^x3-507; (4-78)" x 3-507; {5-28)^x3-507. 
 
 Deflection angles from tangent at P' in minutes : 
 (-191)" X 3-507 {-691)" X 3-507; (1-191)" x 3-507; (I-69i)"x 
 3-507; (2-191)" X 3-507; {2-691)^x3-507; (3-191)" x 
 3-507; {3-691)^x3-507; (4-191)" x 3-507; {4-691)^ x 
 3-507; (5-191)" X 3-507. 
 
FIELD BXEECISES 
 
 53 
 
 Tangential angles from tangent at Q in minutes : 
 57-3' X -803; 57-3'xl-803; 57-3' x 2-803; 57-3'x3-803; 
 57-3' X 4-803; 57-3' x 5-803 . . . 57-3' x 23-803. 
 
 5. Set up the theodolite over the pomt-spwal P, 
 clamp the plates at 0°, and sight the pomt of mtersedion 
 I. Baeksighting thus, set off the several deflection 
 angles, and direct the chainmen accordingly, giving 
 first the deflection for the subchord, then those for 
 
 Pia. 6. 
 
 successive half-chain chords, and finally that for the 
 subchord closing on Q. Check the peg at Q by measur- 
 ing the long chord PQ along the direction given finally 
 by the theodolite, the circle remaining clamped at the 
 total deflection. As a further check, note that the 
 perpendicular distance from Q to the tangent should be 
 4s, or 132 inches. 
 
 6. Set the theodolite over the pomt-spvral-curm Q; 
 clamp the plates at twice O, and, sighting P, lay down 
 a tangent at Q by transiting the telescope and bring- 
 ing the plates to 0°. Proceed to run in the circular 
 
54 THE TEANSITION SPIEAL 
 
 portion QQ', giving the tangential angles and directing 
 the chainmen accordingly in laying down, first the 
 subchord, then successive chain chords, and finally 
 the subchord closing on Q'. Check the position of Q' 
 by measuring the terminal ordinate N'Q', which should 
 be equal to 4s. (See Note vi.) 
 
 7. Set up the theodoUte over the point-spiral- 
 tangent P', clamp the plates at 0°, and sight the 
 point of intersection I. Backsighting thus, lay off an 
 angle II, and observe if the cross hairs of the telescope 
 bisect the point-curve-spiral Q'. Then set off the 
 tabulated deflections in descending order, the first being 
 the value under,s^. At the same time direct the chain- 
 men to measure alfccordingly, first the subchord from 
 Q', then successive half-chain chords, and finally the 
 subchord closing on P'. (See Note vii.) 
 
 Note vii. — In " spiiaHing through," the second spiral Q'P' 
 may be set out from Q' with little additional calculation, 
 the deflections from the tangents at P and P' being invari- 
 ably difierent. 
 
 Here the chainages of spiral points from Q' are '786, 
 1-286, 1-786, etc., for N, while q is the length L, 5477 
 chains, in N(3j - N)A;, k being the deflection per chain, or 
 3-507 minutes. 
 
 The chief difficulty, however, is that of exactly locating 
 Q', since when possible to measure the whole chord QQ' 
 the error of measurement is likely to be appreciable. 
 Nevertheless, when locating Q' from Q with the whole 
 tangential angle J(A - 2$)) arrows coidd be inserted in 
 the line QQ', one on each side of the located curve position 
 of Q', and the instrument could be adjusted in this line 
 untU an angle 2i2 from the tangent at Q' bisects the peg at 
 P', the tangent at Q' being obtained by turning off from Q 
 
\ FIELD EXEECISES 65 
 
 an angle ^(A - ^^)- The procedure of Problem 4 is detailed 
 so as to demonstrate tlie various arti&ces that might be 
 resorted to when working progressively through the ruling 
 points P, Q, C, Q', P', of a compound curve. 
 
 Obsbbvations. 
 
56 THE TEANSITION SPIEAL 
 
 Problem 2. — Connect the given straights with a 
 circular curve of 36 chains radius, introducing suitable 
 spiral approaches in the original location. 
 
 (Assuming the chainage of the point of spiral P 
 to be 76-32, interpolate pegs at even chainage on spirals 
 and curve.) 
 
 Peocbdure. — ^1. Set up the theodolite over the 
 pomt of mtersection I and measure between the straights 
 the angle supplementary to the intersection angle A . 
 (See Fig. 6.) 
 
 For purposes of calculation, let the angle A be 52° 30'. 
 
 2. Calculate the elements and data of the spirals, 
 assuming construction by deflections of one minute 
 per chords 
 
 On inspecting the table on p. 100, it will be seen that the 
 total deflection fl= 95-5 minutes, whereas 12= 100-0 minutes 
 would give ten integral chords without unduly increashig 
 the length L, which is normally 6-000 chains. 
 "RO ^fiOO 
 
 (a) Length L=g^=-g=g-= 6-283 chains, or 28 links in 
 
 excess. 
 
 Adopting this length L, the normal shift s will be in- 
 creased from -04:17 chaia to -0457 chain by the ratio 
 (6-283)' 
 (6-000)'' 
 
 (6) Total tangent distances Ts 
 
 = (R + s)tanjA + iL(^l-^J) 
 
 = (36 + -0457) tan 26° 15' + 3-14:2('l-^'^ '^\ 
 
 \ 5 X 36 / 
 
 = 17-776 + 3-141=20-917 chains. 
 
FIELD EXEECISES 57 
 
 (c) Long chords PQ, FQ'=X sec 12 
 = L(l-lj)secr40' 
 
 = 6-283(l-^^^^) X 1-0004= 6-281 chains. 
 * the total spiral angle= 5° 00'. 
 
 3. Measure the total tangent distances IP, IP', and 
 drive pegs at the points P and P'. Set up the theodolite 
 over the point-spiral P, and, sighting I, set off successive 
 angles of 1', 4', 9', etc., the chainmen laying down 
 successive chord lengths c accordingly. 
 
 4. Set out from the tangent the whole deflection 
 fl, and along the direction thus given measure the long 
 chord PQ. Drive a peg at Q. (Check by measuring 
 the perpendicular from Q to the tangent, which dis- 
 tance should be 4s.) 
 
 5. Set up the theodohte over the pomt-spwal-curve 
 Q, clamp the plates at 212, and, sighting P, lay down a 
 tangent at Q by bringing the plates to 0°. (Transit the 
 telescope.) Proceed to run in the circular curve with 1- 
 chain chords ; if instructed, reducing the chainage of Q 
 by adding the length L to the chainage of P. (See Note v.) 
 
 Chainage of Q= 76-32 + 6-283= 82-603. 
 
 1 1719 
 
 Tangential angle S per chain=sin"i ™, or —o^ — 
 
 47-75 minutes. 
 
 Hence, 1st tangential angle= -397x47-75'= 18-96'; 2nd 
 ditto= 1-397 X 47-75= 66-70'; and so on. 
 
 T .T, ^ • 1 ^- nr>' A -2^ 52° 30'- 10 ° 0' 
 Length of circular portion QQ = — sg — = -.o ok,^ — 
 
 = 26-806 chains. 
 
 Hence chainage of Q'= 109-409; and chainage of P'= 
 115-692. 
 
58 THE TEANSITION SPIRAL 
 
 6. Set up the theodolite over the pomt-curve-spiral 
 Q', and lay down a tangent thereat by setting off an 
 angle |(A -2<J>) from Q. (Transit the telescope.) 
 Lay off from this tangent an angle 2fi, and observe if 
 the- cross hairs bisect the peg at the point-spiral- 
 tangent P'. Set out from P' successive deflections 
 in minutes of the normal deflections from P' plus n 
 times the chord number N counted from P', n being the 
 total number of chords. (See Note vi.) 
 
 The foregoing process reduces to one step, wMoli here 
 consists in sighting Q with the plates clamped at ^(52° 30' 
 - 10° 00') plus 2x1° 40', or 24° 35', turning the plates to 
 0°, and transiting the telescope, the cross hairs of which 
 should bisect the peg at P'. 
 
 Incidentally, the back deflections from the long chord 
 Q'F wiU be 171', 144', 119', etc., n being 10 in N(N + m), 
 where N is the number of the point sighted as covmted 
 from P'. 
 
 7. Interpolate pegs at even chamages either (a) with 
 the theodolite or (b) by offsets from the chords. 
 
 (a) Here the pegs may be sighted in with the theo- 
 dolite at either P or Q of the first spiral, and either 
 P' or Q' of the second spiral, the deflections from the 
 tangents at these points being N(3g[-}-N)A; for P and 
 P' and N(3g - N)fe for Q and Q', while N is the distance 
 in chains from q, the point occupied, which is or L 
 respectively. 
 
 Thus, if P and P' be occupied, stations will occur on the 
 Ist spiral PQ at -68, 1-68, 2-68, 3-68, 4-68, and 5-68 chains 
 from P, and on the 2nd spiral P'Q' at -69, 1-69, 2-69, 3-69, 
 4-69, and 5-69 chains from P'; the chordages will be 1-08 
 
FIELD EXBECISES 59 
 
 2-68, 4-27, 5-88, 745, and 9-04 on PQ, and MO, 2-69, 4-28, 
 5-88, 7-47, and 9-06 on P'Q', and the corresponding deflec- 
 tions from the tangents at P and P' will be respectively 
 1-17', 7-18', 18-23', 34-34', 55-50', and 81-72', and 1-21', 
 7-24', 18-32', 34-57', 55-80', and 82-08'. 
 
 (b) Here the offsets z are calculated in inches by the 
 formula J"/(l -t/)(3I+1)c, where c is the chord length 
 in chains and I and / are respectively the integers and 
 fractions in the chordages as reduced from chain dis- 
 tances from P and P'. Altogether this is the more 
 satisfactory method. 
 
 The following are the offsets to the stations of the present 
 example : 
 
 Station : 
 
 77 
 
 78 
 
 79 
 
 80 
 
 81 
 
 82 
 
 Ofiset : 
 
 1 " 
 
 i" 
 
 r 
 
 5 t 
 
 1". 
 
 3 » 
 
 Station : 
 
 no 
 
 111 
 
 112 
 
 113 
 
 114 
 
 115 
 
 Offset : 
 
 r 
 
 ¥ 
 
 i" 
 
 r 
 
 i" 
 
 1 ". 
 
 TV 
 
 N.B. — When interpolating stations the chain should be 
 carried round the spiral points already located. 
 
 The interpolation of stations is separated in order to empha- 
 sise the fact that this operation would normally be subsequent 
 to tracing the spiral, while the latter operation itself might he 
 deferred in the original location, the ruling points P, Q, and 
 Q' P' temporarily sufficing. 
 
 Note V. — As wiU be seen in Problem 3, the circular curve 
 may be set out with the theodolite at the shift tangent 
 point A, neither Q nor Q' being occupied. But even then 
 a check on Q and Q' is desirable. 
 
 Note vi. — As a rule it is better to insert the spirals from 
 P and P', particularly in the case of compound curves; 
 though, on the other hand, construction in order through 
 P, Q, . . . Q', P' is of practical appeal, and might be 
 
60 THE TEANSITION SPIEAL 
 
 adopted when tie spirals are omitted from tlie primary 
 location. (See Note vii. of Problem 1, and, particularly, 
 the special note of Problem 4, where progressive alignment 
 is extended to compound curves.) 
 
 Obsebvations. 
 
FIELD EXEECISES 61 
 
 Problem 3. — ^The pegs a and a' are the tangent points 
 of a circular curve of 30 chains radius, the pickets indi- 
 cating the centre line of the straights. 
 
 It is required to replace the given curve by one of 
 the same radius with transition curves, the speed being 
 limited to 50 iniles per hour, and the gradient of ap- 
 proach to 1 in 500. 
 
 Eun in the shifted curve with the theodolite, inserting 
 suitable transitions by offset measurements. 
 
 Procedueb. — ^1. The magnitude of the intersection 
 angle A not being known, set up the theodolite over the 
 first tangent point a, and, backsighting along the pro- 
 longation of the straight, measure the angle to the 
 second tangent point a', ascertaining thus the whole 
 tangential angle, which is one-half the angle A . If a' 
 is not visible, prolong both straights towards the point ' 
 of intersection I, and, at convenient points x, y in the 
 respective tangents, measure the angles Ixy, lyx, 
 these together being equal to the required angle A. 
 When I is accessible, and is visible from both a and a', 
 the angle A may be ascertained directly. (Pig. 6.) 
 
 For purposes of calculation, let tlie angle A be 43° 20'. 
 
 2. Ascertain the length of transition L for the hmited 
 speed V miles per hour, confining the gradient of ap- 
 proach to 1 in 500. 
 
