UC-NRLF 
 
 SB EflD mi 
 
THE TRANSITION CURVE 
 
 OR 
 
 CURVE OF ADJUSTMENT 
 
 AS APPLIED TO THE ALIGNMENT OF 
 RAILROADS 
 
 BY THE METHOD OF RECTANGULAR 
 
 CO-ORDINATES AND BY 
 
 DEFLECTION ANGLES 
 
 (OR POLAR CO-ORDINATES) 
 
 BY 
 
 N. B. KELLOGG 
 
 M. AM. SOC. C. E. 
 
 THIRD EDITION 
 
 NEW YORK 
 
 McGRAW PUBLISHING COMPANY 
 1907 
 
COPYRIGHTED, 1899, 1904, 
 
 BY 
 N. B. KELLOGG 
 
 Stanhope press 
 
 F. H. GILSON COMPANY 
 BOSTON U. S. A. 
 
PKEFACE 
 
 ERRATA 
 
 Page 8, Equation (17), read: ^p when- = 0. 
 
 Page 22, Line 5, read: With instrument at A, PA - 0, 
 
 since D = 0. 
 Page 23, Second line above footnote: change factor to 
 
 term. 
 Page 37, Last line of Problem III, Case 2, read: as in 
 
 the first part of this problem. 
 Page 46, Table, 5th column, near bottom: change 7.35 to 
 
 6.62. 
 Page 46, Table, 10th column, middle: change 1.07 to 
 
 1.57. 
 
 Page 65, Table, 7th column, top: change 08.5' to 07.5'. 
 Page 69, near bottom of page, read: External Sec. a = 
 
 Sec.a-L 
 
 Nordling equations, the formulae (42-46) were obtained 
 by the writer during the summer of 1884, as were also 
 (29) inclusive, by an independent process, resulting 
 in the verification of the formulae invented by Froude in 
 1841 and published in 1861. Hence, it is thought that 
 
 366471 
 
Stanbopc press 
 
 F. H. GILSON COMP 
 BOSTON U.S.A. 
 
PREFACE 
 
 WHILE the theory of the transition curve as here devel- 
 oped is based upon the methods given by M. Nordling 
 (see "Annales des Fonts et Chausse*es," 1867), the equa- 
 tions for both the cubic parabola and the spiral appear 
 in Professor Rankine's "Civil Engineering" (Ed. 1863), 
 and are accredited to Dr. William Froude, undoubtedly 
 ante-dating those of Nordling. 
 
 Translations of a portion of Nordling's demonstration 
 have appeared from time to time, but only so far as 
 related to connecting a straight line with a circular 
 curve. That portion relating to connecting circular 
 curves of different radii, by means of the cubic parabola, 
 'has not appeared in the form given by Nordling, so far 
 as I am aware. The formulae deduced in the latter case 
 are of general application and equally true for connecting 
 curve with curve, or curve with tangent, when proper 
 values are introduced into the equations. 
 
 That of joining a tangent with a circular curve, by 
 means of the transition curve, is a special case where one 
 of the radii becomes infinitely great. Some of the recent 
 spirals, adopted as curves of adjustment in railroad 
 location, easily develop from the equations of the cubic 
 parabola by making the proper substitutions in them. 
 
 Following the supposition indicated by M. Nordling, 
 i.e., regarding x = L and substituting L for x in the 
 Nordling equations, the formulae (42-46) were obtained 
 by the writer during the summer of 1884, as were also 
 (2-9) inclusive, by an independent process, resulting 
 in the verification of the formulae invented by Froude in 
 1841 and published in 1861. Hence, it is thought that 
 
 360471 
 
IV PREFACE 
 
 the curve herein presented may, not improperly, be 
 called Froude's Spiral. Since the spiral, as an adjust- 
 ment curve, is free from some defects noticeable in the 
 cubic parabola (appearing in the edition of 1899), the 
 latter is omitted from the present edition. 
 
 The Compound Circular Transition Curve,* known also 
 3,s the " Railroad Spiral" (an approximation to the Froude 
 Spiral), though giving good results as an adjustment 
 curve, is defective in not being expressed by a formula 
 following a law of uniform change of curvature for con- 
 secutive points throughout its entire length. A chapter 
 on the compound circular curve was prepared simul- 
 taneously with that of the true spiral and the cubic 
 parabola, but was abandoned for the reason that the 
 deduced formulae were found to be not easily transposed 
 or modified for arbitrary values of the offset distance 
 and for the computation of fractional chords. 
 
 Few engineers, at the present time, will locate an 
 important railway line without the use of some good 
 curve of adjustment. 
 
 A few simple methods of the calculus are used to 
 derive the formulae, but a knowledge of only the ordinary 
 processes of algebra and trigonometry is required for 
 
 * If a compound circular transition curve and a spiral be run for 
 the same offset, the origin of the compound circular curve will be 
 fonnd one-half chord length in advance of that of the true spiral 
 and the terminus of the compound circular curve one-half chord 
 length back of that of the spiral, nearly, i.e., the ends of the chorda 
 of the spiral correspond closely to the middle points of the several 
 arcs of the compound circular transition curve. For instance, a 
 compound circular curve of nine equal chords will have, nearly, 
 the same offset (/) as a spiral of ten equal chords of the same 
 length, each curve terminating in the same radius of curvature as 
 the main curve; the total curvatures, however, will differ, since the 
 spiral is tangent to the main curve nearly one-half chord length 
 beyond where the compound circular curve would become tangent 
 to the main curve. 
 
PREFACE V 
 
 their application. It is seldom necessary to use more 
 than two terms in the formulae involving series. 
 
 The larger type may be read independently of the 
 smaller type. The latter may be used when the curve 
 is extended beyond ordinary limits and greater accuracy 
 in results is sought. 
 
 For field practice, formulae (42-52), (60-61), are the 
 ones with which, taken with the tables, one should 
 become familiar ; and these formulae may be regarded as a 
 summary of the essential part of the book.* 
 
 The "run off" is understood to be equal in length to 
 that of the transition curve and coincident with it. 
 
 The transition or adjustment curve can rarely be 
 applied to switches or turnouts to advantage, except 
 where high speeds are maintained. 
 
 The more important formulae are in full-faced type. 
 
 I am indebted to Professor H. I. Randall, C.E., of the 
 University of California, and also to Mr. D. E. Hughes, 
 C.E., for valuable suggestions. The latter is the author 
 of an excellent paper on this subject. 
 
 N. B. K. 
 
 SAN FRANCISCO, January, 1904. 
 
 * By making all terms containing the factors m or /, representing 
 spiral offsets in Probs. (I-V) equal zero, the spiral disappears from 
 the formula and it stands for circular curves simple or compound as 
 the case may be. 
 
 Except in a few cases, Greek letters have been used to designate 
 angles, small Roman italics for lines and Roman capitals for points. 
 
 S. F. 1907. 
 
CONTENTS 
 
 THE SPIRAL 
 
 SECTION. PAGE. 
 
 1 DEFINITION AND OBJECT OF THE TRANSITION 
 
 CURVE .... ..,.., '^v* 1 
 
 1 FUNDAMENTAL EQUATION OF THE CURVE ... 3 
 
 2 EQUATION FOR COMPOUND CURVES 3 
 
 3 CENTRAL ANGLE FOR COMPOUND CURVES ... 5 
 
 4 DIFFERENCE IN LENGTH OF SPIRAL AND CIRCU- 
 
 LAR ARCS SUBTENDING SAME ANGLE .... 6 
 
 5 TRANSFORMATION TO RECTANGULAR CO-ORDI- 
 
 NATES 7 
 
 6 RECTA-NGULAR CO-ORDINATES AT THE OFFSET 
 
 DISTANCE 11 
 
 7 OFFSET DISTANCE. DISTANCE BETWEEN CEN- 
 
 TERS .... 13 
 
 8 PRACTICAL FORMULAE 15 
 
 9 METHOD BY DEFLECTION ANGLES 18 
 
 10 ORDINATES FROM LONG CHORD 24 
 
 10 REMARKS ON SUPERELEVATION, ETC. TABLE 
 
 OF RADII AND THEIR RECIPROCALS 25 
 
 PROBLEMS 
 
 11 PROBLEM I. SEMI-TANGENTS . .___. . ^ ... 27 
 
 12 EXTERNAL SECANTS 29 
 
 13 PROBLEM II. LOCATION OF OFFSET "/" ... 31 
 
 14 PROBLEM III. COMPOUND CURVES 33 
 
 15 PROBLEM IV. TANGENTS TO Two CURVES . . 37 
 
 16 TRANSITION CURVE IN OLD TRACK 40 
 
 vii 
 
Vlll CONTENTS 
 
 SECTION. PAGE. 
 
 17 EXPLANATION OF TABLES. LAYING OUT BY 
 
 RECTANGULAR CO-ORDINATES 42 
 
 18 LAYING OUT BY DEFLECTION ANGLES .... 43 
 
 TABLES 
 
 LENGTH OF CIRCULAR ARCS AT RADIUS = 1 . . 45 
 MINUTES IN DECIMAL OF DEGREE ...... 46 
 
 TRANSITION CURVE TABLES 46 
 
 APPENDIX 
 MISCELLANEOUS PROBLEMS AND TABLES ... 59 
 
THE TRANSITION CURVE 
 
 i. The true transition curve is one of which the 
 radius of curvature, at its origin, is infinitely great; 
 and at any other of its points, the radius of curvature is 
 inversely proportional to the distance of the point, 
 measured on the curve, from the origin; the product of 
 the radius and distance being a constant for all points 
 of the curve. 
 
 The object of introducing the transition curve between 
 circular curves of different radii, or between a tangent 
 and circular curve, as applied to the alignment of rail- 
 roads, is to give centrifugal force an appreciable time to 
 develop, from that due to one given radius of curvature 
 to that of another, in a moving body passing from a 
 circular path of one rate of curvature to that of another 
 rate of curvature; and to develop simultaneously a force 
 equal and opposed to the centrifugal force, neutralizing 
 it at every point of the curve. If one of the given radii 
 is made infinitely great, its curve becomes a tangent; 
 and centrifugal force develops gradually from zero to 
 that due to the other given radius of curvature, in the 
 time it takes to traverse the transition curve; thus 
 avoiding instantaneous development or "shock." 
 
 The opposing force is the horizontal component of a 
 force due to gravity developed by inclining the vertical 
 axis, passing through the center of gravity of the moving 
 body, from a normal to the plane of rotation and towards 
 the center of rotation. 
 
 From mechanics the expression for centrifugal force 
 1 
 
SITION* CURVE 
 
 / of a weight -io moving iu a circular path with a radius r 
 and a velocity v is : 
 
 , _ wv 2 
 ~ ' 
 
 The opposing horizontal force due to gravity is 
 (Fig. 1): 
 
 we 
 
 w, = , 
 
 g 
 
 in which w is the weight of the moving body, g the width 
 of the path or gauge of the track, and e the inclination 
 of the path, or "cant," towards the center of rotation, 
 in a distance g. 
 
 Since by the hypothesis the opposing forces equal 
 each other in intensity, 
 
 solving for e 
 
 nearly.* 
 
 * More correctly (Fig. 1) e 
 
 hv 2 
 
 V (32.2) 2 r 2 + v* 
 
 in which h gauge of track. 
 
 The resultant of w and w/ due to 
 the effect of gravity = vV + Wl i 
 which is the pressure normal to the 
 plane of the path when centrifugal 
 force is developed. 
 
 It may not be out of place to 
 remark here, that it is the usual 
 custom in fixing superelevation, to 
 depress the inner rail below the 
 grade of the center line of the track 
 
 %e and elevate the outer rail above the grade line %e, thus main- 
 taining the center line at grade. 
 
 Fig. I. 
 
THE SPIRAL 3 
 
 If i denote the distance, measured on the curve, re- 
 quired to change the inclination or "cant" of the path 
 
 one unit, - will denote the change of cant in one unit of 
 
 distance. Therefore, in a distance OB 4 = Z/ y the cant, or 
 superelevation, will be: 
 
 i 32.27-, 
 
 from which 
 
 The second member of Eq. (2) is a constant, since 
 all its factors are assumed constant, and represented by 
 P, whence, the fundamental equation of the curve: 
 
 If, '= P, (3)* 
 
 which is the equation of a spiral, in which OB, = L /} 
 Fig. 2, represents the required length of the transition 
 curve, DB, = r, the radius of the curvature common to 
 the transition curve and the given circular curve where 
 these curves become tangent each other. (See Fig. 2.) 
 
 GENERAL FORMULA. 
 
 2. If h? the figure we let OB, be denoted by L t and 
 OB n by L y/ , then 
 
 OB, =, =f; OB it = L ti = f; 
 
 T i '// 
 
 subtracting, letting L u L, = L, we have the General 
 Formula for compound curves. 
 
 * Froude's Eq. (3), is in the form of L = ei, which is called by 
 Rankine "The Curve of Adjustment." Any particular spiral is 
 designated by the numerical value assigned to its constant P. 
 
TRANSITION CURVE 
 
 which is the ideal equation of all Transition Curves, 
 B t being the first and B lt the second point of compound. 
 
 .A......^ 
 
 ^V- 
 
 E 
 M 
 
 r-TT^v^S 
 
 \__^ v y:i:z::Wa c 
 
 .*>. A <. / / n r >>.. 
 
 Tpv^ 
 
 ' =* ,-^v '2r 
 
 -t^ * 
 
 It fulfills every condition of theory and offers no unusual 
 difficulties in application. 
 
 When the length L of the spiral and the radii r,, and 
 r y are assumed. 
 
 The constant P = ( Lr//r/ } 
 V, - rj 
 
 L 
 
 or 
 
 
 (5) 
 
THE SPIRAL 5 
 
 p 
 
 3. In general let L = ~ = Pr' 1 ; differentiating dL = Pr~ z dr. 
 
 We have from the calculus: dL/ = rda, whence 
 
 rda = Pr-*dr, da = Pr ~* dr ' =- Pr^dr\ or 
 
 hence for any two arcs a tj and a, at radius = 1, we have: 
 P P 
 
 a " = 2^' a ' == 2^ ; 
 p p PI 1 1 
 
 P/l 1\ /I 1 \ 
 
 a// - a =FT --- -- 1 -- * 
 2 lr^ rj \r rj 
 
 substituting in value for a tt a,, we have for the Central 
 Angle for Compound Curves expressed in arc at radius 
 = 1. 
 
 * The values of / &,/ a., a.// and < are expressed in arc at 
 radius = 1. The degrees in an arc of a circle which is equal to the 
 radius in length = 57.295. The arc of 1 degree at radius = 1 is 
 .01745 -f . Hence the expression for any number of degrees in a 
 given circular arc is: A = ^745 + * 
 
6 TRANSITION CURVE 
 
 in which = ft/ ft (9) 
 
 is the angle subtended by the arc L of the spiral as well as 
 the sum of the circular arcs A 4 B, + A tl B n 
 
 the + sign being used when the directions of curvature are 
 
 similar; 
 the sign when the directions of curvature are reversed. 
 
 EXAMPLE No. 1. GivenL = 150; r n = 818.8; r = 2865; 
 
 to find arc (a,, - a,) = = ft, + ft = (-L + i) , 
 
 ~ w/ '// 
 
 arc ( - n/ ) . . if (gjj-g + ;^- 5 ) - .11775 
 
 = arc of 6 45'. 
 
 If in equation (8) we make r t = oo , and a x = 0, then 
 - 
 
 the angle, expressed in arc, subtended by spiral OB /X . 
 
 4. DIFFERENCE IN LENGTH OF THE SPIRAL AND THE 
 SUM OF THE CIRCULAR ARCS SUBTENDING THE SAME ANGLE. 
 
 The rectangular co-ordinates of the center of the circle with 
 radius r,, as >/, referred to the origin O are: OH/ = l t and #/>/ = &,. 
 and of Z>// O'H /f = I,,, H,,D, t = k,,; whence 
 
 = tan (a, + J8.) ; (11) 
 
 I,, x t , r lt sin 
 /?// = r t , + w//, 
 whence 
 
 < 
 
 or if we have the value of / 
 
THE SPIRAL 7 
 
 from either of these equations 
 
 0,, = a,, - (a, + 8,0 ), (14) 
 
 with // reduced to arc, then 
 
 r/A/ - AnBn 
 and 
 
 / = // - (<*/ + /3,,), r,S, = A,B,\ (15) 
 
 denoting the difference between L and AnB fl + A,B, by "d" we 
 have 
 
 d = J/ - U//5// + ^4/B/) = L - (r^jS,, + r^) 
 
 Placing 
 
 (r//^// + r,/3,) = Lc t 
 
 we have 
 
 "d" = L-Lc. (16) 
 
 5. TRANSFORMATION TO RECTANGULAR CO-ORDINATES. 
 
 By the Calculus dy = dL sin ^ 
 
 rfx == dL cos ^ ; 
 
 . 
 
 in which <j> = any angle. By trigonometry, 
 
 * This and the succeeding formula are for expressing the trigo- 
 nometrical function of an arc in terms of the arc itself (see 
 Chauvenet's Trigonometry, Chapter XIII, 1867). 
 
 m and n positive whole numbers, m = even, n odd. 
 
8 TRANSITION CURVE 
 
 in which m may have any value from zero to infinity, n may have 
 any value from one to infinity, and < any value from zero to infinity. 
 Substituting in the equations for sin and cos <, 
 
 we have 
 
 - = sin ^ = sin (,/ + ,) = sin B/T 
 
 
 (I7W 
 
 EXAMPLE No. 2. Given L = 150; r,, = 818.8; r, = 2865; to 
 find <. 
 
 sin ^ = sin O + ^) - i- + -- 
 
 L 24 Vr^ r// J 
 
 150/1 1 \fi (150) V 1 1 \ 
 
 sin - sin & + 0,) - 3^,4- 28^) [1 - -24-(8l87 8 + 2865) 
 
 sin ^ = sin O// + &,) = .11775 X .9977 = .11748. 
 
 If we make r/ = oc, = 0; 
 
 we have, with origin at O, $ = a,, ; and L = L// (17ft) becomes 
 
 l*(l - *L + _^1_ _ etc .) _ sin Bl , DllNll (I7C) 
 
THE SPIRAL 
 
 _ __ _ 
 
 2P 48P3 3840P 5 
 
 i \ 
 
 n ) : 
 A'PV 
 
 -etc. ... ); (18) 
 
 integrating, 
 
 L 
 or, since P= J 1, with its origin at B- t the ordinate of any 
 
 TI, Ti 
 
 point of the spiral, as 
 
 EXAMPLE 3. Given r, = 2865; r/, = 818.8; L = 150, to find 
 V by: 
 
 i 
 
 -. -.00035) [1 - (.00122 - .00035)2 + etc.] 
 
 D OO 
 
 y = 22500 X .00087 
 = 3.26225. 
 
 Or in terms of <b, since L = 2r^ f and = ( ) 
 
 *" \ r,/ r// 
 
 -fj-Hetc. I. (20a) 
 
10 TRANSITION CURVE 
 
 If we make r, = oo, = 0, L = L,, the ordinate of any point 
 of the spiral, as 
 
 B//G// = y// = |p (l ^7 + etc.) with origin at O (21) 
 
 and OG// as abscissa. 
 Similarly, 
 
 etc. (22a) 
 
 integrating, 
 
 x -L [ 1- 4Qp2 + 3456p4 - 599Q40P 6 + 6tC 
 
 (m + 1) MP* 
 
 (23) 
 
 With origin at B/ then the abscissa of any point of the spiral, as 
 
 KB,,.-*- L [l- ^- (- - ^-) 2 + ^ (i- i)' - etc. ( 24 ) 
 L 40 \r/, r// 3456 \r// r// 
 
 or in terms of <j>, since L = 2r<, (25) 
 
 x is laid off on the arc of the circle A/B/ or AnBn (with radii r, or 
 r//) as axis of X with J5/ or B t , as origin, x being the abscissa of y 
 which is laid off normal, or radial, to the arc A,B, or A //#//. 
 
