EflO 13=1 THE VAN NOSTRAND SCIENCE SERIES. IBnio. Boards. Pri 5ft fanta Wool, REESE LIBRARY OF THE UNIVERSITY OF CALIFORNIA. y^ . f&> ./9Q ^Accession No. 7 8 (D / c . Class No.- . TAINING- WALLS. By James^TatT G "*" No. 8.-A TREATISE ON THE COMPOUND ENGINE By Seeond edition revi - . No; 9.-A TREATISE ON FUEL. By Arthur No. lO.-COMPOUND ENGINES. Translated from the w^hrL,if A >^ alle - Sec od edition, revised, Bud C E American Practice by Richard H. No. ll.-THEORY OF ARCHES. By Prof W Allan No. 12.-THEORY OF VOUSSOIR ARCHES. By Prof W C. Cam. Second edition, revised and enlarged* No. 13.-GASES MET WITH IN COAL-MINES By J J No< 14.-FRICTION OF AIR IN MINES. By J. J. Atkinson No. 15.-SKEW ARCHES. By Prof . E. W. Hyde, C. E. II- lustrated. No. I6.-A GRAPHIC METHOD FOR SOLVING CERTAIN ALGEBRAIC EQUATIONS. By Prof Geofg^ No. 17.-WATER AND WATER-SUPPLY. By Prof. W. H. Corfield of the University College, London If*. 18.- SEWERAGE AND SEWAGE UTILIZATION. By THE VAN NOSTRAND SCIENCE SERIES. No. 19.-3TRENGTH OF BEAMS UNDER TRANSVERSE LOADS. By Prof. W. Allan, author of "Theory of Arches. 1 ' No. 20.-BRIDGE AND TUNNEL CENTRES. By John B. McMaster, C.E. No. 21.-SAFETY VALVES. Second Edition. By Richard H. Biiel, C.E. No. 22. HIGH MASONRY DAMS. By John B. McMaster, C.E. No. 23.THE FATIGUE OF METALS UNDER REPEATED STRAINS. With various Tables of Results and Experiments. From the German of Prof. Ludwig Spangenburgh, with a Preface by S. H. Shreve, A.M. No. 24.-A PRACTICAL TREATISE ON THE TEETH OF WHEELS. By Prof. S. W. Robinson. Second ' edition, revised. No. 25. O:N THE THEORY AND CALCULATION OF CON- TINUOUS BRIDGES. By Mansfield Merriman, " Ph.D. No. 26. PRACTICAL TREATISE ON THEv PROPERTIES OF CONTINUOUS BRIDGES. By Charles Bender, C.E. No. 27.-ON BOILER INCRUSTATION AND CORROSION. By F. J. Rowan. No. 28. TRANSMISSION OF POWER BY WIRE ROPES. Second edition. By Albert W. Stahl, U.S.N. No. 29. STEAM INJECTORS. Translated from the French of M. Leon Pocbet. No. 30. TERRESTRIAL MAGNETISM AND THE MAG- NETISM OF IRON VESSELS. By Prof. Fair- man Rogers. No. 31. THE SANITARY CONDITION OF DWELLING- HOUSES IN TOWN AND COUNTRY. By George E. Waring, jun. No. 32. CABLE-MAKING FOR SUSPENSION BRIDGES. By W. Hildebrand, C.E. No. 33.-MECHANICS OF VENTILATION. By George W. Rafter, C.E. No. 34. -FOUNDATIONS. By Prof. Jules Gaudard, C.E. Second edition. Translated from the French. No, 35. THE ANEROID BAROMETER : ITS CONSTRUC- TION AND USE. Compiled by George W. Plympton. Fourth edition. No. 36. MATTER AND MOTION. By J. Clerk Maxwell, M.A. Second American edition. No. 37. GEOGRAPHICAL SURVEYING ; ITS USES, METHODS, AND RESULTS. By Frank De Yeaux Carpenter, C.E. No. 38.-MAXIMUM STRESSES IN FRAMED BRIDGES. By Prof. William Cain, A.M., C.E, THE VAN NOSTRAND SCIENCE SERIES. NO. .-STE|NGTHOF MATERIALS. B y Wi,, iam Kent, No. 43.-WAVE AND VORTEX MOTION. By Dr Thomas Craig, of Johns Hopkins University' No. 44.-TURBINE WHEELS. By Prof. W. R Trowbridre Columb.a College. Second edition Revved S ' No - *-"!5s?saffls Bypr f - H - T - 3S. No. Vffi.1i*l n . l * of BRACED ARCHES P h *-v -t.ju.vjjjj.cvs \^i ttig, tru. i) ^^-^S E a !?oS ITS C N STRUCT IO N, ETC. No. SS.-INDUCTIO/TOgf ' HOW^MADE^ND HOW No. S4.-KINEMATICS OF MACHINERY By Prof No. W.-SEWER ASES ^THEIR NATURE AND ORIGtN. y . ^-THE. VENTILATION OF COAL- MIN ES. By W TRANSITION CURVES. A FIELD BOOK FOR ENGINEERS, CONTAINING RULES AND TABLES FOR LAYING OUT TRANSITION CUEVES. BY WALTER G. FOX, CIVIL ENGINEEK. NEW YORK: D. VAN NOSTRAND COMPANY, PUBLISHERS, 23 MURRAY AND 27 WARREN STREETS. 