 , . T. V^ 2500 „^ , ^ 
 
 Here superelevation &= 9qqt? = oqO x 30 "^ ■ 
 
 or 5". 
 L = 500E = 208'5 feet, which accords with the rule 
 
 the latter giving L as 3-02 chains. Hence L=3 
 
 1381R 
 
 chains conveniently 
 
62 THE TEANSITION SPIEAL 
 
 3. Calculate the remaining elements and the data 
 of the transition curves — viz., (o) the Shift; (&) the 
 Tangent Excess aP; and (c) the offsets y, these pre- 
 ferably for tenths of the length L. 
 
 (a) Shift s^ ^= ^|^= 9-9"= .0125 chain. 
 
 (6) Tangent Excess aP= s tan ^ A - |I'( 1 ~ 5 r ) 
 
 = -0125 X 4487 + 1-5(1 - -00008) 
 •0056 + 1-5000= 1-506 chains. 
 
 ■(c) Ofisets y, preferably from the tangents consis- 
 tently at successive distances of -^ ^' o^ ^^ links. (See 
 Note iii.) 
 
 Here the offsets will run as 1, 8, 27, 64, 12S, 216, 343, 
 
 is 
 512, 729, and 1,000 times y^, where yi= looo"" '^^^^"' 
 
 ojj being ^L, and y= -jj- generally. 
 
 -04" -32" 1-07" 2-53" 4-96" 8-55'^ 13-58" 20-28'^ 
 28-87" 39-6". 
 
 4. Measure the shift increment a to S, and erect a 
 perpendicular SA inwards equal to the shift s. Drive 
 a peg at the shift tangent point A. Measure with the 
 theodolite at I the angle 7 between S and A. 
 
 Measure along the tangent half the length of transi- 
 tion L from S to the point of spiral P, and the other 
 half to the terminal point N. Erect at N the terminal 
 ordinate NQ=4s, and drive a peg at the point of spiral 
 and curve Q. Eepeat the latter operations, deter- 
 mining the ruling points of the second spiral P'Q'. 
 (See Note iv.) 
 
 5. Set up the theodolite over the point A, and lay 
 
FIELD EXEECISE8 63 
 
 down the shift tangent thereat by turning off from I 
 
 the angle 7, thus setting out a parallel by alternate 
 
 angles. Sight along this shift tangent with the plates 
 
 at 0°, and observe the tangential angle to Q. Measure 
 
 the chord AQ, and note if its length accords with the 
 
 AO 
 tangential angle as given by sin~^ ^. This checks 
 
 the position of Q. 
 
 Eun in the circular portion QQ' from the same back- 
 sight, using 1-chain chords, and assuming Q to be 
 of even chainage, unless otherwise instructed. Check 
 the subchord closing on Q' against its tangential angle 
 in order to ascertain if Q' is correctly on the circular 
 curve. (See Note v.) 
 
 6. Insert the transition curves by offset measure- 
 ments, in this case squaring such distances from the 
 tangents, preferably at ten equal intervals. 
 
 Here the offsets will vary as the cubes of unit distances 
 of 30 links from P and P' ; thus — 
 
 P Q 
 
 0-T^'^ ItV" 2A" ^M" 8A'^ 13A" m" 281" 391". 
 
 Note Hi. — When the shift is great or the radius is small, 
 it is usually preferable to insert the spirals, half from the 
 tangent PS, and half from the redundant portion AQ of 
 the curve, the intervals being measured along the latter. 
 Thus in the present case, the following ofisets will occur 
 on either side of the shift line AS: 
 
 at P and Q ; (i)' | at Pts. 1 and 9 ; (l)^ | at Pts. 2 and 8; 
 
 (1)3 1 at Pts. 3 and 7; (!>' | at Pts. 4 and 8; (|)» | at Pt. 5, 
 on the shift line AS. 
 
64 THE TEANSITION SPIEAL 
 
 Note iv. — Wken the foregoing process supersedes tangent 
 offsets the entire circular curve AA' should be run in before 
 the spirals. In this connection, the points on the redundant 
 portions AQ, Q'A' should be abreast the points on the spirals ; 
 otherwise, offsettiag from estimated points on the curve 
 wUl render the method considerably less accurate than that 
 described in the text. However, the redundant portions 
 may be run in with chord's j;\L and corresponding tangential 
 
 angles, 1, 2, 3, . . . 10 times 171-9 ^ minutes. Or, 
 
 if expedient, the portions may be constructed by the 
 platelayer's method with chord offsets o corresponding to 
 
 chords of ^li chain, o being thus equal to tktvd- Of course, 
 
 a template would avoid the use of either artifice. 
 
 Note V. — Though commonly negligible, the chainage of 
 Q will be slightly greater than the chainage of P plus the 
 length of transition L. Hence, in order to run in the pegs 
 on the circular portion at correct chainages, it may be ad- 
 visable to determiae the chainage of Q by adding to the 
 
 chainage of P half the length L plus the quotient ^, where 
 
 S is the tangential angle per chain and ji that for AQ, or 
 
 AQ 
 
 I sin-i -Hw . If, however, offsets from the redundant portion 
 
 are employed, the chainage of Q will be the chainage of 
 P plus the length L ; and the chainage of A, the commence- 
 ment of the circle, will be this siun less half the length L, 
 the length of transition being twice that of the circular 
 portion displaced. 
 
 Thus, if P is 33-60 chains j,nd L 3 chains, the chainage of 
 Q will be 36-60, and that of A this chainage less 1-5 chains, 
 or 35-10. Hence if the first tangential angle from the shift 
 tangent at A be set out as 0-90 S, the pegs wUl be " chained 
 through " on the curve. If, in addition," it is desired to 
 
FIELD EXEECISES 65 
 
 interpolate a peg at Q, tlie tangential angle to Q must be 
 1-5 S, wtere 8, as before, is the tangential angle per 
 chain. 
 
 Chamage pegs on the transitions can he inserted lest 
 by dvrect measurement round the -pomts located by 
 offsets. 
 
 Observations. 
 
66 THE TEANSITION SPIEAL 
 
 Problem 4. — The central angles of a compound with 
 4° and 4° 30' branches scale and reduce respectively 
 as 28° 44' and 21° 23'. It is decided, however, to 
 supersede this compound by one of the same respective 
 radii with spirals, keeping the point of compound 
 curvature as nearly as may be to the point fixed by the 
 given central angles. 
 
 Effect the proposed change, assuming the work- 
 ing speed to be the maxirnum for the sharper 
 branch. 
 
 Procedure. — ^1. Set up the theodolite over the 
 point of intersection I, and measure the angle supple- 
 mentary to the intersection angle A . 
 
 For purposes of calculation let the angle A actually 
 measure 50° 00'. But since the central angles a and /3 sliould 
 together be equal to A; the scaled values must be amended ; 
 say, by subtracting 4' from the larger and 3' from the 
 smaller angle. Thus, a=-- 28° 40' ; /?= 21° 20'. 
 
 2. Calculate and measure the normal tangent lengths 
 T and t, and establish temporary pegs at the correspond- 
 ing tangent points a and a'. (*Also as a check, calcu- 
 late the apex distance Ic, and establish a temporary 
 peg at c. See data of 8.) 
 
 Thus: 
 
 Rver3A-(R-r)vers(A-a) .„ -..oon^ . ^ 
 
 ^ -^^ (R= 1432-7 feet.) 
 
 1432-7(1 - cos 50°) - (1432-7 - 1273-6)(1 - cos 21° 20') 
 
 sin 50 
 1432-7 X -3572- 159-1 X -0685 
 •7660 
 = 653-9 feet. 
 
FIELD EXEECI8ES 
 r vers A + (R - r) vers (A -jS) 
 
 67 
 
 {r= 1273-6 feet.) 
 
 sm A 
 1273-6(1 - cos 50°) + (1432-7 - 1273-6)(1 - cos 28° 40') 
 
 sin 50° 
 1273-6 X -3572 + 159-1 X -1226 
 
 •7660 
 
 ■■ 619-3 feet. 
 
 Vers.c<= — j^_|- 
 
 ,, a Rver&A-T5ira.A 
 Ver&.^= g=7 
 
 Fio. 7. 
 
 Also: Ic^=T2+ac2-2T . ac . cos \a, : where ac= 2R sin |a 
 = 709-5 feet. 
 Ic2= (653-9)= + (709-5)2 _ 2 x 653-9 x 709-5 x -9689 
 = 31,570; and Ic= 177-7 feet. 
 
 Sin a\c= ~ sin |a= ^^ x -2476= -9887 ; and alc= 81° 23'. 
 
68 THE TEANSITION SPIEAL 
 
 3. Assuming the conditions of time approach, the 
 maximum speed of the sharper branch will obtain — ■ 
 viz., lOO-f-V^ miles per hour. Accordingly the length 
 of transition I to this branch will be 600 -f-\/'^ f^et, 
 and s the normal value — viz., 2-618 feet. The length 
 of transition L to the other branch will have to be 
 found tentatively; say, as the above speed multiplied 
 by the superelevation in inches. But since no central 
 transition is to be considered, S sec a must equal s sec /5; 
 and if the corresponding lengths L and I are incompat- 
 ible, the composite curve may have to be shifted bodily 
 along the tangents. 
 
 s sec 13 
 
 Thus, S should equal 
 
 sec a 
 
 ■R774- 
 = 2-618 x-q4ts= 2466 feet. 
 
 Hence L should equal V^i x RS 
 
 = V'24 X U32-7 X 2-466= 291-2 feet. 
 
 But under normal circumstances L can be taken tentatively 
 as 6V„, where V„ is the controlling speed. That is, 
 L=6x47-l=282-6 feet, which also is the time approach 
 length I. 
 
 Hence, shifting the composite curve is not to be con- 
 sidered, and L and I may be taken respectively as 291-2 
 feet and 282-6 feet with corresponding shifts S and s of 
 2-466 feet and 2-618 feet. (* Also, in the present case, 
 it would be inadvisable to reconsider the entire con- 
 struction, keeping the point of compound curvature at c 
 instead of letting it move inwards S sec a along co.) 
 
 4. Calculate the remaining elements and the data 
 of the spirals, assuming the latter to be traced with 
 
FIELD EXEEOISBS 69 
 
 chords C and c corresponding to deflections of 1, 4, 9, 
 etc., minutes. 
 
 (a) Tangent excess aP= S tan a + ^L 
 
 = 247 X -5467 + 14;5-6= 147-0 feet. 
 „ „ a'P'=s tany8 + iZ 
 
 = 2-62 X -3906 + 141-3= 142-3 feet. 
 Hence, total tangent distances, IP, IP' — 
 
 T,= T + aP = 653-9 + 147-0= 800-9 feet. 
 «, = t +a'P'= 619-3+142-3= 761-6 feet. 
 (6) Total deflection angles, i2 and w, 
 
 LD 291-2x4 
 "-10= 10 =^ ^^^^' 
 &)= '\/d= 2° 7-26', being for time approach. 
 Whence *= 5° 49-44'; ^= 6° 21-78'. 
 Long chords PQ, P'Q', or X sec fi and x sec w respectively. 
 
 PQ = L^l - 1 J) sec 12= 290-9 x sec 1° 57'= 291-1 feet. 
 
 P'Q'= l(l-^jj sec (0= 282-2 x sec 2° 7'= 282-4 feet. 
 
 (c) Chord lengths per minute^ deflection, 
 
 „ ^/lOL ,/lOx 291-2 „^„-. ^ 
 V -fy-= V 7 = 26-98 feet. 
 
 . /ToT , /lO x 282-6 „^ „^ , , 
 
 c= V -j-= V 7-.= — =25-06 feet. 
 
 d ^ 4-0 
 
 5. Measure the total tangent distances T„ t„ and 
 drive three pegs at the points P and P'. 
 
 Set up the theodolite over the pomt-spiral P and 
 sight along the tangent with the plates clamped at 0° ; 
 then set off successive deflections of 1', 4', 9', etc., the 
 chainmen measuring the chord lengths accordingly. 
 Sight with the whole deflection X2 and, measuring the 
 
70 THE TEANSITION SPIRAL 
 
 long chord, establish the point-spml-cune Q. Note 
 if the subchord closing on Q agrees "with this angle, 
 its length being the fraction of a chord expressed by the 
 excess of ■\/D, over the number N of the preceding chord. 
 
 Thus: fi being 116'5', the preceding even square is 100', 
 and V116-5 - VIOO gives a subchord of -794:0, or 2142 feet. 
 
