UC-NRLF GIFT OF MICHAEL REESE THE RAILROAD SPIRAL. THE THEORY OF THE COMPOUND TRANSITION CURVE REDUCED TO PRACTICAL FORMUL/E AND RULES FOR APPLICATION IN FIELD WORK; COMPLETE TABLES OF DEFLECTIONS AND ORDINATES FOR FIVE HUNDRED SPIRALS. KY WILLIAM H. SEARLES, C.E., MEMBER AMERICAN SOCIETY OF CIVIL KNG1NKERS, AUTHOR "FIELD ENGINEERING." UNIVERSITY! NEW YORK : JOHN WILEY & SONS. 1882. COPYRIGHT, 1882, BY JOHN WILEY & SONS. PRE FACE. THE object of this work is to reduce the well-known theory of the cubic parabola or multiform compound curve, used as a transition curve, to a practical and con- venient form for ordinary field work. The applicability of this curve to the purpose in- tended has been fully demonstrated in theory and prac- tice by others, but the method of locating the curve on the ground has been left too much in the mazes of algebra, or else has been described as a system of off- sets, or fudging. Where a system of deflection angles has been given,- the range of spirals furnished has been much too limited for ge^r&^nictice. In consequence the great majority of engineers have contented them- selves with locating circular curves only, leaving to the trackman the task of adjusting the track, not to the centres given near the tangent points, but to such an approximation to the spiral as he could give "by eye." The method here described is that of transit and chain, analogous to the method of running circular curves ; it is quite as simple in practice, and as accu- rate in result. No offsets need be measured, and the curve thus staked out is willingly followed by the track- men because it " looks right," and is right. The preliminary labor of selecting a proper spiral for a given case, and of calculating the necessary distances to locate it at the proper place on the line, is here ex- plained, and reduced to the simplest method. Many of iii IV PREFACE. the quantities required have been worked out and tabu- lated once for all, leaving only those values to be found which are peculiar to the individual case in hand. A large number of spirals are thus prepared, and their essential parts are given in Table III. In section 22 is developed the method of applying spirals to existing circular curves, without altering the length of line, or throwing the track off of the road bed, an important item to roads already completed. Table V. contains samples of this kind of work arranged in order, so that, by a simple interpolation, the proper se- lection can be made in a given case. The series of spirals given in Table III. are obtained by a simple variation of the chord-length, while the de- flections and central angles remain constant. This is the converse of our series of circular curves, in which the chord is constantly 100 feet, while the deflections and central angles take a series of values. The multiform compound curve has been chosen as the basis of the system, rather than the cubic parabola, because, while there is no practical difference in the two, the former is more in keeping with ordinary field methods, and is far more convenient for the calculation and tabulation of values in terms of the chord-unit, or of measurement along the curve. While the several com- ponent arcs of the spiral are thus assumed to be circu- lar, yet the chord-points are points of a true spiral, to which the track naturally conforms when laid according to the chord-points given as centres. The " Railroad Spiral " is in the nature of a sequel to " Field Engineering ; " the same system of notation is adopted, and any tables referred to, but not given here, will be found in that work. WM. H. SEARLES, C. E. NEW YORK, July i, 1882. CONTENTS. CHAPTER I. INTRODUCTION. SECTION PAGE 1. Objections to simple circular curves I 2. Office of the spiral 2 CHAPTER II. THEORY OF THE SPIRAL. 3. Description of the spiral 3 4. Co-ordinates of the spiral 3 5. Deflection angles from the main tangent . . . . 5 6. Deflection angles from an auxiliary tangent 6 7. The chord-length as a variable g Construction of Table of Co-ordinates 10 9. Elements of the spiral 10 10. Selection of a spiral 1 1 CHAPTER III. ELEMENTARY PROBLEMS. 11. To find a long chord SL 13 12. To find the tangents SE and EL 13 13. To find a long chord QL. 14 14. To find the tangents QE' and EL 15 15. To find the tangent-distance T s SV 16 16. To find 7' s approximately 17 17. To find the radius R 1 in terms of T s and spiral 17 V VI CONTENTS. SECTION PAGE 18. To find diff. R' in terms of d'iff. Ts 19 19. To find the external distance E s 20 20. To find the radius R' in terms of E s and spiral 21 21. To find diff. R' in terms of diff. x for Es constant 23 CHAPTER IV. SPECIAL PROBLEMS. 22. Given, a simple curve, to replace it by another with spirals ; length of line unchanged 25 a. To find the radius R' 26 b. To find the offset h 26 c. To find the distance d A S. 27 d. To find lengths of old and new lines 27 e. To select a suitable spiral 28 f. To find diff. h in terms of diff. R' 29 23. Given, a simple curve, to apply spirals without change of radius 32 24. Given, a simple curve, to compound it for spirals without disturbing the middle portion 34 25. Given, a compound curve, to replace it by another, with spirals ; length of line unchanged 36 26. Given, a compound curve, to apply spirals without change of radii 40 27. Given, a compound curve, to introduce spirals without dis- turbing the P. C. C 42 CHAPTER V. FIELD WORK. 28. To locate a spiral from S to L 45 29. To locate a spiral from L to S 46 30. To interpolate the regular stations 47 31. Choice of method for locating spirals 47 32. To locate a spiral by ordinates 48 33. Use of spirals on location work 48 34. Description of line with spirals 48 35. Elevation of outer rail on spirals 49 36. Monuments 49 37. Keeping field-notes 49 CONTENTS. TABLES. PAGE I. Elements .of the spiral of chord-length TOO 50 II. Deflection angles for the spiral 52 II. Co-ordinates and Degree of curve of the spiral 58 IV. Functions of the spiral angle s 77 V. Selected spirals for unchanged length of line. 22 78 UNIVERSITY; A ^\^ Sr it C)Jd' ^*t V*w '/ THE RAILROAD SPIRAL. CHAPTER I. INTRODUCTION. I. ON a straight line a railway track should be level transversely ; on a curve the outer rail should be raised an amount proportional to the degree of curve. At the tangent point of a circular curve both of these condi- tions cannot be realized, and some compromise is usually adopted, by which the rail is gradually elevated for some distance on the tangent, so as to gain at the tan- gent point either the full elevation required for the curve, or else three-quarters or a half of it, as the case may be. The consequence of this, and of the abrupt change of direction at the point of curve, is to give the car a sudden shock and unsteadiness of motion, as it passes from the tangent to the curve. The railroad spiral obviates these difficulties entirely, since it not only blends insensibly with the tangent on the one side, and with the circle on the other, but also affords sufficient space between the two for the proper elevation of the outer rail. Moreover, since the curva- ture of the spiral increases regularly from the tangent to the circle, and the elevation of the outer rail does the same, the one is everywhere exactly proportional to the other, as it should be. The use of the spiral allows i 2 THE RAILROAD SPIRAL. the track to remain level transversely for the whole length of the tangent, and yet to fie fully inclined for the whole length of the circle, since the entire change in inclination takes place on the spiral. 2. The office of the spiral is not to supersede the cir- cular curve, but to afford an easy and gradual transition from tangent to curve, or vice versa, in regard both to alignment and to the elevation of the outer rail. A spiral should not be so short as to cause too abrupt a rise in the outer rail, nor yet so long as to render the rise almost imperceptible, and therefore difficult of ac- tual adjustment. Within these limits a spiral may be of any length suited to the requirements of the curve or the conditions of the locality. To suit every case in practice an extensive list of spirals is required from which to select. THEORY OF THE SPIRAL. CHAPTER II. THEORY OF THE SPIRAL. 3. THE Railroad Spiral is a compound curve closely resembling the cubic parabola ; it is very flat near the tangent, but rapidly gains any desired degree of curva- ture. The spiral is constructed upon a series of chords of equal length, and the curve is compounded at the end of each chord. The chords subtend circular arcs, and the degree of curve of the first arc is made the com- mon difference for the degrees of curve of the suc- ceeding arcs. Thus, if the degree of curve of the first arc be o ic/j that of the second will be o 20', of the third, o 30', &c. The spiral is assumed to leave the tangent at the be- ginning of the first chord, at a tangent point known as the Point of Spiral, and designated by the initials P. S. r or on the diagrams by the letter S. 4. To determine the co-ordinates of the sev- eral chord extremities, let the point S be taken as the origin of co-ordinates, the tangent through S as the axis of Y, and a perpendicular through S as the axis of X. Then x, y, will represent the co-ordinates of any point of compound curvature in the spiral, x being the perpendicular offset from the point to the tangent, and y the distance on the tangent from the origin to that offset. For the purpose of calculation let us assume 100 feet as the chord-length, and o 10' as the degree of curve of THE RAILROAD SPIRAL. the first arc of a given spiral. Then, since the degree of curve is an angle at the centre of a circle subtended by a chord of 100 feet, the central angle of the first chord is 10', of the second 20', of the third 30', &c., and the angles which the chords make with the tangent are : For ist chord, Y x 10' = 5' " 2d - u 10' + 1 X 20 = 20' " 3 d " 10' + 20' 4- i x 30' = 45' " 4th " 10' + 20 + 30 + i x 40 = 80' &c., &c., &c., or in general the inclination of any chord to the tan- gent at S is equal to half the central angle subtended by that chord added to the central angles of all the preceding chords. If now we consider the tangent as a meridian, the latitude of a chord will be the product of the chord by the cosine of its inclination, and its depart- ure will be the product of the chord by the sine of its inclination to the tangent. A summation of the several latitudes for a series of chords will give us the required values of _>', and a summation of the several departures will give us the required values of x. By the aid of a table of sines and cosines, we may therefore readily pre- pare the following statement : Chord. Inclin. to tang. Dep. 100 sine. x. Lat. = ioo cosine. y- I o 05' 0.145 145 100.000 TOO.OOO 2 20' 0.582 .727 99.998 199.998 3 o 45 ' 1.309 2.036 99.991 299.989 4 1 20' 2.327 4-3 6 3 99.979 399.968 &c. &c.. &c. In this manner Table I. has been constructed. THEORY OF THE SPIRAL. 5. To calculate the deflection angles of the Spiral ; Inst. at S. If in the diagram, Fig. i, we draw the long chords 82, 83, 84, &c., g we may easily determine the angle /, which any long chord makes with the tangent by means of the co-ordinates of the further extremity of the chord, for x tan / = . y Having calculated a series of values of the angle /, we may lay out the spiral on the ground by transit deflec- tions from the tangent, the transit t> ing at the point S. The statement of the calculation is as follows : FIG Point. X / tan / = - . i y I .145 too.ooo .00145 o 05' oo" 2 .727 199.998 .00364 12' 30" 3 2.036 299.989 .00679 23' 20" 4 4-3 6 3 399.968 .01091 37' 30" &c. &c. The values of / are more readily found by logarithms however, since log tan / = log x logy. By this formula the first part of Table II. (Inst. at S) THE RAILROAD SPIRAL. FIG. 2. has been calculated, and these are the only deflections needed for field use when the entire spiral is visible from S. 6. To calculate the deflection angles when the transit is at any other chord-point than S : Sup- pose the transit at point I, Fig. 2. In the diagram draw through the point i a line parallel to the tangent at S, and also the long chords 1-3, 1-4, &c., and let a { represent the angle between any one of these long chords and the parallel. Then, from the right-angled triangles of the dia- gram we have the following expres- sions : For point 2, tan #, = - ^ = Q = .00582. y* y\ 99-99 8 x z .