1893. COPYRIGHT, 1803. Do VAN NOSTRAND COMPANY. PREFACE. In this work the endeavor has been to condense the essential facts and principles constituting the theory of the transition curve and to demonstrate how the curve can be conveniently laid out in field prac- tice. The same methotl nas'been adopted as is usually employed in running circular curves so that the field operations may be more readily comprehended and performed. WALTER G. Fox. NEW YOKK, October 3d, 1893. CHAPTER I. INTRODUCTION. Near the ends of a railway curve the curvature should be gradually diminished to enable trains at high speed to deviate gradually from the tangent and avoid any shock caused by changing too suddenly from a straight line to a sharp curve. A transition curve is generally used to connect the circular curve with the tangent as it makes easy transition between them. Moreover, as the outer rail of a curved track is elevated above the inner rail, a transition curve will unite better with the tangent ; for the super elevation of the outer rail is proportional to the degree of curve which is least at the initial point. CHAPTER II. A transition curve is a compound curve with many changes of radius. It has nearly the form of the cubic parabola. The curve is laid out on chords with the transit in the same manner as circular curves. A uniform chord length of 10 feet has been arbitrarily assumed to facili- tate calculation, and the degree of curve changed at the end of every chord. The deflection angles have been computed and will be found in the tables. To compute the deflection angles it was necessary to have the co-ordinates of every point of compound curve. The sines and cosines of the different angles between the tangent and the chords of the curves mul- tiplied by 10 will give the required co-or- dinates. In Fig. I. T is the beginning of the curve. The distance measured along the tangent in the direction of L is the latitude and in the direction of D at a right angle ITig. 1.. with the tangent is the departure of the various points of compound curves. For example let us refer to Table L, which begins with a 10' curve and compounds into a sharper curve at the end of each chord, the degree of curve chang- ing to 20', 30', 40', etc. See Fig. 2. 1 ft Degree of curve. Angle of chord. Cosine xlO. Sine xlO. Co-ordinates Lati- tude. Depar- ture. 10 10' O 7 30" 10.000000 0.001454 10.000000 0.001454 20 20' 2' 00" 9.999998 0.005818 19.999998 0.007272 30 30' 4' 30" 9.999991 0.013090 29.999989 0.020362 40 40' 8' 00" 9.999973 0.023271 39.999962 0.043633 50 50' 12' 30" 9.999933 0.036361 49.999895 0.079994 Having calculated the co-ordinates of the different points the deflection angles were determined by - = tan. T. Example: To find the deflection angle for the first point in Table I. C. 20' C. 10 10 Dep., 0.001454 log. 3.1625644 Lat., 10.000000 log. 1.0000000 Ans. Del angle, 0' 30" log. tan. 6.1625644 The difference in the degree of curve between the different arcs of the transition curve should be maintained as nearly as possible between the terminating arc and the circular curve. In the field the transit should be placed at T and the angles deflected from the tangent. In case there should be an ob- stacle making a point invisible at T, the next point should be measured on the lung- chord. The intermediate point can then be easily set. To find the length of the long chord to any point on a curve. Fig. 3. Let C be the length of chord, L the latitude of the