THE SPIRAL 11 
 
 EXAMPLE 4. 
 
 x = L [l - ^ (^ - ^) 2 etc. (other terms) ] 
 
 / 22500 X .0000007569 + . \ 
 U \ 40 / ' 
 
 x = 150 (.9996 ) = 149.94. 
 With origin at O, any abscissa, as 
 
 also when <j> = <*// 
 
 ^=2^(1-^ + ^ -etc.) (26a) 
 
 6. RECTANGULAR CO-ORDINATES AT THE OFFSET 
 DISTANCE. 
 
 i. To find the abscissa and length of the curve. 
 
 Let j/// be the ordinate, xm t/ the abscissa, and L*/, the length 
 of the curve corresponding to xm^ym/r, then a very nearly correct 
 value of ym-n for central angles up to 50 degrees may be obtained 
 as follows: 
 
 OH,, = xm,i = xu r lt sin ^>: whence, with origin at O, (27) 
 
 by Eq. * page 7 and (27). 
 
 OH,, = xm tl = r//4>(l + ... .J or (27a) 
 
 in terms of r,/ and L// 
 
 OH,, = x* - (l - L//2 2 + L//4 ) (28) 
 
 and when the origin is at Bi 
 
 in which L = I/,, L, = BtBu. 
 
12 
 
 TRANSITION CURVE 
 
 A close approximation to Lf or Lm is La which is determined by 
 making 
 
 La- 
 
 L + Wxf 
 12 
 
 whence 
 
 with origin at B/, then 
 
 * B t U = Lf = 
 
 40 Va rj 
 
 (29) 
 
 . (30) 
 
 We may write Lm for Lf and xm for xf, rm for ra, = 0. in 
 
 TI 
 equation (30) when the origin is at O; then 
 
 1 - 
 
 La-' 
 
 (30a) 
 
 2. To find the ordinate of a transition curve measured on the 
 offset m// (or f): 
 
 Fig. 2a. 
 
 * After solving (Eq. 30) for Lf, compute rf, give La and ra, in 
 the denominator, the value just found for Z//and rf, and solve again 
 for L/till the Eq.x/=L/ Fl - ^~ (j^ - ~}~ etc., is true, ^/having 
 a fixed value from Eqs. (27-28a). 
 
THE SPIRAL 13 
 
 The method given in Equation (47) for finding the ordinate y / 
 is approximate and sufficiently close within the limits of the 
 tables. For a large central angle or great length of spiral a closer 
 approximation is necessary. 
 
 Writing ym,,, for y, when origin is at O, we have by (19) : 
 
 - )'<> 
 
 with origin at B, then B,U = Lf = P (- -- ) , and 
 
 
 This method of obtaining // and j8/ may be used instead of 
 Eq. 6 or 8. 
 
 7. OFFSET DISTANCE. 
 In general, by Fig. (2), the offset distance. 
 
 / = y - R ver <j> = y - R (1 - cos <j>\ (33) 
 
 in which R depends on for its value, substituting for cos $ 
 its value (remembering that R = yj and reducing, since 
 
 R (1 - cos # = R [l - (l - j~ + 3^ - etc.)]' 
 / = y -flver^ = L [(^ - ^ +etc.)- 
 
 (i- 31^3+ etc.)]; (34) 
 
14 TRANSITION CURVE 
 
 - : (35) 
 
 since p = f -- V we have for (36) 
 
 the offset with origin at B,\ 
 
 EXAMPLE 5. Given r,, = 818.8; r, = 2865; L = 150; to find 
 the offset /. First, when the circular curves r// and r/ turn in the 
 same direction, by (125): 
 
 - 
 24 " r L 112 W ~ i 
 
 X .00087 [l - 
 
 (150) 2 (.00087)2 
 
 24 112 
 
 With origin at O, when <f> becomes //, L becomes L// and f becomes 
 m//, then 
 
 * N " H " - m - - T, ( l ~ ir? + etc - ) (38) 
 
 Second, when the circular curves turn in opposite directions or are 
 reversed. 
 
 If L,, = L/, then 
 
 For the distance between the centtrs of the two curves, turning 
 in the same direction, we have: 
 
 r/- (r,,HKf)-W>,/-r,-J* w +^(^ - ^)d -etc.)]. (41) 
 
 * For the constant P, when OB, = B,B,, or L, = L, then H,U 
 -4/f/, m/ = /, and 5/G/ =- J5//5///, or y, = 2/ = 5/ 4 (nearly). 
 t In this case there will be two values of P except when r/, = r/. 
 
THE SPIRAL 15 
 
 EXAMPLE 6. Given L = 150; r/ = 2865; r/, = 818.8; to find 
 the distance between the centers of curves with radii r// and r,, 
 First, if curves turn in the same direction (Fig. 2): 
 
 >>, = r, - (r/, + /) = 2865 - siS.S + + .00087 ; 
 
 DD, = 2865 - [818.8 + .815] = 2865 - 819.615 = 2045.385. 
 
 Second, if curves are reversed (or turn in a contrary direction) 
 (Fig. 9), then: 
 
 X sec K,,D,D,, t = [r/ + (r// -f m tl + m/)] sec K lt Di t D,,,; 
 
 tan K, t D,D tl , = "+**'* -- (416) 
 
 r/ + TH + m// + m/ 
 
 PRACTICAL FORMULAE. 
 
 8. If, in the foregoing values of y, x, -pf, and / we 
 
 dLi 
 
 omit terms in the bracket after the first or second, we 
 have for central angles of 20, or less, the following 
 practical formulae for uniting circular curves of differ- 
 ent radii by means of a spiral arc : 
 
 See Eq. (32) 
 
 See Eq. (28a) 
 
 * When 0,O,, = 0. 
 
16 TRANSITION CURVE 
 
 B.B,, = L = x ( i + ~p,J ; nearly (44) 
 
 sinB/WB,, = sin0 =sin (p it + ft) = (44!)) 
 
 L/ i i\f L 2 / i i V 
 
 -[ +- i -- + - } + etc. 
 
 2\T,, rJL 24Vr /7 rj 
 
 arcB y WB /x (atradius = i) = -( ^-+ ^-) = arc 0: (440) 
 
 2 \ r // r / / 
 
 arc X 57-3 - 
 
 A ' A "= f =gfe-y, (45) 
 
 from which latter equation we find: 
 
 identical with Froude's curve of adjustment, as indicated 
 by Professor Rankine (" Civil Engineering," Edition 
 1863). 
 
 | /= (- --- ), and if in the value for y we write JL 
 48 \TM T t j 
 
 for L and cube it, then 
 
 WL_u_u_ . = JL=vs L _i\ 
 
 6P 8_ 48P 4SR 48\r /; rj' v ; 
 6P 
 
 Hence the ordinate y = J f is at the middle point of 
 the transition curve. 
 
THE SPIRAL 17 
 
 For uniting a tangent with a circular curve by the 
 use of the spiral, 
 
 we have = o, in (42-47); 
 
 whence: by Fig. 2, for any distance on the spiral, as L,,, 
 with radius r lly the corresponding 
 
 Ordinate G y/ B y/ = y// = ^ (48) 
 
 r L 2 i 
 
 and the abscissa OG,, = x y/ = L y/ i - -~- 2 + (49) 
 
 L 4 or // J 
 
 The sine of central angle of the transition curve 
 equals 
 
 L / " L 2 \ 
 
 sinN^D^B^ = sin a// = -^ ( 1 - ^-^-J (50) 
 2r // \ 2 4r/y / 
 
 arc N^D^B^ = a// = -^- , arc a /7 X 57-3 = // 
 
 the offset N /y H /x = m y/ = - (51) 
 
 241 tl 
 
 \ the length of spiral = ^ = Vbm^r^ (52) 
 
 OH,, = x m// = x /x - r y/ sin a y/ , or (a) 
 
 (sab) 
 
 O y d or 0,1 = t= x m + (r y/ + m) tan \ I (see Fig. 5) (6) 
 
 If D// and Z>/ denote the " degrees " of the curves (determined 
 by 100 feet of their length) corresponding to the radii, r// and r/; 
 then, since L// and Z>// and L f and Z>/ are inverse functions of r// 
 and r/, we have 
 
 r r r T D " ~ D ' A 1 l L " ( D " ~ D '\ 
 L ,,-L,= L = L tt D// and -- -= (j^-) 
 
18 TRANSITION CURVE 
 
 TO LAY OUT THE TRANSITION CURVE BY DEFLECTIONS. 
 
 9. If at the point B t (see Fig. 2) we imagine r, to 
 increase until it becomes infinitely great, the curvature 
 of B l A i = and the arc B t A. t will be a straight line still 
 preserving its tangency to the transition curve. The 
 curvature at B tl will diminish to the same extent, i.e., 
 the difference between curvature at B / and B tl will be the 
 
 L 2 
 same as when $, = 0, and $ u = op The ordinates x 
 
 y can be computed and laid off from the new tangent as 
 axis of abscissa with B t as origin, the same as if from *. 
 If we now conceive this new axis of x to be curved 
 to a radius r, the curvature of the transition curve at any 
 point will be increased by the same amount and the 
 ordinates may, without serious error, be laid off normal to 
 the arc B,A, and establish points of the transition curve. 
 The same reasoning will apply if > /y =00 and B tl A lt 
 becomes tangent and values of x and y be laid off from 
 it with B tl as origin, except that the resulting transition 
 curve would be convex to B tl A lt . The ordinates would, 
 however, be equal to those of the corresponding distance 
 from B r If r n now resume its original length the cur- 
 vature of the transition curve at any point will equal 
 that of the circular curve with radius r f/ minus the cur- 
 vature it had in a contrary direction when r u was 
 infinitely great and B tl A. lt a straight line. In determin- 
 
 p 
 ing /3 y/ and r = -=- , data may be taken from the tables. 
 
 Li 
 
 Equations for x and y are equally true whether the 
 origin be taken as 0, B n or B lt . 
 
 x will be measured on A,B /t and y normal to A t B f . 
 To lay the transition curve out by the expressions 
 
 * In Fig. 2, O, should be marked B. S. or E. S. and B, or Bi, 
 marked B. C. C. or E. C. C. in staking out a curve on the ground, 
 according to the direction in which the line runs. 
 
THE SPIRAL 
 
 19 
 
 for x and y, their values may be laid out simultaneously 
 with corresponding equal chord measurements along the 
 transition curve. 
 
 The principle enunciated in the paragraph preceding, enables 
 us to prepare a table of deflection angles according to the following 
 method : 
 
 <K 
 
 Referring to Fig. 3. If the deflection angles from AX to any 
 point be denoted by SS,S/,, it will be found by computation that 
 any angle as 
 
 DAX = 5, = - = gp- (nearly), * (53) 
 
 in which r,, = DK and L AD. 
 
 ADC, = DCX - DAX = a,-^=^--g^-= =. (54) 
 
 DD, y,, a, 
 
 * Since -jyr- = = tan (nearly). 
 
 AL/i Xff O 
 
20 TRANSITION CURVE 
 
 the angle which CT, a tangent common to the spiral and circular 
 curve with radius r//, makes with the chord AD produced. To 
 establish the points E, F and G by deflection at D, from tangent 
 DT we have, from the paragraph already referred to, 
 
 EDT = 5 + A, (55) 
 
 and 8 = the deflection from AX to B, and A, = the deflection 
 from DT to the point E, for the circular curve with radius r,,. In 
 the same way with D as origin. 
 
 GDT = 8,, + A //; = a. (56) 
 
 The angle 
 
 DGG,, = 2 ,/ + A,, = (a - a,) - (S,, + A,,), an d (57) 
 if we add (5,, -|- A,,) to both members of the equation, we have: 
 
 GOT = a,, - a, = ^ = 3 fi,, + 2 A //t (58) 
 
 in which A,, = LA, A = the deflection for a unit length of Z>(r on 
 the circular curve with radius r//, and L the units from D of any 
 point laid off on DG. 
 
 FOR CONVENIENCE IN FIELD WORK. 
 
 Eq. (53) may be reduced to degrees and put in the 
 following form : 
 
 5 = 57. 3, whence by Eq. (6) 
 or 
 
 * . o> = DAL ^- 57.3 in which (59) 
 
 or 
 
 the instrument point is the origin of L. 
 
 A = deflection angle per foot from tangent for 1 
 circular curve. 
 
 D == degree, or rate of curvature, at position of in- 
 strument. 
 
 D, = the degree, or rate of curvature, at the point to be 
 located. 
 
 w = deflection from tangent at any point of spiral to 
 locate any other point of spiral (using the + sign for 
 running toward G, sign towards A. ) 
 
 * More nearly 57.2956 
 
THE SPIKAL 21 
 
 This equation can be still further simplified in appli- 
 cation by the following reduction in 5 = 57.3, let r 
 
 = radius due to L> measured from the position of the 
 instrument to the point of spiral to be located. 
 
 N = the number of chords of equal length (each sub- 
 tending one degree change in rate of curvature) in the 
 length of L of the spiral. Then if r = the radius of a 
 one-degree curve, 
 
 r = ^ and 5 = ^J^T = LN * 0.00166 (60) 
 1\ O/oU 
 
 or 8 f = LN 000.1'. 
 
 If we wish the change in rate of curvature to be any 
 other than one degree to the chord, and denote this change 
 by C, C = the change in degree of curvature, due to 
 one-chord length, and may be either a fraction or whole 
 number; then 
 
 8' = LNC X 000.1'; (60a) 
 
 or since C = ^ , 8' = LD, x 000.1' ;* (606) 
 
 and if one of the chords be fractional, and we indicate the 
 fractional part by -= in which F is the number of parts 
 
 into which a chord is divided, and n the number of such 
 parts taken, 
 
 8' = L(N + j\ C X 000.1', hence (60c) 
 
 r / n\ 1 
 
 w' * L I DA Ylf -h =|C X ooo.i' (61) 
 
 is a general formula for the deflection from a tangent, 
 at any point of the transition curve, to locate any 
 other point of the transition curve. 
 
 * See Appendix. 
 
22 TRANSITION CURVE 
 
 J) T) 
 
 In which case C = 
 
 to be computed and substituted in (61). 
 
 If n = o, C = '" 
 
 and ' = L [DA+ (D, - D) ooo.i'] 
 
 With instrument at A, Z)A = 0. 
 
 To place the line of sight on tangent at any point of the spiral. 
 After backsight on last instrument point, formula (61), for deflec- 
 tion to tangent becomes 
 
 a>' = L [z>A+ 2 (N + j^ C000.1'l 
 
 when running towards G (Fig. 3), D being the rate of curvature at 
 the last instrument point, and L the length of spiral to backsight. 
 
 See Eq. (57) of = L \D - 2 \N + j\ C 000.1'J running towards 
 A. 
 
 In applying Eq. (61), the more frequent the change points, the 
 
 more nearly will the resulting curve agree with the theoretical 
 
 spiral ; their distance apart to be not more than 150 to 300 ft. 
 
 nor '* change points " include a central angle of more than 10 to 15. 
 
 A nearer value for the second term in the bracket of (61) is: 
 
 (N + |;) C 00'.0998. 
 
 EXAMPLE 7. Given, D = 2, A = .3', L = 125, N = 2, ~ = J, 
 
 r 
 
 C = 1, Chords = 50 ft., to find the deflection from a tangent at C 
 to locate a point E + 25 (see Fig. 4). By formulae (61): 
 
 at' = L [jDA + (.V + fy X 1 X 000.1'1 
 
 substituting values given above, 
 
 <u' = 125 [2 X .3' + (2 + i) X 1 X 00.1'], or 
 a' = 125 [.6' + .25'] = 125 X .85' = 1 46i'. 
 
 (See table, Spiral 1.) 
 
 The foregoing values for the deflection angles are closely approxi- 
 mate, though the method indicated in connection with Fig. 4 in 
 determining any deflection angle, as fin, while less elegant is more 
 nearly exact. 
 
 By Eqs. (21) and (26) tan fin = ^, (62) 
 
 from which we determine fin. 
 
THE SPIRAL 
 
 23 
 
 Fig. 4. 
 
 " n " having any value, whole number or fractional. 
 
 With A as origin we obtain, by (62) the deflection angles to the 
 points B, C, D, E, etc., to any change point, as C, where the degree, 
 or rate of curvature, is D. From C, with backsight on A, deflect 
 a. S n (in which n = 2, and a == the central angle of the spiral from 
 A to C) to get on tangent at C, then to locate any point, as E (not 
 shown on Fig. 4), from a tangent at C, deflect 
 
 5n (63) 
 
 n being the deflection angle due to n chords from C, corresponding 
 to the same number of chords from A. 
 
 To get on tangent at E: 
 
 With backsight on C, deflect I/Z)A + (a - &), in which n = 2, 
 L = CE, and a corresponds to the central angle of the spiral from A 
 for a length L. 
 
 The process is the same as with Eq. (61), though the second factor 
 of the second member is different. 
 
 * It will be seen that the differences between this method and 
 that of Eq. (61), for a curve with L = 400 ft. and 8 = 16 is 000'03". 
 
 The difference increases with the central angle a. The above 
 method, with change points 200 to 300 ft. apart, is quite accurate 
 arid the best for preparing a set of tables though not so easily ap- 
 plied in field computations as (60) or (61). By assuming "change 
 points " 100 to 250 ft. apart, the deflection tables of this book may be 
 extended indefinitely. 
 
24 TRANSITION CURVE 
 
 The several angles can be computed and tabulated, to any 
 number which is likely to be needed, to conform to any system of 
 "change points" determined upon after #o1/o, etc., have been com- 
 puted for the particular transition curve where value of P has been 
 fixed in conformity with the character of the alignment. 
 
 EXAMPLE 8. Given L = 200; xc = 199.91; yc = 4.65; to locate 
 C from A. 
 
 tan S c = 6 = .023260 or fi fl = 1 19.95'. 
 
 To get on tangent at C, at which point the total curvature is 4 00': 
 4 00' - 1 19.95' = 2 40.05'. Hence with the instrument at C and 
 backsight on A, deflect 2 40.05' to get on tangent at C, where the 
 rate of curvature is D 4 00'. 
 
 From tangent at Cto locate some point E, which is, in this case, 
 200 ft. from C, then 
 
 01 = LDA + 8 n = 200 X 4 X .3' + 1 19.95' = 5 19.95' 
 and to get on tangent at E, with backsight on C, 
 
 o) = LDA + (a - 5 C ) = 200 X 4 X .3' -f 2 40.05' = 6 40.05'. 
 
 ORDINATES. 
 
 10. To determine the ordinates o, 01, etc. From any chord 
 as Z , let aB = o, Bib s, AB^ = XQ, BiB = y , bABi = y. 
 Then from Fig. 4, = tan y, s = X Q tan y, s y = XQ tan y y Qt 
 
 XQ 
 
 = cos y, o = (s yo) cos y, or since s = XQ tan y, 
 
 o = (x tan y - y cos y = x sin y - y cos y : (64) 
 
 For the distances Aa, etc., Aa = ABi, sec y aB tan y = XQ 
 sec y o tan y. In the same way we may obtain o t and Aa\. To 
 determine the point B, C, etc., by measurement alone: First com- 
 pute and lay off the distances Aa, Aai, etc., then lay off AB and aB 
 simultaneously; next BC and a\C, etc.; when y is small the dis- 
 tances bB and biC may be used, distances A b and Abi being com- 
 puted and laid off first. 
 
