RAILROAD ENGINEERING By WILLIAM G. RAYMOND Railroad Field Geometry (Without Tables) i6mo. 242 + ix pages, 138 figures, leather, gilt edges. Price $1.80 net. Elements of Railroad Engineering Second Edition, Revised. 8vo. 408 + xvi pages, 1 10 figures, 18 plates. Cloth, $3.25 net. Railroad Field Manual for Civil Engineers vii + 398 pages, 4% by 7, 31 figures, 83 tables. Morocco, $3.00 net. Railroad Field Manual for Civil Engineers BY WILLIAM G. RAYMOND, C.E., L.L.D. Professor of Civil Engineering and Dean of the College of Applied Science in the State University of Iowa. Member of the American Society of Civil Engineers and of the American Railway Engineering Association FIRST EDITION FIRST THOUSAND NEW YORK JOHN WILEY & SONS, INC. LONDON: CHAPMAN & HALL, LIMITED 1915 4.* COPYRIGHT, 1915, BY WILLIAM G. RAYMOND Stanhope ipress F. H. GILSON COMPANY BOSTON, U.S.A. PREFACE THIS book is for field use rather than for office use, though it is adapted to a large percentage of office work. It is made on a new plan which is not expected to gain immediate favor but which it is hoped will eventually appeal to railroad engineers as sensible and worthy of adoption, because its use will save time and lessen the liability of error. The degree is divided decimally instead of sexagesimally. When the author was a young man engaged on railroad location he knew one or two engineers who had one vernier of their transits gradu- ated to read hundredths of degrees for greater convenience in setting out curves. They would have done all their work in decimals if tables had been available. When the author was planning this book he gave much thought to the question of the division of the degree and the forms of the tables that would be most convenient and time saving for the field men who might use the book. He remembered that in practically every curve problem it is necessary at some stage of the solution to transpose from minutes and seconds to decimals of a degree or vice versa. He re- membered that to lay out subchords would require much less mental effort if the transit were divided to read decimals of degrees rather than minutes. He wrote to a half dozen of the leading instrument makers to learn what would be the cost of changing the verniers on an old transit to read decimals of a degree and to know whether there would be any difference in price between two instruments ordered new, one to be divided in the usual way and the other divided to read deci- mals of a degree. All but one of the makers gave a price in the neighbor- hood of $20 for changing the verniers on an old instrument, and no difference in cost for new instruments. The author then wrote to about fifty engineers, chief engineers of railroads, independent practicing engineers, and professors of railroad engineering in colleges and asked their opinions as to the desirability of a change in practice from sexa- gesimal to decimal division of the degree, and whether or not a table book based on the decimal division would help to bring about the change, if desirable. All but one of these engineers replied that the change is desirable. The one was a professor of railroad engineering. Of the others all but two thought it doubtful if the change could be iii 3O8538 iv PREFACE brought about, owing to the conservatism of the craft. Two chief engi- neers of prominent eastern roads discussed the matter with their assist- ants and were so favorably impressed with the plan as to say that they would adopt it if they had the tables to make it possible. Those who thought it unlikely that the change could be brought about cited the difficulty with the introduction of the metric system as an argument. To this the author replies that the adoption of the metric system in- volves a change of unit. The adoption of the decimal division of the degree involves no change of unit and merely does for angle work what American engineers long since did for their linear work. The book still retains the 90 quadrant. To be sure, minutes and even seconds have become a sort of unit, but so were inches, chains, and links. These are practically done away with for surveyor's use and there would seem to be as good reason for doing away with minutes and seconds. Practically every computation involving trigonometric logarithms re- quires less work by the decimal system than it does by the sexagesimal system. Instruments will be graduated to read to hundredths of degrees directly or 0.6 of a minute. Although the author believes that the "degree" of a curve should be the angle subtended by an arc of one hundred feet instead of a chord he has not adopted that definition, but has adhered to the definition approved by the American Railway Engineering Association. Five-place tables have been adopted as representing as high a degree of precision as is warranted by the field work. Computations of tables and some few other calculations require more extended tables but these practically always~arise in connection with office work where it is assumed that there are, or may be if necessary, six-place, seven-place, and even ten-place tables. The author has used seven-place tables, and occasionally ten-place tables, for the computation of the tables of this book. Persons do not always realize it, but considerable additional time is required to use six-place tables over that required for five-place tables. In his "Plane Surveying for Classroom and Field," the author discusses this question at some length and works examples to show the relative precision of four-place and five-place tables. The conclusions of the discussion are as follows: "i. It is useless to make linear measurements with a precision of more than I in 3500 if angles are to be read to the nearest minute only. "2. It is useless to use tables of more than four places for angles read to the nearest minute only. " It is difficult for many persons to bring themselves to use the smaller tables because they seem to see a greater precision in the use of tables PREFACE V giving results that are true to five and six significant figures, and fail to realize that the field work on which the computations depend does not warrant any such degree of precision, which is therefore only a seeming precision that is misleading and does not exist in fact. It is true that linear measurements can in general be made with greater pre- cision than the angle work gives and, hence, it is the angle work that fixes the precision and the tables to be used. The following rules may be formulated: " For angles read to the nearest minute use four-place tables. " For angles read to less than o oo' 30" use five-place tables. 11 For work in general requiring certainty in the third significant figure use four-place tables, in the fourth significant figure five-place tables, and in the fifth significant figure six-place tables. " But it must be remembered that no ordinary surveying work is pre- cisely enough done to warrant results certain to more than four signifi- cant figures, and that five-place tables are as extensive as are warranted by any land, topographic, railroad, or other surveys except the most refined city, bridge, and geodetic surveys. " Computation labor is increased about 50 per cent by using five-place tables instead of four-place tables, and about one-third by using six- place tables instead of five-place tables." Before using the logarithmic tables even persons somewhat familiar with the use of logarithms should read the explanatory text preceding the tables. The text concerning spirals and the spiral tables are based on the American Railway Engineering Association's ten-chord spiral. The author is indebted to Mr. Jenks B. Jenkins, Valuation Engineer for the Baltimore and Ohio Railroad and Chairman of the Track Committee of the A.R.E.A., and who devised the ten-chord spiral, for assistance with this part of the work. The author has endeavored to include beside bare tables many of which have been computed for this book and are not found in other books just so much of explanation of common field problems as would seem necessary to refresh the memory of young engineers who have not had these drilled into them by long experience. A few tables have been taken from other books. Acknowledgment is due Mr. Shelby S. Roberts for courteous permission to use tables from his "Track Formulae and Tables" and to the American Book Company for permission to use plates from the author's "Plane Sur- veying for Classroom and Field " for Tables I, XXIV, and LXXXIII, and to use the matter of Tables LXX, LXXI, and LXXVII. vi PREFACE Great care has been taken with the computations and the proof- reading but it is incredible that so many new computations should have been made and the results printed without error. The author will esteem it a favor if persons will report any errors that may be discovered to him or to the publishers. This book may be used about as conveniently as other books based on the sexagesimal division of the degree by those who do not care to have their instruments changed or to adopt the decimal division for their final records, and it has some features not found in existing field books that may commend it to field men. Therefore, it is put forth under the hope that it may find immediate approval in a few places; that it may be tried in some other places; and that familiarity and experience with it will convince users that the author is not a mere faddist but has contributed something of real use to the fraternity. WILLIAM G. RAYMOND STATE UNIVERSITY OF IOWA, IOWA CITY, IOWA, CONTENTS CHAPTER I PAGE SIMPLE, COMPOUND, AND VERTICAL CURVES i CHAPTER II THE SPIRAL 39 CHAPTER III LOGARITHMS AND TRIGONOMETRIC FUNCTIONS 70 CHAPTER IV LOCATION THEORIES AND TABLES 249 CHAPTER V ESTIMATING AND CONSTRUCTION TABLES 262 CHAPTER VI TURNOUTS AND CROSSOVERS 297 CHAPTER VII AZIMUTH, LATITUDE, AND TIME 305 CHAPTER VIII TABLES FOR METRIC CURVES 314 CHAPTER IX MISCELLANEOUS TABLES 320 CHAPTER X ADJUSTMENT OF INSTRUMENTS 333 CHAPTER XI SEXAGESIMAL TRIGONOMETRIC FUNCTIONS 337 Vll RAILROAD FIELD MANUAL CHAPTER I SIMPLE, COMPOUND, AND VERTICAL CURVES SIMPLE CURVES Fundamental notations and equations. The curve running from A to B in Fig. I , A = T. C. (tangent-curve). B = C. T. (curve-tangent). V = P. I. (point of intersection of tangents). I = central angle or deflection angle of tangents. T = tangent distance = R tan 5 7. C = long chord = 2 R sin 1 7. M = middle ordinate = R vers 5 7. E = external distance = R exsec 5 7. E = r tan * 7. C = 2 sec RAILROAD FIELD MANUAL Definition. The "degree" of a curve in American practice is the angle subtended at the center of a circular arc by a chord of 100 feet. In Latin American states where the metric system is used the "degree" is the angle subtended by a chord of 20 meters. If R be the radius and D the degree of a curve, then, in American practice, 7? = 50 sinfZ?' Table I gives R and its logarithm for various degree curves. C 2 Tangent offset / = R vers / = ^ . The tangent offset for one station is tabulated in Table I. For a subchord c the tangent offset is c 2 /c ~ /100 T^o" Approximate Fundamental Relations. Approximately, radii are in- versely as the degrees or R D' . -^> = -p (approx.). Radius of a i curve is 5729.65 = 5730 (approx.). RD 5730 (approx.). Tangent distance for a D curve of central angle 7 is T D = - (approx.). Table II gives tangent distances for a i curve and various values of /, and Table III gives corrections to TD =~^~ for more precise re- D suits. Values of C, M, and E, for D curves are also found approximately by dividing the values for a i curve for a given / by D. Table IV gives values of E for a i curve. A curve departs from a tangent approxi- mately thus: O = | n*D, n being the number of stations from the tangent point and D the degree of curve. Two curves of degrees D T.C. FIG. 3. and A depart from each other by the same approximate law, sub- stituting the difference of degrees D A or D\ D of the D of the foregoing formula. SIMPLE, COMPOUND, AND VERTICAL CURVES 3 The deflection angle A, for a chord of 100 feet is |Z); for a sub- chord, c, it is given by or, with sufficient exactness for all curves under about 8, - 0.5 cD, in hundred 2 5 = 0.3 cD, in minutes. 5 = X - - 0.5 cD, in hundredths of degrees, IOO 2 The ordinate from a chord at any point given by its distance from the center is K = or, approximately, Whence, for - a) 16 a = lc, K = ^ 4 (approx.). If the point is given by the distance from one extremity, QXS K = (approx.) Approximately = 872 Q X S X D 10,000,000 O_ = M 8R 4 RAILROAD FIELD MANUAL Location by Offsets from Long Chord. AG BF Ab 2 R sin - D, n = number stations A to G, D, AG^-BF CE = 100 or 2 R sin BF-CE 2 , etc. Bb be = ef = Be' = e'F Ff = Hh- Hh" = R (vers \nD - vers \(n - 2) D), etc., c h e FIG. 6. or Ab fG = IOOCQS D. Be' = e'F Bb = Ff - Cc' = Ee' IOO COS 100 sin 100 sin -3 2 n i . D, etc. Use natural functions and move decimal. In the particular figure Hh' = 50 tan j D because is odd. If the number'of chords is not more than 8 or the degree more than 20 and of an even number of tenths, the long chords and middle ordinates may be taken from Tables V and VI, thus: AG = long chord of n stations. . , _ ._ long chord of n stations long chord of (n2) stations etc Bb = Ff = middle ordinate for n stations middle ordinate for (n 2) stations, etc. Location from Chord Produced. Bb = R vers D, or 100 sin ^ D for natural functions. Bb = tangent offset of Table I. AV or XA gives line of tangent. FIG. 7. Stretch the tape from A to B so that Bb shall measure as above. Produce AB to c', one tape length, and swing about B until c'C = 2 Bb. Produce BC to e' and swing about C until e'E = 2 Bb, etc. If the curve begins with SIMPLE, COMPOUND, AND VERTICAL CURVES 5 a subchord, /, swing / feet above A to B (Fig. 8) until Bb I sin Swing 100 / about A to F until Ff = (100 /) sin IO ~ . Pro- duce FB to c f , 100 feet, and swing about B until c'C = 2 X 100 sin D, or twice the tangent offset of Table I. Produce BC, etc., as FIG. 9. before. For ending use the same method reversed. Supposing the curve to be running from C to B to A, locate F by producing CB, measure over Ff and Bb, and establish A between / and b at / feet from B. Problems. Suggestions for passing obstacles. If TC only is in- accessible: (a) Run to V to C T and run curve backwards. (b) Assume a point on the curve beyond the obstacle; compute the tangent distance for the point, as TC K\ run to K-, deflect angle at K and run to P and run the curve backward and forward. If CT is inaccessible the same methods with obvious modifications may be used. To pass an obstacle on a curve one of the methods suggested by the figure may be used. The line AbC is run as ABC would be run only with the center on the opposite side. To change TC or CT. First method: Assume or know the necessary change in tangent distance and compute a new degree of curve. Second method : Assume a new D a round number, probably such as to accomplish the O RAILROAD FIELD MANUAL desired result; change in CT or TC = change in tangent distances, or m = T 2 - Ti = (R 2 - RJ tan \ I. /I FIG. ii. FIG. 12. A curve ends in VB or V'B' ; required to end in V'B' or VB. Degree does not change. It is necessary to find change in A . K A A' = VV = BB' = 00' sin / J o'.o o" FIG. 13. 1 FIG. 14. A curve ends in VB; required to end in V'B' or V'B" without changing A . Degree changes. R r = K vers / that is, change in R K vers / A curve ends in VB; required to end in V'B'. A and D change. gr R' = R db ; , according as B' is inward or outward from B. exsec / SIMPLE, COMPOUND, AND VERTICAL CURVES A A' = (R R') tan /, or K cot \ 7, A being moved ahead or back according as R' is less or greater than R. A curve ends in VB; required to end in VB'. Without changing degree A A' = R (tan | I tan \ /'), and A is moved forward or back accord- ing as 1' is less or greater than /. Changing degree and keeping A fixed, T T is unchanged. .*. R' = 377 or T cot \ I'. A curve ends in VB; required to end in V'B. Fig. 16. D and A change. 7?' p vers 7 "* * 1C 77 vers / ' yl^4 = (R sin 7 R' sin /') and ^4 is moved forward or back accord- ing as /' is greater or less than /. A curve ends in VB; required to end in V'B'. Fig. 17. i. Assume new R' < R according as B' is inside or outside VB. Then K K cos a = I R-R' or i R' -R n stations = -= Begin at n stations from B and run in curve of D' for n' stations = !&.-.. FIG. 1 6. FIG. 17. 2. Or, retaining A and letting n be total number of stations A to 5, K D' approx. 8 RAILROAD FIELD MANUAL D' will be greater or less than D according as V'B' lies inside or outside of VB. Take D' nearest round number that will be sufficiently exact and run the curve D r for n' stations = ~-, 3. Or, assuming a new degree less or greater than the original accord- ing as the tangent is to be thrown out or in, find n of Method I by t/ V V 7 (>-')' (D D') is to be taken as the difference of degrees, subtracting the smaller from the larger, a = nD, ' = , to make the tangents par- allel. Run curve of degree D' from C.C. FIG. 1 8. FIG. 19. COMPOUND CURVES Given RI, R 2 , AI, and A 2 ; required 7\ and T 2 . Fig. 18. FiF 2 = /i+* 2 . Solve triangle FFiF 2 for FiFand F 2 F. Given I, 7i, T 2 , and RI; required R 2 , AI, and A 2 . Fig. 19. T 2 -f Ti cos / - Ri sin 7 cot \ A 2 TI sin / RI vers / /-A 2 , TI sin I R 2 vers I vers A 2 (a) SIMPLE, COMPOUND, AND VERTICAL CURVES Given I, 7i, T 2 , and R z ; required RI, AI, and A 2 . RZ sin I T\ Ti cos / l * 2 vers 7-T 2 sin/ A 2 = I - Ai, C0t Rl = Rt _ l^J^ !ili_i. (b) vers AI Given 7, 7\, J?i, and AI; required A 2 , JR 2 , awJ J" 2 . A 2 = 7 AI. Rz is given by equation (a) above. 7" 2 = (Rz RI) sin A 2 + RI sin 7 T\ cos 7. Given 7i, 7" 2 , .R 2 , and A 2 ; required AI, 7?i, awd 7\. AI = 7 A 2 . /? is given by equation (b) above. TI = Rz sin 7 TZ cos I (R 2 RI) sin AI. To end a compound curve in a new tangent, parallel to that first located. (a) Move the curve parallel to itself along the first tangent, a distance A A' = ~. j- (b) Retaining the first branch, changing only the degree of the second, K vers A 2 ' RI' = RI vers A! according as the curve ends with the larger or shorter radius. (c) Retaining both degrees, changing the A's and the station of CC. If the new tangent is inside and the longer radius ends the curve the CC is advanced; if the new tangent is outside the CC is moved back; if the shorter radius ends the curve the movement of the CC is reversed. The new values for the final As are had from vers A 2 ' = vers A 2 -p ^~ ' vers AI' = vers AI -5 5- IO RAILROAD FIELD MANUAL To change the direction of the final tangent by a given amount. Solve the triangle VV'C for the new final tangent and the change in the initial tangent. The new / being known, retain the first radius and solve for the new final radius and the two central angles, by preced- ing equations. VERTICAL CURVES The rate of change of grade in passing sags and summits in the grade line, as recommended by the American Railway Engineer- ing Association, is: For first-class railways change not more than o.i foot per station on summits and not more than 0.05 foot per station in sags; for second-class roads not more than double these rates. How to determine the elevation of the several stations on the vertical curve will be shown by two examples. The first step is to determine the length of the curve; it will be convenient to adopt a rate of change that will give an even number of stations for the length. Example I. Two grades, 0.8 and +0.6, meet at station 462 where the elevation is 723. Required a vertical curve to connect with a change of o.i per station. The total change in rate is 0.8 + 0.6 = 1.4, giving 14 stations =1.4 -5- o.i, of curve, or 7 stations each side of the vertex. The beginning of the curve is therefore at sta. 462 7 = 455 and the end is at 455 + 14 = 469. The elevation of 455 is 723.0 + 7 X 0.8 =728.6; the elevation of sta. 469 is 723.0 + 7 X 0.6 = 727.2. The rate of change for the first station is taken at half the station rate of change or 0.05. Therefore the elevation of FIG. 21. sta. 456 = 728.6 457 == elev. 456 - 458 = elev. 457 - 459 = elev. 458 - 460 = elev. 459 461 = elev. 460 462 = elev. 461 463 = elev. 462 464 = elev. 463 - (0.8 - 0.05) (0.8 - 0.15) = (0.8 - 0.25) = (0.8 - 0.35) = (0.8 - 0.45) = (0.8 - 0.55) = (0.8 - 0.65) = (0.8 - 0.75) = (0.8 - 0.85) = = 728.6 - 727.85 - 727.20 - 726.65 726.20 725.85 - 725.60 - 72545 - 725.40 + 0.75 = 727.85 0.65 = 727.20 0-55 = 726.65 0.45 = 726.20 0-35 = 725.85 0.25 = 725.60 0.15 = 725.45 0.05 = 72540 0.05 = 72545 SIMPLE, COMPOUND, AND VERTICAL CURVES II 465 = elev. 464 - (0.8 - 0.95) = 72545 + 0.15 = 725.60 466 = elev. 465 - (0.8 - 1.05) = 725.60 + 0.25 = 725.85 467 = elev. 466 - (0.8 - 1.15) = 725.85 + 0.35 = 726.20 468 = elev. 467 (0.8 1.25) = 726.20 + 0.45 = 726.65 469 = elev. 468 (0.8 1.35) = 726.65 + 0.55 = 727.20 It will be noticed that the final elevation agrees with that computed above; this proves the work. It will also be noticed that after the bottom of the sag is passed the elevations repeat themselves in reverse order. The bottom of the sag is not always the same station as the apex. This depends on the relative rates of the grades. Example 2. Two grades, 0.2 and i.o, meet at station 867.0, where the elevation is 466.0. To connect the grades with a vertical curve changing at the rate of o.i per station. Total change in rate 0.8. Length of curve 8 stations. Beginning of curve sta. 867 4 = 863; end = sta. 871. Elev. sta. 863 = 466 + 4 X 0.2 = 466.8 Elev. sta. 871 = 466 4 X i.o = 462.0 Elev. sta. 864 = elev. 863 (0.2 + 0.05) = 466.8 0.25 = 466.55 Elev. sta. 865 = elev. 864 (0.2 +0.15) = 466.55 0.35 = 466.20 Elev. sta. 866 = elev. 865 (0.2 + 0.25) = 466.20 0.45 = 465.75 Elev. sta. 867 = elev. 866 (0.2 + 0.35) = 465.75 0.55 = 465.20 Elev. sta. 868 = elev. 867 (0.2 + 0.45) = 465.20 0.65 = 464.55 Elev. sta. 869 = elev. 868 (0.2 + 0.55) = 464.55 0.75 = 463.80 Elev. sta. 870 = elev. 869 (0.2 + 0.65) = 463.80 0.85 = 462.95 Elev. sta. 871 = elev. 870 (0.2 + 0.75) = 462.95 0.95 = 462.00 The work is proved since 462.0 is the elevation first found for station 871. There is no summit or bottom in this case as both grades are of the same sign. TABLE I Radii and their logarithms, and tangent offsets,- and middle ordinates, for IOO feet chords of curves of degrees given. The degrees are given in de- grees and decimals of a degree. Deg. D Radius R Logarithm LogR Tan. Off. t Mid. Ord. m Deg. D Radius R Logarithm LogR Tan. Off. t Mid. Ord. m 0.00 00 00 .000 .000 1. 00 5729-65 3-75813 .873 .218 .02 286478.90 5.45709 .017 .004 .02 5617-31 -74953 .890 -223 .04 143239.45 .15606 .035 .009 .04 5509-29 .74110 .908 .227 .06 95492.97 4-97997 .052 .013 .06 5405-34 -73283 925 231 .08 71619.73 85503 .070 .017 .08 5305.24 .72471 942 236 .IO 57295.79 4-75812 .087 .022 .IO 5208.79 3.71674 .960 .240 .12 47746.49 .67894 .105 .026 .12 5115-78 .70891 977 244 .14 40925.57 .61199 .122 .031 .14 5026.03 .70122 995 .248 .16 35809.87 55400 .140 035 .16 4939-38 69367 I.OI2 253 .18 31831.00 .50285 157 .039 .18 4855.66 .68625 030 -257 .20 28647.90 4-45709 175 .044 .20 4774-74 3-67895 1.047 .262 .22 26043.55 .41570 .192 .048 .22 4696.46 .67177 -06 S .266 .24 23873.26 37791 .209 .052 .24 4620.72 .66471 .082 .271 .26 22036.86 34315 .227 057 .26 4547.38 .65776 .100 -275 .28 20462.80 31097 244 .061 .28 4476.33 65092 .117 .279 30 19098.61 4.28100 .262 .065 30 4407.46 3.64419 I-I34 .284 32 17904.95 .25297 .279 .070 32 4340.69 .63756 .152 .288 34 16851.73 .22664 297 074 34 4275-90 63103 .169 .292 .36 I59I5-52 .20182 314 .079 .36 4213.02 62459 .187 .297 38 15077.86 17834 332 083 38 4I5I-97 .61825 .204 301 .40 14323.97 4.15606 349 .087 .40 4092.66 3.61201 1.222 305 .42 13641.88 13487 .367 .092 .42 4035-02 60585 239 310 44 13021.80 .11467 384 .096 44 3978.98 59977 257 314 .46 12455-64 09537 .401 .100 .46 3924.47 59378 .274 319 .48 11936.66 .07688 .419 .105 .48 3871.44 58787 .292 323 SO 11459.19 4.05915 436 .109 50 3819-83 3.58204 1.309 327 52 11018.46 .04212 454 113 52 3769.57 .57629 .326 332 54 10610.37 02573 .471 .118 54 3720.62 .57o6i 344 .336 56 10231.43 .00994 .489 .122 56 3672.92 .56501 361 340 .58 9878.62 3-99470 .506 .127 58 3626.43 55948 379 345 .60 9549-34 3-97997 524 .131 .60 3581.10 3-55402 1.396 349 .62 9241-30 96573 541 135 .62 3536.89 .54862 .414 353 .64 8952.51 95194 559 .140 .64 3493.76 54329 431 358 .66 8681.26 93858 576 .144 .66 345I.67 53803 449 362 .68 8425.90 .92562 593 .I 4 8 .68 3410.59 53283 .466 -367 70 8185.16 3-91303 .611 153 .70 3370.46 3-52769 1.483 371 72 7957.8o .90079 .628 157 72 333L28 .52261 501 375 74 7742.73 .88889 .646 .l6l 74 3292-99 51759 .518 -380 76 7539-97 -8773I 663 .166 76 3255-57 .51263 .536 -384 78 7345.67 .86603 .681 .170 78 3218.99 .50772 553 -388 .80 7162.03 3-85504 .698 175 .80 3183.23 3.50287 I-57I 393 .82 6987-35 .84431 .716 .179 .82 3148.25 .49807 .588 397 84 6820.99 .83385 733 I8 3 -84 3114-03 49332 .606 .401 .86 6662.36 .82363 750 .188 .86 3080.55 .48863 .623 .406 .88 6510.95 .81364 .768 .192 .88 3047-V8 .48398 .641 .410 .90 6366.26 3-80388 .785 .196 .90 3015-71 3-47939 1.658 415 .92 6227.87 79434 .803 .2OI .92 2984.29 .47484 .675 .419 94 6095.36 78500 .820 .205 94 2953-53 -47034 .693 423 .96 5968.38 .77586 .838 .209 .96 2923.40 .46589 .710 .428 .98 5846.58 .76690 .855 .214 .98 2893-87 .46148 .728 432 1. 00 5729.65 3-758i3 .873 .218 2.OO 2864.93 3-457II 1-745 .436 (12) TABLE I. (Continued] Deg. Radius R Logarithm LogR Tan. Off.' t Mid. Ord. m Deg. D Radius R Logarithm LogR Tan. Off. t Mid. Ord. m o 2.OO 2864.93 3-457" 1-745 436 3.00 1910.08 3.28105 2.618 .655 .02 2836.57 45279 763 .441 .02 1897.43 .27817 635 659 .04 2808.76 44852 .780 445 04 1884.95 27530 653 663 .06 2781.50 .44428 .798 449 .06 1872.63 .27245 .670 .668 .08 2754.76 .44008 .815 454 .08 1860.48 .26962 .687 .672 .10 2728.52 3-43593 1.832 458 .10 1848.48 3.26681 2.705 .676 .12 2702.79 43181 .850 463 .12 1836.63 .26402 .722 .681 .14 2677-53 42773 .867 467 .14 1824.93 .26125 740 .685 .16 2652.74 42369 885 .471 .16 1813-39 25849 757 .689 .18 2628.40 .41969 .902 476 .18 1801.99 25575 775 694 .20 2604.51 3-4I573 1.920 .480 .20 1790.73 3-25303 2.792 .698 .22 2581.05 .41180 937 484 .22 1779-61 25032 .810 .702 .24 2558.01 .40790 955 .489 .24 1768.62 .24764 .827 .707 .26 2535-38 .40404 .972 493 .26 1757-78 .24496 845 .711 .28 2513-14 .40022 990 497 .28 1747-06 24231 .862 .716 30 2491.29 3-39642 2.007 502 30 1736.48 3-23967 2.879 .720 32 2469.81 .39266 .024 506 32 1726.02 23705 897 724 34 2448.71 38894 .042 5ii 34 1715-69 23444 .914 .729 36 2427.96 38524 059 515 36 1705.48 23185 932 733 38 2407.56 38158 077 519 38 1695-39 .22927 949 737 .40 2387-50 3-37794 2.094 524 .40 1685.42 3.22671 2.967 .742 .42 2367.77 37434 .112 528 .42 1675.56 .22416 984 .746 44 2348.36 37077 .129 532 44 1665.83 .22163 3.002 751 .46 2329.28 .36722 .147 537 .46 1656.19 .21911 .019 755 .48 2310.49 36370 .164 541 .48 1646.68 .21661 036 759 SO 2292.01 3-36022 2.181 545 50 1637.28 3-21412 3-054 764 52 2273.83 .35676 .199 550 52 1627.98 .21165 .071 .768 54 2255.92 35332 .216 554 54 1618.78 .20919 .089 .772 56 2238.30 .34992 234 559 56 1609.69 .20674 .106 777 58 2220.95 34654 .251 563 .58 1600.70 .20431 .124 .781 .60 2203.87 3.343I9 2.269 .567 .60 1591.81 3.20189 3-I4I .785 .62 2187.05 .33986 .286 572 .62 1583.02 .19949 159 .790 .64 2170.49 .33656 .304 576 .64 1574-32 .19709 .176 794 .66 2154.17 .33328 .321 580 .66 1565.72 .19472 193 799 .68 2138.10 33003 339 .585 .68 1557-22 19235 .211 .803 .70 2122.26 3-32681 2.356 589 70 1548.80 3.19000 3-228 .807 72 2106.66 32359 373 593 .72 1540.48 .18766 .246 .812 74 2091.29 32041 391 .598 74 1532.24 18533 263 .816 76 2076.13 .31726 .408 .602 76 1524.10 .18301 .281 .820 78 2061.20 31412 .426 .607 78 1516.14 .18071 .298 .825 .80 2046.48 3-3IIOI 2-443 .611 .80 1508.06 3-17842 3.316 .829 .82 2031.97 .30792 .460 615 .82 1500.17 .17614 333 833 .84 2017.66 30485 478 .620 .84 1492.36 .17387 351 838 .86 2003.56 .30180 .496 .624 .86 1484.63 .17162 368 .842 .88 1989-65 .29878 513 .628 .88 1476.98 .16937 385 .847 .90 1975-93 3-29577 2-530 633 .90 1469.41 3.16714 3-403 851 .92 1962.40 .29279 548 637 92 1461.91 .16492 .420 855 94 1949.05 .28982 .565 .641 94 1454-49 .16271 438 .860 .96 1935-88 .28688 .583 .646 .96 1447-15 .16051 455 .864 .98 1922.89 .28396 .600 650 .98 1439-88 15833 473 .868 3-oo 1910.08 3.28105 2.618 655 4-00 1432.69 3-15615 3-490 873 (13) TABLE I. (Continued) Deg. Radius R Logarithm LogR Tan. Off. t Mid. Ord. m Deg. D Radius R Logarithm LogR Tan. Off. t Mid. Ord. m o 4-00 1432.69 3-15615 3-490 .873 5.00 1146.28 3-05929 4.362 1.091 .02 1425-56 15399 507 .877 .02 1141.72 -05756 379 095 .04 1418.51 .15183 525 .881 04 1137-19 05583 397 .100 .06 1411-52 .14969 .542 .886 .06 1132.70 05411 .414 .104 .08 1404.60 14755 .560 .890 .08 1128.24 .05240 .432 .108 .10 1397.76 3-14543 3.577 .895 .10 1123.82 3-05070 4-449 1.113 .12 1390.98 14332 595 .899 .12 1119.43 .04900 467 .117 .14 1384.26 .14122 .612 .903 14 1115.08 04731 484 .122 .16 1377.61 13912 .629 .908 .16 IIII.OO 04571 .501 .126 .18 1371-02 13704 647 .912 .18 1106.47 04394 .519 .130 .20 1364.49 3-13497 3-664 .916 .20 IIO2.22 3-04227 4.536 I-I35 .22 1358.03 .13291 .682 .921 .22 1098.00 .04060 554 139 .24 1351-31 .13076 699 925 .24 1093.81 .03894 571 143 .26 1345-28 .12881 .717 .929 .26 1089.66 .03729 589 .I 4 8 .28 1339-00 .12678 734 934 .28 1085.53 03564 .606 .152 30 1332.77 3.12476 3-752 .938 30 1081.44 3.03400 4-623 I.I56 32 1326.61 .12274 769 943 32 1077.38 03237 .641 .l6l 34 1320.49 .12074 .786 947 34 1073-34 03074 .658 .165 .36 1314.44 .11874 .804 951 .36 1069.34 .02912 .676 .170 38 1308.44 .11675 .821 956 .38 1065.37 .02750 693 .174 .40 1302.50 3.11478 3.839 .960 .40 1061.43 3-02589 4-7II I.I78 42 1296.61 .11281 .856 .964 .42 1057.51 .02429 .728 .183 44 1290.77 .11085 874 .969 44 1053.63 .02269 746 .I8 7 46 1284.98 .10890 .891 973 .46 1049.77 .02109 763 .191 .48 1279.25 .10696 .909 .978 .48 1045.94 .01951 .780 .196 So 1273-57 3.10502 3-926 .982 50 1042.14 3-01793 4.798 I.2OO 52 1267.93 .10310 943 .986 52 1038.37 .01635 .815 .205 54 1262.35 .10118 .961 .991 54 1034.62 .01478 833 .209 56 1256.82 .09927 .978 995 56 1030.90 .01322 .850 .213 58 1251.33 09737 996 999 58 1027.21 .01166 .868 .218 .60 1245.89 3-09548 4.013 1.004 .60 1023.55 3.01011 4-885 1.222 .62 1240.50 .09360 .031 .008 .62 1019.91 .00856 .902 .226 .64 1235.16 .09172 .048 .012 .64 IOl6.29 .00702 .920 .231 .66 1229.86 .08986 .065 .017 .66 IOI2.7O .00548 -937 235 .68 1224.61 .08800 .083 .021 .68 1009.14 00395 955 -239 70 1219.40 3-08615 4.100 1.026 70 1005.60 3.00243 4-972 1.244 72 1214.24 .08430 .118 .030 72 1002.09 .00091 .990 .248 74 1209.12 .08247 135 034 74 998.60 2-99939 5.007 .253 76 1204.04 .08064 .153 039 76 995-14 .99788 .024 257 78 1199.00 .07882 .170 043 78 991.68 .99638 .042 .261 .80 1194.01 3-07701 4.188 1.047 .80 988.28 2.99488 5-059 1.266 .82 1189.06 .07520 .205 .052 .82 984.89 99339 077 .270 .84 1184.15 07341 .222 .056 .84 981.52 .99190 .094 274 .86 1179.28 .07162 .240 .060 .86 978.17 .99041 .112 .279 .88 1174-45 .06983 257 .065 .88 974.66 .98885 .129 .283 .90 1169.66 3.06806 4-275 1.069 .90 971-54 2.98746 5.146 1.287 .92 1164.91 .06629 .292 .074 .92 968.26 .98599 .164 .292 94 1160.19 06453 310 .078 94 965.01 .98453 .I8l .296 .96 1155-52 .06278 327 .082 .96 961.77 .98307 .199 .301 98 1150.88 .06103 345 .087 .98 958.56 .98162 .216 305 S.oo 1146.28 3-05929 4-362 I.09I 6.00 955-37 2.98017 5.234 1.309 (14) TABLE I. (Continued} Deg. D Radius .R Logarithm LogR Tan. Off. t Mid. Ord. m Deg. D Radius R Logarithm LogR Tan. Off. t Mid. Ord. m o o 6.00 955-37 2.98017 5-234 1.309 7.00 819.02 2.91329 6.105 1.528 .02 952.20 97873 251 314 .02 816.69 .91206 .122 532 .04 949-05 97729 .268 318 .04 814-37 .91082 .140 536 .06 945-92 97585 .286 322 .06 812.07 .90959 157 541 .08 942.81 97442 303 327 .08 809.78 90837 175 545 .10 939-72 2.97300 5-321 1-331 .10 807.50 2.90714 6.192 1-550 .12 936.65 97158 .338 335 .12 805.23 .90592 .209 554 .14 933-60 .97016 -356 -340 14 802.98 .90471 .227 558 .16 930-57 -96875 373 -344 .16 800.74 90349 244 -563 .18 927.58 96734 390 -349 .18 798.51 .90228 .262 .567 .20 924.58 2.96594 5-408 1-353 .20 796.30 2.90108 6.279 1-572 .22 921.61 -96455 .425 357 .22 794.10 .89987 .296 576 .24 918.66 96315 443 362 24 791.90 .89867 314 S8o .26 915.72 .96176 .460 366 .26 789-73 .89748 331 584 .28 912.81 .96038 .478 370 .28 787-56 .89628 349 589 30 909.92 2.95900 5-495 1-375 30 785.40 2.89509 6.366 1-593 32 906.48 95736 512 379 32 783-26 .89391 .384 -598 34 904.18 95625 530 .384 34 781.13 .89272 .401 .602 .36 901.34 .95489 547 -389 .36 779-01 .89154 .418 .606 .38 898.52 95353 .565 392 38 776-90 .436 .611 .40 895.7I 2.95217 5-582 1-397 .40 774-81 2.88919 6-453 1.615 .42 892.92 .95081 .600 .401 .42 772.72 .88802 .471 .619 44 890.15 94947 .617 405 44 770.65 .88685 .488 .624 46 887.40 .94812 634 .410 .46 768.58 .88569 SOS .628 48 884.67 .94678 652 .414 .48 766.53 .88453 .523 -632 50 881.95 2-94544 5-669 1.418 .50 764.49 2-88337 6.540 1-637 52 879.24 .94411 .687 .423 52 762.46 .88222 .558 .641 54 876.56 .94278 .704 .427 .54 760.44 .88106 575 .646 56 873-89 .94146 .722 432 .56 758.45 87993 593 .650 58 871.24 .94014 739 436 .58 756.43 .87877 .610 -654 .60 868.60 2.93882 5.756 1.440 .60 754-44 2.87763 6.627 1.659 .62 865.98 93751 774 445 .62 752-47 .87649 645 .663 .64 863.37 .93620 .791 449 .64 750.50 87535 .662 .667 .66 860.78 .93489 .809 453 .66 748.54 .87422 .680 .672 .68 858.21 93359 .826 -458 .68 746.60 .87309 .697 .676 70 855.65 2.93230 5.844 1.462 70 744.66 2.87196 6.714 i. 680 72 853-10 93100 .861 .466 .72 742-74 .87083 732 .685 74 850.58 .92971 .878 .471 -74 740.82 .86971 749 .689 .76 848.06 92843 .896 475 .76 738.91 .86859 .767 .694 78 845-56 92715 913 .480 78 737.02 .86748 784 .698 .80 843.08 2.92587 5-931 1.484 .80 735-13 2.86636 6.802 1.702 .82 840.61 .92459 .948 .488 .82 733-25 .86525 .819 .707 .84 838.16 .92332 .965 .493 .84 731.38 .86415 .836 .711 .86 835.71 .92206 -983 497 .86 729.53 .86304 854 .715 .88 833-29 .92080 6.000 1.501 .88 727.68 .86194 .871 .720 .90 830.88 2.91954 6.018 1-505 .90 725-85 2.86084 6.889 1.724 .92 828.48 .91828 035 .510 .92 724.01 .85974 .906 729 .94 826.09 -91703 053 .515 .94 722.19 .85865 923 -733 .96 823.72 .91578 .070 519 .96 720.38 .85756 .941 737 .98 821.36 -91454 .087 523 .98 7i8.57 .85647 958 742 7.00 819.02 2.91329 6.105 1.528 8.00 716.78 2.85539 6.976 1.746 (15) TABLE I. (Continued} Deg. D Radius R Logarithm LogR Tan. Off. t Mid. Ord. m Deg. D Radius R Logarithm LogR Tan. Off. t Mid. Ord. m o 8.00 716.78 2.85539 6.976 1.746 9.00 637-27 2.80433 7.846 1.965 .02 714.99 85430 993 750 .02 635-86 .80336 863 969 .04 713-22 -85322 7.010 755 .04 634-46 .80240 .881 973 .06 7H-45 .85215 .028 759 .06 633-06 .80145 .898 978 .08 709.69 .85107 045 -764 .08 631-67 .80049 .916 .982 .10 707.94 2.85000 7-063 1.768 .10 630.29 2-79954 7-933 1.987 .12 706.20 .84893 .080 772 .12 628.91 .79859 950 .991 .14 704.47 .84786 .098 777 .14 627.53 79764 .968 995 .16 702.75 .84680 "5 .781 .16 626.17 .79669 -985 999 .18 701-03 .84574 132 .785 .18 624.81 79574 8.002 2.004 .20 699-33 2.84468 7.I50 1.790 .20 623.45 2.79480 8.020 2.008 .22 697-63 .84362 .167 794 .22 622.10 .79386 037 .013 .24 695.94 .84257 185 798 -24 620.76 .79292 .055 .017 .26 694-25 .84152 .202 803 .26 619.42 .79198 .072 .021 .28 692.58 .84047 .219 .807 .28 618.09 79105 .090 .026 30 690.91 2.83942 7-237 1.811 30 616.76 2.79012 8.107 2.030 32 689.26 .83838 254 .816 .32 615.44 .78919 .124 034 34 687.61 83734 .272 .820 34 614.12 .78826 .142 039 36 685.96 .83630 .289 825 36 612.82 .78733 .159 043 .38 684.33 83527 -306 .829 38 611.51 .78640 .176 .048 .40 682.70 2.83423 7-324 1-833 40 610.21 2.78548 8.194 2.052 .42 681.09 .83320 341 838 42 608.92 .78456 .211 .056 44 679.47 83217 359 .842 44 607.63 78364 .229 .O6l .46 677-87 83115 376 -847 46 606.35 .78272 .246 06 5 .48 676.27 .83012 393 851 .48 605.08 .78181 .263 .069 50 674-69 2.82910 7.411 1-855 So 603.80 2.78090 8.281 2.074 52 673-11 .82808 .428 .860 .52 602.54 77999 .298 .078 54 671-53 .82707 446 .864 .54 601.28 .77908 .316 .083 56 669.97 .82605 463 .868 56 600.02 .77817 333 .087 58 668.41 .82504 .480 -873 .58 598.77 .77726 350 .091 .60 666.86 2.82403 7.498 1.877 .60 597-53 2.77636 8.368 2.096 .62 665.33 .82303 515 .881 .62 596.29 77546 385 .IOO .64 663.77 .82202 533 .886 .64 595-o6 77456 403 .104 .66 662.24 .82102 550 .890 .66 593-83 .77366 .420 .109 .68 660.72 .82002 -567 -895 .68 592.60 .77276 437 113 .70 659-21 2.81902 7.585 1.899 .70 591-38 2.77187 8-455 2.II7 .72 657.70 .81803 .602 -903 .72 590.17 .77098 472 .121 74 656.19 -81703 .620 .908 .74 588.96 .77009 490 .126 .76 654.70 .81604 637 .912 .76 587-76 .76920 507 .131 .78 653-21 .81505 655 .916 .78 586.56 .76831 524 .135 .80 65I-73 2.81407 7.672 1.921 .80 585-36 2.76743 8.542 2.139 .82 650.25 .81308 .689 925 .82 584-17 76654 559 .144 .84 648.79 .81210 .707 930 .84 582.99 76566 576 .I 4 8 .86 647.32 .81112 .724 934 .86 581.81 -76478 594 .152 .88 645.87 .81014 742 .938 .88 580.64 76390 .611 157 .90 644.42 2.80917 7-759 1-943 .90 579.46 2.76303 8.629 2.161 .92 642.98 .80820 -776 947 .92 578.30 .76215 .646 .166 94 641.54 .80723 -794 951 94 577-14 .76128 -663 .170 .96 640.11 .80626 .811 .956 96 575-99 .76041 .681 .174 .98 638.69 .80529 .829 .960 .98 574-83 75954 .698 179 9,00 637.27 2.80433 7.846 1.965 10.00 573.69 2.75867 8.716 2.183 (16) TABLE I. (Continued) Deg. D Radius R Logarithm LogR Tan. Off. t Mid. Ord. m Deg. D Radius R Logarithm) LogR Tan. Off. t Mid. Ord. m o o IO.Q S73-69 2.75867 8.716 2.183 15.0 383-06 2-58327 13-053 3-277 .1 568.02 .75436 .803 .205 .1 380.54 .58040 139 299 .2 562.47 .75010 .889 .227 .2 378.05 57755 .226 .321 3 557-02 .74587 .976 234 3 375.6o 57472 .312 343 .4 551-68 .74169 9-063 .270 4 373.17 .57191 399 365 .5 546.44 2-73754 9-150 2.293 .5 370-78 2.56912 13-485 3.387 .6 54L30 73344 237 314 .6 368.42 56634 572 409 .7 536.25 .72937 .324 .336 .7 366.09 56358 .658 431 .8 53I-30 72534 .411 .358 .8 363-78 .56084 744 452 9 526.44 72135 .498 .380 9 361.51 55812 831 474 II. 521.67 2.71740 9-585 2.402 16.0 359-26 2-55541 I3.9I7 3.496 .1 516.99 .71348 .671 423 .1 357-05 55273 14.004 5i8 .2 512.38 .70960 758 445 .2 354-86 .55006 .090 540 3 507-86 .70575 .845 .467 -3 352.70 54740 .177 .562 .4 503-42 .70193 932 .489 4 350.56 54476 263 584 5 499.06 2.69815 10.019 2.511 S 348.45 2.54214 14-349 3.606 .6 494-77 .69441 .106 533 .6 346.37 53953 .436 .628 7 490.56 .69069 .192 555 7 344-31 53694 522 650 .8 486.42 .68701 279 577 .8 342.27 53437 .608 .672 9 482.34 .68336 .366 .598 9 340.26 53i8i .695 694 12.0 478.34 2.67974 10.453 2.620 17.0 338-27 2.52927 14.781 3.7i6 .1 474-40 .67614 540 .642 .1 336.31 .52674 .867 738 .2 470-53 67258 .626 .664 .2 334-37 52423 954 .760 3 466.72 .66905 713 .686 3 332-45 52173 15.040 .781 4 462.96 66555 .800 .708 4 330.56 51924 .126 .803 5 459-28 2.66207 10.887 2.730 5 328.69 2.51677 15.212 3-825 .6 455-65 .65863 973 752 .6 326.83 51432 299 .847 .7 452.07 .65521 1 1 .060 774 7 325.00 .51188 .385 .869 .8 448.56 .65182 147 -795 .8 323.i8 .50945 .471 .891 9 445.09 64845 .234 .817 9 321.39 .50704 557 913 13-0 441.68 2.64511 11.320 2.839 18.0 319-62 2.50464 15-643 3-935 .1 438.33 .64180 .407 .861 .1 317.87 .50225 730 957 .2 435-02 -63851 494 .883 .2 316.14 .49988 .816 979 3 43I-76 63525 .580 90S 3 3I4.43 49752 .902 4.001 4 428.56 .63201 .667 927 4 312.73 49517 .988 .023 5 425-40 2.62879 H-754 2-949 5 311.06 2.49284 16.074 4-045 .6 422.28 .62560 .840 971 .6 309-40 .49052 .160 .067 .7 419.22 .62244 .927 992 7 307.76 .48821 .246 .089 .8 416.19 .61929 12.014 3.014 .8 306.14 .48591 333 .in 9 413-21 .61617 .100 .036 9 304.53 48363 .419 -133 14.0 410.28 2.61308 12.187 3-058 19.0 302.94 2.48136 16.505 4-155 .1 407.38 .61000 274 .080 .1 301.37 .47910 591 .177 .2 404.53 .60695 .360 .102 .2 299.82 .47686 677 .199 3 401.71 60391 447 .124 3 298.28 .47462 763 .221 4 398.94 .60090 -533 .146- 4 296.75 .47240 .849 243 5 396.20 2.59791 12.620 3-i68 S 295.25 2.47019 16.935 4-265 .6 393-50 59495 .706 .190 .6 293.76 46799 17.021 .287 7 390.84 .59200 793 .211 7 292.28 .46580 .107 .308 .8 388.21 .58907 .880 233 .8 290.82 .46362 193 330 9 385-62 .58616 .966 255 9 289-37 .46145 279 352 15.0 383-06 2.58327 13-053 3-277 20. o 287.94 2-45930 17-365 4-374 (17) TABLE I. (Continued) D3g. D Radius R Logarithm LogR Tan. Off. t Mid. Ord. m Deg. D Radius R Logarithm LogR Tan. Off. t Mid. Ord. m o 20.0 287.94 2-45930 17-365 4-374 25.0 231.01 2.36363 21.644 5-476 .1 286.52 .457i6 451 .396 .1 230.11 .36193 .729 498 .2 285.12 45502 537 .418 .2 229.21 .36023 .814 .520 3 283.73 .45290 623 440 3 228.32 35854 899 .542 4 282.35 45079 .708 .462 4 227.43 -35685 985 564 5 280.99 2.44869 17-794 4.484 5 226.55 2.35517 22.070 5-586 .6 279.64 .44660 .880 .506 .6 225.68 35350 ^SS .608 7 278.30 .44452 .966 .528 7 224.82 -35184 .240 .630 .8 276.98 44245 18.052 .550 .8 223.96 .35018 325 .653 9 275.67 .44039 .138 572 9 223.11 -34853 .410 .675 21.0 274-37 2.43834 18.224 4-594 26.0 222.27 2.34688 22.495 5-697 .1 273-08 43630 309 .616 .1 221.43 34524 .580 .719 .2 271.81 43427 395 .638 .2 220.60 .34361 .665 .741 3 270.55 .43225 .481 .660 3 219.78 34199 .750 .763 4 269.30 .43024 .567 .682 4 218.96 34037 .835 .785 5 268.06 2.42824 18.652 4.704 5 218.15 2.33875 22.920 5-807 .6 266.84 .42624 .738 .726 .6 217-34 33715 23-005 .829 7 265.62 .42426 .824 -748 7 216.54 33555 .090 .852 .8 264.42 .42229 .910 770 .8 215-75 33395 175 .874 9 263.22 42033 995 .792 9 214.96 .33237 .260 .896 22.0 262.04 2.41837 19.081 4.814 27.0 214.18 2.33078 23-345 5.918 .1 260.87 .41643 .167 .836 .1 213.41 .32921 .429 .940 .2 259-71 .41449 .252 .858 .2 212.64 .32764 514 .962 .3 258.56 .41256 .338 .881 3 211.87 .32608 599 984 4 257-42 .41064 423 903 4 211. II 32452 .684 6.006 .5 256.29 2.40873 19.509 4-925 5 210.36 2.32297 23.769 6.029 .6 255-17 .40683 595 947 .6 209.6l .32142 .853 .051 .7 254-06 .40494 .680 -969 7 208.87 .31988 .938 073 .8 252.96 .40306 .766 .991 .8 208.14 3i835 24.023 095 9 251.87 .40118 .851 5-013 9 207.40 .31682 .108 .117 23.0 250.79 2.39931 19-937 5-035 28.0 206.68 2.31529 24.192 6.139 .1 249.72 39746 20.022 057 .1 205.96 31378 277 .161 .2 248.66 3956i .108 .079 .2 205.24 31227 .362 .184 3 247.61 .39376 193 .101 3 204.53 31076 446 .206 4 246.56 39193 279 123 4 203.83 .30926 531 .228 .5 245-53 2.39010 20.364 5-145 5 203.13 2.30776 24-615 6.250 .6 244.50 38829 .450 .167 .6 202.43 .30627 .700 .272 .7 243.49 .38647 535 .189 7 201.74 30479 .784 294. .8 242.48 38467 .620 .211 .8 201.05 30331 .869 .316 9 241.48 .38288 .706 233 9 200.37 .30184 954 339 24.0 240.49 2.38109 20.791 5.255 29.0 199.70 2.30037 25-038 6.361 .1 239-50 .37931 877 .278 .1 199.02 .29891 .122 .383 .2 238.53 37754 .962 299 .2 198.36 -29745 .207 /405 3 237-56 37578 21.047 321 3 197.70 .29600 .291 427 4 236.60 -37402 .132 343 4 197.04 29455 .376 450 5 235-65 2.37227 21.218 5-366 5 196.38 2.29311 25.460 6.472 .6 234-71 -37053 303 .388 .6 195-74 .29167 545 494 .7 233-77 .36879 .388 .410 .7 195-09 .29024 .629 .516 .8 232.84 .36707 474 .432 .8 194-45 .28881 713 -538 9 231-92 36535 559 -454 9 193-82 28739 .798 -560 25.0 231.01 2.36363 21.644 5.476 30.0 193.19 2.28597 25.882 6.583 (18) TABLE I. (Concluded) Deg. D Radius R .ogarithm LogR Tan. Off. t Mid. Ord. m Deg, D Radius R Logarithm LogR Tan. Off. t Mid. Ord. m 30.0 193-19 2.28597 25.882 6.583 35.0 166.28 2.22083 30.071 7.696 .1 192.56 .28456 .966 .605 .1 165.82 .21963 154 .718 .2 191.94 .28315 26.050 .627 .2 165.36 .21843 237 .740 3 191-32 .28175 135 649 3 164.91 .21724 .320 .763 4 190.70 .28036 .219 .671 4 164.46 .21605 403 785 5 190.09 2.27896 26.303 6.694 5 164.01 2.21486 30.486 7.807 .6 189.49 27757 387 .716 .6 163.56 .21368 570 .830 7 188.88 .27619 .471 .738 7 163.12 .21250 .653 .852 .8 188.28 .27481 556 .760 .8 162.68 .21133 736 -874 9 187.69 27344 .640 .782 9 162.24 .21016 .819 897 31-0 187.10 2.27207 26.724 6.805 36.0 161.80 2.20899 30.902 7.919 .1 186.51 .27071 .808 .827 .1 161.37 .20782 985 942 .2 185-93 26935 .892 .849 .2 160.94 .20666 3i'.o68 964 3 185.35 .26799 976 .871 3 160.51 20550 151 .986 4 184.77 .26664 27.060 .894 4 160.08 .20435 233 8.009 5 184.20 2.26530 27.144 6.916 5 159.66 2.20320 31-316 8.031 .6 183.63 26395 .228 .938 .6 159.24 .20205 399 053 7 183.07 .26262 .312 .960 7 158.82 .20091 .482 .076 .8 182.51 .26128 396 983 .8 158.40 .19977 .565 .098 9 181.95 .25996 .480 7.005 9 157-99 .19863 .648 .121 32.0 181.40 2.25863 27.564 7.027 37-0 157.58 2.19749 3I-730 8.143 .1 180.85 25731 .648 049 .1 I57.I7 .19636 .8x3 .165 .2 180.30 .25600 731 .072 .2 156.76 19523 .896 .188 3 179.76 -25469 .815 094 3 156.35 .19411 979 .210 4 179.22 25338 899 .116 4 155-95 .19299 32.061 233 5 178.68 2.25208 27.983 7-138 5 155-55 2.19187 32.144 8.255 .6 178.15 .25078 28.067 .161 .6 155.15 .19076 .227 277 7 177.62 .24949 .150 .183 7 154-75 .18964 309 .300 .8 177.09 .24820 234 .205 .8 154-36 .18854 392 .322 9 176.57 .24691 .318 .227 9 153-97 18743 474 345 33-0 176.05 2.24563 28.402 7.250 38.0 153.58 2.18633 32.557 8.367 .1 175-53 24435 485 .272 .1 I53-I9 .18523 639 390 .2 175.02 .24308 569' 294 .2 152.80 .18413 .722 .412 3 I74-5I .24181 .652 316 3 152.42 .18304 .804 434 4 174.00 .24054 .736 339 4 152.04 18195 .887 457 5 173-49 2.23928 28.820 7.361 5 151.66 2.18086 32.969 8.479 .6 172.99 .23802 903 383 .6 151.28 .17978 33.051 .502 7 172.49 .23677 987 .406 7 150.90 .17870 134 524 .8 172.00 23552 29.070 .428 .8 150.53 .17762 .216 547 9 171.50 .23428 i54 450 9 150.16 -17655 .298 569 34-0 171.02 2.23303 29.237 7-473 39-o 149.79 2.17547 33.38i 8.592 .1 170-53 .23180 .321 495 .1 149-42 .17441 463 .614 .2 170.04 .23056 .404 .517 .2 149.05 17334 545 636 3 169.56 .22933 .487 539 3 148.69 .17228 .627 659 4 169.09 .22811 571 .562 4 148.33 .17122 .710 .681 5 168.61 2.22688 29.654 7.584 5 147-97 2.17016 33.792 8.704 .6 168.14 .22567 737 .606 .6 147.61 .16911 .874 .726 7 167.67 .22445 .821 .629 7 147.25 .16805 .956 749 .8 167.20 .22324 .904 .651 .8 146.89 .16701 34-038 .771 9 166.74 .22203 987 .673 9 146.54 .16596 .120 794 35-0 166.28 2.22083 30.071 7.696 40.0 146.19 2.16492 34.202 8.816 (19) TABLE II. TANGENT DISTANCES FOR A i CURVE FOR VARYING /'s T D = Tv/D + C of Table III / i 2 3 4 5 6 7* / .00 o.oo 50.00 IOO.OI 150.04 200.08 250.16 300.28 350.44 .00 .02 I.OO 51.00 101.01 151-04 201.08 251.16 301 . 28 351-44 .02 .04 2.OO 52.00 102.01 152.04 202.09 252.17 302.28 352.45 .04 .06 3-00 53-00 103 . 01 153-04 203.09 253 17 303.29 353-45 .06 .08 4.00 54-00 IO4.OI 154.04 204.09 254.17 304.29 354.46 .08 .10 S.oo 55-00 105.01 155-04 205.09 255.17 305.29 355.46 .10 .12 6.00 56.00 106.01 156.04 206.09 256.17 306.30 356.46 .12 .14 7.00 57-00 107.01 157.04 207.09 257-18 307-30 357-47 .14 .16 8.00 58.00 108.01 158.04 208.09 258.18 308.30 358.47 .16 .18 9.00 59-00 109.01 159.04 209.10 259-18 309.30 359-48 .18 20 10. OO 60.00 I 10. 01 160.04 210.10 260.18 310.31 360.48 .20 .22 11.00 61.00 III. 02 161.04 211. IO 261.18 311.31 361.48 .22 .24 12. OO 62.00 112.02 162.05 212.10 262.19 312.31 362.49 .24 .26 13.00 63.00 113.02 163.05 213.10 263.19 313-32 363.49 .26 .28 14.00 64.00 114.02 164.05 214.10 264.19 314.32 364.50 .28 30 15.00 65.00 115.02 165.05 215.10 265.19 315-32 365.50 30 32 16.00 66.00 116.02 166.05 2l6.II 266.19 316.32 366.50 32 34 17.00 67.00 117.02 167.05 217.11 267.20 317-33 367.51 34 .36 18.00 68.00 118.02 168.05 2l8.II 268.20 318.33 368.51 .36 .38 19.00 69.00 119.02 169.05 219.11 269.20 319.33 369.52 .38 .40 20.00 70.00 120. O2 170.05 220.11 270.20 320.34 370.52 .40 .42 21.00 71.00 121. O2 171.05 221. II 271.21 321.34 371.52 42 44 22.OO 72.00 122. O2 172.05 222.11 272.21 322.34 372.53 44 .46 23.0O 73-00 123.02 173-05 223.12 273.21 323.35 373-53 -46 .48 24.00 74-01 124.02 174.06 224.12 274.21 324.35 374-54 48 SO 25.OO 75-01 125.02 I75.o6 225.12 275.21 325.35 375-54 SO .52 26.OO 76.01 126.02 176.06 226.12 276.22 326.36 376.55 52 54 27.00 77-01 127.02 177-06 227.12 277.22 327.36 377-55 54 -56 28.00 78.01 128 . O2 178.06 228.12 278.22 328.36 378.55 .56 .58 29.OO 79-01 129.02 179.06 229.12 279.22 329.37 379.56 .58 .60 30.00 80.01 I3O.02 180.06 230.13 280.23 330.37 380.56 .60 .62 3I.OO 81.01 131 . O2 181.06 231.13 281.23 331-37 381.57 .62 .64 32.00 82.01 132.03 182.06 232.13 282.23 332.38 382.57 -64 .66 33-00 83.01 133.03 183.06 233.13 283.23 333.38 383.58 .66 .68 34.00 84.01 134.03 184.07 234.13 284.24 334-38 384.58 .68 .70 35-00 85.01 135.03 185.07 235.13 285.24 335.39 38S.S9 .70 72 36.00 86.01 136.03 186.07 236.14 286.24 336.39 386.59 72 74 37-00 87.01 137-03 187.07 237.14 287.24 337-39 387.59 74 .76 38.00 88.01 138.03 188.07 238.14 288.25 338.40 388.60 .76 .78 39-00 89.01 139-03 189.07 239.14 289.25 339-40 389.60 78 .80 40.00 90.01 140.03 190.07 240.14 290.25 340.40 390.61 .80 .82 41.00 91.01 141.03 191.07 24LI5 291.25 341-41 39i.6i .82 .84 42.00 92.01 142.03 192.07 242.15 292.26 342.41 392.62 84 .86 43.00 93-01 143.03 193.07 243.15 293.26 343-42 393.62 .86 .88 44.00 94.01 144-03 194.08 244-IS 294.26 344.42 394.63 .88 .90 45-00 95.01 145.03 195-08 245-15 295.26 345.42 395.63 .90 .92 46.00 96.01 146.03 196.08 246.15 296.27 346.43 396.64 92 94 47.00 97-01 147.03 197.08 247.16 297.27 347-43 397.64 94 .96 48.00 98.01 148.03 198.08 248.16 298.27 348.43 398.65 96 98 49.00 99.01 149.04 199-08 249.16 299.28 349-44 399.65 .98 1.00 50.00 100.01 150.04 200.08 250.16 300.28 350-44 400.66 1.00 (20) TABLE II. (Continued) I 8 9 10 11 12 13 14 15 I .00 400.00 450.93 501.28 551-70 602.21 652.81 703.51 754-32 .00 .02 401.66 451-94 502.29 552.71 603.22 653.82 704.53 755-34 .02 .04 402.67 452.95 503.29 553-72 604.23 654.84 705-54 756.36 .04 .06 403.67 453-95 504.30 554-73 605.24 655.85 706.56 757.38 .06 .08 404.68 454-96 505.31 555-74 606.26 656.86 707-57 758.39 .08 .10 405-68 455.96 506.32 556.75 607.27 657-88 708.59 759-41 .10 .12 406.69 456.97 507.33 557.76 608.28 658.89 709.60 760.43 .12 .14 407.69 457-98 508.33 558.77 609.29 659.90 710.62 76i.45 -14 .16 408.70 458.98 509.34 559.78 6lO.3O 660.92 711.63 762.46 .16 .18 409.70 459-99 510.35 560.79 6II.3I 661.93 712.65 763.48 .18 .20 410.71 461.00 5H.36 561.80 612.32 662.94 713.67 764.50 .20 .22 411.71 462.00 512.37 562.81 613.33 663.96 714.68 765-52 .22 .24 412.72 463.01 513.37 563-82 614.35 664.97 715.70 766.53 .24 .26 413-72 464.02 514.38 564-83 615.36 665.98 716.71 767.55 .26 .28 414.73 465.02 515-39 565.84 6l6.37 667.00 717-73 768.57 .28 30 415.73 466.03 516.40 566.85 617.38 668.01 718.74 769.59 .30 32 416.74 467.04 5I7.4I 567.86 618.39 669.02 719-76 770-61 3* 34 417.74 468.04 518.41 568.87 619.40 670.04 720.78 771-62 34 .36 418.75 469.05 519.42 569-88 62O.42 671.05 721-79 772.64 -36 38 419-75 470.06 520.43 570.89 621.43 672.07 722.81 773-66 38 .40 420.76 471-06 521.44 571-90 622.44 673-08 723-82 774-68 .40 42 421.76 472.07 522.45 572.91 623.45 674-09 724.84 775-70 .42 44 422.77 473.08 523-46 573-92 624.46 675-11 725.86 776.72 44 46 423.78 474-08 524.46 574-93 625.47 676.12 726.87 777-73 .46 .48 424.78 475-09 525.47 575-94 626 . 