 6. Set up over the point-spiral-curve Q, and lay down 
 a tangent thereat by sighting P with the plates clamped 
 at 2X1, and then bringing them to zero. Transit the 
 telescope. [Sometimes it is convenient to establish 
 a temporary intersection peg i by measuring along this 
 tangent a distance of E tan ^{a - <!>)]. Prom this tan- 
 gent run in the circular portion QC, setting off succes- 
 sive deflections of 2° per 100 feet chord. Locate the 
 P.G.C. — C — by setting off the whole deflection angle 
 ^(a-*) and measuring the chord QC equal to 2Rx 
 sin ^{a - <1>). (If the latter measurement be impossible, 
 line in two arrows pi, p^, one on each side of C as deter- 
 mined in the following step.) Observe if the subchord 
 closing on C agrees with the difference between the 
 last two deflection angles, (a-0)-HD being the length 
 of the portion QC in hundreds of feet. 
 
 Length of circular portion QC 
 
 = (28° 40' - 5° 49-5')^4:°= 571-0 feet. 
 
 7. Set up over the point of compound curvature C, 
 and lay down a tangent thereat by sighting Q with the 
 plates clamped at ^(a - <E>), and then bringing them to 0. 
 (Observe if the cross wires bisect the temporary peg 
 at i. Otherwise, obtain the correct position by shifting 
 the instrument in the line of the arrows pi and p^.) 
 Transit the telescope. [Sometimes it is convenient to 
 
FIELD EXEEGISES 71 
 
 establish a temporary intersection peg i' by measuring 
 along this tangent a distance r tan ^(/3 - <^)].' Proceed 
 to run in the circular portion OQ', setting off deflec- 
 tions of 2° 15' per 100 feet chord. Locate the end Q' 
 of the curve CQ' by sighting with the whole deflection 
 angle |(/8-<^) and measuring the whole chord CQ' 
 equal to 2r sin |(/3-^). (If the'latter measurement 
 be impossible, line in two arrows, qj, ga, one on each 
 side of Q' as determined in the following step. See 
 Note vii.) Observe if the subchord closing on Q' 
 agrees with the difference between the last two deflec- 
 tion angles, {^-<f>)-i-d being the length of the portion 
 CQ' in hundreds of feet. 
 
 Lengtli of circular portion CQ' 
 
 = (21° 20' - 6° 21-^')^4° 30'= 332-7 feet. 
 
 8. Set up over the pomt-cune-spiral Q', and lay down 
 a tangent thereat by sighting C with the plates clamped 
 at ^(/3 - (j)) and then bringing them to 0°. (Observe 
 if the cross wires bisect the temporary peg at *'.) 
 Transit the telescope. Prom this tangent lay off twice 
 the total deflection w of the second spiral, and observe 
 if the cross wires bisect the peg at the pomt-spwalr 
 tangent P'. If such is not the case, first check the 
 position of Q' by measuring carefully the terminal or- 
 dinate Q'N'. (If error is evident, shift the instrument 
 in the line of the arrows gji, ^2 until the correct deflection 
 gives P' together with Q'N' equal to 4s. Here the 
 temporary intersection peg i' serves in the case of 
 obstructions in the whole chord OQ'.) 
 
 9. Set up the theodolite over the pomt-spiral- 
 tangent P', and lay off from the tangent successive 
 
72 THE TEAN8ITI0N SPIEAL 
 
 deflections of 121', 100', 81', etc., the chainmen 
 measuring first the subchord from Q', and then succes- 
 sive chords of length c. 
 
 Here the first descending deflection will be 121', the even 
 square under u, or 127', and the corresponding subchord 
 will be (^127-^/121)0 or -292 x 25-06 feet. 
 
 Interpolating Stations. — Stations at 100 feet intervals 
 on the spirals may be inserted by the methods described 
 in Problem 2, the chordages being the ratios of the station 
 distances in feet from P and P' to the chord lengths C and c, 
 also in feet. 
 
 Thus, if P, the P.S., is at 21+62, Q, the P.S.C., will be 
 at 24 + 53-2, C, the P.G.G., at 30 + 24-2, Q', the P.O. 8., at 
 33 + 56-9, and F, the P.S.T. at 36 + 39-5. 
 
 Hence stations on the spirals will occur at the following 
 chordages: 
 
 38-0 138-0 238-0 
 
 "° ^^'- 26-98 26-98 26-98" 
 
 OnP'O'. -^ i^ 260;5 
 
 "^^ ^ "^ • 25-06 25-06 25-06' 
 
 the latter values appertaining to observations from P'. 
 
 N.B.— The occupation of the pomts P, Q, C, Q', P', 
 in order is given on purpose to demonstrate the artifices by 
 which tangency may be obtained at these ruling points 
 in progressive construction. However, in compounding 
 generally, it is usually more satisfactory to run in the first 
 branch through P and the second branch through P', 
 the position of C being exactly located by intersections from 
 Q and Q'. (See Problem 5.) 
 
 Observations. 
 
FIELD EXEEOISES 73 
 
 Problem 5. — ^The pegs a, a', and c indicate respec- 
 tively the tangent points and the point of compound 
 curvature of a compound curve of 36 and 25 chains 
 radii, the pickets indicating the centre line of the 
 straights. 
 
 Eeplace the given compound by curves of the same 
 respective radii, inserting suitable transition curves. 
 
 N.B. — Since the radii E and r differ by about 80 
 per cent, it may be necessary to consider a transition 
 replacing the point of compound curvature. 
 
 Peocedueb. — ^1. Set up the theodolite over the 
 tangent point a, and measure the angle between the 
 tangent and the point of compound curvature c. This 
 will be one-half the central angle a of the first branch, 
 the radius of which is E (Eig. 7). Set up over the point 
 of compound curvature c, and measure the interior 
 angle between a and a'. This will be the supplement 
 of half the sum of the central angles a and /8. 
 
 For purposes of calculation, let a and /8 be 36° 52' and 
 20° 40' respectively, the former subtending the 36 chain 
 curve, the latter the 25 chain curve. 
 
 2. Investigate the compound as to suitable transi- 
 tions, basing the work on the approaches from the 
 straights and the maximum speed of the sharper branch. 
 
 fa) Lengths of transition, L, l^, and I. Assuming the 
 conditions of time approach, the length I of the sharper 
 curve will be ^/r, or 5 chains, while that L of the flatter one 
 
 will be given by los-fp as S-SdB, V being 55 m.p.h. in 
 
 accordance with the rule IIV*"- 
 
 Accordingly the end shifts S and s will be 10'22''' and 
 33-0'' respectively. These will produce central displace- 
 
74 THE TEANSITION SPIEAL 
 
 ments of S sec a and s sec p, and the difierence of these will 
 
 33'< 
 be the gap d of the central transition. Thus d= ^^gg^ - 
 
 10-22'' 
 
 -7gQQQ = 22-5 inches. Now the shift s, in accordance with 
 
 the formula «o=tt(1-s) would only require to be 0-94", 
 
 and the length l, accordingly 1-494 chains by the formula 
 
 M 1-^j. This result accords with the value of I, given by 
 
 multiplying the difference in superelevation (0-5' -•347') 
 by 660, which is the gradient of approach of the sharper 
 curve — i.e., l„= 1-53 chains. On the other hand, a shift 
 under Z" could not be conveniently negotiated at the point 
 of compound curvature. 
 
 There are three ways of considering the problem, except- 
 ing that of dispensing with the central transition, as 
 doubtless would be done in the case in question. 
 
 1. Assuming the shift s^ at a minimum of 2>"', say, finding 
 ?,= 2-727 chains, and finally shifting the sharper curve 
 along the tangent, thus approximating the original location 
 as nearly as may be. 
 
 2. Inserting an absurdly long transition l„ which for the 
 gap of 22J inches would be 7-47 chains, giving a gradient 
 of approach of 1 in 3220. 
 
 3. Lengthening the end transition to the flatter curve 
 such that a convenient gap of, say, 4 inches would result. 
 But this would mean both inordinately long central and 
 end transitions. 
 
 Hence it is evident that the conditions that demand 
 central transitions result normally in a large gap, while 
 such required by radii differing by 50 per cent, are only 
 about 4 inches as far as difference in curvature is concerned. 
 However, other considerations may warrant a large gap 
 even though the transition \ is consequently excessive. 
 
 In order to introduce shifting a half of the composite 
 
FIELD EXEEOISES 75 
 
 curve, let the first assumption be followed, a gap of 3" being 
 introduced. Then the shift S for the larger radius R should 
 be such that S= (s sec /3-3") cos a, or (as in Fig. 4) AA' 
 should be (35-27-3) cos 36° 52', or 17-99 inches. Hence, 
 since AA' is greater than S, the point B must be slid along 
 the tangent a distance B6', which is expressed by 
 
 (gsecj8-3)cosa-S _ 17-99 - 10-22 „ 
 
 8in(a-y8) ~ -8437 -^^'^'■■ 
 
 4. Calculate the remaining elements and data of the 
 transition curves, deciding to spiral through with half- 
 chain chords and to insert the central transition with 
 offsets. 
 
 (a) Total end deflections 12 and u, and long chords PQ, P'Q' : 
 
 fl= 57*^^= 53-3'; and *= 2° 39-8'. 
 
 o)= 573 ^= 1° 54-6'; and <^= 5° 43-8'. 
 
 PQ=--= X sec fl= l('i - g I) sec 53' 
 
 = 3-345 X 1-00012= 3-345 chains. 
 
 P'Q'= X sec 0)= l(l -^fl sec 1° 55' 
 
 = 4-995 X l-00058=,,4-998 chains. 
 
 (6) Shift increment for PQ 
 
 = S tan a= 10-22 x -7499= 7-66", or -01 chain. 
 
 Shift increment for P'Q' 
 
 = s tan /3= 33 x -3772= 12-45". 
 
 But this wUl be reduced by 9-21", and will thus be equal 
 
 to 31", or -004 chain. 
 
 / 1 S\ 
 (c) Spiral extension SP=JLfl-g-gj= 1-673 chains. 
 
 Spiral extension 8'P'=|z(l-g ■^)= 2-499 chains. 
 
76 THE TEANSITION SPIEAL 
 
 6. Set up the theodolite over the fomtrSfwal P, 
 and locate Q by the long chord PQ and the total angle 
 fi. Eun in the transition PQ, using half-chain chords 
 in the manner described in Problem 1. Set up over 
 Q, and run in the circular portion QCi of radius E. 
 Observe if the end of this curve is at S sec a inwards 
 from the point of compound curvature c. 
 
 6. Set up over the pomtrspiral-tangent P', and locate 
 Q' by the long chord P'Q' and the total deflection m. 
 If the chainage of Q' has been computed, the spiral may 
 be run in from this station. Set up over Q', and run 
 in the circular portion O2 Q' of radius r, setting off from 
 the tangent at Q' the differences between j(/3-^) 
 and the tangential angles for an instrument at Ga- 
 
 7. Eun in the central portion EE' by offsets from the 
 displaced circular portions ECi, O^E', varying the 
 ordinates as the cubes of the distances from the pomt- 
 spiral-compound C, where the transition bisects the 
 shift. (Pig. 4) 
 
 Thus with five intervals of 14'9 links on each side of 0, 
 the ofisets from R and R' will be respectively: 
 
 0; (i)» W; (|)» W, (I)' 14"; (I)' Wi and li". 
 
 Incidentally, the chainages of the compound from Q to 
 Q' will be the same as though no transition were inserted. 
 
 Observations. 
 
FIELD EXEECI8ES 77 
 
 Problem 6. — The pegs a and a' are the tangent points 
 of a circular curve of 30 chains radius, the pickets indi- 
 cating the centre line of the straights. It is required 
 to replace the given curve by one provided with transi- 
 tion curves, the greater portion of the curve remaining 
 undisturbed. 
 
 Peocbdure. — ^1. Assume the conditions of time ap- 
 proach, making s thus constant and equal to 33 inches. 
 
 Also assume a slightly smaller radius, 29 chains, 
 say, for the amended radius E,,. Then 7, the angle 
 cut out of each half of the given curve, will be such 
 
 that vers 7==p — ^, E^ being the radius of the given 
 curve in chains. 
 
 Thus: Vers y= ^(30^29)= i= -0*1666; 
 whence y= 16° 36'. 
 
 Had A been less than iy, it would have been advisable 
 to have assumed a larger value for R„. 
 
 For purposes of calculation, let the intersection angle 
 A be 67° 22'. Hence the undisturbed portion CC sub- 
 tends a central angle of 35° 10', and is equal in length to 
 35° 10^1° 54-6', or 18-325 chains, 57-3' being the tan- 
 gential angle S per chain. 
 
 2. Calculate the elements and the data of the transi- 
 tion curves — viz., (a) Whole chord, aO or a'C; (&) Tan- 
 gent excess, aP or a'P'; (c) Total deflection, fl, and long 
 chord, PQ or P'Q'; and {d) Spiral deflection fc per chain^ 
 or per chord c\ 
 
 (a) Whole chords, aC, C'a'= 2R„ sin ^7 
 
 = 2 X 30 X sin 8° 18' = 8-662 chains. 
 (6) Tangent excesses. Pa, a'P' 
 
 = X - R„ sin * - (E„ - R„) sin 7, 
 
78 
 
 THE TEANSITION SPIEAL 
 
 where X is the terminal abscissa and $= 3J2, the total spiral 
 angle. 
 
