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 .016871 58 1.012291 58 .96667 58 .01611 59 .000286 59 .017162 59 1.029744 59 .98333 59 .01639 60 000291 60 .017453 60 1.047198 60 1.00000 60 .01667 TRANSITION CURVE >o o <N io o o c co <M int- o <N o . to q q O O O O. ""I ^ ^ . Id d d >n <N 6 t^ <4 co* vd I 1-1 mCO <S <N M r-l M l-H M q o q i> In i> "b i> ~m i> o b- >o o rj o q q ,<N o q o rj O O' O M rH Cj CO* 4 IO* VO o o o o a oo F^ co a~ q o o q ON os a os oq idoiooto'aas a-* a Nnt> O<N ^t -<t a<N rj- qqqqi-t ^^'O^l M o oooooo aoo i> q So ^^ T *t * ^5 - *1 o q <N in i-j o\ r-n \q w q SOO OCSOOO M (N ff) q q q os q\ os a oq t^. ^ in d d d Tt? a -^ a -* a B. O iO O O O iO O C O iC O CN>mt>o(Mint> o<Nm THE SPIRAL 47 O II o *b ?H *b O r-H O CO C4 M ^H M P <N M 1-1 *b o> "in t ? H % + He "b <S b- ro ^- ^ co CO M rH -1 M * s u -HO^Hinr-tOi-iin Sfl m t> o 48 TRANSITION CURVE 1 1 5 O O O tO O*OO tO OO O POvOTT" O* J> NO in O 00 ~< o p ^ w ~ S o~ o o b CO CO CO 000 ^ C0 C^ ^ O M <s ro ^ m O q q q q ^ in co q oq q~~ o* in <N dr>^ ri eo p *N in ro\OO5 MCO PO1> Tf-^ in 1>00<N O\CO TtC^J MO Ci inSC< MTH M T-H HT-H q q q ino>o into oto in to o H\qoi "J 00 . '^' < -J J: 7' 1 ^ "*? ooo MrHNcopo^in _ o 000*00 t^to OO OOO>O5 O*Oi PO o'^to dtodvTj? o\ <& ^ q .^ . *"! 1. ^"^.^T ooo ooooooot>- ino't^ inc^i di> rtc^ o\ M POCO ^"tOvOCO 1>0000 ~1w in o oo oo M os <s os M q q^ *i v* 1"^Q O . "* r-l M rrjMO MC<II>00 l>iO m q pj -^ i> I-H \q co co 'vq . i i M o$ PO ^ in q qq qcs OAOS ooooo HOOtOO^O^O^fON pOvOI>" O\O MCO ^*CO 1^- vot^">-i^-i" ;< tco'inco"'<t POtO (StO fSO ^00 <N M ^-i co PO -* m b o b b co co co go ooo^o^o^o^o o M N <*j ^- in co OOOtO O>OOtoOtO O POOI> o\o <sco mcooo THE SPIKAL 49 jj o c oo u M t- \ 8 c 4 H r- 1 a K > . vo 5 o * *5 < PC 1 U > c H > x ii o O || o 5 o i ^ O H * N Jj 10 vO VO U ) 00 fr < Tt C< 1 O CC 5 M IT O O o 10 ?J O V r* ~"* fO N (> 1 N r- * ll O T I vO i "10 r^ > Tf C> i ? to (S ^ "in (S 1 & M ^ + O r 00 r ) M O O fs 5 * 1 (N . i * -|N *. > "N o 1 Tfr t> r "o! M3 J CO fc ( O T f o" 4 M r- > IO C? 1 ' (N 5 3 % ? 4 O t> < O rj ; 8 t w % i-H O Tf H 3 "2 jj M i 5 6 5 t ) "lO " 5 vb CO V H. s o - g o 32 c -(j -t t < Tf C< ; i J-^ vO CD 00 '-2 10 - i CO 1 P *' IH : M CC n PQ He* H HN <. O ^ | M v 5 a i ? 1 ^ "io H CO -IN IO IO h ** f-l M <t v (S o c I r- 5 M r (NO J -t M M o o o p fS C M C 5 3 ^ <I ro T h IO M E i n N C (5 t^- ^ 00 5^ 10 j^ 00 ti 49 <J < b ' M T- -1 3 < ^ - o M CO, 3 H o) ^ iH M - H 60 iV ^ q <! P 5 u i ? H 1 W 8 P 1 o jS o o C "* o fr ^> * o iqj p OJTJH O H N CO nj IO vO ** I 1 ^ D O - O vo t 2 ^ ^ Bo M CO 1 | & o 50 TRANSITION CURVE 2 <u g So S v ft 111 * i i " o 8 I * M M II i 5 a O 8 o 00 8 o o (S r}i o o o vO X O q O 1 W 0^ o pq o o O oo a> M o in C O 00 ? ^ . ; o . co { 00 <N M 1 in oo a GO M C<l O O 3- g 2 o i O vO o o VO O5 o o o Tf t-H O cS o c (M CO TP vO 00 O 8 g 8 a t^ O5 00 00 l> vO ^ R it O 00 8 a as M CO in t- o o o 00 (S O ^ co in t^ o * * 8 8 o o 8 S o in co o * S S o (S o co * S in o O\ O5 O t^ 00 O* S 8 M 8 r*J Q vO 00 1-1 O* *" 1 co i* in o> in t M ON ^ O > CO m oo w M 8 8 8 a i s i O <* O oo co rf n 8 o i o t> a M ^ S S a * 1 (N o vO M o i ? ? fr CO O o N o o vO 00 O M Q o (N S w to E 00 O5 ** ^ O O (N I * 8 o o o o o VO 00 O THE SPIRAL 51 . 