point, and A the deflection angle. c= L cos A Example : To find the length of the long chord to the 8th point of the curve in Table III. 11 12 Lat. 79.991 log. 1.9030411 Def. angle, 38' 15" log. cos. 9.9999731 Ans. Long chord, 79.996 log. 1.9030680 CHAPTER III. EXPLANATION OF TABLES. The curvature increases more rapidly in each succeeding table. Table I. begins with a 10' curve which is the difference in curvature between the first arc of each table, the degrees of curve being 10', 20', 30', etc. The de- gree of curve of the first arc is equal to- the difference between all the other arcs in the same table, so that the sixth arc in Table I. is a 1 curve and the sixth arc in Table VI. is a 6 curve. Column I. gives the length of curve ac- cording to the central angle consumed which will be found on the same line in a different column, and also the other corres- ponding parts of the tables under their respective headings. Column II. has the degree of curve of each separate arc. 14 Column III. contains the deflection an- gles. It will be observed that the value of the angles has been calculated to the nearest second. Such accuracy is not ex- pected in practice but was necessary in or- der to correctly determine the long chords. Column IV. gives the central angles of the curves. Columns V. and VI. have the co-ordi- nates of every point of compound curve, By using the co-ordinates in place of the deflection angles the curve may be laid out with offsets from the tangent. Column VII. has the long chords to ev- ery point on the curves. Column VIII. contains numbers by which the tangents of half the intersection angles of the curves should be multiplied and the product added to the tangent distances as a correction. CHAPTER IV. PROBLEMS. Given, a 5 degree curve, to run a transi- tion curve connecting it with a tangent having an intersection angle of 20 degrees, If we select the curve from Table VL its length will be 40 feet, consisting of 4 arcs, and the central angle will be 1 de- gree. As this must be repeated at the other end of the curve, the central angle should be multiplied by 2 and the sum sub- tracted from the intersection angle. The remainder will be the central angle of the circular curve. To find its length divide the central angle by the degree of curve. Example : Intersection angle, 20 Central angle of trans, curve X 2, 2 5/ 18 Length of circular curve, 360 ft. 16 To find the tangent distance V. Let M. be the length of circular curve that the central angle of transition curve consumes, a d the apex distance of cir- cular curve and L the latitude of the ter- minal point of the transition curve. M, Example : Apex. dist. of 5 curve for 20, 202 . 120 Lat. of ter'l point of trans, curve, 39 . 998 242.118 Length of 5 curve to consume 1, 20.000 Ans. V = Tan. distance, 222.118 A correction should be added to the pro- duct which will not materially change the result unless the length of curve exceeds 70 feet, or the intersection angle contains more than 50 degrees. Rule: Multiply the tangent of half the intersection angle by the distance in column 8 corresponding to the terminal point of the curve. ' OF THE UNIVERSITY 17 Example : 1/2 intersection angle 10 Tan. = .1763 X .086 ~ .015 then Y = - 222.118 Correction .015 Tan. dist. - 222.133 With curves of short radius there is a slight error in the preceding formula, as M does not always have exactly its true value. The error, however, is too small to take into account in field practice. In the above example it is only