 As a check it will be an advantage to compute the length of the 
 long chord as well as the angle it makes with the axis of x, thus: 
 
 ; (65) 
 
 in which (n) equals the number of increments or stations between 
 
THE SPIRAL 25 
 
 A and D. Let the length of any chord be c 2 , c 4 , en, in which 2, 4, n 
 indicate the 2d, 4th and nth increment. 
 
 c 2 = (x 2 EI) sec y 2 ; c 4 = (.r 4 z 3 ) sec Y 4 ; 
 
 c = (*n - x n-i> sec y n - C66) 
 
 EXAMPLE 9. Given x 2 = 209.65; x v = 120; y 2 = 4 37', to find 
 the length of chord, c 2 = (x 2 - a*) sec 7 2 = 89.65 X 1.0032 = 89.93. 
 For the length of long chord (from origin), x 2 = 209.65; 7=2 27'; 
 x 2 sec y = 209.65 X 1.009 = 209.84. 
 
 It is to be observed that the superelevation of the 
 outer rail, in the use of the transition curve, may be 
 made greater or less than that which has been assumed 
 in computing the tables; the only effect it will have is to 
 diminish or increase the assumed value of "i" which is 
 equivalent to increasing or diminishing the velocity, 
 since i and v are inverse functions of each other in the 
 constant P, i.e., it makes the rate of rise of the outer 
 rail to effect superelevation a little greater or less. It 
 is, however, best to introduce the average velocity of the 
 express or fast passenger trains in constructing the tables. 
 Where the location is so constrained that the EC's and 
 BC's of the circular curves are quite close together, it 
 may be necessary to give "i" a smaller value than would 
 be otherwise desirable. A value of 300 or 400 is suffi- 
 cient for adjustment, and good results may be obtained 
 with a value of 200 when the radius is not greater than 
 573 feet, since v usually is made to decrease with r. 
 
 The beginning and end of the transition curve should 
 be marked by permanent points. 
 
26 
 
 TRANSITION CURVE 
 
 Degree 
 of 
 Curve 
 
 Radius 
 
 Reciprocal 
 
 Degree 
 of 
 Curve 
 
 Radius 
 
 Reciprocal 
 
 D 
 
 r 
 
 1 
 r 
 
 D 
 
 r 
 
 1 
 r 
 
 030' 
 
 11460. 
 
 .00008726 
 
 800 
 
 716.3 
 
 .00139626 
 
 100' 
 
 5730. 
 
 .00017453 
 
 900 
 
 636.7 
 
 .00157079 
 
 130' 
 
 3820. 
 
 .00026179 
 
 1000 
 
 573. 
 
 .00174533 
 
 200 / 
 
 2865. 
 
 .00034906 
 
 1100 
 
 520.9 
 
 .00191986 
 
 230 / 
 
 2292. 
 
 .00043633 
 
 1200 
 
 477.5 
 
 .00209439 
 
 300' 
 
 1910. 
 
 .00052359 
 
 1300 
 
 440.8 
 
 .00226893 
 
 330' 
 
 1637.1 
 
 .00061086 
 
 1400 
 
 409.3 
 
 .00244346 
 
 400' 
 
 1432.5 
 
 .00069808 
 
 1500 
 
 382. 
 
 .00261799 
 
 430' 
 
 1273.3 
 
 .00078540 
 
 1600 
 
 358.1 
 
 .00279253 
 
 500' 
 
 1146. 
 
 .00087267 
 
 1700 
 
 337. 
 
 .00296706 
 
 530' 
 
 1041.8 
 
 .00095993 
 
 1800 
 
 318.3 
 
 .00314159 
 
 600' 
 
 955. 
 
 .00104712 
 
 1900 
 
 301.6 
 
 .00331613 
 
 700' 
 
 818.8 
 
 .00122173 
 
 2000 
 
 286.5 
 
 .00349066 
 
THE SPIBAL 
 
 27 
 
 PROBLEM I. 
 TO FIND THE SEMI-TANGENTS. 
 
 ii. Given a circular curve whose radius = r tl \ the 
 intersection angle = /; the semi-tangent = T, to unite 
 it with the tangents by means of transition curves whose 
 lengths are L t and L and offsets are m t and m respectively. 
 
 flr. 5. 
 
 CASE 1. 
 When m y > m (by Fig. 5), if I < 90, 
 
 = ^m/ -}- T 7 w / cot 7 + m cose 7. 
 
28 TRANSITION CURVE 
 
 If 7 > 90, 
 
 O,d = I H I + N,c + ab + bd = 
 
 x m/ + T - m, (- cot 7) + TO cose 7; (2) 
 
 0,d = x m + T + m/ cot 7 + m cose 7; (3) 
 
 or in general calling, 
 
 O^d, or O,d = t, ; t, = x m/ +T =p m, cot I+m cose I, (4) 
 
 the + sign being used when 7 > 90 and the sign when 
 7 < 90. 
 
 If 7 < 90, 0,d, = O I H I + AT,c, + c,&, - d,gr, = 
 
 # m -f T 7 + m, cose 7 m cot 7; (5) 
 
 If 7 > 90, 0,d = (),#, + N,c + cb + dg = 
 
 ^ m + T 7 + m y cose 7 + m cot 7. (6) 
 
 In general calling, 
 
 O.d, or O,d = t; t = X m -f T + m, cose I m cot I, (7) 
 using + when 7 > 90 and - when 7 < 90. 
 
 CASE 2. 
 If m = o in equation 4, we have 
 
 t, = T + m, cose I, t = X M + T + m / cot I. (8) 
 
 CASE 3. 
 If m = m in equation 4, 
 
 t t = z m/ + T 7 4- TO, (cose 7 cot 7), = 
 
 = z w + !T + m (cose 7 cot 7). (9) 
 
 If 7 < 90, the last term of t, is m, (cose 7 - cot 7) and 
 
 by trigonometry (see Chauvenet's, p. 35). 
 
 , T I cos 7 1 cos 7 , _ 
 
 cose 7 cot 7 = -. r : - r = : = == tan J 7 
 
 sin 1 sin jf sm L 
 
 .'. t t = x m/ + T -f m, tan i 7 
 
 or ^ = x m/ + (r,, + m t ) tan } 7 ; (10) 
 
 similarly, if 7 > 90, 
 
 T , T 1 + COS 7 
 
 cose 7 + cot 7 = : = ; 
 
 sin / 
 
THE SPIRAL 29 
 
 but if / > 90, 
 
 cos / = cos / and cot / = cot 7 
 
 T , . TN 1 COS 7 
 
 cose 7 + ( cot 7) = - : 7 , 
 sin 7 
 
 which we have seen = tan J7. Hence when m y = m, 
 
 t y =t =x m + T + m tan JI = x m + (r,, +m) tan JI (n)* 
 (See App. I) 
 
 is a general equation whether 7 be greater or less than 
 90. If in (11) we make x m and m each = o, then 
 = T = r /x tan J7 = the semi-tangent for a simple circu- 
 lar curve. 
 
 EXAMPLE 10. Given L = 210; r = 818.8; m/ = 2.23; / = 40; 
 O/#/ = 104.93; to find t (Fig. 5): 
 * = * m/ + (*" + w/) tan 7 = 104.93 + 821.03 X .36397 =403.75 
 
 With the same data 
 
 12. TO FIND THE EXTERNAL SECANT. 
 
 CASE 1. 
 When m, > m, 
 
 H.d, T + m, cose 7 + m cot 7 
 
 ^ry = - = tan ri.Dd. ; 
 
 DH, r u + m 
 
 eA = (r y/ + m)secH / Dd / - r y/ . (12) 
 
 * If we wish to unite two grades by a vertical curve, then, if 
 
 in Eq. 44, we make = o 
 
 TI 
 and sin < = sin 7 
 
 L = 2r// sin 7 
 * - L (1 -) 
 
 1,2 
 
 y= 6^, 
 and Eq. 11, Prob. 1 t = x m + (r// + m) tan 7 
 
 in which 7 = J the angle at which the grades intersect, r// should 
 have a value of from 5730 to 11460, and represents the minimum 
 radius of curvature at the middle point of the vertical curve, x and y 
 ordinates from the origin or terminus of the vertical curve to its 
 middle point or to any other point. 
 
30 TRANSITION CURVE 
 
 CASE 2. 
 If m = O, 
 
 N,b, T + m, cose / , r ~, 
 
 ~ = - ^r = tan A^ ; 
 
 e^b, = r,, sec N,Db, - r y/ . (13) 
 
 CASE 3. 
 m, = m, e.d, = (r,, + m) sec JI - r,, (14) 
 
 If, in Eq. 14, we make m = o, we have for the external 
 secant of a simple circular curve : 
 
 ed = r it (sec \l 1) = ec. 
 
 If we wish the transition curves B Ui O ul and Bf) u to 
 become a tangent to each other at some point A with a 
 minimum radius of curvature = r u = the radius of the 
 circular portion B ltl B n with B, 4 ,B, elided > then B UI is 
 coincident with B 4 . 
 
 B,DN, - = / - A-^ZXY,, = a/ = L- , (15) 
 
 a, + a = / (16) 
 
 /// B /// =L / =2r // (I- ) 
 
 O / B / = L = 2r /y (I a,) = v 24T,,m (18) 
 
 N lf H tt -m t =^ (19) 
 
 L 2 
 
 AV 
 
 94., 
 rifi*t / /y 
 
 If 77 ^/ = m then L / = L = 2r /y J7, the point A is at 
 the middle point of the curve and L, J the total length 
 of the curve from .., through C. C. to #. S. 
 
 Since a x and a are expressed in arc in the above for- 
 mulas, while the intersection angle is measured in degrees 
 and minutes, the latter will have to be reduced to arc at 
 
THE SPIRAL 
 
 31 
 
 radius = 1 before entering the formulas, and after calcu- 
 lations are made the result expressed again in degrees 
 and minutes to apply formulas for semi-tangents, ex- 
 ternal secants, etc. 
 
 PROBLEM II. 
 TO FIND THE LOCATION OF THE OFFSET "/". 
 
 13. Given two curves with radii r, and r in a distance 
 D,D,, = d joining the points D, and D lt also the angles 
 BD,D,, = p and CD,,D, = 0, to find the points A, and 
 A (l at which a line drawn through the centers r t and r tl 
 at B and C will cut the curves A 4 D, and A,,D,,. 
 
 Fig. 6. Fig. 7. 
 
 From Fig. 6 we have 
 
 PC = h = d sin p - r,, sin (180 - (p + 9)), 
 BF = r, - [d cos p + r,, cos (180 - (p + 0))] 
 d sin p r,, sin (180 (p + 0)) 
 
 = tan SF 
 
 BF 
 
 and since 
 
 '' r, - [d cos p + r tl cos (180- (p + 
 
 (1) 
 
 (2) 
 
32 TRANSITION CURVE 
 
 FC h dBmp-r,,sm(lSO-(p+0)) 
 
 BC-r,-(r,, + f)- r/ _ (r// + /) 
 
 D,,CD 4 = S = 180 - ( P + 0), * - S = u = Z^CA,, ; (4) 
 
 whence r y/ w = A y/ D /x , 
 
 in which w is the arc of a circle with radius = 1. (5) 
 
 If S > *, then the point A yy lies between D y/ and D 3 
 and the distance A /y Z) 3 is measured from D 3 towards 
 D it to A /y . 
 
 If SE' > S, the point A yy lies beyond J> /y and the dis- 
 tance A yy D yy is measured from Z) yy to A /y whence A yy is 
 established. On a perpendicular to a tangent at A y/ lay 
 off / y and establish A r 
 
 When / is small, the direction of the radial line can 
 be estimated near enough. The method of fixing B y 
 and J5 /y , in Fig. 2, has already been indicated. If r y = oc, 
 Z> y A y is tangent. Fig. 7 applies. 
 
 EXAMPLE 11. r// = 1000; r /y = 600; d = 300; P = 70; / - 
 96; / = 50/: 
 
 First, to find h = d sin P - r// sin (180 - (p 4- /)) = 300 X 
 .93969 - 60 X .24192 = 136.75 = FC. 
 
 _ h 136.75 
 
 Second, to findsm * - r 
 
 K,00 - (600 + 50) 
 
 Third, to find 180 - [70 + 96] = 14 = 2; * - 2 = w = 23 
 o 
 
 - 14 = 9 <a reduced to arc = z=~SS = - 1571 - 
 o < .^y 
 
 rw = 600 X .1571 = 94.26 = ^1//Z>//, and since * > 2, ^//Z),/ is 
 measured from >// towards Z> 3 . 
 
 If 2 = D4/CD 3 , then 2 > * and yl//Z>3 would be measured from 
 Z> 3 towards D// to establish point An\ A/ is on a perpendicular to a 
 tangent to the curve with radius r// at A//. 
 
THE SPIRAL 
 
 33 
 
 PROBLEM III. 
 COMPOUND CURVES. 
 
 14. Given two circular curves with radii r, and r 3 , respec- 
 tively, whose centers are apart a distance AC = b = r x (r 3 + ///), 
 and which are separated from each other a distance Z>/D// = ///. 
 It is desired to introduce between them a third curve with radius r//, 
 less than r/ and greater than r 3 and to join the curve with radius r// 
 with those having radii r, and r 3 by means of transition curves: 
 
 Fig. 8. 
 
 By the figure AD, = A A, = AB, = r, 
 = b + r 3 + ///; whence 
 
 AC - b - r, - (r 8 + /); (1) 
 
 AS = c = r , - (r,, + f,); (2) 
 
 BC = a = r,/ - (r s -f / 3 ); (3) 
 a, 6 and c form the sides of a triangle A5C in which 
 
 _ L/2 / i i^\ _ W /j^ i \ 
 f/ ~ 2T U, " rj ; f3 ~ 24 Vr 3 r^j : (4) 
 
34 TRANSITION CURVE 
 
 Any angle A may be found by the formulae, 
 
 VerA = 2 (s ~ b) ^ (s ~ C) in which a = Ko+b + c); (6) 
 
 Sin = sin A; (7) 
 
 Sin C = I sin A. (8) 
 
 Reducing each of the above angles to an arc as indicated in another 
 part of this book, we have 
 
 A,D, = r,A\ AJ),, = r 3 C; A,,A* = r,, (180 - ); (9) 
 
 _i4_ iL_JL). 
 
 r/// 
 
 the arc 
 
 _ 
 
 2r 3 2r 3 Vr 3 
 
 L 3 P /i i\ 
 
 P3 = -- ~~ = - I - I 
 2t// 2f// \T 3 r/// 
 
 in the same way, 
 
 / 
 
 2r/ 2r/\r// r// 
 A/B/ = r/^/; A// B// = r//^// ; A 3 B 3 = r/,/3 3 ; 
 
 A 4 B 4 = r 3 ^ 4 ; (14) 
 
 B^Bs = r,, t ; t i = (180 - B) - (// + j3 3 ); (15) 
 
 B/D, = r/A + r/j8/ = r/ (A + /); (16) 
 
 D^B 4 = r 3 (C + /3 4 ). (17) 
 
 EXAMPLE 12. Given r/ = 2865; r// = 1146; r 3 = 716.3; /// = 20; 
 P = 171900; L, = 90; L 3 = 90 .'. // = f 3 = .17 to join the circular 
 curves B^Di and B 4 D/t by Prob. III. First finding a, b and c by 
 (1 - 3) we have by (6) ver A = 3 44'; arc A = .6516; by (7) 
 (180 - B) = 18 48'; arc (180 - B) = .32811; by (8) C = 15 06'; 
 
 * For determining large central angles of /, 0//, 3 and /3 
 the method of 4 is to be preferred. 
 t See Appendix I. 
 
THE SPIRAL 35 
 
 arc C = .26054; AJ>, = 2865 X .06516 = 186.68; AJ> tl = 716.3 X 
 
 Qrt 
 
 .26054 = 186.77; A,,A 3 = 1146 X .3287 = 376.69; ft = ft/ .: 
 
 .6282; j3 3 = /3 2 = 2( ^ 46) = .03926 ; /3, = 2 (2865 = - 01571 AtBl 
 =A,,B it = ^3^3 = ^4#4 = 45.0 ; i = 18 48' - (2 15' + 2 15') 
 = 14 18' arc <- = .25017; B tl B z = 1146 X .25017 = 286.7; B,D, = 
 2865 (.06516 + .0157) = 231.67; B*D lt = 716.3 (.26054 +.06282) 
 = 231.62. 
 
 If we make // = and r/ = r//, then c = 0, C 0, /// and // = 
 0, b = a and is coincident with it, A = 180 B and the problem 
 reduces to uniting A,/A 3 with Z>//# 4 by means of the transition 
 curve. If TII = r/, c 0, C = 0, in which case // and f 3 = 0; a is 
 equal to and coincides with b and A = 180 B = equations for 
 vers and sines = 0. The problem reduces to uniting curves B/D, and 
 
 B^D// by means of a transition curve whose length Z/2 = 
 
 _ J. 
 
 If f 3 and // = 0, while r/, r// and r 3 retain their values, the transi- 
 tion curves disappear and the curve with radius r// compounds at 
 A* and A, with the curves having radii r, and r 3 . 
 
 180 - B is the central angle of A tl A 3 ; (18) 
 
 C = that of 7)//A 4 ; 4 = that of A,D,. (19) 
 
 If the compound circular curve, Prob. Ill, be of two centers 
 with radii r// and r/, then Eq. 12 and 13 give the value of /3/ and // 
 if the central angles be not exceeding 10 or 15 degrees and the 
 radii do not exceed 955 feet; otherwise 
 
 Let o-n o-i = $u + ft/ = 7 and r// and r/ be known ; then, Eq. (6) 
 page 5 
 
 P== 2^T_^> (20) 
 
 iv7 2 ~^2 
 and by (6) 
 
 and / computed by Eq. (37) then by 4 
 
 // and / can be obtained, whence r//j3// = the length of one 
 branch of the curve and r/P, the other. Then to find the B. C., 
 C, C. and E. C. of the compound circular curve, having 
 
 a// - a, = + ft, = 7. (21) 
 
36 TRANSITION CURVE 
 
 r// tan $ $,i the semi-tangent for r///3 x/ and n tan \ , = the 
 semi-tangent for r//3/ 
 
 whence T// + T = r// tan ,/ 4- r/ tan B/ = the hypotenuse of 
 a triangle in which one side and three angles are known, viz.: 
 Tt, + I 1 /; 0// and ft/ and 7 = ft// 4- /3, and for any side opposite // 
 we have 
 
 b, = (IT,, + T) sin j8//. 
 
 By a similar process we may find 6,/ whence T,/ 4- &// = the 
 semi-tangent adjacent ft// and I 7 // + &/ = the semi-tangent adjacent 
 /3, which fixes the B. C. and E. C. and r/jS, the distance on the 
 curve to C. C. The distance r//3, assumed to be measured on the 
 curve.* 
 
 If it be desired to introduce transition curve at the extremities 
 of this compound curve, the formulas of Problem I are applicable 
 by writing 
 
 T,, 4 b// and T + b, for T. 
 
 When the rate of curvature of the branches of compound curve 
 differ from each other by less than two degrees per station of 100 
 feet the transition curves may be omitted at the compound points 
 and introduced only at the ends where the curve merges into the 
 tangents. 
 