\\ 1.891 " 3, tan a, = - -~~ = .00945. y, -yi I99*9 8 9 4, tan #! = &c., 4.218 299.968 &c. = .01411. But these are better worked by logarithms, and the values of a l found directly from the logarithmic tan- gent. Let s the spiral angle = the angle subtended by any number of spiral chords, beginning at S. Then s = the sum of the central angles of the several chords considered ; and it therefore equals the angle between OO 10' 10' 20' 3; 3' 1 00- 40' o t I 40 &c. THEORY OF THE SPIRAL. 7 the tangent at S and a tangent at the last point consid- ered. The series of values of the angle s is as follows : Point. Angle under single chord. Angle f. S I 2 3 4 &c. Since the values of a\ found above are deflections at point i from a parallel to the main tangent, it is evi- dent that if we subtract from each the value of s for point i, or 10', we shall have the deflections, /, from an auxiliary tangent through the point i, which we require for use in the field. The statement is as follows : Instrument at point i ; (s = 10'). Point. Angle ,. Angle/. 2 20" I0 f 3 32' 30" 22' 30" 4 48' 20" 38' 20" &c., &c., &c. The instrument will read zero on the auxiliary tan- gent through point i where it stands, and of course the back deflection over the circular arc Si is 05'. Hence we have the complete table of deflections when the instrument is at point i. Similarly, if we suppose the instrument to be at point 2, we shall have the statement : Point. _ 3 tan a> 2 = = = .01018. - y* _f* 99-99 1 4 tan a* -3^3 _ OI r- 2 7. y y* 199-97 &c., &c., 8 THE RAILROAD SPIRAL. and since for point 2, s = 20', we have : Point. Angle a^. Angle i. 3 55' ' . o 15' 30 &c., 32 30 &c. The instrument will read zero on the auxiliary tangent through the point 2, the back deflection to the point i is half the central angle under the second chord, or 10', and the back deflection to S is the difference between S? and the deflection at S for point 2, or 30' 12' 30" 17' 30". We thus may complete the table of deflections for the instrument at point 2. By a similar process the deflections required at any other chord-point may be deduced. It should be noted, however, in forming the table, that the back deflection 5 \ , y to any point is equal to the value of s for the place of the instru- ment, less the value of s for the back-point, less the forward de- flection at the back-point for the place of the instrument. This is obvious from an inspection of the triangle formed by the two auxil- iary tangents and the chord join- ing the two points in question. Thus, Fig. 3, when the instru- ment is at point 4, the back de- flection for point 2 is equal to 100' 30' 32' 30" = 37' 30." In the manner above described has been calculated the complete table of deflections from auxiliary tangents at chord-points, for every chord-point of the spiral up to point 20, Table II. It is evident, that by FIG. THEORY OF THE SPIRAL. 9 means of this table the entire spiral may be located, the transit being set over any chord-point desired, while the chain is carried around the curve in the usual manner ; also, that the curve may be laid out in the reverse direc- tion from any chord-point not above the 2oth, since all the back deflections are also given. 7. Variation in the chord-length. We have thus far assumed the spiral to be constructed upon chords of 100 feet, but it is evident that such a spiral would be entirely too long for practical use ; it would be 1700 feet long before reaching a 3 curve. We must, therefore, assume a shorter chord ; but in so doing it will not be necessary to recalculate the angles and deflections, for these remain the same whatever be the chord-length. By shortening the chord-length we merely construct the spiral on a smaller scale. The values of x and y and of the radii of the arcs at corre- sponding points are proportional to the chord-lengths, and the degrees of curve for corresponding chords are (nearly) inversely proportional to the same. Thus for any chord-length c we have : c x : x 1QO :: c : 100, or x = ^ 100 - 100 c y : Vioo :: c : 100, or y y 100 . 100 c R s : jR iM :: c : 100, or R 3 ^? 100 . 100 Let D s = the degree of curve due to radius JR S , and .Z} 100 = the degree of curve due to radius ^ 10 o? then, 100 100 A ~ : TTv and -*MOO 2 sin whence IOO sin 1 D, = sin 4- Z> JO o, T* C TO THE RAILROAD SPIRAL. in which D s is the degree of curve upon any chord in a spiral of chord-length c, and Z> 100 is the degree of curve upon the corresponding chord in the spiral of chord- length 100. Accordingly, if we assume a chord-length of 10 feet the values of x and y will be of those calculated for 100 a chord-length of 100 feet, while the degree of curve on each chord will be (nearly) 10 times as great as be- fore. 8. In the construction of Table III., we have as- sumed the chord to have every length successively from 10 feet to 50 feet, varying by a single foot, and have calculated the corresponding values of x, y and Z) 8 . The logarithm of x is also added, and the, length of spiral nc. We are thus furnished with 41 distinct spirals, but since the same spiral may be taken with a different number of chords (not less than three) to suit different cases, the variations which the tables furnish amount to no less than 500 spirals, some one or more of which will be adapted to any case that can arise. The maxi- mum length of spiral has been taken at 400 feet ; the shortest spiral given is 3x10 feet = 30 feet. Be- tween these limits may be found spirals of various lengths. 9. The elements of a spiral are : D^ The degree of curve on the last chord, , The number of chords used, c y The chord-length, n x c y The length of spiral, s, The central angle of the spiral, x y y, The coordinates of the terminal point. Every spiral must terminate, or join the circular curve THEORY OF THE SPIRAL. II at a regular chord-point of which the coordinates are known. 10. To select a spiral. The terminal chord of a spiral must subtend a degree of curve less than that of the circular curve which fol- lows, but the next chord beyond (were the spiral pro- duced) must subtend a degree of curve equal to or differing but a little from that of the circular curve. Thus, if the circle were a 10 degree curve, the spiral may consist of 5 chords 10 feet long (the degree of curve on the 6th chord being 10 oo' 45"), or of 15 chords 26 feet long (the degree of curve on the i6th chord being 10 16' 09"), the length of 'spiral is 50 feet in one case and 390 in the other ; between these limits the tables furnish 15 other spirals of intermediate length, all adapted to join a 10 degree curve. We may therefore introduce one more condition which will fix definitely the proper spiral to employ. If the length of spiral be assumed, we seek in the tables those values of n and c which are consistent with the required value of D s for (;/ -h i), at the same time that their product, nc, equals as nearly as may be the assumed length of spiral. Thus, if with a 10 degree curve a length of about 130 feet were desirable, we should select either n = 8, f= 15, D 9 10 oo' 45"; nc 120 ft; or ;z =-9, c= 16, D s 10 25' 51"; nc 144 ft. D s is always taken for (n.+ i). When circumstances permit, a chord-length of about 30 feet will give the best proportioned spirals. With a 30 foot chord-length the length of spiral will be about 770 times the super- elevation of the outer rail at a velocity of 35 miles per hour. 12 THE RAILROAD SPIRAL. The value of s depends on the number of chords (n) and is independent of the chord-length. If the angle s were selected from the table, this would fix the number n, and we must then choose the chord-length c so as to give the proper value of D s . Thus, if s were assumed = 9 10' then n = 10, and = 18 ft. or 19 ft., giving D s 10 ii r 54" or 9 39' 36" to suit a 10 degree curve, and making the length (nc) of the spiral either 170 or 1 80 ft., according to the spiral selected. The coordinates (' = 7 20', and the angle between tangents, A =42. Let the spiral values be c = 2.3 ; n 9 . ' . s = 7,3o' Then by the last equation and the tables, y x 21 206.627 log log tan 0.978743 9-5 8 4i77 3,6,55 0.562920 ELEMENTARY PROBLEMS. 17 R' 7 20' C log 2.893118 iA s 13 30' log sin 9.368185 j-A 21 a. c. log cos 0.029848 195.502 2.291151 . ' . T a = 405-784 l6. When an approximate value of T, is only re- quired we may employ a more convenient formula derived from the fact that the line OI produced bisects the spiral SL very nearly, and that the ordinate to the spiral on the line OI, being only about -g- x, may be neg- lected. Thus, Approx. T, If tan |A + i nc. . . (6.) Example. Same as above. R' 7 20' C log 2.893118 I A 21 log tan 9.5,84177-' 300.1. 2.477295 i^ = ix9X23 103.5 . ' . 71 == approx. 403.6 > Remark. This formula, eq. (6) when K is" taken equal to the radius corresponding to the degree of curve D s for (n -f i), gives practically correct results. But as in practice, the value of R 1 will differ somewhat frcm the radius of D# so the value of T 8 derived from this formula will differ more or less from the true value, as in the last example. 1/17. Given : the tangent distance T y SV, and the angle A , and the length of spiral SL, to find the radius K of the circular curve, LH, Fig. 5. The length Ib- THE RAILROAD SPIRAL. of spiral is expressed by nc, hence we have from the last equation. approx., ' = (T a \nc) cot -JA. . . .(7.) After R' is thus found, the values of n and c are to be determined, such that, while their product equals the given length of spiral as nearly as may be, the value of Z> 4 for (n -f i) shall correspond nearly with R '. The values of n and c are quickly found by reference to Table III. Examble. Let T s = 406, A = 42, and nc = 170. T 9 \nc 321 log 2.5065 lA 21 log COt. 0.4158 Jt' = say, 6 51' curve, 2.9223 By reference to Table III., we find that when n = 8 and c = 22, the product nc being 176, the value of D 3 for (n + i) is 6 49' 19", and this is the best spiral to use in this case. But as this spiral is longer than our assumed one, we should decrease the value of Jt' some- what, if we would nearly preserve the given value of T s . For instance, assume R' radius of 6 54' curve, and using the same spiral, calculate by eq. (4) the re- sulting value of T S9 and we shall find T = 408.646. As this is an exact value of T* for the values of R\ n and c last assumed, and is also a close approximation to the value first given, it will probably answer the purpose completely. If, however, for any * reason the precise value of T s 406 is required, we may find the precise radius which will give it by the following problem. l8. Given: a curve, and spiral, and tangent-distance, ELEMENTARY PROBLEMS. 19 T to find the difference in ' corresponding to any small difference in the value of T s . If in eq. (5) we assume a constant spiral, and give to K two values in succession and subtract one resulting value of T s from the other, we shall find for their dif- ference, diff. Ts = "h (**"'). diff. jf. . ( 8 .) cos fa Hence d i ff . jf = . COS * A diff. T v . (9.) sin (iA j) Example. When J?' = rad. 6 54' curve, n = 8, c 22, 7^ = 408.646 ; what radius will make T s = 406 with the same spiral ? Eq. (9) diff. T,= 2.646 log 0.422590 3- A, 21 log COS 9.970152 (-3- A s), 15 a. c. log sin 0.587004 .'. diff. 1? 9-544 0.979746 *' 6 54' 830.876 . *. Required radius 821.332, or 6 58' 49" curve. Remark. Care must be taken to observe whether in thus changing the value of 7?', the value of Z>', the de- gree of curve, is so far changed as to require a different spiral according to the rule for the selection of spiral^ 10. Should this be the case (which is not very likely), we may adopt the new spiral, and proceed with a new calculation as before. 1 9- Given : a circular curve with spirals joining two tangents, to find the external distance . VH, Fig- 5- 20 THE RAILROAD SPIRAL. Let SL be the spiral, LH one-half the circular curve, and O its centre. Then VH = VG + GO - OH. But VG = ^T^r - ~~TT > and in the tria ngle cos VGN cos A rni rn T n sin OLI -- & cos s GOL, GO = LO . ^-^7^ Jt sin LGO ( cos x _ ; cos s s -- 1 - T> . COS-^A COS^A or for computation without logarithms ^ f ^ (cos ^ cosJ-A ) / x ^ "cosfA" Example. Let Z> f = 7 20', A = 42, and for the spiral let n 9, ^ = 23, giving ^ = 7 30', and for (0 + i), .A - 7 T 5' 4"- Eq. (10) a; log 0.978743 ^A 21 a. c. log cos 0.029848 10.200 1.008591 K 7 20' log 2.893118 s 7 30' log cos 9.996269 -JA2I a. c. log cos 0.029848 830.300 2.919235 sum 840.500 ./?' 