one-thousandth of a foot. In a ten degree curve the error is about three-tenths. The true value of M can be determined by multiplying the cosine of half the cen- tral angle of transition curve by the length of circular curve it has consumed. Example : 1/2 central angle of trans. curve = 30' log. cos. 9.9999835 Length of circular curve con- sumed = 20 feet log. 1.3010300 Ans. M, = 19.999 ft. loff. 1.3010135 18 Having measured the tangent distances, the transition curves should be first laid out and then connected by running the cir- cular curve. See Fig. 4. Fig. If there are two circular curves of dif- ferent radii, their ends will be a short dis- tance apart, the sharper curve being on the inner side, which can then be eased off to unite with the other curve. See Fig. 5. On railway location it may be desirable to first lay out the circular curve, neglect- ing the transition curves until the time of construction. 19 In this case the N two terminal points of the transition curves should be located with their co-ordinates, the transit set over either point and a foresight taken on the other and the curve run in. Compound curves should not be located in this manner, as a backsight is then re- quired. TRANSITION CURVE TABLES. 23 TABLE I. 1 Length of 1 1 Curve. j Degree of 1 Curve. I "> 1.1. ii w Co-ordinates Long Chord. | Corr'n for f Tang.Dist. Lati- tude. Depar ture. 10 10' 0' 30" 1' 0" 10.000 0.001 10.000 .000 20 20' I' 11" 3' 0" 20.000 0.007 20.000 .004 30 30' 2' 17" 6' 0" 30.000 0.020 30.000 .007 40 40' 3' 41" 10' 0" 40.000 0.043 40.000 .014 50 50' 5' 30" 15' 0" 50.000 0.080 50.000 .029 60 1 00' V 33" 21' 0" 60.000 0.132 60.000 .045 70 1 10' 9' 58" 28' 0" 69.999 0.203 69.999 .061 80 1 20' 12' 43" 36' 0" 79.999 0.296 79.999 .086 90 1 30' 15' 49" 45' 0" 89.998 0.414 89.999 .126 100 1 40' O c 19' 15" 55' 0" 99.997 0.560 99.998 .168 110 1 50' 23' 0" 1 6' 0" 109.995 0.736 109.997 .208 120 2 00' 37' 4" 1 18' 0" 119.993 0.945 119.996 .265 130 2 10' 31' 30" 1 31' 0" 129.990 1.191 129.995 .341 140 2 20' 36' 15" 145' 0" 139.986 1.476 139.994 .417 150 2 30' 41' 19" 2 0' 0" 149.981 1.803 149.992 .494 160 2 40' 46' 44" 2 16' 0" 159.974 2.175 159.989 .593 170 2 50' 52' 30" 2 33' 0" 169.965 2.596 169.985 .717 180 3 00' 58' 35" 2 51' 0" 179.954 3.067 179.980 .843 190 3 10' 1 4' 59" 3 10' 0" 189.940 3.591 189.974 .967 200 3 20' 1 11' 45" 3 30' 0" 199.923 4.173 199. 966 1.119 210 3 30' 1 18' 50" 3 51' 0" 209.903 4.814 209.958 1.302 220 3 40' 1 26' 14", 4 13' 0" 219.878 6.517 219.947 1.487 230 3 50' 1 34' 0" 4 36' 0" 229.848 6.286 229.934 1.6T1 240 4 00' 1 42' 5" 5 0' 0" 239.814 7.123 239.920 1.889 24 TABLE II. Length of 1 Curve. | !| g>3 ft Deflection Angle. Central Angle. Co-ordinates Long Chord. Corr'tJ for Tang.Dist. ( Lati- tude. Depar ture. 10 20' o i' o" 2' 0" 10.000 0.003 10.000 .002 20 40' 2' 24" 6' 0" 20.000 0.014 20.000 .005 30 1 00' 4' 35" 012' 0" 30.000 0.040 30.000 .014 40 1 20' 7' 28" 020' 0" 40.000 0.087 40.000 .029 50 1 40' 11' 0" 030' 0" 50.000 0.160 50.000 .051 60 2 00' 15' 7" 042' 0" 59.999 0.264 59.999 .081 70 2 20' 19' 59" 056' 0" 69.998 0.407 69.999 .122 80 2 40' 25' 29" 112' 0" 79.996 0.593 79.998 .174 90 3 00' 31' 40" 130' 0" 89.993 0.829 89.996 .240 100 3 20' 38' 30" 150' 0" 99.989 1.120 99.995 .320 110 3 40' 45' 58" 212' 0" 109.983 1.471 109.993 .415 120 4 00' 54' 9" 236' 0" 119.974 1.890 119.989 .529 130 4 20' 1 3' 0" 302' 0" 129.962 2.382 129.983 .662 140 4 40' 1 12' 29" 330' 0" 139.946 2.951 139.977 .813 150 5 00' 1 22' 39" 4 0' 0" 149.924 3.605 149.967 .988 160 5 20' 1 33' 29" 432' 0" 159.897 4.349 159.956 1.185 170 5 40' 1 44' 59" 5 6' 0" 169.861 5.189 169.940 1.407 180 6 00' 1 57' 9" 542' 0" 179.817 6.130 179.921 1.655 190 6 20' 2 9' 58" 620' 0" 189.762 7.178 189.897 1.930 200 6 40' 2 23' 28" 7 0' 0" 199.694 8.339 199.868 2.234 2K 1 - 7 00' 2 37 39" 742' 0" 209.612 9.619 209.832 2.569 220 7 20' 2 52' 28" 826' 0" 219.513 11.022 219.789 2.934 230 7 40' 3 1' 58" 9<>i2/ 0" 229.395 12.555 229.738 3.332 240 8 00' 3 24' 6" 10 0' 0" 239.255 14.222 239.677 3.763 25 TABLE III. * S d o Co-ordinates * tl > "o T! n Long Tft So O ?-i j> "3 g> o 5 Lati- Depar Chord. *f |Sr ft A Q ^ tude. ture. as 10 30 1' 30" 3' 0" 10.000 0.004 10.000 .003 20 1 00' 3' 47" 9' 0" 20.000 0.022 20.000 .010 4 30 1 30' 6' 59" 18' 0" 30.000 0.063 30.000 .022 40 2 00' 11' 15" 30' 0" 40.000 0.131 40.000 .044: 50 2 30' 16' 30" 45' 0" 49.999 0.240 49.999 .080 60 3 00' 22' 45" 1 3'0" 59.998 0.397 59.999 .126 70 3 30' 30' 0" 1 24' 0" 69.995 0.611 69.997 .183 80 4 00' 38' 15" 1 48' 0" 79.991 0.890 79.996 .262 90 4 30' O o 477 29" 2 15' 0" 89.985 1.243 89.993 .366 100 5 00' 57' 44" 2 45' 0" 99.975 1.679 99.989 .487 110 5 30' 1 8' 59" 3 18' 0" 109.962 2.207 109.984 .624 120 6 00' 1 21' 14" 3 54' 0" 119.942 2.835 119.975 .793 130 6 30' 1 34' 30" 4 33' 0" 129.915 3.572 129.964 1.001 140 7 00' 1 48' 44" 5 15' 0" 139.878 4.426 139.948 1.231 150 7 30' 2 3' 59" 6 0' 0" 149.830 5.406 149.927 1.480 160 8 00' 2 20' 14" 6 48' 0" 159.768 6.521 159.901 1.776 170 8 30' 2 37' 28" 7 39' 0" 169.688 7.778 169.866 2.130 180 9 00' 2 55' 42" 8 33' 0" 179.589 9.187 179.824 2.491 190 9 30' 3 14' 57" 9 30' 0" 189.465 10.756 189.770 2.889 200 10 00' 3 35' 11" 10 30' 0" 199.313 12.492 199.704 3.342 210 10 30' 3 56' 25" 11 33' 0" 209.128 14.405 209.623 3.855 220 11 OO/ 4 18' 39'' 12 39' 0" 218.. 906 16.501 219.527 4.399 230 11 30' 4 41' 52" 13 48' 0" 228.641 18.V89 229.411 4.973 240 12 OO' 5 6' 6" 15 O'O" 238.327 21.276 239.275 5.612 26 TABLE IV. Length of Curve. Degree of Curve. Deflection Angle. Central Angle. Co-ordinates Long Chord. Corr'n for Tang.Dist. Lati- tude. Depar ture. 10 40' 2' 0" 4' 0" 10.000 0.006 10.000 .003 20 1 20' 4' 59" 12' 0" 20.000 0.029 20.000 .012 30 2 00' 9' 17" 24' 0" 30.000 0.081 30.000 .029 40 2 40' 14' 57" 40' 0" 39.999 0.174 39.999 .058 50 3 20' 22' 0" 1 0' 0" 49.998 0.320 49.999 .102 60 4 00' 30' 19" 1 24' 0" 59.996 0.529 59.998 .162 70 4 40' 39' 59" 1 52' 0" 69.992 0.814 69.996 .244 FO 5 20' 50' 58" 2 24' 0" 79.985 1.186 79.993 .348 90 6 00' 1 3' 18" 3 0' 0" 89.974 1.657 89.989 .479 100 6 40' 1 16' 59" 3 40' 0" 99.957 2.239 99.982 .639 110 7 20' 1 31' 58" 4 24' 0" 109.932 2.942 109.971 .831 120 8 00' 1 48' 19" 5 12' 0" 119.897 3.779 119.956 1.057 130 8 40' 2 5' 59" 6 04' 0" 129.849 4.761 129.936 1.321 140 9 20' 2 24' 58" 7 O 7 0" 139.784 5.898 139.908 1.625 150 10 00' 2 45' 18" 8 0' 0" 149.698 7.204 149.871 1.972 160 10 40' 3 6' 58" 9 4' 0" 159.588 8.688 159.824 2.365 170 11 20' 3 29' 57" 10 12' 0" 169.446 10.361 169.762 2 805 180 12 00' 3 54' 15" 11 24' 0" 179.269 12.235 179.686 3.296 190 12 40' 4 19' 54" 12 40' 0" 189.050 14.320 189.591 3.841 200 13 20' 4 46' 52" 14 0' 0" 198.780 16.626 199.474 4.439 210 14 00' 5 15' 9" 15 24' 0" 208.453 19.163 209.332 5.094 220 14 40' 5 44' 45" 16 52' 0" 218.059 21.942 219.160 5.809 230 15 20' 6 15' 41" 18 24' 0" 227.589 24.971 228.955 6.585 240 16 00' 6 47' 56" 20 0' 0" 237.033 28.260 238.711 7.422 Y v* Or i *a r. \ UNIVERSITY ] 27 TABLE V. I Length of 1 Curve. Degree of Curve. I & "
s
Be
p
Deflection
Angle.