 The semi-tangent to a compound curve of more than two cen- 
 ters may be computed by latitudes and departures. 
 
 CASE 2. 
 
 If it be desired to introduce a third circular curve, exterior to 
 and joining the other two fixed circular curves, by means of spiral 
 arcs, all turning in the same direction, then the conditions are 
 indicated in Fig. (8a) and the solution similar to the first case. P 
 has fixed value from which //, /a, L/ = B/B// and 3 == B^B^ are 
 computed. If, however, we wish to displace the third circular 
 curve by spiral arcs, tangent each other at some point O, having a 
 maximum radius of curvature = r//, then the first approximate 
 solution will be obtained as follows : By Fig. (8a) in the triangle 
 ABC let the side AB = r,, - r/, BC = r,, - r s and AC = r, + r 3 
 f, the sign depending on whether AC is greater or less than 
 r/ + rs. Solving for angle B, we have, approximately, the central 
 angle of arc AnA% and r//B arc A//A& Let LO/ and LOS the 
 
 * The chord of a 10 curve with radius = 573 ft. = 49.99 ft- 
 for an arc of 50 feet. 
 
THE SPIRAL 
 
 37 
 
 length of the spiral arcs computed from this data. Since : /3 a = 
 r 3 : r, then ^j- = r//|3// = A//O/, ^ = r//0 3 = AgO//. 
 
 A^On + A/,O, < A,,A 3 , generally, by an amount O/O//. Make 
 
 ' - and " = '" ~ ' then 2 + ' = L ' 
 
 Fig. 8a. 
 
 2 (~ + O//O/J = 1/3 nearly, compute f/ and / 3 and make the sides 
 of the triangle, AB and AC, respectively = r,, (r/ + f/) and 
 TH (r 3 + fs) and compute OA,, and OA 3 repeatedly till = xf, and 
 x/s, or O/O// = (or iota = 0). If we make r// infinitely great, then 
 A uA 3 will be a straight line and O the common origin of the^spirals 
 in which r/ + r 3 /// = zw/ + xm z and we would have a special 
 value of P for these particular curves. For a compound circular 
 curve f/ and fs = as in the first part of this case. 
 
 PROBLEM IV. 
 
 15. Given two curves turning in the same direc- 
 tion, whose centers are a distance apart = D,D,, and 
 whose radii are r, and r fl . To fix the position of a tan- 
 gent A. t A i4l and connect it with the circular curves A tt C t 
 and A 4 C,, by means of transition curves having a fixed 
 value of P (Fig. 9). Let C,C /t be a line joining any two 
 points C f and C /y of the circular curves with radii r t 
 
38 
 
 TRANSITION CURVE 
 
 and r /y ; measure the angles T> 4 C t C in C^C^D^ and line 
 <?,<?. By traverse we find the angles C,D,D,,, C^D^D, 
 and distance D^D,, between the centers of the curves r t 
 
 and r yy . If from D y and D /y we let fall perpendiculars to an 
 imaginary tangent passing through the origins of the 
 transition curves, by the conditions of the problem the 
 length of these perpendiculars will be: 
 
 D,A, = r y + m t and D yy A /yy = r yy + ^//, 
 and any distance D 4 K, = (r y + m t ) (r y/ + m y/ ), (1) 
 and if we denote the distance D,D,, by D 4) then 
 
 IQ n _ 
 = 
 
 !,)- (r /y + m /x ) 
 ~ 
 
 (2) 
 
THE SPIRAL 
 
 39 
 
 C,D,A /f = difference of C,D,D,, and 0. C t D t A tl reduced 
 to arc and multiplied by r t = the distance C f A ti to 
 establish A lt \ from A lt lay off m y normal to the circle and 
 establish A r In the same way C,,D,,A 4 = the differ- 
 ence between 180 - and />!>,. C^D f/ A 4 mul- 
 tiplied by r lt = arc distance C /7 A 4 from C /x to A 4 to 
 establish A 4 , from w r hich lay off m tl and establish A UI \ 
 then a line through A t A tll is the required tangent. 
 
 The distance Af) t to the origin 0, = x t r, sin a, 
 and the distance A I4I O I{ = r tl sin a /x ; whence the 
 distance 
 
 0,0,, ^D.D,, sin e - [(* -r /7 sin a y/ ) + ( X/ -r, sin a,)]. (3) 
 If we wish the distance O t O lt = zero, make 
 D/ D // = [(x /y - r /y sin a v ) + (x x - r, sin a,)] cose 0. (4) 
 
 This gives the shortest distance possible between the 
 centers of the circular curves when transition curves 
 are introduced. The transition curves may have differ- 
 ent values of P provided Afl t + A tli O tl = or < than 
 A t A til . If the curves are reversed, D,K it (r / + m y ) 
 + (r /y 4- w /y ) and we find the value of ^ by completing 
 the traverse C t C tii D, ti D t C t . 
 
 EXAMPLE 13. Given Z)/C/C/ y = 85; C/C//D/ = 100; C/C// - 300: 
 r/ = 1146; r// = 573. First, to find D/Z>// = D 4 . By traverse with 
 Z)/ as initial point. 
 
 From 
 
 Course 
 
 Distance 
 
 Lat. JV.+ 
 
 Lat.S- 
 
 Dep E + 
 
 DepTF- 
 
 To 
 
 D, 
 
 South 
 
 1146.0 
 
 
 1146.0 
 
 
 
 c, 
 
 c, 
 
 N85E 
 
 300.0 
 
 26.1 
 
 
 298.9 
 
 
 Cn 
 
 c* 
 
 N 5 E 
 
 573.0 
 
 570 8 
 
 
 49 9 
 
 
 D lt 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 + 5i>6.y - 1146.0 + 348.8 
 
 596.9 
 - 549.1 +348.8 
 
40 TRANSITION CURVE 
 
 Z>,Z>// = D 4 = \/(549.1) 2 + (348.8) 2 = 650.5 
 
 m, = .13 m// = 1.05 r, + m, = 1146.13 r,, + m// = 574.05 
 
 C08 s _ frv + m,) - (r, + m ,,) _ W|0| _ _ 87944 = OQS 2go 25 , 
 
 4 635 - 22 = tan 32 25/ 
 
 32 25' - 28 25' = 4 .'. D,A, = r, + m, bears S 4 E. 
 A t , A, = m/ = .13'. By the table C//Z>// bears ^V 5 ^. 
 Di,A,n = r// + w//; parallel to D,Ar, bears /S 4 E. C//D//J./// = 9 
 ^. 4 A/// = 1.05 A,A, lt = T>,iK, = Z) 4 sin B 
 
 = AA tll = 650.5 X .4759 = 309.5 
 
 O,O// = D/Z>/ sin Q - \(x,, - r,, sin a,,) + (a:/ - r, sin a)]. 
 Let a:// = 119.90; a /7 = 6 a;/ = 60'; a, = 130 / 
 
 O.On = 309.5 - [59.95 + 30] = 309.5 - 89.95 = 219.55 
 to make O/O// = 
 
 M/// = 89.95 cose 8, or Z>/Z>/// = 89.95 X 6.4398 = 579.25. 
 
 PROBLEM V. 
 OLD TRACK. 
 
 1 6. To introduce the transition curve in align- 
 ment where circular curves have been run. 
 
 In Fig. 10, suppose ABC, a simple circular curve, to 
 have been run tangent to the line OG tl at A with radius 
 r t = DjA = D,B; it is desired to introduce a spiral whose 
 greatest curvature has a radius r u = D^A,, = &,,& = 
 D lt B < r,. From the tables or by computation we have 
 m depending on the value of r u and Z/ /7 ; with given 
 values of m, r tl and r, we have from the figure 
 
 AD, = A,A,/ + AiPu + D D < cos D D < K '> 
 or r f =m l + r tl + (r, r /y ) cos /. 
 
 ^ = r, - r tt - (r, ~ r,,) cos 
 m = r/ - r,, (1 - cos 0) = (r, - r y/ ) ver 
 
 ver == - ; r, <f> = AB ; r y/ = A y/ B 
 
THE SPIKAL 
 
 41 
 
 AB = the distance to measure from A = PC to locate 
 B; BA lt the distance to measure from B on BA U to 
 locate A lt . From m and r u we have 
 
 
 
 4, 
 Fig. 10. 
 
 2r y/ a y/ = L = \ / 24mr,,, squaring, 4r // 2 a // 2 = 
 or 2 6m /6m L 
 
 "fW ; "" V^ ;r " a// = " " = 2 
 r// (0,, - <*) = BB //; (r, - r y/ ) sin ^ = A y A. 
 
 The rate of curvature of r y/ should not be more than 
 from 1 to 2 greater than that of r, (when possible) for 
 curves of a curvature less than 10; 2 to 3 difference 
 for 10 to 15 rate of curvature; 3 to 5 difference for 
 15 to 20 rate of curvature. 
 
 EXAMPLE 14. Given r/ = 1146, r,, 955, m 
 a,,, AB, BB, t and L 
 
 m 142 
 
 1146 - 955 
 
 1.52, to find < ; 
 =0.00743= ver659/ ; arc # = .12188 
 
42 TRANSITION CUKVE 
 
 r/0 = .1219 X 1146 = AB, r//0 = 1219 X 955 = 115.8 = A,,B 
 
 arc = . _ . 0944 _ 5 o 24 , 
 
 /// yoo 
 
 / (0 - a) = 955 (.1216 - .0944) = 25.7 = B,,B 
 L = 2/v/a,, = 2 (955 X .0944) = 180.2 
 
 EXPLANATION OF THE TABLES. 
 
 ,. ' BY RECTANGULAR CO-ORDINATES. 
 
 17. Tables 1 to 7 give values for laying out the 
 transition curve by the method of rectangular co-ordi- 
 nates. They are equally applicable for uniting a tangent 
 with a circular curve, or curves having different radii, 
 by means of the transition curve. L tl and L t may be 
 taken separately from the same column, as also may a /y 
 and a,, and their difference will be the value of L and < 
 for the length and central angle respectively. The sev- 
 eral ordinates, x, y, x f , y f , are laid off from J3, or E lt as 
 origin (Fig. 2) with arc B t A, or J5 /y A y/ as axis, the 
 same as if B / or B lt were written for A in Fig. 3, and the 
 successive stations were B, C, D, etc., and B,, C 4 , D /} 
 etc., successive points x, x n x lti , etc., with corresponding 
 y> y<) 2///> etc., values normal to the curved axis A 4 B, 
 A u B tl in the same manner as if A 4 B, were tangent. 
 P and v are taken of such values as to avoid introducing 
 fractions in L. L is supposed to be measured on the 
 curve, but since the chords are generally quite short, 
 the sum of their lengths is but little less than that 
 of the curve, hence no allowance is made for the length 
 of the curve being in excess of the sum of the lengths of 
 the chords. 
 
 EXAMPLE 1. Given L = 120; r, = 1432.5; r// = 716.3 (Table 
 II), then // - a/ = 9 36' - 2 24' = 7 12' = <j> with E as origin. 
 At the end of the first chord length from E towards / we have 
 x = 30', y = .03 = co-ordinates for F; x, = 60, y, = .21 = co- 
 ordinates for G\ Xn = 90, y// = .70 = co-ordinates for H\ x$ = 
 119.98, 2/3 = 1.67 = co-ordinates for /; x f = 60; y f = .21; / = .42. 
 
THE SPIRAL 43 
 
 The method of laying out these co-ordinates is shown in Fig. 4, 
 in which the origin A corresponds to the point E in this example, 
 and AD i becomes a curved axis with a radius of 1432,5. 
 
 If the transition curve were laid off from BuAti, Fig. 2 as axis 
 and Bu as origin, the above values x, etc., and y, etc., would be 
 just the same except they would be laid off from the convex side of 
 AnBii instead of from the concave side, as was the case with A/B/ 
 as axis. If the curvature of the circular curve is of fractional 
 degree the value of / and the last values of x, y and; a will have 
 to be computed by the formulae at the head of the respective 
 columns in Spiral Table I. 
 
 BY DEFLECTION. 
 (Or Polar Co-ordinates.) 
 
 1 8. Given a tangent at any point of a transition 
 curve as D to locate any other points as A, B, C, E, F, 
 G and H. As in the case of the Tables for .rectangular 
 co-ordinates they are equally applicable for locating the 
 points of the transition curve uniting a tangent to the 
 circular curve or circular curves of different radii with 
 each other. (Fig. 3.) 
 
 EXAMPLE 2. Let L,, - L, = L = 120 be the length of the 
 transition curve; r/ = 1910; r// = 716.3 the radii of the circular 
 curves to be united by L by deflections from a tangent at Z>, where 
 the curvature corresponds to r/. The tangent will be common to 
 the circle and transition curve whose rate of curvature " D" (Table 
 II) = 3. DE, EF, FG, etc., being chords of the transition curve 
 each = 30 feet. The tangent at D is a tangent to the circular arc 
 with r t = 1910. Then by the formula w = L[D& + N X 000.1'1 
 in which, if we write L = 30, 60, 90 and 120; and N = 1,2, 3 and 4 
 successively, D = 3, A = 0.3', then the deflection from tangent at 
 
 D to locate E is o> = 30 [3 X .3' + 1 X OO.t'] =* 30' 
 D to locate F is <w = 60 [3 X .3' + 2 X OO.l'l = 106' 
 D to locate G is a = 90 [3 X .3' -f 3 X OO.l'l = 1 48' 
 D to locate H is < = 120 [3 X .3' -4- 4 X OO.l'l = 2 36' 
 
44 TRANSITION CURVE 
 
 with measurement from D to E thence E to F, etc. If we wish to 
 locate the points C, B and A from a tangent at D, then the deflec- 
 tion for any point C, for example, = the deflection for 30 feet for a 
 3-degree curve minus the deflection for the transition curve from 
 A to B by formula (61). 
 
 hi = L [DA - N X OO.l'l, whence from 
 D to locate C is aj = 30 [3 X .3' - 1 X 00.1'] = 24' 
 D to locate B is w = 60 [3 X .3' - 2 X 00.1'] = 42' 
 D to locate A is w = 90 [3 X .3' - 3 X 00.1'] = 54'. 
 
 If one of the chords be fractional and the change of curvature 
 per chord be fractional also. 
 
 Let j, = 15' - i, C = 1 30' = U 
 then by (61 ) 
 
 at = L[Z>A( A' + J;) C X000.1']. 
 
 To locate any point, as G +15 from tangent at D, then 
 at = 105 [3 X .3 4- (3 + i) X H X 00.1'] = 2 29'. 
 
 To locate any point, as A + 15' from tangent at D, then 
 <o = 75 [3 X .3' - (2 + |) H X 00.1'] = 39'. 
 
 If we have run the curve from A to D and changed the instrument 
 to D in order to place the line of sight tangent to jD.take a back- 
 sight on A and deflect 54' and we have a tangent at Z>. To 
 facilitate the use of the tables it is best to set the vernier at 54' 
 and set the telescope on line AD, turn the vernier to O and con- 
 tinue deflection as tabulated, reading downward from D, locating 
 the points E, F, G, etc. If the curve is being run from D 
 towards A then set the vernier at the angle indicated for any angle 
 G, when backsight is on G from D deflect from zero and continue 
 to deflect the angles tabulated in succession, reading up the column 
 from D to locate C, B and A. The degree of the curvature at the 
 instrument point controls the deflections either way. The above 
 explanation enables us to run the transition curve from the point 
 of greatest radii to that of its least radius, and vice versa. 
 
 If we take the curvature "Z>" at the position of the instrument 
 as the basis of calculation, then equation (61) can be applied 
 directly to get on tangent at the position of instrument (after back- 
 sight on the last instrument point), using the sign for running 
 towards G and the 4- sign for running towards .4. 
 
THE SPIRAL 45 
 
 TABLE OF CIRCULAR ARCS 
 
 Length of Circular Arcs at Radius = i 
 
 Decimals of a Degree 
 
 Sec. 
 
 Length 
 
 Min. 
 
 Length 
 
 Deg. 
 
 Length 
 
 Min. 
 
 Decimal 
 
 .Sec. 
 