7 20' 781.840 s 58.660 ELEMENTARY PROBLEMS. 21 20. Given : The angle A at the vertex and the dis- tance VH = to determine the radius R' of a circular curve with spirals connecting the tangents and passing through the point H. Fig. 5. Solving eq. (n) for R' we have K _ E, cos -j- A x cos s cos 4- A \ ) But as this expression involves x and s of a spiral de- pendent on the value of R' we must first find R' approxi- mately, then select the spiral, and finally determine the exact value of R' by eq. (12). The radius R of a simple curve passing through the point H is a good approxima- tion to R '. It is found by eq. (27) Field Engineering: R exsec -J-A ' or the degree of curve D may be found by dividing the external distance of a i curve for the angle A by the given value of E s . But evidently the value of D' will be greater than D y and we may assume D' to be from 10' to i c greater according to the given value of A, the difference being more as A is less. We now select from Table III. a value of D K suited to D' so assumed, and corresponding at the same time to any desired length of spiral. Since D s so selected corresponds to (n 4- i) we take the values of n and x from the next line above D s in the table, find the value of s from Table IV., and by substituting them in eq. (12) derive the true value of R 1 for the spiral selected. Example. Let A 42 and E s = 70, to find the value of R' with suitable spirals. From table of externals for i curve, when A = 42 E 407--64, which divided by 70 gives 5.823 ; or D = 22 THE RAILROAD SPIRAL. 5 50'. Assume D' say 20' greater, giving D' 6 10' approx. If we desire a spiral about 300 feet long we find, Table III., n = 10, c = 30, and for (n -f i) D s 6 06' 49". For 72 = 10, s = 9 10'. Eq. (12) cos^A, 21 -9335 8 ^ 7 65-35 o6 x 16.768 48.5826 log 1.686481 cos s 9 10' .98723 COsiA 21 .93358 .05365 log 8.729570 .*. R' = rad. (say) 6 20' curve. 905.55 2.956911 Proof. Take the exact radius of a 6 20' curve and the above spiral and calculate E a by eq. (10) or (n). We shall obtain E s = 69.97. Again : if we desire a spiral of 200 feet, we find, Table III., n = 8, c = 25, and for (n + i) D, 6, and by eq. (12) R' rad. of (say) 6 02' curve ; and by way of proof we find E, = 69.96. Again : if we desire a spiral of about 400 feet, we find, Table III., n = 12, c = 33, s 13, and for (n -f i) D, 6 34' 07". Hence by eq. (12) R' rad. of (say) 6 50' curve. By way of proof we find eq. (TO) E, = Remark. It is thus evident that a variety of curves with suitable spirals will satisfy the problem, but D' is increased as the spiral is lengthened for in the ex- ample, with a 200 ft. spiral, D' 6 02' ; with a 300 ft. spiral, D' = 6 20'; and with a 396 ft. spiral, I)' 6 50'. Therefore the length of spiral, as well as the value of A, must be considered in first assuming the value of I}' as compared with D of a simple curve. ELEMENTARY PROBLEMS. 23 21. In case the value of R ', as calculated by eq. (12), should give a value to D' inconsistent with the spiral assumed, we may easily ascertain by consulting the table what spiral will be suitable. Choosing a spiral of the same number of chords, but of a different chord- length c, we may calculate R' (a new value) as before ; or the work may be somewhat abbreviated by the fol- lowing method : Given : a change in the value of x^ eq. (12) to find the corresponding change in the value of R'\ n being con- stant. If the values of E s , A, and s remain unchanged, we find, by giving to x any two values, and subtracting one resulting value of R' from the other, - - COS S COS - A that is, R' increases as x decreases, and the differences bear the ratio of - = . cos s cos YA Example. Let A = 42, E s = 70, and for the spiral let n 10, c ~ 30, s = 9 10', as in the last example, giving R' = 905.55 ; to find the change in R' due to changing c from 30 to 29. Eq. (13) for c 30, x = 16.768 for c 29, x = 16.209 diff. x .559 log 9.7474 cos s COS^A (as before) .05365 log 8.7296 .'. diff. R' 10.42 1.0178 old value 95-55 . *. new R' 9 T 5-97 -&' ( sa y) 6 J 6', 24 THE RAILROAD SPIRAL. which agrees well with D s 6 19' 29" for (n 4- i) in the new spiral. If we prove this result by calculating the value of E s for these new values by eq. (10) we shall find E, = 69.93. The slight discrepancy between these calculated values of E s and the original is due solely to assuming the value of D' at an exact minute instead of at a fraction. SPECIAL PROBLEMS. CHAPTER IV. SPECIAL PROBLEMS. 22. Given : two tangents joined by a simple curve, to find a circular arc with spirals joining the same tan- gents, that will replace the simple curve on the same ground as nearly as may be, and preserve the same length of line. Fig. 6. To fulfill these conditions it is evident that the new curve must be outside of the old one at the middle point H, since the spirals are inside of the simple curve at its tan- gent points ; also, the radius of the new "curve must be less than that of the old one, otherwise the cir- cle passing out- ox side of H would cut the given tan- gents. Let SV, Fig. 6 be one tangent, and V the vertex. FlG - 6 - Let AH be one half the simple curve, and O its centre. Let SL be one spiral, LH' one half the new circular 26 THE RAILROAD SPIRAL. arc, and O' its centre. Draw the bisecting line VO, the radii AO = R and LO' = R ', and the perpendicular LM = x. Then MS =y. Produce the arc H'L to A' to meet the radius O'A' drawn parallel to OA, and let | A = the angle AOH = A'O'H'. Let s = the angle A 'OX = the angle of the spiral SL. Let h the radial offset HH' at the middle point of the curve. Draw O'N and LF perpendicular to OA, LF intersecting O'A at I. a. To find the radius R 1 of the new arc LH' in terms of a selected spiral SL. We have from the figure AO = ML -f FN + NO. But AO = R, ML = x, FN = LO' cos s = R' cos s and NO = O'O cos A = (OH' - O'H') cos A = (h + R R'} cos \ A ; and substituting we have R = x + R' cos s + (h + R R') cos A . (14.) whence , _ R vers ^ A h + cos \ A 4- x . , cos s cos A cos ^ cos \ A It is found in practice that h bears a nearly constant ratio to x for all cases under the conditions assumed in this problem. Let k = the ratio and the last equa- tion may be written , _ R vers \ A __ (/EcosjA + i)x , , x cos s cos 4 A cos s cos -J A which gives the radius of the new arc LH' in terms of s, ~v and k. SPECIAL PROBLEMS. 27 b. To find the offset h = HH' : From eq. (14) we derive h cos \ A R ( i cos 4- A ) R ( i vers s) 4- R' cos Y A x = jB(i - cos| A) ^'O- cosiA)-h -#' vers s x (R R') vers J A + -#' vers .$ #. Hence *g=X - *') exsec 4- A + -- (17.) cos i A cos i A which gives the value of h in terms of s, x and -#'. C. 70 /// ///^ ^//^ of d AS : We have from the figure SM = SA + NO' + IL. But SM=j;, SA = 4 NO'= OO ; sin | A and IL = LO' sin s, and by substitution, y d + (h + R R') sin -J- A +7?' sin j. Hence ^=7 ~ [( ;/ + R R') sin| A + ^'sin^] (18.) \ which gives the distance on the tangent from the point of curve A to the point of spiral S. d. To compare the lengths of the new and old lines : SAH = SA + AH = d+ 100^-, . .(19.) in which D is the degree of curve of AH ; SLH' = SL + LH' = n . c + 100 * in which D' is the degree of curve of LH'. 28 THE RAILROAD SPIRAL. If the spiral and arc have been properly selected, the two lines will be of equal length or practically so. The last two equations assume the circular curves to be measured by 100 foot chords in the usual manner, but when the curves are sharp it is often desirable that they should agree in the length of actual arcs, especially where the rail is already laid on the simple curve. For this purpose we use the formulae SAH(arc) = s (n + r), thus found; also of the four values of k ; we thus find approx. k = .0885, and D' = 6 18' . Now looking in Table III^ opposite n 9, we find that when c = 26, D s ( n + i) 6 24' 48", we therefore assume c = 26, and proceed to calculate R' by eq. (16). SPECIAL PROBLEMS. 31 Eq. (16) cos j 7 30' -99 X 44 COS^A 25 06' -9055^ .08587 a. c. log 1.066159 R 6 log 2.980170 vers^-A 25 o6 r log 8.975116 1050.6 log 3.021445 cos s cos -JA a. c. log 1.066159 I '-f y^COS 4- A = 1. 080 0.033424 x 1.031989 135-4 2.131572 .'. R' (say 6 16') 9 Z 5- 2 Eq.(i 7 ) JK6 955-366 R' 6 1 6' 914-75 (R R') 40.616 log 1.608697 exsec ^A 25 06' log 9.018194 4.235 log 0.626891 R' 6 16' log 2.961303 vers s , 7 30' log 7.932227 cos -}A 25 06' a. c. log 0.043079 8.642 log 0.936609 12.877 log 1.031989 i 25 .06' a. c. log 0.043079 11.887 1-075068 .*. h 0.990 Eq. (18) (R - R') 40.616 41.606 log 1.619156 sin ^-A 25 06' log 9.627570 17.649 log 1.246726 3 2 THE RAILROAD SPIRAL. R r 6 16' sin s 7 30' 119.399 log log log 2.961303 9.115698 2.077001 137.048 y .'. d n ( r r\ I - . * 233-579 A T R 1 TJ .'. SAH 514.864 Eq. (20) (^A s) 1056' X 100 D' 376' n . c 9 x 26 .--. SLH' Difference h actual k = = 0.092 x 280.851 234- SM-^i -.013 log 5.023664 lo g 2.575188 log 2.448476 Comparison of actual arcs. Eq. (21) 25.1 log 1.399674 i log 8.241877 R 6 log 2.980170 418.525 log 2.621721 96-531 Eq. (22) 17.6 log 1.245513 i log 8.241877 R' 6 i6 7 log 2.961303 280.991 log 2.448693 n.c 234. 5T4-99 1 Difference = 0.065 SPECIAL PROBLEMS. 33 23 Given : a simple curve joining two tangents, to move the curve inward along the bisecting line VO so that it may join a given spiral without change of radius. Fig. 7. Let SL be the given spiral, AH one-half of the given curve, and HL a portion of the same curve in its new position, and compounded with the spiral at L. To find the distance h = HH' = OO 7 : Since the new radius is equal to the old one, or ^?'= R, we have from eq. (17) by changing the sign of h, since it is taken in the opposite direction, x R vers s COS To find the distance d = AS : Changing the sign of h in eq. (18) and making R! R we have d y (R sin s h sin -J A) ( 2 4-) This problem is best adapted to curves of large radius and small central angle. Example. Given, a curve D = i 40' and A 26 40', and a spiral s = i, n = 3, and c = 40, to find // and d and the length LH'. Eq. (23) R i 40' log 3-5363 vers s i log 6. 1 82 7 cos i A 13 20' a. c. log 0.0119 34 - THE RAILROAD SPIRAL. 538 log 9-7309 x log 9.9109 cos-J-A* a. c. log 0.0119 .837 9.9228 .'.// .299 Eq. (24) R i 40' log 3.536289 sin s i u " 8.241855 59.999 x -77 8l 44 - 2 99 log 9-4757 sin 4 A 13 20 9.3629 .069 8.8386 59-93 y 119.996 . * . d 60.066 H'O'L ^ (| A - s) = 12 20' .-. H'L = 740 feet. 24. Given, a simple curve joining two tangents, to compound the curve near each end with an arc and spiral joining the tangent without disturbing the middle portion of the curve. Fig. 8. Let H be the middle point of the given curve, Q the point of compounding with the new arc, and L the point where the new arc joins the spiral SL. Let s = the spiral angle, and let = AOQ. Now in this figure AOQS will be analogous to AOH'S of Fig. 6, if in the latter we suppose H' to coincide with H or // = o. If, therefore, in eq. (15) we write for -J- A and make // = o, we have for the new radius O'Q, , _ R vers x . . ~ cos s cos 0' ' SPECIAL PROBLEMS. 35 in terms of and the spiral assumed. But as the value of D' resulting is likely to be fractional and must be adhered to, it is preferable to as- sume jR' a little less than R, select a suit- able spiral and cal- culate the angle 0. Resolving eq. (17) after making h = o and replacing \ A by 0, we have FIG. 8. vers := r-JT' vers s (26.) The angle so found must be less than -3- A , and in- deed for good 'practice should not exceed 3- A. If too large, may be reduced by assuming a smaller value of ^', and repeating the calculation with a suitable spiral. Otherwise it will be preferable to use one of the forego- ing problems in place of this. This problem is specially useful when the central angle is very large. To find the distance d AS, we have only to write for 4 A and make h = o in eq. (18), whence dy - \(R - J?') sin + R' sin^] . . . (27.) Example. Given a curve D ~ 2 30', A =35, to compound it with a curve D' = 2 40' and a spiral s = 2 30', n = 5, < = 37. THE RAILROAD SPIRAL. Eq. (26) R 2 30' 2292.01 R' 2 40' 2148.79 R- R 1 X 143.22 R- R' vers s 2 30' o r 2 40 R' .