1
M
p rt
Co-ordinates
Long
Chord.
Corr'n for
Tang.Dist.
Lati-
tude.
Depar
turs.
1 oo'
3' 0"
G' 0"
10.000
0.008
10.000
.004
20
2 00'
7' 23"
18' 0"
20.000
0.043
20.000
.017
30
3 00'
13' 58"
36' 0"
29.999
0.122
29.999
.043
40
4 00'
22' 26"
1 0' 0"
39.998
0.261
39.999
.086
50
5 00'
33' 0"
1 30' 0"
49.996
0.480
49.998
.153
60
6 00'
43' 30"
2 6' 0"
59.991
0.794
59.996
.244
70
7 00'
59' 58"
2 48' 0"
69.982
1.221
69.992
.366
80
8 00'
1 16' 28"
3 36' 0"
79.966
1.779
79.985
.523
90
9 00'
1 35' 0"
4 30' 0"
89.941
2.486
89.975
.719
100
10 00'
1 55' 28"
5 30' 0"
99.903
3.357
99.959
.958
110
11 00'
2 17' 58"
6 36' 0"
109.848
4.411
109.936
1.245
120
12 00'
2 42' 29"
7 48' 0"
119.769
5.665
119.903
1 584
130
13 00'
3 8' 57"
9 06' 0"
129.660
7.134
129.856
1.978
140
14 00'
3 37' 26"
10 30' 0"
139.514
8.836
139.793
2.431
150
15 00'
4 V 55"
12 00' 0"
149.322
10.787
149.711
2.947
160
16 00'
4 40' 23"
13 36' 0"
159.073
13.003
159.603
3.531
170
17 00'
5 14' 49"
15 18' 0"
168.757
15.498
169.467
4.183
180
18 00'
5 51' 15"
17 06' 0"
178.360
18.288
179.295
4.908
190
19 00'
6 29' 39"
19 0' 0"
187.868
21.386
189.081
5.706
200
20 00'
7 10' 2"
21 0' 0"
197.265
24.806
198.818
6.582
210
21 00'
7 52' 24"
23 6'0"
206.534
28.561
208.499
7.538
220
22 00'
8 36' 42"
25 18' 0"
215.655
32.660
218.114
8.571
230
23 00'
9 22' 58"
27 36' 0"
224.608
37.114
227.654
9.683
240/24 00'
10 11' 10"
30 0' 0"
233.371
41.932
237.108
10.874
TABLE OF TANGENT DISTANCES
FOB
CIRCULAR RAILROAD CURVES.
This table contains the tangent distances
to a 1 degree curve for every two minutes
of intersection angle to 90 degrees.
The tangent distance to any other de-
gree of curve may be determined by divid-
ing the number corresponding to the inter-
section angle by the degree of curve.