 Decimal 
 
 1 
 
 .000005 
 
 1 
 
 700029T 
 
 1 
 
 .017453 
 
 1 
 
 .01667 
 
 1 
 
 .00028 
 
 2 
 
 .000010 
 
 2 
 
 .000582 
 
 2 
 
 . 034906 
 
 2 
 
 .03333 
 
 2 
 
 .00055 
 
 3 
 
 .000015 
 
 3 1.000873 
 
 3 
 
 .052360 
 
 3 
 
 .05000 
 
 3 
 
 .00083 
 
 4 
 
 .000019 4 
 
 .001164 
 
 4 
 
 .069813 
 
 4 
 
 .06667 
 
 4 
 
 .00111 
 
 5 
 
 .000024 
 
 5 
 
 .001454 
 
 5 
 
 .087266 
 
 5 
 
 .08333 
 
 5 
 
 .00139 
 
 6 
 
 .000029 
 
 6 
 
 .001745 
 
 6 
 
 . 104720 
 
 6 
 
 . 10000 
 
 6 
 
 .00167 
 
 7 
 
 .000034 
 
 7 
 
 .002036 
 
 7 
 
 .122173 
 
 7 
 
 .11667 
 
 7 
 
 .00195 
 
 8 
 
 .000039 
 
 8 
 
 .002327 
 
 8 
 
 . 139626 
 
 8 
 
 .13333 
 
 8 
 
 .00222 
 
 9 
 
 .000044 
 
 9 
 
 .002618 
 
 9 
 
 . 157080 
 
 9 
 
 . 15000 
 
 9 
 
 .00250 
 
 10 
 
 .000048 
 
 10 
 
 . 002909 
 
 10 
 
 . 174533 
 
 10 
 
 . 16667 
 
 10 
 
 .00278 
 
 11 
 
 .000053 
 
 11 
 
 .003200 
 
 11 
 
 .191986 
 
 11 
 
 . 18333 
 
 11 
 
 .00306 
 
 12 
 
 .000058 
 
 12 
 
 .003491 
 
 12 
 
 . 209439 
 
 12 
 
 . 20000 
 
 12 
 
 .00333 
 
 13 
 
 .000063 
 
 13 
 
 .003781 
 
 13 
 
 . 226893 
 
 13 
 
 .21667 
 
 13 
 
 .00361 
 
 14 
 
 .000068 
 
 14 
 
 .004072 
 
 14 
 
 . 244346 
 
 14 
 
 .23333 
 
 14 
 
 .00389 
 
 15 
 
 .000073 
 
 15 
 
 .004363 
 
 15 
 
 .261799 
 
 15 
 
 . 25000 
 
 15 
 
 .00417 
 
 16 
 
 .000078 
 
 16 
 
 .004654 
 
 16 
 
 . 279253 
 
 16 
 
 . 26667 
 
 16 
 
 .00445 
 
 17 
 
 .000083 
 
 17 
 
 .004945 
 
 17 
 
 . 296706 
 
 17 
 
 . 28333 
 
 17 
 
 .00477 
 
 18 
 
 .000087 
 
 18 
 
 .005236 
 
 18 
 
 .314159 
 
 18 
 
 .30000 
 
 18 
 
 .00500 
 
 19 
 
 .000092 
 
 19 
 
 .005527 
 
 19 
 
 .331613 
 
 19 
 
 .31667 
 
 19 
 
 .00528 
 
 20 
 
 .000097 
 
 20 
 
 .005818 
 
 20 
 
 .349066 
 
 20 
 
 .33333 
 
 20 
 
 .00556 
 
 21 
 
 .000102 
 
 21 
 
 .006109 
 
 21 
 
 .366519 
 
 21 
 
 .35000 
 
 21 
 
 .00583 
 
 22 
 
 .000107 
 
 22 
 
 .006399 
 
 22 
 
 .383972 
 
 22 
 
 .36667 
 
 22 
 
 .00611 
 
 23 
 
 .000111 
 
 23 
 
 .006690 
 
 23 
 
 .401426 
 
 23 
 
 . 38333 
 
 23 
 
 .00639 
 
 24 
 
 .000116 
 
 24 
 
 .006981 
 
 24 
 
 .418879 
 
 24 
 
 .40000 
 
 24 
 
 .00667 
 
 25 
 
 .000121 
 
 25 
 
 .007272 
 
 25 
 
 .436332 
 
 25 
 
 .41667 
 
 25 
 
 .00695 
 
 26 
 
 .000126 
 
 26 
 
 .007563 
 
 26 
 
 . 453786 
 
 26 
 
 .43333 
 
 26 
 
 .00722 
 
 27 
 
 .000131 
 
 27 
 
 .007854 
 
 27 
 
 .471239 
 
 27 
 
 .45000 
 
 27 
 
 .00750 
 
 28 
 
 .000136 
 
 28 
 
 .008145 
 
 28 
 
 .488692 
 
 28 
 
 .46667 
 
 28 
 
 .00778 
 
 29 
 
 .000141 
 
 29 
 
 .008436 
 
 29 
 
 .506145 
 
 29 
 
 .48333 
 
 29 
 
 .00806 
 
 30 
 
 .000145 
 
 30 
 
 .008727 
 
 30 
 
 .523599 
 
 30 
 
 .50000 
 
 30 
 
 .00833 
 
 31 
 
 .000150 
 
 31 
 
 .009017 
 
 31 
 
 .541052 
 
 31 
 
 .51667 
 
 31 
 
 .00861 
 
 32 
 
 .000155 
 
 32 
 
 .009308 
 
 32 
 
 .558505 
 
 32 
 
 .53333 
 
 32 
 
 .00888 
 
 33 
 
 .000160 
 
 33 
 
 .009599 
 
 33 
 
 .575959 
 
 33 
 
 .55000 
 
 33 
 
 .00916 
 
 34 
 
 .000165 
 
 34 
 
 .009890 
 
 34 
 
 .593412 
 
 34 
 
 .56667 
 
 34 
 
 .00944 
 
 35 
 
 .000170 
 
 35 
 
 .010181 
 
 35 
 
 .610865 
 
 35 
 
 .58333 
 
 35 
 
 .00972 
 
 36 
 
 .000175 
 
 36 
 
 .010472 
 
 36 
 
 .628318 
 
 36 
 
 . 60000 
 
 36 
 
 .01000 
 
 37 
 
 .000179 
 
 37 
 
 .010763 
 
 37 
 
 .645772 
 
 37 
 
 .61667 
 
 37 
 
 .01028 
 
 38 
 
 .000184 
 
 38 
 
 .011054 
 
 38 
 
 .663225 
 
 38 
 
 .63333 
 
 38 
 
 .01055 
 
 39 
 
 .000189 
 
 39 
 
 .011345 
 
 39 
 
 . 680678 
 
 39 
 
 .65000 
 
 39 
 
 .01083 
 
 40 
 
 .000194 
 
 40 
 
 .011635 
 
 40 
 
 .698132 
 
 40 
 
 . 66667 
 
 40 
 
 .01111 
 
 41 
 
 .000199 
 
 41 
 
 ,011926 
 
 41 
 
 .715585 
 
 41 
 
 .68333 
 
 41 
 
 .01139 
 
 42 
 
 .000204 
 
 42 
 
 .012217 
 
 42 
 
 . 733038 
 
 42 
 
 . 70000 
 
 42 
 
 .01166 
 
 43 
 
 .000209 
 
 43 
 
 .012508 
 
 43 
 
 .750492 
 
 43 
 
 .71667 
 
 43 
 
 .01194 
 
 44 
 
 .000213 
 
 44 
 
 .012800 
 
 44 
 
 .767945 
 
 44 
 
 .73333 
 
 44 
 
 .01222 
 
 45 
 
 .000218 
 
 45 
 
 .013090 
 
 45 
 
 .785398 
 
 45 
 
 .75000 
 
 45 
 
 .01250 
 
 46 
 
 .000223 
 
 46 
 
 .013381 
 
 46 
 
 .802851 
 
 46 
 
 .76667 
 
 46 
 
 .01278 
 
 47 
 
 .000228 
 
 47 
 
 .013672 
 
 47 
 
 .820305 
 
 47 
 
 . 78333 
 
 47 
 
 .01306 
 
 48 
 
 .000233 
 
 48 
 
 .013963 
 
 48 
 
 .837758 
 
 48 
 
 .80000 
 
 48 
 
 .01333 
 
 49 
 
 .000238 
 
 49 
 
 .014253 
 
 49 
 
 .855211 
 
 49 
 
 .81667 
 
 49 
 
 .01361 
 
 50 
 
 .000242 
 
 50 
 
 .014544 
 
 50 
 
 .872665 
 
 50 
 
 .83333 
 
 50 
 
 .01389 
 
 51 
 
 .000247 
 
 51 
 
 .014835 
 
 51 
 
 .890118 
 
 51 
 
 .85000 
 
 51 
 
 .01417 
 
 52 
 
 .000252 
 
 52 
 
 .015126 
 
 52 
 
 .907571 
 
 52 
 
 .86667 
 
 52 
 
 .01444 
 
 53 
 
 .000257 
 
 53 
 
 .015417 
 
 53 
 
 .925024 
 
 53 
 
 .88333 
 
 53 
 
 .01472 
 
 54 
 
 .000262 
 
 54 
 
 .015708 
 
 54 
 
 .942478 
 
 54 
 
 .90000 
 
 54 
 
 .01500 
 
 55 
 
 .000267 
 
 55 
 
 .016000 
 
 55 
 
 .959931 
 
 55 
 
 .91667 
 
 55 
 
 .01528 
 
 56 
 
 .000272 
 
 56 
 
 .016290 
 
 56 
 
 .977384 
 
 56 
 
 .93333 
 
 56 
 
 .01556 
 
 57 
 
 .000276 
 
 57 
 
 .016580 
 
 57 
 
 .994838 
 
 57 
 
 .95000 
 
 57 
 
 .01583 
 
 58 
 
 .000281 
 
 58 
 
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 Tf 1C 
 
 VO 
 
 
 
 <P 
 
 i 
 
 H 
 
 
 
 * 
 fN 
 
 O vO ci 
 iO <*i C 
 <N PO 
 
 5 S 2S 
 
 H 1O CO 
 
 "b 
 
 
 
 vO 
 
 
 g 
 
 p 
 
 J 
 
 S ? i 
 
 3 r^ 
 
 2 
 
 
 
 
 
 
 
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 N 
 
 CO CO T 
 
 f W CO 
 
 t " 
 
 
 M a & 
 
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 04 
 
 
 
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 (N O C 
 ^ N C 
 
 ^ oo oo 
 
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 C3 C? 
 
 *} 
 
 
 
 
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 C^ ro T 
 
 M rf 10 
 
 VO 
 
 <0 
 
 H H 
 
 
 
 
 
 
 
 
 
 
 
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 o 
 
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 w 
 
 ^ M >- 
 
 -* 3 
 
 O 
 
 
 q 
 
 
 
 
 
 
 
 
 
 a) ti 
 
 
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 ^ 
 
 00 
 
 C4 
 
 
 
 s s i 
 
 * V & 
 
 1 C* (35 
 
 
 
 oo 
 
 35 
 
 2 
 
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 3 O O 
 H N CO 
 
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58 
 
 TRANSITION CUKVE 
 
 o 
 o 
 
 SI 
 SB 
 
 
 
 52 ^ w 
 
 asi 
 
 ^ 5 
 g 
 
 ^ H 
 
 HS 
 
 O 
 
 s 
 
 W3 
 
 5 
 
 a 
 
 CO 
 
 O Oi 00 *> O iO "^ CO C^ i-f CONDON 
 
 rHi-H(N(NCO^^OiOCOl^OOa>O 
 
 
 O O <~i f~i r- (COCO'^ | iOOCOI>t > -QOO5G5O' iC^C^COC^T^O'OOCOt^-QOXCS 
 
 oo co os o <N co 10 co i^ oo oi o -I d eo ^' o o o r^ x 06 CD' o o ^ ^ <N ^ co 
 
 s o <N co 10 co i^ oo oi o -I d eo ^' o o o r^ 
 
 r^rHf^i^^i^rHr^fNC^C^C^C^C^iN^lC^ 
 
 83 
 
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 -' 
 
 
 <N CO CO CO CO CC 
 
 -. 
 
 CO 01 H O 00 ^ O <O W OD CO O> ^ H WO 
 
 CiOOO^Cv>(NfCC 
 
 )N o eo w eo c 
 
 - 00 CO GO ffl O O5 O> O O O O O -i --* H 
 
 O OiCXDt>COiO^COC<lr-( 
 
 MOO-^ COM 
 ' t'-HJNCOCO'* 
 
 o co eo eo co co co 
 
APPENDIX I 
 
 PROBLEM. 
 
 Given: The altitude of two circumpolar stars, nearly opposite 
 each other, the local time of Meridian passage of either star, their 
 difference in time of Meridian passage and the polar distance of 
 each star, 
 
 TO FIND THE LATITUDE, THE LOCAL TIME AND THE 
 MERIDIAN OF THE PLACE. 
 
 Fig. 77. 
 
 For convenience, let Polaris and Alioth be the stars observed; 
 O the position of the observer and center of the celestial sphere. 
 ONEW be the north half of that portion of the celestial sphere 
 above the horizon WNE, with A as zenith. 
 
 Let GC = c or G//C// = c,, be the altitude of Alioth at the time 
 of observation and BS = g or //*/. = g// the altitude of Polaris * 
 35 minutes later; then the plane passing through the points S, G 
 and O will pass through the pole P and cut the celestial sphere in 
 the line r + r, SG or //G//, r t and r/ being the polar distance 
 respectively of Polaris and Alioth and AP = s the co-latitude of 
 the place O. Then the plane ASO or AS//O will cut the side of the 
 
 * At this date. 
 
 t The position of these stars have a slight annual change with 
 reference to the pole, which must be taken into account from year 
 to year. 
 
 59 
 
60 APPENDIX 
 
 spherical triangle AS = AB - BS = (90 - 0) or A8 lt = AB lt - 
 //,/ from the celestial sphere. Similarly, AGO or AG,,O will cut 
 (90 - c) or (90 - c,/). The sides /, r + r/ and (90 - c) or (90 
 c//) form the spherical triangle ASG or J.*S//G//, in which the 
 sides are known; from which we find the angle S or // by the 
 spherical formula: 
 
 (cos 90 - c) - cos (90 - g) cos (r + r) 
 
 COS AoOr = COS o = ; , nn r r 7 r 1 
 
 sin (90 - g} sin (r + n) 
 but cos (90 c) = sin c and cos (90 g) = sin 0, hence: 
 
 sin c sin g cos (r 4- r/) 
 cos sin (r 4- r/) 
 
 The angle is common to the two triangles ASG and ASP. 
 
 We now have r, S and (90 g) to determine A the Azimuth, 
 Z = the Latitude and P = the Hour Ang e (expressed in degrees). 
 
 cos (90 - I) = cos (90 - g} cos r sin (90 - g) sin r cos S, 
 or sin 1 = sin g cos i cos g sin r cos S, (a) 
 
 using sign when S > 90; and since 
 
 sin (90 - Z): sin (90 0):: sin S: sin P, 
 or cos Z: cos g:: sin /S: sin P 
 
 . _ cos g sin S /t v 
 
 sin P = ^ : (b) 
 
 cos 1 
 
 P represents the angle the plane OSP makes with the plane 
 OPA at the time of observation on S and P X 4 the Hour Angle, 
 expressed in minutes of sidereal time, from the plane ONPA. Also 
 in the triangle ASP we have 
 
 sin r sin P sin r sin P , N 
 
 sin A = - ; = (c) 
 
 sin (90 g) cos g 
 
 in which A is the angle between the planes ABO and ANO, which 
 is the Azimuth Angle for the Meridian Plane ANO. 
 
 A similar solution applies to the triangles AS//P a.ndASG/i 
 Equations (a), (b) and (c) solve the problem. 
 
 When the altitude of Alioth is much in excess of that of Polaris, 
 the observations are not easily made in higher latitudes with the 
 ordinary engineer's transit, unless equipped with a prismatic eye- 
 piece. 
 
 Sidereal hours X .9972696 = mean solar hours. 
 
MERIDIAN 61 
 
 EXAMPLE. Given g = 40; c = 20; r = 33 30' to find the lati- 
 tude I. We first find the angle S by the formula, 
 
 c = sin c - sin g cos (r + r, ) = .3292 - .64279 X .82181 
 cos g sin (r + r) .76604 X .56976 
 
 To find the latitude we have (a) 
 
 sin I = sin fir cos r cos g sin r cos S 
 using the sign (since S > 90), 
 
 sin I = .64279 .9998 - .76604 X .0215 X 42667 = .642661 - 00702 
 = .63564 = sin 39 28', whence the Latitude 39 28'. 
 
 To find the local time we have (6) 
 sin P i. 'M|iHjS _ -76604^90445 = ^^ _ ^ ^ ^ p ^ fm 
 
 = 63 49' X 4 = 4 h 15 m = 255 sidereal minutes; 255 X .99727 = 254 
 mean solar minutes, or P = 4" 14 m . 
 
 The time of observation was at 1 o'clock A.M., Dec. 1st, 1900, 
 i.e. 13 h - 4 h 14 m = 8 h 46 m the time of meridian passage by the 
 clock. 
 
 The local time of meridian passage was 8 h 40 m i.e. 8 h 46 m - 
 8 h 4Q m = Q h Q6 m {e the clock was 6 m fast> 
 
 To find the Azimuth Angle SAP = BON = A, we have (c) 
 
 sin r sin P .0215 X .7974 no _ 1c , 
 
 sin A = = _ Qgr . , = .02518 whence ^4=1 27' 
 
 cos g .76604 
 
 SAN FRANCISCO, 1898. 
 
62 
 
 APPENDIX 
 
 1 
 
 - 
 
 lO M^ 
 OS OS 
 
 2SgS 
 
 5| 
 
 CO 
 
 CO CO 
 
 cococococococococo 
 
 O 
 
 j| 
 
 
 S* S : 
 
 OS 
 
 1* 
 
 CO 00 
 
 OOO^fOcOWiOCOO 
 
 1 
 
 S TH 
 
 Bco 
 
 : s 
 
 osoocoost^cooocooo 
 
 OOOOCO^IMOOOO 
 
 1 
 
 aj 
 
 fu 
 
 oo co 
 
 : S : : . . : . 
 
 3 
 
 S co 
 
 o o 
 
 ooco^t^co^r-iot- 
 
 I 
 
 B'CO 
 
 * <N 
 
 ooooco^^oox 
 
 00 
 
 s 
 
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 s s . 
 
 8 
 
 OH 
 
 co os 
 
 5 a< 
 
 00 1-1 13 it^COCOt^cO 
 
 u 
 * 
 
 10 
 
 co oo 
 
 coococo^cooco>-o 
 
 
 Bco 
 
 <N 
 
 ooooco^^oooo 
 
 ^ 
 
 S 
 
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 s s . 
 
 s 
 
 .CO 
 
 CO rt* 
 
 < a< 
 
 c8 
 
 s ^' 
 
 -H r-< 
 
 osr-cooot^ooocooo 
 
 * 
 
 B*0 
 
 Tf (N 
 
 ooooco^^oooo 
 
 
 
 jj 
 
 _ 
 
 s s , 
 
 i-H 
 
 J 
 
 ^ 
 
 cs^o^^coot^co 
 
 1 
 
 : 
 
 ! 
 
 o o : ! * : o :. 
 
 1 
 
 sj 
 
 PH* 
 
 
 
 4 PH 
 
 03 
 
 S ^H 
 
 OS 00 
 
 CO^COO^<NiOCOO 
 
 h 
 
 B'CD 
 
 <N 
 
 OOOOCO^^OOOO 
 
 I 
 
 S 
 
 Bt 
 
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 ^ &l 
 
 ol 
 
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 rH !> 
 
 lOCO i -^ (M H CO 1-1 CO 
 
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 ooooco^c^oooo 
 
 
 
 
 
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 4 
 
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 Ct, J^ 
 
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MISCELLANEOUS PROBLEMS, SUPPLEMENTS 63 
 
 METHOD OF COMPUTING RADIUS. 
 
 The usual method of computing the radius of curvature of 
 circular curves used in railroad location, is by the formula 
 
 in which c represents a chord of 50 feet in length, and D is the degree 
 of the curve or central angle for a chord of 100 feet. By this method 
 the radius does not vary inversely as the degree of the curve; while 
 in the method of this book it is assumed to do so, and r is assumed 
 to equal the radius of a one-degree curve divided by the degree of 
 the proposed curve, or 
 
 r = 5 -f?. (2) 
 
 These formulae give the same result for a five-degree curve for 
 50-foot chords, thus: 
 
 from which we see, by the method of this book, the radius of a five- 
 degree curve is the same computed by either formula. For less 
 than 5 rate of curvature r eq. (1) < r eq. (2) and for a rate of curv- 
 ature greater than 5 r eq. (1) > r eq. (2) for 50-foot stations. 
 
 Since the length of the chords is usually less than 50 feet, in the 
 spiral the radius of curvature is assumed to vary inversely as the 
 degree of the curve, and the same radius may be used for the main 
 curve, by making its chords less than 50 feet, the proper length to 
 be determined, by computation, and tabulated for field use. It 
 is scarcely worth notice (= .002 per degree of curve less than 50 
 feet for each degree in rate of curvature per 100 feet of arc). 
 
 TO DETERMINE THE NUMERICAL CONSTANT IN FORMULA (60): 
 
 Place 000.1' = k, then we have 8' = LNCk 
 or k 
 
 We find 6' by dividing eq. (21) by eq. (26), whence, 
 tan 5 
 
 y 6r\ 56r 2 / 6r V 56r2 + ) 
 '- * - ; F7T +. <> r ^n = =- / - 
 
 zr 
 
 ~ 
 
64 APPENDIX 
 
 5 should be reduced to minutes (') before entering the formula 
 for finding the value of k. 
 
 The above value of k is for running in a direction from A towards 
 B, Fig. 3, or D towards E, etc. 
 
 For running in a direction from G towards F, or D towards C: 
 Having a' (minutes), the total number of minutes in the central 
 angle of any given length L of the spiral, corresponding to 5' in the 
 above equation, if we substitute (a' 8') in eq. (1), of this supple- 
 ment, for S and 2L for L t then calling the numerical coefficient k', 
 we have 
 
 . a/ ~ 5 ' 
 
 for running in a contrary direction from the same points as in eq. 
 (1) of this supplement. k / should always come out greater than k. 
 
 The value of k, will be found to increase and k decrease, 
 slightly, with a. For L = 200 feet, or less, and a = 10, or less, 
 we may make k, = k = 000.1'. 
 
 It will be seen by a trial example that the difference between 
 this method and that of eq. (61) for a curve with L = 400 (with 
 change point at 200 feet) and a = 16, is 000'03". 
 
 The difference increases with the central angle a. The above 
 method, with change points 200 to 300 feet apart, is quite accu- 
 rate and best for preparing a set of tables, though not so easily 
 applied in field computations as eqs. (60) or (61). 
 