-. versO 6 28' 30" Eq. (2 1 }R-R' sin <* 6 28' 30" R' sin s 2 U 40' 2 30' .-. d AH log 2.156004 log 0.471203 0020663 log 8.315199 a. c. log 7.843996 log 6.978536 * lo g 3-33 2I 93 log 8. 154725 .014280 .006383 16.151 93.729 109.880 184.962 log 2.156004 9.052192 1.208196 3-33 2I 93 8.639680 1.971873 775.082 SL, n.c 185.00 LQ, - s = 3 58' 30" 149.06 QH, i A = n oi' 30" 441.00 775- 6 Difference .022 SPECIAL PROBLEMS. 37 25. Given: a compound curve joining two tan- gents, to replace it by another with spirals, pre- serving the same length of line. Fig. 9. Let A 2 = AO 2 P, the angle of the arc AP, and A, = PdB, the angle of the arc PB. Let ^ 2 A O 2 , and R, = BO,. Adopting the method of 22, the offset h must be made at the point of compound curve P instead of at the middle point. Cons idering first the arc of the larger radius AO 2 , the formulae of 22 will be made to FIG. 9. apply to this case by writing A 2 in place of \ A , and Ri in place of R, whence eq. (16) vers A g cos s cos A 2 (k cos A 2 + i) x cos s cos A 2 , and eq. (17) 7 / r> r> f\ A , * vers s X t \ h (R^ R^) exsec A 2 + - (20.) COS A 2 COS A 2 and eq. (18) d=y. - [(k + R, - RJ) sin A 2 + R* sin s] . . (30.) 38 THE RAILROAD SPIRAL. But in considering the second arc PB, we must retain the value of // already found in eq. (29) in order that the arcs may meet in P'. We therefore use eq. (15) which, after the necessary changes in notation, becomes n , R l vers A l h cos A , + x , x . . . cos s cos A j cos s cos A ! which value of lt\ must be adhered to. The spiral selected for use in the last equation is in- dependent of the spiral just used in connection with J?./. It should be so selected that while suitable for ^?/ its value of x may be equal to as nearly as may be, the K value of k being inferred from Table V. for D' and 2 A!. Assuming the value of 7?/ found by eq. (31)? even though DI be fractional, we may verify the value of h by h = (JP, - ft) exsec A , + L - - ( 32 .) cos A j cos A j and then proceed to find d' = BS' by d j -y- [(h + ^ -J?,') sin A, f.tf/sin.f] (33.) Example. Given the compound curve D == 8., A , 29 and Z> 2 = 6, A 2 = 25o6' : to replace it by an- other compound curve connected with the tangents by spirals. Considering first the 6 branch of the curve, we may assume the spiral s = 73o', n = 9, c = 26. This part of the problem is then identical with the example given in 22, by which we find h .990 and d 96.531. To select a spiral for the 8 branch, having reference at the same time to this value of h ; we find in Table V. SPECIAL PROBLEMS. 39 under D = 8 and opposite A 2 A l = 58 or say 60, that the given value of h falls between the tabular values of 7* for nc = 9 x 20, and //^ = io"x 22. We there- fore infer that the spiral ^ 9x21 is most suitable to this case. Adopting this, we have Eq. (31) COS.T 73o'. 99144 COS A! 29. 87462 .11682 log 9. 0675 1 7 a>c - l R, 8 " 2.855385 vers A^ " 9.098229 769.302 " 2.886097 h cos 29 .866 x 8.694 9.560 :< 0.980458 cos s cos A! a.c. " 0.932483 8i.8 3 5 " 1.912941 \ " 1.481471 "9-685571 .-. US 82o'3o r 687.467 [. (33) (h + .#,) 717.769 30.302 sin A ! 29 14.691 J?/ 687.467 sin s 73o f 89-732 ' 1.167042 " 2.837251 9. 115698 1.952949 104.423 188.660 y '. d 84.237 40 THE RAILROAD SPIRAL. For the methods of computing the lengths of lines, see 22. 26. Given : a compound curve joining two tangents, to move the curve inward along the line PO 2 so that spirals may be introduced without changing the ra- dii. Fig. 10. The distance h = PP' is found for the arc of larger Fig. 10. radius AO 2 by the following formula derived by analogy from eq. (23): , ___ x R* vers s . , ^ cos A 2 and for the distance d = AS we have analogous to eq. d y ( z sin, /fcsin A 2 ) . (35.) SPECIAL PROBLEMS. 41 . Now the same value of /*, found by eq. (34) must be used for the arc PB, and a spiral must be selected which will produce this value. To find the proper spiral, we have from eq. (34) after changing the subscripts, x R l vers s + h cos A ^ . . (36.) The last term is constant. The values of x and s must be consistent with each other, and approximately so with the value of R^. Assume s at any probable value, and calculate x by eq. (36). Then in Table III. look for this value of x opposite n corresponding to j, and note the corresponding value of the chord-length c. Com- pare D s of the table with Z>i and if the disagreement is too g^eat select another value of s and proceed as be- fore. The term JR^ vers s may be readily found, and with sufficient accuracy for this purpose, by dividing the value of R i versj Table IV. by Z> lm If the calculated value of x is not in the Table III., it may be found by inter- polating values of c to the one tenth of a foot, since for a given value of s or/n the values of x and y are pro- portional to the values of c. When the proper spiral has been found and the value of c determined, it only remains to find the value of d = BS'by d y (Ri sin s h sin A i), . (37.) in which the value of y will be taken according to the values of c and s just established. Example. Given: Z> 2 i4o', A 2 = 13 20', Z> 1 = 3, and A = 224o', to apply spirals without change of radii. Fig. 10. Assume for the i 40' arc the spiral s i, n = 3, c 40. This part of the problem is then identical with the example given in 23, from which we find h = 0.299. THE RAILROAD SPIRAL. For the second part, if we assume s i 40', n = 4, and find by Table IV. ^ vers s = ^ = 0.808, we o have by eq. (36) x = 0.808 -f 0.277 1.085, the nearest value to which in Table III. is under c = 25, giving Ds =2 40', or for (n + i), D = 3 20', which is consistent with D l 3. By interpolation we find that our value of x corresponds exactly to c = 24.85, n = 4, and therefore the spiral should be laid out on the ground by using this precise chord. In order to find d = BS' we first find the value of y by interpolation for c =24.85, when by eq. (37) we^ have *T= 99-39 1 - (55-554 ~ 0.115) = 43-95 2 - 27. Given : a compound curve joining two tan- gents, to introduce spirals without disturbing the point of ^^ B compou nd curvature P. Fig. ii. a. The radius of each arc may be shortened, giv- ing two new arcs compounded at the same point P. Having se- lected a suitable spiral, we have for the arc AP s by analogy from eq- (15), since Fig. SPECIAL PROBLEMS. _ Rv vers A 2 x cos ^ cos A 2 ' 43 (38.) and, similarly, after selecting another spiral for the arc PB, _ ^, vers A , x cos s cos Aj " " V39-J From eq. (18) we have for the distance AS, d~y \_(R* RI'] sin A 2 + RJ sin s], . (40.) and for the distance BS', / d ~ y [(XL Ri') sin A! + RI sin s] . (41.) The values of DI and Dj re- sulting from eq. (39) and (40) must be adhered to, even though involving a frac- tion of a minute. b. Either arc may be again com- pounded at some point Q, leaving the portion PQ undisturbed, as explained in 24. Fig. 12. Let e = the an- Fig. 12. gle AO 2 Q, and we have from eq. (26), after selecting a suitable spiral and assuming .#./, vers vers s 44 THE RAILROAD SPIRAL. For the distance AS, we have from eq. (27) d = y - [(. - A') sin + RJ sin j] . (43.) Similar formulae will determine the angle = BOjQ' and the distance BS' for the other arc PB in terms of a suitable spiral : thus, x R\ vers s d y - [(^ - ^/) sin + ^/ sin s] . (45.) The method a may be adopted with one arc and the method b with the other if desired, since the point P is not disturbed in either case. The former is better adapted to short arcs, the latter to long ones. These methods apply also to compound curves of more than two arcs, only the extreme arcs being altered in such cases. FIELD WORK. 45 CHAPTER V. FIELD WORK. 28. HAVING prepared the necessary data by any of the preceding formulae, the engineer* locates the point S on the ground by measuring along the tangent from V or from A. He then places the transit at S, makes the verniers read zero, and fixes the cross-hair upon the tan- gent. He then instructs the chainmen as to the proper chord c to use in locating the spiral, and as they meas- ure this length in successive chords, he makes in succes- sion the deflections given in Table II. under the heading "Inst. at S," lining in a pin or stake at the end of each chord in the same manner as for a circle. When the point Las reached by (n) chords, the tran- sit is brought forward and placed at L ; the verniers are made to read the first deflection given in Table II. under the heading " Inst. at n " (whatever number n may be), and a backsight is taken on the point S. If the verniers are made to read the succeeding deflections, the cross-hair, should fall successively on the pins already set, this being merely a check on the work done, until when the verniers read zero, the cross-hair will define the tangent to the curve at L. From this tangent the cir- cular arc which succeeds may be located in the usual manner. In case it became necessary to bring forward the tran- sit before the point L is reached, select for a transit- point the extremity of any chord, as point 4, for 46 THE RAILROAD SPIRAL. example, and setting up the transit at this point, make the verniers read the first deflection under " Inst. at 4," Table II., and take a backsight on the point S. Then, when the reading is zero, the cross-hair will define the tangent to the curve at the point 4, and by making the deflections which follow in the table opposite 5, 6, &c., those points will be located on the ground until the desired point L is reached by n chords from the begin- ning S. The transit is then placed at L, and the verniers set at the deflection found under the heading " Inst. at n " (whatever number // may be), and opposite (4) the point just quitted. A backsight is then taken on point 4, and the tangent to the curve at L found by bringing the zeros together, when the circular arc may be proceeded with as usual. 29. To locate a spiral from the point L running toward the tangent at S : we have first to consider the number of chords (n) of which the spiral SL is composed. Then,, placing the transit at L, reading zero upon the tangent to the curve at L, look in Table II. under the heading " Inst. at #," and make the deflection given just above o oo' to define the first point on the spiral from L toward S ; the next deflection, reading up the page, will give the next point, and so on till the point S is reached. The transit is then placed at S ; the reading is taken from under the heading "Inst. at S," and on the line n for a backsight on L. Then the reading zero will give the tangent to the spiral at the point S, which should coincide with the given tangent. If S is not visible from L, the transit may be set up at any intermediate chord-point, as point 5, for example. The reading for backsight on L is now found under the FIELD WORK. 47 heading " Inst. at 5," and on the line n corresponding to L ; while the readings for points between 5 and S are found above the line 5 of the same table. The transit being placed at S, the reading for backsight on 5, the point just quitted, is found under " Inst. at S " and opposite 5, when by bringing the zeros together a tan- gent to the spiral at S will be defined. 30. Since the spiral is located exclusively by its chord-points, if it be desired to establish the regular 100- foot stations as they occur upon the spiral, these must be treated asflusses to the chord-points, and a deflection angle will be interpolated where a station occurs. To find the deflection angle for a station succeeding any chord- point : the differences given in Table II. are the deflec- tions over one chord-length, or from one point to the next. For any intermediate station the deflection will be assumed proportional to the sub-chord, or distance of the station* from the point. We therefore multiply the tabular difference by the sub-chord, and divide by the given chord-length, far the deflection from that point to the station. This applied to the deflection for the point will give the total deflection for the station. This method of interpolation really fixes the station on a circle passing through the two adjacent chord- points and the place of the transit, but the consequent error is too small to be noticeable in setting an ordinary stake. Transit centres will be set only at chord-points, as already explained. 