49 677.13 727.89 778-75 48 .50 425.79 476.10 526.48 576.95 627.50 678.15 728.90 779-77 SO 52 426.79 477-10 527.49 577.96 628.51 679.16 729.92 780.79 52 54 427.80 478.ii 528.50 578.97 629.52 680.18 730.94 781.81 54 .56 428.80 479-12 529.51 579-98 630.54 681 . 19 731-95 782.83 56 .58 429.81 480.13 530.51 580.99 631.55 682.21 732.97 783-85 58 .60 430.81 481 . 13 531.52 582.00 632.56 683.22 733-99 784-86 .60 .62 431-82 482.14 532.53 583.01 633.57 684.23 735-00 785-88 .62 .64 432.83 483.15 533-54 584-02 634-58 685.25 736.02 786.90 64 .66 433.83 484.15 534-55 585.03 635-60 686.26 737-03 787.92 .66 .68 434.84 485.16 535.56 586.04 636.61 687.28 738.05 788.94 .68 .70 435.84 486.17 536.57 587-05 637.62 688.29 739-0? 789.96 .70 .72 436.85 487.18 537-57 588.06 638.63 689.31 740-08 790.98 72 74 437-85 488.18 538.58 589.07 639.65 690.32 741 - io 792-00 74 .76 438.86 489.19 539-59 590.08 640.66 691.33 742.12 793-02 .76 .78 439.87 490.20 540.6o 591.09 641.67 692.35 743-13 794-04 78 80 440.87 491.20 54i.6i 592.10 642.68 693.36 744-15 795-05 .80 .82 441-88 492.21 542.62 593.ii 643.70 694.38 745-17 796.07 .82 84 442.88 493-22 543.63 594-12 644.71 695.39 746.19 797.09 84 .86 443.89 494-23 544.64 595-13 645.72 696.41 747-20 798.ii .86 .88 444-90 495.23 545.65 596.14 646.73 697.42 748.22 799-13 .88 90 445-90 496.24 546.66 597.16 647.75 698.44 749-24 800.15 .90 92 446.91 497-25 547-67 598.17 648.76 699.45 750.25 801 . 17 .92 94 447-91 498.26 548.67 599-18 649.77 700.47 751-27 802.19 94 .96 448.92 499-26 549-68 600.19 650.79 701.48 752.29 803.21 .96 98 449-93 500.27 550.69 601 . 20 651.80 702.50 753-31 804.23 .98 1.00 450.93 501.28 551.70 602.21 652.81 703-51 754-32 805.25 1.00 (21) TABLE II. (Continued) I 16 17 18 19 20 21 22 23 7 .00 805.25 856.30 907.49 958.81 1010.29 1061 . 93 1113-73 1165.71 .00 .02 806.27 857.32 908.51 959-84 1011.32 1062.96 HI4-77 1166.75 .02 .o/ 807.29 858.35 909.54 960.87 1012.35 1064.00 III5.8I 1167.79 .04 .06 808.31 859.37 910.56 961.90 1013-39 1065.03 III6.85 1168.83 .06 .08 809.33 860.39 911-59 962.93 1014.42 1066 . 07 1117.88 1169.88 .08 .10 810.35 861.41 912.61 963.96 ioiS-45 1067.10 III8.92 1170.92 .10 .12 8ii.37 862.44 913-64 964.98 1016.48 1068.14 1119.96 1171.96 .12 .It 812.39 863.46 914.66 966.01 1017.51 1069.17 1121.00 1173.00 14 .16 813.41 864.48 915.69 967.04 1018.54 1070.21 1122.04 1174.04 .16 .18 814.43 865.51 916.72 968.07 1019.58 1071.24 1123.07 1175.09 .18 .20 815.45 866.53 917.74 969-10 1020.61 1072.27 1124. II 1176.13 .20 .22 816.47 867.55 9i8.77 970.13 1021.64 1073.31 H25.I5 1177.17 .22 ,2i 817.49 868.57 919.79 971.16 IO22 . 67 1074-35 1126.19 1178.21 .24 .26 818.51 869.60 920.82 972.19 IO23 . 7O 1075.38 1127.23 1179.25 .26 .28 819.53 870.62 921.84 973-21 1024.73 1076.42 1128.27 1180.30 .28 30 820.55 871.64 922.87 974-24 1025.77 1077-45 1129.31 1181.34 30 32 821.57 872.67 923.90 975-27 1026.80 1078.49 1130.35 1182.38 32 34 822.59 873.69 924.92 976.30 1027.83 1079.52 1131.38 1183.43 34 .36 823.61 874.71 925.95 977-33 1028.86 1080.56 1132.42 1184.47 36 .38 824.63 875.74 926.97 978.36 1029.89 1081.59 1133.46 1185.51 .38 .40 825.66 876.76 928.00 979-39 1030.93 1082.63 1134.50 1186.55 .40 .42 t 826. 68 877.78 929-03 980.42 1031.96 1083.66 1135-54 1187.60 42 44 827.70 878.81 930.05 981.45 1032.99 1084.70 1136.58 1188.64 44 .46 828.72 879.83 93i.o8 982.48 1034.02 1085.74 1137.62 1189.68 .46 48 829.74 880.85 932.ii 983.50 1035.06 1086.77 1138.66 1190.73 .48 SO 830.76 881.88 933-13 984.53 1036 . 09 1087.81 1139.70 1191.77 SO 52 831.78 882.90 934.i6 985.56 1037-12 1088.84 1140.74 1192.81 52 54 832.80 883.92 935-19 986.59 I038.I6 1089.88 1141.78 1193.86 54 56 833.82 88-4.95 936.21 987-62 I039.I9 1090.92 1142.82 1194.90 .56 .58 834.84 885.97 937-24 988.65 IO4O.22 1091.95 1143-86 1195.94 .58 .60 835.87 887.00 938.27 989.68 1041.25 1092.99 1144.90 1196.99 .60 .62 836.89 888.02 939-29 990.71 1042.29 1094.03 1145-94 1198.03 .62 .64 837.91 889.04 940.32 991-74 1043.32 1095.06 1146.98 1199.07 .64 .66 838.93 890.07 941-35 992-77 1044-35 1096.10 1148.02 1200.12 .66 .68 839-95 891.09 942.37 993.80 1045-39 1097.14 II49.O6 I2OI.I6 .68 .70 840.97 892.12 943-40 994.83 1046.42 1098.17 II50.IO I2O2.2I .70 .72 841.99 893.14 944-43 995-86 1047-45 1099.21 II5I.I4 I2O3 . 25 72 74 843.02 894.17 945.46 996.89 1048.49 1100.25 II52.I8 1204.29 74 .76 844-04 895.19 946.48 997-92 1049.52 1101.28 1153-22 1205.34 76 .78 845.06 896.21 947-51 998.95 1050.55 1102.32 1154.26 1206.38 .78 .80 846.08 897-24 948.54 999.98 1051.59 1103.36 1155.30 1207-43 .80 .82 847.10 898.26 949-57 1001.01 IO52.62 1104-39 1156.34 1208.47 .82 .84 848.12 899.29 950.59 1002.04 1053.66 1105-43 H57.38 1209.52 .84 .86 849.15 900.31 951-62 1003.08 1054.69 1106.47 1158.42 I2IO.56 .86 .88 850.17 901.34 952.65 1004.11 1055.72 1107.51 H59.46 I2II.6O .88 90 85LI9 902.36 953-68 1005.14 1056.76 1108.54 1160.50 1212.65 .90 92 852.21 903.39 954-70 1006.17 1057-79 1109.58 II6I.55 1213.69 .92 94 853-24 904.41 955-73 1007.20 1058.83 1110.62 II62.59 1214.74 94 .96 854.26 905.44 956.76 1008.23 1059.86 mi. 66 1163.63 1215.78 .96 98 855.28 906.46 957-79 1009.26 1060.89 1112.69 1164.67 I2I6.83 98 1.00 856.30 907.49 958.81 1010.29 1061.93 1113-73 II65.7I 1217.87 .00 (22) TABLE II. (Continued} I 24 25 26 27 28 29 30 3i e I .00 1217.87 1270.23 1322.79 1375-57 1428.56 1481.79 1535.25 1588.97 .00 .02 1218.92 1271.28 1323-85 1376.62 1429.62 1482.86 1536.33 1590.05 .02 04 1219.97 1272.33 1324.90 1377-68 1430.69 1483.92 1537.40 1591-13 .04 .06 I22I.OI 1273.38 1325.95 1378.74 I43I-7S 1484.99 1538.47 1592.20 .06 .08 1222 . 06 1274.43 1327.01 1379.80 1432.81 1486.06 1539-54 1593-28 .08 .10 1223.10 1275.48 1328.06 1380.86 1433.87 1487.12 1540.62 1594.36 .10 .12 1224.14 1276.53 1329.12 1381.91 1434.94 1488.19 1541.69 1595-44 .12 .14 1225.19 1277.58 1330.17 1382.97 1436.00 1489.26 1542.76 1596.51 14 .16 1226.24 1278.63 1331.22 1384-03 1437-06 1490.33 1543.83 1597-59 .16 .18 1227.28 1279.68 1332.28 1385-09 1438.13 1491.39 1544-91 1598.67 .18 .20 1228.33 1280.73 1333.33 1386.15 1439-19 1492.46 1545-98 1599-75 .20 .22 1229.38 1281.78 1334.39 1387-21 1440.25 1493-53 1547.05 1600.82 .22 .24 1230.42 1282.83 1335-44 1388.27 1441.32 i494.6o 1548.12 1601.90 .24 .26 1231-47 1283.88 1336.50 1389.32 1442.38 1495-67 1549-20 1602.98 .26 .28 1232.52 1284.93 1337-55 1390.38 1443-44 1496.73 1550.27 1604.06 .28 30 1233-56 1285.98 1338.60 1391-44 1444.51 i497.8o I55L34 1605.14 30 32 I234.6I 1287.03 1339-66 1392.50 1445-57 1498.87 1552.42 1606.22 32 34 1235.65 1288.08 1340.71 1393.56 1446.63 1499-94 1553-49 1607.30 34 .36 1236.70 1289.13 1341-77 1394-62 1447.70 1501.01 I554-56 1608.37 .36 .38 1237-75 1290.18 1342.82 1395-68 1448.76 1502 . 08 1555.64 1609.45 38 .40 1238.79 1291.23 1343-88 1396.74 1449.82 1503.15 1556.71 1610.53 .40 .42 1239.84 1292.28 1344-93 1397.80 1450.89 1504.21 1557-79 1611.61 .42 -44 1240.89 1293.33 1345-99 1398.86 1451.95 1505.28 1558.86 1612.69 44 -46 1241.93 1294.39 1347-04 1399-92 1453-02 1506.35 1559-93 i6i3.77 .46 .48 1242.98 1295.44 1348.10 1400.98 1454.08 1507-42 1561.01 1614.85 .48 50 1244.03 1296.49 1349.15 1402.04 1455.15 1508.49 1562.08 i6i5.93 50 -52 1245.08 1297.54 1350.21 1403.10 1456.21 1509-56 1563.16 1617.01 52 -54 1246.12 1298.59 1351.26 1404.16 1457.27 1510.63 1564.23 1618.09 54 -56 1247.17 1299.64 1352.32 1405.22 1458.34 1511.70 1565.31 1619.17 .56 -58 1248.22 1300.69 1353-38 1406.28 1459-40 1512.77 1566.38 1620.25 .58 .60 1249.27 1301.74 1354-43 1407.34 1460.47 1513-84 1567.46 1621.33 .60 .62 1250.31 1302.80 1355-49 1408.40 1461.53 1514-91 1568.53 1622.41 .62 -64 1251.36 1303-85 1356.54 1409.46 1462.60 1515.98 1569.60 1623.49 .64 .66 1252.41 1304.90 1357.60 1410.52 1463.66 1517-05 1570.68 1624.57 .66 .68 1253.46 1305.95 1358.66 1411-58 1464.73 1518.12 1571.76 1625 . 65 .68 .70 1254-50 1307.00 1359.71 1412.64 1465.80 1519.19 1572.83 1626.73 .70 -72 1255.55 1308.06 1360.77 1413.70 1466.86 1520.26 I573.9I 1627.81 .72 -74 I256.6O 1309.11 1361.83 1414-76 1467.93 1521.33 1574.98 1628.89 74 -76 1257.65 1310.16 1362.88 1415.82 1468.99 1522.40 1576.06 1629.97 -76 -78 1258.70 1311.21 1363-94 1416.88 1470.06 1523.47 1577-13 1631.05 .78 .80 1259-75 1312.27 1365.00 1417.94 1471-12 1524-54 1578.21 1632 . 14 .80 .82 1260.79 1313.32 1366.05 1418.01 1472.19 1525.61 1579-28 1633-21 .82 -84 I26I.84 1314.37 1367-11 1420.07 1473.26 1526.68 1580.36 1634.30 .84 .86 1262.89 1315.42 1368.17 1421.13 1474-32 1527.76 1581.44 1635.38 .86 .88 1263.94 1316.48 1369.22 1422.19 1475-39 1528.83 1582.51 1636 . 46 .88 .90 1264.99 1317.53 1370.28 1423-25 1476.45 1529.90 1583.59 1637.54 90 92 1266.04 1318.58 1371.34 1424.31 1477-52 1530.97 1584-67 1638.62 92 -94 1267 . O9 1319.63 1372.39 1425.38 1478.59 1532.04 1585.74 1639.70 94 .96 1268.14 1320.69 1373-45 1426.44 1479-66 i533.li 1586.82 1640.79 96 98 1269.18 1321.74 I374-5I 1427.50 1480.72 1534.18 1587.90 1641.87 98 1.00 1270.23 1322.79 1375-57 1428.56 1481.79 1535.25 1588.97 1642.95 1.00 (23) TABLE II. (Continued^ I 32 33 34 35 36 37 38 39 7 .00 1642.95 1097 . 20 1751.73 1806.55 1861.68 I9I7-II 1972.88 2028.98 .00 .02 1644.04 1698.29 1752.82 1807.65 1862.78 1918.23 1974-00 2030 . 10 .02 .04 1645-12 1699.37 I753.92 1808.75 1863.89 1919.34 I975.il 2031.23 .04 .06 1646 . 20 I7O0.46 1755-01 1809.85 1864.99 1920.45 1976.23 2032.35 .06 .08 1647 . 28 1701.55 1756.10 1810.95 1866.10 1921.56 1977-35 2033.48 .08 .10 1648.36 1702.64 1757-20 1812.05 1867.21 1922.68 1978.47 2034.60 .IO .12 1649.45 I703.73 1758.29 1813.15 1868.31 1923.79 1979-59 2035-73 .12 .14 1650.53 1704.82 1759-39 1814.25 1869.42 1924.90 1980.71 2036.86 14 .16 1651.61 1705.91 1760.48 I8I5.35 1870.53 1926.01 1981.83 2037.98 .16 .18 1652 . 70 1706.99 1761.58 1816.45 1871.63 1927.13 1982.95 2039.11 .18 .20 1653.78 1708.08 1762.67 1817-55 1872.74 1928.24 1984-07 2040.24 .20 .22 1654.86 1709.17 1763.77 1818.65 1873.85 1929.35 1985.19 2041.36 .22 .24 1655.95 I7IO.26 1764.86 I8I9.75 1874.95 1930.47 1986.31 2042.49 24 .26 1657.03 1711-35 1765.95 1820.85 1876.06 1931.58 1987.43 2043.62 .26 .28 1658.11 1712.44 1767.05 1821.96 1877.17 1932.69 1988.55 2044.74 .28 30 1659.20 1713.53 1768.14 1823.06 1878.27 I933.8I 1989.67 2045.87 30 32 1660.28 1714.62 1769.24 1824.16 1879.38 1934.92 1990.79 2047 . oo 3* 34 1661.36 1715.71 1770.34 1825 . 26 1880.49 1936.04 I99L9I 2048.13 34 .36 1662.45 I7I6.80 1771-43 1826.36 1881.60 1937.15 1993.03 2049.26 .36 .38 1663.53 1717.89 1772.53 1827.46 1882.71 1938.26 I994-I5 2050.38 .38 .40 1664.62 1718.98 1773.62 1828.56 1883.81 1939.38 1995.28 2051.51 .40 .42 1665.70 I72O.O7 I774.72 1829.67 1884.92 1940.49 1996.40 2052.64 42 44 1666.79 I72I.I6 I775.8I 1830.77 1886.03 1941.61 1997.52 2053.77 44 .46 1667.87 1722.25 1776.91 1831.87 1887.14 1942.72 1998.64 2054.90 .46 .48 1668.96 1723.34 1778.00 1832.97 1888.25 1943.84 1999.76 2056.03 48 So 1670.04 1724.43 1779-10 1834-08 1889.36 1944-95 2000.88 2057.15 50 52 1671.13 1725.52 1780.20 1835-18 1890.46 1946.07 20O2 . OI 2058.38 52 54 1672.21 I726.6I 1781.30 1836.28 1891.57 1947-18 2003.13 2059.41 54 .56 1673.30 1727.70 1782.39 1837.38 1892.68 1948.30 2004.25 2060.54 .56 58 1674-38 1728.79 1783.49 1838.49 1893.79 1949.42 2005.37 2061.67 -58 .60 1675.47 1729.88 1784.59 1839.59 1894.90 1950.53 2006.49 2062.80 .60 .62 1676.55 1730.97 1785.68 1840.69 1896.01 1951.65 2OO7 . 62 2063.93 .62 .64 1677-64 1732.07 1786.78 1841.80 1897.12 1952.76 2008.74 2065.06 .64 .66 1678.72 I733.I6 1787.88 1842.90 1898.23 1953.88 2OO9.86 2066.19 .66 .68 1679.81 1734.25 1788.98 1844-00 1899.34 1955.00 2010.99 2067.32 .68 .70 1680.90 1735-34 1790.07 1845.11 1900.45 1956.12 2OI2.II 2068.45 .70 .72 1681.98 1736.43 I79LI7 1846.21 1901.56 1957.23 2013.23 2069.58 72 74 1683.07 1737.52 1792.27 I847.3I 1902.67 1958.35 2014.36 2070.71 74 .76 1684.15 1738.62 1793-37 1848.42 1903.78 1959-47 2015.48 2071.84 .76 .78 1685.24 1739-71 1794.46 1849.52 1904.89 1960.58 2OI6.60 2072.97 .78 .80 1686.33 1740.80 1795.56 1850.63 1906.00 1961 . 70 2017.73 2074.10 .80 .82 1687.41 1741.89 1796.66 I85I.73 1907.11 1962.82 2018.85 2075.24 .82 .84 1688.50 1742.99 1797.76 1852.84 1908.22 1963.93 2019.98 2076.37 .84 .86 1689.59 1744-08 1798.86 1853.94 1909.33 1965-05 2O2I . 10 2077.50 .86 .88 1690.67 I745-I7 1799.96 1855.04 1910.44 1966.17 2022.22 2078 . 63 .88 .90 1691.76 1746.26 1801.06 1856.15 1911.56 1967.28 2023.35 2079.76 90 92 1692.85 1747.36 1802.15 1857.25 1912.67 1968.40 2024-47 2080.89 -92 94 1693.94 1748.45 1803.25 1858.36 I9I3.78 1969.52 2O25 . 60 2082.03 94 .96 1695.02 1749-54 1804.35 1859.46 1914-89 1970.64 2026.72 2083.16 .96 98 1696.11 1750.64 1805.45 1860.57 1916.00 1971.76 2027.85 2084.29 98 1.00 1697.20 1751-73 1806.55 1861.68 1917-11 1972.88 2028.98 2085.42 1.00 (24) TABLE II. (Continued) I 40 41 42 43 44 45 46 47 / .00 2085.42 2142.23 2199.41 2256.97 2314.93 2373 30 2432.09 249L32 .00 .02 2086.55 2143.37 2200.55 2258.12 2316.09 2374-47 2433.27 2492.51 .02 .04 2087.69 2144.51 2201 . 70 2259.28 2317.26 2375.64 2434-45 2493.70 .04 .06 2088.82 2145.65 2202.85 2260.44 2318.42 2376.81 2435.63 2494.89 .06 .08 2089.95 2146.79 2204 . oo 2261.59 23I9-58 2377-99 2436.71 2496.08 .08 .10 2091.09 2147.93 2205.15 2262.75 2320.75 2379-16 2438 . oo 2497.27 .10 .12 2092.22 2149.07 2206.29 2263.90 2321.91 2380.33 2439.18 2498.46 .12 14 2093.35 2150.21 2207.44 2265.06 2323.08 2381.50 2440.36 2499.65 14 .16 2094.49 2151.35 2208.59 2266 . 22 2324.24 2382.68 2441.54 2500.84 .16 .18 2095 . 62 2152.49 2209.74 2267.37 2325.40 2383.85 2442.72 2502.03 .18 .20 2096.75 2153.63 2210.89 2268.53 2326.57 2385.02 2443.90 2503-22 .20 .22 2097.89 2154-78 22I2.O4 2269.69 2327.73 2386.20 2445.08 2504.41 .22 .24 2099.02 2155.92 2213.19 2270.84 2328.90 2387.37 2446.27 2505.60 .24 .26 2100.16 2157.06 2214.33 2272.OO 2330.06 2388.54 2447.45 2506.80 .26 .28 2101.29 2158.20 2215.48 2273.16 2331.23 2389.72 2448.63 2507.99 .28 30 2102.43 2159.34 2216.63 2274.31 2332.40 2390.89 2449.82 2509.18 30 32 2103.56 2160.48 2217.78 2275-47 2333.56 2392.07 2451.00 2510.37 32 34 2104.70 2161.63 2218.93 2276.63 2334-73 2393.24 2452.18 2511.56 34 .36 2105.83 2162.77 2220.08 2277-79 2335.89 2394-41 2453.36 2512.75 .36 .38 2106.97 2163.91 2221 . 23 2278.95 2337.06 2395-59 2454.55 2513.95 .38 .40 2108.10 2165.05 2222.38 228O.IO 2338.23 2396.76 2455.73 2515.14 .40 .42 2109.24 2166.20 2223.53 2281.26 2339-39 2397.94 2456.91 2516.33 .42 -44 2110.37 2167.34 2224.68 2282.42 2340.56 2399-H 2458.10 2517.53 44 .46 2111.51 2168.48 2225.84 2283.58 2341.73 2400.29 2459.28 2518.72 .46 -48 2112.64 2169.63 2226.99 2284.74 2342.89 2401.47 2460.47 2519.91 .48 50 2113.78 2170.77 2228.14 2285.90 2344.06 2402 . 64 2461.65 2521.11 50 52 2114.92 2171.91 2229.29 2287.06 2345.23 2403.82 2462.84 2522.30 52 54 2116.05 2173.06 2230.44 2288.22 2346 . 40 2404.99 2464 . 02 2523.49 54 .56 2117.19 2174.20 2231.59 2289.38 2347.56 2406.17 2465.21 2524.69 56 .58 2118.32 2175.34 2232.74 2290.54 2348.73 2407.35 2466.39 2525.88 58 GO 2119.46 2176. 49 2233.90 2291 . 70 2349.90 2408 . 52 2467.58 2527.08 .60 .62 2120.60 2177-63 2235.05 2292.86 2351.07 2409.70 2468.76 2528.27 .62 .64 2121.74 2178.78 2236.20 2294 . O2 2352.24 2410.88 2469.95 2529.47 .64 .66 2122.87 2179.92 2237-35 2295.18 2353.41 2412.05 2471.13 2530.66 .66 .68 2124.01 2181.07 2238.51 2296.34 2354.58 2413-23 2472.32 2531-86 .68 .70 2125.15 2182.21 2239.66 2297.50 2355-74 2414.41 2473.51 2533-05 .70 72 2126.29 2183.36 2240.81 2298.66 2356.91 2415.59 2474.69 2534.25 72 74 2127.42 2184.50 2241.97 2299.82 2358.08 2416.76 2475.88 2535-44 74 -76 2128.56 2185.65 2243.12 2300.98 2359.25 2417.94 2477.07 2536.64 .76 .78 2129.70 2186.79 2244.27 2302.14 2360.42 2419.12 2478.25 2537.83 78 .80 2130.84 2187.94 2245-43 2303.30 2361.59 2420.30 2479-44 2539-03 .80 .82 2131.98 2189.09 2246.58 2304.47 2362.76 2421.48 2480.63 2540.23 .82 .84 2133.11 2190.23 2247.73 2305.63 2363.93 2422.66 2481.82 2541.43 .84 .86 2134.25 2191.38 2248.89 2306.79 2365.10 2423.84 2483.00 2542.62 .86 .88 2135.39 2192.52 2250.O4 2307.95 2366.27 2425.01 2484.19 2543.82 .88 90 2136.53 2193-67 2251.20 2309.12 2367.44 2426 . 19 2485.38 2545.02 90 92 2137.67 2194.82 2252.35 2310.28 2368.61 2427.37 2486.57 2546.21 92 94 2138.81 2195.97 2253-50 2311.44 2369.78 2428.55 2487.76 2547-41 -94 96 2139.95 2197.11 2254-66 2312.60 2370.96 2429.73 2488.95 2548.61 -96 .98 2141.09 2198.26 2255.81 23*3.77 2372.13 2430.91 2490.13 2549.81 -98 1.00 2142.23 2I99-4I 2256.97 2314.93 2373.30 2432.09 2491.32 2551-00 1.00 (25) TABLE II. (Continued) I 48 49 50 5i 52 53 54 55 / 00 2551-00 2611.15 2671.78 2732.90 2794.54 2856.70 2919.40 2982.67 .00 .02 2552.20 2612.36 2672.99 2734.13 2795.77 2857 '95 2920.66 2983.94 .02 .04 2553.40 2613.57 2674.21 2735.36 2797.01 2859.20 2921.92 2985.21 .04 .06 25S4.6o 2614.78 2675-43 2736.59 2798.25 2860.44 2923.18 2986.48 .06 .08 2555.8o 2615.98 2676.65 2737.81 2799.49 2861.69 2924.44 2987.75 .08 .10 2557.00 2617.19 2677.86 2739.04 2800.73 2862.94 2925.70 2989.02 .10 .12 2558.20 2618.40 2679.09 2740.27 2801.97 2864.19 2926.96 2990-30 .12 .14 2559.40 2619.61 2680.31 2741.50 2803.21 2865.44 2928.23 2991.57 14 .16 2560.60 2620.82 2681.53 2742-73 2804.45 2866.69 2929.49 2992.84 .16 .18 2561 . 80 2622 . 03 2682.74 2743.96 2805.69 2867.94 2930.75 2994.12 .18 .20 2563 . oo 2623.24 2683.96 2745 19 2806.93 2869.20 2932.01 2995-39 .20 .22 2564 . 20 2624.45 2685.18 2746.42 2808.17 2870.45 2933.27 2996.66 .22 .24 2565.40 2625.66 2686.40 2747.65 2809.41 2871.70 2934-53 2997-94 24 .26 2566.60 2626.87 2687.62 2748.88 2810.65 2872.95 2935.8o 2999.21 .26 .28 2567.80 2628.08 2688.84 2750.11 2811.89 2874.20 2937.06 3000.48 .28 30 2569 . oo 2629.29 2690.06 2751.34 2813.13 2875-45 2938.32 3001.76 30 32 2570.20 2630.50 2691.28 2752.57 2814.37 2876.70 2939-59 3003.03 32 34 2571.40 2631.71 2692.50 2753.80 2815.61 2877.96 2940.85 3004.31 34 36 2572.60 2632.92 2693.73 2755.03 2816.85 2879.21 2942.11 3005.58 .36 .38 2573.80 2634.13 2694.95 2756.26 2818.10 2880.46 2943.38 3006.86 38 .40 2575.01 2635-34 2696.17 2757-49 2819.34 2881.71 2944.64 3008.13 .40 .42 2576.21 2636.56 2697.39 2758.73 2820.58 2882.97 2945.90 3009.41 42 44 2577.41 2637.77 2698.61 2759.96 2821.82 2884.22 2947.17 3010.68 44 .46 2578.61 2638.99 2699.83 2761.19 2823.06 2885.47 2948.43 3011.96 .46 .48 2579.82 2640.19 2701.06 2762.42 2824.31 2886.73 2949.70 3013-24 .48 So 258l . 02 2641.40 2702.28 2763.65 2825.55 2887.98 2950.96 3014-51 50 52 2582.22 2642.62 2703.50 2764.89 2826.79 2889.24 2952.23 3015-79 52 54 2583.43 2643-83 2704.72 2766.12 2828.04 2890.49 2953-49 3017-07 54 .56 2584.63 2645.04 2705.95 2767.35 2829.29 2891.74 2954.76 3018.35 56 .58 2585.83 2646.26 2707.17 2768.59 2830.52 2893.00 2956.03 3019.62 .58 .60 2587.04 2647.47 2708.39 2769.82 2831.77 2894.26 2957.29 3020.90 .60 .62 2588.24 2648.68 2709.62 2771.05 2833.01 2895.51 2958.56 3022.18 .62 .64 2589.44 2649.90 2710.84 2772.29 2834.26 2896.77 2959.83 3023.46 .64 .66 2590.65 2651.11 2712.06 2773.52 2835.50 2898.02 2961.09 3024.74 .66 .68 2591.85 2652.33 2713.29 2774.76 2836.75 2899.28 2962.36 3026.02 .68 .70 2593.06 2653.54 2714-51 2775-99 2837.99 2900.53 2963-63 3027.29 .70 72 2594.26 2654.76 2715.74 2777-23 2839.24 2901.79 2964.90 3028.57 72 74 2595-47 2655.97 2716.96 2778.46 2840.48 2903.05 2966.16 3029.85 -74 .76 2596.67 2657-18 2718.19 2779-70 2841.73 2904.30 2967.43 3031 13 .76 -78 2597.88 2658.40 2719.41 2780.93 2842.98 2905.56 2968.70 3032.41 -78 .80 2599.08 2659.62 2720.64 2782.17 2844.22 2906.82 2969.97 3033.69 .80 .82 26OO.29 2660.83 2721.86 2783.41 2845.47 2908.07 2971.24 3034-97 .82 .84 26OI . 5O 2662.05 2723.09 2784.64 2846.72 2909.33 2972.51 3036.25 .84 .86 26O2.7O 2663.26 2724.32 2785.88 2847.96 2910.59 2973.78 3037.54 .86 .88 2603.91 2664.48 2725.54 2787.11 2849.21 2911.85 2975.05 3038.82 .88 90 2605.12 2665.69 2726.77 2788.35 2850.46 2913-11 2976.31 3040.10 .90 .92 2606.32 2666.91 2727.99 2789-59 2851.71 2914.37 2977.58 3041-38 -92 -94 2607.53 2668.13 2729.22 2790.82 2852.95 2915.63 2978.86 3042.66 94 .96 2608.74 2669.34 2730.45 2792 . 06 2854.20 2916.88 2980.13 3043.94 96 .98 2609.94 2670.56 2731.68 2793.30 2855.45 2918.14 2981.48 3045-23 98 1.00 26II.I5 2671.78 2732.90 2794-54 2856.70 2919-40 2982.67 3046.51 1.00 (26) TABLE II. (Continued) I 56 57 58 59 60 61 62 63 / .00 3046.51 3110.95 3176.00 3241.68 3308.01 3375-02 3442.72 3511-14 .00 .02 3047.79 3112.24 3177.30 3243-00 3309.35 3376.37 3444-08 3512.51 .02 .04 3049.07 3H3-54 3178.61 3244-32 3310.38 3377-72 3445-44 3513-88 .04 .06 3050.36 3H4.83 3179.92 3245.64 3312.02 3379.06 3446.81 3415.26 .06 .08 3051-64 3116.13 3181.23 3246.96 3313.35 338o.4i 3448.17 3516.64 .08 .10 3052.92 3H7.42 3182.54 3248.28 3314.69 3381.76 3449-53 3518.02 .IO .12 3054.21 3118.72 3183.85 3249-61 3316.02 3383-11 3450.89 3519-39 .12 14 3055.49 3120.02 3I85.I5 3250.93 3317.36 3384.46 3452.26 3520.77 14 .16 3056.78 3121.31 3186.46 3252.25 3318.69 3385.81 3453-62 3522.15 .16 .18 3058.06 3122.61 3187.77 3253.57 3320.03 3387.16 3454-98 3523.53 .18 .20 3059.35 3I23-9I 3189.08 3254-89 3321.36 3388.51 3456.35 3524.90 .20 .22 3060.63 3125-20 3190.39 3256.22 3322.70 3389.85 3457-71 3526.28 .22 .24 3061.92 3126.50 3191.70 3257.54 3324.03 3391-21 3459.07 3527.66 24 .26 3063.20 3127-80 3I93.0I 3258.86 3325.37 3392.56 3460.44 3529.04 .26 .28 3064.49 3129.10 3194.32 3260- .19 3326.71 3393-91 3461.80 3530.42 .28 30 3065-78 3130.40 3195.63 3261.51 3328.05 3395.26 3463.17 3531.80 30 32 3067.06 3131.69 3196.95 3262.83 3329.38 3396.61 3464.53 3533.18 32 34 3068.35 3132.99 3198.26 3264.16 3330.72 3397-97 3465.90 3534.56 34 .36 3069.64 3U4.29 3199-57 3265.48 3332.06 3399-31 3467.27 3535-94 -36 .38 3070.92 3135.59 3200.88 3266.81 3333-40 3400.66 3468.63 3537-32 .38 .40 3072.21 3136.89 3202.19 3268.14 3334-74 3402.02 3470.00 3538.70 .40 .42 3073.50 3138.19 3203.51 3269.46 3336.08 3403-37 3471-37 3540.09 42 44 3074.79 3139-49 3204.82 3270.79 3337-41 3404.72 3472.73 3541-47 44 .46 3076.08 3140.79 3206.13 3272.11 3338.75 3406.08 3474-10 3542.85 .46 .48 3077.36 3142.09 3207.44 3273.44 3340.09 3407.43 3475-47 3544-23 -48 50 3078.65 3143.39 3208.76 3274.76 3341-43 3408.78 3476.84 3545.62 So 52 3079-94 3144.69 3210.07 3276.09 3342.77 3410.14 3478.20 3547.00 52 -54 3081.23 3146.00 3211.39 3277-42 3344-11 34H.49 3479-57 3548.38 54 .56 3082.52 3147.30 3212.70 3278.75 3345.46 3412.85 3480.94 3549.76 56 .58 3083.81 3148.60 3214.01 3280.07 3346.80 3414.20 3482.31 3551.15 58 .60 3085.10 3149.90 3215.33 3281.40 3348.14 3415.56 3483.68 3552.53 .60 .62 3086.39 3151.20 3216.64 3282.73 3349-48 34I6.9I 3485.05 3553-92 .62 .64 3087.68 3152.51 3217-96 3284.06 3350.82 3418.27 3486.42 3555-30 .64 .66 3088.97 3I53.8I 3219.27 3285.39 3352.16 3419.62 3487.79 3556.69 .66 .68 3090.26 3i55.li 3220.59 3286.72 3353-51 3420.98 3489.16 3558.07 .68 70 3091.55 3156.42 3221.91 3288.04 3354-85 3422.34 3490.53 3559.46 .70 72 3092.84 3157.72 3223.22 3289.37 3356.19 3423-69 349L90 3560.84 72 74 3094.13 3159-02 3224.54 3290.70 3357-53 3425.05 3493-27 3562.23 74 .76 3095.43 3160.33 3225.86 3292.03 3358.88 3426.41 3494.65 3563.62 76 .78 3096.72 3161.64 3227.17 3293.36 3360.22 3427.77 3496.02 3565.01 .78 .80 3098.01 3162.94 3228.49 3294.69 3361.57 3429.12 3497-39 3566.39 .80 .82 3099-30 3164.24 3229.81 3296.03 3362.91 3430.48 3498.77 3567.78 .82 .84 3100.60 3165.55 3231-13 3297.36 3364.26 3431.84 3500.14 3569.17 .84 .86 3101.89 3166.85 3232.45 3298.69 3365.60 3433-20 3501.51 3570.56 .86 .88 3103.18 3168.16 3233.76 3300.02 3366.94 3434-57 3502.89 3571-95 .88 .90 3104.48 3169.46 3235.08 3301.35 3368.29 3435.92 3504.26 3573-33 90 .92 3105-77 3170.77 3236.40 3302.68 3369.64 3437-28 3505.63 3574-72 92 94 3107 . 06 3172.08 3237.72 3304.02 3370.98 3438.64 3507.01 3576.li 94 .96 3108.36 3173.38 3239-04 3305.35 3372.33 3440.00 3508.38 3577-50 96 -98 3109-65 3174.69 3240.36 3306.68 3373.67 3441.36 3509.76 3578.89 98 1.00 3110.95 3176.00 3241.68 3308.01 3375-02 3442.72 35II-I4 3580.28 1.00 (27) TABLE II. (Continued) I 64 65 66 67 68 69 ?o 71 / .00 3580.28 3650.19 3720.88 3792.37 3864.70 3937-88 4011.94 4086.92 .00 .02 3581.67 3651-60 3722.30 3793-81 3866.15 3939-35 4013.43 4088.43 .02 .04 3583-06 3653.00 3723.72 3795-25 3867.61 3940.82 4014.93 4089.94 .04 .06 3584-46 3654.41 3725.15 3796.69 3869.06 3942.30 4016.42 4091.45 .06 .08 3585.85 3655.82 3726.57 3798.13 3870.52 3943-77 4017-91 4092.96 .08 .10 3587.24 3657.22 3727.99 3799-57 3871-98 3945.25 4019.40 4094-47 .10 .12 3588.63 3658.63 3729.41 3801.01 3873.43 3946.72 4020.89 4095.98 .12 .14 3590.02 3660.04 3730.84 3802.45 3874.89 3948.19 4022.39 4097.49 14 .16 3591-42 3661.45 3732.26 3803.89 3876-35 3949-67 4023.88 4099.00 .16 .18 3592.81 3662 86 3733-69 3805.33 3877-81 3951.15 4025.37 4100.51 .18 .20 3594-20 3664.26 3735.11 3806.77 3879.27 3952.62 4026.86 4102.03 .20 .22 3595.6o 3665.67 3736.54 3808.21 3880.72 3954-10 4028.36 4103.54 .22 .24 3596.99 3667.08 3737.96 3809.65 3882.18 3955-57 4029.85 4105.05 .24 .26 3598.39 3668.49 3739-39 3811.10 3883.67 3957-05 4031.35 4106.57 .26 .28 3599.78 3669.88 3740.81 3812.54 3885-10 3958.53 4032.84 4108.08 .28 30 3601 . 17 3671.31 3742.24 3813.98 3886.56 3960.00 4034-34 4109.59 30 32 3602.57 3672.72 3743.67 3815.42 3888.02 3961.48 4035-84 4111. ii 32 34 3603.96 3674.13 3745-09 3816.87 3889-48 3962.96 4037-33 4112.62 34 36 3605.36 3675.55 3746.52 3818.31 3890.94 3964.44 4038.83 4114.14 .36 .38 3606.76 3676.96 3747-95 3819.76 3892.40 3965.92 4040.33 4115.65 .38 .40 3608.15 3678.37 3749.38 3821.20 3893.87 3967.40 4041.82 4"7.I7 .40 .42 3609.55 3679.78 3750.81 3822.65 3895.33 3968.88 4043-32 4118.69 42 44 3610.95 368l . 20 3752.24 3824.09 3896.79 3970.36 4044.82 4120.20 -44 .46 3612.34 3682.61 3753-66 3825.54 3898.25 3971.84 4046.32 4121.72 .46 -48 3613.74 3684.02 3755.09 3826.98 3899.73 3973-32 4047.82 4123.24 -48 50 3615-14 3685.43 3756.52 3828.43 3901 . 18 3974-So 4049.32 4124.76 So 52 3616.54 3686.85 3757-95 3829.88 3902.64 3976.28 4050.82 4126.28 52 54 3617.94 3688.26 3759.38 3831.32 3904.li 3977.76 4052.32 4127.80 54 .56 3619.34 3689.68 3760.81 3832.77 3905.57 3979-24 4053-82 4129.31 .56 .58 3620.74 3691.09 3762.24 3834-22 3907-04 3980.73 4055.32 4130.83 .58 .60 3622.14 3692.51 3763.68 3835.67 3908.50 3982.21 4056.82 4132.35 .60 .62 3623.54 3693.92 3765." 3837.12 3909.97 3983.69 4058.32 4133.87 .62 .64 3624.93 3695.34 3766.54 3838.56 3911-43 3985.18 4059-82 4135.40 .64 .66 3626.34 3696.76 3767.97 3840.01 3912.90 3986.66 4061.32 4136.92 .66 .68 3627.74 3698.17 3769.41 3841.46 3914.37 3988.15 4062.83 4138.44 .68 .70 3629.14 3699.59 3770.84 3842.91 3915.83 3989.63 4064.33 4139-96 70 .72 3630.54 3701 . oi 3772.27 3844.36 3917.30 3991-12 4065.83 4141.48 .72 .74 3631.94 3702.42 3773-71 3845.81 3918.77 3992.60 4067.34 4143.00 .74 .76 3633.34 3703.84 377S- 14 3847-26 3920.24 3994.09 4068.84 4144-53 .76 -78 3634.74 3705.26 3776.57 3848.71 3921.70 3995-57 4070.34 4146.05 .78 .80 3636.15 3706.68 3778.01 3850.17 3923.17 3997-06 4071.85 4148.57 .80 .82 3637.55 3708.10 3779-44 3851-62 3924.64 3998.55 4073.36 4149.10 .82 .84 3638.95 3709.52 3780.88 3853.07 3926.li 4000.03 4074-86 4150.62 .84 .86 3640.36 3710.93 3782.31 3854.52 3927.58 4001.52 4076.37 4152.15 .86 .88 3641-76 3712.35 3783.75 3855.97 3929.05 4003 . oi 4077-87 4153.67 .88 .90 3643-16 3713.77 3785.19 3857.43 3930.68 4004.50 4079.38 4155.20 .90 .92 3644.57 3315.19 3786.62 3858.88 3931-99 4005.99 4080.89 4156.73 .92 94 3645.97 3716.61 3788.06 3860.33 3933.46 4007.48 4082.39 4158.25 -94 .96 3647.38 3718.04 3789.50 3861.79 3934-94 4008.96 4083.90 4159.78 .96 98 3648.78 3719.46 3790.93 3863.24 3936.41 4010.45 4085.41 4161.31 .98 1.00 3650.19 3720.88 3792.37 3864.70 3937-88 4011.94 4086.92 4162.83 1.00 (28) TABLE II. (Continued) I 72 73 74 75 76 77 78 / .00 4162.83 4239.72 43i7.6o 4396.52 4476.50 4557.56 4639.78 .00 .02 4164.36 4241.27 43I9.I7 4398.11 4478.10 4559-20 4641.44 .02 04 4165-89 4242.81 4320.74 4399-69 4479-72 4560.84 4643.09 .04 .06 4167.42 4244 .,36 4322.31 4401.28 4481.33 4562.47 4644.75 .06 .08 4168.95 4245.9I 4323.88 4402.87 4482.94 4564.10 4646.41 .08 .10 4170.48 4247.46 4325.45 4404.46 4484.55 4565.74 4648.06 .10 .12 4172.01 4249.01 4327.02 4406.06 4486.16 4567.37 4649.72 .12 .14 4173-54 4250.56 4328.59 4407.65 4487.78 4569-01 4651.38 14 .16 4175.07 4252.11 4330.16 4409.24 4489.39 4570.65 4653-04 .16 .18 4176.60 4253.66 4331-73 4410.83 4491-01 4572.28 4654.70 .18 .20 4178.13 4255.21 4333-30 4412.42 4492.62 4573-92 4656.36 .20 .22 4179-66 4256.77 4334.87 4414.02 4494-23 4575.56 4658.02 .22 .24 4181.20 4258.32 4336.45 4415.63 4495-85 4577-19 4659-68 34 .26 4182.73 4259.87 4338.02 4417.21 4497-47 4578.83 4661.34 .26 .28 4184 . 26 4261.42 4339-59 4418.80 4499-09 4580.47 4663.01 .28 30 4185.80 4262.98 4340.97 4420.39 4500.70 4582.11 4664.67 30 33 4187.33 4264.53 4342.74 4421-99 4502.32 4583.75 4666.33 33 34 4188.87 4266.09 4344-32 4423.59 4503.94 4585.39 4668.00 34 -36 4190.40 4267.64 4345.89 4425-18 4505.55 4587.03 4669.66 -36 38 4I9L93 4269.19 4347-47 4426.78 4507.17 4588.67 4671.32 .38 .40 4193.47 4270.85 4349-04 4428.38 4508.79 4590.31 4672.99 .40 43 4195.01 4272.31 4350.62 4429.97 4510.41 4591.96 4674.65 43 44 4196.54 4273-86 4352.19 4431-57 4512.03 4593.6o 4676.32 44 .46 4198.08 4275.42 4353-77 4433-17 4513.65 4595.24 4677.99 .46 .48 4199.62 4276.98 4355-35 4434-77 4515.27 4596.89 4679.65 .48 50 4201.15 4278.53 4356.93 4436.37 4516.89 4598.53 4681.32 So 53 4202 . 69 4280.09 4358.51 4437-97 45I8.5I 4600.18 4682.99 53 54 4204.23 4281.65 4360.08 4439-57 4520.14 4601.82 4684.66 54 .56 4205.77 4283.21 4361.66 444LI7 4521.76 4603.47 4686.32 -56 58 4207.31 4284.77 4363.24 4442.77 4523.38 4605.19 4687.99 .58 .60 4208.85 4286.33 4364.81 4444-37 4525.01 4606.76 4689.67 .60 .62 4210.39 4287.89 4366.40 4445-97 4526.63 4608 . 40 4691.33 .62 -64 42H.93 4289.45 4367.98 4447.58 4528.25 4610.05 4693-00 .64 .66 4213.47 4291.01 4369.56 4449-18 4529.88 4611.70 4694-68 .66 .68 4215.01 4292.57 4371.15 4450.78 4531.50 4613.35 4696.35 .68 .70 4216.55 4294.13 4372.73 4452-39 4533-13 4614.99 4698.02 .70 .73 4218.09 4295.69 4374-31 4453-99 4534.76 4616.64 4699.69 .73 74 4219.63 4297.25 4375-90 4455-59 4536.38 4618.29 4701.37 74 .76 4221.18 4298.82 4377.48 4457-20 4538.01 4619.94 4703.04 -76 78 4222.72 4300.38 4379-06 4458.81 4539.64 4621.59 4704.71 .78 .80 4224.26 4301.94 4380.65 4460.41 4541.27 4623.24 4706.39 .80 .82 4225.81 4303.51 4382.23 4462 . 02 4542.89 4624.90 4708.06 .82 .84 4227.35 4305.07 4383-82 4463.62 4544-52 4626.55 4709.74 .84 .86 4228.90 4306.64 4385.40 4465.23 4546.15 4628.20 47II-4I .86 .88 4230.44 4308.20 4386.99 4466.84 4547.78 4629.85 4713.09 .88 .90 4231.98 4309-77 4388.58 4468.45 4549.41 4631.51 4714-77 .90 93 4233.53 43H.33 4390.16 4470.05 4551.04 4633-16 4716.45 93 94 4235.08 4312.90 4391-75 4471-66 4552.67 4634-81 4718.12 94 .96 4236.62 4314.46 4393-34 4473-27 4554-31 4636.47 4719.80 96 .98 4238.17 4316.03 4394-93 4474.88 4555-94 4638.12 4721.48 -98 1.00 4239.72 4317.60 4396.52 4476.50 4557.56 4639.78 4723-16 1.00 (29) TABLE II. (Continued) I 79 80 81 82 83 84 85 I .00 4723.16 4807.85 4893.58 4980.71 5069.17 5I59.00 5250.26 .00 .02 4724.84 4809.45- 4895-31 4982.47 5070.95 5160.80 5252-10 .02 .04 4726.52 4811.16 4897.04 4984-22 5072.73 5162.62 5253.94 .04 .06 4728.20 4812.86 4898.78 4985.98 5074.52 5164.43 5255.78 .06 .08 4729-88 4814.57 4900.51 4987.74 5076.30 5166.25 5257.62 .08 .10 4731.56 4816.27 4902.24 4989-49 5078.09 5168-06 5259.46 .10 .12 4733-25 4817.98 4903.97 4991-25 5079.87 5169.88 5261.30 .12 .14 4734-93 4819.69 4905.70 4993-01 5081.66 5171-69 5263.15 14 .16 4736.6i 4821.40 4907.44 4994-77 5083.45 5173.50 5264.99 .16 .18 4738.29 4823.10 4909.17 4996-53 5085.23 5175-32 5266.84 .18 .20 4739-98 4824.81 4910.90 4998-29 5087.02 5I77-I4 5268.68 .20 .22 4741.66 4826 . 52 4912.64 5000.05 5088.81 5178-95 5270.53 .22 .24 4743-35 4828.23 4914.37 5001.82 5090.60 5180.77 5272.38 24 .26 4745-03 4829.94 4916.11 5003.58 5092.39 5182.59 5274.22 .26 .28 4746.72 4831-65 4917.83 5005.34 5094.18 5184-41 5276.07 .28 -30 4748.41 4833.36 4919.58 5007.10 5095.97 5186.23 5277.92 30 32 4750.09 4835.08 4921.32 5008.87 5097.76 5188.05 5279.77 32 34 4751.78 4836.79 4923-06 5010.63 5099.55 5189.87 5281.62 34 .36 4753-47 4838.50 4924-80 5012.40 5101.34 5191-69 5283.47 .36 .38 4755-16 4840 . 22 4926.54 5014.16 5103.14 5193.51 5285.32 38 .40 4756.85 4841.93 4928.28 5015.93 5104.93 5195.33 5287.17 .40 .42 4758.54 4843.64 4930.02 5017.70 5106.73 5197.15 5289 . 02 .42 44 4760.23 4845.36 493I-76 5019.46 5108.52 5198.98 5290.87 44 46 4761.92 4847.08 4933-50 5021.23 5110.32 5200.80 5292.73 .46 .48 4763-61 4848.79 4935.24 5023.00 5112.11 5202.62 5294.58 48 50 4765.30 4850.51 4936.98 5024.77 5H3.9I 5204.45 5296.42 50 52 4766.99 4852.22 4938.73 5026.54 5H5.70 5206.27 5298.29 -52 54 4768.69 4853.94 4940.47 5028.31 5H7.50 5208.10 5300.13 -54 56 4770.38 4855.66 4942.21 5030.08 5II9.30 5209.93 5302 . oo -56 -58 4772.07 4857.38 4943.96 5031.85 5121.10 5211.75 5303.86 -58 .60 4773-76 4859.10 4945-70 5033.62 5122.90 5213.58 5305.71 .60 .62 4775-46 4860.82 4947-45 5035.39 5124.70 52I5.4I 5307.57 .62 .64 4777-15 4862.54 4949-19 5037.17 5126.50 5217.24 5309.43 -64 .66 4778.85 4864.26 4950.94 5038.94 5128.30 5219-07 5311.29 .66 .68 4780.55 4865.98 4952.68 5040.71 5130.10 5220.90 5313.15 .68 .70 4782.24 4867.70 4954-43 5042.49 5131.90 5222.73 5315.01 -70 72 4783.94 4869.42 4956.18 5044-26 5133.70 5224.56 5316.87 -72 74 4785.64 4871.14 4957-93 5046.04 5135.51 5226.39 5318.73 -74 .76 4787.33 4872.87 4959-68 5047.81 5I37.3I 5228.22 5320.59 -76 .78 4789.03 4874.59 4961.63 5049.59 5139-12 5230.06 5322.46 -78 .80 4790.73 4876.31 4963-18 5051.37 5140.92 5231.89 5324.32 .80 .82 4792.43 4878.04 4964.93 5053.14 5142-73 5233.72 5326.18 .82 .84 4794-13 4879.76 4966.68 5054.92 5144.53 5235.56 5328.05 -84 .86 4795.83 4881.49 4968.43 5056.70 5146.34 5237-39 5329.91 .86 .88 4797-53 4883.22 4970.18 5058.48 5148.15 5239-23 5331.78 .88 90 4799-23 4884.94 4971.94 5060.26 5149-95 5241.07 5333.65 -90 .92 4800.93 4886.67 4973.69 5062.04 5151.76 5242.90 5335.51 -92 94 4802 . 64 4888.40 4975-44 5063.82 5153-57 5244.74 5337.38 -94 96 4804.34 4890.13 4977.20 5065.60 5155.38 5246.58 5339.25 -96 .98 4806 . 04 4891.85 4978.95 5067-38 5157.19 5248.42 5341-12 -98 1.00 4807.85 4893.58 4980.71 5069.17 5159-00 5250.26 5342.99 1 00 (30) TABLE II. '(Continued) I 86 8 7 88 8 9 90 9i 92 / .00 .02 .04 .06 .08 .10 .12 .14 .16 .18 .20 .22 .24 .26 .28 30 32 34 .36 .38 .40 .42 44 .46 -48 So 52 54 .56 58 .60 .62 .64 .66 .68 -70 .72 74 -76 .78 .80 .82 .84 .86 .88 .90 .92 94 .96 .98 1.00 5342.99 5437-24 5533-06 5630.51 5729.65 5830.53 5933-23 .00 .02 .04 .06 .08 .10 .12 .14 .16 .18 .20 .22 24 .26 .28 30 32 34 .36 .38 .40 42 44 .46 .48 SO -52 54 56 58 .60 .62 .64 .66 .68 70 .72 74 .76 .78 .80 .82 .84 .86 .88 .90 .92 94 96 98 1.00 5344.8s 5346.73 5348.6o 5350.46 5352.34 5354-21 5356.09 5357.96 5359.84 5439-14 544L04 5442.94 5444.84 5446.75 5448.65 5450.55 5452.46 5454-37 5534-99 5536.92 5538.86 5540.79 5542-73 5544.67 5546.6o 5548.54 5550.48 5632.48 5634.44 5636.41 5638.38 5640.35 5642.32 5644.29 5646.26 5648.23 5731.65 5733-65 5735.65 5737-65 5739-66 5741-66 5743.67 5745.67 5747-68 5832.57 5834-61 5836.64 5838.68 5840.72 5842.76 5844.80 5846.84 5848.88 5935-30 5937-37 5939-45 5941.52 5943-60 5945.67 5947-75 5949-83 5951-91 536I.7I 5456.27 5552.42 5650.20 5749.69 5850.93 5953-99 5363.59 5365.46 5367.34 5369.22 5371-10 5372.98 5374-86 5376.74 5378.62 5458.18 5460.09 5462 . oo 5463.91 5465.81 5467.72 5469-64 5471-55 5473.46 5554.36 5556.30 5558.24 5560.18 5562.12 5564.06 5566.01 5567.95 5569.90 5652.18 5654.15 5656.12 5658.10 5660.07 5662.05 5664.03 5666.00 5667.98 5751.69 5753-70 5755-71 5757-72 5759-73 5761.74 5763.75 5765.76 5767.78 5852.97 5855-01 5857-05 5859.10 5861.15 5863.20 5865.24 5867.29 5869.34 5956.07 5958.15 5960.23 5962.31 5964.40 5966.48 5968.56 5970.6s 5972.74 5380.50 5475.37 5571.84 5669.96 5769.79 5871.38 5974-82 5382.38 5384-26 5386.15 5388.03 5389.92 539i.8o 5393.69 5395-57 5397.46 5474.29 5479-20 5481.12 5483.03 5484.95 5486.86 5488.78 5490.70 5492.62 5573-79 5575-73 5577-68 5579.63 5581.58 5583.53 5585.48 5587.43 5589.38 5671.94 5673.92 5675.90 5677-88 5679.87 5681.85 5683.83 5685.82 5687.80 5771.80 5773-82 5775-84 5777.85 5779-87 5781.89 5783.91 5785-93 5787.95 5873.44 5875.49 5877.54 5879-60 5881.65 5883.71 5885.76 5887.82 5889.87 5976.91 5979-00 5981-09 5983.18 5985.27 5987.36 5989.46 5991-55 5993.64 5399-35 5494-54 5591-33 5689.79 5789.97 5891.93 5995-74 5401.24 5403.13 5405.02 5406.91 5408.80 5410.69 5412.58 5414.47 5416.37 5496.46 5498.38 5500.30 55O2 . 22 5504.14 5506.07 5507.99 5509.92 55H.84 5593.29 5595.24 5597-19 5599-15 5601.10 5603.06 5605.01 5606.97 5608.93 5691-78 5693.76 5695.75 5697.74 5699.73 5701-72 5703.71 5705.70 5707.69 5791-99 5794-01 5796.03 5798.06 5800.08 5802.11 5804.13 5806.16 5808.19 5893.98 5896.05 5898.10 5900.16 5902.23 5904-29 5906.35 5908.41 5910.48 5997.83 5999-93 6002.02 6004.12 6006.22 6008.32 6010.42 6012.52 6014.62 5418.26 5513.77 5610.89 5709.68 5810.21 5912-54 6016.72 5420.16 5422.05 5423.95 5425.84 5427.74 5429.64 5431-54 5433-44 5435-34 5515.69 5517.62 5519.55 5521.48 5523.40 5525.33 5527.26 5529.19 5531.13 5612.85 5614.81 5616.77 56i8.73 5620.69 5622.65 5624.62 5626.58 5628.55 5711-68 5713.67 5715.67 5717.66 5719-66 5721.65 5723.65 5725.65 5727.65 5812.24 5814-27 5816.30 5818.33 5820.37 5822.40 5824.43 5826.47 5828.50 59i4.6i 5916.67 5918.74 5920.81 5922.87 5924.94 5927.01 5929-08 5931.15 6018.83 6020.93 6023 . 04 6025.14 6027.25 6029.35 6031.46 6033-57 6035.68 5437-24 5533.06 5630.51 5729.65 5830.53 5933-23 6037.79 (31) TABLE It. (Continued) I 93 94 95 96 97 98 99 7 .00 .02 .04 .06 6037.79 6144.30 6252.82 6363.42 6476.19 6591.21 6708 . 56 .00 .02 .04 .06 6039.90 6042 . 01 6044.12 6146.45 6148.60 6150.75 6255 . 01 6257.20 6259.39 6365.66 6367.89 6370.13 6478.47 6480.75 6483.03 6593.53 6595-86 6598.18 6710.93 6713.30 6715.67 .08 .10 .12 6046.25 6048.35 6050.47 6152.90 6155.06 6157-21 6261.59 6263.78 6265.98 6372.36 6374.60 6376.84 6485.31 6487.59 6489.87 6600.51 6602.84 6605.16 6718.05 6720.42 6722.80 .08 .10 .12 .14 .16 .18 .20 .22 .24 .26 6052.58 6054.70 6056.82 6i59.37 6161.52 6163.68 6268.16 6270.37 6272.57 6379-08 6381.32 6383.56 6492.16 6494.44 6496.73 6607.49 6609.83 6612.16 6725.18 6727.56 6729.93 14 .16 .18 .20 .22 .24 .26 6058.96 6165.84 6274.77 6385.80 6499-01 6614.49 6732.32 6061.05 6063.17 6065.29 6168.00 6170.15 6172.32 6276.97 6279.17 6281.37 6388.04 6390-29 6392.53 6501.30 6503.59 6505.88 6616.82 6619.17 6621.49 6734.70 6737.08 6739.46 .28 30 32 6067.42 6069.55 6071.66 6174-48 6176.64 6178.80 6283.57 6285.78 6287.98 6394.77 6397-02 6399.27 6508.17 6510.46 6512.75 6623.83 6626.16 6628.50 6741-85 6744.23 6746.62 .28 30 32 34 36 38 .40 .42 44 46 6073.78 6075.91 6078.03 6180.96 6183.13 6185.29 6290.18 6292.39 6294.60 6401 . 52 6403.76 6406.01 6515.04 6517.33 6519.63 6630.84 6633.18 6635 . 52 6749.00 6751.39 6753.78 34 .36 -38 .40 42 44 .46 6080.16 ' 6187.46 6296.80 6408.26 6521.92 6637.86 6756.17 6082.28 6084.41 6086.54 6189.63 6191.79 6193.96 6299.01 6301.22 6303.43 6410.52 6412.77 6415.02 6524.22 6526.62 6528.81 6640.21 6642.55 6644.90 6758.56 6760.96 6763.35 48 50 52 6088.67 6090.80 6092.93 6196.13 6198.30 6200.47 6305.64 6307.85 6310.07 6417.28 64I9.53 6421.79 6531-11 6533.41 6535.71 6647.24 6649.59 6651.93 6765.74 6768.14 6770.53 .48 50 52 54 -56 58 .60 .62 64 .66 6095.06 6097.19 6099.32 6202.64 6204.81 6206.99 6312.28 6314.49 6316.71 6424 . 04 6426.30 6428.56 6538.02 6540.32 6542.62 6654.28 6656.63 6658.98 6772.93 6775.33 6777.73 54 56 .58 .60 .62 .64 .66 6101.46 6209.16 6318.92 6430.82 6544.93 6661.33 6780.12 6103.59 6105.73 6107.86 6211.34 6213.51 6215.69 6321.14 6323.36 6325.58 6433-08 6435.34 6437.60 6547.23 6549.54 6551.85 6663.69 6666.04 6668.39 6782.53 6784.93 6787.33 .68 70 .72 6110.00 6112.14 6114.28 6217.87 6220.04 6222.22 6327.80 6330.02 6332.24 6439.86 6442.13 6444.39 6554.15 6556.46 6558.77 6670.75 6673.10 6675.46 6789.73 6792.14 6794.55 .68 .70 72 74 76 -78 .80 .82 .84 .86 6116.42 6118.55 6120.70 6224.40 6226.58 6228.77 6334.46 6336.68 6338.90 6446.66 6448.93 6451-19 6561.08 6563-39 6565.71 6677.82 6680.18 6682.54 6796.95 6799.36 6801.77 74 .76 .78 .80 .82 .84 .86 6122.84 6230.95 634LI3 6453.46 6568.02 6684.90 6804.18 6124.98 6127.12 6129.27 6233.13 6235-31 6237.50 6343.35 6345.58 6347-81 6455.73 6458 . oo 6460.27 6570.34 6572.65 6574.97 6687.26 6689.62 6691.99 6806.59 6809.00 6811.41 .88 .90 92 6131.42 6133.56 6I35-70 6239-69 6241.87 6244.06 6350.04 6352.26 6354.49 6462.54 6464.81 6467.09 6577.29 6579-6o 6581.92 6694.35 6696.72 6699.08 6813.83 6816.24 6818.66 .88 .90 .92 94 96 98 1.00 6137.85 6140.00 6142.15 6246.25 6248.44 6250.63 6356.72 6358.96 6361 . 19 6469.36 6471.64 6473.91 6584.24 6586.56 6588.89 6701.45 6703.82 6706.19 6821.07 6823.49 6825.91 94 96 98 1.00 6144.30 6252.82 6363.42 6476.19 6591-21 6708.56 6828.33 (32) TABLE III. TANGENT DISTANCE CORRECTIONS T D = Tio/D + tabular correction X; 2 4 6 8 10 12 14 16 18 20 D / / I i o.oo 0.00 o.oo O.OI O.OI O.OI O.OI O.OI O.OI O.OI i 2 .00 .00 .01 .01 .01 .02 .02 .02 .02 03 2 4 .00 .01 .01 .02 .03 03 .04 .04 05 .05 4 6 .01 .01 .02 03 .04 OS 05 .06 .07 .08 6 8 .01 .02 .03 .04 .05 .06 07 .08 .09 .10 8 10 O.OI O.02 0.04 0.05 0.06 O.O8 0.09 O.IO O.II 0.13 IO 12 .01 .03 .04 .06 .08 .09 .11 .12 .14 15 12 14 .01 .03 .05 .07 .09 .11 .13 14 .16 .18 14 16 .02 .04 .06 .08 .10 .12 .14 .16 .18 .20 16 18 .02 .04 .07 .09 .11 14 .16 .18 .21 .23 18 20 0.02 0.05 0.07 O.IO 0.13 0.15 0.18 0.21 0.23 0.26 20 22 .02 .05 .08 .11 .14 .17 .20 23 .25 .29 22 24 .02 .06 .09 .12 .15 .18 .22 25 .28 .31 24 26 03 .06 .10 .13 .17 .20 .24 .27 30 34 26 28 .03 .07 .11 .14 .18 .22 25 29 33 36 28 30 0.03 0.07 O.II 0.15 0.19 0.23 0.27 0.31 0.35 0.39 30 32 .03 .08 .12 .16 .21 25 .29 33 38 42 32 34 .03 .08 .13 .18 .22 .27 31 .36 .40 .45 34 36 .04 .09 14 .19 .23 .28 .33 38 43 47 36 38 .04 .09 .15 .20 .25 30 -35 40 .45 50 38 40 0.04 O.IO 0.15 O.2I O.26 0.32 o.37 0.42 0-47 0.53 40 42 .04 .10 .16 .22 .28 33 39 45 50 .56 42 44 .04 .11 .17 .23 29 35 41 47 53 59 44 46 .05 .12 .18 .24 -31 .37 43 49 56 .62 46 48 .05 .12 .19 .26 32 39 .45 .52 59 65 48 So 0.05 0.13 0.20 0.27 0-34 0.40 0.47 0.54 0.61 0.68 50 52 .05 .13 .21 .28 35 .42 50 57 64 71 52 54 .06 .14 .22 .29 .37 44 52 59 67 74 54 56 .06 .15 .23 .30 .38 .46 54 .62 .70 77 56 58 .06 .15 .24 .32 .40 48 56 .64 73 .80 58 60 0.06 0.16 0.24 0.33 0.42 0.50 0.58 0.67 0.75 0.84 60 62 .07 .16 .25 34 43 52 .61 .70 79 .87 62 64 .07 .17 .27 .36 45 54 .64 73 .82 91 64 66 .07 .18 .28 37 47 .56 .66 75 .85 95 66 68 .07 .18 .29 39 .49 59 69 78 .88 98 68 70 0.08 0.19 0.30 0.40 0.50 0.61 0.71 0.81 0.91 .02 70 72 .08 .20 .31 42 52 .63 74 .84 95 .06 72 74 .08 .21 .32 43 54 .65 77 .87 99 .10 74 76 .09 .21 33 45 .56 .68 79 91 .02 14 76 78 .09 .22 34 .46 58 70 .82 -.94 .06 .18 78 80 0.09 0.23 0.36 0.48 0.60 0-73 0.84 0-97 .IO .22 80 82 .09 .24 37 .50 .63 75 .88 I.OI 14 .27 82 84 .10 25 .38 52 .65 .78 91 1.05 .18 31 84 86 .10 25 .40 53 .67 .81 95 1. 08 .22 36 86 88 .11 .26 41 55 70 .84 98 1. 12 .26 41 88 90 O.II 0.27 0.42 0.57 0.72 0.87 1.02 1.16 -31 45 90 92 .11 .28 44 59 75 .90 1.05 1.20 36 50 92 94 .12 .29 .46 .61 77 93 1.09 1.24 40 56 94 96 .12 .30 47 .64 .80 96 1. 13 1.29 45 .62 96 98 .13 .31 49 .66 .83 I.OO 1. 17 1-34 51 .68 98 100 0.13 o.33 3.51 0.68 0.86 1.03 1. 21 1.38 1.56 1-74 100 (33) TABLE IV. EXTERNAL DISTANCES FOR A i CURVE ED = Ei/D (approx.) > I o 10 20 30" 40 50 60 70 80 90 .0 .0 21.9 88.4 202.1 367.7 592.3 886.4 1265 . o 1749.9 2373.3 .a .0 22.8 90.2 04.9 71.6 97-5 93-1 73-5 60.9 87.5 4 .0 23-7 92.0 07-7 75-5 602.7 99-8 82.1 71-9 2401.8 .6 .1 2 4 .6 93-8 10.5 79-4 07-9 906.5 90.8 83.0 16.1 .8 .1 25-5 95-7 13-4 83.4 13-1 13-3 99-5 94-1 30.5 I.O .2 26.5 97-6 216.2 387-4 618.4 920.1 1308.2 1805.3 2444-9 .2 3 27-5 99-5 I9-I 91-4 23-7 27.0 17.0 16.6 59-5 4 .4 28.5 101.4 22.0 95-4 29.0 33-9 25.8 27-9 74.2 .6 .6 29-5 103-3 25.0 99-5 34-4 40.8 34-7 39-3 88.8 .8 .7 30.5 105-3 27-9 403.5 39-8 47-8 43-6 50-7 2503.6 2.0 9 31.6 107.2 230.9 407-6 645.2 954-8 1352.6 1862.2 2518.5 .a i.i 32.6 9-2 33-9 n. 8 50.6 61.8 61.6 73-8 33-4 4 1-3 33-7 II. 2 36.9 15-9 56.1 68.8 70.6 85-4 48.5 .6 1-5 34-8 13-3 39-9 20.1 61.6 75 9 79-7 97-0 63.6 .8 1-7 35-9 15-3 43-0 24-3 67.1 83.1 88.9 1908.7 78.7 30 2.0 37-1 II7.4 246.1 428.5 672.7 990.2 1398.0 1920.5 2594-0 .a 2.2 38.2 19-5 49-2 32-7 78.2 97-4 1407.3 32.4 2609.4 4 2-5 39-4 21.6 52-3 37-0 83.9 1004.7 16.5 44-3 24.8 .6 2.8 40.6 23-7 55-4 41-3 89-5 12.0 25-9 56.3 40-3 .8 3.2 41.8 25.8 58.6 45-6 95-2 19-3 35.3 68.3 55-9 4.0 3.5 43-0 128.0 261.8 450.0 700.9 1026.6 1444.6 1980.4 2671.6 .2 3.9 44-3 30.2 65.0 54-3 06.6 34-0 54-1 92.5 87.4 .4 4.2 45-5 32.4 68.2 58.7 12.4 41-4 63.6 2004.7 2703.3 .6 4.6 46.8 34-6 71-5 63.2 18.2 48.9 73-2 17.0 19-2 .8 S.o 48.1 36.9 74.8 67.6 24.0 56.4 82.8 29 3 35-2 50 5-5 49-4 I39-I 278.1 472.1 . 729.9 1063.9 1492.4 2041.7 2751.3 .2 5-9 50.8 41-4 81.4 76.6 35.7 71-5 1502.1 54.2 67-5 .4 6.4 52.1 43-7 84.7 81.1 41.6 79-1 II. 8 66.7 83.8 .6 6.8 53-5 46.0 88.1 85.6 47.6 86.8 21.7 79-3 2800.1 .8 7-3 54-9 48.4 91-5 90.2 53-6 94-5 31.5 91-9 16.6 6.0 7-9 56.3 150.7 294-9 494-8 759-6 II02.2 I54L4 2104.6 2833.2 .2 8.4 57-7 53-1 98-3 99-4 65.6 09-9 51-3 17-4 49-8 4 8.9 59 2 55-5 301-7 504.1 71-7 17-7 61.3 30.3 66.5 .6 9-5 60.6 57-9 05.2 08.8 77-8 25.6 71-3 43-2 83-4 .8 10. 1 62.1 60.4 08.7 13-5 83.9 33-5 8L 4 56.2 2900.3 7-0 10.7 63.6 162.8 312.2 518.2 790.1 1141.4 I59I-6 2169.2 2917.3 .2 II. 3 65.2 65-3 15.8 22.9 96-3 49-3 1601.8 82.4 34-4 4 12.0 66.7 67-8 19-3 27.7 802.5 57-3 12.0 95-5 51.6 .6 12.6 68.3 70.3 22.9 32.5 08.8 65-4 22.3 2208.8 68.9 .8 13 3 69.8 72.8 26.5 37-4 15-2 73-4 32.6 22.1 86.3 8.0 14.0 71-4 175-4 330.1 542.2 821.4 1181.6 1643.0 2235.5 3003.8 .2 14-7 73-0 78.0 33.8 47-1 27-7 89-7 63-5 48-9 21.4 .4 15-4 74-7 80.6 37-5 52.0 34.1 97-9 64.0 62.5 39-1 .6 16.2 76.3 83.2 41.2 57-0 40.5 1206. I 74-5 ?6.I 56.8 .8 16.9 78.0 85.8 44-9 61.9 47-0 14-4 85.1 8 9 .8 74-7 9.0 17.7 79-7 188.5 348.6 566.9 853.5 1222.7 1695.8 2303.5 3092.7 .2 18.5 81.4 91.2 52-4 72.0 60.0 31. 1 1706.5 17-3 3110.8 4 19.3 83.1 93-9 56.2 77-0 66.5 39-5 17-3 3L2 28.9 .6 20.2 84-8 96.6 60.0 82.1 73-1 48.0 28.1 45- 1 47-2 .8 21. 86.6 99-4 63.9 87.2 79-7 56.4 39-0 59-2 65.6 10.0 21.9 88.4 202. 1 367.7 592-3 886.4 1265.0 1749-9 2373-3 3184.1 / 10 20 30 40 50 60 70 80 90 (34) TABLE V. LONG CHORDS AND ACTUAL ARCS Degree of curve Actual arc, one station 2Sta. 3 Sta. 4 Sta. 5 Sta. 6 Sta. 7 Sta. 8 Sta. Degree of curve 0.2 4 .6 .8 I.O .2 4 .6 .8 2.0 .2 .4 .6 .8 30 .2 4 .6 .8 4.0 .2 4 .6 .8 5-0 .2 4 .6 .8 6.0 .2 .4 .6 .8 7.o .2 '.6 .8 8.0 .2 4 .6 .8 90 .2 4 .6 .8 10.0 IOO.OOO 200. o 300.0 400.0 500.0 600.0 700.0 800.0 0.2 4 .6 .8 I.O .2 4 .6 .8 2.0 .2 4 .6 .8 30 .2 4 .6 .8 4.0 .2 4 .6 .8 S.o .2 4 .6 .8 6.0 .2 4 .6 .8 7.o .2 4 .6 .8 8.0 .2 .6 .8 9.o .2 4 .6 .8 10.0 .000 .000 .001 IOO.OOI .002 .002 .003 .004 oo.o oo.o oo.o 2OO.O oo.o oo.o oo.o oo.o oo.o oo.o oo.o 300.0 oo.o 299.9 99-9 99-9 oo.o oo.o oo.o 399.9 99-9 99-9 99-8 99-8 00.0 499.9 99.9 499-8 99-8 99-7 99-6 99-5 oo.o 599.9 99-8 599-7 99-6 99-5 99-3 99-1 699.9 699-8 99-7 699-6 99-4 99-2 98.9 98.6 799-9 799-8 99-6 799-4 99-1 98.8 98.4 97-9 100.005 2OO.O 299.9 399-7 499-4 598.9 698.3 797-4 .006 .007 .009 .010 IOO.OII .013 .015 .016 .018 oo.o oo.o oo.o 199.9 199.9 99.9 99.9 99 9 99.9 99-9 99-8 99-8 99-8 299-7 99-7 99-6 99-6 99-6 99-6 99-6 99-5 99-4 399-3 99-2 99-1 99-0 98.9 99-3 99-i 99-0 98.8 498.6 98.4 98.2 98.0 97-8 98-7 98-5 98.2 97-9 597-6 97-3 96.9 96.6 96.2 97-9 97-5 97.1 96.7 696.2 95.6 95-1 94.5 93-9 96.9 96.3 95-7 95-0 794-3 93-5 92.6 91.7 90.8 IOO . O20 199.9 299-5 398.8 497-6 595-7 693.2 789.8 .022 .025 .027 .029 100.032 034 .037 .040 043 99-9 99-9 99-8 99-8 199.8 99-8 99-8 99-8 99-7 99-5 99-4 99-4 99-3 299.2 99-2 99-1 99-0 99-0 98.7 98.5 98-4 98.2 398.1 97-9 97.8 97-6 97.4 97-3 97-1 96.8 96.5 496.2 95-9 95.6 95.2 94-9 95-3 94-9 94-4 93-9 593-4 92.8 92.3 91.7 91. i 92-5 91.8 91.0 90.2 689.4 88.5 87.6 86.7 85". 7 88.8 87-7 86.5 85-3 784.1 82.8 8i.5 80. i 78.7 100.046 199-7 298.9 397-3 494-5 590.4 684.7 777-2 .049 .052 055 059 100.062 .066 .070 073 .077 99-7 99-7 99-7 99-6 199-6 99-6 99-6 99-6 99-5 98.8 98.8 98-7 98.6 298.5 98.4 98-3 98.2 98.1 97-1 96.9 96.7 96-5 396-3 96.1 95.8 95.6 95-4 94.2 93.8 93-4 93-0 492.6 92.1 91-7 91.2 90.8 89.8 89.1 88.5 87-7 587.0 86.3 85.5 84.7 83.9 83-7 82.7 81.6 80.4 679.3 78.1 76.9 75.6 74-3 75-6 74-0 72.4 70.7 769.0 67.2 65.4 63.5 61.6 100.081 -085 .090 094 .098 100.103 .108 .112 .117 .122 199-5 298.1 395-1 490.3 583-1 673.0 759-7 99-5 99-5 99-4 99-4 199-4 99-4 99-3 99-3 99-3 98.0 97-9 97-8 97-6 297-5 97-4 97-3 97-2 97.1 94-9 94-6 94-4 94-1 393-9 93-6 93-3 93.o 92.7 89.8 89-3 88.8 88.3 487.7 87.2 86.6 86.1 85-5 82.2 81.4 80.5 79-6 578.6 77-7 76.7 75-7 74-7 71.7 70.3 68.9 67.4 666.0 64.4 62.9 61.3 59-7 57-7 55.6 53-5 51-4 749-2 46.9 44-6 42.3 39-9 IOO.I27 199.2 297.0 392.4 484-9 573-7 658.1 737-5 Degree Actual arc 2Sta. 3 Sta. 4 Sta. 5 Sta. 6 Sta. 7 Sta. 8 Sta. Degree (35) TABLE V. (Continued) Degree of curve Actual arc one station 2 Sta. 3 Sta. 4 Sta. 5 Sta. 6 Sta. 7 Sta. 8 Sta. Degree of curve 10.0 100.127 199.2 297.0 392.4 484.9 573-7 658.1 737-5 10.0 .2 .132 99-2 96.8 92.1 84-3 72.6 56.5 35-1 .2 4 .137 99-2 96-7 91.8 83.7 71.6 54-8 32.5 .4 .6 .143 99-1 96.6 91-5 83.1 70.5 53-0 30.0 .6 .8 .148 99-1 96.5 91.2 82.4 69.4 51-3 27-4 .8 II. 100.154 I99-I 296.3 390.8 481.8 568.2 649.5 724.8 II. O .2 159 99-0 96.2 90.5 81.1 67.1 47-7 22.1 .2 4 .165 99-0 96.1 90.2 80.4 65-9 45-8 19-4 4 .6 .171 99-0 95-9 89.8 79-7 64.8 44-0 16.6 .6 .8 .177 98.9 95.8 89-5 79-0 63-5 42.1 13-8 .8 12.0 100.183 198.9 295-6 389.1 478-3 562.3 640.1 710.9 12.0 .2 .189 98-9 95-5 88.7 77-6 61.1 38.2 08. .2 4 -195 98.8 95-3 88.4 76.9 59-8 36.2 05- 4 .6 .202 98.8 95-2 88.0 76.2 58.5 34-2 702. .6 .8 .208 98.8 95-0 87.6 75-4 57-2 32.1 699. .8 13 o 100.215 198.7 294-9 387.2 474-6 555-9 630.1 696. 13 o .2 .222 98.7 94-7 86.8 73-9 54-6 28.0 93- .2 4 .228 98.6 94-6 86.5 73-1 53-2 25.8 89-9 .4 .6 235 98.6 94-4 86.0 72.3 Si-9 23-7 86.7 .6 .8 .242 98.5 94.2 85.6 71-5 50.5 21.5 83.5 .8 14.0 IOO.249 198.5 294.1 385.2 470.6 549-1 619.3 680.3 14.0 .2 .256 98-5 93-9 84.8 69.8 47-6 17.0 77.0 .2 4 .264 98-4 93-7 84.4 69.0 46.2 14.8 73-7 .4 .6 .271 98.4 93-5 83.9 68.1 44-7 12.5 70-3 .6 .8 .279 98.3 93-4 83.5 67-3 43-2 10.2 66.9 .8 15.0 100.286 198.3 293-2 383.1 466.4 541-7 607.8 663.5 15.0 .2 294 98.2 93-0 82.6 65-5 40.2 05-4 60.0 .2 4 .302 98.2 92.8 82.2 64.6 38.7 O3.0 56-5 .4 .6 .310 98.2 92.6 81.7 63.7 37-1 600.6 53-0 .6 .8 .318 - 98.1 92.4 81.2 62.8 35-6 598.2 49-4 .8 16.0 100.326 198.1 292.3 380.8 461.9 534-0 595-7 645-8 16.0 .2 334 98.0 92.1 80.3 60.9 32-4 93-2 42.2 .2 4 .342 98.0 91.9 79-8 60.0 30.7 90.7 38.5 4 .6 351 97-9 91-7 79-3 59 o 29.1 88.1 34-8 .6 .8 359 97-9 91-5 78.8 58. 27-5 85-5 31-1 .8 17.0 100.368 197.8 291-3 378.3 457- 525-8 582.9 627.3 17.0 .2 .376 97-8 9I-I 77-8 56- 24.1 80.3 23-5 .2 4 .385 97-7 90.8 77-3 55- 22.4 77-7 19.6 4 .6 394 97-6 90.6 76.8 54- 20.7 75-0 15-8 .6 .8 .403 97-6 90.4 76.3 53-0 18.9 72.3 11.9 .8 18.0 100.412 197-5 290.2 375-7 452.0 517.2 569-6 608.0 18.0 .2 .422 97-5 90.0 75-2 51-0 15-4 66.8 04.0 .2 4 431 97-4 89.8 74-7 49-9 13-6 64.1 600.0 4 .6 .440 97-4 89.6 74-1 48.9 II. 8 61.3 596.0 .6 .8 450 97-3 89-3 73-6 47-8 10. 