 By assuming "change points" from 100 to 250 feet apart the 
 values in the deflection tables of this book may be substituted for 
 the second term in eq. (61) and the spiral extended indefinitely. 
 
 The following table shows the results of computations by the 
 above formulae. 
 
 The value of the constant C in each particular case is determined 
 
 by ~ ~ C, in which D, represents the degree, or rate of curvature, 
 
 at the point to be located; this substituted in eq. (60b) reduces it 
 to the form of 8' = LD,k. The above deduced value of C should 
 always be applied in eq. (60) or (60b). 
 
 k is not a constant, but for small deflection and central angles 
 its successive differences are inappreciable, but for large central 
 angles these differences may not be disregarded. 
 
 If 8' is constant 
 
 k = LD 7 ' L = W ; D ' = kL 
 
 , , , a' - 8' a' _ 
 
 and for fc,= orr . ; L - ., n ; D, = 
 
 2k,D, ' " 2k,L 
 
MISCELLANEOUS PROBLEMS, SUPPLEMENTS 65 
 
 ^ 
 
 CO CO 
 
 8 8 i i 
 
 
 
 
 
 . 
 
 ^ 
 
 41 
 
 O O O O O Oi C3 
 
 r-( r- i-l r-l t-H O O 
 
 fill 
 
 
 
 
 
 
 1 
 
 ^ LO CO 
 C W <N CO 
 
 CO 
 CO 
 
 ^H cq <N co 
 
 
 
 00 W 00 O 1> *-H 
 
 CO 
 
 r-* 00 CD O? 
 
 o o o" 1 o 
 00 O CO CO 
 
 5ft| H 
 
 
 O O W CO <N O <N 
 O C^ Oi <N CN cs ^* 
 
 SO <N CO 00 1-1 
 O O O O O i-i 
 
 1111 
 
 
 O O O O O O O 
 
 o o o o 
 
 ,, 
 
 O t *> 9 O 9 
 
 " * d 3 S 
 
 O5 CO 00 b 
 CD CD OS b-j 
 
 
 8 i 8 9 
 
 t- t~ Oi CO 
 
 H 
 
 O Gi O5 O5 OS 00 O 
 
 sill 
 
 
 
 
 d 
 
 ^* o o ^ o ^ o 
 
 N o o o* P o^ o 
 H CO O O ^ Oi 
 
 W CO "* "t 
 
 - 
 
 rt oo M o o ^ oo 
 
 itri 
 
 - 
 
 W "f O 00 O N ^ 
 
 CO 00 O N 
 
 - r-l CQ W 
 
66 APPENDIX 
 
 in which L is in feet, D t in degrees (), 8' in minutes; m being 
 
 determined by whatever value is assumed for L and D = 
 
 and substituted in eq. (37) or (38). 
 
 SUPPLEMENT TO PROBLEM I. 
 
 The total intersection angle is 
 
 / = a + a + t. 
 
 t is the angle due to the circular portion of the curve or B/B titt 
 with fixed radius r// and 
 
 B.DB,,, = I - (a, + a), 
 
 whence, by making the second term in eq. (61) = 0, we have: 
 
 a) LZ)A = the deflection from a tangent at any point of the 
 circular curve, as Bi or Btn to locate any other point of the circular 
 curve between Bi and Bin inclusive of the last point Bi or .B///as 
 the case may be. 
 
 SUPPLEMENT TO PROBLEM III. 
 
 t = the arc of that portion of the curve, between spirals, that 
 is circular and laid out by the method of circular curves, the deflec- 
 tion from tangent being computed by eq. (61), in which the second 
 term, in brackets, is made = 0, whence, it becomes <D = LZ)A. t will 
 have to be reduced from arc to degrees and minutes to determine 
 the total deflection. 
 
 THE PROCESS OF FIELD WORK. 
 
 Beginning at #/, as instrument point, deflect from tangent OtX 
 for successive stations by formula (60) including Bi (= "j.then, 
 placing the instrument at J5/, with backsight on O/, deflect 
 6' = 2LNC X 00.1' (= | a ) . 
 
 The line of sight will then be a tangent common to both the 
 spiral OtBi and the circular curve B t Bm at /. Deflect thence by 
 
 formula 
 
 (a = LDA 
 
 to locate stations on the circular curve BiBm fixing JS///. 
 
 Set up the instrument at B,, t and, with backsight on #/, deflect 
 w LZ>A for tangent at Bm, thence locate points of spiral J5///O/// 
 by applying formula (61) with sign or by (54), any deflection 
 
 6' = 2LNC X 00.1'. 
 
MISCELLANEOUS PROBLEMS, SUPPLEMENTS 67 
 
 Having located O///, set up instrument at O/// and with back* 
 sight on Bui deflect 
 
 5' = LNC X 00.1', 
 
 and the line of sight should coincide in direction and position with 
 the tangent O///X, or the semi-tangent On id,. 
 
 Some prefer to run the spirals first, i.e., from O/ to Bi and from 
 Oui to B///, and the circular portion B<Bm last, thus throwing the 
 closing error, if any, at Bi or B,n. 
 
 Undoubtedly this makes a better adjustment when the points 
 Oi and On, have been fixed previously by semi-tangents. 
 
 All change points should be established by double centers to 
 eliminate errors of adjustment in the transit instrument. 
 
TRIGONOMETRIC TABLES 
 
 ^ "c trigonometric functions of any angle intermediate those given 
 in the tables may be found by interpolation, thus: 
 
 What is the natural tangent of 1243' ? 
 
 From the Table, tan. 1250' = 0.22781 
 tan. 1210' = 0.22475 
 
 Diff. for 010' 
 
 " " 003' 
 Add " 124U' 
 
 Hence for tan. 1243' = 
 What is the natural cotangent 
 
 T3Vr>r-i flip T*aT1*> rr>t 1VO/i/V 
 
 0.0030G 
 .3 
 
 
 0.000918* *"+ 
 \F coti 
 
 0.22475 ^^ 
 
 Si 
 
 0.22567 / * + 
 ->fl904:^'? Y 
 
 xfi 
 
 " Z 
 
 A AAQAO \ ^f 
 
 ccs t ^Nn 
 
 tl / 
 
 /x-_ / 
 
 Diff. for 010' 
 
 cot. 1250' = 4.38969 
 
 = 0.05973 
 .3 
 
 " 003' = 0.017919* 
 
 Subtract from cot. 1240' = 4.44942 
 
 Hence 
 
 cot. 1243' = 4.43150 
 
 To obtain functions not given in- the tables: 
 Vers. a 1 cos. a ; External Sec. a = 1 Sec. a. 
 
 *The computation being additive or subtractive according as the 
 function increases or decreases with the increase of the angle a. 
 
TRIGONOMETRIC TABLES 
 
 1 
 
 SINE 
 
 
 J? 
 
 0' 
 
 icy 
 
 2O' 
 
 30' 
 
 4O> 
 
 50' 
 
 6O / 
 
 
 
 
 0.00000 
 
 0.00291 
 
 0.00582 
 
 0.00873 
 
 0.01164 
 
 0.01454 
 
 0.01745 
 
 89 
 
 1 
 
 0.01745 
 
 0.02036 
 
 0.02327 
 
 002618 
 
 0.02908 
 
 0.03199 
 
 0.03490 
 
 88 
 
 2 
 
 0.03490 
 
 0.03781 
 
 0.04071 
 
 0.04362 
 
 0.04653 
 
 0.04943 
 
 0.05234 
 
 87 
 
 3 
 
 0.05234 
 
 0.05524 
 
 0.05814 
 
 0.06105 
 
 0.06395 
 
 0.06685 
 
 0.06976 
 
 86 
 
 4 
 
 0.06976 
 
 0.07266 
 
 0.07556 
 
 0.07846 
 
 0.08136 
 
 0.08426 
 
 0.08716 
 
 85 
 
 6 
 
 0.08716 
 
 0.09005 
 
 0.09295 
 
 0.09585 
 
 0.09874 
 
 0.10164 
 
 0.10453 
 
 84 
 
 6 
 
 0.10453 
 
 0.10742 
 
 0.11031 
 
 0.11320 
 
 0.11609 
 
 0.11898 
 
 0.12187 
 
 83 
 
 7 
 
 0.12187 
 
 0.12476 
 
 0.12764 
 
 0.13053 
 
 0.13341 
 
 0.13629 
 
 0.13917 
 
 82 
 
 8 
 
 0.13917 
 
 0.14205 
 
 0.14493 
 
 0.14781 
 
 0.15069 
 
 0.15356 
 
 0.15643 
 
 81 
 
 9 
 
 0.15643 
 
 0.15931 
 
 0.16218 
 
 0.16505 
 
 0.16792 
 
 0.17078 
 
 0.17365 
 
 80 
 
 10 
 
 0.17365 
 
 0.17651 
 
 17937 
 
 0.18224 
 
 0.18500 
 
 0.18795 
 
 0.19081 
 
 79 
 
 11 
 
 0.19081 
 
 0.19366 
 
 0.19652 
 
 0.19937 
 
 0.20222 
 
 0120507 
 
 0.20791 
 
 78 
 
 12 
 
 0.20791 
 
 0.21076 
 
 0.21360 
 
 0.21644 
 
 0.21928 
 
 0.22212 
 
 0.22495 
 
 77 
 
 13 
 
 0.22495 
 
 0.22778 
 
 0.23062 
 
 0.23345 
 
 0.23627 
 
 0.23910 
 
 0.24192 
 
 76 
 
 14 
 
 0.24192 
 
 0.24474 
 
 0.24756 
 
 0.25038 
 
 0.25320 
 
 0.25601 
 
 0.25882 
 
 75 
 
 15 
 
 0.25882 
 
 0.26163 
 
 0.26443 
 
 0.26724 
 
 0.27004 
 
 0.27284 
 
 0.27564 
 
 74 
 
 16 
 
 0.27564 
 
 0.27843 
 
 0.28123 
 
 0.28402 
 
 0.28680 
 
 0.28959 
 
 0.29237 
 
 73 
 
 17 
 
 0.29237 
 
 0.29515 
 
 0.29793 
 
 0.30071 
 
 0.30348 
 
 0.30625 
 
 0.30902 
 
 72 
 
 18 
 
 0.30902 
 
 0.31178 
 
 0.81454 
 
 0.31730 
 
 0.32006 
 
 0.32282 
 
 0.32557 
 
 71 
 
 19 
 
 0.32557 
 
 0.32832 
 
 0.33106 
 
 0.33381 
 
 0.33655 
 
 0.33929 
 
 0.34202 
 
 70 
 
 20 
 
 0.34202 
 
 0.34475 
 
 0.34748 
 
 0.35021 
 
 0.35293 
 
 0.35565 
 
 0.358-37 
 
 69 
 
 21 
 
 0.35837 
 
 0.36108 
 
 0.36379 
 
 0.36650 
 
 0.36921 
 
 0.37191 
 
 0.37461 
 
 68 
 
 22 
 
 0.37461 
 
 0.37730 
 
 0.37999 
 
 0.38268 
 
 0.38537 
 
 0.38805 
 
 0.39073 
 
 67 
 
 23 
 
 0.39073 
 
 0.39341 
 
 0.39608 
 
 0.39875 
 
 0.40142 
 
 0.40408 
 
 0.40674 
 
 66 
 
 24 
 
 0.40674 
 
 0.40939 
 
 0.41204 
 
 0.41469 
 
 0.41734 
 
 0.41998 
 
 0.42262 
 
 65 
 
 25 
 
 0.42262 
 
 0.42525 
 
 0.42788 
 
 0.43051 
 
 0.43318 
 
 0.43575 
 
 0.43837 
 
 64 
 
 26 
 
 0.43837 
 
 0.44098 
 
 0.44359 
 
 0.44620 
 
 0.44880 
 
 0.45140 
 
 0,45399 
 
 63 
 
 27 
 
 0.45399 
 
 0.45658 
 
 0.45917 
 
 0.46175 
 
 0.46433 
 
 0.46690 
 
 0.46947 
 
 62 
 
 28 
 
 0.46947 
 
 0.47204 
 
 0.47460 
 
 0.47716 
 
 0.47971 
 
 0.48226 
 
 0.48481 
 
 61 
 
 29 
 
 0.48481 
 
 0.48735 
 
 0.48989 
 
 0.49242 
 
 0.49495 
 
 0.49748 
 
 0.50000 
 
 60 
 
 30 
 
 0.50000 
 
 0.50252 
 
 0.50503 
 
 0.50754 
 
 0.51004 
 
 0.51254 
 
 0.51504 
 
 59 
 
 31 
 
 0.51504 
 
 0.51753 
 
 0.52002 
 
 0.52250 
 
 0.52498 
 
 0.52745 
 
 0.52992 
 
 58 
 
 32 
 
 0.52992 
 
 0.53238 
 
 0.53484 
 
 0.53730 
 
 0.53975 
 
 0.54220 
 
 0.544C4 
 
 57 
 
 33 
 
 0.54464 
 
 0.54708 
 
 0.54951 
 
 0.55197 
 
 0.55436 
 
 0.55678 
 
 0.55919 
 
 56 
 
 34 
 
 0.55919 
 
 0.56160 
 
 0.56401 
 
 0.56641 
 
 0.56880 
 
 0.57119 
 
 0.57358 
 
 55 
 
 85 
 
 0,57358 
 
 0.57596 
 
 0.57833 
 
 0.58070 
 
 0.58307 
 
 0.58543 
 
 0.58779 
 
 54 
 
 36 
 
 0.58779 
 
 0.59014 
 
 0.59248 
 
 0.59482 
 
 0.59716 
 
 0.599i9 
 
 0.60182 
 
 53 
 
 37 
 
 0.60182 
 
 0.60114 
 
 0.60645 
 
 0.60876 
 
 0.61107 
 
 0.61337 
 
 0.61566 
 
 52 
 
 38 
 
 0.61566 
 
 0.61795 
 
 0.62024 
 
 0.62251 
 
 O.G2479 
 
 0.62706 
 
 0.62932 
 
 51 
 
 89 
 
 0.62932 
 
 0.63158 
 
 0.63383 
 
 0.63608 
 
 0.63832 
 
 0.64056 
 
 0.64279 
 
 50 
 
 40 
 
 0.64279 
 
 0.64501 
 
 0.64723 
 
 0.64945 
 
 0.65166 
 
 0.65386 
 
 0.65606 
 
 49 
 
 41 
 
 0.65606 
 
 0.65825 
 
 0.66044 
 
 0.68262 
 
 0.66480 
 
 0.66697 
 
 0.66913 
 
 48 
 
 42 
 
 0.66913 
 
 0.67129 
 
 0.67344 
 
 0.67559 
 
 0.67773 
 
 O.G7987 
 
 0.68200 
 
 47 
 
 43 
 
 0.68200 
 
 0.68412 
 
 0.68624 
 
 0.68835 
 
 0.69046 
 
 0.69256 
 
 0.69466 
 
 46 
 
 44 
 
 0.69466 
 
 0.69675 
 
 0.69883 
 
 0.70091 
 
 0.70298 
 
 0.70505 
 
 0.70711 
 
 45 
 
 
 e<y 
 
 60' 
 
 40' 
 
 3O' 
 
 2O' 
 
 1O' 
 
 <y 
 
 i 
 
 
 COSINE 
 
 1 
 
TRIGONOMETRIC TABLES 
 
 
 COSINE 
 
 
 
 <y 
 
 icy 
 
 2O* 
 
 SCK 
 
 40* 
 
 60> 
 
 Qfy 
 
 
 