31. It is important that the spiral should join the main tangent perfectly, in order that the full theoretic advantage of the spiral may be realized. In view of this fact, and on account of the slight inaccuracies inseparable from field work as ordinarily performed, it is usually preferable to establish carefully the two points 48 THE RAILROAD SPIRAL. of spiral S and S' on the main tangents, and beginning at each of these in succession, locate the spirals to the points L and L'. The latter points are then connected by means of the proper circular arc or arcs. Any slight inaccuracy will thus be distributed in the body of the curve, and the spirals will be in perfect condition. 32. A spiral may be located without deflection angles, by simply laying off in succession the abscissas y and ordinates x of Table III. corresponding to the given chord-length c. The tangent EL at any point L, Fig. 4, is then found by laying off on the main tangent the dis- tance YE = x cot s, and joining EL. In using this method the chord- length should be measured along the spiral as a check. 33. In making the final location of a railway line through a smooth country the spirals may be introduced at once by the methods explained in Chapter III. But if the ground is difficult and the curves require close ad- justment to the contour of the surface, it will be more convenient to make the study of the location in circular curves, and when these are likely to require no further alterations, the spirals may be introduced at leisure by the methods explained in Chapter IV. The spirals should be located before the work is staked out for con- struction, so that the road-bed and masonry structures may conform to the centre line of the track. 34, When the line has been first located by circular curves and tangents, a description of these will ordi- narily suffice for right of way purposes ; but if greater precision is required the description may include the spirals, as in the following example : " Thence by a tangent N. 10 i5'E., 725 feet to station 1132 + 12; thence curving left by a spiral of 8 chords, 288 feet to station 1 135; thence by a 4 12' curve (radius FIELD WORK. 49 1364.5 feet), 666.7 feet to the station 1141 +66.7; thence by a spiral of 8 chords 288 feet to station 1144 + 54.7 P.T. Total angle 40 left. Thence by a tangent N. 29 45' W.," &c. 35. When the track is laid, the outer rail should re- ceive a relative elevation at the point L suitable to the circular curve at the assumed maximum velocity. Usu- ally the track should be level transversly at the point S, but in case of very short spirals, which sometimes can- not be avoided, it is well to begin the elevation of the rail just one chord-length back of S on the tangent. 36. Inasmuch as the perfection of the line depends on adjusting the inclination of the track proportionally to the curvature, and in keeping it so, it is extremely im- portant that the points S and L of each spiral should be secured by permanent monuments in the centre of the track, and by witness-posts at the side of the road. The posts should be painted and lettered so that they may serve as guides to the trackmen in their subsequent efforts to grade and "line up " the track. The post op- posite the point S may receive that initial, and the post at L may be so marked and also should receive the figures indicating the degree of curve. 37. The field notes may be kept in the usual manner for curves, introducing the proper initials at the several points as they occur. The chord-points of the spiral may be designated as plusses from the last regular sta- tion if preferred, as well as by the numbers i, 2, 3, &c., from the point S. Observe that the chord numbers always begin at S, even though the spiral be run in the opposite direction. TABLE ELEMENTS OF THE SPIRAL Inclina- Point Degree of curve Spiral angle tion of chord Latitude of each chord. Sum of the lati- tudes, to axis i n. Ds. s. of Y. TOO x cos Incl. ?- o oo' o oo' o oo' I 10' 10' 05' 99.99989423 99.99989423 2 20' 30' 20' 99.99830769 199.99820192 3 30' 1 45' 99.99143275 299.98963467 4 40' 1 40' 1 20' 99.97292412 399.96255879 5 50' 2 30' 2 05' 99-93390 07 499 89645886 6 1 3 30' 3 99.8629535 599.7594123 7 1 10' 4 4' 4 05' 99.7461539 699.5055662 8 1 20' 6 5 20' 99.5670790 799.0726452 9 I' 30' 7 30' 6 45' 99.3068457 898.3794909 10 I 4 0' g 10 8 2o| 98.944164 997-3236549 ii I' SO' 11 10 05' 98.455415 1095.779070 12 2 13 12 97.814760 1193.593830 13 2 10' 15 10' 14 05' 96.994284 1290.588114 14 2 20' I7 I 3 1 6 20' 95.964184 1386.552298 15 2 30' 20 i84 5 ' 94.693014 1481.245312 16 2 40' 22 40' 21 20' 93-147975 1574-393287 17 2 50' 25 30 24 05' 91.295292 1665.688579 18 3 28 30'' 27 89.100650 1754.789229 19 3 10' 31 40 3o 05' 86.529730 1841.318959 20 3 20' 35 33 20' 83.548730 1924.867739 Point. Log^ = Deflection angle, . log tan /. it I 7.1626964 o 05' oo. 'oo 2 7.5606380 o 12' 30. 'oo 3 7.831709! 23' 20. '00 4 8.0377730 o 37' 29. '99 5 8.2041217 o 54' 59- '97 6 8.3436473 i 15' 49. '90 7 8.4638309 i 39' 59- '75 8 8.5694047 2 07' 29. '45 9 8.6635555 2 38' IS. '90 10 8.7485340 3 12' 27. '95 OF CHORD-LENGTH, 100. Departure of Sum of the depart- Logarithm, Logarithm, Point each chord. ures, 100 x sin Incl. X. logjj/. log jr. n. .1454441 .1454441 1.9999995 9.1626960 I .5817731 .7272172 2.3010261 9.8616641 2 I.308Q593 2.0361765 2.4771063 0.3088154 3 2.3268960 4.3630725 2.6020194 0.6397924 4 3.6353009 7.9983734 2.6988800 0.9030017 5 5-23359 6 13.231969 2.7779771 1.1216244 6 7.120730 20.352699 2.8447911 1.3086220 7 9. 29499 [ 29.647690 2.9025862 .4719909 8 11-75374 41.40143 2.9534598 .6170153 9 14.49319 55.89462 2.9988361 .7473701 10 17.50803 73.40265 3.0397231 .8657117 n 20.79117 94.19382 3.0768567 .9740224 12 24.33329 118.52711 3.1107877 2.0738177 13 28.12251 146.64962 3.1419362 2.I6628II 14 32.14395 . 178.79357 3.1706269 2.2523519 15 36.37932 215.17289 3.1971131 2.3327875 16 40. 80649 255.97938 3.2215938 2.4082049 17 45.39905 301.37843 3.2442250 2.4791121 18 50.12591 35L50434 3.2651291 2.5459307 19 54.95090 406.45524 3.2844009 2.6090128 20 T * Deflection an- Point L S y = gle, n. log tan i. t. II 8.8259886 349'56."39 12 8.8971657 43o'43."95 13 8.9630300 5 14' 50."28 14 9.0243449 6 02' I4."93 15 9.0817250 652'57."3* 16 9.1356744 746'56."7i 17 9.1866111 8 44' I2."26 IS 9.2348871 9 44' 42. "92 19 9.2808016 10 48' 27. "44 20 9.3246119 n55'24."34 TABLE IT. DEFLECTION ANGLES, FOR LOCATING SPIRAL CURVES IN THE FIELD. Rule for finding a Deflection. Read under the heading corresponding to the point at which the instrument stands, and on the line of the number of the point observed. INSTRUMENT AT S. s = o. No. of Point, Deflection from Tangent, Difference of Deflec- n. i. tion. oo' I 05 05 2 3 4 5 6 7 8 9 10 ir 12 13 14 15 16 17 18 19 20 i i 2 2 3 3 4 5 6 6 7 8 9 10 ii 12 23 37 55 15 40 07 38 12 49 30 14 02 52 4 6 44 44 48 55 30" 20 30 00 50 00 29 19 28 56 .44 SG 15 57 57 12 43 27 24 07 10 14 17 20 24 27 30 34 37 40 44 47 50 54 57 60 63 66 30' 50 10 30 50 10 29 50 09 28 - 48 06 25 42 oo 15 31 44 57 5 2 TABLE II. DEFLECTION ANGLES. INST. AT i. s = o 10'. INST. AT 2. s = o 30'. No. of Deflection 'from Diff . of No. of Deflection from Diff. of Point. aux. tan. Deflection. Point. aux. tan. Deflection. O I 05' 00 05' I 17' 30" 10 7' 30" IO 10 2 IO 2 00 12 30' 15 3 4 5 6 7 8 9 10 ii 12 13 14 15 16 17 18 - 19 20 22 30" 38 20 57 3o' I 9 20 00 i 45 50 2 15 00 2 47 29 3 23 18 4 02 27 4 44 55 5 30 42 6 19 47 7 12 ii 8 07 51 9 06 49 10 09 01 ii 14 28 12 23 08 15 50 19 10 22 30 25 50 29 10 32 29 35 49 39 09 - 42 28 45 47 49 05 52 24 55 40 58 58 62 12 65 27 68 40 3 4 5 6 8 9 10 ii 12 13 14 15 16 17 18 19 20 15 32 30 53 20 i c 17 30 i 45 oo 2 15 50 2 49 59 3 27 29 4 08 18 4 52 26 5 39 54 6 30 40 7 24 44 8 22 06 9 22 45 10 26 39 ii 33 49 12 44 12 17 30 20 50 24 10 27 30 30 50 34 09 37 30 40 49 44 08 47 28 50 46 54 04 57 22 60 39 63 54 67 10 70 23 INST. AT 3. s= i oo'. INST. AT 4. j 1 40'. No. of Deflection from Diff. of No. of Deflection from Diff. of Point. aux. tan. Deflection. Point. aux. tan. Deflection. 36' 40" Q' 10" O 1 O2' 30" 10' 50" I 27 30 12 30 I 51 40 14 10 2 3 15 00 15 2O 2 3 37 30 20 17 30 2O 4 20 4 00 22 30 2 5 5 6 7 8 9 10 ii 12 13 14 15 16 42 30 l p 08 20 i 37 30 2 10 OO 2 45 50 3 24 59 4 07 28 4 53 17 5 42 25 6 34 52 7 30 37 8 29 40 25 50 29 10 32 30 35 50 39 09 42 29 45 49 49 08 52 27 55 45 59 3 62 21 5 6 8 9 10 ii 12 13 M 15 16 25 52 30 I 23 20 i 57 30 2 35 oo 3 ID 50 3 59 59 4 47 28 5 38 16 6 32 24 7 29 50 8 30 34 27 30 30 50 34 10 37 30 40 50 44 09 47 29 50 48 54 08 57 26 60 44 64 02 17 18 19 20 9 32 01 10 37 37 ii 46 29 12 58 35 6 5 36 68 52 72 06 17 18 19 20 9 34 36 10 41 55 ii 52 29 13 06 i 8 67 19 70 34 73 49 53 TABLE II. DEFLECTION ANGLES. INST. AT 5. s = 2 30'. INST. AT 6. j = 3 30'. No. of Deflection from Diff. of No. of Deflection from Diff. of Point. aux. tan. Deflection. Point. aux. tan. Deflection. I i 35' oo" I 22 30 . .12' 30" I 2 14' 10" 2 OO OO 14' 10" 2 3 4 5 I 06 40 47 30 25 00 15 50 19 10 22 30 25 2 3 4 5 I 42 30 I 21 40 57 30 3 17 30 20 50 24 10 27 30 6 3 30 6 oo 30 7 8 9 10 ii 12 13 14 15 16 17 18 20 1 02 30 I 38 20 2 17 30 3 oo oo 3 45 50 4 34 59 5 27 28 6 23 15 7 22 23 8 24 48 9 30 31 10 39 32 ii 51 48 13 07 20 32 3O 35 50 39 10 42 30 45 50 49 09 52 29 55 47 59 08 62 25 65 43 69 01 72 16 75 32 7 8 9 10 ii 12 13 14 15 16 17 18 19 20 35 I 12 30 i 53 20 2 37 30 3 25 oo 4 15 49 5 09 58 6 07 27 7 08 15 8 12 21 9 19 46 10 30 28 ii 44 27 13 oi 41 35 37 30 , 40 50 44 10 47 30 50 49 54 09 57 29 60 48 64 06 67 25 70 42 73 59 77 14 INST. AT 7. ,$ = 4 40'. INST. AT 8. s = 6 co'. No. of Deflection from Diff. of No. of Deflection from Diff. of Point. aux. tan. Deflection. Point. aux. tan. Deflection. I 2 3 4 6 7 3 00' 00" 2 44 10 2 25 00 2 O2 30 I 36 40 I 07 30 35 00 * 5' 50" 19 10 22 30 25 50 29 10 32 30 35 f 2 3 4 5 6 7 352 / 3i // 3 35 oo 3 14 10 2 50 OO 2 22 30 I 51 40 I 17 30 40 17' 31" 20 50 24 10 27 30 30 50 34 10 37 30 8 4 40 8 OO 40 9 10 it 12 13 14 15 16 17 18 I 22 30 2 08 20 2 57 30 3 50 oo 4 45 49 5 44 58 6 47 26 7 53 14 9 02 19 10 14 43 42 30 45 50 49 10 52 30 55 49 59 9 62 28 65 48 69 05 72 24 75 4i 9 10 ii 12 13 14 15 16 17 18 45 i 32 30 2 23 20 3 17 30 4 15 oo 5 15 49 6 19 58 7 27 26 8 38 13 9 52 18 45 47 30 50 50 54 10 57 30 60 49 64 09 67 28 70 47 74 05 77 22 19 20 ii 30 24 12 49 21 78 57 19 20 ii 09 40 12 30 20 80 40 J 54 TABLE II. DEFLECTION ANGLES. INST. AT g. ^ = 7 30'. INST. AT 10. j = 9 10'. No. of Deflection from Diff. of No. of Deflection from Diff. of Point. aux. tan. , Deflection. Point. aux. tan. Deflection. I 2 3 4 5 6 7 8 9 4 5i' 41" 4 32 31 4 10 oi 3 44 10 3 15 oo 2 42 30 2 06 40 I 27 30 45 00 19' 10" 22 30 25 51 29 10 32 30 35 50 39 10 42 30 45 I 2 3 4 6 7 8 9 557 / 32 // 5 36 42 5 12 31 4 45 oi 4 14 10 3 40 oo 3 02 30 2 21 40 i 37 30 50 20' 50" 24 II 27 30 30 51 34 10 37 30 40 50 44 10 47 30 10 50 50 10 CO 5 ii 12 13 14 15 16 17 18 19 20 i 42 30 2 38 20 3 37 30 4 40 oo 5 45 49 6 54 57 8 07 25 9 23 ii 10 42 16 12 04 38 52 30 55 50 59 i 62 30 65 49 69 08 72 28 75 46 79 5 82 22 ii 12 13 15 16 17 18 19 20 55 i 52 30 2 53 20 3 57 30 5 05 oo 6 15 49 7 29 57 8 47 24 10 08 10 ii 32 14 55 57 30 60 50 64 10 67 30 70 49 74 08 77 27 80 46 84 04 INST. AT ii. s = 11 oo'. INST. AT 12. j = 13 oo'. No. of Deflection from Diff. of No. of Deflection from Diff. of Point. aux. tan. Deflection. Point aux. tan. Deflection. I 2 3 4 7 jo' 04" 6 47 33 6 21 42 5 52 32 5 20 oi 22' 31" 25 51 29 10 32 31 1? c T O I 2 3 4 8 29' 16" 8 05 05 7 37 34 7 06 43 6 32 32 24' Ii" 27 31 30 51 34 ii 5 6 4 44 10 4 05 oo 39 I0 6 5 55 oi 5 T 4 IT 40 50 8 9 10 ii 3 22 30 2 36 40 i 47 30 55 00 42 3 45 50 49 10 52 30 55 7 8 9 10 ii 4 30 oo 3 42 30 2 51 40 i 57 30 I OO OO 44 ii 47 30 50 50 54 10 57 30 Co 60 12 I OO OO 12 ^ oo 62 30 65 13 14 15 16 17 18 20 2 02 30 3 08 20 4 17 30 5 29 59 6 45 48 8 04 57 9 27 24 10 53 09 65 50 69 10 72 30 75 49 79 9 82 27 85 45 13 14 16 17 18 19 20 i 05 oo 2 12 30 3 23 20 4 37 30 5 54 59 7 15 48 8 39 56 10 07 23 67 30 70 50 74 10 77 29 80 49 84 08 87 27 55 TABLE II. DEFLECTION ANGLES. INST. AT 13. j= 15 10'. INST. AT 14. j = 17 30'. No. of Deflection from; Diff. of No. of Deflection from Diff. of Point. aux. tan. Deflection. Point. aux. tan. Deflection. O I 2 9 55' 10" 9 29 18 9 oo 06 25' 52" 29 12 O I 2 n 27' 45" ii oo 13 10 29 20 27' 32" 30 53 3 4 5 6 7 8 9 10 8 27 35 7 5i 44 7 12 32 6 30 02 5 44 ii 4 55 oo 4 02 30 3 06 40 32 31 35 5i 39 12 42 30 45 5i 49 ii 52 30 55 50 59 1 3 4 5 6 7 8 9 10 9 55 08 9 17 36 8 36 45 7 52 33 7 05 02 6 14 ii 5 20 oo .4 22 30 34 12 37 32 4 5i 44 12 47 31 50 51 ' 54 ii 57 30 60 50 ii 2 07 30 62 ^o ii 3 21 40 . 