58.5 92.0 .8 19.0 100.460 197-3 289.1 373-0 446.7 508.1 555-6 587.9 19.0 .2 .469 97-2 88.9 72-5 45-6 06.3 52.8 83.8 .2 4 479 97-1 88.6 71-9 44-5 04-4 49-9 79-7 4 .6 .489 97-1 88.4 71-3 43-4 02.5 47-0 75-5 .6 .8 499 97-0 88.2 70.7 42-3 500.6 44.1 71-3 .8 20.0 100.510 197.0 287.9 370.2 441.1 498-7 541. 1 567.1 20.0 Degree Actual arc 2 Sta. 3 Sta. 4 Sta. 5 Sta. 6 Sta. 7 Sta. 8 Sta. Degree (36) TABLE VI. MIDDLE ORDINATES Degree of curve iSta. 2Sta. 3 Sta. 4 Sta. 5 Sta. 6 Sta. 7 Sta. 8 Sta. Degree of curve 0.2 .04 17 39 70 1.09 1-57 2.14 2-79 0.2 4 .09 35 .78 1.40 2.18 3-14 4.28 5.58 4 .6 13 52 1.18 2.09 3-27 4-71 6.41 8.38 .6 .8 I? 70 1.57 2.79 4.36 6.28 8.55 11.17 .8 I.O .22 .87 1.96 3-49 5-45 7-85 10.69 13-96 I.O .2 .26 1.05 2.36 4-19 6.54 9-42 12.82 16.75 .2 4 30 1.22 2.75 4.89 7-63 10.99 14-96 19-53 .4 .6 35 1.40 3-14 5.58 8.72 12.56 17.09 22.32 .6 .8 39 1-57 3-53 6.28 9.81 I4-I3 19.22 25.10 .8 2.0 44 1.75 3 93 6.98 10.90 15-69 21.36 27.88 2.0 .2 .48 1.92 4-32 7.68 11.99 17.26 23-48 30.66 .2 4 52 2.09 4-71 8.37 13.08 18 83 25.61 33-43 4 .6 57 2.27 5-10 9-07 14.17 20.39 27.74 36.21 .6 .8 .61 2.44 5-50 9-77 15.25 21.95 29.86 38.98 .8 30 .65 2.62 5-89 10.46 i6.34 23 52 31.98 41-74 30 .2 .70 2.79 6.28 ii. 16 17-43 25.08 34-10 44-50 .2 -4 74 2.97 6.67 11.86 18.51 26.64 36.22 47.26 .4 .6 .78 3-14 7.06 12.55 19.60 28.20 38.34 50.01 .6 .8 .83 3-32 7.46 13-25 20.68 29-75 40.45 52.76 .8 4.0 .87 3-49 7.85 13-94 21-77 31 31 42.56 55-50 4.0 .2 .92 3-66 8.24 14.64 22.85 32.86 44-66 58.24 .2 4 .96 3-84 8.63 15-33 23.93 34-41 46.77 60.97 4 .6 1. 00 4.01 9.02 16.03 25.01 35.96 48.87 63.69 .6 .8 .05 4-19 9.42 16.72 26.09 37-51 50.96 66.42 .8 S.o .09 4.36 9.81 17.42 27.17 39 06 53-05 69.13 50 .2 .14 4-54 10.20 18.11 28.25 40.60 55-14 71.84 .2 4 .18 4-71 10.59 18.80 29-33 42.15 57-23 74-54 .4 .6 .22 4-89 10.98 19.49 30.40 43.69 59-31 77-23 .6 .8 .27 5.o6 11-37 20.19 31.48 45-22 61.38 79-92 .8 60 31 5 23 11.76 20.88 32.55 46.76 63-46 82.60 6.0 .2 35 5-41 12.15 21.57 33.63 48.29 65-52 85-27 .2 4 .40 5.58 12.54 22.26 34-70 49.82 67.58 87.93 4 .6 .44 5.76 12-93 22.95 35-77 51.35 69.64 90-59 .6 .8 48 5-93 13-32 23.64 36.84 52.88 71.70 93-23 .8 7-0 S3 6. ii 13 71 24 33 37.91 54-40 73-74 95.87 7-0 .2 57 6.28 14.10 25.02 38.97 55.92 75-79 98.50 .2 4 .62 6-45 14-49 25-71 40.04 57-44 77-82 101.12 .4 .6 .66 6.63 14.88 26.39 41.10 58-95 79-85 103-73 .6 .8 70 6.80 15.27 27.08 42.17 60.46 81.88 106.33 .8 8.0 75 6.98 15-66 27-77 43 23 6i.97 83 90 108.92 8.0 .2 79 7-iS 16.05 28.45 44-29 63.47 85.92 111.50 .2 .4 .83- 7-32 16.44 29.14 45-35 64.97 87.92 II4.o6 4 .6 .88 7-50 16.83 29.82 46.40 66.47 89-92 II6.62 .6 .8 .92 7-67 17.22 30.51 47.46 67.97 91.92 119.17 .8 9.0 97 7-85 17.61 3i 19 48.51 69.46 93-91 121.71 9.0 .2 2.00 8.02 18.00 31.87 49-56 70-95 95-89 124.23 .2 .4 2.05 8.19 18.38 32.56 50.61 72.43 97.87 126.75 4 .6 2.10 8.37 18.77 33.24 51-66 73-91 99-83 129.25 .6 .8 2.14 8.54 19.16 33.92 52.71 75-39 101.80 131-74 .8 10 2.18 8.72 19-55 34.6o 53-75 76.86 103-75 134 22 10.0 Degree I Sta. 2 Sta. 3 Sta. 4 Sta. 5 Sta. 6 Sta. 7 Sta. 8 Sta. Degree (37) TABLE VI. (Continued) Degree of curve I Sta. 2 Sta. 3 Sta. 4 Sta. 5 Sta. 6 Sta. 7 Sta. 8 Sta. Degree of curve 10.0 2.18 8.72 19 55 34.6o 53 75 76.86 103-75 134 22 10.0 .2 2.23 8.89 19.94 35.28 54-79 78.33 105 . 70 136.68 .2 -4 2.27 9.06 20.32 35.96 55-83 79-79 107.64 139.14 4 .6 2.31 9-24 20.71 36.63 56.87 81.25 109.57 141.58 .6 .8 2.36 9-41 21.10 37-31 57-91 82.71 in. 49 144.00 .8 II. O 2.40 9 59 21. 4 8 37 99 58.94 84.16 113-41 146.41 II. O .2 2.44 9-76 21.87 38.66 59-98 85.61 115-32. 148.81 .2 4 2.49 9-93 22.25 39-34 61.01 87-05 117.21 151.20 4 .6 2.53 IO.II 22.64 40.01 62.04 88.49 119.11 153.57 .6 .8 2.58 10.28 23.03 40.68 63.06 89-92 120.99 155-93 .8 12.0 2.62 10.45 2341 41 36 64-09 91.36 122.87 158.27 12 .2 2.66 10.63 23.80 42.03 65.11 92.78 124-73 160.59 .2 4 2.71 10.80 24.18 42.70 66.13 94.20 126.59 162.91 4 .6 2.75 10.97 24-57 43-37 67-14 95-62 128.43 165.21 .6 .8 2.80 II. IS 24-95 44-03 68.16 97-03 130.27 167.49 .8 13-0 2.84 11.32 2533 44.70 69.17 98.43 132.10 169.75 13 .0 .2 2.88 11-49 25-72 45-37 70.18 99-83 133.92 172.01 .2 4 2.93 11.67 26.10 46.03 71.19 101.23 135-73 174.24 4 .6 2.97 11.84 26.48 46.70 72.19 102.62 137-53 176.46 .6 .8 3-01 12.01 26.87 47.36 73-20 104.00 139-33 178.67 .8 14.0 3-o6 12.19 27-25 48.02 74.20 105-38 141.11 180.85 14.0 .2 3.io 12.36 27.63 48.69 75-20 106.76 142.88 183.02 .2 4 3-15 12.53 28.01 49-35 76.19 108.12 144.64 185-17 4 .6 3-19 12.71 28.40 50.01 77-18 109.49 146.40 187-31 .6 .8 3-23 12.88 28.78 50.66 78.17 110.85 148.14 189.43 .8 15 o 3 28 13 05 29.16 5i 32 79-16 112. 2O 149 87 191-53 IS o .2 3-32 13-23 29-54 51-98 80.14 113-54 151-59 193-62 .2 4 3.36 13-40 29.92 52.63 81.12 114.88 153.30 195-68 4 .6 3 41 13-57 30.30 53.29 82.10 II6.22 I55.oo 197-73 .6 .8 3-45 13-74 30.68 53-94 83.08 117-55 156.69 199.76 .8 16.0 3 50 13 92 31.06 54-59 84-05 II8.87 158.37 201.77 16.0 .2 3-54 14.09 31-44 55-24 85.02 120.19 160.03 203.77 .2 4 3-58 14.26 31.82 55-89 85-99 121.50 161.69 205.74 .4 .6 3.63 14-44 32.20 56-54 86.95 122.80 163.33 207.70 .6 .8 3.67 14.61 32-57 57-19 87.91 124.10 164.96 209.64 .8 17.0 3 72 14.78 32.95 57-83 88.87 125-39 166.59 211.55 17.0 .2 3-76 14-95 33-33 58.48 89.83 126.68 168.20 213.46 .2 .4 3-8o 15-13 33-71 59-12 90.78 127.96 169.79 215-33 4 .6 3-85 15-30 34.08 59.76 91-73 129.23 171-38 217.20 .6 .8 3-89 15-47 34.46 60.40 92.67 130-49 172.95 219.03 .8 18.0 3 94 15 64 34-84 61.04 9362 I3L75 174-52 220.86 18.0 .2 3-98 15.82 35-21 61.68 94-55 133-01 176.07 222.65 .2 4 4-02 15-99 35-59 62.32 95-49 134-25 177-60 224.43 -4 .6 4.07 16.16 35.96 62.96 96.42 135-49 179-13 226.20 .6 .8 4- II 16.33 36.34 63-59 97-35 136.72 180.64 227.93 .8 19.0 4.16 16.51 36.71 64.22 98.28 137-95 182.15 229.65 19.0 .2 4.20 16.68 37-09 64-85 99-20 I39-I7 183.64 231.35 .2 4 4-24 16.85 37.46 65.48 100.12 140.38 185.11 233.03 -4 .6 4.29 ' 17.02 37.83 66.ii 101.03 141.58 186.57 234.69 .6 .8 4 33 17-19 38.20 66.74 101.95 142.78 188.02 236.33 .8 20 4 37 17 37 38.58 67 37 102.86 143-97 189-46 237.94 20.0 Degree i Sta. 2 Sta. 3 Sta. 4 Sta. 5 Sta. 6 Sta. 7 Sta. 8 Sta. Degree (38) CHAPTER II THE SPIRAL THE railroad spiral is a curve of varying radius connecting a main or central curve with a tangent or connecting the two branches of a compound curve. If it is a true spiral its change in "degree" is pro- portionate to its length. Thus at the point of junction with the tan- gent (T. S. in Fig. 22) the degree is zero; at the junction with the main or central curve of degree D (S. C. in Fig. 22) it is D; at its mid- point it is | Z) ... etc. The radius is closely inversely as the length. Knowing the degree of the central curve the first quantity to determine is the length L, the second is A, the central angle consumed by the spiral. If these two can be conveniently chosen the other functions can be had from tables. The length of the spiral is determined as follows: For curves of 6 or over on which track is canted 8 inches, L = 240 feet as a minimum. _0_ For curves flatter than 6 likely sometime to limit speed, L = -=feet, D being the degree of the central curve. For minor curves not likely to limit speed, L = f SE or 30 E in feet, in which S is speed in miles per hour and E is the difference in elevation of the two rails in inches. If the maximum allowable cant is 6 inches, L = ^-7= feet for curves VD flatter than 4.5 likely to limit speed. For curves of 4.5 and over the minimum length will be 180 feet. For minor curves, L = f SE or 30 E, as above. It will be well to select lengths that are round numbers of feet not less than the required minimums and such that the resulting A may be whole degrees or a whole number of tenths of a degree divisible by 3. This is merely for convenience in computing. 200 39 40 RAILROAD FIELD MANUAL Notation: Referring to Fig. 22 S.T. U ' ' 1 x ! 1 T, FIG. 22. T. S. = Tangent-Spiral. 5. C. = Spiral-Curve. C. S. = Curve-Spiral. S. T. = Spiral-Tangent. T. C. = Tangent-Curve. C. T. = Curve-Tangent. A = central angle of the spiral. A = deflection angle T. S. to S. C. B = deflection angle S. C. to T. S. C = the long chord of the spiral. X = abscissa with T. S. as origin. Y = ordinate with T. S. as origin. U = spiral long tangent. V = spiral short tangent. o = curve offset. Z = distance on tangent from T. S. to T. C. of the offset curve. T t = tangent distance of the spiraled curve. E, = external distance of the spiraled curve. R = radius of central curve. THE SPIRAL 41 The Chord Spiral. If the spiral be laid out by equal chords it is approximately true that the deflection angles from the tangent at the T. S. to the several chord points are as the squares of the chord num- bers, and the final deflection A of the long chord is one-third the central angle A. If the spiral be divided into N chords, the deflection angle for the first chord will closely approximate ^= and the angles that the several chords (produced) will make with the tangent will be approximately A J 7 J 9 37 61, 91, 127, 169, 217, 271, etc., times - The chord spiral is a curve passing through the chord points of a series of equal chords for which the relations of the preceding para- graph are exactly true. It is not a curve of uniformly varying radius or degree, but differs from such a curve by an inappreciable amount. The spiral may be divided into any number of chords. Tables for a division into ten chords have been devised by Mr. Jenks B. Jenkins for the American Railway Engineering Association, and have been approved as good practice by that association. Table VIII gives values for the quantities noted below. Excepting o and Z the linear quantities vary as L and hence the tabular quanti- ties are coefficients by which the L's of any given cases are to be multi- plied to find the functions C, X, F, 7, and V. The use of the table to get o and Z will be evident from the table headings. C X Y Referring to Fig. 22, the tabulated quantities are A, A, T > y-> T . Lt Li Li U V j-i y, coefficients for o and Z. The following formulas give these quantities: A = - , A = \ A 0.000000825 A 3 , B = A A. 2OO T = cos 0.3 A -f- 0.004 exsec f A (approx.). X = CcosA. Y = CsinA. Z = X -tfsinA. sin A sin A o = 7-^versA. 42 RAILROAD FIELD MANUAL Exactly, X = ^ (cos -*- + 7 - + cos ^ + cos &* + cos 10 V 300 300 300 n 300 n 300 9iA I27A . . + cos _ + COS + COS -- + COS - + COS Exactly, F = ( + s!n ^ + . . . etc .). y Exactly, tan A = = ^L The following formulas give the tangent distance and external dis- tance of the curve: E 8 = (R + 0} exsec \I + o. The central 5 subtended by any portion, / feet, of the spiral is the average "degree" of the portion multiplied by - . The average "degree" is the initial and final degree of the portion divided by 2. The degree at any point distant / feet from the T, S. is y>. The jL degree at the end of any chord p (p being the number of the chord) is D for the lo-chord spiral or^-,D for the TV-chord spiral. 10 N For precise computation of positions of points on line, A should be computed from A = I A 0.000000825 A 3 or taken from Table VIII, or Tables IX to XXIII. The deflection 01 for the first chord is always - for the lo-chord spiral. For field use the deflection from the T. S. to any chord point should be taken as 01 times the square of the number of the chord point to be located so long as 5 does not exceed 15. A may be taken as 1/3 A for A^ 15. When A is more than 15 one or more intermediate transit points should be used. Such points should be so chosen that the 5 from the T. S. to the first intermediate point shall not exceed 15, and so that 5 from any occupied point to the next transit point less the 5 from the T. S. to the occupied point shall not exceed 15. With this procedure the deflections from the tangent at any intermediate point may be taken as in Table VII, which gives the coefficients by which a\ is to be multiplied to give deflections to points both forward and back as indicated. This procedure is not exact but results in angular errors less than can be measured by the transit. It is probable that 90 or more per cent of the cases in practice will involve As of less than 10 and an error of 3 seconds or less than 0.001 if A is taken as \ A. THE SPIRAL 43 For A = 15 the error is 10 seconds or less than 0.003. H it is desired to find the deflection from the T. S. to any point to which the 5 exceeds 15 it may be done by finding the 5 and then the corresponding A from Table VIII. If convenient spirals have been chosen, Tables IX to XXIII may be used. To Select and Lay Out the Spiral. Knowing 7 and D, determine L and A; from Table VIII find coefficients for and determine o and Z and substitute in the equation for T 8 . Knowing the P. I. the station of the T. S. can be found. There are now four ways of locating the curve. 1. The curve may be run from the T. S. to the S. C. by deflection angles and chord measurements, using Table VII for multiples of a\ and setting on intermediate points if necessary as advised in the pre- ceding article. To lay out by 5 chords use the deflection coefficients for every second point computing a\ as for 10 chords. 2. The tangent may be continued from the T. S. for Z feet, o laid off and the curve D run in for the full central angle 7, using an offset back sight for the direction of the offset tangent, locating the S. C. at a distance corresponding to A, and which will be nearly L Z, and the C. S. at a distance from the S. C. corresponding to 7 2 A, and off- setting o to the forward tangent and proceeding, locating the spirals later by deflection angles, or by offsets as in 4 below, when staking out for construction. 3. Measure U from the T. S.', establish a transit point, turn the angle A and measure V and establish the S. C. ; run the spiral by de- flections from either end; continue the central curve to the C. S.; lay out V and U to get the S. T. and run in the final spiral by deflections from either end. 4. Many times it will be sufficiently exact after running the central curve as in 2, to bisect the offset o for a point on the spiral and then set over such points as may be desired from tangent and curve respectively, making the offsets from the tangent half proportional to the cube of the distance from T. S. and from the curve half proportional to the cube of the distance from the S. C. Thus the quarter points would be offset 1/8 of \ o or 0/16 from tangent and curve respectively. Time may be saved if a spiral that can be found in one of the Tables IX to XXIII can be chosen. Thus if a i.5-degree curve is to be con- nected and the speed to be considered is 90 miles an hour or less, Table X gives all required quantities for the necessary spiral which is 500 feet long. It may be laid out in ten chords of 50 feet each, for which deflection angles are found under the A column* If pluses are to be 44 RAILROAD FIELD MANUAL located they may be interpolated between the tabular values which are given for each 10 feet on the spiral. But if the speed considered is only 65 miles an hour Table XI 1 1 may be used, the length of the spiral being 200 feet. It may be laid out in 10 chords of 20 feet for which the deflection angles or coordinates are given in the table or it may be laid out by 5 chords of 40 feet. Pluses may be interpolated as indicated above. If the speed is 60 miles an hour, Table XIV may be used and the length of spiral will be found to be 150 feet. It may be laid out in 10 chords of 15 feet interpolating in the table, or better, by 5 chords of 30 feet, with quantities taken directly from the table. Thus each table is good for many different spiral lengths connecting curves of various degrees. Any table may be used in which the degree of the central curve can be found in the column headed D provided the speed for which the table is adapted is that for which the curve is to be spiraled or the track canted. Shorter spirals found in tables for slower maximum speed than that for which the track is canted may be used but are not recommended except where it is necessary materially to save cost of construction or to fit cramped situations in cities or on high embank- ments or in deep cuts when relining old track. Spiraling a Compound Curve. The length of spiral to connect the two branches of a compound curve may be found just as for a tangent and simple curve by substituting the difference of "degrees" (A D 2 ) in the equation for length. It is also prac- tically true that the deflections at the junction points with the two branches to the various chord points of the spiral are the same as for corresponding chord points of the spiral between tangent and simple P IG T 2 - curve of degree (D t D 2 ) if the deflections are considered as turned from equivalent chord points on the respective curves rather than from the tangent. Thus in Fig. 23, the deflection from chord AB to chord AC of the spiral is \ A (A being the central angle for spiral of length L connecting a tangent and (Di D 2 ) curve) if arc AB = L, and from chord CE to chord CA it is the same. These relations are closely approximate. The deflection from the tangent at A is, if D 2 be the degree of the larger radius curve and DI of the shorter, - + f A C, when A = - and C is the correction to be used when necessary or when 200 A is more than 15. Or the curve may be considered as part of a spiral from a tangent to the curve of degree Dt and the deflection coefficients of Table VII may be used. THE SPIRAL 45 Illustrative Example. A 4 curve is to connect with an 8 curve the offset coming at Sta. 464 of the 4 curve. D 2 = 4; D\ = 8; For computing o and Z we use D = A D 2 = 4, and L = 300, whence A = 6, and from the tables for D = 4, L = 300, and A = 6 we find o = 2.62 and Z = 149.92. Therefore C. S. = Sta. 464 149.92 ft. = Sta. 462 + 50.08. If the 300 feet is part of a spiral connecting a tangent and an 8 curve and covers a difference of 4 in o 7" its length the whole spiral would be L 8 = =: - ^- = 2 L = 600 feet JJi L>2 covering a A of 24. Therefore Sta. 462 + 50.08 = C. S. is the fifth chord point of the spiral. Since the central angle consumed by the spiral to be run is 18 and the central angle up to the fifth point of the 6oo-foot spiral is 6, the difference is 12 or less than 15 and it will be proper to use Table VII with the transit at the fifth chord point. Hence the deflections to the several chord points are 16, 34, 54, 76, and 100 times a\. which is - = 0.08 or: 300 Chord point 6 7 8 9 10 Deflection ................. 1.28 2.72 4.32 6.08 8.00 The chords are 60 feet and the deflections from tangent at A to the corresponding points on the 4 curve extended would be: i. 20 2.40 3.60 4.80 6.00 The differences ............. 0.08 0.32 0.72 1.28 2.00 which would be the deflections from corresponding chord points on the DI curve, are the deflections for the chord points of a five-chord spiral of 300 feet connecting a tangent and 4 = (Di DZ) degree curve. Setting up on the S. C., the compounding point with the 8 curve, the deflection from the long chord of the spiral to the tangent is from Table VII, 125 X 01 or 10. It is also - 2 or, stated otherwise, the angle between the long chord of stations of curve DZ and the chord of the spiral is 2, which is the deflection angle of a 3OO-foot spiral connecting a tangent and 4 = (D\ DZ) degree curve. Considering the whole spiraled curve, and referring to Fig. 24, in which RI is the shorter radius and RZ is the longer radius, the com- pound curve is supposed connected first directly to the tangents. To introduce the spirals, the arc of RI must be thrown in along the line of common radius PP\ = o\ sec I\. The arc of RZ must be thrown PPz = oz sec 7 2 . PPi PPz must equal o', the proper offset ^>r the connecting spiral. Therefore determine L* to find 0% and PP*', t*> \ RAILROAD FIELD MANUAL previously found and find a trial value for o\ = - - ; sec /i too small, o z or o' or both may be increased, which will or L' PP 2 add o' if found necessitate increase in or both; if found too large no harm will result unless in relin- ing track the shift is too great. The tangent distances are in- creased by o 2 tan A 2 and 01 tan AI, respectively. To lessen the movement of existing track the following pro- cedure may be had : Conceive the larger radius curve moved out- ward along the common radius line, and the smaller radius curve moved inward, each by \ o', the proper offset for the chosen spiral ; find new radii for the curves such that I o' cos 7 2 + 2 r> / r> J\\ Jvi COS - - vers 7i Both tangent points will be moved toward the vertex by / 2 = (R* - R* + I o'} sin 7 2 /i = (R, - Ri f - % o') sin /L TABLE VII. COEFFICIENTS OF a\ FOR DEFLECTION ANGLES TO CHORD POINTS Deflection Transit at chord-point number angle to chord-point o IO number T. S. i 2 3 4 5 6 7 8 9 S. C. oT. 5. 2 8 18 32 50 72 98 128 162 200 I i O 5 14 27 44 65 90 119 152 189 2 4 4 o 8 20 36 56 80 108 140 176 3 9 10 7 o II 26 45 68 95 126 161 4 16 18 16 IO 14 32 54 80 no 144 5 25 28 27 22 13 17 38 63 92 125 6 36 40 40 36 28 16 20 44 72 104 7 49 54 55 52 45 34 19 o 23 50 81 8 64 70 72 70 64 54 40 22 o 26 56 9 81 88 9i 90 85 76 63 46 25 29 10 S. C. IOO 108 112 112 108 IOO 88 72 52 28 TABLE VIII. GENERAL FUNCTIONS TEN-CHORD SPIRAL A C X Y U V o=mLnD Z=mL-nD A A L L L\ L L m n m 0.0 O.OOO I.OOOOO I.OOOOO .00000 .66667 33333 .00000 .0000 .50000 .0000 0.0 .1 .033 .00000 .00000 .00058 .66667 .33333 .00015 .0000 .50000 .0001 .1 .2 .067 .00000 .00000 .00116 .66667 .33333 .00029 .0000 .50000 .0003 .2 3 .100 .00000 .00000 .00175 .66667 .33333 .00044 .0000 .50000 .0004 3 4 .133 .00000 .00000 .00233 .66667 .33333 .00058 .0000 . 50000 .0005 .4 5 .167 .00000 0.99999 .00291 .66667 .33334 .00073 .0000 .50000 .0006 .5 .6 .200 .00000 99999 00349 .66667 33334. .00087 .0000 .50000 .0008 .6 .7 .233 99999 99999 .00407 .66667 33334 .00102 .0000 .50000 .0009 .7 .8 .267 99999 .99998 .00465 .66667 .33334 .OOII6 .0000 .50000 .0010 .8 9 .300 99999 99998 .00524 .66668 33334 .00131 .0000 .50000 .0011 9 1.0 333 99999 99997 . 00582 .66668 .33334 .00145 .0000 .50000 .0013 1.0 .1 .36? .99998 .99996 .00640 .66668 .33335 .00160 .0000 49999 .0014 .1 .2 .400 99998 .99996 '.00698 .66668 .33335 .00175 .0000 49999 .0015 .2 3 433 99998 99995 .00756 .66668 33335 .00189 .0000 49999 .0017 3 4 .467 99997 99994 .00814 .66669 .33335 .00204 .0000 -49999 .0018 4 5 .500 99997 99993 .00873 .66669 .33336 .00218 .0000 49999 .0019 5 .6 533 99997 99992 .00931 .66669 .33336 .00233 .0000 49999 .0020 .6 7 .567 .99996 99991 .00989 .66670 .33336 .00247 .0000 49999 .0022 .7 .8 .600 .99996 .99990 .01047 .66670 .33337 .00262 .0000 .49998 .0023 .8 9 .633 99995 .99989 .01105 .66671 .33337 .00276 .0000 49998 .0024 9 2.0 .667 99995 .99988 .01163 .66671 33337 . 00291 .0000 .49998 .0025 2.0 .1 .700 99994 99987 .01222 .66671 .33338 .00305 .0000 .49998 .0027 .1 .2 733 99994 99985 .01280 .66672 .33338 .00320 .0001 49998 .0028 .2 3 .767 99993 .99984 .01338 .66672 .33338 .00335 .0001 49997 .0029 3 .4 .800 .99992 .99983 .01396 .66673 33339 .00349 .0001 49997 .0031 4 5 .833 .99992 .99981 .01454 .66673 33339 .00364 .0001 49997 .0032 -5 .6 .867 .99991 .99980 .01512 .66674 33340 .00378 .0001 49997 0033 .6 .7 .900 .99990 99978 .01571 .66674 33340 00393 .0001 49996 .0034 7 .8 .933 .99990 99976 .01629 .66675 33341 .00407 .0001 49996 0036 .8 9 .967 .99989 99975 .01687 .66676 33342 .00422 .0001 .49996 .0037 9 3.0 I.OOO .99988 99973 .01745 .66676 33342 00436 .0001 49996 .0038 3.0 .1 .033 99987 99971 .01803 .66677 33343 .00451 .0001 49995 .0039 .1 .2 .067 .99986 .99969 .01861 .66678 33343 .00465 .0001 49995 .0041 .2 3 .100 .99985 .99967 .01919 .66678 33344 .00480 .0001 49995 .0042 3 .4 .133 .99985 .99965 .01978 .66679 33345 .00494 .0001 49994 0043 4 5 .167 .99984 .99963 .02036 .66680 33345 .00509 .0001 49994 .0045 5 .6 .200 .99983 .99961 .02094 .66681 33346 .00524 .0001 49994 .0046 .6 .7 .233 .99982 99959 .02152 .66681 33347 .00538 .0002 49993 .0047 .7 .8 .267 .99981 99956 .02210 .66682 33347 .00553 .0002 49993 .0048 .8 9 .300 .99980 99954 .02268 .66683 .33348 .00567 .0002 49993 0050 9 4.0 .333 .99979 99952 .02326 .66684 33349 .00582 .0002 49992 .0051 4.0 .1 .367 99978 99949 .02384 .66685 33350 .00596 .0002 49992 .0052 .1 .2 .400 .99976 99947 .02443 .66686 33351 .00611 .0002 49991 0053 .2 3 433 99975 99944 .02501 .66686 33351 .00625 .0002 49991 0055 3 4 .467 99974 99941 .02559 .66687 33352 .00640 .0002 49991 .0056 4 5 .500 99973 99939 .02617 .66688 33353 .00654 .0002 49990 .0057 5 .6 .533 99972 .99936 .02675 .66689 33354 .00669 .0002 49990 .0059 .6 .7 .567 .99970 99933 02733 .66690 33355 .00683 .0002 .49989 .0060 7 .8 .600 .99969 99930 .02791 .66691 33356 .00698 .0003 .49989 .0061 .8 9 .633 .99968 99927 .02849 .66692 33357 .00713 .0003 .49988 .0062 -9 5.0 1.667 .99967 .99924 .02907 .66693 33358 .00727 .0003 .49988 .0064 5.0 (47) TABLE VIII. (Continued^ A ^| C X F U V o=mLnD Z=mL-nD A L L L L L m n m n 5^0 1.667 .99967 -99924 .02907 .66693 .33358 .00727 .0003 .49988 .0064 5.0 .1 .700 .99965 .99921 .02965 .66695 .33359 .00742 .0003 .49987 .0065 .1 .2 -733 .99964 .99918 .03023 .66696 .33360 .00756 .0003 .49987 .0066 .2 3 .767 .99962 99915 .03082 .66697 .33361 .00771 .0003 .49986 .0067 3 .4 .800 .9996i .99912 .03140 .66698 .33362 .00785 .0003 .49986 .0069 4 5 .833 .9996o .99908 .03198 .66699 .33363 .00800 .0003 .49985 .0070 5 .6 .867 .99958 99905 .03256 .66700 .33364 .00814 .0003 .49985 .0071 .6 .7 .900 99957 .99902 .03314 .66702 .33365 .00829 .0004 .49984 .0073 .7 .8 933 99955 .99898 -03372 .66703 .33366 .00843 .0004 .49984 .0074 .8 9 .967 99953 .99895 .03430 .66704 .33367 .00858 .0004 .49983 .0075 -9 6.0 2.OOO 99952 .99891 .03488 .66705 .33369 .00872 .0004 .49982 .0076 6.0 .1 2.033 .99950 -99887 .03546 .66707 33370 .00887 .0004 .49982 .0078 .1 .2 2.067 99949 .99884 .03604 .66708 -33371 .00901 .0004 .49981 .0079 .2 3 2.100 99947 .99880 .03662 .66709 33372 .00916 .0004 .49981 .0080 3 4 2.133 99945 .99876 .03720 .66711 33373 .00930 .0005 .4998o .0081 4 -5 2.167 99944 .99872 03778 .66712 33375 .00945 .0005 49979 .0083 5 .6 2.200 99942 .99868 .03836 .66713 .33376 .00959 .0005 49979 .0084 .6 7 2.233 -99940 .99864 .03894 .66715 33377 .00974 .0005 49978 .0085 .7 .8 2.267 .99938 .99860 03952 .66716 33379 .00989 .0005 .49977 .0086 .8 .9 2.300 .99936 .99856 .04010 .66718 .33380 .01003 .0005 49977 .0088 9 7.0 2-333 99935 .99852 .04068 .66719 .33381 .01018 .0005 49976 .0089 7.0 .1 2.367 .99933 .99847 .04126 .66721 .33383 . 01032 .0006 49975 .0090 .1 .2 2.400 99931 .99843 .04184 .66722 33384 .01047 .0006 49975 .0091 .2 3 2.433 .99929 .99839 .04242 .66724 .33385 .01061 .0006 49974 .0093 3 4 2.467 .99927 .99834 .04300 .66725 .33387 . 01076 .0006 49973 .0094 4 5 2.500 .99925 .99830 .04358 .66727 33388 .01090 .0006 49972 .0095 5 .6 2.533 .99923 .99825 .04416 .66729 33390 .01105 .0006 49972 .0097 .6 -7 2.567 99921 .99821 .04474 .66730 33391 .01119 .0007 49971 .0098 -7 .8 2.600 .99919 .99816 .04532 .66732 33393 .01134 .0007 49970 .0099 .8 .9 2.633 99917 .998n .04590 .66734 .33394 .01148 .0007 .49969 .0100 9 8.0 2.666 .99914 .99806 .04648 -66735 .33396 .01163 .0007 .49969 .0102 8.0 .1 2.699 .99912 .99801 .04706 .66737 .33398 .01177 .0007 .49968 .0103 .1 .2 2.733 99910 .99797 .04764 .66739 33399 .01192 .0007 .49967 .OIO4 .2 3 2.766 .99908 99792 .04822 .66741 33401 .01206 .0008 .49966 .OIO5 3 4 2-799 .99906 .99786 .04880 .66742 33402 .OI22I .0008 .49965 .0107 4 5 2.833 .99903 .99781 .04937 .66744 33404 .01235 .0008 49965 .0108 5 .6 2.866 .99901 .99776 .04995 .66746 .33406 .01250 .0008 .49964 .0109 .6 .7 2.899 .99899 99771 .05053 .66748 33407 .01264 .0008 .49963 .0110 7 .8 2.933 .99897 .99766 .05111 .66750 .33409 .01279 .0009 .49962 .0112 .8 .9 2.966 .99894 .9976o .05169 .66752 334II .01293 .0009 .4996i .0113 9 9.0 2.999 .99892 99755 .05227 .66754 33413 .01308 .0009 .4996o .0114 9.0 .1 3-033 .99889 99749 .05285 .66756 33414 .OI322 .0009 49959 .OIl6 .1 .2 3.066 .99887 .99744 .05343 ,66758 .33416 .01337 .0009 49959 .OII7 .2 3 3.099 .99884 .99738 .05400 .66760 33418 .01351 .0010 49958 .0118 3 4 3-133 .99882 99733 .05458 .66762 .33420 .01366 .0010 49957 .0119 4 5 3-166 .99879 .99727 .05516 .66764 33422 .01381 .0010 49956 .OI20 5 .6 3.199 .99877 99721 05574 .66766 .33424 01395 .0010 49955 .OI22 .6 .7 3.232 .99874 -99715 .05632 .66768 .33425 .01410 .0010 49954 .0123 .7 .8 3.266 .99872 .99709 .05690 .66770 33427 .01424 .0011 49953 .OI24 .8 9 3.299 .99869 .99704 .05747 .66772 .33429 .01439 .OOII .49952 .OI26 9 10.0 3-332 .99866 .99698 . 05805 .66774 -33431 .01453 .0011 49951 .OI27 10.0 (48) TABLE VIII. (Continued) A C X F U V o=mLnD Z=*mLnD A /L L L L L L m n m n 10.0 3.332 .99866 .99698 .05805 .66774 -33431 .01453 .0011 49951 .0127 10.0 .1 3.366 .99864 .99691 .05863 .66776 .33433 .01468 .0011 49950 .0128 .1 .2 3.399 .99861 99685 .05921 .66779 33435 .01482 .0012 49949 .0129 .2 3 3.432 .99858 .99679 .05979 .66781 33437 .01497 .0012 .49948 .0131 3 4 3.466 .99856 .99673 .06036 .66783 33439 .01511 .0012 49947 .0132 4 5 3.499 .99853 .99667 .06094 .66785 33441 .01526 .0012 .49946 0133 5 .6 3-532 .99850 .99660 .06152 .66787 33443 .01540 .0012 49945 0134 .6 .7 3-566 .99847 .99654 .06210 .66790 33446 .01555 .OOI3 49944 .0136 7 .8 3.599 .99844 .99647 .06267 .66792 33448 .01569 .0013 49943 0137 .8 9 3.632 .99841 .99641 .06325 66794 33450 .01584 OOI3 49942 .0138 9 11.0 3.666 .99838 .99634 .06383 .66797 .33452 .01598 .OOI3 49941 .0139 11.0 .1 3-699 .99835 .99627 .06441 .66799 33454 .01612 .0014 49940 .0141 .1 .2 3-732 .99832 .99621 .06498 .66802 33456 .01627 .OOI4 49939 .0142 .2 3 3.765 .99829 .99614 .06556 .66804 33459 .01641 .OOI4 49937 0143 3 4 3-799 .99826 .99607 .06614 .66806 .33461 . 01656 .OOI4 49936 .0144 4 .5 3.832 .99823 .99600 .06671 .66809 .33463 .01670 .0015 49935 .0146 5 .6 3-865 .99820 99593 .06729 .66811 .33465 .01685 .0015 49934 .0147 .6 .7 3-899 .99817 .99586 .06787 .66814 .33468 .01699 .OOI5 49933 .0148 7 .8 3-932 .99814 99579 .06844 .66816 33470 .01714 .0015 49932 .0149 .8 .9 3.965 .99811 99572 .06902 .66819 33472 .01728 .OOl6 49931 .0151 9 12.0 3-999 .99808 .99565 .06960 .66822 33475 01743 .OOl6 49929 .0152 12.0 .1 4-032 .99804 .99557 . 07017 .66824 33477 01757 .OOl6 49928 0153 .1 .2 4.065 .99801 99550 07075 .66827 33479 .01772 .OOl6 49927 0154 .2 3 4.098 99798 99543 .07133 .66830 .33482 .01786 .001? 49926 .0156 3 4 4-132 99795 .99535 .07190 .66832 .33484 .01801 .OOI7 49925 .0157 4 5 4.165 99791 .99528 .07248 .66835 .33487 .01815 .0017 49923 .0158 5 .6 4.198 99788 99520 .07305 .66838 33489 .01830 .O0l8 49922 .0159 .6 .7 4-232 .99785 99513 07363 .66840 33492 . 01844 .ooiJ 49921 .0160 .7 .8 4.265 .99781 .99505 .07421 .66843 33494 .01859 .0018 -49920 .0162 .8 9 4.298 .99778 99497 .07478 .66846 33497 .01873 .0018 49918 0163 9 13.0 4-331 99774 .99489 .07536 .66849 33499 .01888 .0019 .49917 .0164 13.0 .1 4.365 99771 .99481 07593 .66852 33502 .01902 .0019 .49916 .0165 .1 .2 4.398 .99767 99474 .07651 .66854 33504 .01917 .0019 49915 .0167 .2 3 4-431 .99764 .99467 .07708 66857 33507 .01931 .0020 49913 .0168 3 4 4.465 .99760 99457 .07766 .66860 33510 .01946 .0020 49912 .0169 4 5 4.498 99757 99449 .07823 .66863 33512 .01960 .OO20 499II .0170 5 .6 4-531 99753 99441 .07881 .66866 33515 .01974 .0020 49909 .0172 .6 .7 4.564 99749 99433 07938 .66869 33518 .01989 .0021 .49908 0173 7 .8 4.598 .99746 99425 .07996 .66872 33520 .02003 .0021 49907 .0174 .8 9 4-631 99742 99416 .08053 .66875 33523 .02018 .0021 .49905 0175 9 14.0 4.664 99738 .99408 .0811 .66878 33526 .02032 .0022 49904 0177 14.0 .1 4.698 99735 99399 .08168 .66881 33529 .02047 .0022 49903 .0178 .1 .2 4-73 99731 99391 .08226 .66884 33531 .02061 .OO22 49901 .0179 .2 3 4.764 99727 99382 .08283 .66887 33534 .02076 .0023 .49900 .0180 3 4 4-797 99723 99374 .0834 .66890 33537 .02090 .0023 .49898 .0182 4 5 4.83 997IS .99365 .0839 .66893 33540 .02105 .0023 49897 .0183 S .6 4.864 997IS 99356 .0845 .66897 -33543 .02119 .OO2* .49896 .0184 .6 .7 4.89 9971 99347 .0851 .66900 .33546 .02134 .0024 .49894 .0185 -7 .8 4-93 .99708 99339 .0857 .66903 33549 . 0214* .OO24 .49893 .0186 .8 9 4.96 9970 99330 .0862 .66906 3355 .02162 .O025 .49891 .0188 9 15.0 4-99 .9970 -9932 .0868 .66909 3355 .0217 .OO2S .49890 .0189 15 (49) TABLE VIII. (Continued) A 15 .1 .2 3 4 5 .6 7 .8 9 16.0 .1 .2 3 4 '.6 .7 .8 9 17.0 .1 .2 3 4 S .6 .7 .8 9 18.0 .1 .2 3 4 S .6 .7 .8 >9 19.0 .1 .2 3 4 5 .6 .7 .8 9 20.0 A 4-997 C X Y u V o=mLnD 1 Z=mLnD A 15 .1 .2 3 4 5 .6 7 .8 9 16.0 .1 .2 3 4 .5 .6 .7 .8 9 17.0 .1 .2 3 4 5 .6 .7 .8 9 18.0 .1 .2 3 4 5 .6 7 .8 9 19.0 .1 .2 3 4 S .6 7 .8 9 20.0 .99700 33555 m .02177 .OO25 m .49890 n 99321 .08685 .66909 .0189 5-030 5.064 .5.097 5.130 5.163 5-197 5.230 5.263 5.296 .99696 .99692 99687 .99683 .99679 .99675 .99671 .99667 .99662 99312 99302 .99293 .99284 .99275 .99265 .99256 .99246 .99237 .08742 .08799 .08857 .08914 .08971 .09028 .09086 09143 .09200 .66913 .66916 .66919 .66923 .66926 .66929 .66933 .66936 .66940 33558 .33561 33564 33567 33570 33573 .33576 33579 -33582 .02191 . 02206 . O2220 .02235 .02249 . O2264 .02278 . O2292 .02307 .OO25 .O026 .0026 .0026 .C027 .O027 .OO27 .0028 .OO28 .49888 .49887 .49885 .49884 .49882 49881 .49879 .49878 .49876 .0190 .0191 0193 0194 .0195 .0196 .0198 .0199 .0200 5-330 .99658 .99227 .09257 .66943 .33585 .O232I .0028 49875 .0201 5.363 5.396 5.429 5.463 5-496 5.529 5.563 5.596 5.629 .99654 .99650 .99645 .99641 .99637 .99632 .99628 .99623 .996i9 .99218 .99208 .99198 .99188 .99178 .99169 99159 .99148 .99138 .09315 .09372 .09429 . 09486 09543 .09600 .09658 .09715 .09772 -66947 .66950 .66954 .66957 .66961 .66964 .66968 .66972 .66975 33588 .33592 33595 33598 .33601 .33605 .33608 .33611 .33615 .02336 .02350 .02365 .02379 .02393 . 024O8 .O2422 .02437 .02451 .O029 .OO29 .O029 .0030 .OO3O .OO3O .0031 .OO3I .0032 .49873 .49872 .49870 .49868 .49867 .49865 .49863 .49862 .49860 .O202 .O204 .0205 .0206 .0207 .0209 .O2IO .O2II .O2I2 5.662 .99614 .99128 .09829 .66979 .33618 . 02466 .0032 .49859 .0213 5.696 5.729 5.762 5.796 5.829 5.862 5.896 5.929 5.962 .99610 .99605 .99601 99596 99591 .99587 .99582 99577 99572 .99"8 .99108 .99097 .99087 .99076 .99066 .99055 .99044 99034 .09886 .09943 .10000 .10057 .10114 .10171 .10228 .10285 . 10342 .66983 .66986 .66990 .66994 .66998 .67002 .67005 .67009 .67013 .33621 .33625 .33628 .33632 .33635 .33639 .33642 33646 .33649 .02480 .02494 .02509 .02523 .02538 .02552 .02567 .02581 .02595 .0032 .0033 0033 .0033 .0034 .0034 .0035 .0035 0035 .49857 .49855 .49853 .49852 .49850 .49848 .49847 49845 .49843 .O2I5 .0216 .O2I7 .0218 .0220 .O22I .O222 .0223 .0224 5-995 .99568 .99023 .10399 .67017 .33653 .O26lO .0036 49841 .0226 6.028 6.062 6.095 6.128 6.161 6.194 6.228 6.261 6.294 .99563 .99558 99553 .99548 .99543 .99538 .99533 .99528 .99523 .99012 .99001 .98990 98979 .98968 .98957 .98946 .98935 .98923 .10456 .10513 .10570 . 10627 .10684 .10741 .10798 .10855 .10912 .67021 .67025 .67029 .67033 .67037 .67041 .67045 .67049 .67053 .33657 .33660 .33664 .33667 .33671 .33675 .33679 .33682 .33686 . 02624 .02639 .02653 .02667 .02682 .02696 .O27II .02725 .02739 .0036 0037 .0037 .0037 .0038 .0038 .0039 .0039 C039 .49840 .49838 .49836 .49834 .49833 .49831 .49829 .49827 .49825 .O227 .0228 .O229 .0230 .0232 .0233 .0234 0235 .0236 6.328 .99518 .98912 . 10968 .67058 -33690 .02754 .0040 .49823 .0238 6.361 6.394 6.427 6.461 6.494 6.527 6.560 6.594 6.627 .99513 .99508 99503 .99498 99493 .99487 .99482 99477 -99472 .98901 .98889 .98878 .98866 .98854 .98843 .98831 .98819 .98807 .11025 .11082 III39 .11196 .11252 .11309 .11366 .11423 .11479 .67062 .67066 .67070 .67074 .67079 .67083 .67087 .67092 .67096 .33694 .33697 33701 .33705 .33709 .33713 .33717 33721 .33725 .02768 .02783 .02797 .02811 .02826 .02840 .02854 .02869 .02883 .0040 .0041 .0041 .0041 .0042 .0042 .0043 .0043 .0044 .49821 .49820 .49818 .49816 .49814 .49812 .49810 .49808 .49806 0239 .0240 .0241 .0242 .0244 .0245 .0246 .0247 .0248 6.660 .99466 .98795 .11536 .67100 .33729 .02898 .0044 .49804 .0250 (50) TABLE VIII. (Continued) A 20 .1 .2 3 4 5 .6 7 .8 9 21.0 .1 .2 3 4 5 .6 7 .8 9 22.0 .1 .2 3 4 5 .6 .7 .8 -9 23.0 .1 .2 3 4 5 .6 7 .8 -9 24.0 .1 .2 3 4 5 .6 7 .8 9 25.0 A 6.660 C X F U V o=mLnD Z=mLnD A 20! o .1 .2 3 4 5 .6 7 .8 -9 21.0 .1 .2 3 4 5 .6 7 .8 9 22.0 .1 .2 3 4 -5 .6 -7 .8 9 23.0 .1 .2 3 -4 -5 .6 7 .8 9 24.0 .1 .2 3 4 5 .6 -7 .8 9 25.0 L .99466 L L .11536 L .67100 L m n .0044 m .49804 n .0250 .98795 .33729 .02898 6.693 6.727 6.759 6.793 6.826 6.859 6.892 6.926 6-959 .99461 -99456 99450 99445 99439 99434 .99428 99423 .99417 .98783 .98771 .98759 .98747 .98734 .98722 .98710 .98697 .98685 .11593 .11649 . 11706 11763 .11819 .11876 .H932 .11989 . 12046 .67105 .67109 .67114 .67118 .67123 .67127 .67132 .67136 .67141 33733 33737 33741 33745 33749 33753 33757 .33762 .33766 .02912 .02926 .02941 .02955 .02969 .02984 .02998 .03013 .03027 .0044 .0045 .0045 .0046 .0046 .0047 .0047 .0048 .0048 .49802 .49800 .49798 .49796 49794 .49792 49790 .49788 49786 .0251 .0252 0253 .0254 .0256 .0257 .0258 0259 .0260 6.992 99412 .98672 . I2IO2 .67145 33770 .03041 .0048 .49784 .0262 7.026 7.059 7.092 7-125 7-158 7.192 7.225 7-258 7.291 .99406 .99400 99395 .99389 .99383 99378 99372 .99366 .9936o .98660 .98647 .98634 .98622 .98609 98596 -98583 .98570 .98557 .98544 I2I59 .12215 . 12272 .12328 .12385 .12441 .12497 .12554 .12610 .67150 .67155 67159 .67164 .67169 67173 .67178 .67183 .67188 33774 .33778 .33783 .33787 33791 .33796 .33800 .33804 .33809 .03056 .03070 .03084 .03099 .03113 .03127 .03142 .03156 .03170 .0049 .0049 .0050 .0050 .0051 .0051 .0052 .0052 .0053 .49782 .49780 49778 .49776 49774 .49772 49770 .49768 .49765 .0263 .0264 .0265 .0266 .0267 .0269 .0270 .0271 .0272 7-324 -99354 .12667 .67193 .33813 .03185 0053 .49763 0273 7-358 7-391 7.424 7-458 7-491 7.524 7-557 7-590 7.623 99349 99343 99337 99331 .99325 99319 99313 .99307 -99301 .98531 .98517 .98504 .98491 .98477 .98464 .98450 .98437 -98423 .12723 .12779 .12836 .12892 .12948 .13005 . 13061 .13117 .13173 .67197 .67202 .67207 .67212 .67217 .67222 .67227 .67232 67237 .33818 .33822 33827 .33831 .33836 .33840 .33845 .33849 .33854 03199 03213 .03228 .03242 .03256 .03271 .03285 .03299 .03314 .0054 .0054 0055 0055 .0056 .0056 .0057 .0057 .0058 .49761 49759 49757 .49755 49752 49750 .49748 .49746 49744 .0275 .0276 .0277 .0278 .0279 .0281 .0282 .0283 .0284 7.657 .99295 .98409 .13230 .67242 .33859 .03328 .0058 49741 .0285 7.690 7-723 7.756 7.789 7.822 7.856 7.889 7.922 7-955 .99288 .99282 .99276 .99270 .99264 99257 .99251 .99245 .99238 .98396 .98382 .98368 .98354 .98340 .98326 .98312 .98298 .98283 . 13286 .13342 .13398 .13454 .13510 .13567 .13623 .13679 .13735 .67247 .67252 .67258 .67263 .67268 67273 .67278 .67284 .67289 .33863 .33868 .33873 .33877 .33882 .33887 .33892 .33896 33901 .03342 .03357 .03371 .03385 .03400 .03414 .03428 .03443 .03457 0059 .0059 .0060 .0060 .0061 .0061 .0062 .0062 .0063 49739 49737 49735 49732 .49730 .49728 .49725 .49723 49721 .0286 .0288 .0289 .0290 .0291 .0292 .0293 0295 .0296 7.989 .99232 .98269 .13791 .67294 .33906 -03471 .0063 .49718 .0297 8.022 8.055 8.088 8. 121 8.154 8.188 8.221 8.254 8.287 .99226 99219 .99213 .99206 .99200 99193 .99187 .99180 99174 .98255 .98240 .98226 .98211 .98197 .98182 .98167 .98153 .98138 .13847 .13903 .13959 .14015 .14071 14127 .14183 .14239 .14295 .67300 .67305 .67310 .67316 .67321 .67327 .67332 .67338 .67343 339II .33916 33921 .33926 33931 33936 33941 .33946 33951 .03485 .03500 .03514 .03528 .03543 03557 .03571 .03585 .03600 .0064 .0064 .0065 .0065 .0066 .0066 .0067 .0067 .0068 .49716 49714 497II .49709 .49707 49704 49702 .49699 .49697 .0298 .0299 .0300 .0302 .0303 .0304 .0305 .0306 .0307 8.321 .99167 .98123 .14350 .67349 .33956 .0361* .0068 .49695 .0309 (51) TABLE VIII. (Continued) A 250 .1 .2 3 4 5 .6 7 .8 .9 26.0 .1 .2 3 4 '.6 .7 .8 9 27.0 .1 .2 3 4 5 .6 7 .8 9 28.0 .1 .2 .3 .4 5 .6 .7 .8 9 29.0 .1 .2 -3 4 5 .6 .7 .8 9 30.0 A 8.321 C L .99167 X Y U V o=mLnD Z=mL-nD A L L L L .33956 m .03614 n m n .0309 .98123 14350 .67349 .0068 .49695 25. .1 .2 3 .4 '.6 7 .8 9 26.0 .1 .2 3 4 5 .6 7 .8 9 27.0 .1 .2 3 4 5 .6 7 .8 9 28.0 .1 .2 3 4 5 .6 7 .8 9 29.0 .1 .2 3 4 5 .6 7 .8 9 30.0 8.354 8.387 8.420 8.453 8.486 8.519 8.553 8.586 8.619 .99160 .99154 .99147 .99140 ..99133 .99127 .99120 .99113 .99106 .98108 .98093 .98078 .98063 .98048 .98033 .98017 .98002 .97987 97971 .14406 . 14462 14518 14574 .14629 .14685 I474I 14797 .14852 .67354 .67360 .67365 .67371 .67377 .67382 .67388 .67394 .67400 33961 33966 -33971 .33976 .33982 .33987 .33992 -33997 .34002 .03628 .03643 .03657 .03671 .03685 .03700 .03714 .03728 .03742 .0069 .0069 .0070 .0071 .0071 .0072 .0072 .0073 .0073 .49692 .49690 49687 .49685 .49682 .49680 .49677 .49675 .49672 .0310 .0311 .0312 .0313 .0314 .0315 .0317 .0318 .0319 8.652 .90099 .14908 .67405 .34008 .03757 .0074 49670 .0320 8.685 8.719 8.752 8.785 8.818 8.851 8.884 8.917 8.951 .99092 .99085 .99078 .99071 .99064 99057 .99050 99043 .99036 .97956 97940 .97925 .97909 .97893 .97878 .97862 .97846 -97830 .14964 .15019 .15075 .15131 .15186 .15242 .15297 15353 .15408 .67411 .67417 .67423 .67429 .67435 .67441 .67447 .67452 .67458 -34013 .34018 .34024 .34029 .34035 .34040 .34045 .34051 .34056 .03771 .03785 .03800 .03814 .03828 .03842 .03857 .03871 .03885 .0074 .0075 .0076 .0076 .0077 .0077 .0078 .0078 .0079 .49667 .49665 .49662 .49660 .49657 .49654 .49652 .49649 .49647 .0321 .0322 .0323 .0325 .0326 .0327 .0328 .0329 .0330 8.984 .99029 .97814 .15464 .67465 .34062 .03899 .0080 .49644 .0331 9.017 9-050 9.083 9.116 9-149 9-183 9.216 9-249 9.282 .99022 99014 .99007 .99000 .98993 98985 .98978 .98971 98963 97798 .97782 .97766 97749 97733 .97717 .97700 .97684 .97667 .15519 15575 .15630 .15686 .15741 .15796 .15852 .15907 .15962 .67471 .67477 .67483 .67489 .67495 .67501 67507 67514 .67520 34067 34073 .34079 .34084 34090 .34095 .34101 34107 .34113 .03913 .03929 .03942 .03956 .03970 .03985 .03999 .04013 .04027 .0080 .0081 .0081 .0082 .0082 .0083 .0084 .0084 .0085 .49641 .49639 .49636 .49633 .49631 49629 .49625 49623 .49620 .0333 .0334 .0335 0336 0337 .0338 0339 .0340 0342 9-315 .98956 .97651 .16018 .67526 .34118 .04041 .0085 .49617 .0343 9-348 9.381 9-415 9-448 9.481 9-514 9-547 9.58o 9.613 .98948 .98941 .98933 .98926 .98918 .98911 .98903 .98895 .98888 .97634 .97617 .97601 .97584 .97567 97550 97533 -97SI6 97499 . 16073 .16128 .16183 .16239 .16294 .16349 .16404 .16459 .16514 .67532 .67539 .67545 .67551 .67558 .67564 .67571 .67577 .67584 .34124 .34130 .34136 .34141 .34147 34153 .34159 .34165 .34171 .04056 .04070 .04084 .04098 .04113 .04127 .04141 .04155 .04169 .0086 .0087 .0087 .0088 .0088 .0089 .0090 .0090 .0091 .49615 .49612 .49609 .49606 .49604 .49601 .49598 49595 49592 .0344 .0345 .0346 0347 .0348 .0349 0351 .0352 .0353 9.647 .98880 .97482 .16569 .67590 .34177 .04184 .0092 .49590 .0354 9.680 9-713 9.746 9-779 9.812 9-845 9.878 9-9II 9-945 .98872 .98865 .98857 .98849 .98841 .98833 .98826 .98818 .98810 .97465 97447 97430 .97413 97395 97378 .97360 97343 97325 .16624 .16679 .16734 .16789 .16844 .16899 .16954 .17009 .17064 .67597 .67603 .67610 .67616 .67623 .67630 .67636 .67643 .67650 .34183 .34189 .34195 .34201 .34207 34213 .34219 .34225 .34232 .04198 .04212 .04226 .04240 .04254 .04269 .04283 .04297 .04311 .0092 .0093 .0093 .0094 .0095 .0095 .0096 .0097 .0097 .49587 .49584 .49581 .49578 49575 49573 49570 .49567 .49564 .0355 .0356 .0357 .0358 .0359 .0361 .0362 .0363 .0364 9-978 ,98802 -97307 .I7H9 .67657 .34238 .04325 .0098 .49561 .0365 (52) TABLE VIII. (Continued^ A 30 .1 .2 3 4 5 .6 7 .8 9 31.0 .1 .2 3 4 5 .6 .7 .8 9 32.0 .1 .2 3 4 5 .6 .7 .8 9 33.0 .1 .2 3 4 '.6 7 .8 9 34.0 .1 .2 3 4 5 .6 7 .8 9 35.0 A 9.978 C L .98802 X F U V omLnD Z=mL-nD A 30 ."0 .1 .2 3 4 5 .6 '.8 9 31.0 .1 .2 3 4 5 .6 7 .8 9 32.0 .1 ..2 3 4 5 .6 7 .8 9 33.0 .1 .2 3 4 5 .6 7 .8 9 34.0 .1 .2 3 .4 5 .6 7 .8 9 35.0 L .97307 L .17119 L -67657 L .34238 m .04325 .0098 m .49561 n .0365 IO.OII 10.044 10.077 10. IIO 10.143 10.176 10.209 10.242 10.276 -98794 .98786 .98778 .98770 .98762 .98754 .98745 .98737 .98729 .97290 .97272 97254 .97236 .97218 .97200 .97182 97164 .97146 .17174 . 17229 .17283 .17338 .17393 .17448 .17502 17557 .17612 .67663 .67670 .67677 .67684 .67691 .67698 .67705 .67712 .67719 .34244 .34250 34257 .34263 34269 34276 .34282 .34288 .34295 04339 04354 .04368 .04382 .04396 .04410 .04424 .04439 04453 .0098 .0099 .0100 .0100 .0101 .0102 .0102 OIO3 .OIO4 .49558 49555 .49552 49549 49546 49543 49540 49537 .49534 .0366 .0367 .0368 .0369 0371 .0372 .0373 0374 0375 10.309 .98721 .97127 .17666 .67726 .34301 .04467 .OI04 49531 0376 10.342 10.375 10.408 10.441 10.474 10.507 10.540 10.573 10.607 .98713 .98704 .98696 .98688 .98680 .98671 .98663 .98654 .98646 .97109 .97091 .97072 .97054 -97035 .97017 .96998 .96979 .96960 . 17721 17776 .17830 .17885 17939 17994 . 18048 . 18103 .18157 .67733 .67740 67747 .67754 .67761 .67768 67775 .67783 .67790 .34308 34314 34321 34327 34334 34340 34347 34353 .34360 .04481 04495 .04509 04523 .04538 04552 .04566 .04580 04594 .0105 .0106 .OIO6 .OIO7 .OIO8 .0108 .0109 .0110 .OIIO 49528 49525 49522 49519 .49516 49513 49510 49507 49504 0377 0378 0379 .0380 .0381 .0383 0384 .0385 .0386 10.640 98637 .96942 .18212 .67797 .34367 .04608 .OIII 49501 .0387 10.673 10.706 10.739 10.772 10.805 10.838 10.871 10.904 10.937 .98629 .98620 .98612 .98603 98595 .98586 .98577 .98569 .98560 .96923 .96904 .96885 .96866 .96847 .96828 .96808 .96789 96770 .18266 .18320 .18375 .18429 .18483 .18538 .18592 .18646 .18700 .67804 .67812 .67819 .67826 67834 .67841 .67849 .67856 .67864 34373 34380 .34387 34394 .34400 34407 .34414 .34421 .34428 .04622 .04636 .04650 .04665 .04679 04693 .04707 .04721 04735 .0112 .0112 .0113 .OII4 .0114 .0115 .0116 .0116 .Oil? .49498 .49495 49491 .49488 .49485 .49482 49479 -49476 49473 .0388 .0389 .0390 .0391 .0392 0393 0394 0395 0397 10.970 .98551 96750 .18754 .67871 34435 04749 .0118 .49469 .0398 11.003 11.036 11.069 11.103 11.136 11.169 n. 202 11.235 11.268 .98543 98534 .98525 .98516 .98507 .98499 .98490 .98481 .98472 .96731 .96712 .96692 .96672 .96653 .96633 .96613 .96593 .96574 .18809 .18863 .18917 .18971 .19025 .19079 .19133 .19187 .19241 .67879 .67886 .67894 .67901 .67909 .67917 .67925 .67932 .67940 34442 -34448 34455 .34462 .34469 34477 .34484 34491 34498 04763 .04777 04791 .04805 .04820 .04834 .04848 .04862 .04876 .0118 .0119 .OI20 .0121 .OI2I .OI22 .OI23 .0123 .OI24 .49466 .49463 .49460 49457 49453 .49450 49447 49443 .49440 0399 .0400 .0401 .0402 .0403 .0404 .0405 .0406 .0407 11.301 .98463 .96554 .19295 .67948 34505 .04890 .OI25 49437 .0408 11.334 11.367 11.400 11.433 11.466 ".499 11-532 II-565 "598 .98454 .98445 .98436 .98427 .98417 .98408 .98399 .98390 .98381 .96534 .96514 .96494 .96473 .96453 .96433 .96413 .96392 .96372 .19349 .19403 19457 I95IO .19564 .19618 .19672 .19726 19779 .67956 .67963 .67971 .67979 .67987 67995 .68003 .68011 .68019 34512 34519 34527 34534 34541 34548 .34556 .34563 34570 .04904 .04918 04932 .04946 .04960 .04974 .04988 .05002 .05016 .0126 .OI26 .0127 .0128 .0128 .0129 .0130 .0131 .0131 .49434 49430 49427 .49424 .49420 .49417 .49414 49410 .49407 .0409 .0410 .0411 .0412 .0413 .0415 .0416 .0417 .041? 11.631 .98371 .96351 .19833 .68027 34578 .05030 .0132 .49404 .0419 (53) TABLE VIII. (Continued} A C X F U V o=mLnD Z=mLnD A L L L L L m n m n A 35.0 11.631 .98371 .96351 19833 .68027 .34578 .05030 .0132 .49404 .0419 35 .1 11.664 .98362 .96331 . 19887 .68035 .34585 .05044 .0133 .49400 .0420 .1 .2 11.698 .98353 .96310 .19940 .68043 34593 .05058 0133 49397 .0421 .2 3 11.731 .98344 .96290 19994 .68051 .34600 .05072 .0134 49393 .0422 3 4 11.764 .98334 .96269 .20047 .68059 .34608 .05086 0135 .49390 .0423 .4 5 11.797 .98325 .96248 .20IOI .68068 .34615 .05100 .0136 .49387 .0424 5 .6 11.830 98315 .96227 20155 .68076 .34623 05115 .0136 .49383 .0425 .6 .7 11.863 .98306 .96207 .20208 .68084 .34630 .05129 .0137 .49380 .0426 .7 .8 11.896 .98297 .96186 . 2O262 .68092 .34638 05143 .0138 49376 .0427 .8 9 11.929 .98287 .96165 .20315 .68101 .34645 05157 0139 49373 .0428 9 36.0 11.961 .98278 .96144 .20368 .68109 .34653 .05171 0139 .49369 .0429 36.0 .1 11.994 .98268 96123 . 2O422 .68117 .34661 05185 .0140 .49366 .0430 .1 .2 12.027 .98259 .96102 20475 .68126 .34668 05199 .0141 .49362 .0431 .2 3 12.061 .98249 .96080 .20529 .68134 .34676 .05213 .0142 49359 .0432 3 4 12.094 98239 96059 .20582 .68142 .34684 .05227 .0142 49355 0433 4 5 12.127 .98230 .96038 .20635 .68151 34692 .05241 0143 49352 0434 5 .6 12.159 .98220 .96017 .20689 .68159 34699 05255 .0144 .49348 0435 .6 7 12.192 .98210 95995 .20742 .68168 34707 .05269 0145 49345 .0436 7 .8 12.226 .98201 95974 20795 .68176 34715 .05282 .0145 49341 0437 .8 -9 12.259 .98191 95952 .20848 .68185 34723 .05296 .0146 .49338 .0438 9 37.0 12 . 292 .98181 95931 . 2O9OI .68194 34731 .05310 .0147 49334 .0439 37.0 .1 12.324 .98171 .95909 20955 .68202 34739 05324 .0148 49330 .0440 .1 .2 12.358 .98162 .95887 .21008 .68211 34747 .05338 .0149 .49327 .0441 .2 3 12.391 .98152 .95866 .21061 .68219 34754 05352 .0149 49323 .0442 3 4 12.424 .98142 .95844 .21114 .68228 .34762 .05366 .0150 49320 0443 4 5 12.456 .98132 .95822 .21167 .68237 34770 .05380 .0151 .49316 0444 5 .6 12.489 .98122 .95800 .21220 .68246 34779 05394 .0152 .49312 0445 .6 7 12.523 .98112 .95778 .21273 .68254 .34787 .05408 .0152 49309 .0446 .7 .8 12-556 .98102 95756 .21326 .68263 34795 .05422 .0153 .49305 .0447 .8 9 12.588 .98092 95734 .21379 .68272 34803 .05436 .0154 -49302 .0448 9 38.0 12.621 .98082 95712 .21432 .68281 34811 .05450 .0155 .49298 .0449 38.0 .1 12.654 .98072 .95690 .21485 . 68290 .34819 .05464 .0156 .49294 .0450 .1 .2 12.687 .98062 .95668 21537 .68299 34827 05478 .0156 .49291 .0451 .2 3 12 . 72O .98052 .95646 .21590 .68307 .34835 05492 .0157 .49287 .0452 3 .4 12.753 .98042 .95623 .21643 .68316 .34844 .05506 .0158 .49283 0453 4 5 12.786 .98032 .956oi .21696 .68325 .34852 .05520 0159 49279 0454 5 .6 12.819 .98022 95578 .21749 .68334 .34860 .05534 0159 .49276 0455 .6 7 12.852 .98011 .95556 . 21801 -68343 34869 .05548 .0160 .49272 .0456 .7 .8 12.885 .98001 95533 .21854 .68353 .34877 .05561 .0161 .49268 .0457 .8 .8 12.918 .97991 95511 .21907 .68362 .34885 05575 .0162 .49265 .0458 9 39.0 12.951 .9798i .95488 .21959 .68371 .34894 .05589 .0163 .49261 .0459 39.0 .1 12.984 97970 .95466 .22012 .68380 .34902 05603 .0163 .49257 .0460 .1 .2 13.017 .9796o 95443 .22065 .68389 349II .05617 .0164 .49253 .0461 .2 3 13-050 97950 95420 .22117 .68398 .34919 .05631 .0165 .49250 .0462 3 .4 13.083 97939 95397 .22I7O .68408 .34928 .05645 .0166 .49246 .0463 4 5 13.116 .97929 95374 .22222 .68417 .34936 .05659 .0167 .49242 .0464 5 .6 I3.U9 .97919 95351 .22275 .68426 34945 .05673 .0168 .49238 .0465 .6 7 13.182 .97908 95328 .22327 68435 34953 .05687 .0168 .49234 .0466 7 .8 13-215 .97898 95305 .22379 .68445 .34962 .05701 .0169 .49231 .0467 .8 9 13-247 .97887 .95282 .22432 .68454 34970 .05714 .0170 . 49227 .0468 9 40.0 13.281 .97877 95259 .22484 .68464 34979 .05728 .0171 .49223 .0469 40.0 (54) TABLE IX. SPIRAL FUNCTIONS FOR CHANGE OF 0.2 PER 100 FEET SUITABLE FOR SPEEDS OF 104 MILES AN HOUR OR LESS AND CURVES OF i OR LESS L D A A z C X Y U V 10 O.O2 O.OOI o.ooo 5.00 o.oo 10.00 10.00 0.00 6.67 3-33 20 .04 .004 .001 IO.OO .00 20.00 20.00 .00 13 33 6.6? 