 Here X - E„ sin * is practically half the length of transi- 
 tion L, which by the ■\/B. rule is ^'v/29, or 2-693 chains. 
 
 Thus, Pa or Fa'=i-v/29-8in 16° 36' 
 
 = 2-693 - -2857- 2-407 chains. 
 
 (c) Total deflection 12= 
 
 = 106-4'; and *= 5° 19-2'. 
 Long chords PQ, P'Q'= X sec fl 
 
 = L^l-g;|-) seer 46-4' 
 
 = 5-380 X 1-0005= 5-383 chains. 
 (d) Deflection k per chain*= 3-6^9 minutes. 
 Deflection per chord^, c = '522 chain. 
 
FIELD EXEECISES 79 
 
 It follows that eact circular portion QC or C'Q' wUl be 
 equal in length to 7 - $-i-2S chains, where S is the tangential 
 angle for the sharpened curve. Thus, QC and C'Q'= 
 6-079 chains. 
 
 3. Measure the tangent excesses aP and a'P', and 
 establish with pegs the points P and P'. Set up over 
 P, and run in the spiral PQ. Establish the point Q 
 with the long chord and total deflection il. 
 
 4. Set up over Q and establish a tangent thereat 
 by setting off from P an angle 211. From this tangent 
 run in the circular portion QO, if possible, estabUshing 
 the point C with the chord QC, which is equal to 
 
 2EaSinK7-*)- 
 
 5. Set up over C, and lay down a tangent thereat by 
 setting off from Q an angle ^(7 - <E>). Observe if the 
 angle between this tangent and a is equal to ^^7. 
 Check the circular portion CC, if possible, establish- 
 ing C with the whole chord 2E<, sin ^( A -27). 
 
 6. Set up over C, and lay down a tangent thereat 
 by setting off from C an angle equal to |(A -27). 
 Verify this position of C by observing if the angle to 
 the given tangent point a' is equal to ^7. Before adjust- 
 ing, measure the whole chord C'a' if possible. (See 
 Note vi.) 
 
 7. From the tangent at C run in the circular portion 
 C'Q', measuring the whole chord C'Q' if possible; 
 or, failing this, adopt P' instead of Q' as the station in 
 constructing the spiral Q'P'. 
 
 Note vi. — It is evident that there are various modes of 
 procedure in constructing the composite curves, and, in 
 consequence, the given routine may have to be amended 
 considerably in order to conform with local conditions. 
 
80 THE TRANSITION SPIEAL 
 
 In general, it is best to locate the points of compound 
 curvature C and C from either a and a' or Q and Q', the 
 latter as located from P and P' respectively. 
 
 N.B. — It is well to note that when it is impossible to 
 measure either of the whole chords C'Q' and O'a', the point 
 C may be fixed by the intersections from Q' and a', or 
 from C and a'. 
 
 Obsbkvations. 
 
FIELD BXEEOISES 81 
 
 Problem 7, — ^The points a, C, and a' have been located 
 respectively as the points of curvature, compound curva- 
 ture, and tangency of a compound of 6° and 7°, the 
 pickets indicating the centre line of the straights. 
 
 Eeplace the given compound by one provided 
 with suitable spirals at the beginning and end, leaving 
 the point of compound curvature undisturbed. 
 
 State the difference in length, between the original 
 and amended routes, the chainage of the tangent point 
 a being recorded as 61+27 and the central angles of the 
 6° and 7° portions as 24° 16' and 20° 40' respectively. 
 
 Peooedueb. — ^1. Set up the theodolite at the point 
 of compound curvature 0, and measure the exterior 
 angle aGa', which should equal the sum of the central 
 angles a and /3. Set up over the tangent point a, and, 
 backsighting along the tangent, measure the angle laC, 
 which should be equal to a. 
 
 For purposes of calculation, let the angles a and /3 check 
 to their respective values of 24° 16' and 20° 40'. 
 
 Since the central angles are small, it may be expedient 
 merely to sharpen the radii E,, and r^ to Eo and ra 
 respectively, instead of compounding at intermediate 
 points E and E', and thus retaining a very considerable 
 portion of the original curves (Pig. 9). On the former 
 basis, calculate the reduced radii E,, and r,, correspond- 
 ing to degrees D,, and da sufficiently increased to give 
 the necessary space shifts S and s. (See Note ii.) 
 
 Thus: vers a= ^5-1^: vers /3(-^. 
 
 2. Assiuning the conditions of time approach, the shift 
 s of the curve Q'C will have its normal value of 2-618 feet 
 
 6 
 
82 
 
 THE TRANSITION SPIRAL 
 
 in accordance with the length 600-:- -\/«? feet, while the con- 
 trolling speed of this sharper branch wUl determine L and 
 consequently S of the approach to the flatter curve CQ. 
 But, since no central transition is involved and the maxi- 
 mum speeds will not differ materially, the 6° branch may 
 
 be elevated by transition for its own speed, and, thus intro- 
 ducing the normal shift s, will avoid the assuming of a 
 value for R„ in determining S. 
 Hence vers 24° 16' 
 
 2-618 
 = (955-37 _ R ) ; ^^^ ^a= 925-74 feet tentatively; 
 
 vers 20° 40' 
 
 2-618 , „„„„,, 
 
 "^ (819-02 -r„) ' '"'•'" ^^^'^ feet tentatively. 
 
FIELD EXEECISE8 83 
 
 Since these values respectively exceed 924'58 feet and 
 778-31 feet, tie radii for 6° 12' and 7° 22', it will be advis- 
 able to increase tte shifts to 2-721 and 2-620 by multiplying 
 
 2-618 by gq „o and An:aa- Accordingly the lengths of 
 transition L and I wiU be — 
 
 L= V24 E,S= ^24^ 924-58 X 2-721= 245-7 feet. 
 
 Z= V'24 r„s= \/24 X 778-31 X 2-619= 221-2 feet, 
 6° 12' and 7° 22' being adopted as the degrees D„ and d^. 
 
 3. Calculate the remaining elements and data of the 
 spirals, assuming the latter to be " run through," 
 with 33| feet chords preferably. 
 
 (a) Tangent excesses. Pa and P'a'; (b) Total deflec- 
 tions, n and (o, and long chords, PQ and P'Q'; and 
 (c) Deflections, k per 100 feet^. 
 
 (a) Tangent excesses : 
 
 aP= (X - R„ sin *) - (R„ - EJ sin a 
 
 = iL(l - g J) - (R» - RJ sin 24° 16' 
 = 122-8 - 30-79 X -41098= 110-1 feet. 
 
 a'P'= ^(l-\ £) - (»•„ - r„) sin 20° 40' 
 = 110-5 - 40-71 X -35293= 96-1 feet. 
 
 (6) Total deflection 12= -^= 2° 32-27'. 
 
 Total deflection w= —=2° 42-85'. 
 *= 7° 36-81'; and ^= 8° 8-55'. 
 
 Long chord PQ= L^l-^ J-) sec 2° 32' 
 
 = 245-3 X 1-00098= 245-5 feet. 
 
84 THE TEANSITION SPIEAL 
 
 Long chord T'Q'=^l(l-^f\ sec 2° i3' 
 
 = 220-8 X 1-00112= 221-0 feet. 
 
 (c) Deflections per 33J ft^ 
 
 /l\n0D 25-234, 
 
 =(l) 
 
 3; L 9 
 
 10^_ 33-301 
 
 I ~ 
 
 2-804 minutes 
 = 3-700 minutes. 
 
 Assuming P to be 27' rear of 61+27— viz., at 61 + 00— 
 Q will be at 63 + 45-7, C at 66 + 14-3, Q' at 67 + 84-3, and 
 F at 70+5-5. 
 
 Hence the stations from P will be at 1-00, 2-00, or chord- 
 ages 3-0 and 6-0, while intermediate points will occur at 
 chordages 1, 2, . . . 4, 5, . . . and 7. Here the deflec- 
 tions will be merely {ly, (2)^, (3)2, etc., times 2-804 
 minutes. 
 
 Similarly the stations from P' wUl be at -945 and 1-945 
 or chordages 2-835 and 5-835, while intermediate points will 
 occur at chordages -835, 1-835 . . . 3-835, 4-835 . . . 
 Here the deflections will be in order, (-835)2, (1-835)S 
 (2-835)2, etc., times 3-700 minutes. 
 
 4. Measure the tangent excesses aP, a'V, and establish 
 the points of spiral P and P'. Set up the theodolite 
 over the pomt-spml P, and, backsighting along the 
 tangent PI, set out the spiral PQ', running through 
 with 33i feet chords. Set out the whole deflection il, 
 and, measuring the long chor^^Q, estabhsh the pomt- 
 sfvralrcurve Q. 
 
 5. Set up over Q, and lay down a tangent thereat 
 by turning off from P an angle 2X2. (Transit the tele- 
 scope). Eun in the circular branch QC from this 
 tangent, chaining through with 100 feet chords and 
 
FIELD EXEECISES 85 
 
 setting out deflections S accordingly. Note if the sub- 
 chord closing on C accords with the difference between 
 the last two deflections. 
 
 6. Set up over the pomt-compound-curvature 0, and 
 lay down a tangent thereat by turning oil from Q an 
 angle equal to ^(a-4>). (Transit the telescope.) 
 Eun in the circular branch CQ' from this tangent, 
 chaining through with 100 feet chords and deflections 
 S' accordingly. Set out the whole deflection angle to 
 Q' as ^ (/8 - (/)), fixing Q' with the corresponding sub- 
 chord. Check the position of Q' by measuring the 
 terminal offset N'Q', which should equal 4s. (See 
 Notes appended to Problems 1, 4, and 6.) 
 
 7. Set up the theodohte over the pomt-tangent P', 
 and, backsighting along the tangent P'l, lay off the 
 total deflection w, checking thus the point Q'. Eun 
 in the second spiral Q'P', setting off the tabulated 
 deflections in descending order, the subchord coming 
 first. 
 
 8. Calculate the length of the original route between 
 the points P and P', first finding the length a to a' 
 by dividing the central angles by their respective 
 degrees D° and d°, and adding the tangent excesses 
 aP and a'P'. Compare this distance with the amended 
 length already determined in the notes: 
 
 Original length a to a'=-- 24° 16'-i-6° + 20° 40'-^?'' 
 
 = 404-4 + 295-2= 699-6 
 Tangent excesses =110-1+ 96-1=206-2 
 
 Total distance a to a' = 905-8 feet. 
 
 Amended distance from difierences P' and P = 905-5 
 feet. 
 
86 TH^ TEANSITION SPIEAL 
 
 Note ii. — Wten the lengths of the branches aC, C'a' are 
 considerable, it is usual to compound the sharpened curves 
 with a considerable portion of the original curve, thus intro- 
 ducing angles Y and v in the expressions. Here Y and v must 
 be assumed, as a rule, such that they are as small as possible 
 consistent with the allowable difEerence in the degrees of 
 curvature. 
 
 Obsebvations. 
 
FIELD EXBECISES 87 
 
 Problem 8. — ^The pegs a, r, and a' indicate respec- 
 tively the poimts of curvature, reverse curvature, and 
 tangency of a true reverse with branches each of 4°. 
 It is required to replace this curve with one having 
 suitable spirals at the straights and at the reverse. 
 
 Give the ruling points of the most efficient improve- 
 ment, the maximum deviation from the existing route 
 being limited to 10 feet. 
 
 Fia. 10. 
 
 There are two ways of dealing with this problem, 
 each making r the common point of spiral of two separ- 
 ate curves, the lengths of transition at the straight being 
 equal to that at the reverse. Of these the _first (A) 
 involves sharpening the radius of each branch, keeping 
 the vertex undisturbed, while the second (B) allows 
 vertex displacement as well as reduction of radius. 
 
 For purposes of calculation, let the recorded values 
 of the central angles a and /3 be assumed — viz., 52° 10' 
 and 56° 20' respectively. 
 
 (A) Vertex Unchanged. Here h=0 in Eq. 3, so 
 that X -Ea sin *=(Ro -Ea) sin JA. Then, time ap- 
 
88 THE TEANSITION SPIEAL 
 
 proach being assumed, Ea is determinate from the 
 equation — 
 
 Solving the quadratic : 
 
 ^ =-,± V 1432-7 + - 
 
 sin 26° 5'- v — ■ ' sin^ 26" 5' 
 ^ =± V 1432-7 + - * 
 
 ~ -4:397- ^ — • ■ .1933 
 
 = - 4-548+ ^1432-7 + 20-69 
 
 = -33-576 feet; and E^= 1127-4 feet. 
 