64 10 12 13 'or r *: 12 13 2 17 30 1 IO OO 67 30 70 70 14 I 10 00 i _ 14 00 15 16 17 18 19 20 2 22 30 3 38 20 4 57 30 6 19 59 7 45 48 9 14 56 72 30 75 50 79 i 82 29 85 49 89 08 15 16 17 IB '9 20 i 15 oo 2 32 30 3 53 20 5 17 30 6 44 59 8 15 48 75 77 30 80 50 84 10 87 29 90 49 INST. AT 15. j 20 co'. INST. AT 16. s = 22 40'. No. of Point. Deflection from aux. tan. Diff. of Deflection. No. of Point. Deflection from aux. tan. Diff. of Deflection. O I 2 3 4 6 8 1 3 07' 03" 12 37 49 12 05 16 ii 29 23 10 50 TO 10 07 37 9 21 45 8 32 34 7 40 02 29' 14" 32 33 35 53 39 J 3 42 33 45 52 49 ii 52 32 O I 2 3 4 5 6 7 8 14 53' 03" 14 22 09 13 47 54 13 10 20 12 29 26 ii 45 12 10 57 39 10 06 46 9 12 34 30' 54" 34 15 37 34 40 54 44 14 47 33 50 53 54 12 9 10 ii 12 13 14 15 6 44 ii 5 45 oi 4 42 30 3 36 40 2 37 30 i 15 oo oo 55 5 1 59 10 62 31 65 50 69 10 72 30 75 So 9 10 ii 12 13 14 15 8 15 03 6 14 ii 6 10 qi 5 02 30 3 5i 40 2 37 30 I 2O OO 57 3 T 60 52 64 10 67 3i 70 50 74 10 77 30 80 16 I 20 OO 16 00 17 18 19 20 2 42 30 4 08 20 5 37 30 7 09 59 85 50 89 10 92 29 17 18 19 20 i 25 oo 2 52 30 4 23 20 5 57 30 87 30 90 50 94 10 56 TABLE II. DEFLECTION ANGLES. INST. AT 17. j = 25 30'. INST. AT 18, -y 28 30'. No. of Deflection from! DiiL of No. of Deflection from Diff. of Point. aux. tan. Deflection. Point. aux. tan. Deflection. I 2 3 4 5 6 7 8 9 10 ii 12 16 45' 48' 16 13 ii 15 37 15 14 57 59 14 15 24 13 29 29 12 40 14 ii 47 41 10 51 47 9 52 35 8 50 03 ' 7 44 12 6 35 oi 32' 37" 36 56 39 i 6 42 35 45 55 49 15 52 33 55 54 59 12 62 32 65 51 69 IT O I 2 3 4 5 6 7 8 9 10 ii 12 IS 10 59 17 33 21 16 52 23 16 08 05 15 20 28 14 29 32 13 35 17 12 37 42 ii 36 49 10 32 36 9 25 03 8 14 12 34' 1 8" 37 38 40 58 44 18 47 37 50 56 54 15 57 35 60 53 64 13 67 33 70 51 13 5 22 30 72 31 13 7 oo oi 74 it 14 15 16 17 4 06 40 2 47 30 i 25 oo oo 75 5 79 10 82 30 85 14 15 16 17 5 4 2 30 4 21 40 2 57 30 i 30 oo 77 3 r 80 50 84 10 87 30. 18 i 30 oo 9 18 00 9 19 20 3 02 30 4 38 20 92 30 95 50 19 20 i 35 oo 3 12 30 95 97 30 INST. AT 19. j = 31 40'. INST. AT 20. j- = 35 oo' . No. of Deflection from Diff. of ! No. of Deflection from Diff. of Point. aux. tan. Deflection . i Point. aux. tan. Deflection. I 2 3 4 5 6 7 8 9 10 n 12 13 20 5 1' 33" 20 15 32 19 36 II 18 53 3i 18 07 31 17 IS 12 16 25 33 15 29 36 14 30 20 13 27 44 12 21 50 II 12 36 10 oo 04 8 44 12 36' 01 " 39 21 42 40 46 oo 49 J 9 52 39 55 57 59 16 62 36 65 54 69 14 75 32 75 52 70 II I 2 3 4 5 6 8 9 10 ii 12 13 23 04' 36" 22 26 52 21 45 48 21 01 25 20 13 42 19 22 40 18 28 19 17 30 39 16 29 40 15 25 23 14 17 46 13 06 51 ii 52 37 10 35 04 37' 44" 41 04 44 23 47 43 51 02 54 21 57 40 60 59 64 17 67 37 7'^> 55 74 -14 77 33 80 52 M 15 16 17 18 19 7 25 oi 6 02 30 4 36 40 3 07 30 i 35 oo 82 31 85.50 89 10 92 30 95 IOO 14 15 16 17 18. 19 9 r 4 12 7 50 oi 6 22 30 4 51 40 3 17 30 i 40 84 ii 87 31 90 50 94 10 97 30 1 60 20 i 40 20 00 57 TABLE III. DEGREE OF CURVE AND VALUES OF THE COORDINATES x AND y, FOR EACH CHORD-POINT OF THE SPIRAL FOR VARIOUS LENGTHS OF THE CHORD. f. CHORD-LENGTH = IO. n. nc. />. y> Xt Log x . j 10 i 40' oo" IO.OOO 0.0145 8.162696 2 20 3 20 02 20. ooo .0727 8.861664 3 30 5 oo 06 29.99-) . 2036 9.308815 4 40 6 40 13 39.996 .4363 9.639792 5 50 8 20 26 49.990 .7998 9.903002 6 60 10 oo 45 59-976 1.323 0.121624 7 70 II 41 12 69.951 2.035 0.308622 8 80 13 21 48 79.907 2.965 0.471991 9 9 ID 02 34 89.838 4.140 0.617015 10 100 16 43 3i 99.732 5-589 0.747370 ii no 18 24 42 109 578 7-340 0.805712 12 120 20 06 07 119-359 9.419 0.974022 13 130 21 47 48 129.059 11.853 1.073818 14 140 23 29 46 138.655 14.665 1.166281 15 150 25 12 02 148.125 17.879 .252352 16 160 26 54 39 157.439 21.517 .332788 17 170 28 37 38 166.569 25-598 .408205 18 1 80 3O 21 01 175-479 30 138 .479112 19 i go 32 04 48 184.132 35.150 545931 20 200 33 49 02 192.487 40.645 .609013 35 33 46 TABLE III. r. CHORD-LENGTH = n. f c. D s . y- x. Log x. I II i 30' 55" 1 1 . CO3 0.0160 8.204089 2 j 22 3 oi 50 22.OOO .0800 8.903057 3 33 4 32 48 32.999 .2240 9.350208 4 44 6 03 48 43.996 4799 9 681185 5 55 7 34 52 54.989 .8798 9.944394 6 66 9 06 01 65.974 1.456 0.163017 7 77 10 37 16 76.946 2.239 0.350015 8 88 12 08 37 87.898 3.261 0-513384 9 99 13 40 06 98.822 4-554 0.658408 JO no 15 ii 44 109.706 6.148 0.788763 IT 121 16 43 3i 120.536 8.074 0.907104 12 132 18 15,29 I3L295 10.361 1.015415 13 143 ! 19 47 39 141.965 13-038 1.115210 14 154 I 21 20 01 152.521 16.131 1.207674 15 165 22 52 38 162.937 19.667 .293745 16 | 176 24 25 29 173.183 23.669 .374180 17 I8 7 25 58 36 183.226 28.158 .449598 18 198 27 32 01 193.027 33.152 .520505 19 20g 29 05 45 202.545 38.665 587323 20 220 30 39 48 2H.735 44.710 .650405 32 14 ii . c. CHORD-LENGTH = 12. ;/. 11C. D s . - y- jr. Log jr. I 12 i 23' 20" 12.000 0.0175 8.241877 2 24 2 46 41 24.OOO .0873 8.940845 3 36 4 10 03 35-999 2443 9o87997 4 4 8 5 33 28 47.996 .5236 9.718974 5 60 6 56 55 59-988 .9598 9.982183 6 72 8 20 26 71.971 1.588 o. 200806 7 8 4 9 44 oi 83.941 2.442 0.387803 8 9 6 ii 07 42 95.889 3.558 0.551172 9 108 T2 31 28 107.806 4.968 0.696196 10 120 13 55 21 119.679 6.707 0.826551 ii 132 15 19 22 I3L493 8.8c8 0.944893 12 144 16 43 31 143.231 11-303 .053204 13 156 18 07 48 154.871 14.223 .152999 14 168 19 32 15 166.386 17-598 .245462 15 180 20 56 53 177-749 21-455 .331533 16 192 22 21 43 188.927 25.821 .411969 17 204 23 46 44 199.883 30.718 .487386 18 216 25 ii 59 210.575 36.165 558293 19 228 26 37 28 220.958 42.181 .625113 20 240 28 03 12 230.984 48.774 .688194 29 29 12 59 TABLE III. c. CHORD-LENGTH = 13. n. nc. Ds. y- X. Log x. l 13 i 16' 55" 13.000 0.0189 8.276639 2 26 2 33 52 26.000 0945 8.975607 3 39 3 50 49 38.999 .2647 9 422759 4 52 5 07 48 51-995 5672 9.753736 5 65 6 24 49 64.987 1.040 0.016945 6 78 7 4i 53 77.969 1.720 0.235568 7 91 8 59 oo 90.936 2.646 0.422565 8 104 10 16 12 103.879 3.854 0.585934 9 117 ii 33 28 116.789 5-382 0.730959 JO 130 12 50 49 129.652 - 7.266 0.861313 ii 143 ! 14 08 16 142.451 9-542 0-979655 12 156 15 25 50 155.167 12.245 .087966 13 169 16 43 30 167.776 15.409 .187761 14 182 i3 01 18 180.252 19.064 .280224 15 195 IQ 19 14 192.562 23.243 .366295 16 208 20 37 20 204.671 27.972 .446731 17 221 21 55 34 216.540 33.277 .522148 18 234 23 14 oo 228.123 39-179 .593055 19 247 24 32 35 239.371 45.696 .659874 20 260 25 5i 23 250.233 52.839 1.722956. 27 10 23 c. CHORD-LENGTH = 14. 71. 11C. A y- X. Log x. I 14 i n' 26" 14.000 0.0204 8.308824 2 28 2 22 52 28. coo .1018 9.007792 3 42 3 34 19 41.999 .2851 9-454943 4 56 4 45 48 55.995 .6108 9.785920 5 70 5 57 18 69.986 I.I2O 0.049130 6 8 4 7 "8 51 83.966 1.852 0.267752 7 9 8 8 20 26 97.931 2.849 0.454750 8 112 9 32 04 111.870 4-I5I 0.618119 9 126 10 43 47 125 773 5.796 0.763M3 10 140 ii 55 33 139.625 7.825 0.893498 TI 154 13 07 24 153-409 10.276 1.011840 12 1.68 14 19 20 167.103 13.187 1.120150 13 182 15 31 22 180.682 16.594 1.219946 14 196 16 43 29 194.117 20.531 1.312409 15 2IO *7 55 44 207.374 25.031 1,398480 16 224 19 c6 05 220.415 30.124 1.478915 17 238 i 20 20 34 233.196 35-837 1-554333 1-8 252 21 33 ii -245-670 42.193 1.625240 J 9 266 22 45 56 257./35 49.211 1.692059 20 280 23 53 51 269.481 56.903 I.755I4I 25 ii 55 60 TABLE III. c. CHORD-LENGTH = 15. ?/. j nc. D s . y. x. Log x. i IS i 06' 40" 15.000 0.0218 8.338787 2 30 2 13 2O 30. ooo .1091 9-037755 3- 45 3 20 02 44.998 .3054 9.484907 4 60 4 26 44 59.994 .6545 9.815884 5 75 5 33 28 74.984 1.200 0.079093 6 90 6 40 13 89.964 1.985 0.297716 7 105 7 47 oi 104.926 3-053 0484713 8 1 20 8 53 5i 119.861 4-447 0.648082 9 135 10 oo 45 134.757 6.2IG 0.793107 10 150 ii 07 41 149-599 8.384 o 923461 ii 165 12 14 41 164.367 II.OIO 1.041803 12 1 80 13 21 47 179.039 14.129 1.150114 13 195 14 28 56 193.588 17-779 .249909 14 210 15 36 09 207.983 21.997 .342372 15 225 16 43 28 222.187 26.819 428443 16 2 4 17 50 54 236.159 32.276 .508879 17 255 18 58 25 249.853 38.397 .584296 18 270 20 06 02 263 218 45.207 .655203 19 285 21 13 47 276.198 52.726 1.722022 20 3 00 22 21 39 288.730 60.968 1.785104 23 29 48 c. CHORD-LENGTH = 16. ;/. nc D s . - y- jr. Log jr. I 16 i 02' 30" 1 6 ooo 0.0233 8.366816 2 32 2 05 00 32.000 .1164 9.065784 3 48 3 07 31 47.998 .3258 9-5I2935 4 64 4 10 03 63.994 .6981 9.843912 5 80 5 12 36 79-983 1.260 0.107122 6 96 6 15 ii 95.961 2.117 0.325744 7 112 7 17 47 111.921 3-256 0.512742 8 128 8 20 26 127.852 4-744 0.676111 9 144 9 23 07 143-74I 6. 624 o 821135 10 1 60 10 25 51 159-572 8-943 0.951490 ii I 7 6 ii 28 37 175.325 11.744 .069832 12 IQ2 12 31 28 190.975 15.071 .178142 13 208 13 34 21 206.494 18.964 .277938 14 224 14 37 20 221.848 23.464 .370401 15 240 15 4 21 236.999 28.607 .456472 16 256 16 43 28 251.903 34.428 53 f >97 17 272 17 46 40 266.510 40.957 .612325 18 288 18 49 57 280.766 48.221 .683232 19 304 19 53 20 294.611 56.241 .750051 20 320 20 56 49 307.979 65.032 813133 22 00 23 61 TABLE III. t. CHORD-LENGTH = 17. //. 11C. >*. y+ -V. Log x. i 17 o 58' 49" 17.000 0.0247 8.393M5 2 34 i 57 33 34.000 .1236 9.092113 3 51 2 56 27 50.998 .3461 9 539 2 64 4 68 3 55 19 67.994 7417 9.870241 5 85 4 54 12 84.982 1.560 O.I3345I 6 IO2 5 53 06 101.959 2.249 0.352073 7 II 9 6 52 oo 118.916 3-460 0.539071 8 I 3 6 7 50 57 1 3 5- -842 5.040 o. 702440 9 153 8 49 55 152.725 7.038 0.847464 10 I 7 9 48 56 169.545 9.502 0.977819 ii I8 7 10 48 oo 186.282 12.478 1.096161 12 204 ii 47 07 202.911 16.013 1.204471 13 221 12 46 15 219.400 20.150 1.304267 14 238 13 45 27 235.7H 24-930 396730 15 255 14 44 44 251.812 30.395 .482801 16 272 15 44 03 267.647 36.579 .563236 17 289 16 43 27 283.167 43.5I6 .638654 18 306 17 42 56 298.314 5L234 .709561 19 323 18 42 29 313.024 59.756 .776380 20 340 19 42 07 327.228 69.097 .839462 20 41 49 f. CHORD-LENGTH = 18. ;/. ftf. $ y. a~. Log^r. I 18 o 55' 33" 18.000 0.0262 8.417968 2 36 i 51 07 36.000 .1309 9.116937 3 54 2 46 40 53.998 .3665 9.564088 4 72 3 42 16 71.993 .7853 9.895065 5 90 4 37 5i 89.981 1.440 0.158274 6 1 08 5 33 28 107-957 2.382 0.376897 7 126 6 29 05 125.911 3-663 0.563894 8 144 7 24 45 143.833 5-337 0.727263 9 162 8 20 26 161.708 7-452 0.872288 10 180 9 16 08 179.518 10. 06 1 1.002643 ii 198 10 ii 54 197.240 13.212 1.120984 12 216 ii 07 41 214.847 16.955 1.229295 13 234 12 03 31 232. 3c6 21-335 1.329090 14 252 12 59 24 249.579 26.397 I.42I554 15 270 13 55 20 266.624 32.183 1.507624 16 288 14 51 18 283.391 38.731 1.588060 17 306 15 47 20 299.824 46.076 1.663477 18 324 16 43 27 315.862 54.248 1.734385 *9 342 17 39 37 33L437 63.271 1.801203 20 360 18 35 5i 346.476 73.161 1.864285 19 32 08 62 TABLE III. c. CHORD-LENGTH = 19. n. lie. D s . y> x. Log^r. i 19 o 52' 38" 19.000 0.0276 8.441450 2 3^ i 45 16 38.000 .1382 9.140418 3 57 2 37 54 56.998 .3869 9-587569 4 76 3 30 34 75-993 .8290 9-9i 8 546 5 95 4 23 13 94.980 1.520 0.181755 6 H4 5 15 54 113-954 2.514 0.400378 7 133 6 08 36 132.906 3.867 0.587376 8 152 7 01 19 151.824 5.633 0.750744 9 171 7 54 03 170.692 7.866 0.895769 10 190 8 46 49 189.491 10.620 1.026124 ii 209 9 39. 36 208.198 13.947 1.144465 12 228 10 32 26 226.783 17.897 1.252776 13 247 ii 25 18 245.212 22.520 .352571 14 266 12 18 12 263.445 27.863 445035 15 285 13 II 09 281.437 33-971 53H05 16 304 14 04 09 299.135 40.883 .611541 17 323 14 57 ii 316.481 48.636 .686958 18 342 15 50 16 333-410 57.262 .757866 19 361 16 43 25 349- 8 5i 66.786 .824684 20 380 17 36 33 365-725 77.226 .887766 18 29 54 c. CHORD-LENGTH = 20. 11. nc. D s . * }'- X. Log x. i 20 o 50' oo" 20.000 0.0291 8.463726 2 40 i 40 oo 40.000 .1454 9.162694 3 60 2 30 01 59.998 .4072 9.609845 4 80 3 20 02 79-993 .8726 9.940822 5 IOO 4 10 03 99-979 i. 600 0.204032 6 120 5 oo 05 119-952 2.646 0.422654 7 140 5 5 8 139.901 4.071 0.609652 8 1 60 6 40 13 159 8i5 5.930 0.773021 9 180 7 30 18 179.676 8.280 0.918045 10 200 8 20 26 199.465 11.179 1.048400 ii 22O 9 i 34 219.156 14.681 1.166742 12 240 10 oo 44 238.719 18.839 1.275052 13 260 10 50 56 258.118 23-705 1.374848 14 1 280 ii 41 10 277-310 29.330 1.467311 15 3OO 12 31 26 296.249 35-759 1.553382 16 320 13 21 45 314.879 43-035 1.633817 17 340 14 12 c6 333.138 51.196 1.709235 18 360 15 02 29 350.958 60.276 1.780142 19 380 15 52 55 368.264 70.301 1.846961 20 400 16 43 25 384.974 81.290 1.910043 17 33 & TABLE III. c. CHORD-LENGTH = 21. i n. nc. D s . }' X. Log. x. I 21 o 47' 37" 2 1 . OOO 0.0305 8.484915 2 42 i 35 14 42.000 .1527 9.183883 3 63 2 22 52 62.998 .4276 9-63I035 4 84 3 10 30 83.992 .9162 9.962012 5 105 3 58 08 104.978 1.680 0.225221 6 126 4 45 47- 125.949 2.779 0.443844 7 147 5 33 27 146. 