30 .06 .009 .003 15.00 .00 30.00 30.00 .00 20.00 10.00 40 .08 .016 .005 20.00 .00 40.00 40.00 .00 26.67 13-33 50 .IO .025 .008 25.00 .00 50.00 50.00 .01 33-33 16.67 60 .12 .036 .012 30.00 .00 60.00 60.00 .01 40.00 20.00 70 14 .049 .016 35-00 .00 70.00 70.00 .02 46.67 23-33 80 .16 .064 .021 40.00 .01 80.00 80.00 03 53-33 26.67 90 .18 .081 .027 45.00 .01 90.00 90.00 .04 60.00 30.OO 100 .20 .100 033 50.00 .01 IOO.OO IOO.OO .06 66.67 33-33 10 .22 .121 .040 55-00 .02 10.00 10.00 .08 73-33 36.67 20 24 144 .048 60.00 03 20.00 20.00 .10 80.00 40.00 30 .26 .169 .056 65.00 .03 30.00 30.00 .13 86.67 43-33 40 .28 .196 .065 70.00 .04 40.00 40.00 .16 93-33 46.67 50 30 .225 075 75-00 .05 50.00 50.00 .20 IOO.OO 50.00 60 32 .256 .085 80.00 .06 60.00 60.00 -24 06.67 53-33 70 34 .289 .096 85.00 .07 70.00 70.00 .28 13.33 56.67 80 36 324 .108 90.00 .08 80.00 80.00 -34 20.00 60.00 90 38 .361 .120 95-00 .10 90.00 90.00 .40 26.67 63.33 200 40 .400 .133 100.00 .12 200.00 200.00 -47 133.33 66.67 10 .42 441 .147 105.00 -13 IO.OO IO.OO -54 40.00 70.00 20 44 484 .161 110.00 .15 20.00 20.00 .62 46.67 73-33 30 -46 529 .176 115.00 .18 30.00 30.00 71 53.33 76.67 40 .48 .576 .192 120. OO .20 40.00 40.00 .80 60.00 80.00 50 SO .625 .208 125.00 -23 50.00 50.00 -91 66.67 83.33 60 52 .676 .225 130.00 .26 60.00 60.00 .02 73.33 86.67 70 54 .729 .243 135.00 .29 70.00 70.00 -14 80.00 90.00 80 56 784 .261 140.00 .32 80.00 79-99 .28 86.67 93-34 90 58 .841 .280 145.00 35 90.00 89.99 42 93.33 96.67 300 .60 .900 .300 150.00 39 300.00 99.99 -57 200.00 IOO.OO 10 .62 .961 .320 155.00 -43 10.00 309.99 -73 06.67 103.34 20 .64 .024 341 160.00 .48 20.00 19.99 91 13.33 106.67 30 .66 .089 .363 165.00 52 29.99 29.99 .09 20.00 110.00 40 .68 .156 .385 170.00 57 39.99 39.99 -29 26.67 113.34 50 .70 .225 .408 175.00 .62 49.99 49.98 49 33.34 116.67 60 72 .296 432 180.00 .68 59.99 59.98 2.71 40.01 120.01 70 74 .369 .456 185.00 74 69.99 69.98 2.95 46.67 123.34 80 .76 444 .481 189.99 .80 79-99 79.98 3.19 53-34 126.67 90 78 .521 j -507 194.99 .86 89.99 89.97 3-45 60. 01 130.01 400 .80 .600 533 199.99 93 99.99 99.97 3-72 266.68 133.34 10 .82 .681 .560 204.99 .00 409.98 409.96 4.01 73-35 136.68 20 .84 764 .588 209.99 .08 19.98 19.96 4-31 80.02 140.01 30 .86 -849 .618 214.99 .16 29.98 29.96 4-63 86.68 143.35 40 .88 .936 .645 219.99 .24 39.98 39.95 4-96 93-35 146.68 So .90 .025 .675 224.99 33 49.98 49.94 5-30 300.02 150 . 02 60 92 .116 70S 229.99 -42 59.97 59-94 5-66 06.69 153.35 70 94 .209 736 234.99 51 69.97 69.93 6.04 13.36 156.69 80 96 .304 .768 239.98 .61 79.97 79-92 6.43 20.03 160.02 90 98 2.401 .800 244.98 71 89.96 89-91 6.84 26.70 163.36 500 I.OO 2.500 .833 249.98 1.82 499.96 499-91 7.27 333-37 166.70 (55) TABLE X. SPIRAL FUNCTIONS FOR A CHANGE OF 0.3 PER 100 FEET. SUITABLE FOR SPEEDS OF 91 MILES AN HOUR OR LESS, OR CURVES OF 1.5 OR LESS L D A A Z o C X Y U V 10 0.03 O.OOI O.OOI 5.00 o.oo 10.00 IO.OO o.oo 6.67 3-33 20 .06 .006 .002 10. OO .00 20.00 20.00 .00 13-33 6.6? 30 .09 .013 .005 15.00 .00 30.00 30.00 .00 20.00 10.00 40 .12 .024 .008 20.00 .00 40.00 40.00 .00 26.67 13.33 50 IS .037 .013 25.00 .00 50.00 50.00 .01 33-33 16.67 60 .18 054 .018 30.00 .00 60.00 60.00 .02 40.00 20.00 70 .21 073 .025 35-00 .01 70.00 70.00 .03 46.67 23-33 80 .24 .096 .032 40.00 .01 80.00 80.00 .04 53-33 26.67 90 27 .121 .041 45-00 .02 90.00 90.00 .06 60.00 3O.OO 100 .30 ISO .050 50.00 .02 100.00 100.00 .09 66.67 33-33 10 33 .I8l .061 55-00 .03 IIO.OO IIO.OO .12 73-33 36.67 20 .36 .216 .072 60.00 .04 120. OO 120.00 .15 80.00 40.00 30 39 253 .085 65.00 .05 130.00 130.00 19 86.67 43-33 40 42 .294 .098 70.00 .06 140.00 140 . oo .24 93-33 46.67 50 45 .337 .113 75-00 07 150.00 150.00 .29 IOO.OO 50.00 60 .48 .384 .128 80.00 09 160.00 160.00 .36 106.67 53-33 70 SI 433 .145 85.00 .11 170.00 170.00 43 113.33 56.67 80 54 .486 .162 90.00 13 180.00 180.00 51 I2O.OO 60.00 90 57 541 .181 95-oo .15 190.00 190.00 .60 126.67 63.33 200 .60 .600 .200 IOO.OO 17 200.00 200.00 70 133.33 66.67 10 .63 .661 .221 105.00 .20 210.00 2IO.OO .81 140.00 70.00 20 .66 .726 .242 IIO.OO -23 220.00 220.00 93 146.67 73.33 30 .69 793 .265 115.00 27 230.OO 230.00 1. 06 153.33 76.67 40 72 .864 .288 120. OO .30 24O.OO 240 . oo 1. 21 160.00 8o.OO 50 75 937 .313 125.00 34 250.00 250 . oo 1.36 166.67 83.34 60 .78 1.014 338 130.00 38 26O.OO 259-99 1-53 173.34 86.67 70 .81 1.093 -365 135-00 43 269.99 269.99 1.72 180.00 90.00 80 84 1.176 392 140 . oo .48 279-99 279.99 1.92 186.67 93-34 90 87 1.261 .421 145.00 53 289.99 289.99 2.13 193.34 96.67 300 .90 1.350 450 150.00 .59 299-99 299.98 2.36 20O.OI 100.01 10 93 1.441 .481 155-00 .65 309-99 309.98 2.60 206.67 103.34 20 96 1.536 512 160.00 7i 319-99 319.98 2.86 213-34 106.67 30 99 1.633 545 165.00 79 329.99 329.97 3-14 22O.OI 110. 01 40 1.02 1.734 .578 170.00 .86 339-99 339-97 3-43 226.68 113.34 50 1.05 1.837 .613 175-00 93 349-99 349-96 3-74 233-35 116.68 60 1. 08 1-944 .648 180.00 1.02 359-98 359-96 4.07 240.02 120.01 70 I. II 2.053 .685 184.99 I. II 369.98 369.95 4-42 246.68 123-35 80 1. 14 2.166 .722 189.99 1.20 379-98 379-94 4-79 253-35 126.68 90 1. 17 2.281 .761 194-98 1.29 389.97 389.94 5-i8 260.02 130.02 400 1.20 2.400 .800 199-98 1.40 399-97 399-93 5-58 266.69 133.36 10 1.23 2.521 .841 204.98 1-50 409.96 409.92 6.01 273-36 136.69 20 1.26 2.646 .882 209.98 1.62 419.96 419-91 6.46 280.03 140.03 30 1.29 2.773 .925 214.98 1-74 429.96 429.90 6-94 286.70 143-37 40 1.32 2.904 .968 219.98 1.86 439-95 439-89 7-43 293-37 146.70 50 1.35 3-037 1.013 224.98 1.98 449-95 449-87 7-95 300.04 I50.0/) 60 1.38 3-174 1.058 229.97 2.13 459-94 459-86 8.49 306.72 153.38 70 1.41 3.313 1.105 234-97 2.27 469.93 469-84 9.06 313-39 156.72 80 1.44 3.456 1.152 239-96 2.41 479-92 479-83 9-65 320.06 160.06 90 1-47 3-6oi I. 201 244.96 2-57 489.92 489.81 10.27 326.74 163.40 500 1.50 3.7<>0 1.250 249.96 2.72 499.90 499-79 10.91 333-41 166.74 (56) TABLE XL SPIRAL FUNCTIONS FOR A CHANGE OF 0.4 PER 100 FEET. SUITABLE FOR SPEEDS OF 83 MILES AN HOUR OR LESS, OR CURVES OF 2.0 OR LESS L D A A Z o C X Y U V 10 0.04 o O.O02 O.OOI S-oo o.oo 10. OO IO.OO o.oo 6.67 3-33 20 .08 .008 .003 10.00 .00 20.00 20.00 .00 13-33 6.67 30^ .12 .018 .006 15.00 .00 30.00 30.00 .00 20.00 10.00 40 .16 .032 .on 20.00 .00 40.00 40.00 .01 26.67 13.33 So .20 .050 .016 25.00 .00 50.00 50.00 .02 33-33 16.67 60 24 .072 .024 30.OO .01 60.00 60.00 .03 40.00 20.00 70 .28 .008 .033 35.00 .01 70.00 70.00 04 46.67 23-33 80 .32 .128 .043 40.00 .01 80.00 80.00 .06 53-33 26.67 90 .36 .162 .054 45-00 .02 90.00 90.00 .08 60.00 30.00 100 .40 .200 .067 50.00 .03 100.00 100.00 .12 66.67 33-33 10 44 .242 .081 55-00 .04 IIO.OO IIO.OO .16 73-33 36.67 20 48 .288 .096 60.00 .05 120.00 I2O.OO .20 80.00 40.00 30 52 .338 .113 65.00 .06 130.00 130.00 .26 86.67 43-33 40 .56 .392 .131 70.00 .08 140.00 140.00 32 93-33 46.67 So .60 450 .150 75-00 .10 150.00 150.00 39 IOO.OO 5O.OO 60 .64 .512 .171 80.00 .12 160.00 160.00 .48 106.67 53-33 70 .68 .578 .193 85.00 14 170.00 170.00 57 113.33 56.67 80 .72 .648 .216 90.00 .17 180.00 180.00 .68 120. OO 60.00 90 76 .722 .241 95-00 .20 190.00 190.00 .80 126.67 63.33 200 .80 .800 .267 IOO.OO .23 200.00 200.00 93 133-33 66.67 10 .84 .882 .294 105 . oo .27 2IO.OO 210. OO i. 08 I40.OO 70.00 20 .88 .968 323 IIO.OO .31 22O . OO 219-99 1.24 146.67 73-33 30 .92 1.058 353 115.00 35 230.00 229.99 1.41 153-34 76.67 40 96 1. 152 .384 I2O.OO .40 24O.OO 239-99 1.61 I60.00 80.00 50 1. 00 1.250 .417 125.00 .46 250.00 249-99 1.82 166.67 83.34 60 1.04 I.3S2 451 130.00 .51 259-99 259.98 2.04 173-34 86.67 70 1. 08 1.458 .486 135.00 57 269.99 269.98 2.29 ISO.OI 90.01 80 1. 12 1.568 523 140.00 .64 279-99 279.98 2.55 186.67 93-34 90 1.16 1.682 .561 144.99 71 289.99 289.97 2.84 193.34 96.67 300 1.20 1.800 .600 149.99 79 299-99 299-97 3.14 2OO.OI IOO.OI 10 1.24 1.922 .641 154.99 .86 309-98 309.97 3-47 206.68 103.34 20 1.28 2.048 .683 159.99 95 319.98 319 96 3.8z 213-35 106.68 30 1.32 2.178 .726 164.99 1.05 329.98 329.95 4-18 220.02 110.02 40 1.36 2.312 771 169.99 1.14 339-98 339.95 4-57 226.68 113-35 So 1.40 2.450 .817 174.98 1.25 349-97 349.94 4-99 233.36 II6.69 60 1.44 2.592 .864 179.98 1.36 359-97 359-93 5-43 240.03 120.02 70 1.48 2.738 .913 184.98 1.48 369.96 369.91 5.89 246.69 123.36 80 I.S2 2.888 .963 189.98 i. 60 379-96 379-90 6.38 253-37 126.70 90 1-56 3.042 1.014 194.98 1.72 389.95 389-89 6.90 260.04 130.03 400 1. 60 3.200 1.067 199.97 1.86 399-94 399-88 7-44 266.71 133-37 10 1.6 4 3.362 1. 121 204.97 2.OO 409.94 409.86 8.02 273.38 136.71 20 1.68 3-528 I.I76 209.97 2. IS 419.93 419.84 8.61 280.06 140.05 30 1.72 3.698 1.233 214.96 2.31 429-92 429.82 9-25 286.73 143-39 40 1.76 3.872 I.29I 219.96 2. 4 8 439-91 439-80 9.90 293.41 146.73 So i. 80 4.050 1.350 224.95 2.65 449-90 449-78 10. 60 300.08 ISO.O? 60 1.84 4.232 1. 411 229.95 2.83 459-89 459-75 11.32 306.76 153-41 70 1.88 4.418 1.473 234.95 3-02 469.88 469.72 12.07 313.43 156.75 80 1.92 4.608 1.533 239.94 3-22 479-87 479.69 12.86 320.11 160.10 90 1.96 4.802 1.601 244.93 3-42 489-85 489.66 13-68 326.79 163.44 500 2.00 5.000 1.667 249.93 3-63 499.84 499.62 14.54 333 47 166.79 (57) TABLE XII. SPIRAL FUNCTIONS FOR A CHANGE OF 0.5 PER 100 FEET. SUITABLE FOR SPEEDS OF 77 MFLES AN HOUR OR LESS, OR CURVES OF 2.0 OR LESS L D A A Z o C X Y U V 10 0.05 O.002 o O.OOI 5-00 o.oo IO.OO 10.00 0.00 6.67 3-33 20 .10 .OIO .003 IO.OO .00 20.00 20.00 .00 13-33 6.67 30 IS .025 .008 15.00 .00 30.00 30.00 .00 20.00 10.00 40 .20 .040 .013 20.00 .00 40.00 40.00 .01 26.67 13-33 So 25 .062 .021 25.00 .00 50.00 50.00 .02 33-33 16.67 60 .30 .000 .030 30.00 .01 60.00 60.00 .03 40.00 20.00 70 35 .122 .041 35-00 .01 70.00 70.00 05 46.67 23-33 80 .40 .100 .053 40.00 .02 80.00 80.00 .07 53.33 26.67 90 45 .202 .067 45-00 03 90.00 90.00 .11 60.00 30.00 100 50 .250 .083 50.00 .04 IOO.OO IOO.OO .15 66.67 33-33 10 55 -302 .101 55-00 .05 IIO.OO IIO.OO 19 73-33 36.67 20 .60 .360 .120 60.00 .06 I2O.OO 120.00 .25 80.00 40.00 30 .65 .422 .141 65.00 .08 130.00 130.00 32 86.67 43-33 40 70 490 .163 70.00 .10 140.00 140.00 40 93 33 46.67 50 75 .562 .I8 7 75-00 .12 150.00 150.00 49 100.00 50.00 60 .80 .640 .213 80.00 IS 160.00 160 . oo .60 106.67 53-33 70 .85 .722 .241 85.00 .18 170.00 170.00 71 113.33 56.67 80 .90 .8lO .270 90.00 .21 180.00 180.00 .85 I2O.OO 60.00 90 95 .002 .301 95-00 .25 190.00 189.99 I.OO 126.67 63.33 200 I.OO I. 000 333 IOO.OO .29 200.00 199.99 1.16 133.34 66.67 10 .05 1. 102 .367 105.00 34 210.00 209.99 .35 140.00; 70.00 20 .10 I.2IO 403 IIO.OO 39 220.00 219.99 .55 146.67 73-34 30 .15 1-322 .441 115.00 44 230.00 229.99 77 153.34 76.67 40 .20 1.440 .480 120. OO So 239-99 239.99 .01 160.01 80.00 50 25 1.562 .521 125.00 57 249-99 249.98 .27 166.67 83.34 60 .30 I.60X) .563 129.99 .64 259-99 259.98 .56 173.34 86.67 70 35 1.822 .607 134.99 .72 269.99 269.97 2.86 180.01 90.01 80 40 1.960 653 139.99 .80 279-99 279.97 3-19 186.68 93-34 90 45 2.IO2 .701 144.99 .88 289.98 289.96 3-55 193.35 96.68 300 1-50 2.250 750 149.99 98 299.98 299.96 3-93 20O . 02 IOO.OI 10 55 2.4O2 .801 154.99 1. 08 309.98 309.95 4-33 206.69 103.35 20 .60 2.560 .853 159.98 1. 19 319.97 319.94 4-76 213.36 106.69 30 65 2.722 .907 164.98 1.31 329.97 329.93 5-23 220 . 02 110.02 40 .70 2.890 .963 169.98 1-43 339.96 339.92 5-72 226.70 113.36 50 75 3.O62 .021 174.98 1.56 349.96 349.90 6.23 233-37 II6.70 60 .80 3.240 .080 179.97 1.70 359-95 359-88 6.78 240.04 120.03 70 .85 3-422 .141 184.97 1.84 369.94 369.87 7-37 246.71 123.38 80 .90 3.6lO .203 189.97 2.OO 379-94 379-85 7-98 253-39 126.71 90 95 3.8O2 .267 194.96 2.16 389.93 389.83 8.63 260 . 06 130.05 400 2.00 4.000 1.333 199.96 2-33 399-92 399-81 9-30 266.74 133-40 (58) TABLE XIII. SPIRAL FUNCTIONS FOR A CHANGE OF 0.75 PER 100 FEET. SUITABLE FOR SPEEDS OF 67 MILES AN HOUR OR LESS, OR CURVES OF 3.0 OR LESS L D A A Z c X Y U V 10 0.075 0.004 O.OOI 5.00 o.oo IO.OO IO.OO o.oo 6.67 3 33 20 ISO .015 .005 10. OO .00 20.00 20.00 .00 13-33 6.67 30 .225 034 .012 15.00 .00 30.00 30.00 .01 20.00 10.00 40 .300 .060 .020 20.00 .00 40.00 4O.OO .01 26.67 13.33 50 375 .094 .031 25.00 .01 50.00 50.OO .03 33-33 16.67 60 450 .135 .045 30.00 .01 60.00 60.00 .05 40.00 20.00 70 .525 .184 .061 35-00 .02 70.00 7O.OO .07 46-67 23-33 80 .600 .240 .080 40.00 .03 80.00 8O.OO .11 53-33 26.67 90 .675 .304 .101 45-00 .04 90.00 90.OO .16 60.00 30.CO 100 750 375 125 50.00 .06 IOO.OO IOO . CO .22 66.67 33-33 10 .825 454 .151 55-00 7 IIC. 00 110.00 29 73-33 36.67 20 .900 .540 .180 60.00 .09 120.00 120.00 .38 80.00 40.OO 30 975 .634 .211 65.00 .12 130.00 130.00 .48 86.67 43-33 40 1.050 .735 245 70.00 .15 140.00 140 . oo .60 93-33 46.67 So 1.125 .844 .281 75-00 .18 150 . oo 150.00 74 100.00 50.OO 60 1.200 .960 .320 80.00 .22 160.00 160.00 .89 106.67 53-33 70 1-275 1.084 .361 85.00 .27 170 . co 169.99 1.07 113.33 56.67 80 1.350 1.215 .405 90.00 .32 180.00 179.99 1.27 120. OO 60.00 90 1.425 1.354 451 95-00 .37 189.99 189.99 1.50 126.67 63.34 200 I.5OO 1.500 .500 IOO.OO 44 199.99 199.99 1-75 133.34 66.67 10 I 575 1.654 551 104.99 Si 209.99 209.98 2. 02 140.01 70.01 20 1.650 1.815 .60S 109.99 -58 219.99 219.98 2.32 146.67 73-34 30 1.725 1.984 .661 114.99 .66 229.99 229.97 2.65 153.34 76.68 40 1.800 2.160 .720 119.99 73 239-99 239.97 3-02 160.01 80.01 So 1.875 2.344 .781 124.99 .85 249.98 249.96 3.41 166.68 83.34 60 1-950 2.535 845 129.99 .96 259-98 259-95 3.84 173.35 86.68 70 2.025 2.734 .911 134.98 1.07 269.97 269.94 4.30 180.02 90.02 80 2.100 2.940 .980 139.98 1.20 279.97 279-93 4-79 186.69 93.36 90 2.175 3-154 I.O5I 144.98 1.33 289.96 289.91 5-32 193.37 96.69 300 2.25O 3-375 1. 125 149.97 1-47 299.96 299-90 5.86 200.04 100.03 10 2.325 3.604 .201 154.97 1.63 309.95 309-88 6.50 206.71 103-37 20 2.400 3.840 .280 159.97 1.79 319.94 319.86 7-15 213.38 106.71 30 2.475 4.084 .361 164.96 1.96 329.93 329-83 7-84 220.06 110.04 40 2.550 4-335 .445 169.96 2.14 339.92 339-81 8.57 226.73 113-39 50 2.625 4-594 531 174.95 2.34 349.90 349.78 9-35 233.41 116.74 60 2.700 4.860 .620 179.94 2.54 359-88 359-74 10.17 240.09 120.09 70 2.775 5-134 .711 184.93 2.76 369-87 369-70 11.04 246.77 123-43 80 2.850 5.415 .805 189.93 2.99 379-85 379.67 11.97 253.45 126.78 90 2.925 5.704 .901 194.92 3-24 389.83 389-62 12-93 260.14 130.12 400 3-000 6.000 2.OOO 199.91 3-49 399-81 399-56 13-95 266.82 138.34 (59) TABLE XIV. SPIRAL FUNCTIONS FOR A CHANGE OF 1.0 PER 100 FEET. SUITABLE FOR SPEEDS OF 61 MILES AN HOUR OR LESS, OR CURVES OF 4.0 OR LESS L D A A Z o C X Y U V 10 O.IO 0.005 O.O02 5-co o.oo IO.OO IO.OO .00 6.67 3-33 20 .20 .020 .007 IO.OO .00 20.00 20.00 .00 13-33 6.66 30 30 .045 .015 15.00 .00 30.00 30.00 .01 20.00 IO.OO 40 .40 .080 .027 20.00 .00 40.00 40.00 .02 26.67 13.33 50 SO .125 .042 25.00 .01 50.00 50.00 .04 33-33 16.67 60 .60 .180 .060 30.00 .02 60.00 60.00 .06 40.00 20.00 70 .70 .245 .082 35.00 .03 70.00 70.00 .10 46.67 23-33 80 .80 .320 .107 40.00 .03 80.00 80.00 .15 53-33 26.67 90 .90 .405 135 45-00 .05 90.00 90.00 .21 60.00 3O.OO 100 I.OO .500 .167 50.00 .07 IOO.OO IOO.OO 29 66.67 33-33 10 1. 10 .605 .202 55-00 .10 no.oo 110.00 39 73-33 36.67 20 1.20 .720 .240 60.00 .13 120. OO I2O.OO So 80.00 40.00 30 1-30 .845 .282 65.00 .16 130.00 130.00 .64 86.67 43-33 40 1.40 .980 .327 70.00 .20 140.00 140.00 .80 93-34 46.67 50 I.SO 1. 125 375 75-00 .25 150.00 149.99 -98 100.00 50.00 60 1. 60 1.280 .427 80.00 30 160.00 159.99 1.19 106.67 53-34 70 1.70 1. 445 .482 85.00 .36 169.99 169.99 1-43 113.34 56.67 80 1. 80 1.620 540 90.00 .42 179.99 179.99 1.70 120. OO 60.00 90 I.QO 1.805 .602 94-99 50 189.99 189.98 1.99 126.67 63.34 200 2.OO 2.000 .667 99-99 .58 199.99 199.98 2.33 133.34 66.67 10 2.10 2.205 735 104.99 .67 209.99 209.97 2.69 140.01 70.01 20 2.20 2.420 .807 109.99 77 219.98 219.96 3-10 146.68 73-35 30 2.30 2.645 .882 114-99 .89 229.98 229.95 3-54 153.35 76.68 40 2.40 2.880 .960 119.98 .01 239.97 239.94 4-02 160.02 80.02 So 2.SO 3-125 1.042 124.98 14 249.97 249.93 4-54 166.69 83.36 60 2.60 3-380 1.127 129.97 .28 259.96 259.91 5- ii 173.37 86.70 70 2.70 3.645 1. 215 134-97 43 269.95 269.89 5-72 180.04 90.03 80 2.80 3-920 1.307 139-97 .60 279.94 279.87 6.38 186.71 93-37 90 2.90 4.205 1.402 144-96 77 289.93 289.85 7.09 193.39 96.72 300 3-00 4-500 1.500 149-95 1.96 299.92 299.82 7-85 200.06 100.06 10 3.10 4-805 1.602 154-95 2.16 309.90 309.78 8.66 206.74 103.40 20 3-20 5.120 1.707 159-94 2.38 319.89 319.75 9-53 213.42 106.75 30 3-30 5-445 1.815 164-93 2.61 329.87 329.70 10.44 220.10 110.09 40 3-40 5.78o 1.927 169.92 2.86 339.85 339-66 11.42 226.79 113-44 50 3-50 6.125 2.042 174.91 3.12 349.82 349-6o 12.46 233-47 116.80 60 3.6o 6.480 2.160 179.89 3-39 359.80 359-54 13.56 240.16 120.15 70 3-70 6.845 2.282 184.88 3.68 369.77 369.47 14.71 246.85 123.50 80 3.8o 7.220 2.407 189.87 3-99 379.73 379-40 15-94 253.54 126.86 90 3.90 7-605 2.535 194.85 4-31 389.70 389-32 17-23 260.24 I3O . 22 400 4.00 8.000 2.667 199.84 4.65 399-66 399-22 18.59 266.94 133-58 (60) THE SPIRAL 61 TABLE XV. SPIRAL FUNCTIONS FOR A CHANGE OF i$ PER 100 FEET. SUITABLE FOR A SPEED OF 55 MILES AN HOUR OR LESS, OR CURVES OF 4.0 OR LESS L D A A Z o C X Y U V 10 0.133 0.007 0.002 5-00 o.oo IO.OO IO.OO o.oo 6.67 3-33 20 .267 .027 .009 10.00 .00 20.00 20.00 .00 13-33 6.6? 30 .400 .060 .O2O 15.00 .00 30.00 30.00 .01 20.00 IO.OO 40 533 .107 .036 20.00 .01 40.00 40.00 .02 26.67 13.33 50 .667 .167 .056 25.00 .01 50.00 50.00 .05 33-33 16.67 60 .800 .240 .080 30.00 .02 60.00 60.00 .08 40.00 20.00 70 933 .327 .109 35-00 .03 70.00 70.00 13 46-67 23-33 80 1.067 .427 .142 40.00 .05 80.00 80.00 .20 53-33 26.67 90 1. 200 540 .180 45-00 .07 90.00 90.00 .28 60.00 30.00 100 1-333 .667 .222 50.00 .10 IOO.OO IOO.OO -39 66.67 33-33 10 1.467 .807 .269 55-00 13 110.00 IIO.OO .52 73-33 36.67 10 1. 600 .960 .320 60.00 .17 120.00 I2O.OO .67 80.00 40.00 30 1-733 1.127 .376 65.00 .21 130.00 129.99 .85 86.67 43-34 40 1.867 I.307 .436 70.00 .27 140.00 139.99 .06 93-34 46.67 So 2.OOO 1.500 .500 74-99 -33 150.00 149.99 -31 100.00 50.00 60 2.133 1.707 .569 79-99 .40 159.99 159.99 -59 106.67 53-34 70 2.267 1.927 .642 84-99 .48 169.99 169.98 91 113.34 56.67 80 2.400 2.160 .720 89-99 57 179-99 179.97 .26 120. OI 60. oi 90 2.533 2.407 .802 94-99 .66 189.98 189.97 2.66 126.68 63.34 200 2.667 2.667 .889 99-98 78 199.98 199.96 3.10 133-35 66.68 10 2.800 2.940 .980 104.98 .90 209.97 209.94 3-59 I4O.O2 70.02 20 2.933 3-227 . 076 109.98 1.03 219-97 219.93 4.12 146.69 73-35 30 3.067 3.527 .176 H4-97 1.18 229.95 229.91 4-71 153.36 76.69 40 3.200 3.840 .280 119-97 1-34 239-95 239.89 5-35 160.04 80.03 So 3-333 4.167 .389 124.96 I. Si 249-94 249.87 6.05 166.72 83.38 60 3.467 4.507 502 129.95 1.70 259-93 259.84 6.81 173-39 86.72 70 3.6oo 4.860 .620 134-95 1.91 269.91 269.81 7.62 180.07 90.06 80 3-733 5-227 742 139-94 2.13 279-90 279.77 8.50 186.75 93-41 90 3-867 5.607 .869 144-94 2.36 289.88 289.72 9-43 193-43 96.76 300 4.000 6.000 2.000 149-92 2.61 299-86 299.67 10.46 200.12 loo.n 62 RAILROAD FIELD MANUAL TABLE XVI. SPIRAL FUNCTIONS FOR A CHANGE OF 1.5 PER 100 FEET. SUITABLE FOR SPEEDS OF 53 MILES AN HOUR OR LESS, OR CURVES OF 4.5 OR LESS L D A A z c X Y U V 10 0.15 0.008 O.O02 S.oo o.oo IO.OO IO.OO o.oo 6.67 3-33 20 -30 .030 .OIO IO.OO .00 20.00 20.00 .00 13-33 6.67 30 45 .068 .022 15.00 .00 30.00 30.00 .01 20.00 IO.OO 40 .60 .120 .040 20,00 .01 40.00 40.00 .03 26.67 13.33 So 75 .188 .062 25.00 .01 50.00 50.00 05 33-33 16.67 60 90 .270 .090 30.00 .02 60.00 60.00 .09 40.00 20.00 70 1.05 .368 .122 35-00 .04 70.00 70.00 .15 46.67 23.33 80 1.20 .480 .160 40.00 .06 80.00 80.00 .22 53-33 26.67 90 1.35 .608 .202 45-00 .08 90.00 90.00 32 60.00 30.00 100 1.50 750 .250 50.00 .11 IOO.OO IOO.OO 44 66.67 33-33 10 1.65 .908 .302 55-00 IS IIO.OO IIO.OO 58 73-33 36.67 20 i. 80 1.080 .360 60.00 19 120. OO 120. OO 75 80.00 40.00 30 1-95 1.268 .422 65.00 .24 130.00 129.99 -96 86.67 43-34 40 2.10 1.470 490 69.99 30 140.00 139.99 1. 20 93-34 46.67 50 2. 25 1.688 .562 74-99 37 149.99 149.99 1.47 IOO.OO 50.OO 60 2.40 1.920 .640 79-99 45 159.99 159.98 1-79 106.67 53-34 70 2.55 2.168 .722 84.99 54 169.99 169.97 2.14 113.34 56.67 80 2.70 2.430 .810 89-99 .64 179.99 179.97 2.54 I2O.OI 60. 01 90 2.85 2.708 .902 94-98 75 189.98 189.96 2-99 126.68 63.35 200 3-00 3.000 I.OOO 99.98 -87 199.98 199-95 3.49 133-35 66.68 10 3-15 3.308 .102 104.98 I.OI 209.97 209.93 4.04 140.02 70.02 20 3-30 3 630 .2IO 109.97 1.16 219.96 219.91 4.64 146.70 73.36 30 3-45 3-968 .322 114-96 1-33 229.95 229.89 5-30 153-37 76.70 40 3-60 4.320 .440 119.96 51 239-94 239-87 6.03 I6O.O5 80.04 50 3-75 4-688 .562 124-95 .70 249-93 249-83 6.8l 166.73 83.39 60 3-90 5.070 .600 129-94 .92 259-91 259-80 7.66 I73-4I 86.73 70 4-05 5.468 .822 134-93 .15 269.89 269.76 8.58 I80.09 90.08 80 4.20 5-880 .960 139 92 39 279-87 279-71 9-55 186.77 93-43 90 4-35 6.308 .102 144.91 2.66 289.85 289.65 10. 61 193.46 96.78 300 4-50 6.750 2.25O 149.90 2.94 299-82 299-59 11-77 200.15 100.13 THE SPIRAL TABLE XVII. SPIRAL FUNCTIONS FOR A CHANGE OF 2.0 PER 100 FEET. SUITABLE FOR A SPEED OF 48 MILES AN HOUR OR LESS, AND CURVES OF 6.0 OR LESS L D A A Z C X Y U V 10 O.2O O.OIO 0.003 5.00 o.oo 10. OO IO.OO o.oo 6.97 3-33 20 .40 .040 .013 10. OO .00 20.00 20.00 .00 13-33 6.67 30 .60 .090 .030 15.00 .00 30.00 30.00 .02 20.00 10.00 40 .80 .160 .053 20.00 .01 40.00 40.00 .04 26.67 13-33 50 I.OO .250 .083 25.00 .02 50.00 50.00 .07 33-33 16.67 60 .20 .360 .120 30.00 03 60.00 60.00 .13 40.00 20.00 70 .40 .490 .163 35-00 .05 70.00 70.00 .20 46.67 23-33 So .60 .640 .213 40.00 .07 80.00 80.00 .30 53-33 26.67 90 .80 .810 .270 45-00 .11 90.00 90.00 .42 60.00 30.00 100 2.OO I.OOO .333 50.00 14 IOO.OO IOO.OO .58 66.67 33-33 10 .20 I.2IO 403 55-00 19 IIO.OO 109.99 77 73-33 36.67 20 .40 1.440 .480 59-99 .25 I2O.OO 119.99 I.OO 80.00 40.00 30 .60 1.690 .563 64.99 .34 129.99 129.99 1.28 86.67 43-34 40 .80 1.960 .653 69-99 40 139.99 139.98 1. 60 93-34 46.67 So 3.00 2.250 750 74-99 49 149.99 149.98 1.96 IOO.OI 50.01 60 .20 2.560 .853 79.98 .60 159.99 159.97 2.38 106.68 53-34 70 .40 2.890 .963 84.98 71 169.98 169.96 2.86 113.35 56.68 80 .60 3.240 1.080 89.98 .85 179.97 179.94 3.39 120. O2 ' 60.02 90 .80 3.6lO 1. 203 94-97 I.OO 189.97 189.93 3.99 126.69 63.36 200 4.00 4.000 1.333 99.96 1.16 199.96 199.90 4.65 133-37 66.70 10 .20 4.410 1.470 104.96 1.35 209.95 209.88 5.38 140.04 70.04 20 .40 4.840 1.613 109.95 1.55 219.93 219.84 6.19 146.72 73.38 30 .60 5.290 1.763 114.94 1.77 229.91 229.80 7.07 153-40 76.73 40 .80 5.760 1.920 "9-93 2.01 239.89 239.76 8.03 l6o.09 80.08 50 S.oo 6.25O 2.083 124.91 2.27 249.87 249.70 9.08 166.77 83.43 60 .20 6.760 2.253 129.90 2.55 259.84 259.64 10.21 173.46 86.78 70 .40 7.290 2.430 134-88 2.86 269.81 269.57 11.44 ISO.IS 90.14 80 .60 7.840 2.613 139-86 3.19 279.77 279.48 12.75 186.85 93.50 90 .80 8.410 2.802 144.84 3-54 289.73 289.38 14.17 193-55 96.87 300 6.00 9-OOO 2.999 I49-8I 3-92 299.68 299.26 15-68 200.26 100.24 RAILROAD FIELD MANUAL TABLE XVIII. SPIRAL FUNCTIONS FOR A CHANGE OF 2.5 PER 100 FEET. SUITABLE FOR SPEEDS OF 45 MILES AN HOUR OR LESS, AND CURVES OF 7.5 OR LESS L D A A Z c X Y u V 10 0.25 O.OI2 0.004 S.oo o.oo 10.00 10.00 0.00 6.67 3-33 20 So .050 .017 10.00 .00 20.00 20.00 .01 13-33 6.67 30 75 .112 .037 15.00 .00 30.00 30.00 .02 20.00 IO.OO 40 1. 00 .200 .067 20.00 .01 40.00 40.00 .05 26.67 13.33 So .25 -312 .104 25.00 .02 50.00 50.00 .09 33-33 '16.67 60 So 450 ISO 30.00 .04 60.00 60.00 .16 40.00 20.00 70 75 .612 .204 35.00 .06 70.00 70.00 .21 46.67 23.33 80 2.0O .800 .267 40.00 .09, 80.00 80.00 37 53-33 26.67 90 .25 1. 012 337 45-00 .13 90.00 90.00 .53 60.00 30.00 100 .50 1.250 417 50.00 .18 IOO.OO 100.00 .73 66.67 33.34 10 .75 I.5I2 .504 54-99 .24 IIO.OO 109.99 97 73-34 36.67 20 3-00 1.800 .600 59-99 31 I2O.OO 119.99 1.26 80.00 40.00 30 25 2. 112 .704 64.99 40 129.99 129.98 i. 60 86.67 43.34 40 50 2.450 .817 69.98 50 139.99 139.97 2.OO 93-34 46.67 50 75 2.812 937 74.98 .61 149.98 149.96 2.45 IOO.OI 50.01 60 4.00 3.200 1.067 79-98 74 159.98 159.95 2.98 106.68 53.35 70 .25 3.612 1.204 84-97 .89 169.97 169.93 3-57 113.36 56.69 80 50 4.050 1.350 89.96 1. 06 179.96 179.91 4.24 120.03 60.03 90 .75 4.512 1.504 94-95 1.25 189.95 189.88 4-99 126.71 63.37 200 S.oo 5.000 1.667 99-94 1.45 199.93 199.85 5.8i 133.39 66.72 10 25 5.512 1-837 104-93 1.68 209.91 209.81 6-73 140.07 70.06 20 .50 6.050 2.017 109.92 1-93 219.89 219.76 7-74 146.76 73.41 30 75 6.612 2.204 114.90 2.21 229.87 229.70 8.84 153.44 76.76 40 6.00 7.200 2.400 119.89 2.51 239.83 239.62 10.04 160.13 80.12 50 .25 7.812 2.604 124.86 2.8 4 249.80 249.54 11.35 166.83 83.48 60 .50 8.450 2.816 129.84 3-19 259.75 259.44 12.76 173.53 86.85 70 75 9.112 3-036 134-81 3-57 269.70 269.32 14.29 180.24 90.22 80 7.00 9.800 3.266 139.78 3.98 279.64 279.19 15-93 186.96 93.60 90 .25 10.512 3.503 144.75 4.42 289.57 289.03 17-69 193.68 96.98 300 7-50 11.250 3-749 149-71 4-89 299.49 298.85 19.58 200.41 100.37 THE SPIRAL TABLE XIX. SPIRAL FUNCTIONS FOR A CHANGE OF 3.0 PER 100 FEET. SUITABLE FOR SPEEDS OF 41 MILES AN HOUR OR LESS, AND CURVES OF 9.0 OR LESS L D A A Z C X Y u V 10 o 0.30 0.015 0.005 S.oo 0.00 10.00 10.00 0.00 6.67 3-33 20 .60 .060 .020 10.00 .00 20.00 20.00 .01 13-33 6.67 30 .90 .135 .045 15.00 .01 30.00 30.00 .02 20.OO IO.OO 40 1.20 .240 .080 20.00 .01 40.00 40.00 .06 26.67 13.33 So 50 375 -125 25.OO .03 50.00 5O.OO .11 33-33 16.67 60 .80 540 .180 30.OO .05 60.00 60.00 -19 40.00 20.00 70 2.10 735 -245 35-00 .07 70.00 7O.OO 30 46.67 23 33 80 .40 .960 -320 40.00 .11 80.00 80.00 45 53-34 26.67 90 70 1. 215 -405 45-00 .16 90.00 9O.OO .64 60.00 30.00 100 3-00 1.500 .500 49-99 .22 100.00 99-99 .87 66.67 33-34 10 30 1.815 .605 54-99 .29 109.99 109.99 1.16 73-34 36.67 20 .60 2.160 .720 59-99 .36 119.99 119.98 I. Si 80.01 40.01 30 .90 2.535 .845 64.98 .48 129.99 129.98 1.92 86.67 43-34 40 4.20 2.940 .980 69-98 .60 139-98 139.96 2.39 93-35 46.68 So So 3-375 1.125 74-97 74 149.98 149-95 2.94 100.02 50.02 60 .80 3.840 1.280 79-97 .89 159-97 159-93 3-57 106.69 53.36 70 5.10 4-335 1-445 84.96 1.07 169.96 169.90 4.28 113-37 56.70 80 40 4.860 1.620 89-95 1.27 179-94 I79-87 5.08 120.05 60.04 90 .70 5-415 1.805 94-93 1.49 189.92 189.83 5-98 126.73 63.39 200 6.00 6.000 2.000 99.92 1-74 199-90 199.78 6.98 133-41 66.74 10 30 6.615 2.205 104.90 2.02 209.88 209.72 8.06 I4O.IO 70.09 20 .60 7.260 2.420 109.88 2.32 219.85 219.65 9.26 146.79 73-45 30 .90 7-935 2.645 114.86 2.65 229.81 229.56 10.59 153-49 76.81 40 7.20 8.640 2.879 119-83 3.01 239.76 239.46 12.04 160.19 80.17 So 50 9-375 3-124 124.80 3-40 249-71 249-34 13-60 166.90 83-55 60 .80 10.140 3-379 129.77 3-82 259-64 259-19 15-30 173-62 86.93 70 8.10 10.935 3.644 134-73 4.28 269.57 269.03 17.11 180.35 90.32 80 .40 11.760 3.919 139.69 4-77 279.48 278.83 19.08 187.08 93-71 90 70 12.615 4-203 144.64 5.30 289.39 288.61 21.20 193.83 97.12 300 9.00 13-500 4.498 149.58 5-86 298.70 298.35 23-47 200.59 100.54 66 RAILROAD FIELD MANUAL TABLE XX. SPIRAL FUNCTIONS FOR A CHANGE OF 4.0 PER 100 FEET. SUITABLE FOR SPEEDS OF 35 MILES AN HOUR OR LESS, OR CURVES OF 10.0 OR LESS L D A A Z o C X Y u V 10 0.40 o O.O20 0.007 5.00 o.oo IO.OO IO.OO o.oo 6.67 3-33 20 .80 .080 .027 IO.OO .00 20.00 20.00 .01 13-33 6.67 30 1.20 .180 .060 15.00 .01 30.00 30.00 .03 20.00 IO.OO 40 .60 .320 .107 20.00 .02 40.00 40.00 .07 26.67 13.33 50 2.OO .500 .167 25.00 .04 50.00 50.00 -IS 33-33 16.67 60 .40 .720 .240 30.00 .06 60.00 60.00 -25 40.00 20.00 70 .80 .980 327 35.00 .10 70.00 70.00 .40 46.67 23.33 80 3-20 1.280 .427 39.99 .15 80.00 80.00 .60 53-33 26.67 90 .60 1.620 540 44.99 .21 90.00 89.99 .85 60.00 30.00 100 4.00 2.OOO .667 49.99 .29 99-99 99.99 1.16 66.67 33.34 10 40 2.420 .807 54.98 39 109-99 109.98 1.55 73-33 36.67 20 .80 2.880 .060 59.98 -50 119.99 119.97 1.84 80.01 40.01 30 5-20 3.380 1.127 64.97 .64 129.98 129.95 2.56 86.68 43.35 40 .60 3.920 307 69.96 .80 139-97 139.93 3-19 93.36 46.69 50 6.00 4.500 .500 74-95 .98 149.96 149.91 3-93 100.03 50.03 60 40 5.120 .707 79-94 1. 19 159-94 159-87 4-76 106.71 53.37 70 .80 5.780 .927 84.92 1.43 169.92 169.83 5-71 113.40 56.72 80 7.20 6.480 .160 89.90 1.77 179.90 179.77 6.78 120.08 60.08 90 .60 7.220 .407 94-88 1.99 189.87 189.70 7-97 126.77 63.43 200 8.00 8.000 2.667 99-86 2.32 199.83 199.61 9-30 133-47 66.79 10 .40 8.820 2.939 104.83 2.68 209.78 209.51 10.76 140.18 70. ie 20 .80 9.680 3.222 109.79 3-09 219.72 219.38 12.38 146.89 73-54 30 9.20 10.580 3.526 114-75 3-52 229.66 229.22 14.11 I53.6I 76.92 40 .60 11.520 3.839 119.70 4.00 239-57 239.03 6.03 160.34 80.31 250 10.00 12.500 4-165 124.65 4-52 249.48 248.82 8.12 167.09 83.72 THE SPIRAL 6 7 TABLE XXL SPIRAL FUNCTIONS FOR A CHANGE OF 5.0 PER 100 FEET. SUITABLE FOR SPEEDS OF 32 MILES AN HOUR OR LESS, OR CURVES OF 12.5 OR LESS L D A A Z o C X Y u V 10 0.50 0.025 0.008 5.00 o.oo 10.00 10.00 0.00 6.67 3-33 20 I.OO .100 .033 IO.OO OO 20.00 20.00 .01 13.33 6.67 30 So .225 .075 15.00 .01 30.00 30.00 .04 20.00 IO.OO 40 3.00 .400 .133 20.00 .02 40.00 40.00 .09 26.67 13.33 So 50 .625 .208 25.00 .05 50.00 50.00 .18 33.33 16.67 60 3-00 .900 .300 30.00 .08 60.00 60.00 .31 40.00 20.00 70 50 1.225 .408 34.99 .13 70.00 70.00 .50 46.67 23-33 80 4.00 1. 600 533 39.99 .19 80.00 79-99 .74 53-34 26.67 90 SO 2.025 .675 44.99 .27 90.00 89-99 1. 06 60. OQ 3O.OO 100 S.oo 2.500 .833 49.98 .36 99.99 99-98 1.45 66.67 33-34 10 SO 3-025 .008 54.97 .48 109.99 109.97 1-93 73-34 36.68 20 6.00 3.600 .200 59.97 .63 119.98 H9.95 2.51 80.02 40.O2 30 So 4-225 .408 64.95 .80 129.97 129.93 3-19 86.69 43.36 40 7.00 4.900 .633 69.94 .00 139.96 139.90 3-99 93-37 46.70 So So 5-625 .875 74.92 .20 149.94 149-86 4.90 100.05 50.05 60 8.00 6.400 2.133 79.90 .48 159.91 159-80 5-94 106.74 53-40 7o 50 7.225 2.408 84.88 .78 169.88 169.73 7-13 113-43 56.75 80 9.00 8.100 2.699 89.85 .11 179-84 179-64 8.46 120.13 6b. 12 90 50 9-025 3.008 94.82 .48 189.79 189.53 9-95 126.83 63.48 200 10.00 IO.OOO 3-332 99.78 2.90 199.73 199.40 ii. 61 133-55 66.86 10 .50 11.025 3.674 104.73 3.35 209 . 66 209.23 13.41 140.27 70.25 20 II. OO 12.100 4.032 109.67 3.82 219-57 219.04 15.42 147-01 73.65 30 .50 13.225 4.406 114.61 4.40 229.46 228.79 17.62 153.76 77-06 40 12. OO 14.400 4-797 119.54 5-99 239-34 238.50 20.02 160.54 80.49 260 12.50 15.625 5-205 124.46 5.64 249.18 248.16 22.62 167.32 83.94 68 RAILROAD FIELD MANUAL TABLE XXII. SPIRAL FUNCTIONS FOR A CHANGE OF 7.5 PER 100 FEET. SUITABLE FOR A SPEED OF 26 MILES AN HOUR OR LESS, OR A CURVE OF 18.0 OR LESS L D A A Z C X Y u V 10 0.75 0.0375 O.OI2 5.00 o.oo IO.OO IO.OO o.oo 6.67 3-33 20 1.50 .1500 .050 10. OO .00 20.00 20.00 .02 13-33 6.67 30 2.25 3375 .113 15.00 .01 30.00 30.00 .06 20.00 10.00 40 3 oo .6000 .200 20.00 .03 40.00 40.00 .14 26.67 13.33 50 3-75 9375 313 25.0O .07 50.00 50.00 .27 33-33 16.67 60 4-50 1.3500 450 29-99 .12 60.00 60.00 .47 40.00 20.00 70 5-25 1-8375 .613 34-99 .19 70.00 69.99 75 46.67 23-34 80 6.00 2.4000 .800 39-98 .28 79-99 79-99 1. 12 53-34 26.67 90 6.75 3-0375 1.013 44-97 .40 89-99 89-97 1.59 60. 01 30.01 100 7-50 3-7500 1.250 49.96 .54 99.98 99.96 2.18 66.68 33-35 10 8.25 4-5375 I.5I3 54-94 7i 109.97 109.93 2.90 73.36 36.69 20 9.00 5.4000 I.Soo 59-92 94 "9-95 119-89 3-77 80.04 40.03 30 9-75 6.3375 2. 113 64.90 1. 19 129-93 129.84 4-78 86.72 43-38 40 10.50 7-3500 2.450 69.86 1.49 139-90 139-77 5-98 93-41 46-74 50 11.25 8.4375 2.812 74-83 1.83 149-86 149-68 7-35 IOO.II 50.10 60 12. OO 9.6000 3.199 79.78 2.22 159-80 159-55 8.92 106.83 53-48 70 12.75 10.8375 3.612 84-73 2.66 169-73 169.40 10.67 H3-55 56.86 80 13-50 12.1500 4.049 89.66 3-15 179-64 179-20 12.66 120.29 60.26 90 14-25 13-5375 4.511 94-59 3-51 189.54 188.95 14-89 127.04 63.68 200 15.00 15.0000 4-997 99-50 4-32 199.40 198.64 17-37 133 82 67.11 10 15-75 16.5375 5.509 104.40 4-99 209.23 208.27 19.86 140.62 70-57 20 16.50 18.1500 6.045 109.27 5-73 219.03 217.82 22.72 147-45 74-05 30 17-25 I9-8375 6.607 114-13 6.54 228.79 227 . 28 25-91 I54-3I 77.56 240 18.00 21.6000 7.192 118.97 7-41 238.51 236.63 30.01 161.22 81.11 THE SPIRAL 6 9 TABLE XXIII. SPIRAL FUNCTIONS FOR A CHANGE OF 10.0 PER 100 FEET. SUITABLE FOR SPEEDS OF 22 MILES AN HOUR OR LESS, OR CURVES OF 25.0 OR LESS L D A A Z o C X Y U V 10 1. 00 0.050 o O.OI7 S.oo o.oo 10.00 IO.OO o.oo 6.67 3-33 20 2.00 .200 .067 10.00 .01 20.00 20.00 .02 13-33 6.67 30 3.00 450 ISO 15.00 .02 30.00 30. co .08 20.00 10.00 40 4.00 .800 .26 7 20.00 .05 40.00 40.00 .19 26.67 13 33 So 5.00 1.250 417 24-99 .09 50.00 50.00 .36 33-33 16.67 60 6.00 1.800 .600 29.98 .16 60.00 59.99 .63 40.00 20.00 70 7.00 2.450 .817 34-98 .25 69.99 69.99 .98 46.67 23-34 80 8.00 3.200 I.O67 39-96 37 79-99 79.97 1.49 53-34 26.67 90 9.00 4.050 1-350 44-95 53 89.98 89.96 2.12 60.02 30.OI 100 IO.OO 5.000 1.667 49-92 72 99-97 99.92 2.91 66.69 33.36 10 II. OO 6.050 2.OI7 54.89 98 109-95 109.88 3-87 73-38 36.71 20 12. CO 7.200 2.400 59-86 1.25 119 92 119.81 5-02 80.07 40.06 30 13-00 8.450 2.816 64.81 1.59 129.88 129.72 6.38 86.77 43-42 40 14.00 9.800 3-266 69.76 1.98 139-82 139.59 7.97 93.48 46.80 50 15-00 11.250 3-749 74.69 2-43 149-74 149.43 9.78 IO0.2O 50.19 60 16.00 I2.80C 4-265 79.6l 2-95 159.65 159.21 11.87 106.95 53-59 70 17.00 14.450 4-814 84.52 3-53 169.53 168.93 14.21 113.72 57-01 80 18.00 16.200 5.396 89.40 4.18 179-37 178.57 16.87 120.45 60.47 90 19.00 18.050 6.012 94.26 4-91 189.18 188.13 19.78 127-34 63.94 200 20.00 20.000 6.660 99-H 5-71 198.93 197.59 23.07 134-20 67.46 10 21.00 22.050 7.341 103.92 6.69 208 . 64 206.93 26.60 I4I.II 71.01 20 22.OO 24.200 8.055 108.71 7-56 218.28 216.13 30.59 148.07 74-62 30 23.00 26.450 8.802 113.46 8.SI 227.86 225.17 34-77 155.09 78.27 40 24.00 28.800 9-58o 118.18 9-76 237-35 234.04 39-50 I62.I8 82.00 250 25.00 31.250 10.392 122.86 10.99 246.75 242.70 . 44.67 169.36 85-79 CHAPTER III LOGARITHMS AND TRIGONOMETRIC FUNCTIONS TABLE XXIV. COMMON LOGARITHMS OF NUMBERS USE OF THE TABLE OF LOGARITHMS THE logarithm of a number is the exponent denoting the power to which some fixed base number must be raised to equal the number whose logarithm is considered. There are two base numbers in use, but almost all operations are performed by one system, known as Com- mon Logarithms, in which the base is 10. The logarithm of I in any system is o because any number raised to the o power is I. Thus _ = i-i = $0= i. The logarithm of 10 in the common system is i; of 100, 2; of 1000, 3; etc., since lo 1 = 10, io 2 = 100, io 3 = 1000, etc. Numbers between i and io, io and 100, 100 and 1000, etc., will have fractional logarithms. Thus, the logarithm of 8 is 0.90309; of 13 is 1.11394; of 126 is 2.10037, etc., since lo - 90309 = 8, jo 1 - 11394 = 13, io 2 - 10037 = 126, etc. The fraction is called the mantissa and is always the same for the same sequence of figures and is incommensurable, being given to 3, 5, 6, 7> or io decimal places, according to the precision required. Thus log 1.263 = 0.10140; log 12.63 = 1.10140; and log 126.3 = 2.10140, all of 5 decimal places. The whole number, called the charac- teristic, varies. The characteristic is always I less than the number of digits to the left of the decimal point of the number whose logarithm is being found. In the tables only the mantissas are given. The whole number must be known by the computer from the number of digits in the number whose logarithm is wanted. To find the logarithm of a number look in the table for the mantissa and prefix the proper characteristic, determined from the number of digits to the left of the decimal point in the number in question. Thus, on page 76, line 27, we find the mantissa for the sequence 1263 in the fifth column to be 0.10140; then for 1.263 the log is 0.10140; for 12.63 and 126.3 as given above. 70 LOGARITHMS AND TRIGONOMETRIC FUNCTIONS 71 In the table the first three figures of a four-figure number appear in the first column, the fourth figure at the heads of the 10 numbered col- umns. The first two figures of the mantissa appear in the second column, the last three in the column under the fourth digit of the num- ber whose logarithm is sought. If the first two figures change in going across the page, that fact is indicated by an asterisk; which, to economize space, stands for the first two figures of the second column in the line below the asterisk. Thus, the logarithm of 1445 (see p. 76) is 3-15987, while that of 1446 is 3.16017. For numbers of 3 figures or less, the mantissa is found in the column headed o, since the mantissas for I, 10, 100, or for 2, 20, 200, or for 15, 150, 1500, etc., are respectively the same. The logarithm of a number of more than four figures is found as in- dicated in the following example: What is the log of 382.568? On page 81, line 33, columns 2 and 7, find log 382.5 to be 2.58263. Note that log 382.6 is 2.59274, a difference of u in the log for a difference of I in the last place of the sequence in the number. Assuming that the log increases in proportion to the number, the log of 382.568 may be said to be T %V of n larger than log 382.5 since 382.568 is ffa of I in the last place (the place of the 5) greater than 382.5. To facilitate this computation, a table of proportional parts is found alongside the log table. Thus, in the case just given, for a difference of n we find, from the table of proportional parts, that an increase of 0.6 of one in the last place of the number makes a difference in the log of 6.6, an increase of 0.08 in the number increases the log by 0.88, found by moving the dec- imal point i digit to the left in the tabular value for 0.8, so that an in- crease of 0.68 of i in the last place increases the log of 382.5 by 6.6 + 0.88, or 7.48; or, since we make the fifth figure the nearest whole num- ber, we find log 382.568 = 2.58263 + 7 = 2.58270. To find the number corresponding to a given logarithm we reverse the process. Thus, the number corresponding to the logarithm 2.58270 is found as follows: The next smaller mantissa 130.58263 for the sequence 3825 and it is smaller than 0.58270 by 7. The difference for a whole unit in the last place is 1 1 and in the table of proportional parts under 1 1 find 6.6, corresponding to O.6 of I in the last place of the number, as the next smaller difference to 7, leaving still a difference of 7 6.6 = 0.4 to be used, which, by moving the decimal point one digit to the left, is found to correspond most nearly to 0.04 of i in the last place of the number. Therefore, the whole number is 382.564. But we found that 2.58270 is the logarithm of 382.568. We see by this that this table cannot be depended on to give more than a sequence of 5 significant 72 RAILROAD FIELD MANUAL figures correctly. This is true of all tables of logarithms: that as many significant figures in sequence may be correctly determined as there are decimal places in the tabular mantissas. A 5-place table gives five significant figures, a 6-place table, six significant figures, etc. To multiply a by & we add the logarithms of a and b, written log a and log b, and find the number corresponding to the sum. To divide a by b, we subtract log b from log a and find the number corresponding to the difference. It may often occur in computations that the fifth place in a resulting logarithm, found by adding two or more logarithms, may be in error by one unit, thus making the quantity determined certain to one less significant figure than the number of places in the logarithm. To find the product a?b 3 . Find the logarithm of a and multiply it by 2; find the logarithm of b and multiply it by 3; add the two results for the logarithm of the product and find the corresponding number. Thus: What is the product 1.2* X 4-3 3 ? p. 76, line 21, col. 2, log 1.2 = 0.07918 X 2 = 0.15836 p. 82, line 31, col. 2, log 4.3 = 0.63347 X 3 = 1.90041 p. 76, line 15, col. 6 and 7 = 2.05877 = log 114.49. Ans. This example, checked by a 7-place table, gives the same result to the fifth significant figure even though both logarithms were multiplied and then added. The following example, worked by 5-place and 7-place logarithms, shows the uncertainty of the last figure in computations made by any set of tables, by showing the uncertainty of the fifth figure in the com- putations made with the 5-place tables. Required the product 1.652 X I.8 3 . 5-Place Log 1.65 = 0.21748 X 2 = 0.43496 Log 1.8 = 0.25527 X 3 = 0.76581 1.20077 = log 15. 877. Ans. 7-Place , Log 1.65 = 0.2174839 X 2 = 0.4349678 Log 1.8 = 0.2552725 X 3 = 0.7658175 1.2007853 = log 15-8776. Ans. Thus 15.878 is nearer right than 15.877, although the error of the latter is only about 0.6 of one in the last place and the error of the former is LOGARITHMS AND TRIGONOMETRIC FUNCTIONS 73 about 0.4 of one in the last place. Greater differences can occur so that the fifth place is not certain within I unit. Logarithms oj jr actions. The logarithm of I = | = io 1-1 = 10 is o.ooooo. Similarly the logarithm of o.i = ^ = lo " 1 is i.ooooo; the logarithm of 3 foj is 2; of T ^u is 3; etc. Therefore, if the sequence is all fractional the characteristic is minus and with a nu- merical value expressed by the number of the place on the right of the decimal point in which the first significant figure appears. Thus the logarithm of 0.00126 is 3 -f 0.10037, usually written 3.10037. The mantissa is not minus, only the characteristic is minus. This some- times gives the beginner trouble, but need not if he remembers that the mantissa is always plus, while the characteristic is plus or minus accord- ing as the number corresponding is equal to or greater than I or is less than i, and that in general the two must be treated separately. The following examples will make the use of the signs clear: Required the product 43.0 X 0.43. Log 43 =1.63347 1-63347 Log 0.43 = 1.63347 0.63347 ~ I Log 18.49 = 1-26694 2.26694 I = 1.26694 Required the product 43.0 X O.43 2 . Log 43.0 = 1.63347 Log 0.432 = 1.63347 X 2 Log 0.432 = 1.26694 = 2 + 1.26694 = 1-26694 Log 7.951 = 0.90041 Required the product 43.0 X 0.43*. Log 0.43* = 1.63347 -4- 2 = 1.816735 found Log 43.0 = 1.63347 thus: Add 10 to the characteristic and Log 0.43* = 7.81674 subtract 10 before dividing by 2, getting Log 28.207 == 1.45021 9 ' 334 J ~ - =4.816735-5, or 1.816735. Instead of adding and subtracting 10 in the preceding example, only enough may be added and subtracted to make the negative character- istic divisible by the divisor. Thus: In the foregoing example add and subtract i, getting '- 6 3347 - I + I _5 + 1.63347 . ; Jl6yMt as before, without the necessity of considering the 4 and 5. If the 1/3 power has been required, the divisor would have been 3, and 2 should be added and subtracted getting 1 ' 6 3347 -2+2 = 3 +2.63347 3 3 = 1.877823 +. This work need not be written out in full in an exam- 74 RAILROAD FIELD MANUAL pie, as the computer can see the figures mentally. Thus, he knows the characteristic will be I so he writes this down and then imagines a 2, or whatever number is to be added, at the left of the decimal and at once writes down the quotient after the I. In handling very small numbers when the negative characteristic is large, it may be necessary to write out the operation to guard against error. In making the tables, when the remainder after the fifth figure was more than 0.5, the fifth figure was increased by i; when less than 0.5, the fifth figure was not changed. When the computer is finding a logarithm, and the operation brings a 5 in the place following the last place of the table (as the sixth place when using the tables of this book) and the 5 is followed by ciphers, a good rule to adopt is to make the last place the nearest even number. Thus in getting log 0.43' in the example above we add 5 in the sixth place. This rule will tend to balance inaccuracies in a long series of computations. To divide 3.68 X 4.21 we find (page 81) log 3.68 = 0.56585 (page 82) log 4.21 = 0.62428 Now to subtract, we may add and subtract 10 from the log 3.68, getting 10.56585 10 o. 62428 9.94157 - 10 = 1.94157 or, we may add the arithmetical complement of log 4.21 instead of subtracting the log 4.21. That is, we may multiply by instead of dividing by 4.21. The logarithm of is o.ooooo 0.62428, or 4.21 1.37572 and is called the arithmetical complement of the log 4.21. It is the logarithm of the reciprocal of 4.21. In a series of operations, where several factors are to be multiplied and the product divided by the product of several factors, the arithmetical complements of the divisor factors are used to avoid the double operation of two additions and one subtraction, since the arithmetical complements may be written down from the logs almost as readily as the logs themselves. Each digit of the mantissa of the log is subtracted from 9 except the last, which is subtracted from 10. The "characteristic is negative and one greater than the characteristic of the log when the log character- istic is positive; and is positive and one less than the characteristic of the log if that characteristic is negative. Thus: Log 8.364 is 0.92241 ; its complement is 1.07759 Log 83.64 is 1.92241 ; its complement is 2.07759 LOGARITHMS AND TRIGONOMETRIC FUNCTIONS 75 Log 836.4 is 2.92241 ; its complement is 3.07759 Log 0.8364 is 1.92241 ; its complement is 0.07759 Log 0.08364 is 2.92241; its complement is 1.07759 Instead of thinking of the characteristic by the rule above, the log may be subtracted from 10 instead of zero when a characteristic of 9 in the complement will correspond to T; 8 to 2, etc. And the computer can as readily think of 9 as indicating that the first significant figure of the result is in the first or tenths' place of the decimal; 8, that it is in the second or hundredths' place, etc., and this will cause no confusion unless very large quantities are being dealt with, so that the 8 or 9 might sometimes be a positive characteristic not standing in the place of a 2 or T; but this difficulty is not likely to arise in using 5- or 6-place tables, since such tables would not be used with such large numbers. When the first figures are small. When the first three figures of a number of five or more places are between 100 and no inclusive use pages 94 and 95, which give 7-place logarithms for five digits directly. N L O 1 2 3 4 5 6 7 8 9 P P 100 087 130 17 217 260 303 346 389 IOI IO2 103 104 i OS 1 06 107 108 109 432 860 oi 284 703 02 119 53i 938 03 342 743 475 903 326 745 1 60 572 979 383 782 945 368 787 202 612 423 822 988 410 828 243 653 *o6o 463 862 604 "030 45 870 284 694 *IOO 503 902 *072 494 912 325 735 *Hi 543 941 689 *H5 536 953 366 776 *i8i 583 981 578 995 407 816 *222 623 *02I 620 *o 3 6 449 857 *262 663 *o6o 817 *2 4 2 662 *0 7 8 490 8 9 8 *302 * 7 3 *IOO 44 43 42 I 4-4 4-3 4-2 2 8.8 8.6 8.4 3 13-2 12.9 12.6 4 17-6 17.2 16.8 522.0 21.5 21.0 626.4 25.8 25.2 730.8 30.1 29.4 835.2 34-4 33-6 110 04 139 179 218 258 297 336 376 415 454 493 III 112 "3 114 "5 116 117 118 119 532 922 05 308 690 06 070 446 819 07 1 88 555 57i 961 346 729 108 483 856 225 59i 610 999 385 767 145 893 262 628 650 *o 3 8 423 805 183 558 930 298 664 689 *o 7 7 461 843 221 595 967 335 700 727 *"S 500 88 1 258 633 *4 372 737 766 *I54 538 918 296 670 *0 4 I 408 773 80 S *I 9 2 576 956 333 707 *o 7 8 445 809 844 *2 3 I 6i 4 994 37i 744 *S 482 8 4 6 * 883 6 5 2 *0 3 2 408 781 *i5i 882 41 40 39 i 4.1 4-0 3-9 2 8.2 8.0 7.8 312.3 12.0 ii. 7 4 16.4 16.0 15.6 5 20.5 20.0 19.5 624.6 24.0 23.4 728.7 28.0 27.3 832.8 32.0 31.2 936.9 36.0 35.1 120 918 954 990 *02 7 uyy '171 '207 *2 4 3 121 122 123 124 125 126 127 128 129 08 279 636 991 09 342 691 10 037 380 721 ii 059 314 672 *026 377 726 072 415 755 093 350 707 *o6i 412 760 106 449 789 126 386 743 *0 9 6 447 795 140 483 823 1 60 422 ^778 482 830 175 517 857 193 458 814 *i6 7 517 864 209 551 890 227 493 849 *202 552 899 243 585 924 261 is *237 587 934 2 7 8 619 958 294 565 * 92 62! 968 312 653 992 327 600 * 9SS 656 "003 346 687 "025 361 38 37 36 i 3-8 3-7 36 2 7.6 7.4 7.2 3 11.4 ii. i 10.8 415-2 14.8 14.4 5 19.0 18.5 18.0 322.8 22.2 21.6 726.6 25.9 25.2 830.4 29.6 28.8 934-2 33-3 32.4 130 394 428 461 494 528 56i 594 628 661 694 131 132 133 134 135 136 137 138 139 727 12 057 385 710 13 033 354 672 988 14 301 760 090 418 743 066 386 * 7 4 333 793 123 450 775 098 418 * 735 364 826 156 483 808 130 450 767 *082 395 860 189 840 162 481 * 799 426 893 222 548 8 7 2 194 513 8 3 *i45 457 926 254 90S 226 545 862 *i 7 6 489 959 287 613 937 258 577 893 *208 520 992 320 6 4 6 969 290 609 925 *239 551 *02 4 352 678 *OOI 322 640 956 *270 582 35 34 33 i 3-5 3.4 3-3 2 7.0 6.8 6.6 3 10.5 10.2 9.9 414.0 13.6 13-2 517.5 17.0 16.5 52i.o 20.4 19.8 724.5 23.8 23.1 3 28 . o 27 . 2 26 . 4 ?3i-5 30.6 29.7 140 613 644 675 706 737 768 799 829 860 891 141 142 143 144 145 146 III 149 922 15 229 534 836 1 6 137 435 732 17 026 319 953 259 564 866 167 465 76! 056 348 983 290 594 897 197 495 791 085 377 *oi4 320 625 927 227 524 820 114 406 *045 35i 655 957 256 554 850 143 435 8i 685 987 286 584 879 173 464 *io6 412 715 *OI7 316 613 909 202 493 *i37 442 746 346 643 938 231 522 *i68 473 776 376 673 967 260 "198 503 806 406 702 997 289 58o 32 31 30 I 3-2 3-1 30 z 6.4 6.2 6.0 J 9-6 93 9-0 [ 12.8 12.4 12. > 16.0 15.5 15.0 ) 19.2 18.6 18.0 J 22.4 21.7 21.0 525.6 24.8 24.0 ) 28.8 27.9 27.0 ISO 17 609 863 667 696 725 754 782 811 8 4 o 869 N L 1 2 3 4 5 6 7 8 9 p p (76) N L O 1 2 3 4 5 6 7 8 9 PP 150 17 609 638 667 696 725 754 782 811 840 869 151 152 153 154 155 156 157 158 159 898 18 184 469 752 19 033 312 590 866 20 140 926 213 498 780 061 340 618 893 167 955 241 526 808 089 368 645 921 194 984 270 554 837 117 396 673 948 222 "013 298 583 865 145 424 700 976 249 *0 4 I 327 611 893 728 "003 276 *0 7 355 639 921 2OI 479 * 756 303 "099 384 667 949 229 507 783 "058 330 *I2 7 412 696 977 257 535 811 *o8 S 358 441 724 "005 285 562 838 *II2 385 I 2 3 4 5 6 7 8 9 2.9 2.8 5.8 5-6 8.7 8.4 II. 6 II. 2 14-5 14-0 17.4 16.8 20.3 19.6 23.2 22.4 26.1 25.2 160 412 439 466 493 520 548 575 602 629 656 97 96 161 162 163 164 165 1 66 167 1 68 169 683 952 21 219 484 748 22 OH 272 531 789 710 978 245 775 037 298 557 814 737 *oo S 272 537 801 063 324 583 840 ,763 *0 3 2 299 564 82 7 089 350 608 866 790 *059 325 590 854 "5 376 634 891 817 *o8 5 352 6 I7 880 141 401 660 917 844 *II2 378 643 906 I6 7 427 686 943 871 *I39 45 669 932 194 453 712 968 898 "165 43i 696 958 220 479 737 994 925 *I 9 2 458 722 985 246 505 2 3 4 5 6 7 8 9 2.7 2.6 5-4 5-2 8.1 7-8 10.8 10.4 13-5 13-0 16.2 15.6 18.9 18.2 21.6 20.8 24.3 23.4 170 23 045 070 096 121 147 172 198 223 249 274 OK 171 172 173 174 175 176 177 178 179 300 553 805 24 055 304 797 25 042 285 325 578 830 080 329 576 822 066 350 603 855 105 353 601 846 091 334 376 629 880 130 378 625 8 7 I 358 401 654 905 155 403 650 895 139 382 426 679 930 1 80 428 674 920 164 406 452 704 955 204 452 699 944 1 88 477 729 980 229 477 724 969 212 455 502 * 754 *00 5 254 502 748 993 237 479 528 779 030 279 527 773 *oi8 261 503 I 2.5 2 5-0 3 7-5 4 io. o 5 12.5 6 15.0 7 17-5 8 20. 