 R„ thus corresponds to D„ 5° 5', the radius of which 
 is 1127-5 feet. 
 
 (B) Vertex Displaced. On the same assumptions, 
 a central displacement h being admissible, E,, is 
 determinate from the equation — 
 
 ^'•+4t^-(^-^)=«- 
 Solving the quadratic : 
 
 VRa=-s^wA±V(R»-«)+ 
 
 ±\/ (R»-« 
 
 tan 4A" ^ ' ' tanHA 
 -M-F± \/(1432.7-2-67) + ^^^4^ 
 
 = - 4-0856 +V'1430 + 16-69 
 
 = 33-95 feet; and E„= 1152-6 feet. 
 
 Ra is thus nearest to D^ 4° 58', the radius of which is 
 1154-0 feet. 
 
FIELD EXEECISBS 89 
 
 The consequent vertex displacement — 
 
 ^_ (Ro-B„) Yers|A-S 
 cos |A 
 _ (1432-7 - 1152-6) vers 26° 5' - 2-67 
 
 cos 26° 5' 
 = 28-78 feet. 
 
 Hence the process (B) is inadmissible, unless the length 
 of transition is reduced, admitting a larger value of R^ and 
 so reducing h. Therefore let it be assumed that the 5° 5' 
 curve may supersede the first branch, and on similar lines 
 determine the amended degree t?„ of the second branch. 
 But since the angle A is less than 8, the radius Ea wUl be 
 less than r^, and, in consequence, the time approach length 
 S-^/Ro feet may be adopted for the four transitions. 
 
 Thus L, L', V, and «= 268-6 feet, as against 266-2 feet 
 by the rule 600h-vT)„. 
 
 Calculate the remaining elements of the spirals : 
 
 (a) Total deflections, 0=573 ^•, <a=573 -. 
 
 l-^=-Jsecfl. 
 
 Long chords, rg, t^a'=x sec o)=Lf 1-^ — ) sec w. 
 
 (c) Eeduced central angles, (A -2$) and (S-2^), 
 <I> and (/) being Sfi and Sm respectively. 
 
 1. Theodolite at Point of Curvature a. Measure 
 along a direction XI to the tangent the long chord aQ, 
 establishing the pomt-spwal-curve Q. 
 
 2. Theodolite at Point of Eeversed Curvature r. 
 Measure along a direction XI to the back tangent the 
 long chord rQ', establishing the pomt-curve-spiral Q'. 
 Measure along a direction w to the forward tangent 
 
90 THE TEANSITION SPIEAL 
 
 the long chord rq, estabhshing the pomt-spiral- 
 curve q. 
 
 3. Theodolite at Point of Tangency a'. Measure 
 along a direction co to the back tangent the long chord 
 a'q', estabhshing the point-curve-spiral q'. 
 
 4. Check the construction by measuring the terminal 
 ordinates QN and Q'N' and qn, q'n', which should be 
 equal to 4S and 4s respectively. If possible, set up 
 over Q and q, and oheclr the ahgnment of Q' and q' 
 by setting off from tangents at the points occupied 
 angles equal to J(A -2$) and |(S -2(^) respectively. 
 
 Note. — Cases of improvement of existing lines will 
 seldom admit of considerable reduction of radii or large 
 vertex displacements ; and, in consequence, if h is not fixed, 
 a shorter transition than a-\/Bi„ will have to be assumed, 
 f a-^/Ro, for instance. 
 
 Obskbvations. 
 
FIELD EXEE0ISE8 91 
 
 Problem 9. — The pegs a, v, and a' indicate respec- 
 tively the poimt of curve, the vertex, and the pomt of 
 tangent of the given 28 chain curve, the centre line of the 
 straights being reproduced with pickets. 
 
 It is required to replace this curve by one with 
 transitions that approximates the present length of 
 route, and is at the same time within 2 feet of the 
 present aUgnment. 
 
 Set out the required improvement, deferring the 
 insertion of transition curves, and state the maximum 
 difference between the lengths of the original and 
 amended routes. 
 
 Peoobdurb. — ^1. Set up the theodolite over the 
 tangent points a and a', and, backsighting along the 
 whole chord or the straights, measure the angle to the 
 vertex v. In either case this angle should be J A, A 
 being the intersection angle. But in effecting improve- 
 ments, it is often impossible to verify satisfactorily the 
 intersection angle, and, in consequence, this angle is to 
 be taken at the value ascertained in the original location. 
 
 Hence, for purposes of calculation, let the angle A have 
 its recorded value of 42° 36'. 
 
 2. Assuming the conditions of time approach, so that 
 s=-0417 chain, determine Ea tentatively from the 
 relation h cos JA=(Eo-Ea) vers ^A -s, where h is 
 the vertex displacement on amending the radius from 
 Eo to Ea. (See Note ii.) 
 
 Hence, on taking h as -0300 chain, 
 •0300 X cos 21° 18'= (28 - E,) vers 21° 18' - -0417; 
 •0300 X •9317 = (28 - E«) -06831 - -0417. 
 
 Whence 28 - B,^= 1-02 chains. 
 
92 THE TEANSITION SPIEAL 
 
 Let Ea be taken finally as 27 chains, thus reducing h to 
 about 22J inches. 
 
 Then L, the length of transition, will be y/^at or 
 5-196 chains. 
 
 3. Calculate the required elements of the spirals 
 and curve — viz., (a) the Tangent excess, aP or a'P'; 
 (&) the Total deflection, O and the long chord, PQ or 
 P'Q'; (c) the Whole tangential angle, 9; (d) the Tan- 
 gential angle, S per chain; and (e) the Length of circular 
 curve. Also compute the difference in length between 
 the original and amended routes. 
 
 (a) Tangent excess aP= ^l(i -p «-) 
 
 = 2-598fl- ., }. „„" ) 
 \ 5 X 24 X 27/ 
 
 = 2-597 chains. 
 (6) Total deflection fi= ^^= ^^= 1° 50-28' ; whence 
 
 *= 5° 30-84'. 
 3 
 
 Long chord PQ= L( 1 - = ^ j sec fl 
 
 = 5-196(1 -#-^-A) sec 1° 50-28' 
 
 = 5-191 X 1-0005= 5-194 chains. 
 
 (c) Whole tangential angle 9= J( A - 2#) 
 
 = ^(42" 36' -11° 1-7') 
 
 = 15° 47'. 
 
 1719 1719 
 {d) Tangential angle S per chain= -^o~= ~ciir= 1° 3-7'. 
 
 (e) Length of Original Half-Eoute • 
 
 = ^L^l - 5 ]!;) - (R - R«) tan J A + s tan J A + E„i A 
 
 = 2-597 - -3899 + -0417 x -3899 + 28 x -3718 
 = 12-633 chains. 
 
FIELD EXEECISES 93 
 
 Length of Amended Half-Route 
 
 = R«(4A + *)= 27(-3718+ -09623)= 12-636 chains. 
 Hence difference in length is -6 link, or df inches. (See 
 Note iii.) 
 
 4. Measure from the tangent points a and a' the 
 equal tangent excesses aP and a'P', and establish with 
 pegs the points P and P'. 
 
 5. Set up over the pomt-spiral P, and, baeksighting 
 along the tangent, set out the total deflection O. 
 Measure along the direction thus given the long chord 
 PQ, and establish the point Q. 
 
 6. Set up over the pomt-spiral-curve Q with the plates 
 clamped at 20, backsight on P, bring the plates to 
 0°, and transit the telescope. Prom the tangent thus 
 established at Q, set out the circular portion QQ' with 
 1 chain chords, reducing the chainage of Q if " chaining 
 through "is stipulated. During progress interpolate 
 a peg at the amended vertex V, setting out a tangential 
 angle of J(A -2$) to the corresponding chainage 
 around the curve. Verify the position of Q', the pomU 
 spvralrcune. 
 
 7. Set up over the point Q', and check the position 
 of the pomlrsp'i/ral-tangent P' by baeksighting on Q 
 with the plates set at |( A - 2<E>), transiting, and then 
 bringing the vernier past 0° to 211. 
 
 Note a. — The mutual relations between Ji, s, and R^ 
 may be opportundy reviewed at this juncture. 
 
 (a) When h= 0, s= (R„ -R,) vers |^A- 
 
 (6) When R<,= R„, s= ~h cos JA> the displacement 
 being inwards. 
 
 (c) When R„=R„ H-s, s=-h. 
 
 (d) When s tan |A=(R<.-R«) (tan ^A-^A) in the 
 
94 THE TEANSITION SPIEAL 
 
 matter of equal routes, h cos 4A=iS) and s=f(R„-Ea)x 
 vers ^A- 
 
 Tie relation s=2h cos J A is derived by expanding tan 
 1^ A and sin J A after substituting s tan ^ A"^ (tan J A - i A ) 
 from the equation of equal routes in tie general expression 
 h cos ^A=(Eo-E„) vers jA-s- The derived results 
 of {d) thus enable the engineer to deal directly with the 
 case of compound curves. (See p. 95.) 
 
 Note in. — It follows from paragraph (d) of the preceding 
 note that the assumption of equal routes would have led 
 to a displacement Ji under 18 inches, and, in general, the 
 construction would have more closely approximated the 
 original alignment. But since E„ would have been taken 
 as 27 chains, the shift s would be necessarily increased to 
 3 feet, in order to jrield the conditions desired. Merely 
 by chance the improvement in the text is to be preferred. 
 
 Observations. 
 
FIELD EXEECISE8 95 
 
 Problem 10. — The peg a is the pomt of curvature 
 of a 3° curve, the peg c the pomt of compound with a 
 3° 30' curve, and the peg a' the pomt of tangency of the 
 latter branch, the pickets indicating the centre hne 
 of the straights. 
 
 It is required to supersede this compound by one 
 provided with spirals that closely approximates the 
 present location, and at the same time affords the same 
 length of route. 
 
 Give the ruling points for the necessary improvement 
 of the track. 
 
 Peoobdure. — ^1. Set up the theodolite over the 
 point-curvature a and the pomt-tangency a'; and, back- 
 sighting on the point-compound-curvature c, measure 
 the deflection angles to the tangent, these being respec- 
 tively half the central angles a and /8 as subtended by 
 the Do and dg branches. But in spiralling old track, it 
 is often impossible to. verify satisfactorily the angles 
 a and /Q, and, in consequence, such are to be taken 
 at the magnitudes determined in the original loca- 
 tion. 
 
 Hence, for purposes of calculation, let tte angles a and /3 
 have their recorded values : 30° 12' for the 3° branch, and 
 24° 18' for the 3° 30' branch. 
 
 R„= 1910-1 feet. 
 r„= 1637-3 feet. 
 
 2. Since the requirement of equal routes precludes 
 the assuming of h, ra, or E^, proceed on this basis, 
 adopting tentatively the time approach shift of 2-667 
 feet for the da curve, the speed on this sharper branch 
 obtaining throughout the composite curve. Thus, 
 
96 THE TEANSITION SPIEAL 
 
 determine the amended radius r^ and the compound 
 displacement h from the relations h cos y8= 
 ('"o -'"o) vers /8 -s=y. (See Note ii., p. 93.) 
 
 Thus : s= f (r„ - r„) vers /?. 
 
 2-667= 1(1637-3-0 vers 24° 18'. 
 
 Wtence (1637-3-/.)= 45-1; and r.= 1592-2 feet. 
 
 Nearest to the foregoing value of r„ is 1591-8 feet to a 
 degree of 3° 36', and, this latter adopted, the shift s 
 will be increased by -02 foot to 2-687 feet, the length of 
 transition I becoming accordingly ^24 x 1591-8 x 2-687, or 
 320-4 feet. 
 
 3. Since the compound displacement h must be the 
 same for the Da curve, no central transition being neces- 
 sary, determine the shift S of this branch from the equa- 
 tion s cos a=S cos /3, and thence ascertain the amended 
 radius E^ from the relation S=|(E„-Eo) vers a. 
 Some modification will be necessary if the shift S 
 leads to an incongruous length L of transition. As 
 a rule, the disparity will accrue from excess length, 
 and so will involve sliding the sharper branch along its 
 tangent. (See p. 43.) 
 
 XT a cos a 
 
 Now S=s w 
 
 cos p 
 
 cos 30° 12' 
 
 =2-687 
 
 cos 24° 18' 
 
 =2-687 ^^5=2-548 feet. 
 •9114 
 
 And 2-.548=f (1910-1 -EJ vers 30° 12'. 
 Whence (1910-1 -E„)= 28-2; and E„= 1881-9 feet. 
 Nearest to E„ is 1878 feet to degree of 3° 3'; but E^ as 
 calculated must be retained, no change in the shift being 
 
FIELD EXERCISES 97 
 
 admissible; and, although the error in the deflection 6^ 
 per 100 feet would here be inappreciable in work of this 
 nature, it may be advisable to calculate the precise value 
 of Sj, more particularly for ascertaining the length of the 
 circular branch. Thus D„= 2Si= 5730-*-1882= 3° 2f'. The 
 length of transition L will be V'24;X 1881-9 x 2-548, or 
 339-2 feet. Incidentally the normal lengths I and L would 
 be 316-2 feet and 268-7 feet for the speed of the sharper 
 branch ; and the excess over the latter length is reasonably 
 admissible with the conditions involved. 
 