896 4.274 0.630841 8 168 6 21 08 167.805 6.226 0.794210 9 189 7 08 50 188.660 8.694 0.939235 TO 2IO 7 56 33 209.438 11.738 .069589 II 231 8 44 18 230.114 15.415 .187931 12 252 9 32 03 250.655 19.781 .296242 13 273 10 19 51 271.023 24.891 396037 M 294 n 07 40 291.176 30.796 .488500 15 315 ii 55 3i 3II.062 37.547 .574571 16 336 12 43 24 330.623 45.186 .655007 17 357 13 31 20 349-795 53.756 .730424 18 378 14 19 17 368. 506 63.289 .801331 IQ 399 15 07 17 386.677 73.816 .868150 15 55 19 r. CHORD-LENGTH = 22. ;/. nc. D s . y- X. Log. x. i 22 45' 27" 22.000 0.0320 . 8.505119 2 44 i 30 53 44. ooo .1600 9 204087 3 66 2 l6 22 65.998 .4480 9.651238 4 88 3 oi 50 87.992 9599 9.982215 5 no 3 47 18 109.977 1.760 0.245424 6 132 4 32 48 I3L947 2.911 0.464047 / 154 5 18 18 153.891 4.478 0.651045 8 176 6 03 48 I75-796 6.522 0.814414 9 198 6 49 19 197.643 9.108 0.959438 10 220 7 34 5i 219.411 12.297 .089793 ii 242 8 20 25 241.071 16.149 .208134 12 264 9 06 oo 262.591 20.7^3 .316445 13 286 9 5i 36 283.929 26.076 .416240 14 308 10 37 13 305042 32.263 .508704 15 330 ii 22 53 325.874 39-335 594775 1.6 352 12 08 34 346.367 47.338 .675210 17 374 12 54 16 366.451 56.3*5 1.750628 18 396 13 40 oi 386.054 66.303 1.821535 14 25 49 0-!- TABLE III. r. CHORD-LENGTH = 23. n. nc. D*. y- X. Log. x. I 23 o'"' 43' 29" 23.000 0.0335 8.524424 2 46 I 26 58 46.000 .1673 9.223392 3 69 2 10 26 68.998 .4683 9-670543 4 92 2 53 56 91.991 1.004 0.001520 5 H5 3 37 26 114.976 .1.840 0.264729 6 138 4 20 56 137-945 3.043 0.483352 7 161 5 04 26 160.886 4.681 0.670350 8 184 5 47 58 . 183.787 6.819 0.833719 9 207 6 31 30 206.627 9.522 0.978743 10 230 7 15 04 229.384 12.856 1.109098 ii 253 7 58 38 252.029 16.883 .227439 12 276 8 42 13 274 527 21.665 .335750 *3 299 9 25 49 296.835 27.261 435545 14 322 10 09 27 318.907 33.729 .528009 15 345 10 53 06 340.686 41.123 .614080 16 368 1 1 36 47 362.110 49.490 694515 17 391 12 20 29 383.108 58.875 .769933 13 04 13 r. CHQRD-LENGTH = 24. 72. nc. D s . ;' X. Log. r. I 24 41' 40" 24. ooo 0.0349 8.542907 2 48 1 23 20 48.000 .1745 9.241875 3 72 2 05 OO 71.998 .4887 9.689027 4 96 2 46 41 95.991 1.047 0.020004 5 1 20 3 28 22 119-975 1.920 0.283213 6 144 4 10 03 143.942 3.176 0.501836 7 1 68 4 5r 45 167.881 4.885 0.688833 8 192 5 33 28 191-777 7.115 0.852202 9 216 6 15 10 215.611 9.936 0.997226 10 240 6 5^ 54 239-358 13.415 1.127581 n 264 7 38 39 262.987 17.617 1.245923 12 288 8 20 25 286.463 22.607 i 354234 13 14 312 336 9 02 12 9 44 oo 309.741 332.773 28.446 35.196 .1.454029 1.546492 15 360 10 25 48 355.499 42.910 1.632563 16 384 Ji 07 30 377.854 51.641 1.712999 17 408 ii 49 31 399-765 61.435 1.788416 12 31 25 65 TABLE III. C. L.tlU IvU-I^l^INLrl n = 25. n. nc. D s . y* X. Log. x. I 25 o 40' oo" 25.000 0.0364 8.560636 2 50 I 2O CO 50.000 .1818 9.259604 3 75 2 00 00 74-997 .5090 9-706755 4 100 2 40 OI 99.991 1.091 0.037732 5 I2 5 3 20 02 124.974 2.OOO 0.300942 6 150 4 oo 03 149.940 3-308 0.519564 7 175 4 40 04 174.876- 5.088 0.706562 8 2GO 5 20 06 199.768 7.412 0.869931 9 225 6 oo 09 224-595 IO.35O 014955 10 250 6 40 13 249-33I 13-974 .145310 ii 275 7 20 17 273-945 18.351 .263652 12 300 8 00 22 298.398 23.548 .371962 13 325 8 40 28 322.647 29.632 .471758 14 350 9 20 35 346.638 36.662 .564221 '15 375 10 oo 43 37-3i i 44-698 .650292 16 400 10 40 52 393- 59 8 53-793 .730727 II 21 03 f. CHORD-LENGTH = 26. n. nc. D s . >' X. Log. x. I 26 o 38' 28" 26.000 0.0378 8.577669 2 52 i 16 56 52.000 .1891 9.276637 3 78 i 55 24 77.997 .5294 9.723789 4 104 2 33 52 103.990 1. 134 0.054766 5 130 3 12 20 129.973 2.080 0.317975 6 156 3 50 48 155.937 3-440 0.536598 7 182 4 29 18 181.871 5.292 0.723595 8 208 5 07 48 207.759 7.708 0.886964 9 234 5 46 18 233.579 10.764 1.031989 10 260 6 24 48 259-304 14.533 1.162343 ii 286 703 20 284.903 19.085 1.280685 12 312 7 4 1 52 310.334 24.490 1.388996 13 338 8 20 25 335-553 30.817 1.488791 14 364 8 58 59 360.504 38.129 1.581254 15 39 9 37 33 385.124 46.486 1.667325 10 1 6 OQ 66 TABLE III. c. CHORD-LENGTH = 27. n. 11C. D t . }' X. Log. x. I 27 o 37' 02" 27.000 0.0393 8.594060 2 54 i 14 04 54.000 .1963 9.293028 3 81 i 5i 07 80.997 .5498 9.740179 4 108 2 28 10 107.990 1.178 0.071156 5 135 3 05 12 I34-972 2.160 0.334365 6 162 3 42 15 161.935 3-573 0.552988 7 189 4 19 19 188.866 5-495 0.739986 8 216 4 56 23 215-750 8.005 0.903355 9 243 5 33 28 242.562 11.178 1.048379 10 270 6 10 32 269.277 15.092 LI78734 n 297 6 47 38 295.860 19.819 1.297075 12 324 7 24 44 322.270 25.432 1.405386 13 35i 8 or 51 348.459 32.002 1.505181 14 378 8 38 59 374-369 39-595 1.597645 15 405 9 16 07 399.936 48.274 1.683716 9 53 16 c. CHORD-LENGTH = 28. n. nc. /& y- jr. Log. jr. I 28 o 35' 42" 28.000 0.0407 8.609854 2 56 i ii 26 55-999 .2036 9.308822 3 84 i 47 08 83.997 .5701 9-755973 4 112 2 22 52 in. 990 1.222 0.086950 5 140 2 58 36 139-97I 2.24O 0.350160 6 168 3 34 19 1 67.933 3-705 ' 0.568782 7 196 4 10 03 195.862 5.699 o.755/So 8 224 4 45 48 223.740 8.301 0.919149 9 252 5 21 32 251.546 H.592 1.064173 10 280 5 57 17 279.251 15.650 1.194528 ii 308 6 33 03 306.818 20.553 1.312870 12 336 7 08 50 334.206 26.374 1.421180 13 364 7 44 36 361.365 33.188 1.520976 14 392 8 20 24 388.235 4I.O62 I.6I3439 8 56 13 67 TABLE III. CHORD-LENGTH - 29. n. nc. D s . y- X. Log. oc. I 29 o 34' 29" 29.000 0.0422 8.625094 2 53 i 08 58 57-999 .2109 9.324062 3 8? I 43 27 86.997 595 9.771213 4 116 2 17 56 115-989 1.265 0.102190 5 145 2 52 26 144.970 2.320 0.365400 6 174 3 26 55 173-930 3-837 0.584022 7 203 4 01 26 202.857 5.902 0.771020 8 232 4 35 56 231.731 8.598 0.934389 9 261 5 10 26. 260.530 12.006 .079413 10 290 5 44 57 289.224 16.209 .209768 ii 319 6 19 29 317.776 21.287 .328110 12 348 6 54 01 346.142 27-316 .436420 13 377 7 28 34 374.271 34-373 .536216 14 406 8 03 07 402.100 42.528 .628679 8 37 40 CHORD-LENGTH = 30. n. nc. Ds. y- jr. Log. x. T 3 o 33' 20" 30.000 0.0436 8.639817 2 60 i 06 40 59-999 .2182 9-338785 3 90 i 40 oo 89.997 .6108 9-785937 4 120 2 13 20 119.989 1.309 0.116914 5 150 2 46 41 149.969 2.400 0.380123 6 1 80 3 20 02 179.928 3-970 0.598746 7 210 3 53 22 209.852 6.106 0.785743 8 240 4 26 44. 239.722 . 8.894 0.949112 9 270 5 oo 05 269.514 12.420 .094137 10 3CO 5 33 27 299.197 16.768 .224491 ii 330 6 06 49 328.734 22.021 .^42833 12 360 6 40 12 358.078 28.258 .451144 13 39 7 13 36 387.176 35.558 .550939 7 47 oo 68 TABLE ,./ ' / c. CHORD-LENGTH = 31. n. nc. E , y *. Log x. i 31 o 32' 15" 31.000 0.0451 8.654058 2 62 i 04 31 61.999 .2254 9.353026 3 93 i 36 47 92.997 .6312 9.800177 4 124 2 09 C2 123.988 1-353 0.131154 5 155 2 41 18 154.968 2-479 0.394363 6 1 86 3 13 34 185 925 4.102 0.612986 7 217 3 45 50 216.847 6.309 0.799984 8 248 4 1 8 07 247-713 9.191 0-963353 9 279 4 50 24 278.498 12.834 1.108377 10 310 5 22 41 309. 1 70 17.327 1.238732 ii 341 5 54 59 339.692 22.755 1.357073 12 372 6 27 17 370.014 29.200 1.465384 13 6 59 35 400.082 36.743 1.565179 7 3i 53 \ CHORD-LENGTH = 32. n. nc. D s . y- X. Log*. I 3 2 o 31' 15" 32.000 0.0465 8.667846 2 64 I 02 30 63 999 .2327 9.366814 3 96 i 33 45 95-997 .6516 9.813965 4 128 2 05 00 127.988 1.396 0.144942 5 1 60 2 36 16 159.967 2-559 0.408152 6 192 3 07 31 101.923 4-234 0.626774 7 224 3 38 47 223.842 6.513 0.813772 8 256 4 10 03 255.703 9.487 0.977141 9 288 4 4i 19 287.481 13.248 1.122165 10 320 5 12 36 319.144 17.886 1.252520 ii 352 5 43 53 350.649 23.489 1.370802 12 384 6 15 10 381.950 30. 142 I.479 r 72 13 416 6 46 28 412.988 37.929 1.578968 7 17 46 69 TABLE III. c. CHORD-LENGTH = 33. n. 12C. D s . y- X. Log. x. i 33 o 30' 19" 33.000 0.0480 8.681210 2 66 I OO 36 65.999 .2400 9.380178 3 99 i 30 55 98.997 .6719 9.827329 4 132 2 OI 13 131.988 1.440 0.158306 5 165 2 3 I 3 2 164.966 2.639 0.421516 6 198 3 oi 50 197.921 4-367 0.640138 7 231 3 32 09 230.837 6.716 0.827136 8 264 4 02 28 263.694 9.784 0.990505 9 297 4 32 48 296.465 13.662 1-135529 JO 33 5 03 07 329.117 18.445 1.265884 n 363 5 33 27 361.607 24.223 1.384226 12 396 6 03 47 393.886 31-084 1.492536 6 3.4 07 c. CHORD-LENGTH = 34. n. nc. &* y. X. Log. x. I 34 o 29' 25" 34.000 0.0495 8.694175 2 68 o 58 49 67.999 .2473 9093I43 3 102 i 28 14 101.996 .6923 9.840294 4 136 i 57 39 135.987 1.483 O.I7J27I 5 170 2 27 04 169.965 2.719 0.434481 6 204 2 56 2() 203.918 4.499 0.653103 7 233 3 25 55 237.832 6 920 0,840101 8 272 3 55 20 271.685 10.080 1.003470 9 306 4 24 46 305.449 14.076 1.148494 10 340 4 54 12 339.090 19.004 1.278849 ii 374 5 23 38 372.565 24-957 1.397191 12 408 5 53 05 405.822 32.026 1.505501 6 22 II 70 TABLE III. c. CHORD-LENGTH = 35. ;/. 11C. D s . y- A\ Log a\ I 35 o 28' 34" 35.000 0.0509 8.706764 2 70 o 57 09 69.999 2545 9.405732 3 105 i 25 43 104.996 .7127 9.852883 4 140 I 54 17- 139.987 1.527 0.183860 5 175 2 22 52 174.964 2.799 0.447070 C 210 2 51 27 209.916 4.631 0.665692 7 245 3 20 01 244.827 7.123 0.852690 8 280 3 4S 36 279.675 10.377 .016059 9 3J5 4 17 12 314-433 14.490 .161083 10 35 4 45 47 349- 6 3 I9-563 .291438 ii 385 5 14 23 383-523 25.691 .409780 12 420 5 43 oo 417.758 32.968 .518090 6 09 36 \ c. CHORD-LENGTH = 36. //. tic. D s . y- X. Log jc. I 36 o 27' 47" 36.000 O.O524 8.718998 2 72 o 55 33 71.999 .26l8 9.417967 3 io3 I 23 20 107.996 7330 9.865118 4 144 I 51 07 143.987 I-57I 0.196095 5 1 80 2 18 54 179.963 2.879 0.459304 6 216 2 46 41 215.913 4- 764 0.677927 7 252 3 14 28 251.822 7.327 0.864924 8 2:8 3 42 15 287.666 10.673 1.028293 9 324 4 10 03 323-417 14.905 1.173318 10 360 4 37 5i 359037 20. 122 1.303673 ii 39 6 5 05 39 394.480 26.425 1.422014 5 33 27 TABLE III. 11. 11C, z>* / x. Log x. I 37 o 27' 02" 37.000 0.0538 8.730898 2 74 o 54 03 73-999 .2691 9.429866 3 in I 21 05 110.996 7534 9.877017 4 148 I 48 07 147.986 1.614 0.207994 5 185 2 1 5 09 184.962 2-959 0.471203 6 222 2 42 II 221.911 4.896 0.689826 7 259 3 09 ] 3 258.817 7-530 0.876824 8 296 3 36 15 295.657 10.970 1.040193 9 333 4 03 17 332.400 15-319 1.185217 10 370 4 30 20 369.010 20.681 L3I5572 ii 407 4 57 23 405-438 27.159 I.4339I3 5 24 26 e. CHORD-LENGTH = 37. c. CHORD-LENGTH = 38. 11. m\ D* * x Log jr. i 38 o'' 26' 19" 38.000 0-0553 8.742480 2 76 o 52 39 75-999 .2763 9.441448 3 114 i 18 57 113.996 7737 9.888599 4 152 i 45 16 151.986 1.658 0.219576 5 190 2 ii 35 189.961 3.039 0.482785 6 228 2 37 54 227.909 5.028 0.701408 7 266 3 04 14 265.812 7-734 0.888406 8 304 3 30 33 303.648 11.266 1.051774 9 342 3 56 53 34L384 15.733 1.196799 10 380 4 23 13 378.983 21.240 I.327T54 ii 418 4 49 33 416.396 27.893 1-445495 5 15 53 72 TABLE III. c. CHORD-LENGTH = 39. //. nc. D*. >' X. Log x . I 39 o 25' 38" 39.000 0.0567 8.753761 2 78 o 51 17 77-999 .2836 9.452729 3 117 I 16 55 116.996 .7941 9.899880 4 156 I 42 34 I55.9 8 5 1.702 0.230857 5 *95 2 08 13 194.960 5.119 0.494066 6 234 2 33 51 233.906 5.160 0.712689 7 273 2 59 30 272.807 7.938 0.899687 8 312 3 25 09 311.638 11.563 1.063055 9 35i 3 50 48 350.368 16.147 1.208080 10 390 4 16 28 388.956 21.799 L338435 4 42 07 c. CHORD-LENGTH ~ 40. n. 1IC. D s . }' X. Log x. I 40 o 25' oo'' 40. ooo 0.0582 8.764756 2 80 o 50 oo 79-999 .2909 9.463724 3 120 I 15 oo 1 1 9. 996 .8145 9.910875 4 1 6O I 40 oo I59-985 1.745 0.241852 5 2OO 2 05 CO 199-959 3.199 0.505062 6 240 2 30 01 239.904 5.293 0.723684 7 280 2 55 oi 279.802 8.141 0.910682 8 32O 3 20 oi 319.629 11.859 1.074051 9 360 3 45 02 359-352 16.561 1.219075 10 4OO 4 10 03 398.929 22.358 1.349430 4 35 03 c. CHORD-LENGTH = 41. n. nc. />* y- X. Log x . i 41 o 24' 24" 41.000 0.0596 8.775480 2 82 o 48 47 81.999 .2982 9.474448 3 123 i 13 10 122.996 .8348 9.921599 4 164 i 37 34 163 985 1.789 0.252576 5 205 2 oi 57 204.958 3-*79 0.515786 6 246 2 26 21 245.901 5-425 o. 734408 7 287 2 50 45 286.797 8.345 0.921406 8 328 3 15 09 327.620 12.156 1.084775 9 369 3 39 33 368.336 16.975 1.229799 10 410 4 03 57 408.903 22.917 1.360154 4 28 21 73 TABLE III. c. CHORD-LENGTH = 42. n. 11 r. D s . / X. Log x . i 42 o 23' 49" 42.000 0.0611 8.785945 2 84 o 47 37 83.999 .3054 9.484913 3 126 i n 26 125.996 .8552 9.932065 4 168 i 35 14 167.984 1.832 0.263042 5 2IO i 59 02 209.957 3-359 0.526251 6 252 2 22 52 251.899 5-557 0.744874 7 294 2 46 41 293.792 8.548 0.931871 8 336 3- 10 30 335.6ii 12.452 1.095240 9 373 3 34 19 377.319 17-389 1.240265 10 420 3 58 08 418.876 23.476 1.370619 4 21 57 f. CHORD-LENGTH = 43. . nc. />*. y> X. Log x. i 43 o 23' 15" 43.000 0.0625 8.796164 2 86 o 46 31 85.999 3127 9-495I33 3 129 I 09 46 128.996 .8755 9.942284 4 172 i 33 02 171.984 1.876 0.273261 5 215 i 56 17 214-955 3-439 0.536470 6 2^8 2 19 33 257-897 5-690 0.755093 7 301 2 42 48 300.787 8-752 0.942090 8 344 3 06 04 343.601 12.749 1.105459 9 3S7 3 29 20 386.303 17.803 1.250484 10 430 3 52 35 428.849 24.035 1.380839 4 15 50 c. CHORD-LENGTH = 44. n. nc. D s . }' X. Logx. i 44 o 22' 44" 44.000 0.0640 8.806149 2 88 o 45 27 87.999 .3200 9-505117 3 132 i 08 ii 131-995 .8959 9.952268 4 176 i 30 55 175.984 1.920 0.283245 5 220 i 53 38 219-954 3-519 0.546454 6 264 2 l6 22 263.894 5.822 0.765077 7 308 2 39 06 307.78, 8.955 0.952075 8 352 3 oi 50^. 