9 22.5 180 527 55i 575 600 624 648 672 696 720 744 04, 23 181 182 183 !8 4 185 186 187 1 88 189 768 26 007 245 482 717 951 27 184 416 646 792 031 269 505 74i 975 207 439 669 816 055 293 529 764 998 231 462 692 840 079 316 553 788 *02I 254 485 715 864 102 340 576 8n *45 277 508 738 888 126 364 600 834 *o68 300 531 761 912 150 387 623 858 *0 9 I 323 554 784 935 174 411 647 881 346 577 807 959 198 435 670 * 9 5 370 600 830 983 221 458 694 928 *i6i 393 623 852 I 2 3 4 5 6 7 8 9 2.4 2.3 4.8 4.6 7.2 6.9 9.6 9.2 12.0 II. 5 14.4 13.8 16.8 16.1 19.2 18.4 21.6 20.7 190 875 898 921 944 967 989 *OI2 *035 *o S 8 *o8i 22 21 191 192 193 194 195 196 197 198 199 28 103 330 556 780 29 003 226 447 667 885 126 353 578 803 026 248 469 688 907 149 375 601 825 048 270 491 710 929 171 398 623 847 070 292 513 732 951 194 421 646 870 092 535 754 973 217 443 668 892 336 557 776 994 240 466 691 914 137 358 579 798 *oi6 262 488 713 937 159 380 601 820 *o 3 8 285 735 959 181 403 623 842 *o6o 307 533 758 981 203 425 645 863 *o8i I 2 3 4 5 6 7 8 9 2.2 2.1 4-4 4-2 6.6 6.3 8.8 8.4 II. O 10.5 13.2 12.6 15.4 14.7 17.6 16.8 19.8 18.9 200 30 103 I2 5 146 168 190 211 233 255 276 298 N L O 1 2 3 4 5 6 7 8 9 P P (77) N L 1 2 3 4 5 6 7 8 9 P P 2OO 201 202 203 204 205 206 207 208 209 2IO 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 23O 231 232 233 234 235 236 237 238 239 24O 241 242 243 244 245 246 247 248 249 250 N 30 103 125 146 1 68 W 600 814 *027 239 450 660 869 Q77 284 190 211 233 255 -276 298 5M 728 942 *I54 366 576 785 994 201 22 I 2. 2 4- 3 6. 4 8. 5 ii. 6 13- 7 IS- 8 17. 9 19. i 2 3 I 6 7 8 9 I 2 3 4 I 7 8 9 i 2 3 4 5 6 7 8 9 i 2 3 4 5 6 7 8 9 21 2 2.1 4 4-2 6 6.3 8 8.4 o 10.5 2 12.6 4 14-7 6 16.8 8 18.9 20 2.0 4.0 6.0 8.0 10.0 12. 14.0 16.0 1 8.0 19 3.1 $ 9-5 11.4 13-3 IS-2 17.1 18 1.8 3-6 5-4 7-2 9.0 10.8 12.6 14.4 16.2 17 i-7 3-4 5-1 6.8 8-5 10.2 ii-9 13-6 15-3 320 535 750 963 3i 175 387 597 806 32 015 222 428" 634 838 33 041 244 445 646 846 34044 341 557 771 984 197 408 618 827 035 363 578 792 *oo6 218 429 639 848 056 406 621 835 *048 260 47i 681 890 098 428 643 856 *o6 9 281 492 702 911 118 325 449 664 878 "091 302 513 723 931 139 346 471 685 899 *II2 323 534 744 952 1 60 492 707 920 *I33 345 555 765 973 181 243 263 469 675 879 082 284 486 686 885 084 305 366 387 593 797 *OOI 203 405 606 806 *oo 5 203 408 449 654 858 062 264 465 666 866 064 490 695 899 102 304 506 706 905 104 5io 715 919 122 325 526 726 925 124 S3 I 736 940 143 345 546 746 945 143 552 756 960 163 365 566 766 965 163 572 777 980 183 385 586 786 985 183 6l 3 818 *C2I 224 425 626 826 *025 223 242 439 635 830 35 025 218 411 603 793 984 262 459 655 850 044 238 430 622 813 *oo 3 282 301 321 34i 537 733 928 122 315 507 698 889 "078 36i 557 753 947 141 334 526 717 908 *097 380 577 772 967 1 60 353 545 736 927 *n6 400 42O 479 674 869 064 257 449 641 832 *02I 211 399 586 773 959 144 328 5ii 694 876 498 694 889 083 276 468 660 851 *040 518 713 9 08 102 295 488 679 870 *OS9 596 792 986 180 372 564 755 946 *I3S 616 811 *005 199 392 583 774 965 "'154 342 36 173 192 229 418 605 791 977 162 346 530 712 894 248 "436" 624 810 996 181 365 548 731 912 267 286 305 324 361 549 736 922 37 107 291 475 658 840 38 O2I 380 568 754 940 125 310 493 676 858 039 455 642 829 *oi4 199 383 566 749 93i 474 661 847 *033 218 401 585 767 949 493 680 866 *o S i 236 420 603 785 967 5ii 698 884 *070 254 438 621 803 985 530 717 903 *o88 273 457 639 822 "003 057 238 417 596 775 952 129 305 480 655 075 093 112 292 471 650 828 "005 182 358 533 707 130 3io 489 668 846 *02 3 199 375 550 724 148 ~3^8" 507 686 863 "041 217 393 568 742 915 1 66 346 525 703 881 *os8 23.5 410 585 759 .933. 8 184 202 382 56l 739 917 39094 270 445 620 220 399 578 757 934 in 287 463 637 256 435 614 792 970 146 322 498 672 274 453 632 810 987 164 340 515 690 364 543 721 899 *O76 252 428 602 777 95Q 9 39 794 811 829 846 863 881 898 L 1 2 3 4 5 6 7 P P (78) N L 1 2 3 4 5 6 7 8 9 F P 25O 39 794 811 829 846 863 881 898 915 933 950 251 252 253 254 255 256 257 258 259 967 40 140 312 483 654 824 993 41 162 330 985 157 329 500 671 841 *OIO 179 347 *002 175 346 518 688 858 *027 196 363 *OI9 192 364 535 70S 875 212 380 *0 37 209 552 722 892 *o6i 229 397 *054 226 398 569 739 909 '078 246 414 *0 7 I 243 415 586 756 926 *095 263 430 *o88 261 432 603 773 943 *m 280 447 *io6 278 449 620 790 960 *I28 296 464 *I2 3 295 466 637 807 976 *I45 313 481 i 2 3 4 S 6 7 8 9 18 1.8 3-6 S-4 7.2 9-0 10.8 12.6 14.4 16.2 260 497 5U S3i 547 564 58i 597 614 631 647 261 262 664 830 681 847 697 863 714 880 731 896 747 913 764 929 780 946 797 963 814 979 I 263 264 265 266 267 268 269 996 42 160 325 488 651 813 975 *OI2 177 341 504 667 830 991 *O29 193 357 521 684 846 *oo8 *045 210 374 537 700 862 *02 4 *062 226 390 553 716 878 243 406 570 732 * 8 P (80) N L O 1 2 3 4 5 6 7 8 9 F > P 35O 54407 419 432 444 456 469 481 494 506 5i8 351 352 353 354 355 356 357 358 359 53i 654 777 goo 55023 145 267 388 509 543 667 790 913 035 157 279 400 522 555 679 802 925 047 169 291 413 534 568 691 814 937 060 182 303 425 546 5 8o 704 827 949 072 194 315 437 558 593 716 839 962 084 206 328 449 570 605 728 851 974 096 218 340 461 582 617 741 864 986 108 230 352 473 594 630 753 876 998 121 242 364 485 606 642 765 888 *OII 133 255 376 497 618 I 2 3 4 I 7 13 1-3 2.6 3-9 IS 7-8 9-1 360 630 642 654 666 678 691 703 715 727 739 8 10.4 361 362 363 364 365 366 367 368 369 75i 871 991 56 no 229 348 467 585 703 763 883 *oo 3 122 241 360 478 597 714 775 895 *ois 134 253 372 490 608 726 787 907 *027 146 265 384 502 620 738 799 919 "038 158 277 396 514 632 750 811 93i *050 170 289 407 526 644 761 823 943 *062 182 301 419 538 656 773 835 955 *074 194 312 43i 549 667 785 847 967 *o86 205 324 443 56i 679 797 859 979 "098 217 336 455 573 691 808 z 2 3 4 12 1.2 it 4.8 6.0 370 820 832 844 855 867 879 891 902 914 926 6 7-2 371 372 373 374 375 376 377 378 379 937 57054 171 287 403 519 634 749 864 949 066 183 299 415 530 646 761 875 961 078 194 310 426 542 657 772 887 972 089 206 322 438 553 669 784 898 984 101 217 334 449 565 680 795 910 996 H3 229 345 461 576 692 807 921 *oo8 124 241 357 473 588 703 818 933 *oi9 136 252 368 484 600 715 830 944 *03i 148 264 380 496 6n 726 841 955 *043 159 276 392 507 623 738 852 967 I 9 i 2 3 4 9.6 10.8 11 i.i 2.2 3-3 380 978 990 *oi3 *035 *047 "058 "070 *o8i 4 4-4 381 382 383 384 385 386 387 388 389 58092 206 320 433 546 659 771 883 995 104 218 33i 444 557 670 782 894 *oo6 US 229 343 456 569 681 794 906 *oi 7 127 240 354 467 580 692 805 917 "028 138 252 365 478 59i 704 816 928 *O4O 149 263 377 490 602 715 827 939 *0 S I 161 274 388 501 614 726 838 950 *062 172 286 399 512 625 737 850 961 *073 184 297 410 524 636 749 861 973 "084 195 309 422 535 647 760 872 984 *095 5 6 I 9 5-5 6.6 1:1 9-9 10 39O 59 106 118 129 140 151 162 173 184 195 207 2 2.0 391 392 393 394 395 396 397 398 399 218 329 439 550 660 770 879 988 60 097 229 340 450 56i 671 780 890 999 108 240 35i 461 572 682 79i 901 *OIO 119 251 362 472 583 693 802 912 *02I 130 262 373 483 594 704 813 923 *032 141 273 384 494 605 715 824 934 *0 43 152 284 395 506 616 726 835 945 "054 163 295 406 517 627 737 846 956 *o6 S 173 306 417 528 638 748 857 966 "076 184 3i8 428 539 649 759 868 977 *o86 195 3 4 I 7 8 9 3-0 4.0 !' 6.0 7.0 8.0 9-0 4OO 60 206 217 228 239 249 260 271 282 293 304 N L ' 1 2 3 4 5 6 7 8 9 F > P (81) N L 1 2 3 4 5 6 7 8 9 P P 400 60 206 217 228 239 249 260 271 282' 293 304 401 402 403 404 405 406 407 408 409 314 423 53i 638 746 853 959 61 066 172 325 433 54i 649 756 863 970 077 183 336 444 552 660 767 874 981 087 1 94 347 455 563 670 778 885 991 098 204 358 466 574 68 1 788 895 *002 109 215 369 477 584 692 799 906 *oi 3 119 225 379 487 595 703 810 917 *02 3 130 236 390 498 606 713 821 927 *034 140 247 401 509 617 724 831 938 *04S 151 257 412 520 627 735 842 949 *055 162 268 i 11 i.i 2.2 3-3 410 278 289 300 310 321 331 342 352 363 374 4 4-4 411 412 413 414 415 416 417 418 419 384 490 595 700 805 909 62 014 118 221 395 500 606 711 8i5 920 024 128 232 405 511 616 721 826 930 034 138 242 416 521 627 73i 836 941 045 149 252 426 532 637 742 847 951 055 159 26 3 437 542 648 752 857 962 066 170 273 448 553 658 763 868 972 076 1 80 284 458 563 669 773 878 982 086 190 294 469 574 679 784 888 993 097 201 304 479 584 690 794 899 *oo 3 107 211 315 I 7 8 9 6.6 7-7 8.8 9-9 420 325 335 346 356 366 377 387 397 408 418 421 422 423 424 425 426 427 428 429 428 531 634 737 839 941 63 043 144 246 439 542 644 747 849 95i 053 155 256 449 552 655 757 859 961 063 165 266 459 562 665 767 870 972 073 175 276 469 572 675 778 880 982 08 3 185 286 480 583 685 788 890 992 094 195 296 490 593 696 798 900 *002 104 205 306 500 603 706 808 910 *OI2 114 215 317 Sii 613 716 818 921 *022 124 225 327 521 62 4 726 829 931 *033 134 236 337 I 2 3 4 ! 7 8 9 I.O 2.0 3-0 4.0 5- 6.0 7.0 8.0 9-0 430 347 357 367 377 387 397 407 417 428 438 431 432 433 434 435 436 437 438 439 448 548 649 749 849 949 64 048 147 246 458 558 659 759 859 959 058 157 256 468 568 669 769 869 969 068 167 266 478 579 679 779 879 979 078 177 276 488 589 689 789 889 988 088 187 286 498 599 699 799 899 998 098 197 296 508 60 9 709 80 9 909 *oo8 108 '207 306 518 619 719 819 919 *oi8 118 217 316 528 629 729 829 929 *028 128 227 326 538 639 739 839 939 "038 137 237 335 1 9 o.Q 1.8 2-7 3-6 4-5 440 345 355 365 375 385 395 404 414 424 434 6 5-4 441 442 443 444 445 446 447 448 449 444 542 640 738 836 933 65031 128 22^ 454 552 650 748 846 943 040 137 234 464 562 660 758 856 953 050 147 244 473 572 670 768 865 963 060 157 254 483 582 680 777 875 972 070 167 263 493 591 689 787 885 982 079 176 273 503 601 699 797 895 992 089 1 86 283 5i3 611 709 807 904 *002 099 196 292 523 621 719 816 914 *OII 108 205 302 532 631 729 826 924 *02I 118 215 312 1 9 6-3 7-2 8.1 450 65 321 33i 34i 350 360 369 379 389 398 408 N L 1 2 3 4 5 6 7 8 9 P P (82) N L I 8 9 P P 45O 65 321 331 34i 350 36o 369 379 389 398 408 4Si 452 453 454 455 456 457 458 459 418 5M 610 706 801 896 992 66087 181 427 523 619 715 811 906 *OOI 096 191 437 533 629 725 820 916 *OII 106 200 447 543 639 734 830 925 *020 us 210 456 552 648 744 839 935 *030 124 219 466 562 658 753 849 944 *039 134 229 475 571 667 763 858 954 *049 143 238 485 58i 677 772 868 963 658 153 247 495 591 686 782 877 973 068 162 257 504 600 696 792 887 982 077 172 266 i 2 3 / 10 I.O 2.0 3-0 460 276 285 295 304 314 323 332 342 35i 361 4 4.0 461 462 463 464 465 466 467 468 469 370 464 558 652 745 839 932 67 025 117 380 474 567 661 755 848 941 034 127 389 483 577 671 764 857 950 043 136 398 492 586 680 773 867 960 052 145 408 502 596 689 783 876 969 062 154 417 5ii 605 699 792 885 978 071 164 427 521 614 708 801 894 987 080 173 436 530 624 717 811 904 997 089 182 445 539 633 727 820 913 006 099 191 455 549 642 736 829 922 015 108 201 6 7 8 9 6.0 7.0 8.0 9-0 470 2IO 219 228 237 247 256 265 274 284 293 47i 472 473 474 475 476 477 478 479 302 394 486 578 669 761 852 943 68034 3ii 403 495 587 679 770 861 952 043 321 413 504 596 688 779 870 961 052 330 422 514 605 697 788 879 970 06 1 339 431 523 614 706 797 888 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504 592 679 767 854 161 249 338 425 513 601 688 775 862 170 25 ^ 346 434 522 609 697 784 871 179 267 355 443 531 618 70S 793 880 1 88 276 364 452 539 627 714 801 888 1 9 5-6 6.4 7.2 5OO 69897 906 914 923 932 940 949 958 966 975 N L 1 2 3 4 5 6 7 8 F P (83) N L O 1 2 3 4 5 6 7 8 9 F P 5OO 69897 906 914 923 932 940 949 958 966 975 501 502 503 504 SOS 506 507 508 509 984 70 070 157 243 329 415 SOT 586 672 992 079 165 252 338 424 509 595 680 *OOI 088 174 260 346 432 5i8 603 689 *OIO 096 183 269 355 441 526 612 697 *oi8 105 191 278 364 449 535 621 706 *02 7 114 2OO 286 372 458 544 629 7M *036 122 209 295 381 467 552 638 723 *044 131 217 303 389 475 56i 646 73i *o 5 3 140 226 312 398 484 569 655 740 *062 148 234 321 406 492 578 663 749 2 3 9 0.9 1.8 2-7 510 757 766 774 783 79i 800 808 817 825 834 4 3-6 5" 512 513 514 SIS Si6 Si? 5i8 519 842 927 71 012 9 6 181 265 349 433 517 851 935 020 105 189 273 357 441 525 859 944 029 H3 198 282 366 450 533 868 952 037 122 206 290 374 458 542 876 961 046 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1093 1094 1095 1096 1097 1098 1099 8248 038 2226 6202 039 0173 4141 8106 040 2066 6023 9977 8646 2624 6599 0570 4538 8502 2462 ^6419 9044 3022 6996 0967 4934 8898 2858 6814 *0 7 6 7 9442 3419 7393 1364 5331 9294 3254 7210 *Il62 9839 3817 7791 1761 5727 9690 3650 7605 *i557 4214 8188 2158 6124 *oo86 4045 8001 *i9S2 *o6 35 4612 8585 2554 6520 4441 8396 *2347 5009 8982 2951 6917 *o878 4837 5407 9379 3348 7313 5232 9187 *3i37 5804 9776 3745 * 77 9 5628 9582 *3532 1100 041 3927 4322 4716 Sin 5506 5900 6295 6690 7084 7479 N L 1 2 3 4 5 6 7 8 9 (95) 96 RAILROAD FIELD MANUAL TABLE XXV. LOGARITHMS OF SINES, TANGENTS, COSINES, AND COTANGENTS FOR EACH 0.01 OF THE QUADRANT In this table whenever the trigonometric function is fractional, as it always is for sines and cosines, except when it is I, the logarithms have been increased by 10 to avoid the negative characteristic. But where the function is greater than I , as are cotangents and tangents for certain parts of the circle, the logarithm is given without the 10. Thus in the column headed cot at the top of the page the logarithms are un- augmented; in all other columns they are augmented. This must be remembered in working with them. To find the square root of the sine of 26.32 we find on page 151, line 33, the log sin to be 9.64678, which must be divided by 2 to give the logarithm of the square root; the operation is performed thus: 2)9.64678 - 10 = 4-82339 - 5 Whence Vsin 26.32 = the number whose log is 1.82339 o r 0.66587+. But to find the square root of the cotangent of 26.32, page 151, line 33, we have 0.30569 -s- 2 = 0.152845, which corresponds to the number 1.4218+. The numbering of the pages. On each page there will be found four- degree numbers which are to be used as follows: the upper left-hand figure is used with the headings at the tops of the columns and the fractions running down the left side of the page. The lower right-hand figure is used with the headings at the bottoms of the columns, and frac- tions running up the right-hand side of the page. The upper right-hand figure is used with headings at the tops of the columns and fractions running up the right-hand side of the page. The lower left-hand figure is used with headings at the bottom of the page and fractions running down the left-hand side of the page. Generally stated the figures are used with the headings and fractions nearest them. Example. See pages 163-164. The log sin of 32.4 is found in the 2nd column, page 163, 4ist line, to be 9.72902, while the log sin of 122.4 is found in the 7th column of the same page and line to be 9.92651. The log sin of 147.4 is found in the 2nd column on page 164, 4ist line from the bottom, to be 9.73140, while the log sin 57.4 is found in the 7th column of the same page and line to be 9.92555. When the function or angle sought is not in the table, as when the log sin of 14.436 or the angle whose sine is 9.34220 is wanted, we must interpolate between the tabular quantities. Thus: LOGARITHMS AND TRIGONOMETRIC FUNCTIONS 97 (a) Log sin 14.43 ' ls found on page 127, line 44, to be 9.39654 with a tabular difference of 30 for a change of o.oi given in the column headed d. Then the addition for 6 tenths of 0.01 is found by multiplying 30 X 0.6 = 18, and adding to 9.39654 giving 9.39672 as the required sine. When the quantities are less simple the proportional part given in the right-hand column of the page may be used. If the angle is 14.4362 the table of proportional parts is convenient even for the simple difference of 30. Thus, from the values in the right-hand column headed "30" we find 18 opposite 6; moving the decimal point one place to the left, we find 0.6 opposite 2; adding we get ,000186, or .00019, to be added to the log sin 14.43 to give log sin 14.4362. The figures are uncertain beyond five places and may be a fraction of one in error in the fifth place. (6) For the angle whose log sine is 9.34220 we find on page 124 the next smaller log sin = 9.34212 = log sin 12.70; the difference is 9.34220 9.34212 = 8 in the last place. The tabular difference is 34. Looking in the column of proportional parts under 34 we find 6.8 next smaller number than 8 corresponding to 0.002 and leaving 1.2 unused; moving the decimal point one place to the left in the tabular quanti- ties, the next smaller and nearest number is 1.02 opposite 3. There- fore the angle is 12.7023. The last figure is uncertain. For fractional angles near o and 90, the differences are changing so rapidly that linear interpolation is not sufficiently exact. The right- hand columns of pages 99 to 104 show what to do. Thus to find the log sin 0.1246 we find (pages 76 and 99) Log. 0.1246 =1.09552 5 = 1.75812 7-33740 To find log cos and log cot we find log sin and tangents of comple- mentary angles. Thus, to get log cos 89.367, we get log sin 0.633. Conversely, to find the angle whose log sin is 7.33740, we refer to page 99, and find that the angle is between 0.12 and 0.13 and hence S = 1.75812 log sin = 7-3374Q io (page 76) log 0.1246 = 9-09552 - 10 = 1.09552. TABLE XXVI. LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS FOR EACH 0.02 OF THE QUADRANT The use of this table will be evident except the interpolation for small angles and for external secants for angles near 90. 98 RAILROAD FIELD MANUAL For small angles use the quantity V for versed sines and E for external secants as follows: Log vers a = 2 log a + V. Log exsec a. = 2 log a -f- E. Interpolate for V and E when necessary. Example. Required log vers 1.354. Log 1.354 = 0.13162 2 0.26324 Ffor 1.36 = 6.18270 Log vers 1-354 = 6.44594- _ log vers a V _ log exsec a E For external secants near 90 the interpolation is as follows: Log exsec A = log vers A log sin (90 A). 179< Sin d. Tan d. c. Cot Cos 00 100 5 01 6.24188 6.24188 3.75812 o.ooooo 99 oa 03 6.54291 6.71900 30103 17609 6.54291 6.71900 30103 17609 3.45709 3.28100 o.ooooo o.ooooo 98 97 J_ 1.758 .0 123 04 05 06 6.84394 6.94085 7.02003 12494 9691 7918 6695 6.84394 6.94085 7.02003 12494 9691 7918 6695 3-15606 3.05915 2.97997 o.ooooo o . ooooo 0.00000 96 95 94 .1 123 .2 124 .3 125 4 126 07 7.08698 7.08698 2.91302 o.ooooo 93 08 7-14497 5799 7-14497 5799 2.85503 o.ooooo 92 09 7.19612 5H5 .- 7.19612 5H5 2.80388 o.ooooo 9i 10 7.24188 7.24188 2.75812 o.ooooo 90 ii 7-28327 7.28327 2.71673 o.ooooo 89 12 13 7.32106 7.35582 3779 3476 7.32106 7.35582 3779 3476 2.67894 2.64418 o.ooooo o.ooooo 88 87 14 7.38800 7.388oi 2.61199 o.ooooo 86 IS 16 7-41797 7.44600 2997 2803 7.41797 7.44600 2996 2803 2677 2.58203 2.55400 o.ooooo o.ooooo 85 84 . | 17 18 7.47233 7-49715 2482 7-47233 7-49715 2482 2.52767 2.50285 o.ooooo o.ooooo 83 82 ' "S ah. g 19 7.52063 2348 7.52063 2348 2.47937 o.ooooo 81 o+ 8 20 7.54291 7.54291 2.45709 o.ooooo 80 7 2* 21 7.56410 7.56410 2.43590 o . ooooo 79 8 -3 rt 22 7.58430 2020 7.58430 2020 2.41570 o . ooooo 78 ^ | J3* 23 7.60360 1930 1840 7.60361 1931 1848 2.39639 o.ooooo 77 24 7.62209 7.62209 2.37791 o.ooooo 76 ' 4) "* ^ .S 25 7-63982 1773 7.63982 1773 2.36018 o.ooooo 75 S ^ + ^ 26 7.65685 1703 1630 7-65685 1703 1639 2.34315 0.00000 74 J % 27 7-67324 7.67324 2.32676 o.ooooo 73 P? 8 'to Q 28 7-68903 1579 7.68904 1580 2.31096 9.99999 72 c -2.2 5s 29 7.70427 1524 7.70428 1524 2.29572 9.99999 71 Jill 30 7.71900 1473 7.71900 2.28100 9.99999 70 u S "S 3i 7.73324 7.73324 2.26676 9-99999 69 15 "in o 1 ^ 32 7.74703 1379 7.74703 1379 2.25297 9-99999 68 f t 1 33 7.76039 1336 1206 7.76040 1337 1296 2.23960 9-99999 67 34 7-77335 7.77336 2.22664 9 99999 66 .S 35 7.78594 1259 7-78595 1259 2.21405 9-99999 65 1 36 7.79818 1224 7.79819 1224 2.20181 9-99999 64 37 7.81008 ' 7.81009 2.18991 9-99999 63 38 7.82166 1158 7.82167 1158 2.17833 9-99999 62 39 7-83294 1128 7-83295 2 . 16705 9-99999 61 40 7-84393 1099 7-84394 1099 2.15606 9-99999 60 41 42 7.85466 7.86512 1046 7-85467 7.86513 1046 2.14533 2.13487 9-99999 9-99999 59 58 T 43 7-87534 7-87535 2.12465 9-99999 57 44 7-88533 999 7.88534 2.11466 9-99999 56 1.758 45 7.89509 970 7.89510 970 2.10490 9-99999 55 .0 123 46 7.90463 954 7-90464 954 2.09536 9-99999 54 .1 122 47 7.91397 934 7.91398 934 2.08602 9-99999 53 .2 121 48 49 7.923H 7.93207 914 896 877 7.92313 7.93208 915 895 878 2.07687 2.06792 9-99998 9-99998 52 5i 3 "9 .4 "6 50 7.94084 7.94086 2.05914 9.99998 50 .5 H2 Cos d. Cot d. c. Tan Sin 90 (99) 179< Sin d. Tan d. c. Cot Cos 50 51 52 53 54 55 56 57 58 59 60 6i 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 92 93 94 95 96 97 98 99 100 7.94084 860 843 828 811 797 783 769 755 742 730 7i8 706 695 684 673 663 653 644 634 624 616 608 599 59i 583 575 568 56o 553 546 540 533 526 520 514 508 502 496 491 485 480 475 469 465 459 455 450 445 441 437 7.94086 860 843 828 811 797 783 769 755 742 730 7i8 707 695 684 673 663 653 644 634 625 616 607 599 591 583 576 567 56i 553 547 539 533 527 520 514 508 502 496 491 486 480 474 470 464 460 455 450 446 441 436 2.05914 9.99998 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 24 23 22 21 20 19 18 17 16 15 14 13 12 II 10 09 08 07 06 05 04 03 02 OI 00 b o oo ^ b 6, 1 To interpolate when angles are small: S i bo C b> 6, 1 !_ log sin n = log n .S; Ing tan n lr>g n T- 1 . M loga = logsina + S = logtana + r. M 00 & Xl M 00 00 7-94944 7.95787 7.96615 7.97426 7-98223 7.99006 7-99775 8.00530 8.01272 7.94946 7.95789 7.96617 7.97428 7.98225 7.99008 7-99777 8.00532 8.01274 2.05054 2.04211 2.03383 2.02572 2.01775 2.00992 2.00223 1.99468 1.98726 9-99998 9.99998 9-99998 9-99998 9-99998 9-99998 9-99998 9-99998 9-99998 8.O2OO2 8 . 02004 1.97996 9-99998 8 . 02720 8.03426 8.04121 8.04805 8.05478 8.06141 8.06794 8.07438 8.08072 8.02722 8.03429 8.04124 8.04808 8.05481 8.06144 8.06797 8.07441 8.08075 1.97278 1.96571 1.95876 I.95I92 I.945I9 1.93856 1.93203 1.92559 1.91925 9.99998 9-99997 9-99997 9-99997 9-99997 9-99997 9-99997 9-99997 9-99997 8.08696 8.08700 1.91300 9-99997 8.09312 8.09920 8.10519 S.iiiio 8.11693 8.12268 8.12836 8.13396 8.13949 8.09316 8.09923 8 . 10522 8.11113 8.11696 8.12272 8.12839 8.13400 8.13953 1.90684 1.90077 1.89478 1.88887 1.88304 1.87728 1.87161 1.86600 1.86047 9-99997 9-99997 9-99996 9.99996 9-99996 9.99996 9.99996 9^9996 9.99996 8.14495 8.14500 1.85500 9.99996 8.15035 8.15568 8.16094 8.16614 8.17128 8.17636 8.18138 8.18634 8.19125 8.IS039 8.15572 8.16099 8.16619 8.I7I33 8.17641 8.18143 8.18639 8.19130 1.84961 1.84428 1.83901 1-83381 1.82867 1.82359 1.81857 1.81361 1.80870 9.99996 9.99996 9-99995 9-99995 9-99995 9-99995 9-99995 9-99995 9-99995 8.19610 8.19616 1.80384 9-99995 8.20090 8.20565 8.21034 8.21499 8.21958 8.22413 8.22863 8.23308 8.23749 8.20096 8.20570 8.21040 8.21504 8.21964 8.22419 8.22869 8.23315 8.23756 1.79904 1-79430 1.78960 1.78496 1.78036 I.7758I I.77I3I 1.76685 1.76244 9-99995 9-99994 9-99994 9-99994 9-99994 9-99994 9-99994 9-99994 9-99994 8.24186 8.24192 i.758o8 9-99993 Cos d. Cot d.c. Tan Sin 90' (ioo) 178' Sin d. Tan d. c. Cot Cos 00 8.24186 8.24192 I.75&CS 9-90993 100 01 8.24618 8.24624 .75376 C- 99993 99 s 02 8.25045 427 8.25052 74948 9-99993 98 i 758 03 8.25469 424 420 8.25476 424 420 74524 9-99993 97 i.o 145 04 05 8.25889 8.26304 415 8.25896 8.26312 416 .74104 .73688 9-99993 9-99993 96 95 .1 149 .2 154 o6 8.26716 408 8.26723 408 .73277 9-99993 94 .3 160 07 8.27124 8.27131 .72869 9-99992 93 .4 166 08 8.27528 404 8.27535 404 .72465 9-99992 92 -5 172 09 8.27928 400 8.27936 401 .72064 9.99992 91 10 8.28324 8.28332 .71668 9-99992 90 ii 8.28717 8.28725 .71275 9-99992 89 12 13 8.29107 8.29493 390 386 382 8.29115 8 . 29501 39O 386 383 .70885 .70499 9-99992 9.99992 88 87 14 8.29875 8.29884 .70116 9-99991 86 15 8.30255 380 8.30263 379 .69737 9-99991 85 s 16 8.30631 376 372 8.30639 37O 373 .69361 9-99991 84 -a e 17 8.31003 8.31012 .68988 9-99991 83 18 8.31373 37O 8.31382 37O .68618 9.99991 82 i 19 8.31739 366 34 8.31749 307 .68251 9.99991 81 ^S 20 8.32103 8.32112 jfir .67888 9.99990 80 g + $ 21 8.32463 3 o 8.32473 .67527 9.99990 79 22 8.32820 357 8.32830 357 .67170 9.99990 78 * 23 8.33175 355 352 8.33185 355 352 .66815 9.99990 77 .. s %'* 24 8.33527 8.33537 .66463 9.99990 76 9 fc| 9 25 26 8.33875 8.34221 348 346' 344 8.33886 8.34232 349 346' 343 .66114 .65768 9-99990 9.99989 75 74 *+"a 27 8.34565 8.34575 .65425 9.99989 73 S i | 28 29 8.34905 8.35243 340 338 8.34916 8.35254 341 338 .65084 .64746 9.99989 9.99989 72 71 fill 30 8.35578 8.35590 33 .64410 9.99989 70 Sr7 * 31 8.359H 8.35922 .64078 9.99989 69 *" a 32 33 8.36241 8.36569 330 328 325 8.36253 8.36581 331 328 325 .63747 .63419 9.99988 9.99988 68 67 m 34 8.36894 8.36906 .63094 9-99988 66 35 8.37217 323 8.37229 323 .62771 9.99988 65 75 8.48485 249 8.48505 249 I-5I495 9.99980 25 s ~ ^ 76 8.48732 247 8.48753 248 I.5I247 9.99980 24 C/3 -f- 8> 77 8.48978 8.48999 1.51001 9-99979 23 78 8.49223 245 8.49244 245 1.50756 9-99979 22 3 " "3 2 79 8.49466 243 8.49487 243 1.50513 9-99979 21 C ~ 80 8.49708 8.49729 1.50271 9-99979 20 S-" " S 81 8.49948 8.49970 I . 50030 9.99978 19 |1 "^ 82 8.50188 240 8.50209 239 I-4979I 9.99978 18 83 8.50425 237 8.50448 239 236 1-49552 9.99978 17 ~ 3 -d 84 8.50662 8.50684 I.493I6 9.99978 16 p< rt 8s 8.50897 235 8.50920 230 1.49080 9.99977 IS O t/3 86 8.51131 234 233 8.5"54 234 233 1.48846 9-99977 14 'o o 87 8.51364 8.51387 1.48613 9-99977 13 H fa 88 8.51596 232 8.51619 232 1.48381 9-99977 12 89 8.51826 230 8.51850 231 1.48150 9.99976 II 90 8.52055 228 8.52079 228 I.4792I 9.99976 10 9i 8.52283 8.52307 1.47693 9.99976 09 92 8.52510 227 8-52534 227 1.47466 9.99976 08 r 93 8.52735 225 225 8.52760 225 1.47240 9-99975 07 i-75 94 95 96 8.52960 8.53183 8.53405 223 222 221 8.52985 8.53208 8.53430 223 222 221 I.470I5 1.46792 1.46570 9-99975 9-99975 9-99975 06 05 04 1.5 8023 .6 8010 7 7995 97 8.53626 8.53651 1.46349 9-99974 03 .8 7980 98 8.53846 220 8.53872 1.46128 9-99974 02 9 7963 99 8.54064 8.54091 219 1.45909 9-99974 01 2.0 7946 100 8.54282 8.54308 1.45692 9-99974 00 Cos d. Cot d. c. Tan Sin 88 c (102) 177' Sin d. Tan d. c. Cot Cos 00 8.54282 216 8.54308 1.45692 9-99974 100 01 02 03 04 05 06 07 08 09 8.54498 8.54714 8.54928 8.55142 8.55354 8.55565 8.55775 8.55985 8.56193 216 214 214 212 211 210 210 208 8.54525 8.54741 8.54956 8.55169 8.55382 8.55593 8.55804 8.56013 8.56222 216 215 213 213 211 211 209 209 45475 45259 45044 44831 .44618 .44407 .44196 .43987 .43778 9-99973 9-99973 9-99973 9.99972 9.99972 9.99972 9.99972 9-99971 9.99971 99 98 97 96 95 94 93 93 5 1.758 2.O 211 .1 22O .2 229 3 239 4 250 .5 260 10 8.56400 8.56429 43571 9-99971 90 II 12 13 14 IS 16 17 18 19 8.56606 8.56811 8.57016 8.57219 8.57421 8.57623 8.57823 8.58023 8.58222 205 205 203 2O2 202 2OO 200 199 197 8.56636 8.56841 8.57046 8.57249 8-57452 8.57654 8.57854 8.58054 8.58253 205 205 203 203 2O2 200 2OO 199 IQ8 43364 43159 .42954 .42751 42548 .42346 .42146 .41946 41747 9-99971 9-99970 9.99970 9-99970 9.99969 9.99969 9.99969 9.99969 9.99968 89 88 87 86 85 84 83 82 81 ;a-r; {T. an of complement. 20 8.58419 107 8.58451 1 08 41549 9.99968 80 21 22 23 24 25 26 27 28 29 8.58616 8.58812 8.59007 8.59201 8.59395 8.59587 8-59779 8.59970 8.60160 I 9 6 195 194 194 192 192 191 190 1 80 8.58649 8.58845 8.59040 8.59235 8.59428 8.59621 8.59813 8.60004 8 . 60194 196 195 195 193 193 192 191 190 4I35I 4II55 .40960 40765 .40572 .40379 .40187 .39996 .39806 9-99968 9.99967 9-99967 9.99967 9-99967 9.99966 9-99966 9.99966 9.99965 79 78 77 76 75 74 73 73 angles are small: loga S; log tan a = log sin a -f- S = log tan gles near 90 use sin an 30 8.60349 188 8.60384 1 88 .39616 9.99965 70 go" S 31 32 33 34 35 36 37 38 39 8.60537 8.60725 8.60911 8.61097 8.61282 8.61467 8.61650 8.61833 8.62015 188 186 186 185 185 183 183 182 181 8.60572 8.60760 8.60947 8.61133 8.61319 8.61504 8.61687 8.61870 8.62053 188 187 186 186 185 183 183 183 181 .39428 .39240 .39053 .38867 .38681 .38496 .38313 .38130 37947 9-99965 9.99964 9.99964 9.99964 9.99963 9-99963 9.99963 9-99963 9.99962 69 68 67 66 65 64 63 62 6z To interpolate wh log sin log a For cos and cot o! 40 8.62196 181 8.62234 181 37766 9.99962 60 41 42 43 44 45 46 47 48 49 8.62377 8.62556 8.62735 8.62914 8.63091 8.63268 8.63444 8.63619 8.63794 179 179 179 177 177 176 175 175 8.62415 8.62595 8.62774 8.62953 8.63131 8.63308 8.63484 8.63660 8.63835 180 179 179 178 177 176 176 175 .37585 .37405 .37226 37047 .36869 .36692 .36516 .36340 .36165 9.99962 9.99961 9.99961 9.99961 9.99960 9.99960 9.99960 9-99959 9-99959 59 58 57 56 55 54 53 52 5i T _1 '-757 2.0 946 .1 928 .2 909 .3 889 .4 869 .5 847 50 8.63968 8.64009 I-3599I 9-99959 50 Cos d. Cot d. c. Tan Sin 92' 87' (103) 177 < Sin d. Tan d. c. Cot Cos ' 50 8.63968 8.64009 1.35991 9-99959 50 51 52 53 54 55 56 57 58 59 8.64141 8.64314 8.64486 8.64657 8.64827 8.64997 8.65166 8.65335 8.65503 173 172 171 170 170 169 I6 9 168 167 8.64183 8.64356 8.64528 8.64700 8.64870 8.65041 8.65210 8.65379 8.65547 173 172 172 170 171 169 169 168 1 68 .35817 .35644 35472 .35300 .35130 .34959 34790 .34621 34453 9-99958 9-99958 9-99958 9-99957 9-99957 9-99957 9-99956 9-99956 9.99956 49 48 47 46 45 44 43 42 41 2.5 .6 .7 .8 9 3-0 5 1.758 260 27* 283 29<5 308 321 60 8.65670 167 8.65715 167 34285 9-99955 40 61 62 63 64 65 66 67 68 69 8.65837 8.66003 8.66168 8.66333 8.66497 8.66660 8.66823 8.66985 8.67147 166 165 165 164 163 163 162 162 161 8.65882 8.66048 8.66214 8.66379 8.66543 8.66707 8.66870 8.67033 8.67195 166 166 165 164 164 163 163 162 161 .34118 33952 33786 .33621 33457 .33293 33130 32967 .32805 9-99955 9-99955 9-99954 9-99954 9-99954 9-99953 9-99953 9-99952 9-99952 39 38 37 36 35 34 33 32 31 E la- 2, -T. m of complement. 70 8.67308 ifin 8.67356 161 .32644 9-99952 30 5- -1-43 y 8 T) 71 72 73 74 75 76 77 78 79 8.67468 8.67628 8.67788 8.67946 8.68104 8.68262 8.68419 8.68575 8.68731 160 160 158 158 158 157 156 156 8.67517 8.67677 8.67837 8.67996 8.68154 8.68312 8.68470 8.68626 8.68783 160 160 159 158 158 158 156 157 32483 .32323 .32163 .32004 .31846 .31688 .31530 31374 3I2I7 9-99951 9-99951 9-99951 9-99950 9-99950 9-99950 9-99949 9-99949 9-99948 29 28 27 26 25 24 23 22 21 angles are small: log a o; log tan a log sin a + 5 = log tan gles near 90 use sin an 80 8.68886 8.68938 .31062 9-99948 20 g ' "S 81 82 83 84 85 86 87 88 89 8.69041 8.69195 8.69349 8.69502 8.69654 8.69806 8.69958 8.70109 8.70259 154 154 153 152 152 152 151 ISO 8.69093 8.69248 8.69402 8.69555 8.69708 8.69860 8.70012 8.70164 8.70314 155 154 153 153 152 152 152 ISO .30907 .30752 .30598 .30445 .30292 .30140 .29988 .29836 .29686 9-99948 9-99947 9-99947 9-99947 9.99946 9.99946 9-99945 9-99945 9-99945 19 18 17 16 15 14 13 12 II To interpolate wh log sin i log a For cos and cot o 90 8.70409 8.70465 29535 9-99944 10 9i 92 93 94 95 96 97 98 99 8.70558 8.70707 8.70856 8.71003 8.71151 8.71298 8.71444 8.71590 8.71735 149 149 147 148 147 146 146 145 8.70614 8.70764 8.70912 8.71061 8.71208 8.71356 8.71502 8.71649 8.71794 150 148 149 147 148 146 147 145 146 .29386 .29236 .29088 -28939 .28792 .28644 .28498 .28351 .28206 9-99944 9-99944 9-99943 9-99943 9-99942 9-99942 9-99942 9-99941 9-99941 09 08 07 06 05 04 03 02 01 2.5 .6 .7 .8 9 3-0 T 1-757 847 824 801 111 752 725 100 8.71880 8.71940 1.28060 9-99940 00 Cos d. Cot d.c. Tan Sin 92' 87' (104) 3 176 < Sin d. Tan d.c. Cot Cos P. P. 145 143 00 8.71880 144 8.71940 144 1.28060 9-99940 100 i 14-5 14-3 OI 02 8.72024 8.72168 144 8.72084 8.72229 145 1.27916 i. 27771 9-99940 9.99940 99 98 2 3 29.0 43-5 58.0 28.6 42.9 57.2 03 8.72312 144 8.72373 1.27627 9-99939 97 5 72.5 71-5 04 05 8.72455 8.72597 142 8.72516 8.72659 143 1.27484 1-27341 9-99939 9-99938 96 95 6 7 8 87.0 101.5 116.0 85.8 IOO.I 114.4 06 8.72739 142 8.72801 1.27199 9-99938 94 9 130.5 128.7 07 8.72881 8.72943 1.27057 9-99938 93 141 139 08 09 8.73022 8.73163 141 141 8.73085 8.73226 142 141 1.26915 1.26774 9-99937 9-99937 92 91 2 3 14.1 28.2 42.3 13-9 27.8 41-7 10 8.73303 139 8.73366 140 1.26634 9.99936 90 4 5 56-4 7o 5 55-6 69-5 iz 8.73442 8.73506 1.26494 9-99936 89 6 84-6 83-4 12 8.73582 140 8.73646 140 1-26354 9-99936 88 7 98.7 97-3 13 8.73721 139 138 8.73785 139 139 1.26215 9-99935 87 8 Q 112. 8 126.9 III. 2 I25.I 14 8.73859 8.73924 1.26076 9-99935 86 138 137 15 8-73997 138 8.74063 139 1-25937 9-99934 85 I 13-8 13-7 16 8.74134 137 8.74201 138 1.25799 9-99934 84 2 27.6 27.4 138 137 3 41.4 41.1 17 8.74272 8.74338 1.25662 9-99933 83 4 54-8 18 8.74408 130 8.74475 137 1.25525 9-99933 82 5 69.0 68.5 19 8-74544 136 iifi 8.74612 137 Tlfi 1.25388 9-99933 81 6 82.8 82.2 20 8.74680 136 8.74748 136 1.25252 9-99932 80 7 8 110.4 95-9 109.6 21 22 23 8.74816 8-74950 8.75085 134 135 134 8.74884 8.75019 8.75154 135 135 135 1.25116 1.24981 1.24846 9-99932 9-99931 9-99931 79 78 77 9 2 124.2 135 13-5 27.0 123-3 133 13-3 26.6 24 8.75219 8.75289 1.24711 9-99931 76 3 40-5 39-9 25 8.75353 134 8.75423 134 1.24577 9-99930 75 4 54-0 fi? 3 8.80149 I2O 8.80237 119 1.19763 9-99913 37 8 9 96.8 108.9 95-2 107.1 64 8.80269 8.80356 1.19644 9-99912 36 118 117 65 8.80388 119 8.80476 I2O 1.19524 9.99912 35 i II. 8 ii. 7 66 8.80506 118 8.80595 119 I . 19405 9.99911 34 2 23.6 23-4 119 H9 3 35.4 35-1 67 8.80625 8.80714 I . 19286 9-999" 33 4 47-2 46.8 68 8.80743 8.80832 I . 19168 9.99910 32 5 59-0 58.5 69 8.80860 117 118 8.80950 118 1.19050 9.99910 31 6 70.8 70.2 70 8.80978 117 8.81068 118 1.18932 9.99909 30 7 8 94.4 81. Q 93-6 71 73 73 8.81095 8.81212 8.81328 ii? 116 116 8.81186 8.81303 8.81420 117 117 117 1.18814 1.18697 I . 18580 9.99909 9-99908 9.99908 29 28 27 9 i 2 106.2 116 ii. 6 23.2 105-3 115 n. 5 23.0 74 75 76 8.81444 8.81560 8.81675 116 us 116 8.81537 8.81653 8.81769 116 116 116 I . 18463 1.18347 I . 18231 9.99907 9.99907 9.99906 26 25 24 3 4 5 6 34-8 46.4 58.0 69.6 34-5 46.0 57-5 69.0 77 78 79 8.81791 8.81905 8.82020 H4 US 8.81885 8.82000 8.82115 US US 1.18115 1.18000 1.17885 9.99906 0.99905 9.99905 23 22 21 7 8 9 81.2 92.8 104.4 114 80.5 92.0 103.5 113 80 8.82134 114 8.82230 114 I . 17770 9.99904 20 i ii. 4 22 8 11.3 22 6 81 8.82248 8.82344 I . 17656 9.99904 19 3 34.2 33-9 82 8.82362 114 8.82458 114 I . 17542 9.99903 18 4 45-6 45.2 83 8.82475 in 8.82572 114 114 I . 17428 9.99903 17 5 6 6M 56-5 67.8 84 8.82588 8.82686 1.17314 9-99902 16 7 79-8 79.1 85 8.82701 113 8.82799 113 I . 17201 9-99902 IS 8 91.2 90-4 86 8.82814 H3 8.82912 113 I . 17088 9-99901 14 9 102.6 101.7 112 113 112 111 87 8.82926 8.83025 I . 16975 9-99901 13 i II. 2 II. I 88 8.83038 8.83137 I . 16863 9.99900 12 2 22.4 22.2 89 8.83149 8.83249 1.16751 9-99900 II 3 33.6 33-3 90 8.83261 III 8.83361 112 I . 16639 9.99899 10 4 5 44-8 56.0 44-4 55-5 91 93 8.83372 8.83482 no 8.83473 8.83584 III 1.16527 1.16416 9.99899 9.99898 09 08 6 7 8 67.2 78-4 89 6 66.6 77-7 88.8 93 8.83593 8.83695 I . 16305 9.99898 07 9 100.8 99-9 94 95 8.83703 8.83813 IIO 8.83806 8.83916 IIO 1.16194 I . 16084 9.99897 9.99897 06 05 I 109 10.9 21 8 108 10.8 21 6 96 8.83923 109 8 . 84026 IIO i. 15974 9.99896 04 3 32-7 32-4 97 8.84032 8.84136 1.15864 9.99896 03 4 43-6 43-2 98 99 8.84141 8.84250 109 109 1 08 8.84246 8.84355 IIO 109 i. 15754 1.15645 9.99895 9.99895 02 01 5 6 7 1! 54 - 64.8 75-6 100 8.84358 8.84464 i . 15536 9.99894 00 8 9 87.2 98.1 86.4 97-2 Cos d. Cot d. c. Tan Sin P. P. 93' (106) 86 c 175 < Sin d. Tan d. c. Cot Cos P. P. 109 108 00 8.84358 IOQ 8.84464 109 I.I5536 9.99894 100 i 10.9 10.8 OI 02 03 8.84467 8.84575 8.84682 108 107 108 8.84573 8.84682 8.84790 109 108 1 08 i 15427 1.15318 1.15210 9.99894 9-99893 9.99892 99 98 97 2 3 4 5 21.8 32.7 43-6 54-5 21.6 32.4 43-2 SJ- 04 OS 8.84790 8.84897 107 8.84898 8.85006 108 I . 15102 I. 14994 9.99892 9.99891 96 95 7 8 65.4 76.3 87.2 75-6 86.4 06 8.85004 107 8.85113 107 1.14887 9.99891 94 9 98.1 97-2 07 08 09 8.85111 8.85217 8.85323 107 106 106 Tnfi 8.85220 8.85327 8.85434 107 107 107 Tflfi I . 14780 1.14673 I . 14566 9.99890 9.99890 9.99889 93 92 9i i 2 3 107 10.7 21.4 32.1 106 10.6 21.2 31-8 10 8.85429 106 8.85540 106 I . 14460 9.99889 90 4 5 42.8 53-5 42.4 53-0 ii 8.85535 8.85646 I 14354 9.99888 8q 6 64.2 63.6 12 8.85640 105 8.85752 106 I . 14248 9-99888 88 7 I 4 '?- 74-2 13 8.85745 105 105 8.85858 106 105 1.14142 9.99887 87 8 9 85.6 96.3 84.8 95-4 14 8.85850 8.85963 I.I4037 9.99887 86 105 104 15 8.85955 105 8.86069 I.I393I 9.99886 85 i 10. s 10.4 16 8.86059 104 8.86173 104 1.13827 9.99885 84 2 21.0 20.8 104 105 3 31.5 31.2 17 8.86163 8.86278 I.I3722 9.99885 83 4 42.0 41.6 18 8.86267 104 8.86383 105 1.13617 9.99884 82 s 52.0 19 8.86370 103 8.86487 104 I.I35I3 9.99884 81 6 63.0 62.4 20 8.86474 103 8.86591 103 I.I3409 9.99883 80 8 73-5 84.0 83.2 21 22 8.86577 8.86680 103 8.86694 8.86798 104 I 13306 1.13202 9.99883 9.99882 79 78 9 94-5 103 93-6 102 23 8.86782 102 8.86901 103 I.I3099 9.99882 77 2 1U.J 20.6 2O.4 24 25 8.86885 8.86987 103 102 8.87004 8.87106 103 102 1.12996 1.12894 9.99881 9.99880 76 75 3 4 30.9 41.2 30.6 40.8 26 8.87089 IO2 101 8.87209 103 I O2 1.12791 9.99880 74 6 61.8 61.2 27 28 29 8.87190 8.87292 8.87393 IO2 101 8.87311 8.87413 8.87515 102 102 i . 12689 1.12587 i . 12485 9.99879 9.99879 9.99878 73 72 71 7 8 9 B:J 92.7 101 71.4 81.0 91-8 100 30 8.87494 IOO 8.87616 IOI 1.12384 9.99878 70 i 2 10. 1 2O 2 10. 20 o 31 8.87594 8.87717 i . 12283 9.99877 69 3 30-3 30.0 32 8.87695 8.87819 i. 12181 9.99876 68 4 40.4 40.0 33 8.87795 IOO IOO 8.87919 IOO IOI i. 12081 9.99876 67 5 6 50.5 60.6 50.0 60.0 34 8.87895 8.88020 1.11980 9.99875 66 7 70.7 70.0 35 8.87995 8.88120 i. 11880 9.99875 65 8 80.8 80.0 36 8.88094 99 8.88220 IOO i . 11780 9-99874 64 9 90.9 90.0 IOO IOO 99 98 37 8.88194 8.88320 i. 11680 9.99874 63 i 9.9 9.8 18 8.88293 99 8.88420 1.11580 9.99873 62 2 IQ.8 19.6 39 8.88392 99 o o 8.88519 99 1.11481 9.99872 61 3 29-7 29.4 40 8.88490 99 8.88618 99 99 1.11382 9.99872 60 4 s 39-6 49-5 39-2 49-0 41 8.88589 8.88717 1.11283 9.99871 59 59-4 60 "* 58.8 68 6 42 8.88687 9 8.88816 99 1.11184 9.99871 58 8 79-2 78.4 43 8.88785 9 Q Q 8.88915 99 QO 1.11085 9.99870 57 9 89.1 88.2 44 8.88883 8.89013 1.10987 9.99869 56 97 96 45 8.88980 97 8.89111 98 1.10889 9.99869 55 i 9-7 9.6 46 8.89077 97 97 8.89209 98 98 i . 10791 9-99868 54 3 29.1 28.8 47 48 8.89174 8.89271 97 8.89307 8.89404 97 1.10693 i . 10596 9-99868 9.99867 53 52 4 i 38.8 48.5 rg 2 38.4 48.0 r ? g 49 8.89368 97 06 8.89501 97 i . 10499 9.99867 5i 7 67-9 67.2 50 8.89464 8.89598 I . 10402 9-99866 60 8 9 77-6 87-3 76.8 86.4 Cos d. Cot d. c. Tan Sin P.P. (107) 86< 176' Sin d. Tan d. c. Cot Cos P. P 50 8.89464 8.89598 i . 10402 9.99866 50 Si 52 53 54 55 56 57 58 59 8.89561 8.89657 8.89752 8.89848 8.89943 8.90038 8.90133 8.90228 8.90323 96 95 96 95 95 95 95 95 8.89695 8.89792 8.89888 8.89984 8.90080 8.90176 8.90272 8.90367 8.90462 97 96 96 96 96 96 95 95 QC 10305 .10208 .10112 .10016 .09920 .09824 .09728 .09633 .09538 9.99865 9.99865 9.99864 9.99864 9.99863 9.99862 9.99862 9.99861 9.99860 49 48 47 46 45 44 43 42 41 i 2 3 4 5 6 7 8 9 97 9-7 19-4 29.1 38.8 48-5 58.2 77:8 87.3 95 96 9-6 19.2 28.8 38.4 48.0 |S 76.8 86.4 94 60 8.90417 QA 8.90557 Qtr .09443 9.99860 40 i 9-5 9-4 18 8 61 62 63 64 65 66 67 68 69 8.90511 8.90605 8.90699 8.90792 8.90885 8.90978 8.91071 8.91164 8.91257 94 94 93 93 93 93 93 93 8.90652 8.90746 8.90841 8.90935 8.91029 8.91122 8.91216 8.91309 8.91402 94 95 94 94 93 94 93 93 93 .09348 .09254 09159 .09065 .08971 .08878 .08784 .08691 .08598 9.99859 9.99859 9.99858 9-99857 9-99857 9.99856 9-99856 9.99855 9.99854 39 38 37 36 35 34 33 32 3i 3 4 5 6 7 8 9 i 2 28.5 38.0 47-5 57-0 66.5 76.0 85-5 93 9-3 18.6 28.2 37-6 47-0 65.8 75-2 84.6 92 9-2 18.4 70 8.91349 8.91495 q-i .08505 9-99854 30 3 4 27.9 37 2 27.6 36.8 71 72 73 74 75 76 77 78 79 8.91441 8.91533 8.91625 8.91716 8.91807 8.91898 8.91989 8.92080 8.92171 92 92 91 91 91 91 91 91 8.91588 8.91680 8.91773 8.91865 8.91957 8.92049 8.92140 8.92231 8.92323 92 93 92 92 92 91 91 92 91 .08412 .08320 .08227 .08135 .08043 .07951 .07860 .07769 .07677 9-99853 9.99852 9.99852 9.99851 9.99851 9.99850 9.99849 9.99849 9.99848 29 28 27 26 25 24 23 22 21 7 8 9 i 2 3 4 46.5 55-8 65.1 74-4 83-7 91 9-1 18.2 3:J 46.0 55-2 6 4 .4 73-6 82.8 90 9.0 18.0 27.0 36.0 80 8.92261 8.92414 90 .07586 9.99847 20 i 45-5 54-6 54-0 81 82 83 84 85 86 87 88 89 8.92351 8.92441 8.92531 8.92621 8.92710 8.92799 8.92888 8.92977 8.93066 90 90 90 89 89 89 89 89 88 8.92504 8-92595 8.92685 8.92776 8.92866 8.92956 8.93045 8.93135 8.93224 91 90 91 90 90 89 90 89 89 .07496 07405 .07315 .07224 .07134 .07044 .06955 .06865 .06776 9.99847 9.99846 9.99846 9.99845 9.99844 9.99844 9.99843 9.99842 9.99842 19 18 17 16 15 14 13 12 II 7 8 9 63.7 72.8 81.9 89 i * 2 I' 3 2( 4 3. 5 4^ 6 s; 7 6: 63.0 72.0 8l.o *.9 i.i u 1-5 J-4 > 3 90 8.93154 8q 8.93313 89 .06687 9.99841 10 8 71 .2 91 9 93 94 95 96 97 98 99 8,93243 8.93331 8.93419 8.93507 8.93594 8.93682 8.93769 8.93856 8.93943 88 88 88 87 88 87 87 87 87 8.93402 8-93491 8.93580 8.93668 8.93756 8.93845 8.93932 8.94020 8.94108 89 89 88 88 89 87 88 88 87 .06598 .06509 .06420 .06332 .06244 .06155 .06068 .05980 .05892 9.99840 9.99840 9.99839 9.99838 9.99838 9.99837 9.99836 9.99836 9.99835 09 08 07 06 05 04 03 02 01 i 2 3 4 5 6 7 8 9 9 & 88 8.8 17.6 26.4 35-2 44-0 52.8 61.6 70.4 79.2 ).I 87 8.7 17-4 26.1 34-8 43-5 52.2 60.9 69.6 78.3 100 8.94030 8.94195 1.05805 9.99834 00 Cos d. Cot d. c. Tan Sin P. P 94' 85< (108) 174 Sin d. Tan d. c. Cot Cos P. P. 00 8.94030 86 8.94195 o 7 1.05805 9.99834 100 01 02 03 04 05 o6 07 08 09 8.94116 8.94203 8.94289 8.94375 8.94461 8.94546 8.94632 8.94717 8.94802 87 86 86 86 85 86 85 85 QS- 8.94282 8.94369 8.94456 8.94543 8.94630 8.94716 8.94802 8.94888 8.94974 87 87 87 87 86 86 86 86 86 1.05718 1.05631 1.05544 1.05457 1.05370 1.05284 1.05198 1.05112 1.05026 9.99833 9.99832 9.99831 9.99830 9.99830 9.99829 9.99828 99 98 97 96 95 94 93 92 91 i 2 3 4 i 7 8 9 8 17 26 34 43 52 60 69 8( 7 4 I 8 5 i i 8.6 17-2 25.8 34-4 43-0 frj 68.8 77-4 84 10 8.94887 8.95060 0- 1.04940 9.99828 90 i 2 h 17 b o 8.4 16 8 ii 12 13 14 15 16 17 18 19 8.94972 8.95057 8.95141 8.95226 8-95310 8.95394 8.95478 8.95562 8.95645 84 85 84 84 84 84 83 81 8.95145 8.95231 8.95316 8.95401 8.95486 8-95570 8.95655 8.95739 8.95823 85 85 85 84 84 84 u- 1.04855 1.04769 1.04684 1.04599 I.04SI4 1.04430 1.04345 1.04261 1.04177 9.99827 9.99826 9.99826 9.99825 9.99824 9.99824 9.99823 9.99822 9.99822 89 88 87 86 85 84 83 82 81 3 4 5 6 7 8 9 25 34 42 59 68 76 i i 2 5 o 5 5 5 53 f i( 25.2 33-6 42.0 50-4 58.8 67.2 75-6 !-3 >.6 20 8.95728 &A. 8.95908 1.04092 9.99821 80 3 i\ 2i T t-9 1.2 21 22 23 24 25 26 27 28 29 8.95812 8.95895 8.95978 8.96060 8.96143 8.96225 8.96308 8.96390 8.96472 83 83 82 83 82 83 82 82 81 8.95991 8.96075 8.96159 8.96242 8.96325 8.96409 8.96492 8.96574 8.96657 84 84 83 83 84 83 82 82 i . 04009 1.03925 1.03841 1.03758 1-03675 I-0359 1 1.03508 1.03426 1.03343 9.99820 9.99820 9.99819 9.99818 9.998I7 9.99817 9.99816 9.998I5 9.99815 79 78 77 76 75 74 73 72 71 i 2 3 4 5 6 7 8 9 8! 8 16 24 32 43 i 6( 7' I 2 4 6 8 5 >.8 !.i >-4 \-l 81 8.1 16.2 24-3 32.4 30 8.96553 82 8.96739 1.03261 9.99814 70 6 4Q 2 48.6 32 33 34 35 36 37 38 39 8.96635 8.96716 8.96798 8.96879 8.96960 8.97041 8.97122 8.97202 8.97283 81 82 81 81 81 81 80 81 80 8.96822 8.96904 8.96986 8.97068 8.97150 8.97231 8.97313 8.97394 8.97475 82 82 82 82 8l 82 81 81 8l 1.03178 1.03096 1.03014 1.02932 1.02850 i . 02769 1.02687 1.02606 1.02525 9.99813 9.99813 9.99812 9.998H 9.998io 9.99810 9.99809 9.99808 9.99808 69 68 67 66 65 64 63 62 61 7 8 9 57 65 73 I 2 3 4 i 4 6 .8 80 ! K 2i 3 4< 4 56.7 64.8 72.9 5.0 >.o J-O z.o 3.0 5.0 5 n 40 8.97363 80 8.97556 81 1.02444 9.99807 60 8 6, l-o 42 43 44 45 46 47 48 49 8.97443 8.97523 8.97603 8.97682 8.97762 8.97841 8.97920 8.98000 8.98078 80 80 79 80 79 79 80 78 8.97637 8.97717 8.97798 8.97878 8.97959 8.98039 8.98119 8.98199 8.98278 80 81 80 81 80 80 80 79 Ro 1.02363 i . 02283 1.02202 I. 02122 I . O2O4I I.OI96I I.OI88I I. OlSOI I.OI722 9.998o6 9.99805 9-99805 9.99804 9-99803 9.99803 9.99802 9.998oi 9-99800 59 58 57 56 55 54 53 52 51 2 3 4 5 6 7 8 9 9 7 7 IS 23 3i 39 47 71 7 9 9 .8 i -5 4 -3 .2 T z.o 78 7-8 15.6 23.4 31.2 8: f g:J 70.2 50 8.98157 79 8.98358 I.OI642 9.998oo 50 Cos d. Cot d. c. Tan Sin 1 Mr > 95' 84* (109) 174 c Sin d. Tan d. c. Cot Cos p.p. 50 8.98157 8.98358 1.01642 9.998oo 50 51 8.98236 Q 8.98437 79 i . 01563 9-99799 49 79 IT ft 78 7 8 52 8.98314 7 8.98516 79 i . 01484 9-99798 48 7 >y 2 15.8 7.0 15.6 53 8.98393 79 78 8.98595 79 7O 1.01405 9-99797 47 3 23.7 23.4 54 55 56 8.98471 8.98549 8.98627 78 78 78 8.98674 8.98753 8.98832 ty 79 79 78 1.01326 1.01247 1.01168 9-99797 9.99796 9-99795 46 45 44 4 31-6 5 39-5 6 47-4 7 55-3 31-2 39-0 46.8 54-6 57 8.98705 / 8.98910 4 1.01090 9-99794 43 8 63.2 62.4 58 8.98782 77 8.98989 79 I.OIOII 9-99794 42 9 71- 1 70.2 59 8.98860 78 8.99067 78 78 1.00933 9-99793 41 77 60 8.98937 77 78 8.99145 7 78 1.00855 9-99792 40 i 7-7 6i 8.99015 / 8.99223 / 1.00777 9-99791 39 2 15-4 62 8.99092 77 8.99301 78 <-rQ 1.00699 9.99791 38 3 4 23. i 30.8 63 8.99169 77 i-i& 8-99379 7 1.00621 9-99790 37 5 38.5 64, 8.99245 70 8.99456 77 1.00544 9.99789 36 6 46.2 65 8.99322 77 8-99534 78 i . 00466 9.99788 35 7 8 53-9 61 6 66 8-99399 77 76 8.99611 77 77 1.00389 9.99788 34 9 69-3 67 8.99475 -./ 8.99688 1.00312 9.99787 33 76 68 8-99551 70 8.99765 77 1.00235 9.99786 32 69 8.99628 77 76 8.99842 77 1.00158 9-99785 31 i 2 y.u 15 2 70 8.99704 70 8.99919 77 76 1.00081 9.99785 30 3 22.8 7i 8-99779 ,-C 8-99995 7" 1.00005 9.99784 29 4 30.4 38.O 72 8.99855 70 n 9.00072 77 *i 0.99928 9.99783 28 6 73 8-99931 70 9.00148 70 0.99852 9.99782 27 7 53-2 74 9.00006 75 9.00225 77 0.99775 9.99782 26 8 60.8 ro . 75 9.00082 76 9.00301 76 ,- 0.99699 9.99781 25 9 oc 4 76 9-00157 75 9-00377 70 0.99623 9-99780 24 75 74 77 78 9.00232 9-00307 75 75 9.00452 9.00528 75 76 _ 0.99548 0.99472 9-99779 9-99779 23 22 i 7-5 2 15.0 3 22.5 7-4 14.8 22.2 79 9.00382 75 74 9 . 00604 70 0.99396 9.99778 21 4 30.0 2 9 .6 80 9.00456 9.00679 76 0.99321 9-99777 20 5 37-5 o 45 . o 37-0 44 4 81 9-00531 75 9-00755 7 0.99245 9.99776 19 7 52-5 Si.8 82 9.00605 74 9.00830 75 0.99170 9-99776 18 8 60. o 59-2 83 9.00680 75 9-00905 75 0.99095 9-99775 17 9 67.5 66.6. 84 9.00754 74 9.00980 75 0.99020 9-99774 16 73 85 9.00828 74 9-01055 75 0.98945 9-99773 IS i 7-3 86 9.00902 74 9.01129 74 0.98871 9.99772 14 2 14-6 87 9.00976 74 9.01204 75 0.98796 9.99772 13 3 4 21.9 29.2 88 9.01049 73 9.01278 74 0.98722 9-99771 12 5 36.5 89 9.01123 74 9-01353 75 0.98647 9-99770 II 6 43-8 90 9.01196 73 9.01427 74 0.98573 9.99769 10 7 8 51 58 .1 4 9i 9.01269 73 9.01501 74 0.98499 9.99769 09 9 65 .7 92 9-01343 74 9-01575 74 0.98425 9.99768 08 72 93 9.01416 73 9.01649 74 0.98351 9.99767 07 i 7 .2 94 9-01489 73 9 . 01722 73 0.98278 9.99766 06 2 4 95 9.01561 72 9.01796 74 o . 98204 9.99765 05 3 21 28 .6 96 9-01634 73 73 9.01869 73 74 0.98131 9-99765 04 4 5 36 .0 97 9.01707 9-01943 0.98057 9.99764 03 6 43 .2 98 9.01779 72 9.02016 73 0.97984 9.99763 02 I g-i 99 9.01851 72 1-1 y 9.02089 73 0.97911 9.99762 01 o 9 6d .u .8 100 9-01923 7-* 9.02162 73 0.97838 9.99761 00 Cos d. Cot d. c. Tan Sin P. P. 95 84* (no) 173 < Sin d. Tan d. c. Cot Cos P. P. 00 9-01923 77 9.02162 0.97838 9.99761 100 OI 9-01996 /O 9.02235 73 0.97765 9.99761 99 73 02 9.02067 71 9.02308 73 0.97692 9.9976o 98 i 7-3 03 9-02139 72 72 9 . 02380 72 73 0.97620 9-99750 97 2 3 14.0 21.9 04 9.02211 9-02453 0.97547 9-99758 96 4 2C ua 05 9.02283 72 9-02525 72 0.97475 9-99757 95 5 3J >-5 06 9-02354 71 71 9.02597 72 73 0.97403 9-99757 94 6 7 43- 51. i 07 9.02425 9.02670 0-97330 9.99756 93 8 58-4 08 9-02497 72 9.02742 72 0.97258 9-99755 92 9 65.7 09 9.02568 71 9.02813 71 0.97187 9-99754 72 71 10 9-02639 7T 9.02885 72 0.97115 9-99753 90 i 7.2 7-1 ii 12 9.02710 9 . 02780 1 * 70 9.02957 9.03028 72 71 0.97043 0.96972 9-99753 9-99752 89 88 2 14.4 3 21.6 4 28.8 14.2 21.3 28.4 13 9.02851 7 1 9 . 03100 7 2 o . 96900 9-99751 87 5 36.0 35-5 14 9.02921 70 9.03171 71 0.96829 9-99750 86 6 43-2 42.6 15 9-02992 71 9-03242 7 1 0.96758 9-99749 85 7 50.4 8 57 6 49-7 eg Q 16 9 . 03062 70 70 9-033I4 72 71 0.96686 9-99749 84 9 6 4 !8 o u o 63.9 17 9-03132 9-03385 0.96615 9-99748 83 70 18 9.03202 70 9-03455 70 0.96545 9-99747 82 19 9.03272 70 9-03526 71 0.96474 9.99746 81 i 2 7.0 14.0 20 9.03342 9-03597 7 1 0.96403 9-99745 80 3 21.0 21 9-03412 f\n 9-03667 70 0.96333 9-99744 79 4 5' 28.O jr n 22 9-03481 09 9.03738 7i 0.96262 9-99744 78 ! 6 OO *-' 42.O 23 9.0355I 70 69 9.03808 7 70 0.96192 9-99743 77 7 1 >.o 24 9.03620 9-03878 0.96122 9-99742 76 8 >.O 25 9-03690 7o ~ 9.03948 70 0.96052 9.99741 75 9 6 03. u 26 9-03759 69 f\f\ 9.04018 70 0.95982 9.99740 74 69 68 t>9 70 i 6.9 6.8 27 9.03828 _ 9.04088 0.95912 9-99739 73 2 13.8 13.6 28 9-03897 69 /-_ 9.04158 70 0.95842 9-99739 72 3 20.7 20.4 29 9-03966 69 68 9 . 04228 70 fio 0.95772 9-99738 71 4 27.6 27.2 30 9.04034 9.04297 09 0-95703 9-99737 70 5 34-5 6 41 -4 34-0 40 8 31 9-04103 09 9-04367 70 / 0.95633 9.99736 69 7 48.3 47-6 32 33 9-04171 9 . 04240 69 68 9.04436 9.04505 69 69 69 0.95564 0.95495 9-99735 9-99734 68 67 8 55-2 9 62.1 54-4 61.2 34 9.04308 9-04574 /_ 0.95426 9-99734 66 67 35 9-04376 /to 9.04643 69 0.95357 9-99733 65 i 6.7 36 9.04444 08 68 9.04712 69 69 0.95288 9.99732 64 2 3 13-4 20.1 37 9.04512 /?o 9.04781 0.95219 9-99731 63 4 26.8 38 9.04580 Oo ro 9-04850 69 0.95150 9-99730 62 5 33 5 39 9.04648 Oo 67 9-04918 68 fio 0.95082 9.99729 61 6 40.2 40 9.04715 O/ 68 9.04987 oy 68 0.95013 9.99728 60 7 8 46.9 53.6 41 9.04783 9-05055 .r\ 0.94945 9-99728 59 9 60.3 42 9.04850 07 JSfl 9-05124 09 0.94876 9.99727 58 66 43 9.04918 Oo 67 9.05192 Oo 68 0.94808 9-99726 57 i 6.6 44 9-04985 9 . 05260 0.94740 9-99725 56 2 13.2 45 9-05052 M 9-05328 Q 0.94672 9.99724 55 3 J 9 .0 46 9-05119 O7 67 9.05396 Oo 67 0.94604 9.99723 54 4 5 33-0 47 9.05186 67 9-05463 0.94537 9.99723 53 6 39-6 48 9.05253 O7 66 9-05531 68 0.94469 9-99722 52 7 46.2 r"- Q 49 9-053I9 6? 9-05599 0.94401 9.99721 Si 52.0 en A 50 9-05386 w 9.05666 0.94334 9 . 99720 50 jy Cos d. Cot d. c. Tan Sin P. P. 96' (in) 173' Sin d. Tan d. c. Cot Cos p P. 60 9-05386 66 9 . 05666 6? 0.94334 9.99720 50 Si 52 53 54 55 56 57 58 59 9-05452 9-05519 9-05585 9-0565I 9.05717 9-05783 9-05849 9-059I5 9-05980 67 66 66 66 66 66 66 65 66 9-05733 9.05801 9.05868 9-05935 9.06002 9.06068 9-06135 9.06202 9.06268 68 67 67 67 66 67 67 66 67 0.94267 0.94199 0.94132 0.94065 0.93998 0-93932 0.93865 0-93798 0.93732 9-99719 9-99718 9.99717 9.99716 9.99716 9-99715 9.99714 9-99713 9.99712 49 48 47 46 45 44 43 42 41 I 2 3 4 1 7 8 9 6 < i; 2C 2' # 4C 4' 7 ).8 .6 M .2 .0 .8 .6 4 .2 66 60 9.06046 66 9-06335 66 0.93665 9.99711 40 i 6 7 6.6 fa 62 63 64 65 66 67 68 69 9.06112 9.06177 9.06242 9.06307 9.06372 9-06437 9 . 06502 9.06567 9.06632 65 65 65 65 65 65 65 65 6,1 9.06401 9.06467 9-06534 9.06600 9.06666 9-06731 9.06797 9-06863 9.06928 66 67 66 66 65 66 66 65 66 0.93599 0.93533 0.93466 0.93400 0-93334 0.93269 0.93203 0.93137 0.93072 9.99710 9.99709 9-99709 9.99708 9.99707 9.99706 9-99705 9-99704 9-99703 39 38 37 36 35 34 33 32 31 2 13 3 20 4 26 5 33 6 40 7 46 8 53 9 60 i 2 -4 .8 5 .2 I -3 65 ( i" 13.2 10.1 26.4 33-0 39-6 46.2 52.8 59-4 >-5 .0 70 9.06696 f.- 9.06994 fie 0.93006 9.99702 30 3 Ij i 7i 72 73 74 75 76 77 78 79 9.06761 9.06825 9.06889 9-06954 9.07018 9.07082 9.07I45 9.07209 9-07273 64 64 65 64 64 63 64 64 6/1 9.07059 9.07124 9.07190 9.07255 9.07320 9.07385 9.07449 9.075I4 9-07579 65 66 65 65 65 64 65 65 6A 0.92941 0.92876 0.92810 0.92745 0.92680 0.92615 0.92551 0.92486 0.92421 9.99701 9.99701 9.99700 9.99699 9.99698 9.99697 9.99696 9.99695 9.99694 29 28 27 26 25 24 23 22 21 4 1 7 8 9 6 i 6 2 12 3 19 4 . 25 32 y. 4! 