 
 1.00000 
 
 1.00000 
 
 0.99998 
 
 0.99996 
 
 0.99993 
 
 0.99989 
 
 0.9&985 
 
 89 
 
 1 
 
 0.99985 
 
 0.99979 
 
 0.99973 
 
 0.99966 
 
 0.99958 
 
 0.99949 
 
 0,99939 
 
 88 
 
 2 
 
 0.99939 
 
 0.99929 
 
 0.99917 
 
 0.99905 
 
 0.99892 
 
 0.99878 
 
 0.99863 
 
 87 
 
 3 
 
 0.99863 
 
 0.99847 
 
 0.99831 
 
 0.99813 
 
 0.99795 
 
 0.99776 
 
 0.99756 
 
 86 
 
 4 
 
 0.99756 
 
 0.99736 
 
 0.99714 
 
 0.99692 
 
 0.99668 
 
 0.99644 
 
 0.99619 
 
 85 
 
 5 
 
 0.99619 
 
 0.99594 
 
 0.99567 
 
 0.99540 
 
 0.99511 
 
 0.99482 
 
 0.99452 
 
 84 
 
 6 
 
 0.99152 
 
 0.99421 
 
 0.99390 
 
 0.99357 
 
 0.99324 
 
 0.99290 
 
 0.99255 
 
 83 
 
 7 
 
 0.99255 
 
 0.99219 
 
 0.99182 
 
 0.99144 
 
 0.99106 
 
 0.99067 
 
 0.99027 
 
 82 
 
 8 
 
 0.9P027 
 
 0.98986 
 
 0.98944 
 
 0.98902 
 
 0.98858 
 
 0.98814 
 
 0.98769 
 
 81 
 
 9 
 
 0.98769 
 
 0.98723 
 
 0.98676 
 
 0.98629 
 
 0.98580 
 
 0.98531 
 
 0.98481 
 
 80 
 
 10 
 
 0.98481 
 
 0.98430 
 
 0.98378 
 
 0.98325 
 
 0.98272 
 
 6.98218 
 
 0.98163 
 
 79 
 
 11 
 
 0.98163 
 
 0.98107 
 
 0.98050 
 
 0.97992 
 
 0.97934 
 
 0.97875 
 
 0.97K15 
 
 78- 
 
 12 
 
 0.97815 
 
 0.97754 
 
 0.97692 
 
 0.97630 
 
 0.97566 
 
 0.97502 
 
 0.97437 
 
 77 
 
 13 
 
 0.97437 
 
 0.97371 
 
 0.97604 
 
 0.97237 
 
 0.97169 
 
 0.97100 
 
 0.97030 
 
 76 
 
 14 
 
 0.97030 
 
 0.96959 
 
 0.96887 
 
 0.96815 
 
 0.96742 
 
 0.96667 
 
 0.96593 
 
 75 
 
 15 
 
 0.96593 
 
 0.96517 
 
 0.96440 
 
 0.96363 
 
 0.96285 
 
 0.96206 
 
 0.96126 
 
 74 
 
 16 
 
 0.96126 
 
 0.96040 
 
 0.95964 
 
 0.95882 
 
 0.95799 
 
 0.95715 
 
 0.95630 
 
 7$ 
 
 17 
 
 0.95630 
 
 0.95545 
 
 0.95459 
 
 0.95372 
 
 0.95284 
 
 0.95195 
 
 0.95106 
 
 72 
 
 18 
 
 0.95106 
 
 0.95015 
 
 0.94924 
 
 0.94832 
 
 0.94740 
 
 0.94646 
 
 0.94552 
 
 71 
 
 19 
 
 0.94552 
 
 0.94457 
 
 0.94361 
 
 0.94264 
 
 0.94167 
 
 0.94068 
 
 0.93969 
 
 70 
 
 20 
 
 0.93969 
 
 0.93869 
 
 0.93769 
 
 0.93667 
 
 0.93565 
 
 0.93462 
 
 0.93358 
 
 69 
 
 21 
 
 0.93358 
 
 093253 
 
 0.93148 
 
 0.93042 
 
 0.92935 
 
 0.92827 
 
 0.92718 
 
 68 
 
 22 
 
 0.92718 
 
 0.92609 
 
 0.92499 
 
 0.92388 
 
 0.92276 
 
 0.92164 
 
 0.92050 
 
 67 
 
 23 
 
 0.92050 
 
 0.91936 
 
 0.91822 
 
 0.91706 
 
 0.91590 
 
 jO.91472 
 
 0.91355 
 
 66 
 
 24 
 
 0.91355 
 
 0.91236 
 
 0.91116 
 
 0.90996 
 
 0.90875 
 
 0.90753 
 
 0.90631 
 
 65 
 
 25 
 
 0.90631 
 
 0.90507 
 
 0.90383 
 
 0.90259 
 
 0.90133 
 
 0.90007 
 
 0.89879 
 
 64 
 
 26 
 
 0.89879 
 
 0.89752 
 
 0.89623 
 
 0.89493 
 
 0.89863 
 
 0.89232 
 
 0.89101 
 
 63 
 
 27 
 
 0.89101 
 
 0.88968 
 
 0.88835 
 
 0.88701 
 
 0.88566 
 
 0.88431 
 
 0.88295 
 
 62 
 
 28 
 
 0.88295 
 
 0.88158 
 
 0.88020 
 
 0.87882 
 
 0.87743 
 
 0.87603 
 
 0.87462 
 
 61 
 
 29 
 
 0.87462 
 
 0.87321 
 
 0.87178 
 
 0.87036 
 
 0.86892 
 
 0.86748 
 
 0.86603 
 
 60 
 
 30 
 
 0.86603 
 
 0.86457 
 
 0.86310 
 
 0.86163 
 
 0.86015 
 
 0.85866 
 
 0.85717 
 
 69 
 
 31 
 
 0.85717 
 
 0.85567 
 
 0.85416 
 
 0.85264 
 
 0.85112 
 
 0.84959 
 
 0.84805 
 
 58 
 
 32 
 
 0.84805 
 
 0.84650 
 
 0.84495 
 
 0.84339 
 
 0.84182 
 
 0.84025 
 
 0.83867 
 
 57 
 
 33 
 
 0.83867 
 
 0.83708 
 
 0.83549 
 
 0.83389 
 
 0.83228 
 
 0.83066 
 
 0.82904 
 
 56 
 
 34 
 
 0.82904 
 
 0.82741 
 
 0.82577 
 
 0.82413 
 
 0.82248 
 
 0.82082 
 
 0.81915 
 
 55 
 
 35 
 
 0.81915 
 
 0.81748 
 
 0.81580 
 
 0.81412 
 
 0.81242 
 
 0.81072 
 
 0.80902 
 
 54 
 
 36 
 
 0.80902 
 
 0.80730 
 
 0.80558 
 
 0.80386 
 
 0.80212 
 
 0.80038 
 
 0.79864 
 
 53 
 
 37 
 
 0.79864 
 
 0.79688 
 
 0.79512 
 
 0.79335 
 
 0.79158 
 
 0.78980 
 
 0.78801 
 
 52 
 
 38 
 
 0.78801 
 
 0.78622 
 
 0.78442 
 
 0.78261 
 
 0.78079 
 
 0.77897 
 
 0.77715 
 
 51 
 
 39 
 
 0.77715 
 
 0.77531 
 
 0.77347 
 
 0.77162 
 
 0.76977 
 
 0.76791 
 
 0.76604 
 
 60 
 
 40 
 
 0.76604 
 
 0.76417 
 
 0.76229 
 
 0.76041 
 
 0.75851 
 
 0.75661 
 
 0.75471 
 
 49 
 
 41 
 
 0.75171 
 
 0.75280 
 
 0.75088 
 
 0.74896 
 
 0.74703 
 
 0.74509 
 
 0.74314 
 
 48 
 
 42 
 
 0.74314 
 
 0.74120 
 
 0.73924 
 
 0.73728 
 
 0.73351 
 
 0.73333 
 
 0.73135 
 
 47 
 
 43 
 
 0.73185 
 
 0.72937 
 
 0.72737 
 
 0.72587 
 
 0.7*337 
 
 0.72136 
 
 0.71934 
 
 46 
 
 44 
 
 0.71934 
 
 0.71732 
 
 0.71529 
 
 0.71325 
 
 0.71121 
 
 0.70916 
 
 0.70711 
 
 45 
 
 
 6O' 
 
 6(X 
 
 4CX | 30' 
 
 20' 
 
 1O' 
 
 0' 
 
 f' 
 
 
 SINE 
 
 
TRIGONOMETRIC TABLES 
 
 1 
 
 TANGENT 
 
 
 f 
 
 0' 
 
 10' 
 
 20' 
 
 30' 
 
 40' 
 
 5O' 
 
 60' 
 
 
 
 
 0.00000 
 
 0.00291 
 
 .0.00582 
 
 O.OC873 
 
 0.01164 
 
 0.01455 
 
 0.01746 
 
 89 
 
 1 
 
 0.01746 
 
 0.02036 
 
 0.02328 
 
 0.02619 
 
 0.02910 
 
 0.03201 
 
 0.03492 
 
 88 
 
 2 
 
 0.03492 
 
 0.03783 
 
 0.04075 
 
 0.04366 
 
 0.04658 
 
 0.04949 
 
 0.05241 
 
 87 
 
 s 
 
 0.05241 
 
 0.05538 
 
 0.05824 
 
 0.06116 
 
 0.06408 
 
 0.06700 
 
 0.06993 
 
 86 
 
 4 
 
 0.06993 
 
 0.07285 
 
 0.07578 
 
 0.07870 
 
 0.08163 
 
 0.08456 
 
 0.08749 
 
 85 
 
 5 
 
 0.08749 
 
 0.09042 
 
 0.09335 
 
 0.09629 
 
 0.09923 
 
 0.10216 
 
 0.10510 
 
 84 
 
 6 
 
 0.10510 
 
 0.10805 
 
 0.11099 
 
 0.11394 
 
 0.11688- 
 
 0.11983 
 
 0.12278 
 
 83 
 
 7 
 
 0.12278 
 
 0.12574 
 
 0.12869 
 
 0.13165 
 
 0.13461 
 
 0.13758 
 
 0.14054 
 
 82 
 
 8 
 
 0.14054 
 
 0.14351 
 
 0.14048 
 
 0.14945 
 
 0.15243 
 
 0.15540 
 
 0.15838 
 
 81 
 
 9 
 
 0. 15.838 
 
 0.16J37 
 
 0.16435 
 
 0.16734 
 
 0.17033 
 
 0.17333 
 
 0.17633 
 
 80 
 
 10 
 
 0.17633 
 
 0.17933 
 
 0.18238 
 
 0.1S534 
 
 0.18835 
 
 0.19136 
 
 0.19438 
 
 7<> 
 
 il- 
 
 0.19438 
 
 0.19740 
 
 0.20042 
 
 0.20345 
 
 0.20848 
 
 0.20952 
 
 0.21256 
 
 78 
 
 ls 
 
 0.21256 
 
 0.21560 
 
 0.21864 
 
 0.22169 
 
 0.22475 
 
 0.22781 
 
 0.23087 
 
 77 
 
 13 
 
 0.23087 
 
 0.23393 
 
 0.23700 
 
 0.24008 
 
 0.24316 
 
 0.24G24 
 
 0.24933 
 
 76 
 
 14 
 
 0.24933 
 
 0.25242 
 
 0.25552 
 
 0.25862 
 
 0.26172 
 
 0.26483 
 
 0.26795 
 
 75 
 
 15 
 
 0.26795 
 
 0.27107 
 
 0.27419 
 
 0.27732 
 
 0.28016 
 
 0.28360 
 
 0.28675 
 
 74 
 
 16 
 
 0.28675 
 
 ,0.28990 
 
 0.29305 
 
 0.29621 
 
 0.29938 
 
 0.30255 
 
 0.30578 
 
 78 
 
 17 
 
 0.30573 
 
 0.30891 
 
 0.31210 
 
 0.31530 
 
 0.31850 
 
 0.32171 
 
 0.32492 
 
 72 
 
 18 
 
 0.32492 
 
 0.32814 
 
 0.33136 
 
 0.33460 
 
 0.33783 
 
 0.34108 
 
 0.3443S 
 
 71 
 
 19 
 
 0.34433 
 
 0.34758 
 
 0.35085 
 
 0.35412 
 
 0.35740 
 
 0.86068 
 
 0.36397 
 
 70 
 
 SO 
 
 0.36397 
 
 0.36727 
 
 0.37057 
 
 0.37388 
 
 0.37720 
 
 0.38053 
 
 0.38386 
 
 69 
 
 21 
 
 0.38386 
 
 0.38721 
 
 0.39055 
 
 0.39391 
 
 0.39727 
 
 0.40065 
 
 0.40403 
 
 68 
 
 22 
 
 0.40403 
 
 0.40741 
 
 0.41081 
 
 0.41421 
 
 0.41763 
 
 0.42105 
 
 0.42447 
 
 67 
 
 23 
 
 0.42447 
 
 0.42791 
 
 0.43136 
 
 0.43481 
 
 0.43828 
 
 0.44175 
 
 0.44523 
 
 66 
 
 24 
 
 0.44523 
 
 0.44872 
 
 0.45222 
 
 0.45578 
 
 0.45924 
 
 0.46277 
 
 0.46631 
 
 65 
 
 25 
 
 0.48631 
 
 0.46985 
 
 0.47341 
 
 0.47698 
 
 0.48055 
 
 0.48414 
 
 0.48773 
 
 64 
 
 26 
 
 0.48773 
 
 0.49134' 
 
 0.49495 
 
 0.49858 
 
 0.50222 
 
 0.50587 
 
 0.50953 
 
 63 
 
 27 
 
 0.50953 
 
 0.51820 
 
 0.51688 
 
 0.52057 
 
 0.52427 
 
 0.52798 
 
 0.53171 
 
 62 
 
 28 
 
 0.53171 
 
 0.53545 
 
 0.58920 
 
 0.54296 
 
 0.54674 
 
 0.55051 
 
 0.55431 
 
 61 
 
 29 
 
 0.55481 
 
 0.55812 
 
 0.56194 
 
 0.56577 
 
 0.56962 
 
 0.57348 
 
 0.57785 
 
 60 
 
 80 
 
 0.57735 
 
 0.58124 
 
 0.58518 
 
 0.58905 
 
 0.59297 
 
 0.59691 
 
 0.60086 
 
 59 
 
 81 
 
 0.60086 
 
 0.60483 
 
 0.60881 
 
 0.61280 
 
 0.61681 
 
 Ot 62083 
 
 0.62487 
 
 58 
 
 82 
 
 0.62487 
 
 0.62892 
 
 0.68299 
 
 0.63707 
 
 0.64117 
 
 0.64528 
 
 0.64941 
 
 57 
 
 33 
 
 0.64941 
 
 0.65355 
 
 0.65771 
 
 0.66189 
 
 0.66608 
 
 0.67028 
 
 0.67451 
 
 56 
 
 84 
 
 0.67451 
 
 0.67875 
 
 0.68301- 
 
 0.68728 
 
 0.69157 
 
 0.69588 
 
 0.70021 
 
 55 
 
 85 
 
 0.70021 
 
 0.70455 
 
 0.70891 
 
 0.718S9 
 
 0.71769 
 
 0.72211 
 
 0.72654 
 
 54 
 
 86 
 
 0.72654 
 
 0.73100 
 
 0.73547 
 
 0.73996 
 
 0.74447 
 
 0.74900 
 
 0.75355 
 
 53 
 
 87 
 
 0.75355 
 
 0.75812 
 
 0.76272 
 
 0.76738 
 
 0.77196 
 
 0.77661 
 
 0.78129 
 
 52 
 
 88 
 
 0.78129 
 
 0.78598 
 
 0.79070 
 
 0.79544 
 
 0.80020 
 
 0.80498 
 
 0.80978 
 
 51 
 
 89 
 
 0.80978 
 
 0.81461 
 
 0.81946 
 
 0.82434 
 
 0.82923 
 
 0.83415 
 
 0.83910 
 
 50 
 
 40 
 
 0.88910 
 
 0.84407 
 
 0.84906 
 
 0.85408 
 
 0.85912 
 
 0.86419 
 
 0.86929 
 
 49 
 
 41 
 
 0.86929 
 
 0.87441 
 
 0.87955 
 
 0.88473 
 
 0.88992 
 
 0.89515 
 
 0.90040 
 
 48 
 
 43 
 
 0.90040 
 
 0.90569 
 
 0.91099 
 
 0.91638 
 
 0.92170 
 
 0.92709 
 
 0.93252 
 
 47 
 
 43 
 
 0.98252 
 
 0.98797 
 
 0.94845 
 
 0.94896 
 
 0.95451 
 
 0.96008 
 
 0.96569 
 
 45 
 
 44 
 
 0.96569 
 
 0.97183 
 
 0.97700 
 
 0.98270 
 
 0.98848 
 
 0.99420 
 
 1.00000 
 
 45 
 
 
 e<y 
 
 50> 
 
 4cx 
 
 sex 
 
 2O' 
 
 icy 
 
 O 7 
 
 
 
 COTiHGSHT 
 
 
TRIGONOMETRIC TABLES 
 
 
 COTANGENT 
 
 
 
 O' 
 
 10' 
 
 20' 
 
 3O' 
 
 40' 
 
 50' 
 
 6O' 
 
 
 o 
 
 00 
 
 343.77371 
 
 171.88540 
 
 114.58865 
 
 85.93979 
 
 68.75009 
 
 57.28996 
 
 89 
 
 1 
 
 57.28996 
 
 49.10388 
 
 42.96408 
 
 38.18846 
 
 84.86777 
 
 81.2415828.68625 
 
 88 
 
 2 
 
 28.63625 
 
 26.43160 
 
 24.54176 
 
 22.903-77 
 
 21.47040 
 
 20.20555 
 
 19.08114 
 
 87 
 
 8 
 
 19.08114 
 
 18.07498 
 
 17.16934 
 
 16.34980 
 
 15.C0478 
 
 14.92442 
 
 14.30067 
 
 86 
 
 4 
 
 14.3006? 
 
 18.72674 
 
 13.19683 
 
 12.70621 
 
 12.25051 
 
 11.82617 
 
 11.48005 
 
 85 
 
 5 
 
 11.43005 
 
 11.05943 
 
 10.71191 
 
 10.88540 
 
 10.07803 
 
 9.78817 
 
 9.51436 
 
 84 
 
 6 
 
 9.51436 
 
 9.25530 
 
 9.00983 
 
 8.77689 
 
 8.55555 
 
 8.84496 
 
 8.14435 
 
 83 
 
 7 
 
 8.14435 
 
 7.95302 
 
 7.77035 
 
 7.59575 
 
 7.42871 
 
 7.26873 
 
 7.11537 
 
 82 
 
 8 
 
 7.11537 
 
 6.93823 
 
 6.82694 
 
 6.69116 
 
 6.56055 
 
 6.48484 
 
 6.81375 
 
 81 
 
 9 
 
 6.31375 
 
 6.19703 
 
 6,08444 
 
 5.97576 
 
 5.87080 
 
 5.76987 
 
 5.67128 
 
 80 
 
 10 
 
 5.67128 
 
 5.57638 
 
 5.48451 
 
 5.39552 
 
 5.30928 
 
 5.22566 
 
 5.14455 
 
 79 
 
 11 
 
 5.14455 
 
 5.05584 
 
 4.98940 
 
 4.91516 
 
 4.84300 
 
 4.77286 
 
 4.70463 
 
 78 
 
 12 
 
 4.70463 
 
 4.63825 
 
 4.57863 
 
 4.51071 
 
 4.44942 
 
 4.88969 
 
 4.38148 
 
 77 
 
 18 
 
 4.83148 
 
 4.27471 
 
 4.21933 
 
 4.16530 
 
 4.11256 
 
 4.06107 
 
 4.01078 
 
 76 
 
 14 
 
 4.01078 
 
 3.96165 
 
 3.91364 
 
 3.86671 
 
 8.82083 
 
 3.77595 
 
 3.73205 
 
 75 
 
 15 
 
 3.73205 
 
 8.68909 
 
 8.64705 
 
 8.60588 
 
 8.56557 
 
 8.52609 
 
 3.48741 
 
 74 
 
 16 
 
 3.48741 
 
 3.44951 
 
 3.41236 
 
 3.37594 
 
 3.34023 
 
 8.30521 
 
 8.27085 
 
 78 
 
 17 
 
 3.27085 
 
 3.23714 
 
 3.20406 
 
 3.17159 
 
 3.18972 
 
 8.10842 
 
 8.07768 
 
 72 
 
 18 
 
 3.07768 
 
 3.04749 
 
 8.01783 
 
 2.98869 
 
 2.96004 
 
 2.93189 
 
 2.90421 
 
 71 
 
 19 
 
 2.90421 
 
 2.87700 
 
 2.85023 
 
 2.82391 
 
 2.79802 
 
 2,77254 
 
 2.74748 
 
 70 
 
 20 
 
 2.74748 
 
 2.72281 
 
 2.69853 
 
 2.67462 
 
 2.65109 
 
 2.62791 
 
 2.60509 
 
 69 
 
 21 
 
 2.60509 
 
 2.58261 
 
 2.56046 
 
 2.53865 
 
 2.51715 
 
 2.49597 
 
 2.47509 
 
 68 
 
 22 
 
 2.47509 
 
 2.45451 
 
 2.43422 
 
 2.41421 
 
 2.39449 
 
 2.37504 
 
 2.35585 
 
 67 
 
 23 
 
 2.35585 
 
 2.33693 
 
 2.81826 
 
 2.29984 
 
 2.28167 
 
 2.26374 
 
 2.24604 
 
 66 
 
 24 
 
 2.24604 
 
 2.2285? 
 