351-592 13.045 1.115444 9 396 3 24 34 395.287 18.217 1.260468 3 47 18 74 TABLE III. c. CHORD-LENGTH = 45. ti. 7l('. z y- jr. Log jr. 45 o 22' 13" 45.000 0.0655 8.815908 2 9 o 44 27 89.999 .3272 9-5I4877 3 135 i 06 40 134-995 .9163 9.962028 4 i So i 28 53 179.983 1.963 o. 293005 5 225 i 5i 07 224.953 3-599 0.556214 6 270 2 13 20 269.892 5.954 0.774837 7 315 2 35 34 314.778 9-159 0.961834 8 360 2 57 43 359-583 I3.34I 1.125203 9 405 3 20 01 404.271 18.631 1.270228 3 42 15 c. CHORD-LENGTH ~ 46. ;/. nc. D s . y- jr. Log x. I 46 2l' 44" 46. ooo 0.0669 8.825454 2 92 o 43 29 91.999 3345 9-524422 3 138 i 05-13 137.995 .9366 9.971573 4 184 i 26 58 183.983 2.007 0.302550 5 230 i 48 42 229.952 3.679 0.565759 6 276 2 10 26 275.889 6.087 0.784382 7 322 2 32 II 321.773 9.362 0.971380 8 368 2 53 56 367.573 13.638 I.I34749 9 414 3 15 40 413.255 I9- 45 1.279773 3 37 24 r. CHORD-LENGTH = 47. . nc. Ds. ' y- jr. Log x. i 47 o 21' 16" 47.000 0.0684 8.834794 2 94 o 42 33 93.999 .3418 9.533762 3 141 i 03 50 140.995 9570 9.980913 4 188 i 25 06 187.982 2.051 0.311890 5 235 i 46 23 234.951 3-759 0.575100 6 282 2 O7 40 281.887 6.219 0.793722 7 329 2 28 57 28.768 9.566 0.980720 8 376 2 50 14 .375^564 13-934 1.144089 9 423 3 ii 31 422.238 -9-459 1.289113 3 32 48 75 TABLE III. f. CHORD-LENGTH - 48. n. nc. zv '? jr. Log^r. i 48 o 20' 50" 48.000 o. 0698 8.843937 2 96 o 41 40 95-999 3491 9- 542905 3 144 I 02 30 143.995 9774 9.990057 4 192 I 23 20 191.982 2.094 0.321034 5 240 i 44 10 239-950 3.839 0.584243 6 288 2 05 OO 287.885 6.351 0.802866 7 336 2 25 51 335-7^3 9.769 0.989863 8 3S4 2 46 41 3S3.555 14.231 1.153232 3 06 31 f. CHORD-LENGTH = 49. ft. nc. D s . y- jr. Log x. I 49 o 20' 25" 49.000 0.0713 8.852892 2 q8 o 40 49 97.999 .3563 9.551860 3 147 i 01 14 146.995 .9977 9.999011 4 196 I 21 38 195.982 2.138 0.329988 5 245 I 42 03 244.949 3.919 0.593198 6 294 2 02 27 293.882 6.484 0.811820 7 343 2 22 52 342.758 9-973 0.998818 8 39 2 2 43 17 39!-546 14.527 1.162187 3 03 31 c. CHORD-LENGTH = 50. n. nc. /?* y- X. Log x. I 50 o 20' oo" 50.000 0.0727 8.861666 2 100 o 40 oo 99-999 .3636 9. 560634 3 150 I OO OO 149.995 1.018 0.007785 4 200' I 20 00 199.981 2.182 0.338762 5 250 i 40 oo 249.948 3-999 0.601972 6 3OO 2 00 00 299.880 6.616 0.820594 7 35 2 2O OO 349-753 10.176 1.007592 8 400 2 40 OO 399.536 14.824 1.170961 3 oo oo 76 TABLE IV. FUNCTIONS OF THE ANGLE s. n. S. cos s. log vers s. R i x vers s. sin s. log sin s. s. I o 10' 99999 4.626422 .024 .00291 7.463726 o 10' 2 o 30 .99996 5.580662 .218 .00873 7.940842 o 30 /? I 00 .99985 6.182714 873 -01745 8.241855 I 00 4 I 40 .99958 6.626392 2.424 . 02908 8.463665 I 40 5 2 30 .99905 6.978536 5-453 . 04362 8.639680 2 30 6 3 30 .99813 7.720726 10.687 .06105 8.785675 3 30 7 4 40 .99668 7.520498 18.994 .08136 8.910404 4 40 8 6 oo .99452 7.738630 31 383 10453 9.019235 6 oo 9 7 30 .99144 7.932227 49.018 13053 9.115698 7 30 10 9 10 .98723 8.106221 73-173 .15931 9.202234 9 10 ii II 00 .98163 8.264176 105.270 .19081 9.280599 II 00 12 13 oo 97437 8 408748 146.857 .22495 9.352088 13 oo 13 15 10 .9651718.541968 199.570 .26163 9.417684 15 10 14 17 30 .953728.665422 265.186 .30071 9.478142 17 30 15 20 oo .93969 8.780370 345-540 . 34202 9-53405 2 20 00 16 22 40 .92276 8.887829 442-543 38537 Q-585877 22 40 17 25 30 90259 8.988625 558.153 43051 9.633984 25 30 18 28 30 .87882 9.083441 694-335 .47716 9.678663 28 30 19 31 40 .85112 9.172846 853-0501 .52498: 9.720140 31 40 20 35 oo .81915 9- 2 573*4 1036.20 1 -57358 9-75859 1 35 oo 77 TABLE SELECTED SPIRALS FOR A 2 CURVE, GIVING A s. n x c. -As(w+l). >'. flC 10 1 00' 3 x 32 2 05' OO" 2 03' 41.12 10 I 40 4 x 39 2 08 13 2 09 61.04 10 2 3O 5 x 43 2 19 33 2 IS 73.69 10 3 30 6 x 45 2 35 34 2 33 78.81 10 4 40 7 x 44 3 oi 50 2 40 70.47 20 I OO 3 x 33 2 01 13 2 OI 45.28 20 I 40 4 x 41 2 oi 57 2 02 73.85 20 2 30 5 x 48 2 05 00 2 05 99.99 20 3 30 6 x 50 2 20 00 2 06 109.52 30 I OO 3 x 34 i 57 39 2 OI 46.14 30 I 40 4 x 41 2 oi 57 2 01 75.16 30 2 30 5 x 49 2 02 27 2 02 109.78 30 3 30 6 x 50 2 2O OO 2 02 115.63 30 3 30 6 x 50 2 20 OO 2 03 110.90 40 I 00 3 x 35 I 54 17 2 01 46.90 40 I 40 4 x 42 i 59 02 2 01 76.96 40 2 30 .5 x 50 2 00 00 2 01 117.87 78 V. EQUAL LENGTHS BY CHORD MEASUREMENT. old line. ^ new line. Diff. X. h. 291.12 291.12 .00 .6516 .040 .061 311-04 311-04 .00 1.702 .187 .110 323.69 323.70 + .01 3.439 354 .103 328.81 328.82 + .01 5.954 59 .099 320.47 320.50 -h .03 8.955 .897 .100 545.28 545.28 .00 .6719 .122 .182 573-85 573.84 .01 1.789 .118 .066 599-99 600. oo + .01 3.839 .527 .137 609.52 609.52 .00 6.616 554 .084 796.14 796.22 + .08 .6923 .566 .082 825.16 825.16 .00 1.789 .227 .127 859.78 859'75 - .03 3.919 377 .096 865.63 865.57 - .06 6.616 .249 .038 860.90 860.98 + .08 6.616 1.013 .153 1046.90 1047.15 + .25 .7127 1.222 1.715 1076.96 1077.09 + .13 1.832 .848 .463 1117.87 1117.77 .10 3.999 .141 .035 79 TABLE SELECTED SPIRALS FOR A 4 CURVE, GIVING A S. 72 X C. ->(n + l). D\ d. 10 i oo' 3 x 16 4 10' 03" 4 07' 20.22 10 I 40 4 - 19 4 23 13 4 16 29.12 10 2 30 5 X 22 4 32 48 4 39 38.75 10 3 30 6 x 23 5 04 26 5 17 41.37 20 I 40 4 x 20 4 10 03 4 04 34.92 2O 2 30 5 x 24 4 10 03 4 09 50.72 20 3 30 6 x 27 4 !9 19 4 H 63-69 20 4 40 7 x 30 4 26 44 4 3i 78.07 20 6 oo 8 x 31 4 50 24 4 46 81.88 2O 7 30 9 x 32 5 12 36 5 16 85.40 30 i 40 4 x 20 4 I0 03 4 02 35-57 30 2 30 5 x 25 4 oo 03 4 04 57-39 30 3 30 6 x 28 4 10 03 4 07 72.37 30 4 40 7 x 32 4 10 03 4 14 93-09 30 6 oo 8 x 35 4 17 12 4 23 110.31 30 7 30 9 x 37 4 30 20 4 34 122. 2O 30 9 10 10 x 38 4 49 33 4 47 126.86 40 2 30 5 x 25 4 oo 03 4 02 58.91 40 3 30 6 x 28 4 10 03 4 04 73-75 40 4 40 7 x 32 4 10 03 4 08 94.65 40 6 GO 8 x 36 4 10 03 4 12 121.33 40 7 30 9 x 39 4 16 28 4 17 142.86 40 9 10 10 x 41 4 28 21 4 26 154.34 60 2 30 5 x 25 4 oo 03 4 01 59-68 60 3 30 6 x 29 4 01 26 4 02 81.04 60 4 40 7 x 32 4 10 03 4 03 99-59 60 6 oo 8 x 36 4 10 03 4 05 125.81 60 7 30 9 x 40 4 10 03 4 08 154-42 80 2 30 5 x 25 4 oo 03 4 or 58.29 80 3 30 6 x 29 4 01 26 4 01 82.82 80 4 40 7 x 33 4 02 28 4 02 106.99 80 6 oo 8 x 37 4 03 17 4 03 I35.6I 80 7 30 9 x 41 4 03 57 4 05 164.79 80 V. f fuinvEESiTY- Vv *,d o:H ' *t ^ EQUAL LENGTHS BY cfeQjto ^JTASUR^kENT. old line. | new line. Diff. X. h. k. 145-22 145.17 - -05 .3258 .045 135 154-12 I54-I3 + .01 -.8290 .080 .100 163.75 163.76 + .01 1.760 .177 .100 166.37 166.39 + .02 3.043 305 .100 284.92 284.92 .00 .8726 .081 .100 300. 72 300.72 .00 1.920 .184 .096 313.69 313.75 + .06 3-573 375 .iQ5> 328.07 328.08 + .01 6.106 .598 .098 332.88 33L92 + .04 9.191 .910 .092 335-40 335-47 + .07 13-248 1.310 .099 410.57 410.57 .00 .8726 137 .157 432-39 432.38 .01 2.000 .147 .074 447-37 447-35 .02 3.705 .284 .077 468.09 468.09 .00 6.513 .687 .105 4S5.3I 485-32 s + .01 10-377 1.091 .105 497.20 497.23 + .03 15.319 1.526 .100 501.86 5QJ.95 H- .09 2T.240 2.126 .100 558.91 558.88 .03 2.000 .109 .054 573-75 573-74 .01 3-705 .361 .097 594.65 594-66 + .01 6.513 977 .150 621.38 621.33 .05 10.673 973 .091 642.86 642.83 - .03 16.147 1. 100 .086 654.34 654-36 + .02 22.917 2.186 .095 809.68 809.67 .01 2.000 .180 .090 831.04 831.03 .01 3.837 .461 .120 849.59 849-52 .07 6.513 572 .088 875.81 875-76 - .05 10.673 1.074 .106 904.42 904.36 - .06 16 561 1.718 .104 1058.29 1058.61 -f- .32 2.OOO 979 .490 1082.82 1082.71 .11 3.837 .295 .074 1106.99 1107.03 + .04 6.716 1. 000 .149 1135-61 H35.5I .10 10.970 1.199 .109 1164.79 1164.92 + .13 16.975 2.440 .144 81 TABLE SELECTED SPIRALS FOR AN 8 CURVE, GIVING A S. n X c. #fi(7l + l). D'. d. 10 2 30' 5x11 9 06' 01" 9 06' 19-95 20 2 30 5 x 12 8 20 26 8 16 25.71 2O 3 30 6 x 14 8 20 26 8 34 34-86 20 4 40 7 x 15 8 53 5i 8 54 39-90 20 6 oo 8 x 16 9 23 07 9 24 45-52 30 2 30 5 x 12 8 20 26 8 07 26.50 30 3 30 6 x 14 8 20 26 8 14 36.16 30 4 40 7 x 16 8 20 26 8 26 47.01 30 6 oo 8 x 17 8 49 55 8 36 53-13 30 7 30 9 x 18 9 16 08 8 46 60.05 30 9 10 10 x 19 9 39 36 9 14 65.70 40 2 30 5 x 12 8 20 26 8 04 26.93 40 3 30 6 x 14 8 20 26 8 08 36.85 40 4 40 7 x 16 8 20 26 8 14 48.25 40 6 oo 8 x 18 8 20 26 8 22 61.35 40 7 30 9 x 19 8 46 49 8 30 68.07 40 9 10 10 X 20 9 10 34 8 40 75-01 40 II 00 II X 21 9 32 03 8 54 82.13 40 13 oo 12 X 22 9 5i 36 9 H 89.81 60 2 30 5 x 12 8 20 26 8 02 27.30 60 3 30 6 x 14 8 20 26 8 03 38.22 60 4 40 7 x 16 8 20 26 8 06 49-75 60 6 oo 8 x 18 8 20 26 8 10 62.87 60 7 30 9 x 20 8 20 26 8 16 77.16 - 60 9 10 10 X 22 8 20 25 8 24 93.05 60 II 00 II X 23 8 42 13 8 31 101.08 60 13 12 X 25 8 40 28 8 48 118.19 60 15 10 13 x 26 8 58 59 9 02 127.21 60 17 30 14 x 27 9 16 07 9 22 136.45 80 4 40 7 x 17 7 50 57 8 04 57.04 80 6 oo 8 x 19 7 54 03 8 06 71.78 80 7 30 9 x 20 8 20 26 8 oSJ- 79.18 80 9 10 IO X 22 8 20 25 8 13 95.23 80 II OO ii x 24 8 20 25 8 19 112.67 80 13 oo 12 X 26 8 20 25 . 8 28 130.86 80 15 10 13 x 27 8 38 59 8 34 140.88 80 17 30 14 x 28 8 56 13 8 42 150.55 82 EQUAL LENGTHS BY CHORD MEASUREMENT. % old line. new line. Diff. X. h. k. 82.45 82.47 + .02 .8798 .051 .058 150.71 150.72 + .01 -9598 .051 .053 159.86 I5Q-88 + .02 1.852 .117 .063 164.90 164.92 + .02 3-C53 .185 .061 170.52 170.55 -f .03 4-744 .221 .047 214.00 214.00 .CO .9598 .049 .051 223.66 223.68 + .02 1.852 .142 .077 234.51 234.53 + .02 3-256 .260 .080 240. 63 240.65 + .02 5.040 .325 .065 247-55 247-55 .00 7-452 .287- 039 253.20 253.18 .02 10.620 590 .056 276.93 276.94 + .01 .9598 .079 .082 286.85 286.87 + .02 1.852 .181 .098 298.25 298.24 .01 3.256 293 .090 3H-35 3X1-33 .02 5.337 .330 .062 318.07 318.06 .01 7.866 .472- .c6o 325-01 325.00 v .OI 11.179 .629 .056 332.13 332.12 .01 15.415 .840 054 339-81 ' 339-81 .00 20.723 I.O24 .049 402. 30 402.32 + .02 .9598 .136 .142 413.22 413.19 - .03 1.852 .083 045 424.75 424.76 + .01 3-256 .317 .097 437.87 437.88 + .01 5-337 539 .101 452.16 452.18 + .02 8.280 .863 - .104 468.05 468.02 - .03 12.297 I-I39 .093 476.08 476.09 + .01 16.883 1.523 .090 493-19 493.18 .01 23.548 2.160 .092 502.21 502.21 .00 30.817 2.613 .085 5H-45 5H.45 .00 39-595 3-157 .080 557.04 557.02 .02 3.460 .366 .106 57L78 57L75 - .03 5*633 .408 .072 579^8 579- 18 .00 8.280 .860 .104 595.23 595.25 H- .02 12.297 1.346 .110 612.67 612.70 + .03 17.617 1.719 - .109 630.86 630.90 + .04 ,24.490 2.738 .112 640. 88 640.88 .00 32.002 3.H9 .098 650.55 650.62 + .07 41.062 3.809 .093 TABLE SELECTED SPIRALS FOR A 16 CURVE, A S. n X c. .>s(n + l). D\ d. 30 4 4o' 7 x 10 13 2l' 48" 18 oo' 33-59 40 6 oo 8 x 10 15 02 34 17 14 36.14 60 7 3o 9 x 10 16 43 3i 16 32 38.47 60 9 10 10 X 11 16 43 3i 16 48 46.40 60 II 00 II X 12 16 43 3i 17 14 54-62 60 13 oo 12 X 12 18 07 48 17 22 54- M 60 15 10 13 x 13 18 oi 18 18 10 62.88 60 17 30 14 x 13 19 19 14 18 12 62.85 60 20 oo 15 x 14 19 06 05 20 00 72.14 80 7 30 9 x 10 16 43 31 16 16 39-74 80 9 10 IO X II 16 43 31 16 26 47-49 80 II OO II X 12 16 43 31 16 38 56.19 80 13 oo 12 X 13 16 43 30 16 56 65.24 80 15 10 13 x 14 16 43 29 17 22 74.72 80 17 30 14 x 14 17 55 44 17 24 75.02 80 20 00 15 x 15 J7 50 54 18 06 85.15 So" 22 40 16 x 15 18 58 25 18 08 85-18 80 28 30 18 x 16 19 53 20 19 42 95.84 84 GIVING EQUAL LENGTHS OF ACTUAL ARCS. old line. new line. Biff. X. h. k. 127.64 127.64 .00 2.035 .388 .191 I6I.55 161.55 .00 2.965 .430 .145 226.58 226.56 .02 4.140 .436 .105 234.50 234.45 - .05 6.148 .576 .094 242.73 246.67 .06 8.808 .860 .099 242.25 242.26 + .01 11-303 1.093 .097 250.99 250.99 \ .00 15.409 1.516 .098 250.96 250.97 + .01 19.064 1-552 .081 260.25 260.25 .00 25-031 2.182 .087 290.55 290.47 - .08 4.140 ,328 305 298.30 298.27 - -03 6.148 .680 1 .Hi 307.01 306.96 - .05 8.808 -943 .107 316.06 316.03 .03 12.245 1.384 113 325.53 325-54 -f .01 16.594 1-973 .119 325.83 325-81 .02 20.531 1-939 ,094 335-97 335-96 .01 26.819 2.657 .099 336.00 335-99 .01 32.276 2.677 .083 346.65 346.66 + .01 48.221 3.748 .078 1 THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW AN INITIAL PINlToF 25 CENTS ^!, L B BE ASSESSED F0 * FAILURE TO RETURN THIS BOOK ON THE DATE DUE. THE PENALTY DAV'-A^ RE T ASE T0 50 CENTS ON OVERDUE^. " N THE 8EVEN I8Apr52l?J JUL 27 1943 Yft UNIVERSITY OF CALIFORNIA LIBRARY