52 5? i 4 .8 .2 .6 5 .0 5 .0 5 63 6.3 12.6 18.9 25.2 80 9-07337 6^ 9-07643 ge 0.92357 9.99693 20 6 38 .4 37-8 81 82 83 84 85 86 87 88 89 9.07400 9.07464 9-07527 9-07590 9-07653 9.07716 9.07779 9.07842 9-07905 64 63 63 63 63 63 63 63 fit 9.07708 9.07772 9-07836 9.07900 9.07964 9.08028 9.08092 9.08156 9.08220 64 64 64 64 64 64 64 64 fil o . 92292 0.92228 0.92164 0.92100 0.92036 0.91972 0.91908 0.91844 0.91780 9-99693 9.99692 9.99691 9.99690 9.99689 9.99688 9-99687 9-99686 9-99685 19 18 17 16 15 14 13 12 II 7 44 8 51 9 57 i 3 3 4 5 6 8 2 6 62 12 Ifi 24 31 37 44-1 50-4 56.7 .2 4 .6 .8 .0 .2 90 9.07968 3 62 9.08283 64 0.91717 9.99684 10 8 4S -.6 9x 92 93 94 95 96 97 98 99 9.08030 9-08093 9-08155 9.08217 9.08280 9-08342 9.08404 9.08466 9.08528 63 62 62 63 62 62 - 62 62 61 9-08347 9.08410 9.08474 9-08537 9.08600 9.08663 9.08726 9-08789 9.08852 63 64 63 63 63 63 63 63 62 0.91653 0.91590 0.91526 0.91463 0.91400 0.91337 0.91274 0.91211 0.91148 9.99683 9.99682 9.99682 9.99681 9.9968o 9.99679 9.99678 9.99677 9.99676 09 08 07 06 05 04 03 02 01 9 i 2 3 4 5 6 7 8 9 55 61 t 12 18 24 3C 36 42 48 S<1 .8 .1 .2 3 4 5 .6 .7 .8 .9 100 9.08589 9.08914 0.91086 9.99675 00 Cos d. Cot d. c. Tan Sin P. P. 96 (112) 172' Sm d. Tan d. c. Cot Cos P . P. 00 9.08589 fio 9.08914 fii 0.91086 9-99675 100 OI 02 03 04 05 06 07 08 09 9.08651 9-08713 9-08774 9.08836 9.08897 9.08958 9.09019 9.09080 9.09141 62 61 62 61 61 61 61 61 fir 9.08977 9.09040 9.09102 9.09164 9.09227 9.09289 9.09351 9.094I3 9-09475 63 62 62 63 62 62 62 62 f.~ 0.91023 0.90960 0.90898 0.90836 0.90773 0.90711 0.90649 0.90587 0.90525 9.99674 9.99673 9.99672 9.99671 9.99670 9.99669 9.99669 9.99668 9.99667 99 98 97 96 95 94 93 92 9i I 2 3 4 I 7 8 9 63 12.6 18.9 25.2 31-5 37-8 44-1 50.4 56.7 62 10 9.09202 61 9-09537 61 0.90463 9-99666 90 I 6.2 ii 12 13 14 15 16 17 18 19 9.09263 9-09324 9-09385 9-09445 9-09506 9.09566 9.09626 9.09686 9-09747 61 61 60 61 60 60 60 61 fio 9.09598 9.09660 9.09722 9.09783 9.09845 9.09906 9.09967 9 . 10028 9.10089 62 62 61 62 61 61 61 61 fir 0.90402 0.90340 0.90278 0.90217 0.90155 0.90094 0.90033 0.89972 0.89911 9.99665 9.99664 9.99663 9.99662 9.99661 9.99660 9.99659 9.99658 9.99657 89 88 87 86 85 84 83 82 81 3 4 I 9 2 18.6 24.8 31.0 37-2 43-4 49-6 55-8 61 6.1 12.2 20 9.09807 60 9.10150 fir 0.89850 9.99656 80 3 18.3 21 22 23 24 25 26 27 28 29 9.09867 9.09926 9.09986 9 . 10046 9.10106 9.10165 9 . 10225 9.10284 9-10343 59 60 60 60 59 60 59 59 9.10211 9.10272 9-10333 9-10394 9.10454 9-10515 9-10575 9.10635 9.10696 61 61 61 60 61 60 60 61 fin 0.89789 0.89728 0.89667 0.89606 0.89546 0.89485 0.89425 0.89365 0.89304 9.99655 9-99654 9.99653 9.9965*2 9.99651 9.99650 9.99649 9.99648 9.99648 79 78 77 76 75 74 73 72 71 5 6 7 8 9 I 2 3 4 30.5 36.6 42.7 48.8 54-9 60 6.0 12.0 18.0 24.0 30 9 . 10402 59 fio 9.10756 fio 0.89244 9.99647 70 I 30.0 36.0 31 32 33 34 35 36 37 38 39 9 . 10462 9.10521 9.10580 9.10638 9-10697 9.10756 9.10815 9-10873 9.10932 59 59 58 59 59 59 58 59 t-0 9.10816 9.10876 9.10936 9.10996 9.11056 9.11115 9.HI75 9-II234 9-II294 60 60 60 60 59 60 59 60 0.89184 0.89124 0.89064 0.89004 0.88944 0.88885 0.88825 0.88766 0.88706 9.99646 9.99645 9.99644 9.99643 9.99642 9.99641 9.99640 9.99639 9.99638 69 68 67 66 65 64 63 62 61 7 8 9 I 2 3 4 42.0 48.0 54-0 59 5-9 ii. 8 17-7 23.6 29-5 35-4 40 9.10990 re 9-II353 59 60 0.88647 9.99637 60 8 41-3 47-2 41 42 43 44 45 46 47 48 49 9.11048 9.11107 9.11165 9.11223 9.11281 9 -II339 9-II397 9 -"454 9.11512 59 58 58 58 58 58 57 58 rQ 9-II4I3 9.11472 9-II53I 9-II590 9.11649 9.11708 9.11767 9.11826 9.11884 59 59 59 59 59 59 59 58 0.88587 0.88528 0.88469 0.88410 0.88351 0.88292 0.88233 0.88174 o. 88116 9.99636 9.99635 9.99634 9.99633 9-99632 9.99631 9.99630 9.99629 9.99628 59 58 57 56 55 54 53 52 51 9 2 3 4 5 6 7 8 9 53-1 58 5-8 ii. 6 17-4 23.2 29.0 34-8 40.6 46.4 52.2 50 9."570 9-II943 o . 88057 9.99627 50 Cos d. Cot d. c. Tan Sin P .P. 97 82< (U3) 172< Sin d. Ian d. c. Cot Cos P. P. 50 9.IIS70 i e , 9-II943 58 0.88057 9.99627 50 M 51 9.11627 58 9.12001 0.87999 9.99626 49 I 5 9 52 9.11685 9.12060 eg 0.87940 9.99625 48 3 II. 8 53 9.11742 57 9.12118 cq 0.87882 9.99624 47 3 17-7 54 9.II799 58 9.12177 58 0.87823 9.99623 46 4 23-6 55 9."857 9-12235 eg 0.87765 9.99622 45 g 35 4 56 9.11914 57 9.12293 58 0.87707 9.99621 44 7 41-3 57 9.11971 9-I235I eg 0.87649 9.99620 43 8 47-2 58 9.12028 9.12409 58 0.87591 9.99619 42 9 53 I 59 9.12085 57 9.12467 58 0.87533 9.99618 41 58 60 9.12142 56 9-12525 58 0.87475 9.99617 40 i 5-8 61 9-I2I98 9-12583 0.87417 9.99616 39 3 17 4 62 9-12255 9.12640 eg 0.87360 9.996IS 38 4 23.2 63 9.12312 56 9.12698 58 0.87302 9.99614 37 5 29.0 64 9.12368 57 9.12756 0.87244 9.996I3 36 34.8 65 9.12425 eg 9-12813 0.87187 9.99612 35 8 46.4 66 9.12481 56 9.12870 .S8 0.87130 9.996II 34 9 52.2 67 9-12537 57 9.12928 0.87072 9.99610 33 57 68 9-12594 56 9.12985 0.87015 9.99609 32 I 5 7 69 9 . 12650 56 9.13042 57 0.86958 9.99608 31 2 II. 4 70 9.12706 56 9.13099 57 0.86901 9.99607 30 3 I7.I 71 9.12762 eg 9.I3I56 0.86844 9.99606 29 5 28.5 72 9.12818 eg 9-13213 0.86787 9.99605 28 6 34-2 73 9.12874 ,S6 9-13270 57 0.86730 9.99604 27 7 i 39-9 74 75 9-12930 9-12985 55 eg 9.13327 9.I3384 57 0.86673 0.86616 9.99603 9.99601 26 25 9 51-3 76 9.13041 56 9.I344I 56 0.86559 9.99600 24 56 77 9.13097 er 9-13497 o . 86503 9-99599 23 5-O 78 9.I3I52 e.6 9.13554 eg 0.86446 9.99598 22 3 16 8 79 9.13208 55 9.13610 57 0.86390 9-99597 21 4 22.4 80 9.13263 55 9-13667 56 0.86333 9.99596 20 | 28.0 -,, /; 81 9.I33I8 9.13723 eg 0.86277 9-99595 19 7 39-2 82 9-13373 r 9-13779 eg 0.86221 9-99594 18 8 44-8 83 9.13429 55 9.I3835 57 0.86165 9-99593 17 9 50.4 84 9.13484 9-13892 eg 0.86108 9-99592 16 55 85 9-13539 9.13948 eg o . 86052 9-99591 IS i 5-5 86 9-13594 55 9.14004 56 0.85996 9-99590 14 2 II. 87 9.13649 9.14060 0.85940 9.99589 13 3 16.5 88 9.13703 9.I4II5 r6 0.85885 9-99588 12 5 27-5 89 9.13758 55 9.I4I7I 56 0.85829 9.99587 II 6 33 -O 90 9-13813 54 9.14227 56 0.85773 9-99586 10 7 g 38.5 91 9.13867 9-14283 0.85717 9-99585 09 9 49-5 92 9.13922 9.I4338 cfi 0.85662 9.99584 08 93 9.13976 55 9-14394 55 0.85606 9.99583 07 04 94 9.14031 54 9-14449 0.85551 9-99582 06 2 10.8 95 9-14085 9.14504 eg 0.85496 9.99581 05 3 16.2 96 9.I4I39 54 9.14560 55 0.85440 9-99580 04 4 21.6 97 9.I4I93 55 9.I46I5 55 0.85385 9.99578 03 27.0 3 2 4 98 9.14248 54 9.14670 55 0.85330 9-99577 02 7 37-8 99 9.14302 54 9.14725 55 0.85275 9.99576 01 8 43-2 100 9.14356 9.14780 0.85220 9-99575 00 9 48.6 Cos d. Cot d. c. Tan Sin P.P. 97< 82 < (114) 8 171 < Sin d. Tan d. c. Cot Cos P. P. 00 9-I43S6 c-j 9.14780 55 0.85220 9-99575 100 OI 9.14409 Do r \ 9.14835 re 0.85165 9-99574 99 55 02 9.14463 o4 54 9.14890 Ov> 55 0.85110 9-99573 98 I 5-5 03 9-I45I7 54 9-14945 55 0.85055 9-99572 97 2 II. 04 9-I457I 9.15000 0.85000 9-99571 96 3 16.5 05 9-14624 53 9.15054 54 0.84946 9-99570 95 4 22. 06 9.14678 54 53 9-15109 55 55 0.84891 9.99569 94 5 6 27-5 33-0 07 9-I473I CA 9.15164 CA 0.84836 9-99568 93 7 38.S 08 9.14785 54 9.15218 54 0.84782 9.99567 92 8 44-0 09 9-14838 53 53 9.15273 55 54 0.84727 9-99566 91 9 49- S 10 9.14891 54 9-I5327 54 0.84673 9-99565 90 II 9-14945 C*J 9.I538I r/t 0.84619 9.99563 89 54 12 9.14998 JO 9-15435 O4 0.84565 9.99562 88 13 9-15051 53 53 9.I5490 55 54 0.84510 9.99561 87 I 2 5-4 10.8 14 9.15104 9-15544 0.84456 9.9956o 86 3 16.2 IS 9.I5I57 53 9-15598 54 CA 0.84402 9-99559 85 4 21.6 16 9.15210 53 53 9-15652 o4 54 0.84348 9-99558 84 1 27.0 32.4 17 9-15263 9.15706 CA 0.84294 9-99557 83 7 37-8 18 9.I53I5 52 9.15760 v>4 c-> 0.84240 9-99556 82 8 43-2 19 9.15368 53 53 9.15813 Oo 54 0.84187 9-99555 81 9 48.6 20 9-I542I 52 9.15867 54 0.84133 9-99554 80 21 9-15473 C7 9.15921 C7 0.84079 9-99553 79 53 22 9.15526 56 9-15974 Do CA o . 84026 9-99552 78 J 5 3 23 9.I5578 52 53 9.16028 o4 53 0.83972 9-99550 77 2 10.6 24 9-15631 9.16081 0.83919 9-99549 76 3 iS-9 25 9-15683 52 9-16135 54 C-2 0.83865 9-99548 75 4 21.2 ~f - 26 9-15735 52 52 9.16188 oo 53 0.83812 9-99547 74 20.5 31-8 27 9.15787 9.16241 CA 0.83759 9-99546 73 7 37-1 28 9-15840 53 9-16295 *>4 e-2 0.83705 9-99545 72 8 42.4 29 9-15892 52 5 2 9-16348 55 53 0.83652 9-99544 7i 9 47-7 30 9-!5944 51 9.16401 53 0.83599 9-99543 70 31 9-15995 e*j 9.16454 C.7 0.83546 9-99542 69 52 32 9-16047 D" 9-16507 oo Cl 0.83493 9-99540 68 I SO 33 9.16099 S 2 52 9.16560 5o 53 0.83440 9-99539 6? 2 10.4 34 35 9.16151 9 . 16203 52 9.16613 9.16665 52 C"2 0.83387 0.83335 9-99538 9-99537 66 6s 3 4 15-6 20.8 oA o 36 9-16254 51 52 9.16718 5o 53 0.83282 9-99536 64 5 6 20. 31.2 37 9.16306 9.16771 C-7 0.83229 9-99535 63 7 36.4 38 9-16357 51 9.16823 5^ 0.83177 9-99534 62 8 41.6 39 9.16409 52 51 9.16876 53 52 0.83124 9-99533 61 9 46.8 40 9.16460 51 9.16928 53 0.83072 9-99532 60 4i 9.16511 CO 9.16981 CO 0.83019 9-99530 59 51 42 9-16563 O-* 51 9-I7033 O-* 52 0.82967 9.99529 58 I 5.1 43 9.16614 51 9.17085 53 0.82915 9-99528 57 2 10.2 44 9-16665 9-I7I38 0.82862 9-99527 56 3 15-3 45 9.16716 5 1 CT 9.17190 52 CO 0.82810 9-99526 55 4 20.4 46 9.16767 51 51 9.17242 o^ 52 0.82758 9-99525 54 5 6 25.5 30.6 47 9.16818 CT 9.17294 co 0.82706 9-99524 53 7 35-7 48 9.16869 O A 9-I7346 D-* 0.82654 9-99523 52 8 40.8 49 9.16919 SO 51 9-17398 52 52 0.82602 9-99521 51 9 45-9 50 9.16970 9-17450 0.82550 9.99520 50 Cos d. Cot d. c. Tan Sin P.P. 98 C 81' 171' Sin d. Tan d. c. Cot Cos P. P. 50 9.16970 51 9-17450 S 2 o . 82550 9.99520 50 51 9.17021 CT 9-17502 51 0.82498 9.99519 49 52 52 9.17072 O 1 - 9-17553 o 82447 9.99518 48 53 9.17122 5O 51 9-17605 52 52 0.82395 9.99517 47 I 2 :>* 10.4 54 9.I7I73 Cn 9.17657 5 1 0.82343 9.99516 46 3 I5 55 9.17223 O u 9.17708 co 0.82292 9-99515 45 4 20. 8 56 9.17273 5 51 9.17760 O^ 51 0.82240 9-99514 44 5 6 26.0 31.2 57 9-17324 cn 9.17811 CO 0.82189 9-99512 43 7 36.4 58 9-17374 5O 9-17863 5* CT 0.82137 9-995II 42 8 41.6 59 9.17424 SO 50 9-I79I4 5* 51 0.82086 9.99510 41 9 46.8 60 9-17474 50 9-I7965 52 0.82035 9-99509 40 6i 9-I7524 CT 9.18017 CT 0.81983 9-99508 39 fil 62 9-17575 O A 9.18068 O A 0.81932 9-99507 38 63 9.17624 49 50 9.18119 5 1 51 o. 81881 9.99505 37 I 2 5-1 10.2 64 9-I7674 9.18170 0.81830 9-99504 36 3 15-3 65 9.17724 50 en 9.18221 51 CT 0.81779 9-99503 35 4 20.4 66 9-17774 O (J 50 9.18272 o 1 51 0.81728 9-99502 34 5 6 25-5 30.6 67 9.17824 9-18323 0.81677 9-99501 33 7 35-7 68 9.17873 49 9-18374 51 0.81626 9-99500 32 8 40.8 69 9.17923 5 5 9.18425 51 50 0.81575 9-99499 31 9 45-9 70 9-17973 49 9-18475 51 0.81525 9-99497 30 71 9.18022 *ry Cft 9.18526 CT 0.81474 9.99496 29 so 73 9.18072 5 9.18577 D* cn 0.81423 9-99495 28 73 9.18121 49 49 9.18627 o u 51 0.81373 9-99494 27 2 5-O 10. 74 9.18170 rn 9.18678 cn 0.81322 9-99493 26 3 iS-o 75 9.18220 5 9.18728 o y 0.81272 9.99492 25 4 20. o 76 9.18269 49 49 9.18778 SO 51 0.81222 9.99490 24 5 6 25.0 30.0 77 9.18318 9.18829 0.81171 9.99489 23 7 35-0 78 9-18367 49 9.18879 SO 0.81121 9.99488 22 8 40.0 79 9.18416 49 AQ 9-18929 SO 5 0.81071 9.99487 21 9 45-0 80 9.18465 *ty 4Q 9.18979 SO 0.81021 9.99486 20 8z 9-18514 ly 9.19029 CT 0.80971 9.99485 19 49 82 9-18563 49 9.19080 J* o . 80920 9-99483 18 j 4O 83 9.18612 49 49 9.I9I30 SO 49 0.80870 9.99482 17 2 y 9-8 84 85 9.18661 9-18709 48 9.19179 9-19229 50 cn 0.80821 0.80771 9.99481 9-99480 16 IS 3 4 14-7 19.6 86 9-18758 49 48 9.19279 o u so 0.80721 9-99479 14 g 24.5 29-4 87 9.18806 9.19329 0.80671 9-99477 13 7 34 3 88 9-18855 49 9-19379 5 AQ 0.80621 9.99476 12 8 39-2 89 9.18904 49 d8 9.19428 VJ 50 0.80572 9-99475 II 9 44.1 90 9.18952 40 48 9.19478 SO 0.80522 9-99474 10 9i 9.19000 9.19528 Af\ 0.80472 9-99473 09 48 92 9.19049 49 4.8 9-19577 4y So 0.80423 9-99472 08 I 4-8 93 9.19097 4 48 9.19627 49 0.80373 9.99470 07 2 96 94 9.I9I45 .0 9.19676 Af\ 0.80324 9.99469 06 3 A 14.4 IQ 2 95 9.I9I93 40 A o 9.19725 4y cn 0.80275 9.99468 05 5 j.y . * 24.O 96 9.19241 40 48 9-19775 5 U 49 0.80225 9-99467 04 6 28.8 97 9.19289 48 9.19824 AQ 0.80176 9.99466 03 7 g 33-6 ,0 A 98 9-19337 40 .0 9.19873 Hy AQ 0.80127 9.99464 02 g 3O.4 43-2 99 9.19385 40 48 9.19922 4y 49 0.80078 9.99463 01 100 9-19433 9.19971 o . 80029 9.99462 00 Cos d. Cot d. c. Tan Sin P.P. 81' (116) 170' Sin d. Tan d. c. Cot Cos P. P. 00 9 15433 48 9.I997I 49 0.80029 9.99462 100 01 .9.19481 9.20020 0.79980 9.99461 99 O2 9.19529 48 9.20069 49 0.79931 9.99460 98 49 03 9-19577 40 47 9.20118 49 0.79882 9-99458 97 i Si 04 05 9.19624 9.19672 48 47 9.20167 9.20216 49 49 0.79833 0.79784 9-99457 9-99456 96 95 2 3 4 98 14.7 19.6 06 9.19719 48 9.20265 48 0.79735 9-99455 94 5 24.5 07 9.19767 9 20313 0.79687 9-99454 93 6 29.4 08 9.19814 48 9.20362 49 0.79638 9-99452 92 7 g 34-3 39 2 09 9.19862 47 9.20411 48 0.79589 9-99451 91 9 44.1 10 9.19909 47 9.20459 49 0.79541 9-99450 90 II 9-19956 A o 9.20508 48 0.79492 9-99449 89 12 9.20004 40 9-20556 40 0.79444 9-99447 88 48 13 9.20051 47 9.20605 48 0.79395 9.99446 87 i 2 4-8 9.6 14 9.20098 47 9-20653 48 0.79347 9-99445 86 3 14-4 15 9-20145 9.20701 0.79299 9.99444 85 4 19.2 16 9 . 20192 47 9-20750 48 0.79250 9-99443 84 i 24-0 28 8 17 9-20239 9.20798 48 0.79202 9.99441 83 7 33 6 a 9.20286 47 9.20846 48 0.79154 9-99440 82 8 38.4 19 9-20333 47 9.20894 48 0.79106 9-99439 81 9 43-2 20 9.20380 47 9.20942 48 0.79058 9-99438 80 21 9.20427 46 9.20990 48 0.79010 9.99436 79 47 22 9-20473 /17 9.21038 48 0.78962 9-99435 78 4y 23 9.20520 4/ 47 9.21086 48 0.78914 9-99434 77 2 7 9.4 24 9.20567 46 9-2II34 48 0.78866 9-99433 76 3 14 1 25 9.20613 9.21182 0.78818 9-99432 75 4 18.8 26 9.20660 46 9.21229 48 0.78771 9-99430 74 5 6 23-5 28.2 27 9.20706 46 9.21277 48 0.78723 9.99429 73 7 32-9 28 9.20752 9-21325 4 0.78675 9-99428 72 8 37.6 29 9-20799 46 9-21372 48 0.78628 9-99427 7i 9 42.3 30 9.20845 46 9.21420 47 0.78580 9.99425 70 31 9.20891 47 9.21467 48 0.78533 9.99424 69 46 32 9-20938 46 9-2I5I5 47 0.78485 9-99423 68 i 4.6 33 9.20984 46 9.21562 48 0.78438 9-99422 67 2 9 o 34 9 . 21030 46 9.21610 47 0.78390 9.99420 66 3 4 13 8 18.4 35 9.21076 46 9-21657 47 0.78343 9.99419 65 5 23.0 36 9.21122 46 9.21704 47 0.78296 9.99418 64 6 27.6 37 9.21168 46 9-2I75I 47 0.78249 9.99417 63 7 8 32.2 38 9.21214 46 9.21798 48 0.78202 9-99415 62 41 .4 39 9.21260 46 9.21846 47 0.78154 9-99414 61 40 9.21306 45 9-21893 47 0.78107 9-99413 60 41 9.21351 46 9.21940 47 0.78060 9.99412 59 45 42 9-21397 46 9-21987 47 0.78013 9.99410 58 i 4-5 43 9-21443 45 9.22034 46 0.77966 9.99409 57 2 3 9-0 13.5 44 9.21488 46 9.22080 47 0.77920 9.99408 56 4 18.0 45 9.21534 45 9.22127 47 0.77873 9.99407 55 5 22.5 46 9-21579 46 9.22174 47 0.77826 9.99405 54 6 7 27.0 31.5 47 9.21625 45 9.22221 46 0.77779 9-99404 53 8 36.0 48 9.21670 46 9.22267 47 0.77733 9-99403 52 9 40.5 49 9.21716 45 9.22314 47 0.77686 9.99402 Si 50 9.21761 9.22361 0.77639 9.99400 50 Cos d. Cot d. c. Tan Sin P.P. 99 80 170' Sin d. Tan d. c. Cot Cos P. P. 50 9.21761 45 9.22361 46 0.77639 9.99460 50 51 9.21806 9.22407 0-77593 9-99399 49 52 9.21851 45 A6 9.22454 47 46 0.77546 9-99398 48 47 53 9.21897 4" 45 9.22500 47 0.77500 9-99396 47 2 4-7 9 a 54 9.21942 45 9.22547 46 0-77453 9-99395 46 3 4 14 I 55 9.21987 AC 9-22593 46 0.77407 9-99394 45 4 18.8 56 9.22032 4o 45 9-22639 46 o.7736i 9-99393 44 5 23-5 57 9. 22077 9.22685 0.77315 9-99391 43 6 28.2 58 9.22122 45 AC 9.22732 47 46 0.77268 9-99390 42 7 8 32-9 37-6 59 9.22167 4o 44 9.22778 46 0.77222 9-99389 41 9 42.3 60 9.22211 45 9.22824 46 0.77176 9-99388 40 6i 9.22256 45 9.22870 46 0.77130 9-99386 39 A.K. 62 9 . 22301 9.22916 46 0.77084 9.99385 38 63 9-22346 45 44 9 . 22962 4" 46 0.77038 9.99384 37 I 2 4.6 9-2 64 9.22390 45 9.23008 46 0.76992 9.99382 36 3 13-8 65 9-22435 AC 9-23054 d6 0.76946 9.99381 35 4 18.4 66 9.22480 4o 44 9.23100 4*J 46 0.76900 9.99380 34 5 g 23.0 27 ft 67 9.22524 44 9-23146 45 0.76854 9-99379 33 7 27.O 32-2 68 9.22568 45 9.23191 46 0.76809 9-99377 32 8 36-8 69 9-22613 44 9.23237 46 0.76763 9-99376 31 9 41.4 70 9.22657 45 9-23283 45 0.76717 9-99375 30 71 9.22702 44 9-23328 46 0.76672 9-99373 29 45 72 9.22746 44 9 23374 45 0.76626 9-99372 28 ! 4-5 73 9.22790 44 9.234I9 46 0.76581 9-99371 27 2 9.0 74 9-22834 44 9-23465 45 0.76535 9.99369 26 3 13 5 T Q -. 75 9.22878 44 9.23510 46 0.76490 9.99368 25 4 lo. O oo c 76 9.22922 45 9-23556 45 0.76444 9-99367 24 6 6Z, . $ 27.0 77 9-22967 44 9.23601 45 0.76399 9.99366 23 7 31.5 78 9-23011 A*l 9-23646 46 0.76354 9.99364 22 8 36.0 79 9.23054 4o 44 9.23692 45 0.76308 9.99363 21 9 1^.3 80 9-23098 44 9-23737 45 0.76263 9.99362 20 81 9-23142 AA 9-23782 45 0.76218 9.9936o 19 44 82 9.23186 44 44 9.23827 45 0.76173 9-99359 18 I 4.4 83 9-23230 44 9-23872 45 0.76128 9-99358 17 2 3 8.8 13 2 84 9.23274 A"t 9.23917 45 0.76083 9 99356 16 4 17^6 85 9.23317 4o 44 9.23962 45 0.76038 9-99355 15 5 22. 86 9.23361 43 9.24007 45 0.75993 9-99354 14 6 26.4 10 8 87 9.23404 44 9-24052 45 0.75948 9-99352 13 7 8 o " 35-2 88 9.23448 43 9.24097 45 0.75903 9-99351 12 9 39-6 89 9-23491 44 9.24142 44 0.75858 9.99350 II 90 9-23535 43 9.24186 45 0.75814 9-99348 10 91 9-23578 44 9-24231 45 0.75769 9-99347 09 so 9a 9.23622 43 9.24276 45 0.75724 9.99346 08 i 2 4-3 8.6 93 9-23665 43 9-24321 44 0.75679 9-99344 07 3 12^9 94 9-23708 44 9.24365 45 0.75635 9-99343 06 4 17.2 95 9 23752 43 9.24410 44 0.75590 9-99342 05 I 21 .5 96 9-23795 43 9-24454 45 0.75546 9-99340 04 o 7 25.8 30.1 97 9.23838 43 9.24499 44 0-75501 9-99339 03 8 34-4 98 9.23881 43 9.24543 45 0-75457 9-99338 02 9 38-7 99 9.23924 43 9.24588 44 0.75412 9-99336 OI 100 9.23967 9.24632 0.75368 9-99335 00 Cos d. Cot d. c. Tan Sin P. P. 99 80' (118) 169< Sin d. Tan d. c. Cot Cos P. P. 00 9.23967 43 9.24632 44 0.75368 9-99335 100 OI 9 . 24010 A1 9.24676 AA 0.75324 9-99334 99 02 9-24053 4o A1 9.24720 44 0.75280 9-99332 98 03 9.24096 4o 43 9.24765 45 44 0.75235 9-99331 97 44 04 9.24139 A 1 ? 9.24809 A A 0.75I9I 9-99330 96 I 4-4 05 9.24181 4* 43 9-24853 44 44 0-75147 9.99328 95 2 8.8 06 9.24224 43 9.24897 44 0.75103 9-99327 94 3 13-2 07 9.24267 A1 9.24941 0.75059 9.99326 93 4 17.6 08 9-24310 4o 9-24985 44 0.75015 9-99324 92 5 22. 09 9.24352 42 43 9.25029 44 44 0.74971 9-99323 91 6 26.4 10 9-24395 42 9.25073 44 0.74927 9.99322 90 7 8 3O.8 35-2 II 9-24437 43 9.25117 44 0.74883 9-99320 89 9 39.6 12 9.24480 A 1 } 9-25161 0.74839 9-99319 88 13 9.24522 {\& 43 9.25205 44 43 0.74795 9-99318 87 14 9.2456s A? 9.25248 0.74752 9-99316 86 15 9.24607 l\4 A? 9.25292 44 0.74708 9-99315 85 43 16 9.24649 4* 43 9-25336 44 43 0.74664 9-99314 84 i 4-3 17 9.24692 42 9-25379 AA 0.74621 9-99312 83 2 8.6 18 9.24734 42 9-25423 44 A1 0.74577 9-993II 82 3 12.9 19 9.24776 42 9-25466 4o 44 0.74534 9-99310 81 4 17.2 20 9.24818 42 9.255io 43 0.74490 9.99308 80 5 21.5 ?c 8 21 9.24860 42 9-25553 44 0.74447 9-9930? 79 7 25. o 30.1 22 9.24902 4 2 9-25597 A1 0.74403 9-99305 78 g 1A A. 23 9-24944 42 9-25640 4o 44 0.74360 9.99304 77 9 o4- 4 38.7 24 9.24986 4 2 9-25684 A1 0.74316 9.99303 76 25 9.25028 42 9-25727 4o 43 0.74273 9-99301 75 26 9.25070 42 9.25770 43 0.74230 9-99300 74 27 9.25112 42 9-25813 A A 0.74187 9-99299 73 42 28 9.25I54 42 9.25857 44 A"2 0.74143 9.99297 72 I 4.2 29 9.25196 41 9.25900 4o 43 0.74100 9.99296 71 2 8.4 30 9.25237 42 9-25943 AT. 0-74057 9.99294 70 3 12.6 31 9-25279 42 9-25986 *K3 A1 0.74014 9-99293 69 4 16.8 32 9-25321 4 1 9.26029 4o A1 0.73971 9.99292 68 5 21 -O 33 9-25362 42 9.26072 4o 43 0.73928 9-99290 67 7 25.2 29.4 34 9.25404 41 9.26115 A1 0.73885 9.99289 66 8 33-6 35 9-25445 42 9.26158 4o 0.73842 9.99288 65 g 37-8 36 9.25487 41 9 . 26201 43 42 0.73799 9.99286 64 37 9.25528 42 9-26243 A1 0.73757 9.99285 63 38 9-25570 4 1 9.26286 4o 0.73714 9.99283 62 39 9.25611 41 9-26329 43 43 0.73671 9.99282 61 41 40 9-25652 42 9-26372 42 0.73628 9.99281 60 i 4-1 41 9.25694 41 9.26414 0.73586 9-99279 59 2 8.2 42 9-25735 41 9.26457 43 0-73543 9.99278 58 3 12.3 43 9.25776 41 9.26500 43 0.73500 9.99276 57 4 16.4 44 9.25817 4 A 41 9-26542 4 2 43 0.73458 9-99275 56 5 6 20.5 24 6 45 9-25858 41 9.26585 42 0.73415 9-99274 55 &i\.\j 28 7 46 9-25899 41 9.26627 43 0.73373 9.99272 54 8 zo. / 32.8 47 9.25940 41 9.26670 0.73330 9.99271 53 9 36. Q 48 9-25981 41 9.26712 4 2 y2 0.73288 9.99269 52 49 9 . 26022 41 9-26754 4~ 43 0.73246 9.99268 Si 50 9.26063 9.26797 0.73203 9.99267 50 Cos d. Cot d. c. Tan Sin P.P. 100 C 79 ' (119) 10' 169' Sin d. Tan d. c. Cot Cos P.P. 50 9.26063 41 9.26797 42 0.73203 9.99267 50 51 9.26104 41 9-26839 42 0.73161 9-99265 49 52 9-26145 41 9.26881 0.73119 9.99264 48 53 9.26186 41 9-26923 43 0.73077 9-99262 47 43 54 9.26227 40 9.26966 0.73034 9.99261 46 I 4-3 55 9.26267 9.27008 0.72992 9.99260 45 2 8.6 56 9.26308 41 9-27050 42 0.72950 9.99258 44 3 12.9 57 9-26349 9-27092 0.72908 9.99257 43 4 17.2 58 9.26389 40 9.27I34 42 0.72866 9.99255 42 5 21.5 OF* & 59 9.26430 4O 9.27176 42 0.72824 9.99254 41 25.8 60 9.26470 41 9.27218 42 0.72782 9.99252 40 8 34-4 6i 9.26511 9.27260 0.72740 9-99251 39 9 38.7 62 9-26551 40 9.27302 42 0.72698 9.99250 38 63 9.26592 4^ 40 9-27343 41 42 0.72657 9.99248 37 64 9.26632 ?9- 27385 0.72615 9.99247 36 65 9.26672 4O 9.27427 0-72573 9.99245 35 42 66 9-26713 41 4O 9-27469 42 41 0.72531 9.99244 34 i 4-2 67 9-26753 9-27510 o. 724*90 9-99243 33 2 8.2 68 9-26793 9.27552 0.72448 9.99241 32 3 12.6 69 9-26833 4O 40 9-27594 42 41 0.72406 9.99240 31 4 16.8 70 9-26873 4O 9-27635 42 0.72365 9.99238 30 5 6 21 .0 25.2 71 9-26913 9.27677 0.72323 .9.99237 29 7 29.4 72 9-26954 ACt 9.27718 0.72282 9-99236 28 8 33-6 73 9-26994 4U 40 9-27760 41 0.72240 9-99234 27 9 37-8 74 9.27034 9.27801 41 0.72199 9-99233 26 75 9.27073 39 9.27842 0.72158 9.99231 25 76 9.27II3 40 40 9.27884 42 41 0.72116 9.99230 24 41 77 9.27153 9.27925 0.72075 9.99228 23 A 78 9.27193 40 9.27966 4 1 0.72034 9.99227 22 I 4.1 79 9-27233 40 40 9.28008 42 41 0.71992 9.99225 21 2 8.2 80 9-27273 39 9-28049 41 0.71951 9.99224 20 3 12.3 81 9.27312 9.28090 0.71910 9-99222 19 20.5 82 9-27352 40 9.28131 4 1 0.71869 9-99221 18 6 24.6 83 9.27392 40 39 9.28172 4i 0.71828 9-99220 17 7 28.7 84 9-27431 9.28213 0.71787 9.99218 16 8 32.8 85 9.27471 40 9.28254 41 0.71746 9.99217 IS 9 36.9 86 9-27510 39 40 9.28295 41 41 0.71705 9.992I5 14 87 9.27550 9.28336 0.71664 9.99214 13 88 9-27589 39 9-28377 41 0.71623 9.99212 12 89 9.27629 40 9.28418 41 41 0.71582 9.99211 II 40 90 9.27668 39 9-28459 41 0.71541 9.99209 10 I 4-0 8.^ 92 9.27707 9.27747 40 9.28500 9-28540 40 0.71500' 0.71460 9.99208 9.99206 09 08 2 3 .0 12.0 16 o 93 9-27786 39 9.28581 41 0.71419 9-99205 07 5 20. 94 9-27825 9 . 28622 0.71378 9.99203 06 6 24.0 95 9.27864 39 9.28662 40 0.71338 9-99202 05 7 28.0 96 9.27904 40 39 9-28703 41 41 0.71297 9.99201 04 8 32.0 97 9-27943 9-28744 0.71256 9-99199 03 9 36.0 98 9-27982 39 9.28784 40 0.71216 9.99198 02 99 9.28021 39 39 9.28825 41 40 0.71175 9.99196 01 100 9.28060 9.28865 0.71135 9.99195 00 Cos d. Cot d. c. Tan Sin P.P. 100 79' (120) ir 168< Sin d. Tan d. c. Cot Cos P.P. 00 9.28060 39 9.28865 41 o.7"35 9.99195 100 01 9-28099 Of\ 9.28906 /in 0.71094 9.99193 99 02 9.28138 oy "3O 9.28946 4U Art 0.71054 9.99192 9 03 9.28177 oy 39 9.28986 4u 41 0.71014 9.99190 97 41 04 9.28216 og 9.29027 An 0.70973 9.99189 96 i 4-1 OS 9.28254 o 3Q 9.29067 4U 40 0.70933 9.99187 95 2 8.2 06 9-28293 oy 39 9.29107 41 0.70893 9.99186 94 3 12.3 07 9-28332 OfJ 9.29148 0.70852 9.99184 93 4 16.4 08 9-28371 oy *8 9.29188 4O 0.70812 9.99183 92 5 20.5 09 9.28409 o 3Q 9.29228 4O 4O 0.70772 9.99i8i 9i 6 24.6 10 9.28448 oy 39 9.29268 40 0.70732 9.99180 90 7 g 28.7 32.8 II 9.28487 38 9-29308 40 0.70692 9.99178 89 9 36.Q 12 9.28525 7Q 9.29348 /in 0.70652 9.99177 88 13 9.28564 oy 38 9-29388 4u 40 0.70612 9-99175 87 14 9.28602 OQ 9.29428 0.70572 9.99174 86 IS 9.28641 oy -,o 9.29468 40 0.70532 9.99172 85 40 16 9.28679 JO 39 9.29508 40 40 0.70492 9.99171 84 i 4-0 17 9.28718 38 9.29548 Af\ 0.70452 9.99169 83 2 8.0 18 9.28756 ^8 9.29588 \\J 0.70412 9.99168 82 3 12.0 19 9.28794 o 39 9.29628 40 40 0.70372 9.99166 81 4 16.0 20 9-28833 38 9.29668 TO 0.70332 9.99165 80 5 20.0 21 9.28871 ?8 9.29707 oy ACt 0.70293 9.99163 79 7 24.O 28.0 22 9.28909 O og 9.29747 4vJ 0.70253 9.99162 78 8 32.O 23 9.28947 O 38 9.29787 40 40 0.70213 9.99160 77 9 36.0 24 9.28985 7Q 9.29827 7ft 0.70173 9-99159 76 25 9.29024 oy 3g 9.29866 M 0.70134 9.99157 75 26 9.29062 o 38 9.29906 40 7O 0.70094 9.99I56 74 27 9.29100 Og 9-29945 oy 0.70055 9-99154 73 39 28 9.29138 o T 9.29985 40 0.70015 9.99153 72 i 3-9 29 9.29176 o 38 9-30024 39 40 0.69976 9.99I5I 71 2 7-8 30 9.29214 38 9-30064 TO 0.69936 9.99150 70 3 II. 7 _- /? 3i 32 9.29252 9-29289 37 T 9-30103 9-30143 Oy 40 0.69897 0.69857 9.99148 9.99147 69 68 4 5 g IS-O 19-5 33 9-29327 o 38 9.30182 39 39 0.69818 9.99I45 67 7 23.4 27-3 34 9.29365 jR 9.30221 0.69779 9-99144 66 8 31-2 35 9.29403 o 38 9.30261 40 OQ 0.69739 9.99142 65 9 35-1 36 9.2944I 37 9-30300 oy 39 o . 69700 9.99141 64 37 9.29478 og 9-30339 0.69661 9.99139 63 38 9-29516 o og 9.30378 39 0.69622 9.99138 62 39 9-29554 o 37 9.30418 40 -jq 0.69582 9.99136 61 38 40 9.29591 38 9-30457 oy 39 0.69543 9-99135 60 i 3.8 41 9.29629 77 9.30496 7Q 0.69504 9.99133 59 2 7-6 42 9.29666 O/ og 9.30535 oy "JO 0.69465 9.99132 58 3 K .4 43 9.29704 o 37 9-30574 oy 39 0.69426 9.99130 57 4 5 15-2 19.0 44 9.29741 og 9-30613 7Q 0.69387 9.99128 56 6 22.8 45 9-29779 O 77 9-30652 oy 7O 0.69348 9.99127 55 7 26.6 46 9.29816 O/ 38 9-30691 oy 39 0.69309 9.99125 54 8 30.4 47 9.29854 -3<7 9.30730 on 0.69270 9.99124 53 9 34-2 48 9.29891 O/ 77 9-30769 3y 18 0.69231 9.99122 52 49 9.29928 O/ 38 9-30807 o 39 0.69193 9.99121 5i 50 9.29966 9-30846 0.69154 9.99H9 50 Cos d. Cot d. c. Tan Sin P.P. 101 ( 78 c (121) 11' 168< Sin d. Tan d. c. Cot Cos p .P. 50 9.29966 37 9.30846 39 0.69154 9.99119 50 Si 52 53 54 55 56 57 58 59 9.30003 9.30040 9.30077 9-30II4 9-30ISI 9.30188 9.30226 9.30263 9-30299 37 37 37 37 37 38 37 36 37 9-30885 9.30924 9-30963 9.31001 9.31040 9.31078 9-3III7 9.3II56 9-3II94 39 39 38 39 38 39 39 38 39 0.69115 0.69076 0.69037 0.68999 0.68960 0.68922 0.68883 0.68844 0.68806 9.99H8 9.99116 9-99II5 9-99II3 9-99II2 9.99110 9.99108 9.99107 9.99105 49 48 47 46 45 44 43 42 41 I 2 3 4 5 6 38 3.8 7-6 II. 4 15-2 19.0 22.8 60 9.30336 37 9-31233 38 0.68767 9.99104 40 7 8 30.4 6z 62 63 64 65 66 67 68 69 9-30373 9.30410 9.30447 9-30484 9.30521 9-30557 9-30594 9-30631 9.30667 37 37 37 37 36 37 37 36 37 9-31271 9.3i3io 9.31348 9.31386 9.31425 9-3I463 9-3I50I 9-3I540 9.31578 39 38 38 39 38 38 39 38 38 0.68729 0.68690 0.68652 0.68614 0.68575 0.68537 0.68499 0.68463 0.68422 9.99102 9-99IOI 9.99099 9.99098 9.99096 9.99094 9.99093 9.99091 9.99090 39 38 37 36 35 34 33 32 31 9 I 2 3 4 34-2 37 3-7 7-4 ii. i 14.8 70 9-30704 37 9.31616 38 0.68384 9.99088 30 5 6 18.5 22 2 71 72 73 74 75 76 77 78 79 9.30741 9.30777 9-30814 9-30850 9.30887 9-30923 9-30960 9.30996 9-31032 36 37 36 37 36 37 36 36 36 9.31654 9.31692 9-3I730 9.31768 9.31806 9-3I844 9.31882 9-31920 9.31958 38 38 38 38 38 38 38 38 og 0.68346 0.68308 0.68270 0.68232 0.68194 0.68156 o. 68118 0.68080 0.68042 9.99087 9.99085 9.99083 9-99082 9.99080 9.99079 9-99077 9.99076 9.99074 29 28 27 26 25 24 23 22 21 7 8 9 I 2 25-9 29.6 33-3 36 3.6 7-2 80 9.31068 37 9.31996 38 0.68004 9.99072 20 3 10.8 8z 83 83 84 85 86 87 88 89 9-3II05 9.3II4I 9-3II77 9.31213 9-31250 9.31286 9.31322 9.31358 9-31394 36 36 36 37 36 36 36 36 36 9-32034 9.32072 9.32110 9-32147 9-32185 9.32223 9.32260 9.32298 9-32336 38 38 37 38 38 37 38 38 0.67966 0.67928 0.67890 0.67853 0.67815 0.67777 0.67740 0.67702 0.67664 9.99071 9-99069 9.99068 9.99066 9.99064 9-99063 9.99o6i 9.99060 9.99058 19 18 17 16 15 14 13 12 XX 5 6 7 8 9 18.0 21.6 25.2 28.8 32.4 35 90 9.3I430 36 9-32373 38 0.67627 9.99056 10 I 3-5 91 92 93 94 95 96 97 98 99 9.31466 9.3I502 9.31538 9.31573 9-31609 9.3I645 9.31681 9.3I7I7 9-31752 36 36 35 36 36 36 36 35 36 9-324II 9-32448 9.32486 9.32523 9-32561 9.32598 9.32636 9-32673 9.32710 37 38 37 38 37 38 37 37 37 0.67589 0.67552 0.67514 0.67477 0.67439 0.67402 0.67364 0.67327 0.67290 9.99055 9.99053 9-99052 9.99050 9.99048 9-99047 9.99045 9.99044 9.99042 09 08 07 06 05 04 03 02 OX 3 4 5 6 7 8 9 7.0 10.5 14.0 17-5 21.0 24-5 28.0 31-5 100 9.31788 9.32747 0.67253 9.99040 00 Cos d. Cot d. c. Tan Sin I >. P. ior 178 (122) 167* Sin d. Tan d. c. Cot Cos P.P. 00 9-31788 36 9-32747 38 0.67253 9.99040 100 01 9-31824 -3C 9.32785 o 77 0.67215 9.99039 99 02 9.3I8S9 OJ jg 9.32822 J/ 0*7 0.67178 9.99037 98 03 9-31895 O u 35 9-32859 J/ 37 0.67141 9.99036 97 38 04 9.31930 16 9.32896 77 0.67104 9-99034 96 I 3-8 05 9.31966 O w 7tr 9-32933 O/ og 0.67067 9.99032 95 2 7-6 06 9.32001 oo 36 9-3297I o 37 0.67029 9.99031 94 3 II. 4 07 9-32037 9.33008 0.66992 9.99029 93 4 15.2 08 9.32072 35 ^6 9-33045 37 77 0.66955 9.99028 92 5 19.0 09 9.32108 J w 35 9.33082 O/ "V7 0.66918 9.99026 91 6 22.8 10 9 32143 35 9-33II9 O/ 37 0.66881 9-99024 90 7 3 26.6 7O A II 9.32178 36 9 33156 37 0.66844 9.99023 89 9 J u *l 34-2 12 9.32214 -2C 9-33193 0*7 0.66807 9.99021 88 13 9-32249 O%> 35 9-33230 O/ 36 0.66770 9.99019 87 14 9.32284 1C 9-33266 0.66734 9.99018 86 IS 9-32319 65 *6 9.33303 37 0.66697 9.99016 85 37 16 9-32355 JO 35 9-33340 37 37 0.66660 9.99014 84 I 3-7 17 9.32390 9-33377 0.66623 9.99013 83 2 7-4 18 9-32425 35 7C 9-33414 37 16 0.66586 9-990II 82 3 II. I 19 9.32460 Jj 35 9-33450 JO 37 0.66550 9.99010 81 4 14.8 20 9-32495 35 9.33487 37 0.66513 9.99008 80 5 18.5 21 22 9.32530 9-32565 35 'JC 9-33524 9-3356o 36 77 0.66476 0.66440 9.99006 9.99005 79 78 7 8 22.2 25.9 29.6 23 9.32600 oa 35 9-33597 Ol 37 0.66403 9-99003 77 9 33.3 24 9-32635 9 33634 ifi 0.66366 9.99001 76 25 9-32670 35 JC 9-33670 O" 0*7 0.66330 9.99000 75 26 9-32705 OJ 35 9.33707 J/ 36 0.66293 9.98998 74 27 9.32740 9-33743 0.66257 9.98996 73 36 28 9.32775 35 JA 9.3378o 37 76 o . 66220 9.98995 72 i 3-6 29 9.32809 J4 35 9.33816 o u 77 0.66184 9.98993 71 2 7-2 30 9-32844 35 9.33853 O/ 36 0.66147 9.98991 70 3 10.8 31 9-32879 35 9-33889 76 0.66111 9.98990 69 4 14.4 18 o 32 9-329I4 1A 9-33925 O' J 77 0.66075 9-98988 68 5 g 21 6 33 9.32948 J4 35 9.33962 O/ 36 0.66038 9.98987 67 7 25.2 34 9-32983 21- 9.33998 76 0.66002 9.98985 66 8 28.8 35 9-330x8 00 7/1 9.34034 O u 77 0.65966 9-98983 65 9 32.4 36 9.33052 OH- 35 9-3407I O/ 36 0.65929 9.98982 64 37 9-33087 9-34107 jfi 0.65893 9.98980 63 38 9-33I2I 34 jC 9-34143 30 ^6 0.65857 9.98978 62 39 9.33156 OJ 34 9.34179 JO 36 0.65821 9.98977 61 35 40 9.33190 35 9-34215 "V7 0.65785 9.98975 60 I 3-5 41 9-33225 7/1 9.34252 o/ 76 0.65748 9.98973 59 2 7-0 42 9.33259 54 35 9.34288 3O 05 0.65712 9.98972 58 3 10.5 43 9.33294 34 9.34324 o^ 36 0.65676 9.98970 57 4 5 14.0 17.5 44 9-33328 Oyf 9.34360 7fi 0.65640 9.98968 56 6 21.0 45 9-33362 o4 35 9.34396 JO 06 0.65604 9.98967 55 7 24.5 46 9-33397 34 9-34432 o u 36 0.65568 9-98965 54 8 28.O 47 9-33431 1A 9.34468 7fi 0.65532 9.98963 53 9 31-5 48 9.33465 o4 34 9.34504 JO ?6 0.65496 9.98962 52 49 9-33499 35 9-34540 O 1 - 1 36 o . 65460 9.98960 51 60 9-33534 9.34576 0.65424 9.98958 50 Cos d. Cot d. c. Tan Sin P.P. 102' 77 (123) 167' Sin d. Tan d. c. Cot Cos P.P. 50 5i 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 9-33534 34 34 34 34 34 34 34 34 34 34 34 34 34 34 33 34 34 34 33 34 34 33 34 33 34 33 34 33 34 33 33 34 33 33 33 34 33 33 33 33 33 33 33 33 33 33 33 33 33 33 9-34576 35 36 36 36 36 35 36 36 35 36 35 36 36 35 36 35 35 36 35 36 35 35 36 35 35 35 35 36 35 35 35 35 35 35 35 35 35 35 35 34 35 35 35 34 35 35 35 34 35 34 0.65424 9.98958 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 IS 14 13 12 ii 10 09 08 07 06 05 04 03 oa 01 00 2 3 4 S 6 7 8 9 i 2 3 4 5 6 7 8 9 I 2 3 4 5 6 7 8 9 i 2 3 4 5 6 7 8 9 36 3-6 7-2 10.8 14-4 18.0 21.6 25.2 28.8 32.4 35 3-5 7-0 10.5 14.0 17-5 21.0 24.5 28.0 31.5 34 3-4 6.8 10.2 13-6 17.0 20.4 23.8 27.2 30.6 33 3-3 6.6 9-9 13-2 16.5 19-8 23.1 26.4 29.7 9-33568 9.33602 9-33636 9.33670 9.33704 9-33738 9-33772 9.338o6 9-33840 9.346II 9-34647 9.34683 9.34719 9-34755 9-3479 9.34826 9.34862 9.34897 0.65389 0.65353 0.65317 0.65281 0.65245 0.65210 0.65174 0.65138 0.65103 9.98956 9-98955 9-98953 9.98951 9.98950 9.98948 9.98946 9.98945 9.98943 9-33874 9-34933 0.65067 9.98941 9.33908 9-33942 9-33976 9-34010 9-34043 9-34077 9-34IH 9.34145 9-34I78 9.34968 9-35004 9-35040 9.35075 9-35III 9-35I46 9.35i8i 9.35217 9.35252 0.65032 0.64996 0.64960 0.64925 0.64889 0.64854 0.64819 0.64783 0.64748 9.98940 9.98938 9.98936 9.98934 9.98933 9-9893I 9.98929 9.98928 9.98926 9-34212 9-35288 0.64712 9.98924 9.34246 9.34279 9-34313 9.34346 9-34380 9.344I3 9-34447 9.34480 9-34514 9-35323 9-35358 9-35394 9-35429 9.35464 9-35499 9-35534 9-35570 9.35605 0.64677 o . 64642 0.64606 0.64571 0.64536 o . 64501 0.64466 0.64430 0.64395 9.98923 9.98921 9.989I9 9.98917 9.98916 9.98914 9.98912 9.98911 9.98909 9-34547 9.35640 0.64360 9.98907 9.34580 9.34614 9-34647 9.34680 9.34713 9-34747 9.34780 9.34813 9.34846 9-35675 9-35710 9-35745 9.3578o 9.35815 9.35850 9-35885 9-35920 9-35955 0.64325 o . 64290 0.64255 0.64220 0.64185 0.64150 0.64115 0.64080 0.64045 9.98905 9.98904 9.98902 9-98900 9.98898 9.98897 9-98895 9.98893 9.98892 9-34879 9-35989 0.64011 9.98890 9-34912 9-34945 9-34978 9-350II 9.35044 9-35077 9 -351 10 9-35143 9-35I76 9.36024 9-36059 9-36094 9.36128 9-36163 9.36198 9-36233 9-36267 9-36302 0.63976 0.63941 0.63906 0.63872 0.63837 0.63802 0.63767 0.63733 0.63698 9.98888 9.98886 9-98885 9-98883 9.98881 9.98879 9.98878 9.98876 9.98874 9-35209 9.36336 o . 63664 9.98872 Cos d. Cot d. c. Tan Sin P.P. 102 77 c (124) 13' 166' Sin d. Tan d. c. Cot Cos I '.P. 00 9.35209 33 9.36336 35 0.63664 9.98872 100 ox 02 03 04 OS 06 07 08 09 9.35242 9-35274 9.35307 9-35340 9-35373 9-35405 9-35438 9-35471 9-35503 32 33 33 33 33 33 32 33 9-3637I 9-36406 9.36440 9.36475 9-36509 9.36543 9.36578 9.36612 9-36647 35 34 35 35 34 35 34 0.63629 0.63594 0.63560 0.63525 0.63491 0.63457 0.63422 0.63388 0.63353 9.98871 9-98869 9-98867 9-98865 9.98864 9.98862 9.98860 9.98816 9-98857 99 98 97 96 95 94 93 92 I 2 3 4 5 6 3S 35 7-0 10.5 14.0 17-5 21.0 10 9.35536 32 9.36681 34 0.63319 9-98855 90 7 8 24-5 28.0 ii 12 13 14 15 16 17 18 19 9-35568 9.356oi 9-35633 9-35666 9.35698 9-35731 9.35763 9.35796 9.35828 33 33 32 32 33 32 32 9-36715 9-36750 9-36784 9.36818 9-36852 9-36887 9-36921 9.36955 9-36989 35 34 34 35 34 34 0.63285 0.63250 0.63216 o . 63182 0.63148 0.63113 0.63079 0.63045 0.63011 9.98853 9.98851 9-98850 9-98848 9-98846 9.98844 9.98842 9.98841 9-98839 89 88 87 86 85 84 83 82 81 9 I 2 3 4 31-5 34 3-4 6.8 IO.2 13-6 20 9.3586o 33 9.37023 34 0.62977 9.98837 80 5 17.0 21 22 23 24 25 26 27 28 29 9.35893 9.35925 9-35957 9.35989 9.36022 9-36054 9.36086 9.36118 9-36I50 32 33 32 32 32 32 3 2 9.37057 9.37091 9.37125 9.37159 9.37193 9-37227 9.37261 9.37295 9.37329 34 34 34 34 34 34 34 0.62943 0.62909 0.62875 0.62841 0.62807 0.62773 0.62739 0.62705 0.62671 9-98835 9.98834 9-98832 9-98830 9.98828 9.98826 9.98825 9.98823 9.98821 79 78 77 76 75 74 73 72 71 7 8 9 I 2 2 3 .8 27.2 30.6 33 3-3 6.6 30 9.36182 32 9.37363 0.62637 9.98819 70 3 9-9 31 32 33 34 35 36 37 38 39 9-36214 9-36246 9-36278 9-36310 9.36342 9-36374 9-36406 9.36438 9.36470 32 32 32 32 32 32 32 9-37397 9-37431 9-37464 9-37498 9-37532 9-37566 9-37599 9.37633 9.37667 34 33 34 34 33 34 o . 62603 0.62569 0.62536 o . 62502 o . 62468 0.62434 o . 62401 0.62367 0.62333 9.98817 9.98816 9.98814 9.98812 9.98810 9.98808 9.98807 9-98805 9-98803 69 68 67 66 65 64 63 62 61 4 5 6 7 8 9 13-2 16.5 19.8 23.1 26.4 29-7 32 40 9.36502 31 9-37700 34 o . 62300 9.08801 60 I 3-2 41 42 43 44 45 46 47 48 49 9.36533 9-36565 9-36597 9-36629 9-36660 9-36692 9.36755 9-36787 32 31 32 32 9-37734 9-37768 9.37801 9.37835 9.37868 9-37902 9-37935 9.37969 9.38002 34 33 34 33 34 33 33 33 0.62266 0.62232 0.62199 0.62165 0.62132 o . 62098 0.62065 0.62031 0.61998 9.98799 9.98798 9-98796 9.98794 9.98792 9.98790 9.98789 9.98787 9-98785 59 58 57 56 55 54 53 52 5i 2 3 4 5 6 7 8 9 6.4 9-6 12.8 16.0 19.2 22.4 25.6 28.8 60 9-36819 9-38035 0.61965 9.98783 50 Cos d. Cot d. c. Tan Sin P P. 103* 76 C (125) 166' Sin d. Tan d. c. Cot Cos , P. P. 50 Si 53 53 54 55 56 57 58 59 60 61 fa 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 93 93 94 95 96 97 98 99 100 9-36819 31 32 31 32 31 32 31 31 32 31 31 32 3i 31 31 32 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 30 31 31 31 30 31 31 30 31 31 30 31 30 31 30 31 30 31 9-38035 34 33 33 34 33 33 34 33 33 33 33 33 34 33 33 33 33 33 33 33 33 33 32 33 33 33 33 33 32 33 33 32 33 33 32 33 33 32 33 32 33 32 33 32 32 33 32 32 33 32 0.61965 9.98783 50 49 48 47 46 45 44 43 43 41 40 39 38 37 36 35 34 33 33 31 30 39 28 37 26 35 34 33 22 21 20 19 18 17 16 IS 14 13 13 XI 10 09 08 07 06 05 04 03 02 01 00 i 2 3 4 5 6 7 8 9 I 2 3 4 5 6 7 8 9 I 2 3 4 5 6 7 8 9 I 2 3 4 5 6 7 8 9 33 3-3 6.6 9-9 13-2 16.5 19.8 23.1 26.4 29-7 32 3-2 6-4 9-6 12.8 16.0 19.2 22.4 25.6 28.8 31 3-1 6.2 9-3 12.4 I5-S 18.6 21.7 24.8 27-9 30 3-0 6.0 9-0 12.0 15-0 18.0 21.0 24.0 27.0 9-36850 9.36882 9-369I3 9.36945 9.36976 9.37008 9-37039 9.37070 9-37102 9.38069 9.38102 9.38I35 9.38169 9.38202 9-38235 9.38269 9-38302 9.38335 0.61931 0.61898 0.61865 0.61831 0.61798 0.61765 0.61731 0.61698 o . 61665 9.98781 9.98780 9.98778 9.98776 9.98774 9.98772 9.98770 9.98769 9.98767 9-37133 9.38368 0.61632 9.98765 9.37I64 9.37196 9-37227 9.37258 9-37289 9-37321 9-37352 9.37383 9-37414 9.38401 9-38434 9-38468 9-38501 9.38534 9.38567 9.38600 9-38633 9.38666 0.61599 0.61566 0.61532 0.61499 0.61466 0.61433 0.61400 0.61367 0.61334 9-98763 9.98761 9-98759 9.98758 9.98756 9.98754 9.98752 9.98750 9.98748 9-37445 9-38699 0.61301 9.98746 9.37476 9-37507 9-37538 9.37569 9.376oo 9.37631 9.37662 9-37693 9.37724 9.38732 9-38765 9.38797* 9-38830 9-38863 9-38896 9-38929 9-38962 9.38994 0.61268 0.61235 0.61203 0.61170 0.61137 0.61104 0.61071 0.61038 0.61006 9.98745 9.98743 9.98741 9.98739 9-98737 9.98735 9.98734 9.98732 9.98730 9-37755 9.39027 0.60973 9.98728 9-37786 9-37817 9.37847 9.37878 9-37909 9-37940 9-37970 9.38001 9-38032 9.39o6o 9.39092 9.39125 9.39158 9.39190 9-39223 9.39256 9.39288 9-39321 0.60940 0.60908 0.60875 0.60842 0.60810 0.60777 0.60744 0.60712 0.60679 9.98726 9.98724 9.98722 9-98720 9.98719 9.98717 9.98715 9.98713 9.98711 9-38062 9-39353 0.60647 9.98709 9-38093 9.38124 9.38154 9-38185 9-38215 9-38246 9-38276 9-38307 9-38337 9-39386 9-39418 9-39451 9.39483 9-39515 9-39548 9.39580 9-39612 9.39645 0.60614 0.60582 0.60549 0.60517 0.60485 0.60452 0.60420 0.60388 0.60355 9.98707 9.98705 9.98704 9.98702 9.98700 9.98698 9-98696 9.98694 9.98692 9.38368 9.39677 0.60323 9.98690 Cos d. Cot d. c. Tan Sin P.P. 103* 76' (126) 166< Sin d. Tan d. c. Cot Cos P.P. 00 9.38368 3 9.39677 32 0.60323 9-98690 100 01 9-38398 or* 9.39709 T* 0.60291 9.98689 99 02 9-38428 o u OT 9-39742 oo 12 0.60258 9.98687 98 03 9-38459 o* 30 9-39744 o-* 2 32 0.60226 9-98685 97 S3 04 9.38489 30 9.398o6 32 0.60194 9.98683 96 i 3-3 05 9.38519 9.39838 0.60162 9.98681 95 2 6.6 06 9-38550 3 1 30 9-39870 32 33 0.60130 9-98679 94 3 9-9 07 9.38580 9 39903 JO 0.60097 9.98677 93 4 13-2 08 9.38610 30 9-39935 o~ 0.60065 9-98675 92 5 16.5 09 9.38640 3 3o 9.39967 32 32 0.60033 9.98673 9i 6 19.8 10 9-38670 31 9-39999 32 0.60001 9.98671 90 7 g 23.1 26.4 it 9-38701 30 9-40031 32 0.59969 9.98670 89 9 29-7 12 9.38731 30 9.40063 32 0.59937 9.98668 88 13 9-3876I 30 9.40095 32 0.59905 9.98666 87 14 9.38791 3 9.40127 32 0.59873 9-98664 86 15 9-38821 -in 9.40159 32 0.59841 9.98662 85 32 16 9-3885I O u 30 9.40191 32 0.59809 9.98660 84 i 3-2 17 9.38881 Ort 9.40223 7-7 0.59777 9.98658 83 2 6-4 18 9.38911 yJ 5O 9 -40255 o* 32 0.59745 9.98656 82 3 9-6 19 9.38941 O u 3O 9.40287 32 0.59713 9.98654 81 4 12.8 20 9-38971 30 9.40319 32 0.59681 9-98652 80 5 g 16.0 21 9-39001 3O 9.40351 31 0.59649 9.98650 79 7 19-2 22.4 22 9.39031 30 9-40382 32 0.59618 9.98648 78 8 25.6 23 9-39061 30 9.40414 32 0.59586 9.98647 77 9 28.8 24 9.39091 30 9.40446 32 0.59554 9.98645 76 25 9-39I2I 20 9.40478 3 2 0.59522 9.98643 75 26 9-39I50 ~y 30 9-40510 31 0.59490 9.98641 74 27 9-39180 30 9.40541 32 0.59459 9.98639 73 31 28 9-39210 30 9.40573 32 0.59427 9.98637 72 i 3-1 29 9-39240 30 9-40605 31 0.59395 9.98635 71 2 6.2 30 9.39270 29 9-40636 32 0.59364 9.98633 70 3 9-3 31 9.39299 30 9.40668 32 0-59332 9.98631 69 4 5 12.4 15.5 32 9.39329 30 9.40700 31 0.59300 9.98629 68 g 18 6 33 9-39359 29 9.4073I 32 0.59269 9.98627 67 7 21.7 34 9.39388 30 9-40763 32 0.59237 9.98625 66 8 24.8 35 9.39418 30 9.40795 31 0.59205 9-98623 65 9 27-9 36 9-39448 29 9.40826 32 0.59174 9.98621 64 37 9-39477 30 9.40858 31 0.59142 9 . 98620 63 38 9-39507 2 9 9.40889 32 0.59111 9.98618 62 39 9-39536 30 9.40921 31 0.59079 9.98616 61 30 40 9-39566 29 9.40952 32 0.59048 9-98614 60 i 3-0 4i 9-39595 3 9.40984 31 0.59016 9.98612 59 2 6.0 42 9.39625 2 9 9-4IOI5 31 0.58985 9.98610 58 3 9-o 43 9.39654 30 9.41046 32 0.58954 9.98608 57 4 5 12 . I5-O 44 9.39684 29 9.41078 31 0.58922 9.98606 56 6 18.0 45 9-39713 30 9.41109 32 0.58891 9.98604 55 7 21 .0 46 9-39743 29 9.41141 31 0.58859 9.98602 54 8 24.0 47 9-39772 29 9.41172 31 0.58828 9.98600 53 9 27.0 48 9.398oi 30 9.41203 32 0.58797 9.98598 52 49 9.39831 29 9-41235 31 0.58765 9.98596 Si 50 9.3986o 9.41266 0.58734 9.98594 60 Cos d. Cot d. c. Tan Sin P.P. 104' 76 (127) 165' Sin d. Tan d.c. Cot Cos , P.P. 60 9.39860 29 9.41266 31 0.58734 9.98594 50 Si 9-39889 in 9.41297 TT 0.58703 9.98592 49 52 9.39919 o*J 20 9-41328 O* -72 0.58672 9.98590 48 53 9-39948 5P 29 9.4i36o O^ 3i 0.58640 9.98588 47 31 54 9-39977 on 9.4I39I or 0.58609 9.98586 46 i 31 55 9.40006 *g on 9.41422 O A 0.58578 9.98584 45 2 6.2 56 9.40035 *9 30 9.4I453 3 1 31 0.58547 9.98582 44 3 9-3 57 9.40065 9.41484 0.58516 9.98580 43 4 12.4 58 9.40094 2 9 on 9.4T5I5 3 1 0.58485 9.98578 42 5 15.5 59 9.40123 29 9.41546 31 32 0.58454 9.98576 41 6 18.6 60 9.40I52 29 9.41578 31 0.58422 9.98574 40 7 8 21.7 24.8 61 9.40181 20 9.41609 OT 0.58391 9.98573 39 9 27.9 62 9.40210 ~y on 9.41640 O A OT 0.58360 9.9857I 38 63 9-40239 *y 29 9.41671 O A 31 0.58329 9-98569 37 64 9.40268 on 9.41702 0.58298 9-98567 36 65 9.40297 *y OQ 9.4I733 31 7T 0.58267 9.98565 35 30 66 9-40326 *y 29 9-41764 o* 3i 0.58236 9-98563 34 i 3-0 67 9.40355 20 9.4I79S on 0.58205 9.98561 33 2 6.0 68 9.40384 ^y on 9-41825 0*"* OT 0.58175 9-98559 32 3 9-o 69 9.40413 *y 29 9.41856 O*- 31 0.58144 9.98557 31 4 12.0 70 9.40442 29 9-41887 31 0.58113 9.98555 30 5 6 iS-o 18.0 7i 9.40471 20 9.41918 3T 0.58082 9.98553 29 7 21. 72 9-40500 ~y 20 9.41949 O- 1 - -2T 0.58051 9.98551 28 8 24.0 73 9-40529 *y 28 9.41980 O- 1 - 31 0.58020 9-98549 27 9 27.0 74 9-40557 20 9.42011 3O 0.57989 9.98547 26 75 9.40586 ~y 9.42041 0.57959 9.98545 25 76 9.40615 2 9 29 9.42072 31 31 0.57928 9-98543 24 OA 77 9.40644 28 9-42103 31 0.57897 9.98S4I 23 29 78 9.40672 20 9-42134 30 0.57866 9-98539 22 I 2-9 79 9.40701 ^y 29 9.42164 31 0.57836 9-98537 21 2 5.8 80 9-40730 2 9 9.42I95 31 0.57805 9-98535 20 3 8.7 TT 6 81 9.40759 28 9.42226 30 0.57774 9.98533 19 4 5 II .O 14.5 82 9.40787 2 9 9.42256 31 0-57744 9-98531 18 6 17.4 83 9.40816 28 9.42287 31 0.57713 9-98529 17 7 20.3 84 9.40844 20 9.42318 3 0.57682 9-98527 16 8 23.2 85 9-40873 *y 2 9 9.42348 31 0.57652 9-98525 IS 9 26.1 86 9-40902 28 9.42379 31 0.57621 9-98523 14 87 9.40930 20 9.42410 3 0.57590 9.98521 13 88 9.40959 ~y 28 9.42440 3 1 0.57560 9-98519 12 89 9.40987 29 9.42471 30 0.57529 9.985I7 II 28 90 9.41016 28 9.42501 31 0-57499 9.98SI5 10 i 2.8 ^ r 91 9.41044 29 9.42532 3O 0.57468 9-985I3 09 2 5-O 8.4 92 9.4I073 28 9.42562 31 0.57438 9-985H 08 II .2 93 9.41101 29 9-42593 30 0.57407 9-98509 07 5 14.0 94 9.4II30 28 9.42623 3O 0.57377 9.98507 06 6 16.8 95 9.4H58 28 9-42653 31 0.57347 9.98505 05 7 19.6 96 9.41186 29 9.42684 30 0.57316 9.98502 04 8 22.4 97 9-4I2I5 28 9.42714 31 0.57286 9.98500 03 9 25.2 98 9-41243 28 9.42745 30 0.57255 9.98498 02 99 9.41271 29 9.42775 30 0.57225 9.98496 OX 100 9.41300 9 . 42805 0.57195 9 . 98494 00 Cos d Cot d. c. Tan Sin P. P. 104' 75 ( (128) 164 < Sin d. Tan d. c. Cot Cos P. P. 00 9.41300 28 9.42805 31 0.57195 9.98494 100 01 02 03 04 05 06 07 08 09 9-41328 9.4I356 9-41384 9.4I4I3 9.41441 9.41469 9.41497 9.41525 9-41553 28 28 29 28 28 28 28 28 2 9 9.42836 9.42866 9.42896 9.42926 9-42957 9.42987 9.43017 9.43047 9.43077 30 30 30 31 30 30 30 30 31 0.57164 0.57134 0.57104 0.57074 0.57043 0.57013 0.56983 0.56953 0.56923 49.98492 9.98490 9-98488 9.98486 9.98484 9.98482 9.98480 9.98478 9.98476 99 98 97 96 95 94 93 92 91 I 2 3 4 5 6 31 3-1 6.2 9-3 12-4 15-5 18.6 10 9.41582 28 9-43108 3O 0.56892 9.98474 90 7 8 21.7 24 8 ii 12 13 14 15 16 17 18 19 9.41610 9.41638 9.41666 9.41694 9.41722 9.41750 9.4I778 9.41806 9-4I834 28 28 28 28 28 28 28 28 27 9.43138 9.43I68 9.43I98 9.43228 9.43258 9.43288 9.43318 9-43348 9.43378 30 30 30 30 30 30 30 30 3 0.56862 0.56832 0.56802 0.56772 0.56742 0.56712 0.56682 0.56652 0.56622 9.98472 9.98470 9.98468 9.98466 9.98464 9.98462 9.98460 9.98458 9.98456 89 88 87 86 85 84 83 82 81 9 I 2 3 4 27.9 30 3-0 6.0 9-0 12. 20 9.41861 28 9.43408 3 0.56592 9.98453 80 5 g 15-0 T o o 21 22 23 24 25 26 27 28 29 9-41889 9.41917 9.4I945 9.4I973 9.42001 9.42029 9.42056 9.42084 9.42112 28 28 28 28 28 27 28 28 28 9-43438 9-43468 9-43498 9.43528 9-43558 9.43587 9-43617 9.43647 9.43677 30 30 30 30 29 30 30 30 3 o . 56562 0.56532 0.56502 0.56472 0.56442 0.56413 0.56383 0.56353 0.56323 9.98451 9.98449 9.98447 9.98445 9.98443 9.98441 9.98439 9.98437 9-98435 79 78 77 76 75 74 73 73 71 7 8 9 I 2 21.0 24.0 27.0 29 2-9 5-8 30 9.42140 27 9.43707 29 0.56293 9.98433 70 3 8-7 3i 32 33 34 35 36 37 38 39 9.42167 9.42195 9-42223 9-42250 9.42278 9-42305 9-42333 9.42361 9-42388 28 28 27 28 27 28 28 27 28 9.43736 9-43766 9.43796 9.43826 943855 9-43885 9-43915 9-43944 9-43974 30 30 30 29 30 30 29 30 3 0.56264 0.56234 0.56204 0.56174 0.56145 0.56115 0.56085 o . 56056 o . 56026 9.98431 9.98429 9.98427 9-98425 9.98422 9.98420 9.98418 9.98416 9-98414 69 68 67 66 65 64 63 62 61 4 5 6 7 8 9 I4-S 17-4 20.3 23.2 26.1 28 40 9-42416 27 9.44004 29 0.55996 9.98412 60 I 2.8 41 42 43 44 45 46 47 48 49 9.42443 9.42471 9.42498 9.42526 9-42553 9.42580 9.42608 9-42635 9-42663 28 27 28 27 27 28 27 28 27 9-44033 9.44063 9.44092 9.44122 9.44I5I 9.44181 9.44210 9.44240 9.44269 30 29 30 29 30 29 30 29 30 0.55967 0.55937 0.55908 0.55878 0.55849 0.55819 0-55790 0.55760 0-55731 9.98410 9.98408 9.98406 9.98404 9.98402 9.98399 9-98397 9.98395 9.98393 59 58 57 56 55 54 53 52 Si 2 3 4 5 6 7 8 9 5-6 8.4 II. 2 14.0 16.8 19.6 22.4 25.2 50 9.42690 9.44299 0.55701 9.98391 50 Cos d. Cot d. c. Tan Sin P P. 105* (129) 16 164 Sin d. Tan d. c. Cot Cos P.P. 50 9.42690 27 9-44299 2Q o.557oi 9.98391 50 51 9.42717 28 9-44328 ~y 0.55672 9.98389 49 52 9-42745 27 9-44358 3 20 0.55642 9-98387 48 53 9.42772 *? 27 9.44387 ~y 29 0.55613 9.98385 47 29 54 9.42799 9.44416 in 0.55584 9-98383 46 I 2.9 55 9.42826 27 28 9.44446 o u 0.55554 9-98381 45 2 5-8 56 9.42854 27 9-44475 29 29 0.55525 9.98378 44 3 8.7 57 9.42881 7*7 9-44504 in 0.55496 9.98376 43 4 ii. 6 58 9-42908 <=7 9-44534 o*-' 0.55466 9.98374 42 5 14-5 59 9-42935 2 7 27 9-44563 29 29 0.55437 9.98372 41 6 17-4 60 9-42962 27 9-44592 30 0.55408 9.98370 40 7 8 20.3 23.2 6i 9.42989 2R 9-44622 0.55378 9.98368 39 9 26.1 62 9-43017 2o 9.4465I 29 0-55349 9.98366 38 63 9-43044 2 7 27 9.44680 29 29 0.55320 9-98364 37 64 9.4307I 9.44709 0.55291 9.98361 36 65 9.43098 27 9.44738 29 0.55262 9.98359 35 28 66 9-43125 2 7 27 9-44768 3 29 0.55232 9.98357 34 i 2.8 6? 9.43I52 9-44797 0.55203 9.98355 33 2 5.6 68 9.43179 27 9-44826 29 0.55174 9.98353 32 3 8.4 69 9-43206 27 27 9.44855 29 2Q 0.55145 9.98351 31 4 II. 2 70 9-43233 27 9.44884 *y 29 0.55116 9-98349 30 5 5 14.0 16.8 71 9.43260 7*7 9-44913 on 0.55087 9-98347 29 7 19.6 72 9.43287 * 9*7 9.44942 ~y 7O 0.55058 9-98344 28 8 22.4 73 9.433I4 *7 27 9-44971 *SP 29 0.55029 9.98342 27 9 25.2 74 9-43341 26 9-45000 20 0.55000 9.98340 26 75 9.43367 jn 9.45029 4y OQ 0.54971 9.98338 25 76 9-43394 f 27 9-45058 *y 29 0.54942 9-98336 24 77 9-43421 9.45087 0.54913 9.y8334 23 27 78 9-43448 27 0*7 9-45ii6 29 20 0.54884 9.98332 22 I 2.7 79 9-43475 1 27 9.45145 ~y 29 0.54855 9-98329 21 2 5-4 80 9-43502 26 9-45174 *y 29 0.54826 9-98327 20 3 8.1 T . Q 81 9-43528 07 9-45203 20 0.54797 9-98325 19 4 IO . o TO r 82 9-43555 *7 9-45232 *y 20 0.54768 9-98323 18 5 A O O 16.2 83 9.43582 27 27 9-4526I vy 29 0.54739 9.98321 17 7 I8.' 9 84 9-43609 96 9-45290 0.54710 9.98319 16 8 21.6 85 9.43635 20 9.45319 29 0.54681 9.98317 *5 9 24.3 86 9.43662 27 27 9-45348 29 28 0.54652 9.983I4 14 87 9.43689 26 9.45376 20 0.54624 9-98312 13 ' 88 9-43715 9-45405 ~y 0.54595 9-98310 12 89 9-43742 27 27 9-45434 29 20 0.54566 9-98308 II 26 90 9.43769 26 9-45463 ~y 29 0.54537 9.98306 10 i 2.6 9i 9-43795 O*T 9-45492 28 0.54508 9.98304 09 2 5-2 7Q 92 9.43822 *7 ?fi 9-45520 0.54480 9-98302 08 3 .8 93 9-43848 20 27 9-45549 29 29 0.54451 9.98299 07 4 5 IO.4 I3.O 94 9.43875 26 9-45578 28 0.54422 9-98297 06 6 15.6 95 9-43901 2*7 9.456o6 2O 0.54394 9-98295 05 7 18.2 96 9.43928 */ 26 9.45635 ~y 29 0.54365 9.98293 04 8 20.8 97 9-43954 27 9.45664 28 0.54336 9.98291 03 9 23 4 98 9.43981 26 9.45692 29 0.54308 9-98289 02 99 9-44007 27 9-45721 2 9 0-54279 9.98286 01 100 9.44034 9-45750 0.54250 9-98284 00 Cos d. Cot d. c. Tan Sin P. P. (130) 16' 163' Sin d. Tan d. c. Cot Cos P. P. 00 OI 02 03 04 05 06 07 08 09 10 ii 12 13 U IS 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 4i 42 43 44 45 46 47 48 49 50 9-44034 26 27 26 26 2? 26 26 27 26 26 27 26 26 26 26 27 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 25 26 26 26 26 25 26 26 26 25 26 26 25 26 26 25 9-45750 28 29 28 29 28 29 29 28 28 29 28 29 28 29 28 28 29 28 28 29 28 28 28 29 28 28 28 28 29 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 27 0.54250 9.98284 100 99 98 97 96 95 94 93 92 91 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 'S 64 63 62 61 60 59 58 57 56 55 54 53 52 Si 50 I 2 3 4 5 6 7 8 9 i 2 3 4 5 6 7 8 9 I 2 3 4 5 6 7 8 9 I 2 3 4 5 6 7 8 9 29 2.9 5-8 8.7 11. 6 14-5 17-4 20.3 23.2 26.1 28 2.8 5-6 8.4 II. 2 14-0 16.8 19.6 22.4 25.2 27 2.7 5-4 8.1 10.8 13-5 16.2 18.9 21.6 24-3 26 2.6 5-2 7-8 10.4 13-0 15.6 18.2 20.8 23-4 9.44o6o 9.44087 9-44II3 9.44139 9.44166 9.44192 9.44218 9-44245 9.44271 9-45778 9-45807 9.45835 9.45864 9-45892 9-45921 9-45950 9-45978 9.46006 0.54222 0.54193 0.54165 0.54136 0.54108 0.54079 0.54050 0.54022 0.53994 9.98282 9.98280 9.98278 9-98275 9-98273 9.98271 9-98269 9.98267 9-98265 9.44297 9-46035 0.53965 9.98262 9-44324 9-44350 9-44376 9.44402 9.44428 9-44455 9.44481 9.44507 9-44533 9-46063 9-46092 9.46120 9.46149 9.46177 9-46205 9-46234 9.46262 9.46290 0.53937 0.53908 0.53880 0.53851 0.53823 0.53795 0.53766 0.53738 0.53710 9.98260 9.98258 9.98256 9-98254 9.98251 9.98249 9.98247 9-98245 9-98243 9-44559 9-46319 0.53681 9.98240 9-44585 9.44611 9-44637 9.44663 9.44689 9-44715 9.44741 9.44767 9-44793 9.46347 9-46375 9-46403 9-46432 9.46460 9.46488 9.46516 9-46544 9.46573 0.53653 0.53625 0.53597 0.53568 0.53540 0.53512 0.53484 0.53456 0.53427 9-98238 9.98236 9-98234 9.98232 9.98229 9.98227 9.98225 9.98223 9.98221 9.44819 9.46601 0.53399 9.98218 9-44845 9.44871 9-44897 9-44923 9.44948 9-44974 9. 45000 9.45026 9.45052 9-46629 9.46657 9.46685 9-467I3 9.46741 9.46769 9.46797 9.46825 9.46853 0.53371 0.53343 0.53315 0.53287 0.53259 0.53231 0.53203 0.53175 0.53147 9.98216 9-98214 9.98212 9-98209 9.98207 9-98205 9.98203 9.98201 9.98198 9-45077 9.46881 0.53H9 9-98196 9-45103 9-45129 9-45155 9.45i8o 9.45206 9-45232 9-45257 9.45283 9.45309 9.46909 9.46937 9-46965 9.46993 9.47021 9.47049 9.47077 9-47105 9-47133 0.53091 0.53063 0.53035 0.53007 0.52979 0.52951 0.52923 0.52895 0.52867 9.98194 9-98192 9.98189 9.98187 9-98185 9.98183 9.98180 9.98178 9.98176 9-45334 9.47160 0.52840 9.98174 Cos d. Cot d. c. Tan Sin P. P. 106< 73 163' Sin d. Tan d. c. Cot Cos , P. P. 50 9-45334 26 9.47160 28 0.52840 9.98174 50 5i 9.4536o 9.47188 28 0.52812 9.98171 49 52 9-45385 26 9.47216 28 0.52784 9.98169 48 53 9-454II 25 9.47244 28 0.52756 9-98167 47 28 54 9-45436 26 9.47272 0.52728 9.98165 46 i 2.8 55 9.45462 9-47299 27 0.52701 9.98162 45 2 5-6 56 9-45487 25 26 9.47327 28 0.52673 9.98160 44 3 8.4 57 9-45513 9-47355 28 0.52645 9-98158 43 4 II. 2 58 9.45538 26 9.47383 07 0.52617 9.98156 42 5 g I4.O 16.8 59 9-45564 25 9.47410 */ 28 0.52590 9.98153 41 7 19.6 60 9.45589 26 9-47438 28 o . 