 4. Calculate the required elements of the spirals 
 and curves — ^viz., {a) the Tangent excesses aP and a'P'; 
 (b) the Total deflections O, and w and the long chords 
 PQ and P'Q'; (c) the Whole /tangential angles @ and 9; 
 (d) the deflections ^i, Sg per 100 feet; and (e) the 
 Lengths L,, and ?,. of the circular portions. Also com- 
 pute the difference in length between the original and 
 amended routes. 
 
 / 1 S \ 
 (a) Tangent excess aP=|Lf 1-= ^j=JL= 169-6 feet. 
 
 Tangent excess aT'= ^l (l-g ^)= il= 160-2 feet. 
 
 T 339-2 
 
 (6) Total deflection 0= 573 g-= 573 1881:9= 1° 43-27'; 
 
 whence *= 5° 9-81'. 
 
 Total deflection <o= 573 -^= 573 ^-^8= 1° 55-33'; 
 
 whence ^= 5° 45-99'. 
 
 l_g ^j sec a. 
 
 = 339.2(i_| 1^9) sec 1=43-3' 
 = 339-0 feet. 
 
98 THE TRANSITION SPIRAL 
 
 Long chord P'Q'=ifl-g— j sec u> 
 
 = 320-1 feet. 
 
 (o) Whole deflection 6 
 
 = 4(a-$)=^(30°12'-5'' 9-8')= 12° 3M'. 
 
 Whole deflection 9 
 
 = M/8-'^)=4(24:° 18' -5° 46')= 9° 16'. 
 
 (d) Deflection angles per 100 feet chord: 
 
 8i=l°31i'; 82=1° 48'. 
 
 (e) Length of circular portion QC = 0-5-8^= 822-3 feet. 
 Length of circular portion CQ'= ^^8^= 514-8 feet. 
 
 Length of Original Half-Route Vac 
 
 = p(l-g ^V(R„-R„) tan a+S tan a + R„a 
 = 169-6- 28-2 X -5820 + 2-548 X -5820 +1910-1 x -52705 
 = 1161-4 feet. 
 Length of Amended Half -Route PQC 
 
 = R.(a+*)= 1881-9{-52705+ -09011)= 1161-4 feet. 
 
 Length of Original Half-Route ca'P' 
 = ilfl -^ *-) - ('•0 - »•<.) tan /3 + s tan /? + r^ 
 = 160-2 - 45-5 X -45152 + 2-687 x -45152 + 1637-3 x -42408 
 = 835-2 feet. 
 
 Length of Amended Half-Route CQ'P' 
 
 = k(P + ^>=- 1591'8(-42408+ -10064)= 835-3 feet. 
 
 5. Measure from the tangent points a and a' the 
 respective tangent excesses aP and a'P', and establish 
 ■with pegs the points P and P'. 
 
FIELD EXEECISES 99 
 
 6. Set up over the pomi-spvral P and, backsighting 
 along the tangent, set out the total deflection 12. 
 Measure along the direction thus given the long chord 
 PQ, and estabhsh the pomt-spiral-cune Q. 
 
 6(a). If possible, set out from the tangent at Q 
 the whole deflection |(a=<E>), and line out this direction 
 near the original point-compound-curvature c. Other- 
 wise, the amended point-compound-ourvature C will 
 have to be verified in the running of the amended- 
 branches. 
 
 7. Set up over the pomt-spiral-tangent P', and, back- 
 sighting along the tangent, set out the total deflection 
 CO. Measure along the direction thus given the long 
 chord P'Q', and estabhsh the pomt-curve-spvral Q'. 
 
 7 (a). If possible, set out from the tangent at Q' 
 the whole deflection J(j8-^), and determine the inter- 
 section C with the direction hned out in 6 (a). Measure 
 as h, the displacement cC, which should be equal to 
 
 s 8 
 
 or , 
 
 2 cos /S 2 cos a 
 
 Observations. 
 
APPENDIX 
 
 Table I. — Spiral Elements 
 
 
 
 Terminal De- 
 
 Terminal 
 
 
 Radius 
 
 Length 
 
 flection Angle 
 
 Abscissa X 
 
 Long Chord 
 
 (B). 
 
 (Ii)=VB. 
 
 <">-^- 
 
 K'-f0 
 
 X sec a. 
 
 Chains. 
 
 Chains. 
 
 Minutes, 
 
 Chains. 
 
 Chains. 
 
 20 
 
 4-472 
 
 128-13 
 
 4-466 
 
 4-469 
 
 21 
 
 4-583 
 
 125-03 
 
 4-578 
 
 4-581 
 
 22 
 
 4-690 
 
 122-17 
 
 4-685 
 
 4-688 
 
 23 
 
 4-796 
 
 119-47 
 
 4-791 
 
 4-794 
 
 24 
 
 4-899 
 
 116-96 
 
 4-894 
 
 4-897 
 
 25 
 
 5-000 
 
 114-59 
 
 4-995 
 
 4-998 
 
 26 
 
 5-099 
 
 112-37 
 
 5-094 
 
 5-097 
 
 27 
 
 5-196 
 
 110-27 
 
 5-191 
 
 5-194 
 
 28 
 
 5-292 
 
 108-27 
 
 5-287 
 
 5-290 
 
 29 
 
 5-385 
 
 106-40 
 
 5-380 
 
 5-383 
 
 30 
 
 5-477 
 
 104-61 
 
 5-472 
 
 5-475 
 
 31 
 
 5-568 
 
 102-90 
 
 5-563 
 
 5-566 
 
 32 
 
 5-657 
 
 101-29 
 
 5-653 
 
 5-656 
 
 33 
 
 5-745 
 
 99-73 
 
 5-741 
 
 5-744 
 
 34 
 
 5-831 
 
 98-26 
 
 5-827 
 
 5-829 
 
 35 
 
 5-916 
 
 96-85 
 
 5-912 
 
 5-914 
 
 36 
 
 6-000 
 
 95-50 
 
 5-996 
 
 5-998 
 
 37 
 
 6-083 
 
 94-20 
 
 6-079 
 
 6-081 
 
 38 
 
 6-164 
 
 92-95 
 
 6-160 
 
 6-162 
 
 39 
 
 6-245 
 
 91-75 
 
 6-241 
 
 6-243 
 
 40 
 
 6-325 
 
 90-60 
 
 6-321 
 
 6-323 
 
 41 
 
 6-403 
 
 89-49 
 
 6-399 
 
 6-401 
 
 42 
 
 6-481 
 
 88-42 
 
 6-477 
 
 6-479 
 
 43 
 
 6-557 
 
 87-24 
 
 6-553 
 
 6-555 
 
 44 
 
 6-633 
 
 86-39 
 
 6-629 
 
 6-631 
 
 45 
 
 6-708 
 
 85-42 
 
 6-704 
 
 6-706 
 
 46 
 
 6-782 
 
 84-49 
 
 6-778 
 
 6-780 
 
 47 
 
 6-856 
 
 83-58 
 
 6-852 
 
 6-854 
 
 48 
 
 6-928 
 
 82-70 
 
 6-924 
 
 6-926 
 
 49 
 
 7-000 
 
 81-86 
 
 6-996 
 
 6-998 
 
 50 
 
 7-071 
 
 81-03 
 
 7-067 
 
 7-069 
 
 100 
 
AND Data: Eadius System. 
 
 APPENDIX 
 
 Spiral 
 
 Deflection 
 
 
 
 Extension 
 
 per Chord* 
 
 Chord Length 
 
 Number of 
 
 •2 V 5 R^ 
 
 
 (c)=A Ri. 
 
 Chords {n). 
 
 Chains. 
 
 Minutes. 
 
 Chains. 
 
 
 2-235 
 
 6-407 
 
 •395 
 
 11-312 
 
 2-291 
 
 5-954 
 
 ■410 
 
 11-182 
 
 2-234 
 
 5-553 
 
 -424 
 
 11-053 
 
 2-397 
 
 5-194 
 
 -439 
 
 10-930 
 
 2-449 
 
 4-873 
 
 •453 
 
 10-815 
 
 2-499 
 
 4-584 
 
 ■467 
 
 10-704 
 
 2-549 
 
 4-332 
 
 •481 
 
 10-600 
 
 2-597 
 
 4-084 
 
 •495 
 
 10-501 
 
 2-645 
 
 3-867 - 
 
 •508 
 
 10-405 
 
 2-692 
 
 3-669 
 
 •522 
 
 10-315 
 
 2-738 
 
 3-487 
 
 ■535 
 
 10-228 
 
 2-784 
 
 3-319 
 
 ■549 
 
 10-144 
 
 2-828 
 
 3-165 
 
 ■562 
 
 10-064 
 
 2-872 
 
 3-022 
 
 ■575 
 
 9-986 
 
 2-915 
 
 2-978 
 
 •588 
 
 9-913 
 
 2-957 
 
 2-767 
 
 •601 
 
 9-840 
 
 3-000 
 
 2-653 
 
 ■614 
 
 9-772 
 
 3-041 
 
 2-540 
 
 ■627 
 
 9-705 
 
 3-082 
 
 2-451 
 
 ■639 
 
 9-640 
 
 3-122 
 
 2-352 
 
 ■652 
 
 9-579 
 
 3-162 
 
 2-265 
 
 •664 
 
 9-517 
 
 3-201 
 
 2-182 
 
 ■677 
 
 9-458 
 
 3-240 
 
 2-105 
 
 ■689 
 
 9-404 
 
 3-278 
 
 2-032 
 
 ■701 
 
 9-340 
 
 3-316 
 
 1-963 
 
 ■714 
 
 9^292 
 
 3-354 
 
 1^898 
 
 ■726 
 
 9-243 
 
 3-391 
 
 1-836 
 
 ■734 
 
 9-191 
 
 3-428 
 
 1-779 
 
 •750 
 
 9-141 
 
 3 464 
 
 1-723 
 
 ■762 
 
 9-095 
 
 3-500 
 
 1-670 
 
 ■774 
 
 9-046 
 
 3-535 
 
 1-621 
 
 •785 
 
 9-002 
 
 101 
 
102 
 
 THE TBAN8ITI0N SPIEAL 
 
 Table I. — Spiral Elements 
 
 
 Lengt;h 
 
 Terminal De- 
 
 Terminal 
 
 
 Badius 
 
 flection Angle 
 
 Abscissa X 
 
 Long Chord 
 
 (E). 
 
 ,a,.(f)-- 
 
 K-|-R> 
 
 Xsecfifl 
 
 Chains. 
 
 Chains. 
 
 Minutes. 
 
 Chains. 
 
 Chains. 
 
 51 
 
 7-141 
 
 80-24 
 
 7-137 
 
 7-139 
 
 52 
 
 7-211 
 
 79-46 
 
 7-208 
 
 7-210 
 
 53 
 
 7-280 
 
 78-71 
 
 7-277 
 
 7-279 
 
 54 
 
 7-348 
 
 77-98 
 
 7-345 
 
 7-347 
 
 55 
 f-f 
 
 7-416 
 
 77-26 
 
 7-413 
 
 7-415 
 
 4 
 
 56 
 
 7-393 
 
 75-64 
 
 7-390 
 
 7-392 
 
 57 
 
 7-261 
 
 73-01 
 
 7-258 
 
 7-260 
 
 58 
 
 7-138 
 
 70-51 
 
 7-135 
 
 7-137 
 
 59 
 
 7-017 
 
 68-14 
 
 7-014 
 
 7-015 
 
 60 
 
 6-900 
 
 65-89 
 
 6-898 
 
 6-899 
 
 61 
 
 6-786 
 
 63-75 
 
 6-784 
 
 6-785 
 
 62 
 
 6-677 
 
 61-71 
 
 6-675 
 
 6-676 
 
 63 
 
 6-571 
 
 59-77 
 
 6-569 
 
 6-570 
 
 64 
 
 6-468 
 
 57-91 
 
 6-466 
 
 6-467 
 
 65 
 
 6-369 
 
 56-15 
 
 6-367 
 
 6-368 
 
 66 
 
 6-277 
 
 54-46 
 
 6-276 
 
 6-277 
 
 67 
 
 6-181 
 
 52-84 
 
 6-180 
 
 6-181 
 
 68 
 
 6-088 
 
 51-30 
 
 6-087 
 
 6-088 
 
 69 
 
 5-990 
 
 49-82 
 
 5-989 
 
 5-990 
 
 70 
 
 5-914 
 
 48-41 
 
 5-913 
 
 5-914 
 
 71 
 
 5-830 
 
 47-05 
 
 5-829 
 
 5-830 
 
 72 
 
 5-749 
 
 45-76 
 
 5-748 
 
 5-749 
 
 73 
 
 5-671 
 
 44-51 
 
 5-670 
 
 5-671 
 
 74 
 
 5-594 
 
 43-32 
 
 5-593 
 
 5-593 
 
 75 
 
 5-519 
 
 42-12 
 
 5-518 
 
 5-518 
 
 76 
 
 5-446 
 
 41-07 
 
 5-445 
 
 5-445 
 
 77 
 
 5-376 
 
 40-01 
 
 5-375 
 
 5-375 
 
 78 
 
 5-308 
 
 38-99 
 
 5-307 
 
 5-307 
 
 79 
 
 5-240 
 
 38-01 
 
 5-239 
 
 5-239 
 
 80 
 
 5-175 
 
 37-06 
 
 5-174 
 
 5-174 
 
APPENDIX 
 AND Data: Radius System — Continued. 
 