. OF THF
UNIVERSITY )
31
t
1
2
3
/
0.00
50.00
100.00
150.07
2
1.67
51.67
101.67
151.74
2
4
3.33
53.33
103 34
153.41
4
6
5.00
55.00
105.01
155.08
6
8
6.67
56.67
106.68
156 . 75
8
10
8.33
58.33
108.35
158.42
10
12
10.00
60.00
110.02
160.09
12
14
11.67
61.67
111.69
161.76
14
16
13.33
63.33
113.36
163.43
16
18
15.00
65.00
115.02
165.09
18
20
16.67
66.67
116.69
166.76
20
22
18.33
68.33
118.36
168.43
22
24
20.00
70.00
120.03
170.10
24
26
21.67
71.67
121.70
171.77
26
28
23.33
73.33
123.37
173.44
28
30
25.00
75.00
125.03
175.10
30
32
26.67
76.67
126.70
176.72
32
34
28.33
78.33
128.37
178.39
34
36
30.00
80.00
130.04
180.06
36
38
31.67
81.67
131.71
181.73
38
40
33.33
83.33
133.38
183.40
40
42
35.00
85.00
135.05
185.07
42
44
36.67
86.6^
136.72
186.74
44
46
38.33
88.33
138.38
188.40
46
48
40.00
90.00
140.05
190.07
48
50
41.67
91.67
141.72
191.74
50
52
43.33
93.33
143.39
193.41
52
54
45.00
95.00
145.06
195.08
54
56
46.67
96.67
146 . 73
196.75
56
58
48.33
98.33
148.40
198.42
58
60
50.00
100.00
150.07
200.09
60
32
/
4
5
6 7
/
200.09
250.17
300.30
350.44
2
201 . 76
251.84
301.97
352.11
2
4
203.43
253.51
303 . 64
353.79
4
6
205.10
255.18
305.31
355.46
6
8
206.77
256 . 85
306.98
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10
208.44
258.52
308.65
358.81
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12
210.11
260.20
310.32
360.48
12
14
211.77
261.86
311.99
362.15
14
16
213.45
263.54
313.66
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16
18
215.11
265.20
315.33
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18
20
216.78
266.87
317.00
367.17
20
22
218.45
268.54
318.67
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22
24
220.12
270.21
320.34
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24
26
221.79
271.88
322.01
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26
28
223.46
273.54
323 . 68
373.86
28
30
225.13
275.21
325.35
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30
32
226.80
276.88
327.02
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34
228.47
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36
230.14
280.23
330.37
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36
38
231.81
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332.04
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40
233.48
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40
42
235.15
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335.38
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44
236.82
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337.05
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238.48
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240.15
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241.82
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342.07
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243.49
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54
245.16
295.27
345.41
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54
56
246.83
296.95
347.08
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58
248.50
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348.76
399.01
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60
250.17
300.30
350.44
400.70
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8
9
10
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400.70
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417.43
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518.12
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519.80
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602.22
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602.22
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56
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2
4
4482.1
4563.3
4645 . 6
4729.0
4
6
4484.8
4566.0
4648.3
4731.8
6
8
4487.5
4568.7
4651.1