52562 9.98151 40 8 22.4 6i 9.45615 9.47466 0.52534 9.98149 39 9 25.2 62 9.45640 25 or 9-47493 27 28 0.52507 9.98147 38 63 9.45665 *O 26 9-47521 28 0.52479 9.98144 37 64 9.45691 9-47549 0.52451 9.98142 36 27 65 9.45716 25 26 9-47576 27 28 0.52424 9.98140 35 I 2.7 66 9-45742 25 9.47604 28 0.52396 9.98138 34 2 5-4 67 9.45767 9.47632 0*7 0.52368 9.98135 33 3 8.1 68 9-45792 9.47659 * / 28 0.52341 9.98I33 32 4 10.8 69 9-458I7 26 9.47687 2 7 0.52313 9-98131 31 5 13-5 70 9.45843 25 9.47714 28 0.52286 9-98129 30 6 16.2 18 o 71 9-45868 9.47742 0.52258 9.98126 29 7 8 10. y 21.6 72 9-45893 9.47769 27 28 0.52231 9.98124 28 24.3 73 9.459I8 26 9-47797 27 0.52203 9.98122 27 74 9-45944 9.47824 28 0.52176 9.98H9 26 75 9.45969 ' 9.47852 0.52148 9.98117 25 76 9-45994 25 9.47879 27 28 0.52121 9.98II5 24 77 9-46019 9-47907 0.52093 9-98113 23 78 9.46044 ' 9-47934 2? 0.52066 9.98110 22 25 79 9.46069 26 9.47961 27 28 0.52039 9.98108 21 i 2-5 80 9-46095 25 9.47989 27 0.52011 9.98106 20 2 5-0 81 9.46120 9 . 48016 0.51984 9-98103 19 3 7-5 82 9.46145 25 9.48044 2 0.51956 9.98101 18 4 ' 83 9.46170 25 25 9.48071 2 7 27 0.51929 9.98099 17 5 6 12.5 15-0 84 9.46195 9-48098 28 0.51902 9.98097 16 7 17-5 85 9.46220 25 9.48126 0.51874 9.98094 IS 8 20. o 86 9.46245 25 25 9.48I53 2 7 2? 0.51847 9-98092 14 9 22.5 87 9.46270 9.48180 28 0.51820 9.98090 13 88 9-46295 ' 9.48208 0.51792 9.98087 12 89 9-46320 25 25 9-48235 27 27 0.51765 9-98085 II 24 90 9.46345 25 9.48262 27 0.51738 9-98083 10 i 2.4 91 9-46370 9.48289 0.51711 9.98080 09 2 4-8 92 9.46395 25 9-483I7 2 0.51683 9.98078 08 3 7-2 93 9.46420 25 9.48344 2 7 0.51656 9-98076 07 4 9-6 94 9.46444 24 9-48371 27 0.51629 9.98074 06 5 12.0 95 96 9-46469 9.46494 25 25 25 9.48398 9-48425 27 27 28 0.51602 0.51575 9.98071 9-98069 05 04 7 8 14.4 16.8 19.2 97 9-46519 9.48453 0.51547 9.98067 03 9 21.6 98 9.46544 25 9.48480 27 0.51520 9.98064 02 99 9-46569 25 25 9-48507 27 27 0.51493 9.98062 OI 100 9.46594 9.48534 0.51466 9.98060 00 Cos d. Cot d. c. Tan Sin P. P. 106 73 (132) 162 < Sin d. Tan d. c. Cot Cos P. P. 00 9-46594 24 9.48534 27 0.51466 9.98060 100 01 9.46618 9-48561 0.51439 9.98057 99 02 9-46643 2 5 9.48588 2? 0.51412 9.98055 98 03 9.46668 25 25 9-48615 2 7 27 0.51385 9.98053 97 27 04 9-46693 9.48642 0.51358 9.98050 96 i 2-7 05 9.46717 24 9.48669 2 7 0.51331 9-98048 95 2 5-4 06 9-46742 25 25 9.48696 27 27 0.51304 9.98046 94 3 A 8.1 10.8 07 08 9.46767 9.46791 24 2S 9.48723 9.48750 27 07 0.51277 0.51250 9.98043 9.98041 93 92 5 g 13 5 16.2 09 9.46816 25 9.48777 f 27 0.51223 9.98039 9i 7 18.9 10 9.46841 24 9.48804 27 0.51196 9-98036 90 8 21.6 ii 9-46865 or 9.48831 27 0.51169 9.98034 89 9 24.3 12 9.46890 *o or 9-48858 */ 07 0.51142 9.98032 88 13 9-469I5 -so 24 9-48885 * / 27 0.51115 9.98029 87 14 9-46939 9.48912 0.51088 9.98027 86 26 IS 9-46964 25 0/1 9.48939 27 07 0.51061 9-98025 85 I 2.6 16 9-46988 d 25 9.48966 ? 27 0.51034 9.98022 84 2 52 17 9-47013 OA 9.48993 27 0.51007 9 . 98020 83 3 7-8 18 9-47037 t\ or 9.49020 26 0.50980 9.98018 82 4 10.4 19 9.47062 ^O 24 9.49046 27 0.50954 9-98015 Si 5 13-0 20 9.47086 25 9-49073 27 0.50927 9.98013 80 6 15-6 18 2 21 9-47III 2/1 9.49ioo 27 0.50900 9.98011 79 7 8 20.8 22 9.47I35 ^4 or 9.49127 f 07 0.50873 9.98008 78 9 23. ,1 23 9-47i6o ^j 24 9.49154 *? 27 0.50846 9.98006 77 24 9.47184 oe 9.49I 81 26 0.50819 9-98004 76 25 9.47209 ^o *?A 9.49207 07 0.50793 9.98001 75 26 9-47233 ^4 24 9-49234 *? 27 0.50766 9-97999 74 27 9-47257 9.49261 0.50739 9-97997 73 28 9.47282 25 9.49288 27 26 0.50712 9-97994 72 25 29 9.47306 24 24 9-49314 27 0.50686 9.97992 71 i 2.5 30 9-47330 25 9-49341 27 0.50659 9.97989 70 2 S.o 31 9-47355 *?A 9-49368 26 o . 50632 9.97987 69 3 7.5 IO O 32 9-47379 XQ 24 9-49394 27 0.50606 9.97985 68 4 5 12.5 33 9-47403 25 9-49421 27 0.50579 9.97982 67 6 15.0 34 9-47428 0,4 9-49448 26 0.50552 9.97980 66 7 17.5 35 9-47452 n 0,4 9-49474 97 0.50526 9.97978 65 8 20. o 36 9.47476 ^4 24 9-49501 *f 27 0.50499 9-97975 64 9 22.5 37 9.47500 oe 9-49528 26 0.50472 9-97973 63 38 9-47525 *o 24 9-49554 27 0.50446 9-97971 62 39 9-47549 24 9-49581 26 0.50419 9.97968 6x 24 40 9-47573 24 9.49607 27 0.50393 9.97966 60 i 2.4 41 9-47597 9.49634 26 0.50366 9.97963 59 2 4-8 42 9.47621 24 9.49660 0.50340 9.97961 58 3 7.2 43 9.47646 25 24 9.49687 27 26 0.50313 9-97959 57 4 9-6 44 9.47670 7/1 9-49713 o . 50287 9.97956 56 5 12. 45 9.47694 ^4 24 9-49740 27 26 o . 50260 9-97954 55 6 14.4 T r o 46 9-47718 24 9.49766 27 0.50234 9-97951 54 7 8 10. o 19.2 47 9-47742 24 9-49793 26 o . 50207 9-97949 53 9 21.6 48 9.47766 24 9-49819 27 0.50181 9-97947 52 49 9.47790 24 9.49846 26 0.50154 9-97944 Si 50 9.47814 9.49872 0.50128 9.97942 50 Cos d. Cot d. c. Tan Sin P.P. 107' 72 (133) 162 ! Sin d. Tan d. c. Cot Cos P. P. 50 9.47814 24 9.49872 27 0.50128 9-97942 50 Si 9.47838 2A 9.49899 26 0.50101 9-97940 49 52 9.47862 "Q 2\ 9.49925 26 0.50075 9-97937 48 53 9.47886 44 24 9-49951 27 0.50049 9-97935 47 27 54 9-47910 2A 9-49978 26 o . 50022 9-97932 46 I 2.7 55 9-47934 A(\ OA 9.50004 2*7 0.49996 9-97930 45 2 5-4 56 9-47958 *4 24 9-50031 */ 26 0.49969 9.97928 44 3 8.1 57 9.47982 9.50057 26 0.49943 9.97925 43 4 10.8 58 9.48006 24 2A 9-50083 27 0.49917 9.97923 42 5 13-5 fC ~ 59 9-48030 AI\ 24 9.50110 26 0.49890 9-97920 41 10.2 T8 Q 60 9.48054 24 9.50136 26 0.49864 9.97918 40 7 8 io.y 21.6 61 9.48078 /t 9.50162 26 0.49838 9.97916 39 9 24-3 62 9.48102 ^4 0-2 9.50188 0*7 0.49812 9.97913 38 63 9-48125 ^O 24 9-50215 ^7 26 0.49785 9.97911 37 64 9.48149 0,1 9-50241 26 0.49759 9.97908 36 26 65 9.48173 ^4 24 9-50267 26 0.49733 9.97906 35 I 2.6 66 9.48197 24 9-50293 27 0.49707 9.97904 34 2 5-2 67 9.48221 ?A 9-50320 26 0.49680 9.97901 33 3 7.8 68 9.48245 4t\ 00 9.50346 26 0.49654 9.97899 32 4 10.4 69 9.48268 ^O 24 9.50372 26 0.49628 9.97896 31 5 13-0 70 9-48292 24 9.50398 26 0.49602 9.97894 30 6 15-6 ,Q r> 7i 9-48316 24 9-50424 27 0.49576 9.97891 29 7 8 lo . 2 20.8 72 9.48340 27. 9.50451 26 0.49549 9-97889 28 9 27 A 73 9-48363 *O 24 9.50477 26 0.49523 9.97887 27 74 9-48387 24 9.50503 26 0.49497 9.97884 26 75 9.48411 9.50529 26 0.49471 9-97882 25 76 9.48434 23 24 9-50555 26 0.49445 9-97879 24 77 9.48458 9-50581 26 0.49419 9.97877 23 78 9.48482 24 o-? 9-50607 26 0.49393 9.97874 22 24 79 9.48505 *J 24 9-50633 26 0.49367 9.97872 21 i 2.4 80 9.48529 23 9-50659 26 0.49341 9.97870 20 2 4-8 81 9.48552 9A 9.50685 26 0.49315 9.97867 19 3 7.2 o 6 82 9.48576 *4 tJA 9.507H 26 0.49289 9-97865 18 4 y.o 83 9.48600 *4 23 9-50737 26 0.49263 9.97862 17 5 6 14.4 84 9.48623 OA 9-50763 26 0.49237 9.97860 16 7 16.8 85 9.48647 *4 0-5 9-50789 26 0.49211 9.97857 15 8 19.2 86 9.48670 A3 24 9-50815 26 0.49185 9-97855 14 9 21.6 87 9.48694 00 9.50841 26 0.49159 9.97853 13 88 9.48717 A5 9.50867 26 0.49133 9.97850 12 89 9.48741 2 4 23- 9-50893 26 0.49107 9.97848 II 23 90 9-48764 24 9.50919 26 0.49081 9-97845 10 I 2.3 9i 9.48788 O-J 9.50945 26 0.49055 9-97843 09 2 4-6 92 9.48811 A3 9.5097I 26 0.49029 9.97840 08 3 6-9 93 9.48835 24 23 9-50997 26 0.49003 9.97838 07 4 9-2 94 9.48858 9.51023 0.48977 9-97835 06 5 II. 5 95 9.48881 23 9.51048 25 ?fi 0.48952 9-97833 05 13-8 96 9.48905 2 4 23 9.51074 2O 26 o . 48926 9.97830 04 7 8 16. i 18.4 97 9.48928 OA 9.51100 26 0.48900 9-97828 03 9 20.7 98 9.48952 ]M| 07 9-51126 26 0.48874 9.97826 02 . 99 9.48975 A3 23 9-5II52 26 0.48848 9.97823 01 100 9.48998 9.5H78 0.48822 9.97821 00 Cos d. Cot d. c. Tan Sin P.P. 107 ( 72 d34) 18' 161' Sin d. Tan d. c. Cot Cos P. P. 00 9.48998 24 9-5H78 25 o . 48822 9.97821 100 OI 9.49022 01 9-51203 26 0.48797 9.97818 99 02 9.49045 *6 9-51229 26 0.48771 9.97816 98 03 9.49068 23 23 9-51255 26 0.48745 9-978I3 97 26 04 9.49091 9.51281 9C 0.48719 9.978H 96 I 2.0 05 9-49IIS 24 9-51306 *i> 26 0.48694 9.97808 95 2 5-2 06 9-49I38 2 3 23 9-5I332 26 0.48668 9.978o6 94 3 7-8 10 4 07 9.49161 27 9.51358 26 0.48642 9.97803 93 4 5 13.0 08 9.49184 *o ?A 9.51384 2C 0.48616 9.97801 92 6 15.6 09 9.49208 Nt 21 9.51409 ^o 26 0.48591 9-97798 9i 7 18.2 10 9.49231 *o 23 9-51435 26 0.48565 9.97796 90 g 20.8 ii 9-49254 o-i 9.51461 25 0.48539 9-97793 89 9 23.4 12 9-49277 ^O 07. 9.51486 26 0.48514 9-97791 88 13 9.49300 ^o 23 9.5I5I2 25 0.48488 9.97788 87 14 9-49323 2A 9-51537 26 0.48463 9.97786 86 15 9-49347 iBQ 07 9.5I563 26 0.48437 9-97784 85 I 2.5 16 9-49370 *O 23 9.5I589 25 0.48411 9.97781 84 2 5-0 17 9-49393 9.5i6i4 26 0.48386 9-97779 83 3 7-5 18 9.49416 23 9-51640 0.48360 9.97776 82 4 IO.O 19 9-49439 23 23 9-51665 25 26 0.48335 9-97774 81 5 g 12.5 1C. O 20 9.49462 23 9.51691 26 0.48309 9-97771 80 7 *O- w 17.5 21 9.49485 07 9.5I7I7 or 0.48283 9.97769 79 8 20.0 22 9-49508 *o 07 9.51742 *O 26 0.48258 9.97766 78 9 22.5 23 9-49531 *O 23 9.51768 25 0.48232 9-97764 77 24 9-49554 07 9-51793 26 0.48207 9.97761 76 25 9-49577 AS 9.51819 or 0.48181 9-97759 75 26 9.49600 23 23 9.51844 ^5 26 0.48156 9-97756 74 27 9-49623 07 9.51870 Or 0.48130 9-97754 73 24 28 9-49646 ^0 9.51895 *O or 0.48105 9-97751 72 29 9.49669 23 23 9.51920 *5 26 0.48080 9-97749 71 i 2.4 . Q 30 9-49692 23 9.51946 25 0.48054 9-97746 70 2 3 4.0 7-2 31 9-49715 07 9-5I97I 26 0.48029 9-97744 69 4 9.6 32 9-49738 **5 07 9-51997 or 0.48003 9-97741 68 5 12. 33 9.49761 *o 22 9.52022 *o 25 0.47978 9-97739 67 6 14-4 34 9-49783 07 9-52047 26 0.47953 9-97736 66 7 16.8 35 9.49806 **5 9-52073 or 0.47927 9-97734 65 8 19.2 36 9.49829 23 23 9.52098 ^o 26 0.47902 9-97731 64 9 21.6 37 9.49852 O*J 9.52124 or 0.47876 9-97729 63 38 9-49875 *6 9-52149 *o or 0.47851 9.97726 62 39 9.49898 23 22 9-52174 *5 26 0.47826 9.97723 61 23 40 9.49920 23 9.52200 25 0.47800 9-97721 60 I 2.3 41 9-49943 0-3 9.52225 or 0-47775 9.97718 59 2 4-6 6 n 42 9-49966 *O 07 9.52250 *O or 0.47750 9.97716 58 3 9 43 9.49989 ^0 22 9-52275 *o 26 0.47725 9-97713 57 4 5 9.2 II. 5 44 9.50011 07 9-52301 or 0.47699 9.97711 56 6 13.8 45 9-50034 ^o 00 9-52326 *5 or 0.47674 9.97708 55 7 16.1 46 9-50057 **i 23 9-52351 *O 25 0.47649 9.97706 54 8 18.4 47 9.50080 22 9.52376 26 0.47624 9-97703 53 9 20.7 48 9.50102 00 9.52402 or 0.47598 9.97701 53 49 9 -50125 A5 23 9.52427 *j 25 0.47573 9.97698 51 50 9.50148 9.52452 0.47548 9.97696 50 Cos d. Cot d. C. Tan Sin P.P. 108 71 ( (135) 18 ' 161' Sin d. Tan d. c. Cot Cos l P. P. 50 9.50148 22 9.52452 25 0.47548 9.97696 50 Si 9.50170 23 9-52477 25 0-47523 9.97693 49 52 9.50I93 23 9.52502 26 0.47498 9.97691 48 53 9.50216 22 9.52528 25 0-47472 9.97688 47 25 54 9.50238 23 9-52553 25 0.47447 9.97686 46 I 2.5 55 9.50261 22 9.52578 2^ 0.47422 9.97683 45 2 S.o 56 9-50283 23 9-52603 ^O 25 0-47397 9.9768o 44 3 7-5 57 9.50306 22 9.52628 OC 0-47372 9.97678 43 4 IO.O 58 9-50328 00 9-52653 ^5 0-47347 9.97675 42 5 12-5 59 9.5035I ^o 23 9-52678 25 25 0.47322 9.97673 41 6 iS.o 60 9-50374 22 9-52703 25 0.47297 9.97670 40 7 g 17-5 20 o 6z 9.50396 2 3 9.52728 25 0.47272 9.97668 39 9 22. 5 62 9.50419 22 9-52753 25 0.47247 9.97665 38 63 9-50441 23 9.52778 26 0.47222 9.97663 37 64 9-50464 22 9-52804 oC 0.47196 9.9766o 36 65 9.50486 22 9.52829 *o 25 0.47I7I 9.97657 35 24 66 9-50508 23 9-52854 25 0.47146 9-97655 34 I 2.4 6? 9-50531 22 9-52879 oe 0.47121 9.97652 33 2 4-8 68 9-50553 07 9-52904 *j or 0.47096 9.97650 32 3 7-2 69 9.50576 *o 22 9-52929 **) 24 0.47071 9.97647 31 4 9-6 70 9.50598 22 9 52953 25 0.47047 9.97645 30 5 g 12. 7i 9.50620 2T 9.52978 or 0.47022 9.97642 29 7 14.4 16.8 72 9-50643 *o 22 9-53003 3 2*r 0.46997 9.97640 28 8 19.2 73 9-50665 23 9.53028 *o 25 0.46972 9.97637 27 9 21.6 74 9.50688 22 9-53053 or 0.46947 9.97634 26 75 9.50710 22 9-53078 *O or o . 46922 9.97632 25 76 9-50732 23 9-53103 *O 25 0.46897 9.97629 24 77 9-50755 22 9-53128 OC 0.46872 9.97627 23 23 78 9.50777 22 9-53153 9 or 0.46847 9.97624 22 i 2.3 79 9.50799 22 9.53178 O 24 o . 46822 9.97621 21 2 4-6 80 9-50821 23 9-53202 25 0.46798 9.97619 20 3 6-9 81 9-50844 22 9-53227 or 0.46773 9.976i6 19 4 9-2 82 9.50866 9-53252 *o 0.46748 9.97614 18 g ii . 5 j-2 Q 83 9.50888 22 9-53277 25 25 0.46723 9.97611 17 7 Ij.O 16.1 84 9 50910 01 9-53302 "?C 0.46698 9.97609 16 8 18.4 85 9.50933 ~ 24 0.46574 9.97596 II 22 90 9-51043 23 ' 9-53450 25 0.46550 9-97593 10 I 2.2 91 9.51066 22 9-53475 or 0.46525 9-97590 09 2 4-4 6/r 92 9.51088 22 9-53500 *O 0.46500 9-97588 08 3 . O 8Q 93 9.51110 22 9-53525 25 24 0.46475 9-97585 07 4 5 .O II. O 94 9-51132 9-53549 oe 0.46451 9-97583 06 6 13.2 95 9-5II54 9-53574 *o or 0.46426 9-97580 05 7 15.4 96 9-5II76 22 9-53599 *5 24 0.46401 9-97577 04 8 17.6 97 9.51198 22 9.53623 or 0.46377 9-97575 03 9 19.8 98 9.51220 22 9-53648 *o or 0.46352 9-97572 02 99 9-51242 22 9.53673 *o 24 0.46327 9-97570 OI 100 9.51264 9.53697 0.46303 9-97567 00 Cos d. Cot d. c. Tan Sin P.P. 108* 71' (136) 19' 160' Sin d. Tan d. c. Cot Cos P. P. 00 9.51264 22 9.53697 25 0.46303 9.97567 100 01 9.51286 22 9-53722 OA 0.46278 9-97564 99 02 9.5I308 9.53746 ~4 7C 0.46254 9-97562 98 03 9-51330 22 9-53771 ^D 25 0.46229 9-97559 97 25 04 9.51352 22 9.53796 24 0.46204 9-97557 96 I 2.5 OS 9-51374 22 9.53820 25 0.46180 9-97554 95 2 5-o 06 9.- 51396 22 9.53845 24 0.46155 9-97551 94 3 A 7-5 IO.O 07 9.5I4I8 22 9.53869 25 0.46131 9-97549 93 5^%, TO r 08 09 9-5I440 9.51462 22 22 9.53894 9-53918 24 25 0.46106 0.46082 9.97546 9-97543 92 91 6 7 1-; . ^> 15.0 17.5 10 9.51484 22 9-53943 24 0.46057 9-97541 90 8 20. o II 9.51506 21 9.53967 25 0.46033 9-97538 89 9 22.5 12 9.5I527 22 9-53992 O/l 0.46008 9.97536 88 13 9-51549 22 9.54016 ^4 25 0.45984 9-97533 87 A4 14 9.5I57I 22 9.54041 *)A 0.45959 9-97530 86 M IS 9-51593 9-54065 *4 0-45935 9-97528 85 I 2.4 16 9.5I6I5 22 22 9.54090 2 5 24 0.45910 9-97525 84 2 4.8 17 9-5I637 9.54II4 0.45886 9-97522 83 3 7.2 /\ A 18 9-51658 9-54139 25 0.45861 9-97520 82 4 9.0 19 9.51680 22 9.54163 24 24 0.45837 9-97517 81 5 g 12.0 TA A 20 9.51702 22 9.54I87 25 0.45813 9-97515 80 7 14.4 16.8 21 9.51724 9.54212 0.45788 9-97512 79 8 19.2 22 9-51745 22 9.54236 24 2*C 0.45764 9-97509 78 9 21.6 23 9.51767 22 9.54261 *O 24 0-45739 9.97507 77 24 9.51789 22 9.54285 2A 0.45715 9-97504 76 25 9.5i8n 21 9.54309 ^4 or 0.45691 9-97501 75 26 9.5I832 22 9-54334 ^O 24 0.45666 9-97499 74 27 9.5I854 22 9-54358 0.4 0.45642 9.97496 73 22 28 9.51876 21 9.54382 ^4 or 0.45618 9-97493 72 29 9.51897 22 9.54407 ^O 24 0.45593 9-97491 71 I 2.2 30 9.5I9I9 22 9-54431 24 0.45569 9-97488 70 2 3 4-4 6.6 31 9.5I94I 21 9-54455 or 0.45545 9-97485 69 4 8.8 32 9.51962 22 9-5448o *J 0,1 0.45520 9-97483 68 5 II. 33 9.51984 22 9-54504 24 24 0.45496 9.97480 67 6 13.2 34 9.52006 21 9.54528 0/1 0.45472 9-97477 66 7 iS-4 35 9.52027 22 9-54552 Zf\ 0.45448 9-97475 65 8 17.6 36 9.52049 21 9-54577 25 24 0.45423 9.97472 64 9 19.8 37 9.52070 22 9.546oi 0-45399 9.97469 63 38 9.52092 21 9-54625 24 0-45375 9.97467 62 39 9.52H3 22 9.54649 24 24 0.45351 9.97464 61 21 40 9-52135 21 9.54673 25 0.45327 9.9746i 60 I 2.X 41 9.52I56 22 9.54698 *?A 0.45302 9-97459 59 2 4.2 6~ 42 9.52178 21 9-54722 ~4 24 0.45278 9-97456 58 3 3 8. 43 9.52199 22 9.54746 24 0.45254 9-97453 57 4 5 4 10.5 44 9-52221 21 9-54770 24 0.45230 9-97451 56 6 12.6 45 9-52242 22 9-54794 24 0.45206 9.97448 55 7 14.7 46 9.52264 21 9.54818 25 0.45182 9-97445 54 8 16.8 47 9.52285 22 9-54843 9A 0.45157 9-97443 S3 9 18.9 48 9-52307 21 9.54867 *4 0.45133 9-97440 52 49 9.52328 22 9.54891 24 24 0.45109 9-97437 Si 50 9.52350 9.549IS 0.45085 9-97435 50 Cos d. Cot d. c. ' Tan Sin P.P. 109 s 70 < (137) 160* Sin d. Tan d. c. Cot Cos- P. P. 50 9-52350 21 9-54915 24 0.45085 9-97435 50 Si 9-52371 21 9-54939 fyjt 0.45061 9-97432 49 52 9-52392 22 9.54963 *4 24 0.45037 9-97429 48 53 9.S24I4 21 9.54987 24 0.45013 9.97427 47 24 54 9-52435 21 9-550II VA 0.44989 9-97424 46 i 2.4 55 9-52456 9.55035 mQ 0.44965 9.97421 45 2 4-8 56 9-52478 9-55059 24 0.44941 9-97419 44 3 7-2 57 9-52499 21 9-55083 24 24 0.44917 9.97416 43 4 5 9.6 12.0 58 9-52520 22 9-55107 24 0.44893 9-97413 42 6 14.4 59 9.52542 21 9-55I3I 24 0.44869 9.97410 41 7 16.8 60 9-52563 21 9-55155 24 0.44845 9.97408 40 8 19.2 61 9.52584 9-55179 0.44821 9.97405 39 9 21.6 62 9.52606 21 9-55203 24 OA 0.44797 9-97402 38 63 9.52627 21 9-55227 ~4 24 0-44773 9.97400 37 An 64 9-52648 21 9-55251 *yA 0.44749 9-97397 36 23 65 9-52669 21 9.55275 f\ *2A 0.44725 9-97394 35 I 2.3 66 9.52690 22 9.55299 *4 24 0.44701 9.97392 34 2 4-6 6- 67 9-52712 21 9-55323 0.44677 9.97389 33 3 9 68 9-52733 9-55347 24 0.44653 9.97386 32 4 9-2 69 9-52754 21 9-55371 24 24 0.44629 9.97383 31 5 g ZX.f 17 8 70 9-52775 21 9-55395 23 0.44605 9.97381 30 7 J.J.O 16.1 71 9.52796 22 9-55418 fylt 0.44582 9-97378 29 8 18.4 72 9.52818 21 9-55442 ~4 0.44558 9-97375 28 9 20.7 73 9.52839 21 9-55466 24 24 0.44534 9-97373 27 74 9.52860 21 9-55490 O1 0.44510 9-97370 26 75 9.52881 21 9-55514 ^4 0.44486 9-97367 25 76 9.52902 21 9.55538 24 24 0.44462 9.97364 24 77 9-52923 21 9-55562 /71 0.44438 9.97362 23 21 78 9-52944 21 9-55585 *O OA 0.44415 9-97359 22 79 9.52965 21 9.55609 ~4 24 0.44391 9-97356 21 I 2.1 80 9.52986 21 9.55633 24 0.44367 9-97353 20 2 3 4-2 6.3 81 9.53007 21 9-55657 0.44343 9-97351 19 4 8-4 82 9.53028 9.55680 23 0.44320 9-97348 18 5 10.5 83 9-53049 22 9-55704 24 24 0.44296 9-97345 17 6 12.6 84 9-5307I 21 9.55728 0.44272 9-97343 16 7 14.7 85 9.53092 9-55752 24 0.44248 9-97340 15 8 16.8 86 9-53H2 21 9-55775 23 24 0.44225 9-97337 14 9 18.9 87 9.53133 21 9-55799 2/1 0.44201 9-97334 13 88 9-53154 21 9-55823 44 24 0.44177 9-97332 12 89 9-53175 21 9.55847 21 0.44153 9.97329 II 20 90 9.53196 21 9.55870 ^o 24 0.44130 9.97326 10 i 2.0 9i 9-53217 9.55894 0.44106 9.97323 09 2 4.0 fi n 92 9-53238 21 9-55918 24 23 0.44082 9-97321 08 3 A o.o 8.0 93 9-53259 21 9-55941 24 0.44059 9-97318 07 5 10. 94 9-53280 21 9-55965 24 0.44035 9-97315 06 6 12.0 95 9-53301 9.55989 0.44011 9-97312 05 7 14.0 96 9-53322 21 9.56012 23 24 0.43988 9-97310 04 8 16.0 97 9-53343 2O 9.56036 91 0.43964 9-97307 03 9 18.0 98 9.53363 21 9-56059 A5 o* 0.43941 9.97304 02 99 9.53384 21 9-56083 ^4 24 0.43917 9-97301 01 100 9-53405 9.56107 0.43893 9.97299 00 Cos d. Cot d. c. Tan Sin P. P. 109 70 < 20 < 169< Sin d. Tan d. c. Cot Cos P. P. 00 9.53405 21 9.56107 23 0.43893 9-97299 100 01 9-53426 21 9-56130 *?A 0.43870 9.97296 99 02 9-53447 21 9.56154 ~*\ 23 0.43846 9-97293 98 03 9-53468 2O 9.56177 24 0.43823 9.97290 97 24 04 9.53488 21 9.56201 23 0.43799 9.97288 96 I 2.4 05 9-53509 9.56224 0.43776 9-97285 95 2 4-8 06 9-53530 21 9.56248 24 23 0.43752 9.97282 94 3 A 7-2 q 6 07 9-53551 20 9.56271 24 0.43729 9.97279 93 y ^ 12 .0 08 9-53571 21 9-56295 27 0.43705 9.97276 92 5 Ty| A 09 9-53592 21 9.56318 *o 24 0.43682 9-97274 9i 7 14 . 4 16.8 10 9.536I3 21 9-56342 23 0.43658 9.97271 90 8 19-2 II 9.53634 2O 9-56365 ?A 0.43635 9.97268 89 9 21.6 12 9.53654 21 9-56389 *4 27 0.43611 9.97265 88 13 9.53675 21 9-56412 *o 24 0.43588 9'97263 87 14 9.53696 2O 9.56436 23 0.43564 9.97260 86 23 IS 9.537I6 9.56459 0.43541 9-97257 85 I 2.3 16 9-53737 21 9-56483 24 27 0.43517 9-97254 84 2 4-6 17 9-53758 2O 9.56506 *O *>A 0.43494 9.9725I 83 3 6-9 18 9-53778 9.56530 *4 0.43470 9.97249 82 4 9.2 19 9-53799 2O 9.56553 23 23 0-43447 9.97246 81 5 ii. 5 T7 R 20 9.53819 21 9-56576 24 0.43424 9.97243 80 7 13-0 16.1 21 9.53840 21 9.56600 0"} 0.43400 9.97240 79 8 18.4 22 9.53861 20 9-56623 *o 23 0.43377 9.97238 78 9 20.7 23 9.53881 21 9.56646 24 0.43354 9.97235 77 24 9-53902 20 9-56670 0-7 0.43330 9.97232 76 25 9-53922 9-56693 ^o 0.43307 9.97229 75 26 9-53943 20 9.56716 23 24 0.43284 9.97226 74 27 9.53963 21 9.56740 0-7 0.43260 9-97224 73 M 28 9.53984 2O 9-56763 J 0.43237 9.97221 72 mlL 29 9.54004 21 9-56786 23 24 0.43214 9.97218 7i i 2.1 30 9-54025 2O 9.56810 23 0.43190 9.97215 70 2 3 4.2 6.3 31 9.54045 21 9-56833 *77 0.43167 9-97212 69 4 8.4 32 9.54066 2O 9-56856 3 2A 0.43144 9-97210 68 5 10.5 33 9.54086 21 9-56880 **r 23 0.43120 9.97207 67 6 12.6 34 9-54107 20 9-56903 0-7 0.43097 9.97204 66 7 14-7 35 9-54127 21 9.56926 Ai 0.43074 9-97201 65 8 16.8 36 9.54148 2O 9.56949 23 24 0.43051 9.97198 64 9 18.9 37 9.54168 20 9-56973 0.43027 9-97195 63 38 9.54188 9.56996 23 0.43004 9.97I93 62 39 9-54209 20 9.57019 23 23 0.42981 9.97190 61 20 40 9-54229 21 9.57042 23 0.42958 9.97187 60 i 2.0 41 9.54250 2O 9.57065 0.42935 9.97184 59 2 4-0 42 9.54270 20 9-57089 24 23 0.42911 9.97i8i 58 3 6.0 8 o 43 9-54290 21 9.57112 23 0.42888 9-97179 57 4 5 IO.O 44 9-543" 20 9.57135 2 3 0.42865 9.97176 56 6 12. 45 9-54331 20 9-57158 00 0.42842 9.97173 55 7 14.0 46 9-54351 21 9.57i8i ^o 23 0.42819 9.97170 54 8 16.0 47 9-54372 2O 9-57204 Oyf 0.42796 9.97167 53 9 18.0 48 9-54392 2O 9.57228 ~4 23 0.42772 9.97164 52 49 9.54412 21 9.57251 23 0.42749 9.97162 51 50 9-54433 9.57274 0.42726 9.97159 60 Cos d. Cot d. c. Tan Sin P. P. 110 ( (139) 20' 159' Sin d. Tan d. c. Cot Cos p.p. 50 9.54433 20 9-57274 23 0.42726 9-97159 50 5i 9-54453 2O 9-57297 23 0.42703 9.97156 49 52 9-54473 20 9-57320 07 o . 42680 9-97153 48 53 9-54493 21 9-57343 A5 23 0.42657 9-97150 47 23 54 9-54514 9.57366 0.42634 9-97147 46 I 2-3 55 9-54534 9.57389 23 0.42611 9-97145 45 2 4-6 56 9-54554 20 9-57412 23 23 0.42588 9.97142 44 3 6-9 90 57 58 59 9-54574 9-54594 9.54615 2O 21 2O 9-57435 9-57458 9.57481 23 23 23 0.42565 0.42542 0.42519 9-97139 9.97136 9-97133 43 42 41 5 6 7 -* ii. 5 13-8 16.1 60 9-54635 20 9-57504 23 0.42496 9.97130 40 8 18.4 6i 9-54655 9-57527 0.42473 9.97127 39 9 20.7 62 9-54675 2O 9-57550 23 OT 0.42450 9-97125 38 63 9-54695 2O 9-57573 A5 23 0.42427 9-97122 37 64 9-54715 20 9.57596 2-1 0.42404 9.97119 36 22 65 9-54735 9.57619 A3 0.42381 9.97116 35 I 2.2 66 9-54756 20 9-57642 23 23 0.42358 9-97H3 34 2 4-4 6 67 9-54776 20 9.57665 23 0.42335 9.97o 33 3 .0 8 8 68 9-54796 9-57688 0.42312 9.97io8 32 4 69 9.548i6 2O 9-577II 23 23 0.42289 9-97105 31 5 II . O T7 o 70 9.54836 20 9-57734 23 0.42266 9-97102 30 7 AO- iS-4 7i 9-54856 2O 9-57757 0-3 0.42243 9.97099 29 8 17.6 73 9.54876 2O 9-57780 A5 23 0.42220 9.97096 28 9 19-8 73 9.54896 20 9-57803 23 0.42197 9-97093 27 74 9.54916 2O 9.57826 ?7 0.42174 9.97090 26 75 9.54936 2O 9.57849 ^O 22 0.42151 9-97087 25 76 9-54956 2O 9-5787I 23 0.42129 9-97085 24 77 9-54976 2O 9.57894 07 0.42106 9.97082 23 20 78 9.54996 9-57917 *o 0.42083 9-97079 22 79 9-55016 2O 9-57940 23 23 0.42060 9.97076 21 i 2.0 80 9-55036 20 9.57963 23 0.42037 9.97073 20 2 3 4- 6.0 81 9-55056 2o 9-57986 07 0.42014 9.97070 19 4 8.0 82 9-55076 2O 9-58009 ^o 22 0.41991 9.97067 18 5 10. 83 9-55096 20 9-58031 23 0.41969 9.97064 17 6 12.0 84 9-55II6 20 9-58054 71 0.41946 9.97062 16 7 14.0 85 9.55136 TO 9.58077 ^O 07 0.41923 9.97059 IS 8 16.0 86 9-55155 *y 20 9.58100 *O 22 0.41900 9.97056 14 9 18.0 87 9-55175 2O 9.58122 27 0.41878 9-97053 13 88 9-55195 20 9-58145 A3 ?-j 0.41855 9.97050 12 89 9-55215 2O 9.58168 A} 23 0.41832 9-97047 II 19 90 9-55235 20 9.58191 22 0.41809 9-97044 10 I 1-9 9i 9-55255 9-58213 0.41787 9.97041 09 2 38 92 93 9-55275 9-55294 2O 19 20 9-58236 9-58259 23 23 23 0.41764 0.41741 9.97038 9.97036 08 07 3 4 5 5-7 7-6 9-5 94 9-55314 2O 9.58282 22 0.41718 9.97033 06 6 ii. 4 95 9-55334 2O 9-58304 97 0.41696 9.97030 05 7 13-3 96 9-55354 20 9-58327 *o 23 0.41673 9.97027 04 8 15-2 97 9-55374 TO 9-58350 22 0.41650 9.97024 03 9 17.1 98 9-55393 Mr 20 9-58372 27 0.41628 9.97021 02 99 9.55413 2O 9-58395 ^O 23 0.41605 9.97018 01 100 9-55433 9-58418 0.41582 9.97015 00 Cos d. Cot d. c. Tan Sin P. P. 110 ' (140) 168 ' Sin d. Tan d. c. Cot Cos d. P. P. 00 9-55433 9.58418 22 0.41582 9.97015 100 J OI 9-55453 9.58440 0.41560 9.97012 3 99 Oft 02 9-55472 19 9-58463 23 0.41537 9.97009 3 98 03 9-55492 20 20 9.58486 23 22 0.41514 9.97006 3 2 97 i 2 46 04 9-55512 9.58508 0.41492 9.97004 96 3 6^9 05 9-55532 2O 9 58531 23 0.41469 9.97001 3 95 4 9-2 06 9-55551 19 20 9.58554 23 22 0.41446 9.96998 3 3 94 I 11. 5 13.8 07 9-55571 9.58576 0.41424 9.96995 93 7 16.1 08 9-55591 20 9-58599 23 0.41401 9.96992 3 92 8 18.4 09 9.556io 19 9.58621 22 0.41379 9.96989 3 9 20.7 10 9.55630 9.58644 23 0.41356 9.96986 3 90 IX 9-55650 20 9.58666 22 0.41334 9.96983 3 89 L 22 12 9.55669 19 9-58689 23 0.41311 9.96980 3 88 I 2.2 13 9.55689 2O 9-58712 23 22 0.41288 9.96977 3 3 87 2 3 4-4 6.6 14 9-55708 9-58734 0.41266 9.96974 86 4 8.8 15 9.55728 20 9-58757 23 0.41243 9.96971 3 85 5 II. 16 9.55748 2O 9.58779 22 0.41221 9.96968 3 84 6 13.2 17 9.55767 19 9.58802 23 0.41198 9-96965 3 83 7 g 15-4 17.6 18 9.55787 2O 9.58824 22 0.41176 .9-96963 2 82 9 19 9.558o6 19 20 9-58847 23 22 0.4H53 9.96960 3 81 20 9.55826 9-58869 0.41131 9.96957 80 21 9.55845 19 9.58892 23 0.41108 9.96954 3 79 20 22 9.55865 20 9.58914 22 0.41086 9-9695I 3 78 2 23 9.55884 19 20 9.58937 23 22 0.41063 9.96948 3 3 77 2 4-0 24 9-55904 9.58959 0.41041 9.96945 76 3 6.0 25 9.55923 19 9-58981 22 0.41019 9.96942 3 75 4 8.0 26 9-55943 20 19 9.59004 23 22 0.40996 9.96939 3 3 74 5 6 10. 12. 27 9-55962 9.59026 0.40974 9.96936 73 7 14-0 28 9-55982 2O 9.59049 23 0.40951 9.96933 3 72 8 16.0 29 9.56001 19 20 9.5907I 22 0.40929 9-96930 3 71 9 18.0 30 9.56021 9.59094 23 0.40906 9.96927 70 31 9-56040 19 9-59II6 22 0.40884 9.96924 3 69 19 32 9.56060 20 9.59138 22 0.40862 9-96921 3 68 I 1*1 33 9-56079 19 19 9.59161 23 22 0.40839 9-96918 3 3 67 2 3 3-8 5-7 34 9.56098 9.59183 0.40817 9.96915 66 4 7-6 35 9.56118 20 9-59205 22 0.40795 9.96912 3 65 9-5 36 9.56I37 19 2O 9.59228 23 22 0.40772 9.96909 3 64 6 II. 4 37 9.56157 9.59250 0.40750 9.96906 63 7 8 I3.3 15.2 38 9.56176 19 9.59272 22 0.40728 9.96904 2 62 9 17. 1 39 9.56195 19 9.59295 23 0.40705 9.96901 3 61 40 9.56215 20 9.593I7 22 0.40683 9.96898 3 60 41 9-56234 19 9-59339 22 o . 40661 9.96895 3 59 3 42 9.56253 Iy 23 0.40638 9.96892 3 58 I O 3 43 9 56273 20 9.59384 22 0.40616 9.96889 3 57 2 06 44 9.56292 19 9.59406 22 0.40594 9.96886 3 56 3 0.9 45 9-56311 T 9 9-59429 23 0.40571 9-96883 3 55 4 1.2 46 9 56330 20 9-59451 22 22 0.40549 9.96880 3 3 54 6 I -5 1.8 47 9 56350 9-59473 0.40527 9.96877 53 7 2.1 48 9-56369 19 9-59495 22 0.40505 9-96874 3 52 8 2-4 49 9.56388 20 9 59518 23 0.40482 9.96871 3 Si 9 2.7 50 9-56408 9 59540 0.40460 9.96868 3 50 Cos d. Cot d. C. Tan Sin d. P. P. iir (HO 168 ( Sin d. Tan d. c. Cot Cos . d. P. P. 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 9i 92 93 94 95 96 97 98 99 100 9-56408 19 19 19 19 20 19 19 19 19 19 20 19 19 19 19 19 19 19 19 19 19 20 19 19 19 19 19 19 18 19 19 19 19 19 19 19 19 19 3 19 19 19 19 19 18 19 19 19 19 9.59540 22 22 22 23 22 22 22 22 22 23 22 22 22 22 22 22 22 23 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 21 22 22 22 22 22 22 22 21 22 22 22 0.40460 9.96868 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 ii 10 09 08 07 06 05 04 03 02 OI 00 I 2 3 4 1 7 8 9 I 2 3 4 5 6 7 8 9 2 3 4 3 7 8 9 I 2 3 4 I 7 8 9 I 2 3 4 I 7 8 9 22 2.2 4-4 6.6 8.8 II. O 13.2 15-4 17.6 19-8 21 2.1 4-2 6.3 8.4 10.5 12.6 14.7 16.8 18.9 19 i-9 3-8 $ 9-5 II. 4 13 3 15-2 17 I 18 1.8 3-6 5-4 7-2 9-0 10.8 12.6 14-4 16.2 4 0.4 0.8 1.2 1.6 2.0 2.4 2.8 1:1 9-56427 9-56446 9-56465 9-56484 9-56504 9-56523 9-56542 9-56561 9.56580 9-59562 9.59584 9.59606 9.59629 9.59651 9.59673 9.59695 9-59717 9-59739 0.40438 0.40416 0.40394 0.40371 0.40349 0.40327 0.40305 0.40283 0.40261 9-96865 9.96862 9.96859 9.96856 9 96853 9.96850 9-96847 9.96844 9.96841 9-56599 9-59762 0.40238 9-96838 9.56619 9-56638 9-56657 9-56676 9.56695 9-567I4 9.56733 9-56752 9.5677I 9.59784 9.59806 9.59828 9.59850 9.59872 9-59894 9.59916 9-59939 9.5996I 0.40216 0.40194 0.40172 o . 40150 0.40128 0.40106 0.40084 0.49061 0.40039 9.96835 9-96832 9.96829 9.96826 9-96823 9.96820 9-96817 9.96814 9.96811 9.56790 9.59983 0.40017 9.96808 9-56809 9-56829 9.56848 9.56867 9.56886 9-56905 9-56924 9.56943 9-56961 9-60005 9.60027 9-60049 9.60071 9-60093 9.60115 9.60137 9.60159 9.60181 0.39995 0-39973 0-39951 0.39929 0.39907 0.39885 0.39863 0.39841 0.39819 9-96805 9.96802 9.96799 9.96796 9.96793 9.96790 9.96787 9.96784 9-96781 9-56980 9.60203 0-39797 9.96778 9.56999 9.57018 9-57037 9.57056 9.57075 9.57094 9-57H3 9 57132 9.57I5I 9.60225 9.60247 9.60269 9.60291 9-60313 9-60335 9.60357 9.60379 9.60400 0-39775 0-39753 0-39731 0.39709 0.39687 0.39665 0.39643 0.39621 0.39600 9.96774 9.96771 9-96768 9.96765 9-96762 9.96759 9-96756 9-96753 9.96750 9-57169 9.60422 0.39578 9-96747 9.57188 9.57207 9.57226 9.57245 9-57264 9-57282 9 57301 9 57320 9-57339 9.60444 9.60466 9.60488 9.60510 9-60532 9-60554 9.60575 9-60597 9.60619 0.39556 0.39534 0.39512 0-39490 0.39468 0.39446 0-39425 0.39403 0.39381 9.96744 9.96741 9-96738 9.96735 9.96732 9.96729 9.96726 9-96723 9-96720 9 57358 9 . 60641 0.39359 9-967I7 Cos d. Cot d. C. Tan Sin d. P. P. 111' 68 < (142) 22 167 ( Sin d. Tan d.c. Cot Cos d. P . P. 00 9-57358 18 9.60641 0.39359 9-96717 100 OI 02 03 04 05 06 07 08 09 9-57376 9-57395 9.57414 9-57433 9-57451 9-57470 9.57489 9-57507 9.57526 19 19 19 18 19 19 18 19 9.60663 9.60685 9.60706 9.60728 9.60750 9.60772 9-60794 9.60815 9-60837 22 21 22 22 22 22 21 22 0.39337 0.39315 0.39294 0.39272 0.39250 0.39228 0.39206 0.39185 0.39163 9.96714 9.96710 9.96707 9.96704 9-96701 9.96698 9-96695 9.96692 9-96689 4 3 3 3 3 3 3 3 99 98 97 96 95 94 93 92 9i I 2 3 4 5 6 7 8 9 22 2.2 4-4 6.6 8.8 II. 13-2 15-4 17.6 19.8 10 9-57545 18 9-60859 0.39I4I 9.96686 90 ii 12 13 14 15 16 17 18 19 9.57563 9.57582 9.57601 9.57619 9.57638 9.57657 9.57675 9.57694 9-57712 19 19 18 19 19 18 19 18 9.60881 9.60902 9.60924 9.60946 9.60967 9.60989 9.61011 9-61033 9.61054 21 22 22 21 22 22 22 21 0.39H9 0.39098 0.39076 0.39054 0.39033 0.39011 0.38989 0.38967 0.38946 9-96683 9.96680 9.96677 9-96674 9.96670 9.96667 9.96664 9.96661 9-96658 3 3 3 4 3 3 3 3 89 88 87 86 85 84 83 82 81 I 2 3 4 1 7 8 9 21 2.1 8.4 10.5 12.6 14-7 16.8 18.9 20 9-57731 18 9 . 61076 0.38924 9.96655 80 21 22 23 24 25 26 27 28 29 9-57749 9-57768 9.57787 9.57805 9-57824 9.57842 9.57861 9-57879 9.57898 19 19 18 19 18 19 18 19 18 9.61098 9.61119 9.61141 9.61162 9.61184 9.61206 9.61227 9.61249 9.61271 21 22 21 22 22 21 22 22 0.38902 0.38881 0.38859 0.38838 0.38816 0.38794 0.38773 0.38751 0.38729 9.96652 9.96649 9.96646 9-96643 9.96640 9-96636 9 96633 9-96630 9.96627 3 3 3 3 4 3 3 3 79 78 77 76 75 74 73 72 71 I 2 3 4 5 6 7 8 9 19 i-9 3-8 |:i 9-5 II. 4 13 3 15-2 17.1 30 9.579I6 9.61292 0.38708 9.96624 70 3i 32 33 34 35 36 37 38 39 9-57935 9-57953 9-57972 9-57990 9.58008 9.58027 9-58045 9.58064 9.58082 18 19 18 18 19 18 19 18 9-61314 9.6I335 9-61357 9.61378 9.61400 9. 61422 9.61443 9-61465 9.61486 21 22 21 22 22 21 22 21 22 0.38686 0.38665 0.38643 0.38622 0.38600 0.38578 0.38557 0.38535 0.38514 9.96621 9.96618 9.96615 9.96612 9.96608 9-96605 9.96602 9.96599 9.96596 3 3 3 4 3 3 3 3 69 68 67 66 65 64 63 62 61 I 2 3 4 7 8 9 18 1.8 3-6 5-4 7-2 9-0 10.8 12.6 14-4 16.2 40 9.58101 T o 9.61508 0.38492 9.96593 60 4i 42 43 44 45 46 47 48 49 9.58H9 9.58I37 9-58156 9-58I74 9-58192 9-58211 9-58229 9-58247 9.58266 18 19 18 18 19 18 18 19 fO 9.61529 9.6i55i 9-61572 9.61594 9.61615 9.61637 9.61658 9 . 61680 9.61701 22 21 22 21 22 21 22 21 0.38471 0.38449 0.38428 0.38406 0.38385 0.38363 0.38342 0.38320 0.38299 9-96590 9-96587 9-96S83 9.96580 9.96577 9-96574 9.96571 9.96568 9.96565 3 3 4 3 3 3 3 3 3 59 58 57 56 55 54 53 52 5i I 2 3 4 7 8 9 3 0.3 0.6 0.9 1.2 1-5 1.8 2.1 2.4 2.7 50 9.58284 9.61722 0.38278 9.96562 50 Cos d. Cot d.c. Tan Sin d. I >. p. 112 67 ( (143) 22 157' Sin d. Tan d. c. Cot Cos d. P. P. 50 9-58284 18 9.61722 0.38278 9-96562 50 Si 9-58302 9.6i744 0.38256 9.96558 4 49 19 52 9-58321 J 9 T Q 9-61765 21 0.38235 9.96555 3 48 S3 9.58339 lo 18 9.61787 22 21 0.38213 9.96552 3 47 I 2 2.2 4-4 54 9.58357 T o 9.61808 0.38192 9.96549 46 3 6.6 55 9.58375 lo 9.61830 22 0.38170 9.96546 3 45 4 8.8 56 9.58394 *9 18 9.61851 21 21 0.38149 9.96543 3 3 44 5 6 II. 13.2 57 9.58412 T Q 9.61872 0.38128 9.96540 43 7 15-4 58 9.58430 lo T Q 9-61894 22 0.38106 9.96536 4 42 8 17.6 59 9-5*448 Jo 10 9-61915 21 21 0.38085 9.96533 3 41 9 19.8 60 9-58467 *JF 18 9.61936 0.38064 9.96530 3 40 61 9.58485 T o 9.61958 0.38042 9.96527 3 39 21 62 9-58503 lo T o 9.61979 21 0.38021 9-96524 3 38 i 2.1 63 9-5852I lo 18 9.62001 22 21 0-37999 9.96521 3 4 37 2 4-2 6 i 64 9.S8539 T o 9.62022 0.37978 9.96517 36 4 ** j 8.4 65 9.58557 lo 9-62043 21 0-37957 9.96514 3 35 5 10.5 66 9.58576 !9 T Q 9 . 62065 22 0.37935 9.965II 3 34 6 12.6 67 9-58594 lo T o 9.62086 21 0.37914 9.96508 3 33 7 8 14.7 16 8 68 9 58612 lo T Q 9.62107 21 0.37893 9.96505 3 32 9 18.Q 69 9-58630 lo 18 9.62128 21 22 0.37872 9-96502 3 31 70 9.58648 18 9.62150 21 0.37850 9.96498 4 30 71 9.58666 18 9.62171 0.37829 9.96495 29 18 72 9-58684 T Q 9.62192 21 0.37808 9.96492 3 28 I 1.8 73 9.58702 lo 9.62214 22 0.37786 9-96489 3 27 2 3-6 74 9-58721 J 9 18 9-62235 21 O.37765 9.96486 3 26 3 5-4 75 9.58739 18 9 . 62256 21 0.37744 9-96483 3 25 4 5 7- 2 g o 76 9.58757 18 9.62277 21 22 0.37723 9^479 4 3 24 6 y v 10.8 77 9.58775 T o 9-62299 0.37701 9.96476 23 7 12.6 78 9.58793 lo 18 9.62320 21 0.37680 9-96473 3 22 8 14.4 16.2 79 9.58811 18 9-62341 21 21 0.37659 9.96470 3 21 80 9-58829 18 9.62362 21 0.37638 9.96467 3 20 4f* 81 9-58847 18 9-62383 0.37617 9.96463 19 XI 82 9-58865 18 9-62405 0.37595 9.96460 3 18 I i-7 83 9-58883 18 9.62426 21 21 0-37574 9-96457 3 3 17 3 3-4 5-1 84 9.58901 18 9.62447 0.37553 9.96454 16 4 6.3 85 9-58919 18 9.62468 0.37532 9.96451 3 IS 5 8.5 86 9.58937 18 9.62489 22 0-375II 9.96447 4 3 14 7 10. 2 11.9 87 9.58955 18 9.62511 21 0.37489 9.96444 13 8 13-6 88 9.58973 18 9-62532 0.37468 9.96441 3 12 9 15-3 89 9-5899I 18 9-62553 21 0.37447 9-96438 3 3 II 90 9-59009 18 9.62574 21 0.37426 9.96435 10 9i 9-59027 18 9-62595 0.37405 9.96432 09 4 92 9.59045 18 9.62616 0.37384 9.96428 4 08 I 0-4 93 9-59063 18 9-62637 22 0.37363 9-96425 3 3 07 2 3 0.8 1.2 94 9.59o8i T*7 9.62659 0-37341 9.96422 06 4 1.6 95 9-59098 */ 18 9.62680 0.37320 9.96419 3 05 5 2.0 96 9.59ii6 18 9.62701 21 0.37299 9.96415 4 3 04 6 7 2-4 2.8 97 9-59134 18 9 . 62722 21 0.37278 9.96412 03 8 32 98 9.59152 18 9-62743 21 0-37257 9-96409 3 02 9 3-6 99 9.59I70 18 9.62764 21 0.37236 9.96406 3 3 01 100 9-59188 9-62785 0.37215 9-96403 00 Cos d. Cot d. c. Tan Sin d. P. P. 112* 67' (144) 23' 156' Sin d. Tan d. c. Cot Cos d. P. P. 00 01 02 03 04 05 o6 o? 08 09 10 ii 12 13 14 IS 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 4i 42 43 44 45 46 47 48 49 50 9.S9I88 18 17 18 18 18 18 18 17 18 18 18 17 18 18 18 17 18 18 18 I? 18 18 17 18 18 17 18 17 18 18 17 18 17 18 18 17 18 17 18 I? 18 17 18 17 18 17 18 17 18 17 9.62785 21 21 21 21 21 22 21 21 21 21 21 21 21 21 21 2O 21 21 21 21 21 21 21 21 21 21 21 21 2O 21 21 21 21 21 21 20 21 21 21 21 20 21 21 21 20 21 21 21 2O 21 0.37215 9.96403 4 3 3 3 3 4 3 3 3 4 3 3 3 4 3 3 3 4 3 3 3 4 3 3 3 4 3 3 3 4 3 3 4 3 3 4 3 3 3 4 3 3 3 4 3 3 4 3 3 100 99 98 97 96 95 94 93 92 9i 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 Si 50 I 2 3 4 5 6 7 8 9 I 2 3 4 5 6 7 8 9 I 2 3 4 I I 9 I 2 3 4 i I 9 I 2 3 4 7 8 9 21 2.1 4.2 6.3 8.4 10.5 12.6 14-7 16.8 18.9 20 2.0 4-0 6.0 8.0 10. 12. IA.O 16.0 18.0 18 1.8 3-6 5-4 7-2 9.0 10.8 12.6 14.4 16.2 17 1-7 3-4 ii 8.5 10.2 ii. 9 13.6 15.3 3 0-3 0.6 0.9 1.2 I-S 1.8 2.1 2.4 2.7 9.59206 9.59223 9-S924I 9-59259 9-59277 9-59295 9-59313 9-59330 9-59348 9.62806 9.62827 9.62848 9.62869 9 . 62890 9.62912 9-62933 9.62954 9-62975 0.37194 0.37173 0.37152 0.37131 0.37110 0.37088 0.37067 0.37046 0.37025 9.96399 9.96396 9-96393 9.96390 9.96387 9-96383 9-96380 9.96377 9.96374 9-59366 9.62996 0.37004 9-96370 9.59384 9.59401 9-59419 9-59437 9-59455 9-59472 9-59490 9.59508 9-59526 9.63017 9.63038 9-63059 9.63080 9.63101 9.63121 9.63142 9-63163 9 63184 0.36983 0.36962 0.36941 0.36920 0.36899 0.36879 0.36858 0.36837 0.36816 9-96367 9.96364 9-96361 9-96357 9-96354 9-9635I 9.96348 9-96344 9.96341 9-59543 9-63205 0.36795 9-96338 9-59561 9-59579 9-59596 9-596I4 9-59632 9-59649 9.59667 9-59684 9-59702 9.63226 9.63247 9.63268 9-63289 9 63310 9-6333I 9-63352 9 63373 9-63393 0.36774 0.36753 0.36732 0.36711 0.36690 0.36669 0.36648 0.36627 0.36607 9-96335 9-96331 9-96328 9-96325 9.96322 9-96318 9-96315 9-96312 9-96309 9.59720 9-59737 9-59755 9-59772 9-5979 9.59808 9.59825 9-59843 9.5986o 9.59878 9 63414 0.36586 9-96305 9-63435 9-63456 9-63477 9.63498 9-63519 9-63539 9-63560 9-63581 9.63602 0.36565 0.36544 0.36523 0.36502 0.36481 0.36461 0.36440 0.36419 0.36398 9-96302 9.96299 9-96296 9.96292 9.96289 9.96286 9.96282 9.96279 9.96276 9-59895 9 63623 0.36377 9-96273 9-59913 9-59930 9-59948 9.59965 9-59983 9.60000 9.60018 9-60035 9-60053 9-63643 9-63664 9-63685 9.63706 9-63726 9-63747 9.63768 9.63789 9-63809 0.36357 0.36336 0.36315 0.36294 0.36274 0.36253 0.36232 0.36211 0.36191 9.96269 9.96266 9.96263 9.96260 9.96256 9-96253 9.96250 9.96246 9-96243 9.60070 9-63830 0.36170 9.96240 Cos d. Cot d. c. Tan Sin d. P. P. 113' 66' (H5) 156' Sin d. Tan d.c. Cot Cos d. p P. 50 9.60070 9.63830 0.36170 9.96240 50 Si 53 53 54 55 56 57 58 59 9.60087 9 . 60105 9.60122 9.60140 9.60157 9.60174 9.60192 9.60209 9 . 60227 18 17 18 17 17 18 17 18 9-63851 9.63872 9-63892 9.639I3 9-63934 9-63954 9.63975 9.63996 9.64016 21 2O 21 21 20 21 21 20 0.36149 0.36128 0.36108 0.36087 0.36066 0.36046 0.36025 0.36004 0.35984 9-96236 9-96233 9-96230 9-96227 9-96223 9.96220 9.96217 9-96213 9.96210 3 3 3 4 3 3 4 3 49 48 47 46 45 44 43 42 41 I 2 3 4 7 8 9 21 2.1 ti 8.4 10.5 12.6 14-7 16.8 18.9 60 9.60244 9.64037 0.35963 9.96207 40 61 62 63 64 65 66 6? 68 69 9.60261 9.60279 9.60296 9-60313 9-60331 9-60348 9-60365 9.60382 9.60400 18 I? 17 18 17 I? 17 18 9.64058 9.64078 9.64099 9 . 64120 9.64140 9.64161 9.64182 9 . 64202 9-64223 20 21 21 2O 21 21 20 21 0-35942 0.35922 0.35901 0.35880 0.35860 0.35839 0.35818 0.35798 0.35777 9-96203 9.96200 9.96197 9.96I93 9.96190 9-96187 9.96184 9.96180 9.96177 3 3 4 3 3 3 4 3 39 38 37 36 35 34 33 32 31 I 2 3 4 5 6 7 8 9 20 2.0 4-0 6.0 8.0 10. 12. 14.0 16.0 18.0 70 9.60417 9.64243 0.35757 9.96174 30 71 72 73 74 75 76 77 78 79 9.60434 9-60451 9-60469 9.60486 9.60503 9.60520 9-60538 9.60555 9-60572 17 18 17 17 17 18 17 17 9.64264 9-64285 9-64305 9-64326 9.64346 9.64367 9-64387 9.64408 9-64429 21 20 21 20 21 20 21 21 0.35736 0.35715 0.35695 0.35674 0.35654 0.35633 0.35613 0.35592 0.35571 9.96170 9-96167 9.96164 9.96160 9.96157 9.96I54 9-96150 9.96147 9.96144 3 3 4 3 3 4 3 3 29 28 27 26 25 24 23 22 21 I 2 3 4 I 7 8 9 18 1.8 3.6 5-4 7-2 9.0 10.8 12.6 14.4 16.2 80 9-60589 9.64449 0.35551 9.96140 -1 20 17 81 82 83 84 85 86 87 88 89 9.60606 9.60624 9.60641 9.60658 9-60675 9.60692 9.60709 9.60726 9.60744 18 17 17 17 17 17 17 18 9.64470 9.64490 9-64511 9.64531 9.64552 9.64572 9.64593 9.64613 9-64634 20 21 20 21 20 21 20 21 20 0.35530 0.35510 0.35489 0.35469 0.35448 0.35428 0.35407 0.35387 0.35366 9-96I37 9.96134 9.96130 9.96127 9-96123 9.96120 9.96117 9.96113 9.96110 3 4 3 4 3 3 4 3 3 19 18 17 16 15 14 13 12 II I 2 3 4 5 6 7 8 9 1.7 3.4 1:1 8.5 IO.2 II. 9 13.6 15.3 90 9.60761 9.64654 2O 0.35346 9.96107 10 9i 92 93 94 95 96 97 98 99 9.60778 9-60795 9.60812 9.60829 9.60846 9.60863 9.60880 9.60897 9.60914 17 17 17 I? 17 17 17 17 17 9-64674 9-64695 9.647I5 9.64736 9.64756 9.64777 9.64797 9.64818 9-64838 21 20 21 2O 21 20 21 20 20 0.35326 0-35305 0.35285 0.35264 0.35244 0.35223 0.35203 0.35182 0.35162 9.96103 9.96100 9.96097 9.96093 9.96090 9.96087 9 96083 9.96080 9-96076 3 3 4 3 3 4 3 4 3 09 08 07 06 05 04 03 02 OI 2 3 4 5 6 7 8 9 4 0.4 0.8 1.2 1.6 2.0 2-4 2.8 32 3-6 100 9-60931 9-64858 0.35142 9-96073 00 Cos d. Cot d.c. Tan Sin d. F . P. 113' 66' (146) 165< Sin d. Tan d. c. Cot Cos d. F .P. 00 9-60931 9-64858 0.35142 9-96073 100 01 02 03 04 05 06 o? 08 09 9.60948 9-60965 9.60982 9.60999 9.61016 9-61033 9.61050 9.61067 9.61084 17 17 17 17 17 17 17 17 9.64879 9.64899 9-649I9 9.64940 9.64960 9.64981 9.65001 9.65021 9-65042 20 20 21 20 21 2O 20 21 0.35121 0.35101 0.35081 0.35060 0.35040 0.35019 0.34999 0.34979 0.34958 9.96070 9.96066 9-96063 9.96060 9.96056 9-96053 9.96049 9.96046 9-96043 4 3 3 4 3 4 3 3 99 98 97 96 95 94 93 92 9i I 2 3 4 5 6 7 8 9 21 2.1 4-2 6.3 8.4 10.5 12.6 14-7 16.8 18.9 10 9.61101 9 . 65062 0.34938 9-96039 90 ii 12 13 14 15 16 17 18 19 9.61118 9-6II35 9.61152 9.61169 9.61186 9.61203 9.61220 9.61236 9 61253 17 17 17 17 17 17 16 17 9.65082 9-65103 9-65123 9-65143 9.65164 9.65184 9.65204 9.65224 9-65245 21 20 20 21 20 20 20 21 0.34918 0.34897 0.34877 0.34857 0.34836 0.34816 0.34796 0.34776 0.34755 9-96036 9-96032 9.96029 9.96026 9.96022 9-96019 9.96015 9.96012 9.96009 4 3 3 4 3 4 3 3 89 88 87 86 85 84 83 82 81 I 2 3 4 7 8 9 20 2.0 4-0 6.0 8.0 10. 12.0 14-0 16.0 18.0 20 9.61270 9-65265 0.34735 9.96005 80 21 22 23 24 25 26 27 28 29 9.61287 9.61304 9.61321 9.61338 9-61354 9.61371 9.61388 9.61405 9.61422 17 17 17 16 17 17 17 17 16 9-65285 9.65306 9-65326 9.65346 9.65366 9-65387 9.65407 9.65427 9-65447 21 2O 20 20 21 20 20 20 0-34715 0.34694 0.34674 0.34654 0.34634 0.34613 0-34593 0-34573 0.34553 9.96002 9-95998 9-95995 9-95992 9-95988 9^5985 9-95981 9-95978 9-95974 4 3 3 4 3 4 3 4 79 78 77 76 75 74 73 72 7i I 2 3 4 I 7 8 9 17 1-7 3-4 5-1 6.8 8.5 IO.2 II.9 13-6 15-3 30 9-61438 9-65467 0.34533 9-95971 3 70 3i 32 33 34 35 36 37 38 39 9.61455 9.61472 9-61489 9.61506 9.61522 9-61539 9-6I556 9-61573 9-61589 17 17 17 16 17 17 17 16 9-65488 9.65508 9-65528 9-65548 9-65568 9.65589 9.65609 9-65629 9-65649 20 20 20 20 21 2O 20 20 0.34512 0,34492 0.34472 0.34452 0.34432 0.34411 0-34391 0.34371 0.34351 9-95968 9.95964 9.9S96I 9-95957 9-95954 9-95950 9-95947 9-95944 9-95940 3 4 3 4 3 4 3 3 4 69 68 67 66 65 64 63 62 61 I 2 3 4 5 6 7 8 9 16 1.6 3-2 4-8 6.4 8.0 9.6 II. 2 12.8 14.4 40 9.61606 9-65669 0-34331 9-95937 3 60 41 42 43 44 45 46 47 48 49 9.61623 9.61639 9.61656 9.61673 9.61689 9.61706 9.61723 9-61739 9.61756 16 I? 17 16 17 17 16 17 9.65689 9-65710 9-65730 9 65750 9.65770 9.65790 9.65810 9.65830 9-65850 21 20 20 2O 2O 20 2O 2O 0.343H 0.34290 0.34270 0.34250 0.34230 0.34210 0.34190 0.34170 0.34150 9-95933 9-95930 9.95926 9-95923 9.95920 9.95916 9-95913 9.95909 9.95906 3 4 3 3 4 3 4 3 59 58 57 56 55 54 53 52 Si I 2 3 4 5 6 7 8 9 3 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 50 9-61773 9-65870 0.34130 9-95902 4 50 Cos d. Cot d. c. Tan Sin d. I '.p. 65< d47) 165< Sin d. Tan d. c. Cot Cos d. I >. P. 50 9-61773 16 9-65870 0.34130 9-95902' 50 Si 52 53 54 55 56 57 58 59 9-61789 9.61806 9.61823 9.61839 9.61856 9.61872 9.61889 9.61906 9.61922 16 16 17 17 16 9-65890 9-65911 9.65931 9.65951 9.65971 9.6599I 9.66011 9.66031 9.66051 21 2O 2O 20 2O 20 20 20 0.34110 0.34089 0.34069 0.34049 0.34029 0.34009 0.33989 0.33969 0.33949 9.95899 9.95895 9-95892 9-95888 9.95885 9.95882 9-95878 9.95875 9.9587I 4 3 4 3 3 4 3 4 49 48 47 46 45 44 43 42 41 i 2 3 4 i 7 8 9 20 2.0 4.0 6.0 8.0 IO.O 12. 14.0 16.0 18.0 60 9.61939 16 9.66071 0.33929 9-95868 40 fa 62 63 64 65 66 67 68 69 9.61955 9.61972 9.61988 9.62005 9.62021 9.62038 9.62054 9.62071 9.62087 16 17 16 17 16 17 16 9 . 66091 9.66111 9.66131 9.66151 9.66171 9.66191 9.66211 9.66231 9.66251 20 20 20 2O 20 20 20 2O 0.33909 o . 33889 0.33869 0.33849 0.33829 0.33809 0.33789 0.33769 0.33749 9.9586 4 9.9586i 9-95857 9.95854 9.95850 9-95847 9.95843 9.95840 9-95836 3 4 3 4 3 4 3 4 39 38 37 36 35 34 33 32 31 i 2 3 4 I 7 8 9 19 19 3-8 & 9-5 ii. 4 13 3 15-2 17.1 70 9.62104 rfi 9.66271 20 0.33729 9.95833 30 71 72 73 74 75 76 77 78 79 9.62120 9-62137 9-62153 9.62170 9.62186 9.62203 9.62219 9-62235 9.62252 17 16 17 16 17 16 16 17 9.66291 9.66311 9.66331 9-6635I 9.66371 9-66391 9.66411 9.66430 9.66450 2O 20 20 2O 20 20 19 2O 0.33709 0.33689 0.33669 0.33649 0.33629 0.33609 0.33589 0.33570 0-33550 9-95829 9-95826 9-95822 9-958I9 9.95815 9-95812 9.958o8 9.95805 9.958oi 3 4 3 4 3 4 3 4 29 28 27 26 25 24 23 22 21 I 2 3 4 I 9 17 1-7 3-4 5-1 6.8 8.5 10.2 ii. 9 13.6 15.3 80 9.62268 9.66470 6.33530 9-95798 20 81 82 83 84 85 86 87 88 89 9.62285 9.62301 9.62317 9-62334 9.62350 9-62367 9 62383 9.62399 9.62416 16 16 17 16 16 16 9.66490 9.66510 9.66530 9.66550 9-66570 9.66590 9.66609 9.66629 9-66649 20 20 20 20 20 19 20 20 0.33510 0.33490 0.33470 0.33450 0.33430 0.33410 0.33391 0-33371 0.33351 9-95794 9-95791 9-95787 9-95784 9.95780 9-95777 9-95773 9-95770 9.95766 3 4 3 4 3 4 3 4 19 18 17 16 15 14 13 12 II i 2 3 4 5 6 I 9 16 1.6 3-2 4-8 6.4 8.0 9-6 ii. a 12.8 14.4 90 9-62432 T r. 9.66669 0.33331 9.95763 10 92 93 94 95 96 97 98 99 9.62448 9.62465 9.62481 9-62497 9-62513 9.62530 9.62546 9.62562 9.62579 i? 16 16 17 16 16 Tfi 9.66689 9-66709 9-66729 9.66748 9.66768 9.66788 9.66808 9.66828 9.66847 20 2O 19 20 20 20 2O 19 0-333II 0.33291 0.33271 0.33252 0.33232 0.33212 0.33192 0.33172 0.33153 9-95759 9-95756 9-95752 9-95749 9-95745 9-95742 9 95738 9-95735 9.95731 3 4 3 4 3 4 3 4 09 08 07 06 05 04 03 02 01 i 2 3 4 I 7 8 9 4 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3-2 100 9-62595 9.66867 20 0.33133 9.95728 00 Cos d Cot d. c. Tan Sin d. F .p. 114 65' (148) 25< 154 ' Sin d. Tan d. c. Cot Cos d. I '. P. 00 9.62595 rfi 9.66867 0.33133 9-95728 100 OI 02 03 04 05 06 07 08 09 9.62611 9 . 62627 9.62644 9.62660 9.62676 9.62692 9.62708 9.62725 9.62741 16 17 16 16 16 16 17 16 16 9.66887 9-66907 9.66927 9.66946 9.66966 9.66986 9.67006 9-67025 9.67045 20 20 19 20 2O 2O 19 2O 0.33H3 0.33093 0.33073 0.33054 0.33034 0.33014 0.32994 0.32975 0-32955 9.95724 9.95720 9-95717 9-95713 9.95710 9-95706 9-95703 9.95699 9-95696 4 3 4 3 4 3 4 3 99 98 97 96 95 94 93 92 9i I 2 3 4 1 7 8 9 20 2.0 4.0 6.0 8.0 IO.O 12.0 14.0 16.0 18.0 10 9-62757 rfi 9-67065 0.32935 9-95692 90 ii 12 13 14 IS if 17 18 19 9-62773 9.62789 9.62806 9.62822 9.62838 9.62854 9.62870 9.62886 9 . 62902 16 17 16 16 16 16 16 16 16 9-67085 9.67104 9.67124 9.67144 9-67163 9-67183 9-67203 9-67223 9.67242 19 20 2O 19 2O 20 2O 19 0.32915 0.32896 0.32876 0.32856 0.32837 0.32817 0.32797 0.32777 0.32758 9-95689 9.95685 9-95681 9.95678 9-95674 9-95671 9-95667 9.95664 9-9566o 3 4 4 3 4 3 4 3 4 89 88 87 86 85 84 83 82 81 I 2 3 4 1 1 9 19 i-9 3-8 1:1 9-5 ii. 4 13-3 15.2 17.1 20 9.62918 9.67262 0.32738 9.95657 80 21 22 23 24 25 26 27 28 29 9-62935 9-62951 9.62967 9-62983 9.62999 9-63015 9-63031 9-63047 9-63063 17 16 16 16 16 16 16 16 16 16 9.67282 9-67301 9-67321 9-6734I 9-67360 9-67380 9.67399 9.674I9 9.67439 19 2O 2O 19 20 19 2O 2O 0.32718 0.32699 0.32679 0.32659 0.32640 0.32620 0.32601 0.32581 0.32561 9-95653 9-95649 9-95646 9-95642 9.95639 9.95635 9-95632 9-95628 9.95624 4 4 3 4 3 4 3 4 4 79 78 77 76 75 74 73 72 71 i 2 3 4 5 6 7 8 9 17 1-7 3-4 5-1 6.8 8.5 10.2 ii. 9 13-6 15-3 30 9-63079 -rf. 9.67458 0.32542 9-95621 70 31 32 33 34 35 36 37 38 39 9-63095 9.63111 9-63127 9.63I43 9-63I59 9.63I75 9.63191 9-63207 9-63223 16 16 16 16 16 16 16 16 16 9.67478 9.67498 9.67517 9.67537 9.67S56 9.67576 9.67596 9-67615 9-67635 2O 19 20 19 20 20 19 20 0.32522 0.32502 0.32483 0.32463 0.32444 0.32424 0.32404 0.32385 0.32365 9.956I7 9-95614 9.956io 9.956o6 9-95603 9-95599 9 95596 9-95592 9-95588 4 3 4 4 3 4 3 4 4 69 68 67 66 65 64 63 62 61 i 2 3 4 5 6 7 8 9 16 1.6 3-2 4.8 6.4 8.0 9.6 II. 2 12.8 14-4 40 9-63239 16 9.67654 0.32346 9.95S85 60 4i 42 43 44 45 46 47 48 49 9-63255 9.63271 9-63287 9-63303 9-63319 9-63335 9.6335I 9-63367 9-63383 16 16 16 16 16 16 16 16 9.67674 9.67693 9.677I3 9.67732 9.67752 9.67772 9.67791 9.67811 9.67830 19 2O 19 2O 2O 19 2O 19 0.32326 0.32307 0.32287 o . 32268 0.32248 0.32228 0.32209 0.32189 0.32170 9.9558I 9-95578 9-95574 9-95570 9-95567 9.95563 9.9556o 9-95556 9-95552 3 4 4 3 4 3 4 4 59 58 57 56 55 54 53 52 Si I 2 3 4 5 6 7 8 9 3 3 .9 .2 5 .8 .1 4 7 50 9-63398 9-67850 0.32150 9-95549 3 50 Cos d. Cot d. c. Tan Sin d. \ . P. 64' (149) 26< 154' Sin d. Tan d. c. Cot Cos d. f . P. 60 9.63398 16 9-67850 0.32150 9-95549 50 Si 52 53 54 55 56 57 58 59 9.63414 9.63430 9-63446 9-63462 9.63478 9-63494 9-63SIO 9.63525 9-6354I 16 16 16 16 16 16 IS 16 16 9.67869 9.67889 9.67908 9.67928 9-67947 9.67967 9.67986 9.68005 9.68025 19 20 19 20 19 2O 19 19 2O 0.32131 0.32111 0.32092 0.32072 0.32053 0.32033 0.32014 0.31995 0.31975 9-95545 9-95542 9-95538 9-95534 9-95531 9-95527 9-95523 9-95520 9.95516 4 3 4 4 3 4 4 3 4 49 48 47 46 45 44 43 42 41 I 2 3 4 5 6 7 8 9 20 2.0 4-o 6.0 8.0 10. 12. 14.0 16.0 18.0 60 9-63557 16 9.68044 0.31956 9-95513 40 61 62 63 64 65 66 67 68 69 9-63573 9.63589 9.63604 9.63620 9.63636 9.63652 9.63668 9.63683 9.63699 16 15 16 16 16 16 IS 16 16 9.68064 9.68083 9.68103 9.68122 9.68142 9.68161 9.68180 9.68200 9.68219 19 2O 19 20 19 19 20 19 0.31936 0.31917 0.31897 0.31878 0.31858 0.31839 0.31820 0.31800 0.31781 9.95509 9-95505 9-95502 9-95498 9-95494 9-95491 9.95487 9.95483 9.95480 4 3 4 4 3 4 4 3 39 38 37 36 35 34 33 32 31 I 2 3 4 5 6 7 8 9 19 i-9 3-8 \l 9-5 II. 4 13-3 IS-2 I7-I 70 9.63715 16 9.68239 0.31761 9-95476 30 71 72 73 74 75 76 77 78 79 9.6373I 9.63746 9.63762 9.63778 9-63794 9.63809 9-63825 9-63841 9-63856 IS 16 16 16 IS 16 16 IS 16 9.68258 9.68277 9-68297 9.68316 9.68336 9-68355 9.68374 9-68394 9-68413 19 20 19 20 19 19 20 19 0.31742 0.31723 0.31703 0.31684 0.31664 0.31645 0.31626 0.31606 0.31587 9-95473 9.95469 9.95465 9.95462 9-95458 9-95454 9-95451 9-95447 9-95443 4 4 3 4 4 3 4 4 29 28 27 26 25 24 23 22 21 I 2 3 4 i 7 8 9 16 1.6 ii 6.4 8.0 9-6 II. 2 12.8 14-4 80 9.63872 16 9-68432 0.31568 9-95440 20 81 82 83 84 85 86 87 88 89 9.63888 9-63903 9.639I9 9-63935 9.63950 9.63966 9-63982 9-63997 9-64013 IS 16 16 IS 16 16 IS 16 9.68452 9.68471 9.68490 9.68510 9.68529 9.68548 9.68568 9.68587 9.68606 19 19 20 19 19 20 19 19 0.31548 0.31529 0.31510 0.31490 0.31471 0.31452 0.31432 0.31413 0.31394 9-95436 9-95432 9-95429 9-95425 9-95421 9.95418 9-95414 9-95410 9-95407 4 3 4 4 3 4 4 3 19 18 17 16 15 14 13 12 II 2 3 4 7 8 9 15 1-5 3-0 4-5 6.0 7-5 9-0 10. s 12.0 13-5 90 9.64028 16 9.68626 0.31374 9-95403 10 9i 92 93 94 95 96 97 98 99 9.64044 9.64060 9.64075 9.64091 9.64106 9.64122 9-64138 9-64I53 9.64169 16 IS 16 IS 16 16 IS 16 9-68645 9.68664 9.68683 9-68703 9.68722 9.68741 9.68760 9.68780 9-68799 19 19 20 19 19 19 20 19 0.31355 0.31336 0.31317 0.31297 0.31278 0.31259 0.31240 0.31220 0.31201 9-95399 9-95396 9-95392 9-95388 9-95384 9-95381 9-95377 9-95373 9-95369 3 4 ' 4 4 3 4 4 4 09 08 07 06 05 04 03 02 01 I 2 3 4 I 7 8 9 4 -4 .2 .6 .0 4 .8 3-2 3-6 100 9.64184 9.68818 0.31182 9-95366 00 Cos d. Cot d. c. Tan Sin d. P .P. 115 64' (150) 26< 153< Sin d. Tan d. c. Cot Cos d. I J . P. 00 9.64184 16 9.68818 0.31182 9.95366 100 OI 02 03 04 OS 06 07 08 09 9.64200 9-64215 9.64231 9.64246 9.64262 9.64277 9-64293 9-64308 9-64324 15 16 15 16 15 16 15 16 9-68837 9-68857 9.68876 9-68895 9.68914 9-68934 9-68953 9-68972 9-68991 20 19 19 19 20 19 19 19 0.31163 0.3H43 0.31124 0.31105 0.31086 0.31066 0.31047 0.31028 o . 31009 9.9S362 9-95359 9-95355 9-95351 9-95348 9-95344 9-95340 9-95336 9-95333 3 4 4 3 4 4 4 3 99 98 97 96 95 94 93 92 9i 2 3 4 7 8 9 20 2.0 4-0 6.0 8.0 IO.O 12. I4.O 16.0 18.0 10 9-64339 16 9 . 69010 0.30990 9-95329 90 ii 12 13 14 IS 16 17 18 19 9.64355 9.64370 9-64386 9.64401 9.64417 9.64432 9-64447 9-64463 9.64478 15 16 IS 16 IS IS 16 IS 16 9-69029 9.69049 9.69068 9.69087 9.69106 9-69125 9.69144 9-69164 9-69183 20 19 19 19 19 19 20 19 0.30971 0.30951 0.30932 0.30913 0.30894 0.30875 0.30856 0.30836 0.30817 9-95325 9-95322 9.95318 9-95314 9-95310 9.95307 9-95303 9.95299 9.95295 3 4 4 4 3 4 4 4 89 88 87 86 85 84 83 82 81 i 2 3 4 i l 9 19 i-9 3-8 5-7 7.6 9-5 11.4 13-3 15.2 17.1 20 9.64494 It; 9.69202 0.30798 9.95292 80 21 22 23 24 25 26 27 28 29 9-64509 9-64524 9-64540 9.64555 9.6457I 9-64586 9.64601 9.64617 9.64632 15 16 IS 16 15 15 16 IS 15 9.69221 9-69240 9-69259 9-69278 9-69298 9.693I7 9.69336 9.69355 9.69374 19 19 19 2O 19 19 19 19 0.30779 0.30760 0.30741 0.30722 0.30702 0.30683 o . 