 2.21182 
 
 2.19430 
 
 2.17749 
 
 2.16090 
 
 2.14451 
 
 65 
 
 25 
 
 2.14451 
 
 2.12832 
 
 2.11233 
 
 2.09654 
 
 2.08094 
 
 2.06553 
 
 2.05030 
 
 64 
 
 28 
 
 2.05030 
 
 2.03525 
 
 2.02039 
 
 2.00569 
 
 1.99116 
 
 1.97680 
 
 1.96261 
 
 68 
 
 87 
 
 .98261 
 
 1.94858 
 
 1.93470 
 
 1.92098 
 
 1.90741 
 
 1.89400 
 
 1.88078 
 
 62 
 
 28 
 
 .88073 
 
 1.86760 
 
 1.85462 
 
 1.84177 
 
 1.82907 
 
 1.81649 
 
 1.80405 
 
 61 
 
 29 
 
 .80405 
 
 1.79174 
 
 1.77955 
 
 1.76749 
 
 1.75556 
 
 1.74375 
 
 1.73205 
 
 60 
 
 80 
 
 .73205 
 
 1.72047 
 
 1.70901 
 
 1.69766 
 
 1.68643 
 
 1.67580 
 
 1.66428 
 
 59 
 
 SI 
 
 .66428 
 
 1.65337 
 
 1.64256 
 
 i.63i as 
 
 1.62125 
 
 1.61074 
 
 1.60033 
 
 58 
 
 82 
 
 .60033 
 
 1.59002 
 
 1.57981 
 
 1.56969 
 
 1.55966 
 
 1.54972 
 
 1.58987 
 
 57 
 
 83 
 
 .53987 
 
 1.53010 
 
 1.52043 
 
 1.51084 
 
 1.50138 
 
 1.49190 
 
 1.48256 
 
 50 
 
 34 
 
 .48256 
 
 1.47330 
 
 1.46411 
 
 1.45501 
 
 1.44598 
 
 1.48703 
 
 1.42815 
 
 55 
 
 35 
 
 1.42815 
 
 1 41934 
 
 1.41061 
 
 1.40195 
 
 1.39336 
 
 .88484 
 
 1.87638 
 
 54 
 
 86 
 
 1.87638 
 
 1.36800 
 
 1.35968 
 
 1.35142 
 
 1.84323 
 
 .88511 
 
 1.82704 
 
 58 
 
 37 
 
 1.32704 
 
 1.31904 
 
 1.31110 
 
 1.30323 
 
 1.29541 
 
 .28764 
 
 .27994 
 
 52 
 
 38 
 
 1.27994 
 
 1.27230 
 
 1.23471 
 
 1.25717 
 
 1.24969 
 
 .24227 
 
 .23490 
 
 51 
 
 39 
 
 1.23490 
 
 1.22758 
 
 1.22031 
 
 1.21310 
 
 1.20593 
 
 .19882 
 
 .19175 
 
 50 
 
 40 
 
 1.19175 
 
 1.18474 
 
 .17777 
 
 1.17085 
 
 1.16398 
 
 .15715 
 
 .15037 
 
 49 
 
 41 
 
 1.15037 
 
 1.14368 
 
 .13694 
 
 1.13029 
 
 1.12369 
 
 .11713 
 
 .11061 
 
 48 
 
 42 
 
 1.11081 
 
 1.10414 
 
 .09770 
 
 1.09181 
 
 1.08496 
 
 1.07864 
 
 .07237 
 
 47 
 
 43 
 
 1.07237 
 
 1.06613 
 
 .05994 
 
 1.05378 
 
 1.04766 
 
 1.04158 
 
 1.08553 
 
 46 
 
 44 
 
 1.03553 
 
 1.02952 
 
 .02355 
 
 1.01761 
 
 1.01170 
 
 1.00583 
 
 1.00000 
 
 45 
 
 
 60' 
 
 50' 
 
 40' 
 
 3O' 
 
 2<y icy 
 
 O / 
 
 
 
 T4NGSNT 
 
 
TRIGONOMETRIC TABLES 
 
 2 
 
 
 
 
 
 SECAMS 
 
 
 
 
 
 H 
 I 
 
 0' 
 
 10' 
 
 20' 
 
 3O' 
 
 40' 
 
 6CK 
 
 6O' 
 
 
 
 
 1.00000 
 
 1.00001 
 
 1.00002 
 
 1.00004 
 
 1.00007 
 
 1.00011 
 
 1.00015 
 
 89 
 
 1 
 
 1.0C015 
 
 1.00021 
 
 1.00027 
 
 1.00034 
 
 1.00042 
 
 1.00051 
 
 1.00061 
 
 88 
 
 a 
 
 1.00061 
 
 1.00072 
 
 1.00083 
 
 1.00095 
 
 1.00108 
 
 1.00122 
 
 1.00137 
 
 87 
 
 3 
 
 1.00137 
 
 .00153 
 
 1.00169 
 
 1.00187 
 
 1.00205 
 
 1.00224 
 
 1.00244 
 
 86 
 
 4 
 
 1.00244 
 
 .00265 
 
 1.00287 
 
 1.00309 
 
 1.00333 
 
 1.00357 
 
 1.00382 
 
 85 
 
 5 
 
 1.00382 
 
 .00408 
 
 1.00435 
 
 1.00463 
 
 1.00491 
 
 1.00521 
 
 1.00551 
 
 84 
 
 6 
 
 1.00551 
 
 .00582 
 
 .00(514 
 
 1.00647 
 
 1.006S1 
 
 1.00715 
 
 1.00751 
 
 83 
 
 7 
 
 1.00751 
 
 .00787 
 
 .00825 
 
 1.00863 
 
 1.00W2 
 
 1.00942 
 
 1.00983 
 
 82 
 
 8 
 
 1.00983 
 
 .01024 
 
 .01067 
 
 1.01111 
 
 1.01155 
 
 1.01200 
 
 1.01247 
 
 81 
 
 9 
 
 1.01247 
 
 .01294 
 
 .01342 
 
 1.01391 
 
 1.01440 
 
 1.01491 
 
 1.01543 
 
 80 
 
 10 
 
 1.01543 
 
 1.01595 
 
 .01649 
 
 1.01703 
 
 1.01758 
 
 1.01815 
 
 1.01872 
 
 79 
 
 11 
 
 1.01872 
 
 1.00930 
 
 .01989 
 
 1.02049 
 
 1.02110 
 
 1.02171 
 
 1.02234 
 
 78 
 
 12 
 
 1.02284 
 
 1.02298 
 
 .02362 
 
 1.02428 
 
 1.02494 
 
 1 .02562 
 
 1.02630 
 
 77 
 
 13 
 
 1.02630 
 
 1.02700 
 
 .02770 
 
 1.02842 
 
 1.02914 
 
 1.02987 
 
 1.03061 
 
 76 
 
 14 
 
 1.03061 
 
 1.03137 
 
 1.03213 
 
 1.03290 
 
 1.03368 
 
 1.03447 
 
 1.03528 
 
 75 
 
 15 
 
 1.03528 
 
 1.03609 
 
 1.03691 
 
 1.03774 
 
 1.03858 
 
 1.03944 
 
 1.04080 
 
 74 
 
 16 
 
 1.04030 
 
 1.04117 
 
 1.04206 
 
 1.04295 
 
 1.04385 
 
 1.04477 
 
 1.04569 
 
 73 
 
 17 
 
 1.04569 
 
 1.04663 
 
 1.04757 
 
 1.04853 
 
 1.04950 
 
 1.05047 
 
 1.05146 
 
 72 
 
 18 
 
 1.05146 
 
 1.05246 
 
 1.05347 
 
 1.05449 
 
 1.05552 
 
 1.05657 
 
 1.05762 
 
 71 
 
 19 
 
 1.05762 
 
 1.05869 
 
 1.05976 
 
 1.06085 
 
 1.06195 
 
 1.06306 
 
 1.06418 
 
 70 
 
 20 
 
 1.06418 
 
 1,06531 
 
 1.06645 
 
 1.06761 
 
 1.06878 
 
 1.06995 
 
 1.07115 
 
 69 
 
 21 
 
 1.07115 
 
 1.07235 
 
 1.07356 
 
 1.07479 
 
 1.07602 
 
 1.07727 
 
 1.07853 
 
 68 
 
 22 
 
 1.07858 
 
 1.07981 
 
 1.08109 
 
 1.08239 
 
 1.08370 
 
 1.08503 
 
 1.08636 
 
 67 
 
 23 
 
 1.08686 
 
 1.03771 
 
 .08907 
 
 1.09044 
 
 1.09183 
 
 1.06323 
 
 1.09464 
 
 66 
 
 24, 
 
 1.09464 
 
 1.09606 
 
 .09750 
 
 1.09895 
 
 1.10041 
 
 1.10189 
 
 1.10338 
 
 65 
 
 25 
 
 1.10338 
 
 1.10488 
 
 .10640 
 
 1.10793 
 
 1.10947 
 
 1.11103 
 
 1.11260 
 
 64 
 
 2a 
 
 1.11260 
 
 1.11419 
 
 .11579 
 
 1.11740 
 
 1.11903 
 
 1.12067 
 
 1.12283 
 
 63 
 
 27 
 
 1.12233 
 
 .12400 
 
 .12568 
 
 1.12738 
 
 1.12910 
 
 1.13083 
 
 1.13257 
 
 62 
 
 23 
 
 1.13257 
 
 .13433 
 
 .13610 
 
 1.13789 
 
 1:13970 
 
 1.14152 
 
 1.14385 
 
 61 
 
 29 
 
 1.14335 
 
 .14521 
 
 1.14707 
 
 1.14896 
 
 1.15085 
 
 1.15277 
 
 1.15470 
 
 60 
 
 30 
 
 1.15470 
 
 .15665 
 
 1.15861 
 
 1.16059 
 
 1.16259 
 
 1.16460 
 
 1.16668 
 
 59 
 
 31 
 
 1.16663 
 
 .16888 
 
 1.17075 
 
 1.17283 
 
 1.17498 
 
 1.17704 
 
 1.17918 
 
 58 
 
 82 
 
 1.17918 
 
 .18133 
 
 1.18350 
 
 1.18589 
 
 1.18790 
 
 1.19012 
 
 1.19286 
 
 57 
 
 33 
 
 1.19236 
 
 .19463 
 
 1.19891 
 
 1.19920 
 
 1.20152 
 
 1.20386 
 
 1.20622 
 
 56 
 
 54 
 
 1.20622 
 
 .20859 
 
 1.21099 
 
 1.21341 
 
 1.21584 
 
 1.21830 
 
 1.22077 
 
 55 
 
 35 
 
 1,22077 
 
 1.22327 
 
 1.32579 
 
 1.22883 
 
 1.23089 
 
 1.23347 
 
 1.23607 
 
 54 
 
 36 
 
 1.23607 
 
 1.23889 
 
 1.24134 
 
 1.24400 
 
 1.24669 
 
 1.24940 
 
 1.25214 
 
 53 
 
 37 
 
 1.25214 
 
 1.25489 
 
 1.25767 
 
 1.26047 
 
 1.26330 
 
 1.26615 
 
 1.26902 
 
 52 
 
 38 
 
 1.26902 
 
 1.27191 
 
 1.27463 
 
 1.27778 
 
 1.28075 
 
 1.28374 
 
 1.28676 
 
 51 
 
 39 
 
 1.28676 
 
 1.28980 
 
 1.29387 
 
 1.29597 
 
 1.29909 
 
 1.30223 
 
 1.80541 
 
 50 
 
 40 
 
 1.80541 
 
 1.3*3861 
 
 1*. 81183 
 
 1.81509 
 
 1.31837 
 
 1.82168 
 
 1.82501 
 
 49 
 
 41 
 
 1.32501 
 
 1.32838 
 
 1.33177 
 
 1.83519 
 
 1.33864 
 
 1.34212 
 
 1.84563 
 
 48 
 
 42 
 
 1.34563 
 
 1.84917 
 
 1.35274 
 
 1.85384 
 
 1.85997 
 
 1.36363 
 
 1.36733 
 
 47 
 
 43 
 
 1.36733 
 
 1.87105 
 
 1.37481 
 
 1.87860 
 
 1.88242 
 
 1.38628 
 
 1.89016 
 
 46 
 
 44 
 
 1.89016 
 
 1.89409 
 
 1.89804 
 
 1.40203 
 
 1*40608 
 
 1.41012 
 
 1.41421 
 
 45 
 
 
 6O' 
 
 6O' 
 
 4O' 
 
 SO' 
 
 2O* 
 
 10' 
 
 O 
 
 1 
 
 
 
 
 
 COSECANTS 
 
 
 
 
 S 
 
TRIGONOMETRIC TABLES 
 
 
 -7- ',:.. iJOSECMW ; 
 
 
 
 O' 
 
 1O' 
 
 20' 
 
 30' 
 
 40> 
 
 50' 
 
 60' 
 
 
 
 
 00 
 
 343.77516 
 
 171.88831 
 
 114.59301 
 
 85.94561 
 
 68.75736 
 
 57.29869 
 
 89 
 
 1 
 
 57.29869 
 
 49.11406 
 
 42.97571 
 
 38.20155 
 
 34.38232 
 
 31.25758 
 
 28.65371 
 
 88 
 
 2 
 
 28.65371 
 
 26.45051 
 
 24.56212 
 
 22.92559 
 
 21.49368 
 
 20.23028 
 
 19.10732 
 
 87 
 
 3 
 
 19.10732 
 
 18.10262 
 
 17.19843 
 
 16.38041 
 
 15.63679 
 
 14.95788 
 
 14.33559 
 
 86 
 
 4 
 
 14 .-33559 
 
 13.76312 
 
 13.23472 
 
 12.74550 
 
 12.29125 
 
 11.86837 
 
 11.47371 
 
 85 
 
 5 
 
 11.47371 
 
 31.10455 
 
 10.75849 
 
 10.43343 
 
 10. 12752 
 
 9.83912 
 
 9.56677 
 
 84 
 
 6 
 
 9.56677 
 
 9.30917 
 
 9.06515 
 
 8.83367 
 
 8.61379 
 
 8.46466 
 
 8.20551 
 
 83 
 
 7 
 
 8.20551 
 
 8.01565 
 
 7.83443 
 
 7.66130 
 
 7.49571 
 
 7.33719 
 
 7.18530 
 
 82 
 
 8 
 
 7.18530 
 
 7.03962 
 
 6.89979 
 
 6.76547 
 
 6.63633 
 
 6.51208 
 
 6.39245 
 
 81 
 
 9 
 
 6.39245 
 
 6.27719 
 
 6.16607 
 
 6.05886 
 
 , 5.95536 
 
 5.85539 
 
 5.75877 
 
 80 
 
 10 
 
 5.75877 
 
 5.66533 
 
 5.57493 
 
 5.48740 
 
 5.40263 
 
 5.32049 
 
 5.24084 
 
 79 
 
 11 
 
 5.24084 
 
 5.16359 
 
 5.08863 
 
 5.01585 
 
 4.94517 
 
 4.87649 
 
 4.80973 
 
 78 
 
 12 
 
 4.80973 
 
 4.74482 
 
 4.68167 
 
 4.62023 
 
 4.56041 
 
 4.50216 
 
 4.44541 
 
 77 
 
 13 
 
 4.44541 
 
 4.39012 
 
 4.33622 
 
 4.28366 
 
 4.23239 
 
 4.18238 
 
 4.13357 
 
 76 
 
 14 
 
 4.13357 
 
 4.08591 
 
 4.03938 
 
 3.99393 
 
 3.94952 
 
 3.90613 
 
 3.86370 
 
 75 
 
 15 
 
 3.86370 
 
 8.82223 
 
 3.78166 
 
 8.74198 
 
 8.70315 
 
 3.66515 
 
 3.62796 
 
 74 
 
 16 
 
 8.62796 
 
 3.59154 
 
 3.55587 
 
 3.52094 
 
 3.48671 
 
 3.45317 
 
 3.42030 
 
 73 
 
 17 
 
 3.42030 
 
 3.38808 
 
 3.35649 
 
 3.32551 
 
 3.29512 
 
 3.26531 
 
 3.23607 
 
 72 
 
 18 
 
 3.23607 
 
 3.20737 
 
 3.17920 
 
 3.15155 
 
 3.12440 
 
 3.09774 
 
 8.07155 
 
 71 
 
 19 
 
 3.07155 
 
 3.04584 
 
 3.02057 
 
 2.99574 
 
 2 97135 
 
 2.94737 
 
 2.92380 
 
 70 
 
 20 
 
 2.92380 
 
 2.90063 
 
 2.87785 
 
 2.85545 
 
 2.83342 
 
 2.81175 
 
 2.79043 
 
 69 
 
 21 
 
 2.79043 
 
 2.76945 
 
 2.74881 
 
 2.72850 
 
 2.70851 
 
 2.68884 
 
 2.66947 
 
 68 
 
 22 
 
 2.66947 
 
 2.65040 
 
 2.63162 
 
 2.61313 
 
 2.59491 
 
 2.57698 
 
 2.55930 
 
 67 
 
 23 
 
 2.55930 
 
 2.54190 
 
 2.52474 
 
 2.50784 
 
 2.49119 
 
 2.47477 
 
 2.45859 
 
 6ft 
 
 24 
 
 2.45859 
 
 2.44264 
 
 2.42692 
 
 2.41142 
 
 2.39614 
 
 2.38107 
 
 2.36620 
 
 65 
 
 25 
 
 2.36620 
 
 2.35154 
 
 2.33708 
 
 2.32282 
 
 2.30875 
 
 2.29487 
 
 2.28117 
 
 64 
 
 26 
 
 2.28117 
 
 2.26766 
 
 2.25432 
 
 2.24116 
 
 2.22817 
 
 2.21535 
 
 2.20269 
 
 63 
 
 27 
 
 2.20269 
 
 2.19019 
 
 2.17786 
 
 2.16568 
 
 2.15366 
 
 2.14178 
 
 2.13005 
 
 62 
 
 28 
 
 2.13005 
 
 2.11847 
 
 2.10704 
 
 2.09574 
 
 2.08458 
 
 2.07356 
 
 2.06267 
 
 61 
 
 29 
 
 2.05267 
 
 2.05191 
 
 2.04128 
 
 2.03077 
 
 2.02039 
 
 2.01014 
 
 2.00000 
 
 60 
 
 30 
 
 2.00000 
 
 1.98998 
 
 1.98008 
 
 1.97029 
 
 1.96062 
 
 1.95106 
 
 1.94160 
 
 59 
 
 31 
 
 .94160 
 
 1.93226 
 
 1.92302 
 
 1.91388 
 
 1.90485 
 
 1.89591 
 
 1.88709 
 
 58 
 
 32 
 
 .88708 
 
 1.87834 
 
 1.86990 
 
 1.86116 
 
 1.85271 
 
 1.84435 
 
 1.83608 
 
 57 
 
 33 
 
 .83608 
 
 1.82790 
 
 1.81981 
 
 1.81180 
 
 1.80388 
 
 1.79604 
 
 1.78829 
 
 56 
 
 34 
 
 .78829 
 
 1.78082 
 
 1.77303 
 
 1 76552 
 
 1.75808 
 
 1.75073 
 
 1.74345 
 
 55 
 
 35 
 
 .74345 
 
 1.73624 
 
 1.72911 
 
 1.V2205 
 
 1.71506 
 
 1.70815 
 
 1.70130 
 
 54 
 
 38 
 
 .70130 
 
 1.69452 
 
 1.68782 
 
 1.68117 
 
 1.67460 
 
 1.66809 
 
 1.66164 
 
 53 
 
 37 
 
 .66164 
 
 1.65526 
 
 1.64894 
 
 1.64268 
 
 1.03648 
 
 1.63035 
 
 1.62427 
 
 52 
 
 38 
 
 .62427 
 
 1.61825 
 
 1.61229 
 
 1.60839 
 
 1.60054 
 
 1.59475 
 
 1.58902 
 
 51 
 
 39 
 
 .58902 
 
 1.58333 
 
 1.57771 
 
 1.57213 
 
 1.56661 
 
 1.56114 
 
 1.55572 
 
 50 
 
 40 
 
 .55572 
 
 1.55036 
 
 1.54504 
 
 1.53977 
 
 1.53455 
 
 1.52938 
 
 1.52425 
 
 40 
 
 41 
 
 .52425 
 
 1.51918 
 
 1.51415 
 
 1.50916 
 
 1.50422 
 
 1.49933 
 
 1.49448 
 
 48 
 
 42 
 
 .49448 
 
 1.48967 
 
 1.48491 
 
 1.48019 
 
 1.47551 
 
 1.47087 
 
 1.46628 
 
 4? 
 
 43 
 
 .46628 
 
 1.46173 
 
 1.45721 
 
 1.45274 
 
 1.44^31 
 
 1.44391 
 
 1.43956 
 
 4'j 
 
 44 
 
 1.43956 
 
 1.43524 
 
 1.43096 
 
 1.42672 
 
 1.42251 
 
 1.41835 
 
 1.41421 
 
 45 
 
 
 60' 
 
 6O' 
 
 40' 
 
 SO' 
 
 20' 
 
 1O' 
 
 O / 
 
 1 
 g) 
 
 
 SECANTS 
 
 
 
YA 0686 
 
 **s >