 103 
 
 Spiral 
 
 Deflection 
 
 
 4 
 
 Extension 
 
 per Chain-* 
 
 Chord Length 
 
 Numherof 
 
 2 V^ 5E; 
 
 <'>=S- 
 
 (c)=aVel. 
 
 Chords (n). 
 
 Chains. 
 
 Minutes. 
 
 Chains. 
 
 
 3-570 
 
 1-573 
 
 •797 
 
 8-958 
 
 3-605 
 
 1-528 
 
 -809 
 
 8-913 
 
 3-640 
 
 1-485 
 
 -821 
 
 8-872 
 
 3-674 
 
 1-444 
 
 -832 
 
 8-818 
 
 3-708 
 
 1-405 
 
 -844 
 
 8-788 
 
 3-696 
 
 1-384 
 
 -850 
 
 8-967 
 
 3-630 
 
 1-384 
 
 -850 
 
 8-545 
 
 3-569 
 
 1-384 
 
 •850 
 
 8-397 
 
 3-508 
 
 1-384 
 
 •850 
 
 8^255 
 
 3-450 
 
 1-384 
 
 •850 
 
 8-117 
 
 3-393 
 
 1-384 
 
 •850 
 
 7-984 
 
 3-338 
 
 1-384 
 
 -850 
 
 7-856 
 
 3-285 
 
 1-384 
 
 -850 
 
 7-731 
 
 3-234 
 
 1-384 
 
 -850 
 
 7-610 
 
 3-184 
 
 1-384 
 
 •850 
 
 7-492 
 
 3-138 
 
 1-384 
 
 •850 
 
 7-379 
 
 3-090 
 
 1-384 
 
 -850 
 
 7-269 
 
 3-044 
 
 1-384 
 
 •850 
 
 7-162 
 
 2-995 
 
 1-384 
 
 •850 
 
 7-059 
 
 2-957 
 
 1-384 
 
 •850 
 
 6-958 
 
 ^ 2-915 
 
 1-384 
 
 -850 
 
 6-860 
 
 2-874 
 
 1-384 
 
 -850 
 
 6-764 
 
 2-835 
 
 1-384 
 
 •850 
 
 6-672 
 
 2-797 
 
 1-384 
 
 •850 
 
 6-582 
 
 2-759 
 
 1-384 
 
 -850 
 
 6-494 
 
 2-723 
 
 1-384 
 
 •850 
 
 6-408 
 
 2-688 
 
 1-384 
 
 -850 
 
 6-325 
 
 2-654 
 
 1-384 
 
 -850 
 
 6-244 
 
 2-620 
 
 l-38i 
 
 -850 
 
 6-165 
 
 2-587 
 
 1-384 
 
 -850 
 
 6-088 
 
104 
 
 THE TEANSITION SPIEAL 
 
 Table II. — Spiral Elements 
 
 
 Length 
 (L) = 1000 2- 
 
 Terminal 
 
 Terminal 
 
 
 Degree 
 
 Deflection 
 
 Abscissa X 
 
 Long Chord 
 
 (D°). 
 
 Angle 
 (fi) = ArLD. 
 
 '(-fi> 
 
 X sec a. 
 
 
 Feet. 
 
 Minutes. 
 
 Feet. 
 
 Feet. 
 
 1° 00' 
 
 333-3 
 
 33-33 
 
 333-3 
 
 333-3 
 
 1° 05' 
 
 36M 
 
 39-12 
 
 361-1 
 
 3611 
 
 1° 10' 
 
 388-9 
 
 45-37 
 
 388-8 
 
 388-9 
 
 1° 15' 
 
 416-7 
 
 52-08 
 
 416-6 
 
 416-7 
 
 1° 20' 
 
 444-4 
 
 59-26 
 
 444-3 
 
 444-4 
 
 1° 25' 
 
 472-2 
 
 66-90 
 
 472-1 
 
 472-2 
 
 1° 30' 
 
 489-8 
 
 73-48 
 
 489-6 
 
 489-7 
 
 1° 35' 
 
 476-6 
 
 75-12 
 
 476-4 
 
 476-5 
 
 1° 40' 
 
 464-9 
 
 77-45 
 
 464-7 
 
 464-8 
 
 1° 45' 
 
 454-5 
 
 79-25 
 
 454-3 
 
 454-4 
 
 1° 50' 
 
 443-2 
 
 81-23 
 
 443-0 
 
 443-1 
 
 1° 55' 
 
 434-7 
 
 82-75 
 
 434-5 
 
 434-6 
 
 2° 00' 
 
 424-3 
 
 84-86 
 
 424-1 
 
 424-2 
 
 2° 05' 
 
 416-6 
 
 86-40 
 
 416-4 
 
 416-5 
 
 2° 10' 
 
 407-6 
 
 88-32 
 
 407-4 
 
 407-5 
 
 2° 15' 
 
 400-0 
 
 90-00 
 
 399-8 
 
 399-9 
 
 2° 20' 
 
 392-8 
 
 91-64 
 
 392-5 
 
 392-6 
 
 2° 25' 
 
 387-0 
 
 93-10 
 
 386-7 
 
 386-8 
 
 2° 30' 
 
 379-5 
 
 94-87 
 
 379-2 
 
 379-3 
 
 2° 35' 
 
 375-0 
 
 96-48 
 
 374-7 
 
 374-8 
 
 2° 40' 
 
 367-5 
 
 97-97 
 
 367-2 
 
 367-3 
 
 2° 45' 
 
 361-4 
 
 99-50 
 
 361-1 
 
 361-3 
 
 2° 50' 
 
 356-5 
 
 101-01 
 
 356-2 
 
 356-4 
 
 2° 55' 
 
 350-9 
 
 102-25 
 
 356-8 
 
 357-0 
 
 3° 00' 
 
 346-4 
 
 103-92 
 
 346-1 
 
 346-3 
 
 3° 05' 
 
 342-8 
 
 105-30 
 
 342-5 
 
 342-7 
 
 3° 10' 
 
 337-2 
 
 106-76 
 
 336-9 
 
 337-1 
 
 3° 15' 
 
 333-3 
 
 108-17 
 
 333-0 
 
 333-2 
 
 3° 20' 
 
 328-6 
 
 109-55 
 
 328-3 
 
 328-5 
 
 3° 25' 
 
 326-0 
 
 110-90 
 
 325-7 
 
 325-9 
 
APPENDIX 
 AKD Data: Degkee System. 
 
 105 
 
 Spiral 
 
 Deflection 
 
 Chord 
 
 
 
 Extension 
 
 per 100 feet" 
 
 Length 
 
 Number of 
 
 Radius 
 
 w-ny 
 
 (.') = ^-g. 
 
 W)=V''^- 
 
 Chords (n). 
 
 (R'). 
 
 Feet. 
 
 Minutes. 
 
 Feet. 
 
 
 Feet. 
 
 166-6 
 
 3-000 
 
 57-74 
 
 5-840 
 
 5729-6 
 
 180-5 
 
 3-000 
 
 57-74 
 
 6-255 
 
 5288-9 
 
 194-4 
 
 3-000 
 
 57-74 
 
 6-736 
 
 4911-2 
 
 208-3 
 
 3-000 
 
 57-74 
 
 7-217 
 
 4583-8 
 
 222-2 
 
 3-000 
 
 57-74 
 
 7-698 
 
 4297-3 
 
 236-1 
 
 3-000 
 
 57-74 
 
 8-179 
 
 4044-5 
 
 245-0 
 
 3-062 
 
 57-14 
 
 8-572 
 
 3819-8 
 
 239-5 
 
 3-320 
 
 54-86 
 
 8-667 
 
 3618-8 
 
 232-4 
 
 3-586 
 
 52-81 
 
 8-801 
 
 3437-9 
 
 227-2 
 
 3-866 
 
 50-98 
 
 8-902 
 
 3274-2 
 
 221-6 
 
 4-126 
 
 49-17 
 
 9-013 
 
 3125-4 
 
 217-3 
 
 4-422 
 
 47-32 
 
 9-096 
 
 , 2989-5 
 
 212-1 
 
 4-714 
 
 46-06 
 
 9-212 
 
 2864-9 
 
 208-3 
 
 5-012 
 
 44-72 
 
 9-295 
 
 2750-4 
 
 203-8 
 
 5-315 
 
 43-37 
 
 9-398 
 
 2644-6 
 
 200-0 
 
 5-615 
 
 42-15 
 
 9-487 
 
 2546-6 
 
 196-4 
 
 5-941 
 
 41-03 
 
 9-573 
 
 2455-7 
 
 193-5 
 
 6-261 
 
 40-02 
 
 9-660 
 
 2371-0 
 
 189-7 
 
 6-580 
 
 38-96 
 
 9-740 
 
 2293-0 
 
 187-5 
 
 6-935 
 
 38-10 
 
 9-822 
 
 2218-1 
 
 183-7 
 
 7-258 
 
 37-12 
 
 9-898 
 
 2148-8 
 
 180-7 
 
 7-601 
 
 36-25 
 
 9-915 
 
 2083-7 
 
 178-2 
 
 7-959 
 
 35-47 
 
 10-049 
 
 2022-4 
 
 175-5 
 
 8-331 
 
 34-65 
 
 10-114 
 
 1964-6 
 
 173-2 
 
 8-660 
 
 33-98 
 
 10-193 
 
 1910-1 
 
 171-4 
 
 9-024 
 
 33-37 
 
 10-261 
 
 1858-5 
 
 168-6 
 
 9-392 
 
 32-63 
 
 10-338 
 
 1809-6 
 
 166-6 
 
 9-765 
 
 32-02 
 
 10-397 
 
 1763-2 
 
 164-3 
 
 10-143 
 
 31-40 
 
 10-458 
 
 1719-1 
 
 162-9 
 
 10-526 
 
 30-89 
 
 10-531 
 
 1667-3 
 
106 THE TEANSITION SPIEAL 
 
 Table II. — Spibal EleSiknts 
 
 
 Length 
 ,j. 600 
 
 Terminal 
 
 Terminal 
 
 
 Degree 
 
 Deflection 
 
 Abscissa X 
 
 Long Chord 
 
 (D°). 
 
 Angle 
 (fl) = 60v^D. 
 
 '('-ID- 
 
 X sec fi. 
 
 
 Feet. 
 
 Minutes. 
 
 Feet. 
 
 Feet. 
 