0000000000 00 vo -* (S O 00 CO ^f <N 1? <UfiWfaOE~^W ooooooooooO o r-> (sco Tfuivor^oooi'-' - - ooo <s <N oo o CQ <N (s oo ,. vOCD U)O *t *& f5 C^ M .4J I 'W Tf^OC^O^Tj-ON ^0 w r ^ ^ ^ ^ voocvboocboocb ^5 w uo ro -H -t O N ^ o ! ! s ? s ' s * s II 3 T+-O dO ^"^rK^ OCOvO <S^H inco oco u ?O o Np o m II 3 1 ^ | OOO MC^oOp^ (SOOOOM O ^" ^ M O ^ ^H M O< rrt S 1 .2 .2 I 2 OO ^C s ?f_i<N 1OCO OO **> o W o o o o M I-H N CN PO cc >J ffl vb oo o *b ^ TP *b o V (*5 <M M Q N Tt <* JS O Q ^t fi IH^H^i-IPOlO lONO^t o o o o \j- OOOvocO OOOOvO PQ o o o o H W 00 00 * O 00 N "b o Q ^jpqOQWfaOK^^W o ^ N SD % fe ^ SL So & 2 5 ^ '^^ScggSgvSgg 52 TRANSITION CURVE II II S a * * ^ o < <J HH iJ H ^ | H 1C & B ^ Io o o c o CO vO l> O o o <s g o in S S Q <J o W fe O m 00 '^ cr. W o ^ (S o in o o 00 ff) i-^ o 00 in k ic N co in co o "* in <N H JH cc oo I S S i ' 2 CO S5 o M o ^ n \ O O >O O CO N l>- t>> H . (4 IQ 1^ CD cc 1 tQ 1^. CO l> OS o . o H 8 8 o OS s | O^ CO X l> t r^J CO vO l> O* M CC a S ^ 8 ^ 2 <S R 5- 8 00 1 ^- * H 8 g ^ a 9 I cc o\ l> H S S % & 9 in CO t> oc Ov 00 M U> 00 2 SO ^ 00 K N 1C N W M O ^t 00 "fr K in g Jt 00 * H f*> r? VO 00 2 8 ^ S &0 in fo H W ^ S H N 3 3 S * 00 w <N *M MOO ! Hfcl * O "b c *)ta "* 00 Tt- o M ^ <J O W PH O o & % Ss ^ t^ -l So "1 cr. o ^_! f> ^ o o o ic o 1 o g S 1C CD 00 THE SPIRAL 53 * O O O iO O 00 *G <* O O O O iO O iO O ^ s*~. << & w Pc< O W ~ ^ w ^ s a II 000 q O C^l Tf O vO t- 00 05 O T-l N - * b o 3* "0 * CO <*) O <*5 <N <s oo iO M (M "N ^' g ^ J ! o cq ro o <*i i <E ^ rf " a H|^ V in ^ % w rH V CO ON 1 * o o <N M M "*> c "in oo oo 1C Ov "N ?- "in o o^ o o^ o ,1 * CO (S iM M ^ 3 % 0** co ,^ co ^ HN ^ 00 1-1 00 o o o M (N II 3 3 (S IM H a * I oo o m *f co M H ^ % 1 a fi o o S c 00 ^ v Hf >HN ^ HN (M O CO - ^ fe a r*> " H r-4 C<l n) 1 in Tf r-4 Tf O <S O ro o in W M T 1 CO fO ^ ^ H* *. ^ CO pq T ~ l ^f i 1 % & 5- S * M (N CO -tf *<N V o "b M -> (N Tt< O S3 ^ S 00 <N *O -* (N T 1 M C^ <S CO t H ^, ^ ^ H* ^ " O ^ I s - ^ ^ o <N cc m S 1 % 5 ^ M ^ % 1 i 00 q <; Q W fe O pq 090 a* 000 O O M ^ O vO t>- 00 O5 M ^H M Oil O O O iO O iO O to O iO O II ^ ?o vo t- a O fs co in co oo 54 TRANSITION CURVE u 5 ii CO P S S s s >-4 d PQ o g ^J H ^ H ^ 1 K^ 8*S<8888*8 j M 5^ IO* <S jrt *O 00 I> O OC^ ?o <*i i> IH i> t- "ti ooo 00 ^"05 I>O ^^f TtCO . (N M ^ a o o o o o tj :*oooo8 ^ww o 00 ^ O O i < fO^O t^OO O i * (S O OO Ol> fOO lOOi O OO O^OSOxCftOOt^ o l> J :S 8 S M S ^^ o> in : 8 oo MMOOO aw < ^- * O OO OO5 O>Ci O*O> fO 0\ H^ ; 2S "'" ss * e o J> O