4734.7
8
10
4490.2
4571.5
4653.9
4737.5
10
12
4492.9
4574.2
4656.7
4740.3
12
14
4495 . 6
4576.9
4659.4
4743.1
14
16
4498.3
4579.7
4662.2
4745.9
16
18
4501.0
4582.4
4665.0
4748.7
18
20
4503.7
4585.1
4667.7
4751.5
20
22
4506.3
4587.9
4670.5
4754.3
22
24
4509.0
4590.6
4673.3
4757.1
24
26
4511.7
4593.3
4676.0
4760.0
26
28
4514.4
4596.0
4678.8
4762.8
28
30
4517.1
4598.8
4681.6
4765.6
30
32
4519.8
4601.5
4684.4
4768.4
32
34
4522.5
4604.3
4687.2
4771.2
34
36
4525.3
4607.0
4689.9
4774.1
36
38
4528.0
4609.8
4692.7
4776.9
38
40
4530.7
4612.5
4695.5
4779.7
40
42
4533.4
4615.3
4698.3
4782.6
42
44
4536.1
4618.0
4701.1
4785.4
44
46
4538.8
4620.8
4703.9
4788.2
46
48
4541.5
4623.5
4706.7
4791.0
48
50
4544.2
4626.3
4709.5
4793.9
50
52
4547.0
4629.0
4712.2
4796.7
52
54
4549 . 7
4631.8
4715.0
4799.5
54
56
4552.4
4634.5
4717.8
4802.4
56
58
4555.1
4637.3
4720.6
4805.2
58
60
4557.8
4640.0
4723.4
4808.0
60
51
'
80
81
82
83 '
4808.0
4893 . 9
4981.0
5069.4
2
4810.9
4896.8
4983.9
5072.4
2
4
4813.7
4899.7
4986.8
5075.4
4
6
4816.6
4902.6
4989.8
5078.4
6
8
4819.4
4905.4
4992.7
5081.4
8
10
4822.3
4908.3
4995.7
5084.4
10
12
4825.1
4911.2
4998.6
5087.3
12
14
4828.0
4914.1
5001.5
5090.3
14
16
4830.8
4917.0
5004.5
5093.3
16
18
4833 . 7
4919.9
5007.4
5096.3
18
20
4836.5
4922.8
5010.3
5099.3
20
22
4839.4
4925.7
5013.3
5102.3
22
24=
4842.2
4928.6
5016.2
5105.2
24
26
48-15.1
4931.5
5019.2
5108.2
26
2S
4847.9
4934.4
5022.1
5111.2
28
30
4850.8
4937.2
5025.0
5114.2
30
32
4853 . 7
4940.2
5028.0
5117.2
32
34
4856.5
4943.1
5031.0
5120.2
34
36
4859.4
4946.0
5033.9
5123.2
36
38
4862.3
4948.9
5036.9
5126.2
38
40
4865.1
4951.8
5039.8
5129.2
40
42
4868.0
4954.7
5042.8
5132.2
42
44
4870.9
4957.6
5045.8
5135.2
44
46
4873.8
4960.6
5048.7
5138.2
46
48
4876.6
4963.5
5051.7
5141.2
48
50
4879.5
4966.4
5054.6
5144.3
50
52
4882.4
4969.3
5057.6
5147.3
52
54
4885.3
4972.2
5060.6
5150.3
54
56
4888.1
4975.1
5063.5
5153.3
56
58
4891.0
49^8.0
5066.5
5156.3
58
60
4893.9
4981.0
5069.4
5159.3
60
/
84
85
86
87
'
5159.3
5250.6
5343.3
5437.5
2
5162.3
5253.6
5346.4
5440.7
2
4
5165.3
5256.7
5349.5
5443 . 9
4
6
5168.4
5259.8
5352.7
5447.1
6
8
5171.4
5262.9
5355.8
5450.3
8
10
5174.4
5266.0
5358.9
5453.4
10
12
51-77.5
5269.0
5362.0
5456.6
12
14
5180.5
5272.1
5365.2
5459.8
14
16
5183.5
5275.2
5368.3
5463.0
16
18
5186.6
5278.3
5371.4
5466.2
18
20
5189.6
5281.4
5374.6
5469.4
20
22
5192.6
5284.4
5377.7
5472.5
22
24
5195.6
5287.5
5380.8
5475 . 7
24
26
5198.7
5290.6
5383.9
5478.9
26
28
5201.7
5293.7
5387.1
5482.1
28 -
30
5204.7
5296.7
5390.2
5485.3
30
32
5207.8
5299.8
5393.4
5488.5
32
34
5210.8
5302.9
5396.5
5491.7
34
36
5213.9
5306.1
5399.7
5494.9
36
38
5216.9
5309.2
5402.8
5498 . 1
38
40
5220.0
5312.3
5406.0
5501.3
40
42
5223.1
5315.4
5409.1
5504.5
42
44
5226.1
5318.5
5412.3
5507.7
44
46
5229.2
5321.6
5415.4
5510.9
46
48
5232.2
5324.7
5418.6
5514.1
48
50
5235.3
5327.8
5421.8
5517.3
50
52
5238.3
5330.9
5424.9
5520.5
52
54
5241.4
5334.0
5428.1
5523.7
54
56
5244.5
5337.1
5431.2
5526.9
56
58
5247.5
5340.2
5434.4
5530.1
58
60
5250.6
5343.3
5437.5
5533.3
60
53
/
88
/
88
/
se
/
89
5533.3
30
5581.9
5630.8
30
5680.2
2
5536.6
32
5585.1
2
5634.1
32
5683.5
4
5539.8
34
5588.4
4
5637.4
34
5686.8
6
5543.1
36
5591.7
6
564C.7
36
5690.2
8
5546.3
38
5594.9
8
5644.0
38
5693.5
10
5549.5
40
5598.2
10
5647.3
40
5696.8
12
5552.8
42
5601.4
12
5650.6
42
5700.1
14[5556.0
44
5604.7
14
5653.9
44
5703.4
165559.2
46
5608.0
16
5657.1
46
5706.8
18
5562.5
48
5611.2
18
5660.4
48
5710.1
20
5565.7
50
5614.5
20
5663.7
50
5713.4
22
5568.9
52
5617.8
22
5667.0
52
5716.7
84
5572.2
54
5621.0
24
5670.3
54
5720.0
2(\
5575.4
56
5624.3
26
5673.6
56
5723.4
2*
5578.6
58
5627.5
28
5676.9
58
5726.7
30
5581.9
60
5630.8
30
5680.2
60
5730.0
TABLE
GIVING BADII OF DEGEEES
OF CURVE.
57
02
S
00
8
Q
r^
o
o
SCOCOCOCOCOCOCOCOCO-^^-'^-^H'**'^
58
CO
s
s
CO CO CO CO CO
r^--
OCOCOlOlOlO^"* *COCOOO