 3° 30' 
 
 320-7 
 
 112-25 
 
 320-4 
 
 320-6 
 
 3° 35' 
 
 317-4 
 
 113-58 
 
 317-1 
 
 317-3 
 
 3° 40' 
 
 313-4 
 
 114-88 
 
 313-1 
 
 313-3 
 
 3° 45' 
 
 310-8 
 
 116-20 
 
 310-5 
 
 310-7 
 
 3° 50' 
 
 306-5 
 
 117-31 
 
 306-2 
 
 306-4 
 
 3° 55' 
 
 303-2 
 
 118-80 
 
 302-9 
 
 303-1 
 
 4° 00' 
 
 300-0 
 
 12000 
 
 299-7 
 
 299-9 
 
 4° 05' 
 
 297-3 
 
 121-15 
 
 297-0 
 
 297-2 
 
 4° 10' 
 
 293-9 
 
 122-48 
 
 293-6 
 
 293-8 
 
 4° 15' 
 
 291-5 
 
 123-69 
 
 291-2 
 
 291-4 
 
 4° 20' 
 
 288-2 
 
 124-91 
 
 287-9 
 
 288-1 
 
 4° 25' 
 
 285-7 
 
 126-01 
 
 285-4 
 
 285-6 
 
 4° 30' 
 
 282-8 
 
 127-28 
 
 282-5 
 
 282-7 
 
 4° 35' 
 
 280-8 
 
 128-40 
 
 280-5 
 
 280-7 
 
 4° 40' 
 
 279-5 
 
 128-78 
 
 279-2 
 
 279-4 
 
 4° 45' 
 
 275-2 
 
 130-76 
 
 274-8 
 
 275-0 
 
 4° 50' 
 
 272-9 
 
 131-90 
 
 272-5 
 
 272-7 
 
 4° 55' 
 
 271-0 
 
 133-00 
 
 270-6 
 
 270-8 
 
 5° 00' 
 
 268-3 
 
 134-16 
 
 267-9 
 
 268-1 
 
 5° 05' 
 
 266-3 
 
 135-25 
 
 265-9 
 
 266-1 
 
 5° 10' 
 
 263-8 
 
 136-40 
 
 263-4 
 
 263-6 
 
 5° 15' 
 
 262-0 
 
 137'52 
 
 261-6 
 
 261-8 
 
 5° 20' 
 
 259-8 
 
 138-56 
 
 259-4 
 
 259-6 
 
 5° 25' 
 
 257-8 
 
 139-64 
 
 257-4 
 
 257-6 
 
 5° 30' 
 
 255-8 
 
 140-70 
 
 255-4 
 
 255-6 
 
 5° 35' 
 
 254-2 
 
 141-81 
 
 253-8 
 
 254-0 
 
 5° 40' 
 
 252-0 
 
 142-82 
 
 251-6 
 
 251-8 
 
 5° 45' 
 
 250-2 
 
 143-91 
 
 249-8 
 
 250-0 
 
 5° 50' 
 
 248-4 
 
 144-86 
 
 248-0 
 
 248-2 
 
 5° 55' 
 
 246-7 
 
 145-92 
 
 246-3 
 
 246-5 
 
APPENDIX 
 AND Data: Degree System — Continued. 
 
 107 
 
 Spiral 
 Extension 
 
 2\ 5RJ 
 
 Deflection 
 per 100 leet^ 
 
 (f) = lD|. 
 
 Cliord 
 Length 
 
 (c')=V D 
 
 Number of 
 Chords (n). 
 
 1 
 
 Radius 
 (E). 
 
 Feet. 
 
 Minutes. 
 
 Feet. 
 
 
 Feet. 
 
 160-3 
 
 10-859 
 
 30-27 
 
 10-593 
 
 1637-3 
 
 158-6 
 
 11-316 
 
 29-76 
 
 10-658 
 
 1599-2 
 
 156-6 
 
 11-702 
 
 29-24 
 
 10-718 
 
 1562-9 
 
 155-3 
 
 12-103 
 
 28-82 
 
 10-778 
 
 1528-2 
 
 153-2 
 
 12-509 
 
 28-28 
 
 10-831 
 
 1495-0 
 
 151-5 
 
 12-963 
 
 27^79 
 
 10-899 
 
 1463-2 
 
 149-9 
 
 13-333 
 
 27-39 
 
 10-954 
 
 1432-7 
 
 148-6 
 
 13-752 
 
 26-90 
 
 11-007 
 
 1403-5 
 
 146-9 
 
 14-178 
 
 26-56 
 
 11-067 
 
 1375-4 
 
 145-7 
 
 14-603 
 
 26-19 
 
 11-122 
 
 1348-4 
 
 144-0 
 
 15-032 
 
 25-75 
 
 11-176 
 
 1322-5 
 
 142-8 
 
 15-470 
 
 25-44 
 
 11-225 
 
 1297-6 
 
 141-3 
 
 15-909 
 
 25-07 
 
 11-293 
 
 1273-6 
 
 140-3 
 
 16-343 
 
 24-75 
 
 11-331 
 
 1250-4 
 
 139-7 
 
 16-803 
 
 24-47 
 
 11-349 
 
 1228-1 
 
 137-5 
 
 17-255 
 
 24-07 
 
 11-437 
 
 1206-6 
 
 136-4 
 
 17-710 
 
 23-76 
 
 11-485 
 
 1185-8 
 
 135-4 
 
 18-170 
 
 23-48 
 
 11-533 
 
 1165-7 
 
 134-1 
 
 18-634 
 
 23-16 
 
 11-583 
 
 1146-3 
 
 133-1 
 
 19-101 
 
 22-89 
 
 11-629 
 
 1127-5 
 
 131-8 
 
 19-573 
 
 22-60 
 
 11-679 
 
 1109-3 
 
 130-9 
 
 20-048 
 
 22-34 
 
 11-727 
 
 1091-7 
 
 129-8 
 
 20-527 
 
 22-07 
 
 11-771 
 
 1074-7 
 
 128-8 
 
 21-010 
 
 21-82 
 
 11-817 
 
 1058-2 
 
 127-8 
 
 21-498 
 
 21-57 
 
 11-862 
 
 1042-1 
 
 127-0 
 
 21-998 
 
 21-33 
 
 11-908 
 
 1026-6 
 
 125-9 
 
 22-483 
 
 21-09 
 
 11-951 
 
 1011-5 
 
 125-1 
 
 22-980 
 
 20-87 
 
 11-996 
 
 996-9 
 
 124-1 
 
 23-483 
 
 20-64 
 
 12-036 
 
 982-6 
 
 123-3 
 
 23-986 
 
 20-43 
 
 12-080 
 
 968-8 
 
INDEX 
 
 Abscissa, terminal, 6, 32 
 Adjustment, gradient of, 6, 11, 
 
 12, 61 
 Alignment, 35-50 
 Amended length of route, 47, 92, 
 
 98 
 Amended radius or degree, 36, 
 
 44, 47, 77, 81, 87, 91, 95 
 Angle, intersection, 5, 36 
 
 total deflection, 6, 16, 21, 
 
 32, 100 
 total spiral, 5, 15, 31 
 whole tangential, 54, 58, 93 
 Angles, central, 40, 48, 81 
 
 deflection, 15, 17, 18, 21, 
 
 24, 32, 100 
 tangential, 53, 57, 64 
 Apex of curve, 4, 36, 40, 44, 77, 
 
 93 
 Appendix, 100 
 
 Backsighting, 21, 53, 58, 70 
 Bogey, I 
 
 Cant, 1, 2, 12 
 Cartesian co-ordinates, 15 
 Cases of alignment, 35 
 Central angles, 40, 58, 81 
 Central shifts, 39, 40, 74 
 Central transition, 38, 76 
 Centrifugal force, 1 
 Chainage, 20, 51, 56, 64, 65, 72 
 " Chaining through," 19, 61 
 Chord, 17-25, 32 
 long, 6, 25, 32 
 whole, 54, 70, 77, 80 
 Chordage of station points, 25- 
 
 28, 59, 72, 84 
 Chord lengths, 17, 18, 23, 56, 69 
 
 Chord offsets, 27, 59, 64 
 
 Clothoid, 13 
 
 Compound curves, 37-43, 45, 48, 
 
 66, 73, 95 
 Construction, 17-30 
 
 by chord-deflections, 18, 19, 
 23, 32, 51, 66, 66, 73, 77, 
 81 
 by ofisets, 28, 61, 76 
 Cubic parabola, 16, 17, 28, 61, 
 76 
 
 Data of spirals, 32, 62, 56, 62, 
 68, 75, 77, 83, 89, 92, 97, 
 100 
 Data of time approaches, 11, 12, 
 
 32, 100 
 Deflection angle, total, 6, 16, 21, 
 
 32, 100, 104 
 Deflections, 15, 17, 18, 21, 24, 
 
 32, 100 
 Degree of curvature : degree 
 
 system, 2, 3, 12, 18, 19, 22, 32, 
 
 104 
 Development, 13-34 
 Displacement, 36, 40, 91, 93, 
 
 99 
 
 Elements of spirals, 5, 30, 100 
 Elevation of outer rail, 1, 2, 6, 7, 
 
 11, 12 
 End shifts, 40, 73 
 Equation of clothoid, 13 
 Equation of equal routes, 48, 94, 
 
 96 
 Excess, tangent, 5, 31, 37, 40, 
 
 45 
 Extension, spiral, 5. 31, 37, 40, 
 
 100, 104 
 
 109 
 
110 
 
 THE TEANSITION SPIEAL 
 
 Field exerciseE, 51-99 
 Froude's curve of adjustment, 
 
 28, 43 
 Fundamental case of spiralling, 
 
 4, 39 
 
 Gap in shifting, 41, 74 
 Gauge, standard, 2 
 General case of alignment, 36 
 General construction, 36, 49 
 Gradient of adjustment, or 
 approach, 6, 11, 12, 61 
 
 Inaccessible intersection, 67 
 Increment, shift, 5, 11, 40 
 Initial deflections, 18, 19, 32, 33 
 Interpolating stations, 25-28, 59, 
 
 72, 84 
 Intersection angle, 5, 36 
 Intrinsic equation of clothoid, 
 
 13 
 Introduction, 1-12 
 
 Length of circular curves, 16, 47, 
 
 52, 57, 92, 98 
 Length of transition, 6-12, 32, 
 
 38, 74, 100 
 Limiting speeds, 9, 10 
 Long chord, 6, 25, 32, 100 
 
 Main tangent, 4, 36 
 Maximum speed, 2, 11, 12, 32 
 Maximum superelevation, 2, 3 
 Minimum shift, 74 
 Minute-reading theodolite, 33 
 Modes of construction, 17-30 
 with chain, 28 
 with theodolite, 17 
 
 Normal deflections, 21, 24 
 Normal shift, 11, 12, 32 
 Notation, 33 
 
 Objects of transition curve, 4 
 GfEsets from the chord, 27, 59 
 
 from the tangent, 28, 62 
 Oflsetting a transition curve, 28, 
 
 62, 76 
 Ordinate, terminal, 6, 16, 32 
 Original length of route, 47, 92, 
 
 98 
 
 Original radius or degree, 36, 44, 
 47, 77, 81, 87, 91, 95 
 
 Points, spiral, 17, 21, 25 
 
 station, 25-28, 69, 72, 84 
 Polychord spirals, 17 
 Problems, field, 51-99 
 Problems in alignment, 35 
 
 Radius of curvature : radius 
 system, 2, 9, 11, 20, 
 100 
 amended radius, 36, 44, 
 
 47 
 original radiup, 36, 44, 
 47 
 Redundant portion of curve, 30, 
 
 64, 76 
 Reverse curves, 43, 46, 49, 87 
 Routes, amended and original, 
 47, 92, 98 
 equation of equal, 48, 94, 
 96 
 Rules, for length of transition, 
 6, 32, 38 
 for maximum speed, 3, 11, 
 
 12 
 for superelevation, 1, 2, 11, 
 
 12 
 Shortt's rule, 8, 11, 39 
 
 Shift, ,5, 11, 12, 16, 30, 40, 44, 49, 
 
 63, 74 
 Shift centre, 74, 76 
 
 increment, 5, 11, 31, 40 
 tangent, 4, 63 
 "Shifting forward," 21 
 Shortt's rule, 8, 11, 39 
 Simple curves, 39, 44, 47, 51, 56, 
 
 61, 77, 91 
 Speed on curves, 3, 11, 12 , 
 "Spiralling in," 19, 23, 56, 66 
 "Spiralling through," 19, 51, 81 
 Spiral angle, 5, 15, 31, 100, 104 
 Spiral elements, 5, 30, 100, 104 
 Spiral extension, 5, 40, 31, 37, 
 
 100, 104 
 Spiral notation, 33 
 Spiral reverse, 43, 49, 87 
 Standard gauge, 2 
 Superelevation, 1, 2, 6, 7, 11, 12 
 
INDEX 
 
 111 
 
 Tables, 11, 12, 24, 32, 100, 104 
 Tangent, 3, 4, 35, 43 
 
 at any point, 21, 24, 25 
 
 at origin, 13, 21 
 
 shift tangent, 4, 62 
 Tangential angles, 53, 57, 64 
 
 whole tangential angle, 64, 
 58, 93 
 Tangent distances, 6, 31, 51, 56, 
 
 62, 66 
 Tangent excess, 5, 31, 37, 40, 45 
 Tangent offsets, 28, 62 
 Terminal abscissa, 6, 32 
 Terminal deflection angle, 6, 15, 
 
 21, 32, 100, 104 
 Terminal ordinate, 6, 16, 32 
 
 Time approach, 7, 11, 12, 32 
 Total spiral angle, 5, 15, 31 
 Total tangent lengths, 6, 31, 51, 
 
 57, 69 
 Transition, 4 
 
 Uniform approach, 6, 61 
 
 Velocity on curves, 1, 3, 11, 12, 
 
 32 
 Vertex of curve, 4, 36, 40, 44, 77, 
 
 93 
 
 Whole cord, 54, 70, 77, 80 
 Widening of gauge, 4 
 
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