O 1 " 1 <*J I> S O & *& in a "** ; ' M rH M C4 N <N aco ao 2Jo <3 O MCO ^Oi OCC ONOO fci M C^ IO O l> O^ M '.8 oiaSw^vo^ <o H M i ( M rH IS * | C<l 00 00 *N O (S t^ 00 LO . r-t ^ ^ M O M o 0* 0^ o" M Q o O o o o oooooONCO-'friO OW TJ-OOO^ MT-H Mr^ M M) oo oo oo oo oo * TJ- cc 00 C r CO ^ 10 o SO THE SPIRAL 55 o o o \O "* N 8 o o ^ CO o o o N i-H K} 3I> < o w o H ^ ^ S Z 0^0 If C; o o %t o a O N CO M t2 "2 N CO (S b "c 5 CNI Ib (S l> ^ +> (V) 1Q M 2 2^ ^ OO I> r- C^l f CO * o ^ O M ro -^ O 1 H H O CO TJ- fO to n 1^ d iO ; g 2 g V -b * ^ 000 f ; 5 (M O T-H o co o s ^ CO 00 ^ o^ 1 I * CO i 3 00 00 +:*<',* O (N vO ii 3 ^ S ? 8 "? 00 t^- 3 CO vo t^ O M IQ Tj- 3 X fr rj< <*) CO t i*4 M H1-IP| V" 1 g O CO * N iO N ^ 1 3 *! O5 O CO vO iO "O M LO fO 5 <O <N N _, hH M r-l N C* f> 03 9 o .2 o 0) 00 00 O O T^ (N i CN (S LO IQ I/) (N O CO M Q 2 H M C^ *J CO % 3 <s CO r> ( 3 oo Tt in 00 CO O PQ H IQ f) * <M (s in M O ~ N * % > A g s ; r ^ O CO 00 (4 H rH (S CO o oo ^o o S <N O l> CO O ^t 000 ! < *M <o CO * $ 0* $ H T-l N <N <*) CO ^ *e H H ^J <! O W O " H w S J5 PH q 000 &s o ^ V CO O N CO 1 e3"t * <N 0_ C 5 O O O <S CO S 1 O II 56 TRANSITION CURVE * 53 I H 8 <J ^ 8 oo O 00* vo' O 00 O 4 in 1-1 00 CD ro C^l O f*i CO <S iM fS <N N M co in oo NO o o "o o "o o o o o o o rJ^vOcO ^ r ~ l . I " H ""t ^ ^ O M CO* vo' OO' O* <N 4 CO O o in co t>> co (scoinoot^- O OO OJO5 O>O50000 1> i-( O\ C^ V) O in CO M 00 t" o oco J>o tos u? 1 -? o; 10 F^ co < TM >< ^ M o f*T~ rH fs^'int> dc<iin O 00 O O O O O O O~ OO5 O\ 00 ^T 1 ^ f}O5 * O OOS O\O5 O\O5 ONOOOO* N ^10 J>00 OO Md ro p 8 oo o jN O Q CO V co a oooooooo TtcOOOCTS O^H tSCO THE SPIRAL . * M 8 o o o < o ^f D * =C < Q W o s -D M 3 S JZJ o^ Q Jl 9 00 2 HI oo o | || ' V 1 S N (M So rj "Vj. Q f "N O ^ ' ^ "b tN ."S *> 2 S : 1 H CO lO I f5 o^ ^ O d -H 1*1 | CO f 1 $ 10 O vO C Tt* PO * o o <1 M ^ M vO ro P % S vO 00 vO C 1it-tr B 3 fO a 00 ** * CO M -1 I 1 M Q J- 2 * o O O O * vO 11 00 ^ * ** ^ M H ^ t, * 3 o S? 5 CO fS 10 M 00 C C M 00 00 15 o r 1 H N CO *C 3 Tf O c^ (N 1O t ? O rs S 2 05 in ^ g <N *5 -1 f> * ir *H a O O 8 M s ^a |i S 5 = <i -<t o 11 o* o*> 1 2 * CO r* H ^ ro * ^ & PQ W s * (N o u> S ? ^ S v^ PQ N ^ o M !N N C 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 1-1 N CO CO T f W CO t " M a & < a 04 J! (N O C ^ N C ^ oo oo ro C3 C? *} n C^ ro T M rf 10 VO <0 H H < o H O w ^ M >- -* 3 O q a) ti O ^ 00 C4 s s i * V & 1 C* (35 oo 35 2 o o i f g 5 M r- 3 O O H N CO 1 j 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 *5 Oft CO 00 - ^ 00 N O O> CO t *- 1$ "* < -' <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 . 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 . 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 iO O ^ &l ol *9 rH !> lOCO i -^ (M H CO 1-1 CO h ooooco^c^oooo B fe O 4 4 s Ct, J^ i 1 1 1 ! 1 1 1 1 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 >