30664 0.30645 0.30626 9-95288 9-95284 9.95281 9.95277 9.95273 9.95269 9.95266 9.95262 9.95258 4 4 3 4 4 4 3 4 4 79 78 77 76 75 74 73 72 71 i 2 3 4 i 7 8 9 16 1.6 3-2 4-8 6.4 8.0 9-6 II. 2 12.8 14.4 30 9.64647 16 9.69393 0.30607 9-95254 70 3i 32 33 34 35 36 37 38 39 9-64663 9.64678 9.64693 9.64709 9-64724 9-64739 9-64755 9.64770 9.64785 is is 16 IS IS 16 IS IS 15 9.69412 9.69431 9-69450 9.69469 9.69488 9.69507 9-69526 9.69545 9-69565 19 19 19 19 19 19 19 20 0.30588 0.30569 0.30550 0.30531 0.30512 0.30493 0.30474 0.30455 0.30435 9.95251 9.95247 9-95243 9.95239 9.95236 9-95232 9.95228 9-95224 9-95221 3 4 4 4 3 4 4 4 3 69 68 67 66 65 64 63 62 61 i 2 3 4 5 6 7 8 9 15 i.S 3-0 4-5 6.0 7-5 9-0 10. S 12. I3-S 40 9.64800 16 9.69584 0.30416 9.95217 4 60 41 42 43 44 45 46 47 48 49 9.64816 9.64831 9.64846 9.64861 9-64877 9.64892 9.64907 9.64922 9.64938 IS IS IS 16 IS IS IS 16 9-69603 9.69622 9-69641 9.69660 9.69679 9-69698 9.697I7 9.69736 9.69755 19 19 19 19 19 19 19 19 0.30397 0.30378 0.30359 0.30340 0.30321 0.30302 0.30283 0.30264 0.30245 9-95213 9.95209 9.95206 9-95202 9.95198 9.95I94 9.95190 9.95187 9.95183 4 4 3 4 4 4 4 3 4 59 58 57 56 55 54 53 52 51 i 2 3 4 7 8 9 3 0-3 0.6 o.9 1.2 i-S 1.8 2.1 2.4 2.7 50 9.64953 9-69774 0.30226 9.95I79 4 50 Cos d. Cot j d. C. Tan Sin d. p. p. (151) 163' Sin d. Tan d. c. Cot Cos d. P. P. 50 51 52 53 54 55 56 57 58 59 60 6i 62 63 64 65 66 67 68 69 70 7i 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 9i 92 93 94 95 96 97 98 99 100 9-64953 15 15 15 16 15 IS 15 IS 15 15 16 15 15 15 IS 15 IS IS IS 15 16 15 IS 15 15 IS IS IS IS IS IS IS IS IS IS 15 15 15 IS IS IS 14 IS IS IS IS IS 15 15 IS 9-69774 19 19 19 19 18 19 19 19 19 19 19 19 19 19 19 19 19 18 19 19 19 19 19 19 19 18 19 19 19 19 19 19 18 19 19 19 19 18 19 19 19 19 18 19 19 19 18 19 19 19 0.30226 9-95179 4 3 4 4 4 4 3 4 4 4 4 3 4 4 4 4 3 4 4 4 4 3 4 4 4 4 4 3 4 4 4 4 3 4 4 4 4 4 4 3 4 4 4 4 4 3 4 4 4 4 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 3i 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 II 10 09 08 07 06 05 04 03 02 OI 00 i 2 3 4 5 6 7 8 9 I 2 3 4 5 6 7 8 9 I 2 3 4 5 6 7 8 9 I 2 3 4 5 6 7 8 9 I 2 3 4 1 9 19 1-9 3-8 5-7 7.6 9-5 11. 4 13-3 IS- 2 I7.I 18 1.8 3-6 5-4 7-2 9-0 10.8 12.6 14-4 16.2 15 i-S 3-0 4-5 6.0 7-5 9-0 10.5 12.0 13-5 14 1-4 2.8 4-2 5-6 7-0 8.4 9-8 II. 2 12.6 4 0.4 0.8 1.2 1.6 2.O 2.4 2.8 1:1 9.64968 9-64983 9.64998 9.65014 9-65029 9-65044 9-65059 9-65074 9-65089 9.69793 9.69812 9-69831 9-69850 9.69868 9.69887 9.69906 9-69925 9-69944 0.30207 o . 30188 0.30169 0.30150 0.30132 0.30113 0.30094 0.30075 0.30056 9-95175 9.9SI72 9.95168 9.95164 9.95i6o 9.95I56 9-95153 9-95149 9-95145 9-65104 9.69963 0.30037 9.95I4I 9.65120 9-65I35 9-65I50 9.65165 9.65180 9-65I95 9.65210 9-65225 9-65240 9.69982 9.70001 9 . 70020 9.70039 9-70058 9.70077 9.70096 9.70114 9-70133 0.30018 0.29999 0.29980 0.29961 0.29942 0.29923 0.29904 0.29886 0.29867 9.95137 9-95134 9-95I30 9-95126 9-95122 9-95II8 9-95115 9-95III 9-95107 9-65255 9.70152 0.29848 9-95103 9-65271 9.65286 9-6530I 9-653I6 9.65331 9-65346 9-6536I 9-65376 9-65391 9.70171 9.70190 9-70209 9.70228 9.70247 9.70265 9.70284 9-70303 9-70322 0.29829 0.29810 0.29791 0.29772 0.29753 0.29735 0.29716 0.29697 0.29678 9-95099 9-95096 9.95092 9-95088 9-95084 9.9So8o 9-95076 9-95073 9-95069 9.65406 9-70341 0.29659 9-95065 9.65421 9-65436 9-65451 9.65466 9-65481 9-65496 9-655II 9.65526 9-6554I 9-70360 9-70379 9-70397 9.70416 9.70435 9-70454 9.70473 9.70491 9.70510 0.29640 0.29621 0.29603 0.29584 0.29565 0.29546 0.29527 0.29509 0.29490 9.95o6i 9-95057 9.95054 9.950SO 9.95046 9.95042 9.95038 9-95034 9.95030 9-65556 9-70529 0.29471 9-95027 9-6557I 9-65585 9.65600 9-65615 9-65630 9-65645 9.65660 9-65675 9-65690 9.70548 9-70567 9-70585 9.70604 9-70623 9.70642 9.70660 9.70679 9-70698 0.29452 0.29433 0.29415 0.29396 0.29377 0.29358 0.29340 0.29321 0.29302 9-95023 9.95019 9-95015 9 -9501 I 9.95007 9.95004 9-95000 9.94996 9.94992 9.65705 9.70717 o . 29283 9-94988 Cos d. Cot d. c. Tan Sin d. P.P. 116 63' (152) 27' Sin d. Tan d. c. Cot Cos d. P. P. 00 9-65705 9.70717 18 0.29283 9.94988 100 01 9-65720 15 9-70735 0.29265 9.94984 4 99 10 02 9-65734 14 9.70754 19 0.29246 9.94980 4 98 03 9-65749 15 15 9-70773 19 19 0.29227 9-94976 4 3 97 I 2 19 3-8 04 9.65764 9.70792 _Q 0.29208 9-94973 96 3 5-7 05 9-65779 15 9.70810 lo 0.29190 9.94969 4 95 4 7-6 ob 9-65794 15 15 9-70829 19 0.29171 9-94965 4 4 94 6 9-5 11.4 07 9-65809 9.70848 0.29152 9.94961 93 7 13 3 08 9-65823 14 9.70866 18 0.29134 9-94957 4 92 8 IS-2 09 9-65838 15 9.70885 19 0.29115 9-94953 4 91 9 I7.I 10 9.65853 15 9.70904 T Q 0.29096 9-94949 4 90 ii 9.65868 15 9.70922 lo 0.29078 9.94946 3 89 18 12 9-65883 15 9.70941 19 0.29059 9.94942 4 88 I 1.8 13 9-65898 15 9.70960 19 TQ 0.29040 9.94938 4 87 2 3-6 14 9-65912 14 9.70978 IS o . 29022 9-94934 4 86 3 4 5-4 7-2 15 9.65927 15 9.70997 19 0.29003 9-94930 4 85 5 9.0 16 9-65942 15 9.71016 19 0.28984 9-94926 4 84 6 10.8 17 9-65957 15 9-71034 18 0.28966 9.94922 4 83 7 g 12.6 18 9.65971 14 9-71053 19 0.28947 9.94918 4 82 14.4 16.2 19 9.65986 15 9.71072 19 T ft 0.28928 9.94914 4 81 20 9.66001 15 9.71090 lo 0.28910 9.949H 3 80 21 9.66016 15 9.71109 !9 0.28891 9.94907 4 79 IB 22 9.66030 14 9.71128 19 0.28872 9.94903 4 78 23 9-66045 15 15 9.71146 18 0.28854 9.94899 4 A 77 2 1.5 3-0 24 9.66060 9.7H65 0.28835 9.94895 76 3 4-5 25 9-66075 15 9.71184 19 0.28816 9.94891 4 75 4 6.0 26 9.66089 14 15 9.71202 18 19 0.28798 9-94887 4 4 74 1 7-5 9.0 27 9 . 66104 9.71221 0.28779 9.94883 73 7 10-5 28 9.66119 15 9-71239 18 0.28761 9.94879 4 72 8 12.0 29 9 66133 14 IS 9-71258 19 0.28742 9.94875 4 71 9 13-5 30 9.66148 9.71277 T o 0.28723 9.94871 4 70 31 9.66163 IS 9-7I295 lo 0.28705 9-94868 3 69 14 32 9.66177 14 9.7I3I4 19 0.28686 9.94864 4 68 i M 33 9.66192 IS 9-7I332 18 0.28668 9.94860 4 67 2 2.8 34 9.66207 15 9.7I35I 19 0.28649 9.94856 4 66 3 4 4.2 5-6 35 9.66221 14 9.71370 19 0.28630 9.94852 4 65 7-0 36 9.66236 IS 9.71388 18 0.28612 9.94848 4 64 6 8.4 37 9.66251 15 9.71407 19 0.28593 9.94844 4 63 7 9-8 II. 2 38 9.66265 14 9.71425 18 0.28575 9.94840 4 62 9 12.6 39 9.66280 15 T C 9.71444 19 18 0.28556 9.94836 4 61 40 9.66295 T A 9.71462 0.28538 9.94832 4 60 4i 9.66309 14 9.7I48I J 9 0.28519 9.94828 4 59 3 42 9-66324 15 9.71499 18 o . 28501 9.94824 4 58 I O.3 43 9-66338 14 9-7I5I8 19 0.28482 9.94820 4 57 2 0.6 44 9-66353 15 9.7IS37 19 0.28463 9.94817 3 56 3 0.9 45 9-66368 15 9.7I5S5 18 0.28445 9.94813 4 55 4 1.2 46 9.66382 14 IS 9-71574 19 18 o . 28426 9.94809 4 4 54 5 6 1-5 1.8 47 9.66397 9.71592 0.28408 9.94805 53 7 2.1 48 9.66411 14 9.71611 19 0.28389 9.94801 4 52 8 2.4 49 9.66426 15 9.71629 18 0.28371 9-94797 4 5i 9 2.7 50 9.66441 15 9.71648 19 0.28352 9-94793 4 50 Cos d. Cot d. c. Tan Sin d. P.P. 117* 62' d53) 27< 162 * Sin d. Tan d. c. Cot Cos d. P. P. 50 9.66441 T A 9.71648 18 0.28352 9-94793 50 5i 9-66455 14 9.71666 0.28334 9.94789 4 49 1Q 52 9.66470 15 9.71685 19 ,Q 0.28315 9.94785 4 48 53 9.66484 14 15 9.7I703 lo 19 0.28297 9.94781 4 4 47 I 2 !-9 3-8 54 9.66499 9.71722 T Q 0.28278 9-94777 46 3 5-7 55 9.66513 14 9.71740 lo 0.28260 9-94773 4 45 4 7-6 56 9.66528 15 14 9-71759 19 18 0.28241 9-94769 4 4 44 I 9-5 11.4 57 9-66542 9.71777 0.28223 9.94765 43 7 13-3 58 9.66557 15 9.71796 19 T o 0.28204 9.94761 4 42 8 15-2 59 9.66571 14 TC 9.71814 lo TO 0.28186 9-94757 4 41 9 17.1 60 9.66586 lo 9.71833 x y T Q 0.28167 9-94753 4 40 61 9.66600 14 9.7I85I lo -rO 0.28149 9-94749 4 39 18 62 9.66615 15 9.71869 lo 0.28131 9-94745 4 38 i 1.8 63 9.66629 14 9.71888 19 T o 0.28112 9-94741 4 37 2 3-6 64 9.66644 15 9-71906 lo 0.28094 9-94737 4 36 3 4 5-4 7-2 65 9.66658 14 9.71925 19 o . 28075 9 94734 3 35 5 9.0 66 9-66673 IS 9-71943 18 0.28057 9-94730 4 34 6 10.8 67 9.66687 14 9-71962 19 T o 0.28038 9.94726 4 33 7 g 12.6 14 4 68 9.66702 15 9.71980 lo T Q o . 28020 9.94722 4 32 9 16^2 69 9.66716 14 9.71998 lo o . 28002 9.94718 4 31 70 9-66731 15 T A 9.72017 19 18 0.27983 9.94714 4 30 71 9-66745 A 4 9-72035 0.27965 9.94710 4 29 15 72 9.66759 14 9-72054 19 -rQ 0.27946 9.94706 4 28 J Ie 73 9.66774 15 14 9.72072 lo 19 0.27928 9.94702 4 4 27 2 5 3-0 74 9.66788 9-72091 0.27909 9.94698 26 3 4-5 75 9.66803 15 9.72109 18 TQ 0.27891 9.94694 4 25 4 6.0 76 9.66817 14 14 9.72127 lo 19 0.27873 9.94690 4 4 24 1 7-5 9-o 77 9.66831 9.72146 T Q 0.27854 9.94686 23 7 10. s 78 9.66846 15 9.72164 lo 0.27836 9.94682 4 22 8 12.0 79 9.66860 14 9.72182 18 0.27818 9 94678 4 21 9 13-5 80 9-66875 15 9 . 72201 J 9 18 0.27799 9.94674 4 20 81 9.66889 14 9-72219 0.27781 9.94670 4 19 14 82 9.66903 14 9-72238 19 o . 27762 9-94666 18 I 1-4 83 9.66918 15 14 9.72256 18 18 0.27744 9.94662 17 2 3 2.8 4-2 84 9-66932 9.72274 0.27726 9.94658 16 4 5-6 85 9.66946 14 9.72293 19 0.27707 9.94654 15 s 7.o 86 9.66961 15 14 9-723II 18 18 0.27689 9-94650 14 6 n 8.4 o 8 87 9-66975 9.72329 0.27671 9.94646 13 8 y- II 2 88 9.66989 14 9.72348 19 0.27652 9.94642 4 12 9 12^6 89 9.67004 15 T A 9.72366 18 18 0.27634 9.94638 4 II 90 9.67018 A 4 T A 9.72384 TO 0.27616 9.94634 4 10 9i 9-67032 14 9-72403 *9 rQ 0.27597 9.94630 4 09 4 92 9.67047 15 9.72421 lo 0.27579 9-94626 4 08 j 0.4 93 9.67061 14 9-72439 18 0.27561 9.94622 4 07 2 0.8 94 9.67075 14 9-72458 *9 0.27542 9.94618 4 06 3 1.2 _ r 95 9-67090 IS 9.72476 18 0.27524 9.94614 4 05 4 1.0 96 9.67104 14 9.72494 18 0.27506 9.946io 4 04 6 2.4 97 9.67118 14 9.725I3 19 0.27487 9.94606 4 03 7 2.8 98 9-67132 14 9-72531 18 0.27469 9.94602 4 02 8 I'l 99 9.67147 IS 9-72549 18 18 0.27451 9-94598 4 OI 9 3-0 100 9.67161 14 9.72567 0.27433 9-94593 5 00 Cos d. Cot d. c. Tan Sin d. P. P. 117' 62 < (154) 161' Sin d. Tan d. c. Cot Cos d. P. P. 00 9.67161 9.72567 0.27433 9-94593 100 01 9-67I75 14 9-72586 T Q 0.27414 9.94589 99 10 O2 9.67189 14 9.72604 lo 0.27396 9.94585 4 98 03 9.67204 15 14 9.72622 18 19 0.27378 9.94581 4 4 97 I 2 1-9 3-8 04 9.67218 9.72641 TO 0.27359 9-94577 96 3 5-7 05 9-67232 14 9-72659 18 0.27341 9-94573 4 95 4 7.6 06 9-67246 14 15 9.72677 18 18 0.27323 9.94569 4 4 94 5 6 9-5 11.4 07 9.67261 9-72695 0.27305 9-94565 93 7 13-3 08 9.67275 14 9.72714 19 0.27286 9.94561 4 92 8 15-2 09 9-67289 14 9-72732 18 18 0.27268 9-94557 4 9 17.1 10 9.67303 14 9-72750 18 0.27250 9-94553 4 90 ii 9.67317 14 9.72768 0.27232 9-94549 4 89 18 12 9.67332 15 9.72787 19 0.27213 9-94545 4 88 i 1.8 13 14 9-72805 18 0.27195 9-94541 4 87 2 3-6 14 9-67360 14 9.72823 18 _Q 0.27177 9-94537 4 86 3 4 5-4 7-2 IS 9.67374 14 9-72841 lo 0.27159 9-94533 4 85 5 9-o 16 9-67388 14 18 0.27141 9-94529 4 84 6 10.8 17 9.67402 14 9.72878 19 rQ 0.27122 9-94525 4 83 7 a 12.6 18 9.67417 15 9.72896 lo 0.27104 9-94521 4 82 16^2 19 9.6743I 14 9.72914 18 18 0.27086 9-94517 4 81 20 9.67445 14 9.72932 18 0.27068 9-94513 4 80 21 9-67459 14 9.72950 0.27050 9-94508 79 IB 22 9-67473 T 4 9.72969 19 0.27031 9-94504 4 78 If* 23 9.67487 14 14 9.72987 18 18 0.27013 9.94500 4 4 77 2 5 3-0 24 9.67501 9-73005 0.26995 9.94496 76 3 4-5 25 9.675I5 14 9-73023 18 0.26977 9-94492 4 75 4 6.0 26 9-67530 15 14 9.7304I 18 19 0.26959 9.94488 4 4 74 5 6 7-5 9.0 27 9-67544 9.73o6o x y 0.26940 9-94484 73 7 10.5 28 9.67558 14 9.73078 18 0.26922 9.94480 4 72 8 12.0 29 9-67572 14 9.73096 18 18 0.26904 9-94476 4 71 9 13-5 30 9-67586 14 9-73II4 18 0.26886 9.94472 4 70 31 9.67600 14 9.73132 _0 0.26868 9.94468 4 69 14 32 9.67614 14 9-73150 lo 0.26850 9.94464 4 68 I 1-4 33 9.67628 14 14 9-73I69 19 18 0.26831 9.94460 4 5 67 2 3 2.8 4-2 34 9.67642 9.73187 T Q 0.26813 9-94455 66 4 5-6 35 9-67656 T 4 . 9-73205 lo 0.26795 9-94451 4 65 5 7-0 36 9.67670 14 9-73223 18 0.26777 9-94447 4 64 6 8.4 37 9.67684 14 9-73241 18 _Q 0.26759 9-94443 4 63 7 g 9-8 II. 2 38 9-67698 14 9-73259 lo 0.26741 9-94439 4 62 9 12.6 39 9.67712 14 9-73277 18 18 0.26723 9-94435 4 61 40 9.67726 T A 9.73295 o . 26705 9-94431 4 60 41 9.67740 "4 9-73314 T Q 0.26686 9.94427 4 59 4 42 9-67754 14 9-73332 lo T Q 0.26668 9-94423 4 58 I 4 43 9.67768 I 4 9-73350 lo 18 0.26650 9.94419 4 57 2 o'.S 44 9.67782 9-73368 o . 26632 9-94415 56 3 1.2 45 9.67796 14 9-73386 18 0.26614 9.94410 5 55 4 1.6 2 O 46 9.67810 14 14 9-73404 18 0.26596 9.94406 4 4 54 6 2.4 47 9.67824 9-73422 0.26578 9.94402 53 7 2.8 48 49 9-67838 9.67852 14 14 9-73440 9-73458 18 18 18 0.26560 0.26542 9-94398 9-94394 4 4 52 Si 8 9 1:1 50 9-67866 9-73476 0.26524 9-94390 4 50 Cos d. Cot d. c. Tan Sin d. P.P. 118' (155) 28 ( 161 ' Sin d. Tan d. c. Cot Cos d. P. P. 60 9.67866 . 9-73476 0.26524 9-94390 60 Si 9.67880 14 9-73495 T Q 0.26505 9-94386 4 49 1R 52 9.67894 14 9-73513 lo T Q 0.26487 9-94382 4 48 IQ 53 9-67908 14 14 9-73531 lo 18 0.26469 9-94377 5 4 47 I 2 .5 3-6 54 9.67922 9-73549 1-Q 0.26451 9-94373 46 3 5-4 55 9-67936 14 9-73567 lo 0.26433 9-94369 4 45 4 7.2 56 9-67950 14 14 9.73585 18 18 0.26415 9-94365 4 4 44 1 9-0 10.8 57 9.67964 9-73603 ,0 0.26397 9-94361 43 7 12.6 58 9.67978 14 9-73621 Xo _Q 0.26379 9-94357 4 42 8 14.4 59 9.67992 14 9.73639 lo 18 0.26361 9-94353 4 41 9 16.2 60 9.68006 9-73657 18 0.26343 9-94349 4 40 6i 9.68020 4 9.73675 T o 0.26325 9-94344 39 17 62 9-68033 13 9.73693 lo T o o . 26307 9-94340 4 38 i 1.7 63 9.68047 14 14 9-737H lo 18 0.26289 9-94336 4 4 37 2 3 3-4 5 I 64 9.68061 9.73729 T 0.26271 9-94332 36 4 6^8 65 9-68075 14 9-73747 lo ,0 0.26253 9-94328 4 35 5 8.5 66 9.68089 14 9-73765 lo T Q 0.26235 9-94324 4 34 6 10.2 67 9.68103 14 9.73783 IB T Q 0.26217 9.94320 4 33 7 g ii. 9 13.6 68 9-68117 14 9.738oi lo T o 0.26199 9-94315 5 32 9 69 9.68130 13 9.73819 lo 18 0.26181 9-943H 4 3i 70 9.68144 9.73837 18 0.26163 9-94307 4 30 71 9.68158 9-73855 T o 0.26145 9-94303 29 14 73 9.68172 14 9-73873 lo 18 0.26127 9.94299 4 28 i 1-4 73 9.68186 14 14 9-7389I 18 0.26109 9-94295 4 4 27 2 2.8 74 9.68200 9-73909 T o 0.26091 9.94291 26 3 JJ 75 9.68213 13 9-73927 lo T o 0.26073 9.94286 5 25 4 c 5.0 7.0 76 9.68227 14 14 9-73945 lo 18 0.26055 9.94282 4 4 24 6 77 9.68241 9.73963 T Q 0.26037 9.94278 23 7 9-8 78 79 9-68255 9.68269 14 14 14 9-73981 9-73999 lo 18 18 0.26019 0.26001 9.94274 9.94270 4 4 4 22 21 8 9 II. 2 12.6 80 9.68283 9.74017 18 0.25983 9.94266 20 4 A 81 9.68296 3 9-74035 18 0.25965 9.94261 19 M 82 9.68310 14 14 9-74053 18 0.25947 9.94257 4 4 18 I 2 1:1 83 9-68324 14 9.74071 18 0.25929 9-94253 4 17 3 3.9 84 9.68338 9.74089 18 0.25911 9.94249 16 4 5-2 85 9 68351 13 9.74107 18 0.25893 9-94245 4 15 5 6.5 86 9-68365 14 14 9-74125 17 0.25875 9.94241 4 5 14 7 7-8 87 9-68379 9.74142 18 0.25858 9 . 94236 13 8 10.4 88 9.68393 14 9.74160 18 0.25840 9.94232 4 12 9 II. 7 89 9.68406 14 9.74178 18 0.25822 9.94228 4 4 ii 90 9 . 68420 9.74196 18 o . 25804 9.94224 10 9i 9-68434 9-74214 18 0.25786 9.94220 09 6 92 9.68448 9.74232 18 0.25768 9.94215 08 i o.S 93 9.68461 14 9.74250 18 0.25750 9.94211 4 4 07 2 I.O 94 9.68475 14 9.74268 18 0.25732 9.94207 4 06 3 4 1-5 2.O 95 9-68489 T7 9-74286 18 0.25714 9.94203 4 05 5 2-5 96 9.68502 A O 14 9-74304 18 0.25696 9.94199 4 04 6 3-0 97 9.68516 9.74322 0.25678 9.94I95 03 7 3-5 98 99 9.68530 9.68543 13 14 9-74339 9-74357 18 18 0.25661 0.25643 9.94190 9.94186 5 4 4 02 01 8 9 4-0 4-5 100 9-68557 9-74375 o . 25625 9.94182 00 Cos d. Cot d. c. Tan Sin d. P. P. 118* 61' (156) 29< 150' Sin d. Tan d. c. Cot Cos d. P. P. 00 9.68557 9-74375 18 0.25625 9.94182 100 01 9-68571 14 9-74393 T o 0.25607 9.94178 99 18 02 9-68584 13 9.74411 lo TO 0.25589 9.94174 4 98 T K 03 9-68598 14 14 9-74429 lo 18 0.25571 9.94169 5 4 97 X 2 I .0 3-6 04 9.68612 9-74447 T Q 0.25553 9.94I6S 96 3 5-4 05 9.68625 13 9.74465 lo 0.25535 9.94161 4 95 4 7-2 06 9-68639 14 14 9.74482 17 18 0.25518 9-94157 4 s 94 I 9.0 10.8 07 9-68653 9.74500 T o 0.25500 9-94152 93 7 12.6 08 9.68666 13 9.74518 lo 1-0 0.25482 9.94148 4 92 8 14.4 09 9.68680 14 9.74536 lo 18 0.25464 9.94144 4 9i 9 16.2 10 9.68694 J-4 T7 9-74554 18 0.25446 9.94140 90 ii 9-68707 $ 9-74572 0.25428 9.94I36 89 17 12 9.68721 14 9-74589 17 _Q 0.25411 9-94I3I 5 88 I 1-7 13 9-68734 13 TA 9.74607 lo 18 0.25393 9.94127 4 4 87 2 3-4 ST ' 14 9.68748 *n 9-74625 , 0.25375 9-94123 86 4 . X 6.8 15 9.68762 14 9-74643 lo _Q 0.25357 9-94II9 4 85 8.5 16 9.68775 13 14 9.74661 lo 18 0.25339 9.94114 5 4 84 IO.2 TT O 17 9.68789 9-74679 0.25321 9.94110 83 8 11. y 13-6 18 9.68802 13 9.74696 I? T Q 0.25304 9.94106 4 82 9 15-3 19 9.68816 14 T7 9.74714 lo 18 0.25286 9-94102 4 81 20 9.68829 lO Tyt 9-74732 18 0.25268 9.94098 80 21 9-68843 4 9-74750 T Q 0.25250 9-94093 79 14 22 9-68857 14 9-74768 lo 0.25232 9.94089 4 78 I 1.4 23 9.68870 13 14 9.74785 17 18 0.25215 9.94085 4 4 77 2 2.8 24 25 9.68884 9-68897 13 9.74803 9.7482I 18 T o 0.25197 0.25179 9.94081 9.94076 5 76 75 3 4 5 4-2 5-6 7 o 26 9.68911 J-4 13 9.74839 . 1 17 0.25161 9.94072 4 4 74 i 8.4 27 9.68924 9.74856 18 0.25144 9.94068 73 Z 9.8 28 9-68938 14 9.74874 18 0.25126 9.94064 4 72 II. 2 12.6 29 9-68951 13 TA 9.74892 18 0.25108 9-94059 71 30 9-68965 ^ T-7 9.74910 17 0.25090 9-94055 4 70 4O 31 9-68978 1J 14 9-74927 18 0.25073 9.94051 4 69 j * T 1 32 9.68992 T-J 9-74945 18 0.25055 9.94047 68 2 \l 33 9-69005 lO 14 9-74963 18 0.25037 9-94042 4 67 3 3-9 34 9.69019 9.74981 0.25019 9.94038 66 4 5-2 35 9.69032 13 9-74998 I? T o 0.25002 9.94034 4 65 5 6.5 36 9.69046 J-4 13 9-75016 IO 18 0.24984 9.94030 4 5 64 6 7 7.8 9-1 37 9.69059 T A 9.75034 18 0.24966 9.94025 63 8 10.4 38 9-69073 14 9.75052 0.24948 9-94021 62 9 II. 7 39 9.69086 13 TA 9.75069 17 18 0.24931 9.94017 4 61 40 9.69100 IH 13 9.75087 18 0.24913 9.94012 4 60 41 9-69113 9-75105 18 0.24895 9-94008 59 4 42 9.69127 *4 9-75123 TT 0.24877 9.94004 4 58 X 0.4 43 9.69140 13 13 9-75I40 f 18 0.24860 9.94000 4 s 57 2 0.8 44 9.69IS3 9-75158 18 o . 24842 9-93995 56 3 1.2 1.6 45 9.69167 J-4 9-75I76 TT o . 24824 9-93991 55 4 5 2.0 46 9.69180 13 14 9-75193 *7 18 0.24807 9.93987 4 4 54 6 2-4 47 9.69194 T-2 9 -7521 I 18 0.24789 9.93983 53 7 2.8 48 9.69207 o 13 9-75229 18 0.24771 9-93978 4 52 8 3.2 3.6 49 9 . 69220 14 9-75247 17 0.24753 9-93974 4 Si BO 9-69234 9.75264 0.24736 9.93970 50 Cos d. Cot d. c. Tan Sin d. P. P. 119* 60 e (157) 150< Sin d. Tan d. c. Cot Cos d. ] 3 . P. 50 9.69234 9-75264 18 0.24736 9-93970 50 Si 52 53 54 55 56 57 58 59 9-69247 9.69261 9.69274 9-69287 9-69301 9.69314 9-69328 9.69341 9-69354 14 13 13 14 13 14 13 13 14 9.75282 9-75300 9-75317 9-75335 9-75353 9-75370 9.75388 9.75406 9-7542.3 18 17 18 18 17 18 18 17 18 0.24718 0.24700 0.24683 0.24665 0.24647 0.24630 0.24612 0-24594 0.24577 9.93965 9.9396i 9-93957 9-93953 9.93948 9-93944 9-93940 9-93935 9-93931 4 4 4 5 4 4 5 4 49 48 47 46 45 44 43 42 41 I 2 3 4 1 7 8 9 18 1.8 3-6 5-4 7.2 9-o 10.8 12.6 14.4 16.2 60 9-69368 9-75441 18 0.24559 9-93927 40 61 62 63 64 65 66 67 68 69 9-69381 9.69394 9-69408 9-69421 9.69434 9.69448 9.69461 9.69474 9-69487 13 14 13 13 14 13 13 13 9-75459 9-75476 9-75494 9-75511 9.75529 9-75547 9.75564 9-75582 9.756oo 17 18 17 18 18 17 18 18 0.24541 0.24524 0.24506 0.24489 0.24471 0.24453 0.24436 0.24418 0.24400 9-93922 9.93918 9.93914 9.93909 9.93905 9-93901 9.93897 9.93892 9-93888 4 4 5 4 4 4 5 4 39 38 37 36 35 34 33 32 31 I 2 3 4 1 I 9 17 1.7 3-4 5-1 6.8 8.5 IO.2 ii. 9 13.6 15.3 70 9-69501 9.75617 18 0.24383 9.93884 30 71 72 73 74 75 76 77 78 79 9-69514 9-69527 9.69541 9.69554 9-69567 9-69580 9.69594 9.69607 9.69620 13 14 13 13 13 14 13 13 I"? 9.75635 9.75652 9.75670 9-75688 9-75705 9-75723 9-75740 9-75758 9.75776 17 18 18 17 18 17 18 18 I? 0.24365 0.24348 0.24330 0.24312 0.24295 0.24277 0.24260 0.24242 0.24224 9.93879 9.93875 9.93871 9-93866 9.93862 9.93858 9-93853 9.93849 9-93845 4 4 5 4 4 5 4 4 29 28 27 26 25 24 23 22 21 I 2 3 4 5 6 7 8 9 14 1-4 2.8 4.2 5-6 7-0 8.4 9-8 II. 2 12.6 80 9-69633 9-75793 18 0.24207 9-93840 20 81 82 83 84 85 86 87 88 89 9.69647 9.69660 9.69673 9.69686 9.69699 9.69713 9.69726 9.69739 9.69752 13 13 13 13 14 13 13 13 J-7 9.758U 9.75828 9-75846 9.75863 9.75881 9.75899 9.759!6 9-75934 9-75951 17 18 17 18 18 17 18 17 18 0.24189 0.24172 0.24154 0.24137 0.24119 0.24IOI 0.24084 0.24066 0.24049 9-93836 9.93832 9-93827 9.93823 9.93819 9.938I4 9.93810 9.93805 9.938oi 4 5 4 4 5 4 5 4 i9 18 17 16 IS 14 13 12 II 2 3 4 I 7 8 9 IS 1-3 2.6 3-9 5-2 6.5 7-8 9-1 10.4 ii. 7 90 9-69765 14 9.75969 17 0.24031 9-93797 5 10 91 92 93 94 95 96 97 98 99 9.69779 9.69792 9-69805 9.69818 9.69831 9.69844 9-69858 9.69871 9.69884 13 13 13 13 13 14 13 13 13 9.75986 9.76004 9.76021 9.76039 9.76056 9.76074 9-76091 9.76109 9.76126 18 17 18 17 18 17 18 17 18 0.24014 0.23996 0.23979 0.23961 0.23944 0.23926 0.23909 0.23891 0.23874 9-93792 9-93788 9.93784 9-93779 9-93775 9-93771 9.93766 9.93762 9-93757 4 4 5 4 4 5 4 5 09 08 07 06 05 04 03 02 OI I 2 3 4 5 6 7 8 9 5 0.5 I.O i.S 2.0 2.5 3-0 35 4-0 4-5 100 9.69897 9-76144 0.23856 9-93753 00 Cos d. Cot d. c. Tan Sin d. P. P. 119' (158) 30 C 149< Sin d. Tan d. c. Cot Cos d. P.P. 00 9.69897 9.76144 T*7 0.23856 9-93753 100 01 9.69910 13 9.76161 A 7 T-Q 0.23839 9-93749 4 99 10 02 9.69923 13 9.76179 lo 0.23821 9-93744 5 98 T ft 03 9-69936 13 13 9.76196 17 18 0.23804 9-93740 4 4 97 I 2 I.O 3-6 04 9.69949 9-76214 0.23786 9.93736 96 3 5-4 OS 9.69963 14 9-76231 17 0.23769 9-93731 5 95 4 7.2 06 9.69976 13 13 9-76249 18 17 0.23751 9-93727 4 5 94 5 6 9.0 10.8 07 9.69989 9.76266 T o 0.23734 9.93722 93 7 12.6 08 9.70002 13 9.76284 lo 0.23716 9-93718 4 92 8 14.4 09 9-70015 13 T1 9.76301 17 18 0.23699 9-93714 4 9i 9 16.2 10 9 . 70028 A o 9-76319 TT 0.23681 9-93709 90 ii 9.70041 13 9.76336 A 7 T Q 0.23664 9-93705 4 89 17 12 9-70054 13 9.76354 lo 0.23646 9-93700 5 88 i 1-7 13 9.70067 13 9.76371 17 T Q 0.23629 9.93696 4 87 2 3-4 14 9.70080 13 9-76389 lo 0.23611 9.93692 4 86 3 4 1:1 IS 9.70093 13 9.76406 I? -Q 0.23594 9.93687 5 85 5 8.5 16 9.70106 13 9.76424 lo 0.23576 9-93683 4 84 6 10.2 17 9.70119 13 9.76441 17 0.23559 9.93678 5 83 7 g ii. 9 T-I 5 18 9-70132 13 9.76458 17 T Q 0.23542 9-93674 4 82 9 AO.U IS..1 19 9-70145 13 Tyi 9-76476 lo 17 0.23524 9.93670 4 81 20 9-70159 A 4 9-76493 * i 18 0.23507 9.93665 80 21 9.70172 13 9.76511 0.23489 9.93661 4 79 14 22 9.70185 13 9-76528 17 18 0.23472 9-93656 5 78 I 1-4 23 9-70198 13 13 9.76546 17 0.23454 9-93652 4 A 77 2 2.8 24 9.70211 9-76563 0.23437 9.93648 76 3 4 i 25 9.70224 13 9-76580 i? 0.23420 9-93643 5 75 4 5-6 26 9.70237 13 13 9-76598 18 17 0.23402 9-93639 4 5 74 I 7.0 8.4 27 9.70250 9-76615 TQ 0.23385 9.93634 73 7 9-8 28 9-70263 13 9 76633 IS 0.23367 9.93630 4 72 8 II. 2 29 9.70276 13 12 9.76650 17 18 0.23350 9.93625 5 71 9 12.6 30 9.70288 9.76668 TT 0.23332 9.93621 4 70 31 9.70301 13 9-76685 A 7 0.23315 9.93617 4 69 13 32 9.70314 13 9.76702 17 T Q 0.23298 9.93612 5 68 I 1-3 33 9.70327 13 9.76720 lo TT 0.23280 9.936c8 4 67 2 2.6 1 Q 34 9.70340 13 9.76737 "7 0.23263 9-93603 5 66 4 o-y 5-2 35 9-70353 13 9.76754 17 T Q 0.23246 9-93599 4 65 5 6.5 36 9.70366 13 9.76772 lo 0.23228 9-93594 5 64 6 7.8 37 9-70379 13 9.76789 J 7 T o 0.23211 9-93590 4 63 7 8 9-1 10.4 38 9.70392 13 9.76807 lo 0.23193 9-93585 5 62 9 II. 7 39 9.70405 13 TO 9.76824 17 T*7 0.23176 9-93581 4 61 40 9.70418 *-o 9.76841 */ T Q 0.23159 9-93577 4 60 41 9-70431 13 9-76859 lo 0.23141 9-93572 5 59 4 42 9.70444 13 9.76876 17 0.23124 9-93568 4 58 f\ A 43 9-70457 13 13 9.76893 17 18 0.23107 9-93563 5 A 57 2 u -4 0.8 44 9.70470 9.76911 0.23089 9-93559 56 3 1.2 45 9.70482 12 9.76928 17 0.23072 9-93554 5 55 4 1.6 46 9-70495 13 13 9-76945 17 18 0.23055 9-93550 4 5 54 g 2.O 2-4 47 9-70508 9-76963 0.23037 9-93545 53 7 2.8 48 9.70521 13 9.76980 I? 0.23020 9-93541 4 52 8 3-2 49 9-70534 13 T^ 9.76998 18 0.23002 9-93537 4 51 9 3.6 50 9-70547 *J 9-77015 ^7 0.22985 9-93532 5 50 Cos d. Cot d. c. Tan Sin d. P.P. 120 69 (159) 30< 149 ' Sin d. Tan d. c. Cot Cos d. P. P. 50 9.70547 9-77015 0.22985 9-93532 60 Si 9.70560 13 T7 9.77032 J 7 18 0.22968 9.93528 4 49 18 52 9-70573 A O 9.77050 0.22950 9-93523 48 T ft 53 9-70585 12 13 9.77067 17 17 0.22933 9-93519 4 5 47 i 2 I . O 3-6 54 9.70598 9.77084 0.22916 9.93514 46 3 5-4 55 9.70611 13 9.77101 17 0.22899 9-935IQ 4 45 4 7-2 56 9.70624 13 13 9.77119 18 17 0.22881 9.93505 5 4 44 | 9.0 10.8 57 9.70637 9-77I36 0.22864 9-93501 43 7 12.6 58 9.70650 13 9-77153 I? 0.22847 9-93496 5 42 8 14.4 59 9.70662 12 T1 9-77I7I 18 0.22829 9-93492 4 41 9 16.2 60 9.70675 *O TO 9.77188 ^7 T7 0.22812 9-93487 5 40 61 9.70688 *O 9.77205 * / 0.22795 9.93483 39 17 62 9.70701 13 9.77223 18 0.22777 9-93478 5 38 i 1-7 63 9.70714 13 TO 9.77240 17 T7 0.22760 9-93474 4 37 2 3-4 5T 64 9.70727 A O 9-77257 */ 0.22743 9.93469 36 4 A 6.8 65 9-70739 12 9.77274 17 0.22726 9.93465 4 35 5 8.5 66 9-70752 13 9.77292 18 0.22708 9.93460 5 34 6 10.2 6? 9.70765 13 9-77309 17 0.22691 9.93456 4 33 I II. 13.6 68 9.70778 13 9-77326 17 T Q 0.22674 9-93451 5 32 9 15-3 69 9.70790 12 9-77344 Io 0.22656 9-93447 4 31 70 9-70803 13 T7. 9.77361 17 T*7 0.22639 9-93442 5 80 7i 9.70816 A O 9-77378 !/ 0.22622 9-93438 4 29 13 72 9.70829 13 9-77395 I? T Q 0.22605 9-93433 5 28 i I 3 73 9.70842 13 9-77413 Io 0.22587 9 93429 4 27 2 2.6 74 9-70854 12 9-77430 17 0.22570 9-93424 5 26 3 3-9 75 9.70867 13 9-77447 17 0.22553 9.93420 4 25 4 e 1.1 76 9.70880 13 12 9.77464 17 18 0.22536 9-93415 5 4 24 6 V. o 7-8 77 9.70892 9.77482 0.22518 9 -9341 i 23 I 9-i 78 9-70905 13 9-77499 i? 0.22501 9.93406 5 22 8 10.4 79 9-70918 13 9.77516 17 0.22484 9-93402 4 21 9 11.7 80 9.70931 13 9-77533 17 18 0.22467 9-93397 5 20 81 9.70943 12 9-77551 0.22449 9-93393 4 19 12 82 9-70956 13 9-77568 i? 0.22432 9-93388 5 18 i 1.2 83 9.70969 13 12 9-77585 17 17 0.22415 9-93384 4 5 17 2 3 11 84 9.70981 9.77602 0.22398 9-93379 16 4 4-8 85 9-70994 13 9.77619 i? T Q 0.22381 9-93375 4 15 6.0 86 9.71007 13 9-77637 Io 0.22363 9-93370 5 14 " 7.2 8, 87 9.71020 13 9.77654 17 0.22346 9.93366 4 13 I 4 9-6 88 9.71032 12 9.77671 17 o . 22329 9.93361 5 12 9 10.8 89 9-71045 13 9.77688 17 18 0.22312 9-93357 4 II 90 9.71058 13 9.77706 TT 0.22294 9-93352 10 9i 9.71070 12 9-77723 A 7 0.22277 9-93347 09 5 92 9-71083 13 9-77740 17 o . 22260 9-93343 4 08 i 0.5 93 9.71096 13 9-77757 17 0.22243 9-93338 5 07 2 I.O 94 9.71108 12 9-77774 I? 0.22226 9-93334 4 06 3 1-5 95 9.71121 13 9-77791 17 _Q o . 22209 9-93329 5 05 4 5 2.O 2.5 96 9-7II33 12 17 9.77809 Io I? 0.22191 9-93325 4 5 04 6 3-0 97 9-7II46 o 9.77826 0.22174 9-93320 03 7 35 98 9-7II59 13 9.77843 17 0.22157 9-93316 4 02 8 4.0 4r 99 9.71171 12 9.77860 I? 0.22140 9-933H 5 OI 9 5 100 9.71184 J 3 9.77877 ^7 0.22123 9-93307 4 00 Cos d. Cot d. c. Tan Sin d. P. P. 120 < 59 ' (160) 31' 148 ' Sin d. Tan d. c. Cot Cos d. P. P. 00 9.71184 9-77877 18 0.22123 9.93307 100 OI 9.71197 13 9-77895 0.22105 9-93302 99 1ft 02 9.71209 12 9.77912 I? 0.22088 9.93297 5 98 I I 8 03 9.71222 13 12 9.77929 17 0.22071 9.93293 5 97 2 3-6 04 9-71234 9-77946 0.22054 9.93288 96 3 5-4 05 9.71247 13 9.77963 17 0.22037 9.93284 4 95 4 7-2 06 9.71260 13 12 9.77980 17 17 O.2202O 9.93279 5 4 94 1 9-o 10.8 07 9.71272 9.77997 O.22O03 9-93275 93 7 12.6 08 9-71285 13 9-78015 18 0.21985 9.93270 5 92 8 14.4 09 9.71297 12 9-78032 17 T7 0.21968 9-93265 5 9 16.2 10 9.7i3io 12 9.78049 *- 1 0.2I95I 9.93261 90 ii 9-71322 9.78066 17 0.21934 9-93256 89 17 12 9.7I335 13 9-78083 17 O.2I9I7 9.93252 4 88 I 1-7 13 9.71348 13 9.78100 17 O.2I9OO 9.93247 5 87 2 3-4 14 9.71360 12 9.78117 17 0.21883 9-93243 4 86 3 4 11 15 9.71373 13 9-78I35 18 0.21865 9.93238 5 85 5 8.S 16 9.7I385 12 9.78I52 i? 0.21848 9-93233 5 84 6 10.2 17 9.71398 13 9.78169 17 0.2I83I 9.93229 4 83 J ii. 9 13.6 !8 9.71410 12 9.78186 17 O.2I8I4 9-93224 5 82 9 15.3 19 9-7I423 13 12 9.78203 17 T*7 0.21797 9-93220 4 81 20 9-71435 9.78220 * / O.2I78O 9-93215 80 21 9.71448 13 9.78237 i? 0.21763 9-932II 4 79 18 22 9.71460 12 9-78254 17 O.2I746 9.93206 5 78 i X 3 23 9-71473 13 12 9.78271 i? 18 0.21729 9-93201 5 77 2 Ii 24 9.71485 9.78289 0.2I7II 9-93197 76 3 3-9 25 9.71498 13 9.78306 17 O.2I694 9.93192 5 75 4 K 26 9-7i5io 12 13 9-78323 17 17 0.21677 9.93188 4 s 74 i 0.5 7.8 27 9-7I523 9.78340 0.21660 9.93183 73 7 9- 1 28 9-71535 12 9.78357 17 0.21643 9.93I78 5 72 8 10.4 29 9-71548 13 12 9-78374 i? 0.21626 9-93174 4 5" 71 9 11.7 30 9-7i56o 9.7839I 17 0.2I6O9 9.93I69 70 31 9-71573 13 9.78408 17 0.21592 9.93165 4 69 12 32 9-7I585 12 9.78425 i? 0.21575 9.93i6o 5 68 i i.a 33 9.7I598 13 12 17 0.21558 9-93155 5 4 67 2 3 2.4 3-6 34 9.71610 9.78459 0.2I54I 9-93I5I 66 4 4-8 35 9.71622 12 9.78476 17 0.21524 9.93I46 5 65 5 6.0 36 9.7I635 13 12 9.78493 17 17 0.21507 9-93I4I 5 4 64 6 7 7.2 8-4 37 9.71647 9.78510 O.2I49O 9-93137 63 8 9-6 38 9.71660 13 9-78528 18 O.2I472 9.93I32 5 62 9 10.8 39 9-71672 12 9.78545 i? 0.21455 9.93128 4 61 40 9-71685 13 9-78562 17 0.21438 9-93123 60 4i 9.71697 9.78579 17 0.21421 9.93118 5 59 4 42 9.71709 12 9.78596 17 0.21404 9-93II4 4 58 0.4 43 9.71722 13 9-78613 i? 0.21387 9.93I09 5 57 2 0.8 44 9-71734 12 9.78630 17 0.21370 9.93104 5 56 3 1.2 r 45 9.71747 13 9.78647 17 0.21353 9-93100 4 55 4 I.O 46 9-71759 12 12 9.78664 i? i? 0.21336 9-93095 5 4 54 I 2.4 47 9.71771 9.78681 0.21319 9.93091 53 7 2.8 48 9.71784 13 9.78698 17 0.21302 9-93086 5 52 8 11 49 9.71796 12 9.78715 17 0.21285 9.93081 5 Si 9 3-6 50 9.71809 13 9-78732 0.21268 9-93077 4 50 Cos d. Cot d. c. Tan Sin d. P.P. 121* 58 c (161) ar 148 ' Sin d. Tan d. c. Cot Cos d. P. P. 50 9.71809 9.78732 0.21268 9.93077 50 51 9.71821 9.78749 ^7 0.21251 9.93072 5 49 <"7 52 9.71833 12 9.78766 17 0.21234 9-93067 5 48 At 53 9.71846 13 12 9.78783 17 17 0.21217 9-93063 4 5 47 I 2 1-7 3-4 54 9-7I858 9.78800 O.2I2OO 9.93058 46 3 5-1 55 9.71870 12 9-78817 17 0.2II83 9-93053 5 45 4 6.8 56 9-71883 13 12 9-78834 17 i? O.2II66 9-93049 4 5 44 j? 8.5 IO.2 57 9.7I89S 9-78851 0.2II49 9.93044 43 7 II.9 58 9.71907 12 9.78868 17 0.2II32 9-93039 5 42 8 13-6 59 9.71920 13 9-78885 17 TT 0.21115 9-93035 4 41 9 15-3 60 9.71932 12 9.78902 ** TT 0.21098 9.93030 5 40 61 9.71944 9.78919 - 1 / 0.21081 9-93025 5 39 16 62 9-71957 13 9-78936 17 o?2io64 9-93021 4 38 I 1.6 63 9.VI969 12 9.78953 I? 0.21047 9.93016 5 37 2 32 64 9.71981 12 9.78970 17 0.21030 9-930II 5 36 3 4 4-8 6.4 65 9.7I994 13 9.78987 17 0.21013 9.93007 4 35 5 8.0 66 9.72006 12 9.79004 17 0.20996 9-93002 5 34 6 9.6 67 9.72018 12 9.79021 17 0.20979 9.92997 5 33 1 II. 2 12.8 68 9-72030 12 9.79038 I? 0.20962 9.92993 4 32 9 14.4 69 9.72043 13 12 9.79055 17 T7 0.20945 9-92988 5 31 70 9-72055 12 9.79072 A / T7 0.20928 9-92983 5 30 71 9.72067 9.79089 A / 0.20911 9.92979 4 29 13 73 9.72079 12 9.79io6 17 0.20894 9-92974 5 28 J T Q 73 9.72092 13 9.79122 16 0.20878 9.92969 5 27 2 2.1 74 9.72104 12 9-79139 J 7 0.20861 9-92965 4 26 3 3-9 75 9.72116 12 9.79156 i? 0.20844 9-92960 5 25 4 5-2 6 r 76 9.72128 12 13 9.79173 17 17 0.20827 9-92955 5 4 24 5 6 5 7.8 77 9.72141 9.79190 0.20810 9.92951 23 7 9-1 78 9-72153 12 9.79207 i? 0.20793 9.92946 5 22 8 10.4 79 9-72165 12 12 9.79224 17 17 0.20776 9.92941 5 21 9 11.7 80 9.72177 9.79241 * / T7 0.20759 9.92936 20 81 9.72190 13 9.79258 A 7 0.20742 9.92932 19 12 82 9.72202 12 9-79275 17 o . 20725 9.92927 5 18 I 1.2 83 9.72214 12 12 9.79292 17 17 0.20708 9.92922 5 4 17 2 3 3i6 84 9.72226 9-79309 0.20691 9-92918 16 4 4-8 85 9-72238 12 9.79326 17 0.20674 9.92913 5 15 5 6.0 86 9-72251 13 12 9-79343 17 16 0.20657 9-92908 5 5 14 6 7 7-2 8.4 87 9.72263 9-79359 0.20641 9.92903 13 8 9-6 88 9.72275 12 9.79376 J-7 0.20624 9.92899 4 12 9 10.8 89 9.72287 12 12 9-79393 i? T7 0.20607 9-92894 5 II 90 9-72299 9-79410 * / 0.20590 9-92889 10 9i 9-72312 13 9.79427 !7 0.20573 9.92885 4 09 5 92 9.72324 12 9-79444 17 0.20556 9.92880 5 08 i 0.5 93 9.72336 12 9.79461 I? 0.20539 9.92875 5 07 2 I.O 94 9.72348 12 9.79478 I? 0.20522 9.92870 5 06 3 1-5 95 9.72360 12 9-79495 17 _/ 0.20505 9.92866 4 05 4 2.0 2.5 96 9.72372 12 13 9-795II It) 17 0.20489 9.92861 5 5 04 6 3-0 97 9-72385 9-79528 0.20472 9.92856 03 7 35 98 9-72397 12 9-79545 I? 0.20455 9-92852 4 02 8 4-0 99 9.72409 12 9.79562 I? 0.20438 9.92847 5 01 9 4-5 100 9.72421 12 9-79579 17 0.20421 9.92842 00 Cos d. Cot d. C. Tan Sin d. P. P. 121 ' 58' (162) 32 ' 147' Sin d. Tan d. c. Cot Cos d. P. P. 00 9.72421 9-79579 0.20421 9.92842 100 ox 9 72433 12 9-79596 I? 0.20404 9-92837 99 17 02 9-72445 12 9.79613 i? 0.20387 9.92833 4 98 i 1-7 03 9-72457 12 12 9.79630 17 16 0.20370 9.92828 5 5 97 2 3-4 04 9-72469 9-79646 0.20354 9.92823 96 3 5-1 05 9.72482 13 9-79663 i? 0.20337 9.92818 5 95 4 6.8 06 9-72494 12 12 9.79680 i? 17 0.20320 9-92814 4 5 94 5 6 8.5 10.2 07 9.72506 9.79697 0.20303 9-92809 93 7 ii. 9 08 9-725I8 12 9-79714 17 0.20286 9.92804 5 92 8 13-6 09 9-72530 12 9-79731 17 16 0.20269 9.92799 5 9i 9 15-3 10 9.72542 12 12 9-79747 TT 0.20253 9-92795 90 ii 9-72554 9.79764 -I/ 0.20236 9.92790 89 16 12 9.72566 12 9.79781 17 0.20219 9.92785 5 88 i 1.6 13 9.72578 12 12 9-79798 I? 17 0.20202 9-92780 5 4 87 2 3 \l U 9.72590 9-798I5 O.2OI85 9.92776 86 4 6.4 15 9.72602 12 9-79832 17 T 0.20168 9.92771 5 85 5 8.0 16 9.72614 12 9.79848 10 0.20152 9.92766 5 84 6 9.6 17 9.72627 13 9.79865 17 0.20135 9.92761 5 83 7 8 II. 2 12.8 18 9-72639 12 9.79882 I? 0.20II8 9-92756 5 82 9 14.4 19 9.72651 12 9.79899 17 0.20IOI 9.92752 4 81 20 9-72663 12 9.79916 I? 16 0.20084 9.92747 80 21 9.72675 9-79932 0.20068 9.92742 79 13 22 9-72687 12 9-79949 i? 0.20051 9-92737 5 78 J I 3 23 9.72699 12 12 9.79966 17 17 0.20034 9-92733 4 5 77 2 2^6 24 9.72711 9-79983 0.20017 9.92728 76 3 3-9 25 9-72723 12 9.80000 17 O.20OOO 9-92723 5 75 4 52 6~ 26 9-72735 12 12 9.80016 16 17 0.19984 9.92718 5 5 74 i 5 7-8 27 9.72747 9-80033 0.19967 9.92713 73 7 9-i 28 9-72759 12 9.80050 17 0.19950 9.92709 4 72 8 10.4 29 9.72771 12 12 9.80067 17 17 0.19933 9.92704 5 71 9 11.7 30 9.72783 12 9.80084 */ 16 O.I99I6 9.92699 70 31 9-72795 9.80100 0.19900 9.92694 69 12 32 9.72807 12 9.80117 17 0.19883 9-92690 4 68 i 1.2 33 9.72819 12 12 9.80134 i? 17 0.19866 9-92685 5 5 67 2 3 2.4 3-6 34 9-72831 9.80151 0.19849 9.92680 66 4 4-8 35 9-72843 12 9.80168 17 0.19832 9.92675 5 65 5 6.0 36 9-72855 12 12 9.80184 16 17 O.I98I6 9.92670 5 4 64 6 7 7-2 8.4 37 9.72867 9.80201 0.19799 9.92666 63 8 9-6 38 9.72879 12 9.80218 17 O.I9782 9.92661 5 62 9 10.8 39 9.72890 II 9-80235 17 rfi 0.19765 9-92656 5 61 40 9.72902 12 9.80251 10 0.19749 9.92651 5 60 4i 9.72914 9.80268 ^7 0.19732 9.92646 5 59 4 42 9.72926 12 9.80285 17 O.I97I5 9.92641 5 58 T O A 43 9.72938 12 12 9.80302 I? 16 o . 19698 9 92637 4 5 57 2 u.4 0.8 44 9-72950 9.80318 0.19682 9.92632 56 3 1.2 45 9.72962 12 9-80335 i? 0.19665 9.92627 5 55 < 1.0 46 9-72974 12 12 9.80352 i? 17 0.19648 9.92622 5 5 54 I 2. 2.4 47 9-72986 9.80369 T 0.19631 9.92617 53 7 2.8 48 9.72998 12 9-80385 10 0.19615 9.92613 4 52 8 3 * 49 9-73010 12 12 9.80402 17 0.19598 9.92608 5 51 9 36 50 9.73022 9.80419 ^7 0.19581 9-92603 5 50 Cos d. Cot d. c. Tan Sin d. P.P. 122 57 < (163) 32' 147' Sin d. Tan d. c. Cot Cos d. 3 . P. 50 9-73022 12 9.80419 16 0.19581 9-92603 50 51 9.73034 9.80435 0.19565 9.92598 5 49 17 52 9.73045 II 9-80452 17 0.19548 9.92593 5 48 53 9-73057 12 12 9.80469 17 17 O.I953I 9.92588 5 4 47 I 2 1-7 3-4 54 9.73069 9.80486 0.19514 9.92584 46 3 5-1 55 9-73081 12 9 . 80502 16 0.19498 9-92579 5 45 4 6.8 56 9.73093 12 12 9-80519 17 17 0.19481 9.92574 5 5 44 1 8.5 IO.2 57 9-73105 9-80536 0.19464 9 92569 43 7 II. g 58 9-73II7 12 9.80552 16 0.19448 9-92564 5 42 8 13-6 59 9.73129 12 9-80569 17 TT 0.19431 9.92559 5 41 9 15 3 60 9-73140 12 9.80586 A / 0.19414 9-92555 4 40 6i 9-73152 9.80603 _/ 0.19397 9-92550 5 39 16 62 9-73I64 12 9.80619 10 0.19381 9.92545 5 38 I 1.6 63 9.73176 12 12 9.80636 17 17 0.19364 9.92540 5 5 37 2 32 64 9.73188 9-80653 _/ 0.19347 9.92535 36 4 64 65 9-73200 12 9.80669 lO o.i933i 9.92530 5 35 5 So 66 9-732H II 9.80686 17 0.19314 9-9252S 5 34 6 9-6 67 9-73223 12 9-80703 17 0.19297 9.92521 4 33 7 g II. 2 12 8 68 9-73235 12 9.80719 16 0.19281 9.92516 5 32 9 14.4 69 9-73247 12 12 9-80736 17 0.19264 9.92511 5 31 70 9-73259 12 9.80753 17 16 0.19247 9.92506 5 30 71 9.7327I 9-80769 0.19231 9.92501 29 12 72 9.73282 II 9.80786 17 0.19214 9.92496 5 28 I 2 73 9-73294 12 12 9.80803 16 0.19197 9-92491 5 5 27 2 2.4 74 9-73306 9.80819 0.19181 9-92486 26 3 3-6 75 9-73318 12 9.80836 17 0.19164 9.92482 4 25 4 I-j 76 9-73329 II 12 9-80853 17 16 0.19147 9.92477 5 5 24 5 6 0.0 7.2 77 9-73341 9.80869 0.19131 9.92472 23 7 8.4 78 9-73353 12 9.80886 17 0.19114 9.92467 5 22 8 9-6 79 9 73365 12 12 9.80903 17 16 0.19097 9.92462 5 S. 21 9 10.8 80 9-73377 9.80919 0.19081 9 92457 20 81 9-73388 9.80936 17 0.19064 9-92452 19 11 82 9.73400 12 9-80953 17 _/ 0.19047 9.92447 5 18 I i.i 83 9-73412 12 12 9.80969 10 17 0.19031 9.92443 4 5 17 2 3 2.2 3-3 84 9-73424 9-80986 0.19014 9.92438 16 4 4.4 85 9-73435 II 9.81003 17 _r 0.18997 9.92433 5 IS 5 5.5 86 9-73447 12 12 9.81019 10 0.18981 9.92428 5 5 14 7 6.6 7-7 87 9-73459 9.81036 T 0.18964 9.92423 13 8 8.8 88 9-73470 II 9.81052 ID 0.18948 9.92418 5 12 9 9-9 89 9.73482 12 9.81069 17 0.18931 9.92413 5 II 90 9-73494 12 9.81086 t? 16 0.18914 9.92408 5 10 91 9-73506 12 9.81102 0.18898 9-92403 09 5 92 9-73517 II 9.81119 17 o. 18881 9.92398 5 08 I 5 93 9-73529 12 12 9.81136 16 0.18864 9.92394 4 5 07 2 .0 94 9-73541 9.81152 0.18848 9-92389 06 3 .5 95 9-73552 II 9.81169 _/? 0.18831 9.92384 5 05 4 e .5 96 9-73564 12 12 9.81185 10 17 0.18815 9.92379 5 5 04 6 .0 97 9-73576 9.81202 0.18798 9.92374 03 7 5 98 9-73588 12 9.81219 17 0.18781 9-92369 5 02 8 .0 99 9-73599 II 9-81235 16 0.18765 9.92364 5 01 9 5 100 9.736H 12 9.81252 17 0.18748 9-92359 00 Cos d. Cot d. c. Tan Sin d. P . P. 122 ( 57' (164) 146' Sin d. Tan d. c. Cot Cos d. P. P. 00 9 736ii 12 9.81252 16 0.18748 9-92359 100 01 9-73623 9.81268 0.18732 9.92354 99 M 02 9-73634 II 9.81285 i? 0.18715 9.92349 5 98 03 9-73646 12 12 9.81302 17 16 0.18698 9-92344 5 5 97 I 2 1-7 3-4 04 9-73658 9.81318 0.18682 9.92339 96 3 5-1 05 9-73669 II 9.8I335 17 0.18665 9-92335 4 95 4 6.8 o6 9.73681 12 II 9-8I35I 16 17 0.18649 9.92330 5 5 94 i 8.5 IO.2 07 9.73692 9.81368 0.18632 9-92325 93 7 II.9 38 9-73704 12 9.81384 16 o. 18616 9-92320 5 92 8 13-6 9 9-73716 12 jj 9.81401 i? T*7 0.18599 9-92315 5 9i 9 15 3 10 9-73727 12 9.81418 A / 16 0.18582 9-92310 90 ii 9-73739 9.81434 0.18566 9-92305 89 16 12 9-73751 12 9.8I45I 17 0.18549 9-92300 5 88 i 1.6 13 9-73762 II 12 9-81467 16 17 0.18533 9.92295 5 5 87 2 i 3-2 4 8 14 9-73774 9-81484 T 0.18516 9-92290 86 O 4 6^4 IS 9-73785 II 9.81500 ID o . 18500 9-92285 5 85 5 8.0 16 9-73797 12 9-8I5I7 17 -f 0.18483 9.92280 5 84 6 9-6 17 9.73809 12 9.81533 10 0.18467 9-92275 5 83 7 g II. 2 T2 8 18 9-73820 II 9 81550 17 0.18450 9.92270 5 82 12. o TA A 19 9-73832 12 j j 9-81567 17 16 0.18433 9-92265 5 81 14 . n 20 9 73843 12 9-81583 1*7 0.18417 9.92260 5 80 21 9-73855 9.81600 A / rfi o . 18400 9-92255 79 12 22 9.73867 12 9.81616 10 0.18384 9-92250 5 78 23 9-73878 II 12 9.81633 17 16 0.18367 9-92245 5 5 77 I 2 1 . 2 2.4 24 9-73890 9.81649 0.18351 9-92240 76 3 3-6 25 9-73901 II 9.81666 17 T 0.18334 9 92235 5 75 4 4.8 26 9-73913 12 II 9.81682 ID 17 0.18318 9-92231 4 5 74 i 6.0 7.2 27 9-73924 9-81699 -fC 0.18301 9.92226 73 7 8.4 28 9-73936 12 9 81715 IO 0.18285 9.92221 5 72 8 9-6 29 9-73947 II 12 9.81732 17 16 0.18268 9.92216 | 71 9 10.8 30 9-73959 12 9.81748 f7 0.18252 9.92211 5 70 3i 9-73971 9 81765 * I tf\ 0.18235 9.92206 5 69 11 32 9-73982 II 9.81781 10 0.18219 9.92201 5 68 i i.i 33 9-73994 12 II 9-81798 i? 16 0.18202 9.92196 5 5 67 2 3 2.2 3-3 34 9-74005 9.81814 o. 18186 9-92191 66 4 4-4 35 9.74017 12 9.81831 i? T/? 0.18169 9.92186 5 65 5 5-5 36 9.74028 II 12 9.81847 IO 17 0.18153 9.92181 5 5 64 6 6.6 37 9.74040 9.81864 T 0.18136 9.92176 63 7 g 7-7 8.8 38 9-74051 II 9.81880 10 o. 18120 9.92171 5 62 9 9-9 39 9.74063 12 jj 9-81897 17 16 0.18103 9.92166 5 61 40 9.74074 12 9-81913 T*7 0.18087 9.92161 5 60 41 9-74086 9.81930 * / -,C 0.18070 9.92156 5 59 6 42 9.74097 II 9.81946 IO 0.18054 9.92151 5 58 I 0.5 43 9.74109 II 9-81963 17 16 0.18037 9.92146 5 5 57 2 I.O 44 9.74120 9.81979 o. 18021 9.92141 56 3 1-5 45 9.74132 12 9-81996 i? T/T 0.18004 9-92136 5 55 4 2.0 46 9-74143 II 12 9.82012 IO 17 0.17988 9.92131 5 5 54 i 2.5 3-o 47 9.74155 9 . 82029 rfi. 0.17971 9.92126 53 7 3-5 48 9.74166 II 9-82045 IO 0.17955 9.92121 5 52 8 4-0 49 9.74177 II 12 9.82062 17 16 0.17938 9.92116 5 Si 9 4-5 50 9-74I89 9 . 82078 o. 17922 9.92111 5 50 Cos d. Cot d. c. Tan Sin d. P. P. 123' 56' (165) 146 Sin d. Tan d. c. Cot Cos d. P. P. 50 9.74189 9.82078 0.17922 9.92111 50 Si 9.74200 9-82095 T 0.17905 9.92106 5 49 1T 52 9-74212 12 9.82111 IO 0.17889 9.92101 5 48 53 9-74223 II 12 9.82128 17 16 0.17872 9-92096 5 47 I 2 1-7 3-4 54 9-74235 9.82144 0.17856 9.92091 46 3 5-1 55 9.74246 II 9.82161 T 0.17839 9 . 92086 5 45 4 6.8 56 9-74258 12 9.82177 io 0.17823 9.92081 5 6 44 5 8.5 10.2 57 9.74269 9.82194 0.17806 9.92075 43 7 II.9 58 9.74280 II 9.82210 _/? 0.17790 9.92070 5 42 8 13.6 59 9.74292 12 9.82226 ID 0.17774 9-92065 5 41 9 15-3 60 9-74303 9-82243 ifi 0.17757 9 . 92060 5 40 6i 9.74315 9-82259 IO 0.17741 9.92055 5 39 16 62 9-74326 ii 9.82276 i? 0.17724 9-92050 5 38 i 1.6 63 9-74337 ii 9.82292 16 0.17708 9.92045 5 37 2 32 64 9-74349 12 9-82309 -r 0.17691 9.92040 5 36 3 4 4-8 6.4 65 9-7436o II 9-82325 10 0.17675 9.92035 5 35 5 8.0 66 9-74372 12 9-82341 16 0.17659 9-92030 5 34 6 9-6 67 9.74383 II 9-82358 17 _ 0.17642 9.92025 5 33 7 II. 2 12 8 68 9-74394 II 9.82374 10 0.17626 9 . 92020 5 32 9 14.4 69 9.74406 12 9-82391 17 Tfi 0.17609 9-92015 5 31 70 9-74417 9.82407 IO 0.17593 9.92010 5 30 71 9.74428 9.82424 17 0.17576 9-92005 29 12 73 9-74440 12 9.82440 IO 0.17560 9.92000 5 28 73 9-74451 II 12 9-82456 16 17 0.17544 9.91995 5 5 27 I 2 1 . 2 2-4 74 9.74463 9-82473 0.17527 9.91990 26 3 3-6 75 9-74474 II 9-82489 16 0.17511 9.91985 5 25 4 iS 76 9-74485 II 12 9.82506 17 16 0.17494 9-91980 5 5 24 5 6 o.o 7.2 77 9-74497 9-82522 0.17478 9.91975 23 7 8.4 78 9-74508 II 9-82538 16 0.17462 9.91969 6 22 8 9-6 79 9.74519 II 9' 82555 rfi 0-17445 9.91964 5 21 9 10.8 80 9-74531 12 9.82571 IO 0.17429 9 91959 5 20 81 9-74542 9.82588 17 0.17412 9.91954 19 11 83 9-74553 II 9.82604 16 _/ 0.17396 9.91949 5 18 i i.i 83 9-74565 12 II 9.82620 10 17 0.17380 9-9*944 5 5 17 2 3 2.2 3-3 84 9-74576 9-82637 T 0.17363 9-91939 16 4 4-4 85 9.74587 II 9.82653 10 0.17347 9.91934 5 15 5 5-5 86 9-74598 II 12 9.82670 17 16 0.17330 9-9I929 5 14 6 6.6 87 9.74610 9.82686 O.I73I4 9.91924 13 7 8 7-7 8.8 88 9.74621 II 9.82702 16 0.17298 9.91919 5 12 9 9-9 89 9-74632 II 9.82719 17 Tfi 0.17281 9.91914 5 II 90 9.74644 12 9-82735 IO rfi 0.17265 9.91908 10 9 1 9-74655 9.82751 10 0.17249 9-91903 09 6 93 9-74666 II 9.82768 17 -r 0.17232 9.91898 5 08 i 0.6 93 9-74677 II 9.82784 IO 0.17216 9.91893 5 07 2 1.2 94 9.74689 12 9.82801 17 O.I7I99 9.91888 5 06 3 1.8 95 9.74700 II 9.82817 16 T 0.17163 9-91883 5 05 4 5 2-4 3 96 9-747H II II 9-82833 IO 17 0.17167 9.91878 5 5 04 6 3^6 97 9.74722 9.82850 0.17150 9.91873 03 7 4-2 98 9-74734 12 9.82866 16 O.I7I34 9.91868 5 02 8 4-8 99 9-74745 II 9 . 82882 16 0.17118 9 91863 5 g 01 9 5-4 100 9-74756 II 9.82899 17 0.17101 9.91857 00 Cos d. Cot d. c. Tan Sin d. P. P. 123 ( 66 (i 66) 34' 145' Sin d. Tan d. c. Cot Cos d. P . P. 00 9.74756 9.82899 16 0.17101 9-9I857 100 01 02 03 04 OS 06 o? 08 09 9.74767 9-74779 9-74790 9.74801 9.74812 9-74824 9.74835 9.74846 9.74857 12 II II II 12 II II II 11 9.82915 9-82931 9-82948 9.82964 9.82980 9.82997 9.83013 9-83029 9-83046 16 17 16 16 17 16 16 17 16 0.17085 0.17069 0.17052 0.17036 0.17020 0.17003 0.16987 0.16971 0.16954 9-91852 9.91847 9.91842 9-9I837 9-91832 9.91827 9.91822 9.91816 9.91811 5 5 5 5 5 5 6 5 99 98 97 96 95 94 93 92 9i I 2 3 4 I 7 8 9 17 17 3-4 5-1 6.8 8-5 10.2 ii 9 13-6 15 3 10 9.74868 9.83062 ,.-, 0.16938 9.91806 90 ii 12 13 14 15 16 I? 18 19 9.74880 9.74891 9.74902 9 74913 9.74924 9-74935 9-74947 9-74958 9.74969 II II II II II 12 II II 9.83078 9-83095 9.83111 9-83127 9-83I44 9.83160 9.83176 9.83I93 9.83209 17 16 16 17 16 16 i? 16 16 0.16922 0.16905 0.16889 0.16873 0.16856 0.16840 0.16824 0.16807 0.16791 9.91801 9.91796 9-9I79I 9.91786 9.91781 9-91775 9.91770 9.91765 9-91760 5 5 5 5 6 5 5 5 89 88 87 86 85 84 83 82 81 I 2 3 4 i 7 8 9 16 1.6 ?;J 6.4 8.0 9-6 II. 2 12.8 14.4 20 9.74980 11 9-83225 0.16775 9.91755 80 21 22 23 24 25 26 27 28 29 9.74991 9-75002 9-75014 9-75025 9.75036 9-75047 9-75058 9.75069 9.75o8o II 12 II II II II II II 9.83242 9.83258 9-83274 9-83290 9-83307 9.83323 9-83339 9.83356 9.83372 16 16 16 17 16 16 17 16 16 0.16758 0.16742 0.16726 0.16710 0.16693 0.16677 o. 16661 0.16644 0.16628 9.9I750 9-91744 9-91739 9-91734 9.91729 9.9I724 9.91719 9.91714 9-91708 6 5 5 5 5 5 5 6 79 78 77 76 75 74 73 72 71 i 2 3 4 5 6 7 8 9 12 1.2 2.4 3-6 4-8 6.0 7-2 8.4 9-6 10.8 30 9.75091 9-83388 0.16612 9.91703 70 3i 32 33 34 35 36 37 38 39 9.75103 9 75H4 9.75I25 9.75I36 9.75I47 9.75I58 9-75169 9.75i8o 9-75I9I II II II II II II II II 9-83405 9-83421 9-83437 9-83453 9.83470 9-83486 9-83502 9-83518 9-83535 16 16 16 17 16 16 16 17 16 o.i6595 0.16579 0.16563 0.16547 0.16530 0.16514 0.16498 0.16482 0.16465 9-91698 9.91693 9.91688 9.91682 9.91677 9.91672 9.91667 9.91662 9-91657 5 5 6 5 5 5 5 5 5 69 68 67 66 65 64 63 62 61 I 2 3 4 I 9 11 i.i 2.2 33 4-4 i! 7-7 8.8 9-9 40 9-75202 9.83551 16 0.16449 9-91651 60 4i 42 43 44 45 46 47 48 49 9 75213 9.75224 9.75236 9-75247 9.75258 9.75269 9.75280 9-7529I 9-75302 II 12 II II II II II II 9-83567 9-83583 9.83600 9.83616 9-83632 9-83648 9.83665 9.83681 9-83697 16 17 16 16 16 17 16 16 16 0.16433 0.16417 0.16400 0.16384 0.16368 0.16352 0.16335 0.16319 0.16303 9.91646 9.91641 9.91636 9.91631 9.91625 9.91620 9-9i6i5 9.91610 9.91605 5 5 5 6 5 5 5 5 5 59 58 57 56 55 54 53 52 Si i 2 3 4 5 6 7 8 9 5 o.S I.O i-5 2.0 2.5 3-0 3-5 4-0 4-5 50 9 75313 9.837I3 o . 16287 9-91599 50 Cos d. Cot d. c. Tan Sin d. I '. P. 124' 56' (167) 34' 145< Sin d. Tan d. c. Cot Cos d. P. P. 50 9-75313 jj 9.83713 o 16287 9-91599 50 Si 9.75324 9.83730 *7 rfi 0.16270 9-91594 5 49 17 52 9-75335 11 9.83746 ID tfi 0.16254 9.91589 ' 5 48 53 9-75346 II II 9.83762 IO 16 0.16238 9.91584 5 ^ 47 i 2 1-7 34 54 9-75357 9 83778 0.16222 9-91579 46 3 5-1 55 9.75368 ii 9-83795 17 -r& o . 16205 9.9I573 6 45 4 6.8 56 9-75379 ii II 9.83811 10 16 0.16189 9.91568 5 5 44 5 6 8.5 10.2 57 9-7539 9-83827 T 0.16173 9.91563 43 7 II-9 58 9-75401 ii 9-83843 10 ifi 0.16157 9-9I558 5 42 8 13-6 59 9-75412 jj 9 83859 10 T*7 0.16141 9 91552 41 9 15-3 60 9-75423 JT 9-83876 *7 16 0.16124 9-91547 5 40 61 9-75434 9-83892 _/ o. 16108 9-91542 5 39 16 62 9-75445 II 9-83908 10 -f 0.16092 9-91537 5 38 1. 1.6 63 9-75456 II 9-83924 ID 0.16076 9-9I53I 6 37 2 3-2 64 9.75467 II 9.83941 17 if. 0.16059 9.91526 5 36 3 4 64 65 9-75478 II 9-83957 ID ._./ 0.16043 9-9I52I 5 35 5 8.0 66 9.75489 II TT 9-83973 10 16 0.16027 9.9I5I6 5 34 6 9-6 67 9-75500 9.83989 t. o. 16011 9.9I5H 5 33 7 g II. 2 12 8 68 9-755II II 9-84005 IO 0.15995 9.9I505 32 9 14.4 69 9-75522 II J 9.84022 17 16 0.15978 9.91500 5 31 70 9-75533 J J 9-84038 16 o. 15962 9.91495 5 30 7i 9-75544 9-84054 rA 0.15946 9.91490 29 11 72 9-75554 IO 9.84070 10 T 0.15930 9.91484 28 j 73 9.75565 II II 9.84086 ID 17 0.15914 9.91479 5 5 27 2 2.2 74 9-75576 9.84103 0.15897 9.91474 26 3 33 75 9.75587 II 9.84119 16 0.15881 9.91469 5 25 4 4-4 76 9-75598 II II 9-84I35 16 16 0.15865 9.9I463 6 5 24 5 6 5-5 6.6 77 9.75609 9-84I5I T y- 0.15849 9.91458 23 7 7-7 78 9.75620 II 9.84167 10 0.15833 9.91453 5 22 8 8.8 79 9.75631 II 9-84183 16 0.15817 9.91447 6 21 9 99 80 9.75642 II JT 9.84200 J 7 16 0.15800 9.91442 5 20 81 9-75653 9.84216 T 0.15784 9-91437 19 10 82 9.75664 II 9.84232 10 _/- 0.15768 9.91432 5 18 i I.O 83 9.75675 II IO 9 . 84248 10 16 0.15752 9-91426 5 17 2 3 2.0 3.O 84 9.75685 9.84264 T /r 0.15736 9.91421 16 4 4.0 85 9.75696 II 9 . 84280 ID 0.15720 9.91416 5 15 5 5-0 86 9.75707 II II 9-84297 I? 16 0.15703 9.91411 5 6 14 6 n 6.0 7 87 9.75718 9.84313 0.15687 9.9I405 13 8 8.0 88 9-75729 II 9.84329 16 T r 0.15671 9.91400 5 12 9 9.0 89 9-75740 II 9.84345 ID ifi 0.15655 9.91395 5 g II 90 9-75751 II JJ 9 84361 ID 16 0.15639 9.91389 10 9i 9.75762 9.84377 0.15623 9.9I384 09 6 92 9-75772 IO 9-84394 17 Tfi 0.15606 9.91379 5 08 I 0.6 93 9.75783 II II 9.84410 ID 16 0.15590 9.9I374 5 6 07 2 1.2 94 9-75794 9-84426 Wl 0.15574 9.91368 06 3 1.8 95 9.75805 II 9.84442 ID ,/; 0.15558 9.91363 5 05 4 2.4 3f\ 96 9.758I6 II II 9-84458 10 16 0.15542 9.9I358 5 6 04 6 u 3-6 97 9-75827 9-84474 rfi 0.15526 9.9I352 03 7 4-2 98 9-75837 IO 9.84490 10 0.15510 9.9I347 5 02 8 4-8 99 9.75848 II 9.84507 17 16 0.15493 9.91342 5- g OI 9 5-4 100 9-75859 II 9-84523 0.15477 9.91336 00 Cos d. Cot d. c. Tan Sin d. P.P. 124< 55 (168) 144' Sin d. Tan d. c. Cot Cos d. P. P. 00 9-75859 9-84523 16 0.15477 9.91336 100 OI 9-75870 11 9-84539 16 0.15461 9-9I33I 99 17 02 9-75881 ii j j 9-84555 16 0.15445 9-91326 C 98 i 1.7 03 9-75892 10 9.84571 16 0.15429 9.91321 D 6 97 2 3-4 04 9-75902 ii 9-84587 r O.I54I3 9-9I3I5 96 3 Tl 05 9-75913 9-84603 10 16 0.15397 9.91310 5 95 4 6.8 8 06 9-75924 ii ii 9-84619 17 0.15381 9-9I305 5 6 94 5 6 5 10.2 07 9-75935 9-84636 T/\ 0.15364 9.91299 93 7 ii. 9 08 9-75946 ii 9-84652 IO 0.15348 9-91294 5 92 8 13-6 09 9-75956 IO 9.84668 16 16 0.15332 9.91289 5 g 9i 9 15-3 10 9.75967 11 9.84684 Tfi 0.15316 9-91283 90 ii 9-75978 II 9.84700 IO Tfi 0.15300 9.91278 5 89 16 12 9-75989 II 9.84716 10 rA 0.15284 9.9I273 5 88 i 1.6 13 9.76000 II IO 9.84732 10 16 0.15268 9.91267 $ 87 2 3^Ky^ 3-2 4 8 14 9 . 76010 9.84748 rfi 0.15252 9.91262 86 4 6^4 15 9 . 76021 II 9-84764 10 0.15236 9-91257 5 85 5 8.0 16 9.76032 II II 9.84781 17 16 0.15219 9-9I25I 5 84 6 9-6 II 2 17 9-76043 9-84797 T 0.15203 9.91246 83 g 12^8 18 9-- 76053 IO 9-84813 IO vt\ 0.15187 9.91241 5 82 9 14.4 19 9.76064 II 11 9-84829 ID 16 0.15171 9-91235 81 20 9-76075 9.84845 16 O.I5I55 9.91230 80 21 9.76086 XI 9.84861 ifi O.I5I39 9.91225 5 79 11 22 9-76096 10 9.84877 IO T 0.15123 9.91219 78 I.I 23 9-76107 II II 9.84893 10 16 0.15107 9.91214 5 5 77 2 2.2 24 9.76118 9-84909 _/ 0.15091 9-91209 76 3 3-3 25 9.76129 II 9-84925 10 -rf\ 0.15075 9-91203 75 4 4-4 26 9-76I39 IO II 9.8494I ID iy 0.15059 9.9H98 5 6 74 i ill 27 9.76150 9.84958 -rf\ 0.15042 9-9II92 73 7 7.7 28 9.76161 II 9-84974 IO 0.15026 9.91187 5 72 8 8.8 29 9.76171 IO JT 9.84990 16 16 0.15010 9.91182 5 g 71 9 9-9 30 9.76182 9.85006 16 0.14994 9-9II76 70 3i 9 76i93 9.85022 T& 0.14978 9.91171 5 69 10 32 9.76203 IO 9.85038 ID 0.14962 9.91166 5 68 i I.O 33 9.76214 II II 9.85054 16 16 0.14946 9.91160 6 5 67 2 3 2.O 3-0 34 9.76225 9-85070 rfi 0.14930 9-9II55 66 4 4-0 35 9-76236 II 9.85086 10 0.14914 9.91149 65 5 5-0 36 9.76246 IO II 9.85102 16 16 0.14898 9.91144 5 5 64 7 6.0 7.0 37 9-76257 9-85118 Tfi 0.14882 9.91139 63 8 8.0 38 9.76268 II 9.85134 10 T 0.14866 9.9II33 62 9 0.0 39 9-76278 IO 9-85150 ID Tfi 0.14850 9.91128 5 61 40 9.76289 II JJ 9.85166 xo 16 0.14834 9-9II23 5 60 4i 9-76300 9.85182 T /* 0.14818 9.91117 59 5 42 9.76310 IO 9.85198 IO 0.14802 9.91112 5 58 J ft c 43 9-76321 II II 9-852I5 17 16 O.I478S 9.91106 6 $ 57 2 u - 5 I.O 44 9-76332. 9-85231 T 0.14769 9.91101 56 3 1-5 45 9-76342 10 9-85247 10 T 0.14753 9.91096 5 55 4 2.O 46 9-76353 II II 9.85263 10 16 0.14737 9-91090 6 5 54 i 2-5 3-0 47 9-76364 9.85279 T 0.14721 9.91085 53 7 3-5 48 9-76374 10 9.85295 10 T 0.14705 9.91079 52 8 4-0 49 9.76385 II 10 9.853H 10 16 0.14689 9.91074 5 Si 9 4-5 50 9-76395 9-85327 0.14673 9.91069 5 50 Cos d. Cot d. c. Tan Sin d. P.P. 125 ' 64 (169) 36< 144< Sin d. Tan d. c. Cot Cos d. P. P. 50 9.76395 jj 9-85327 16 0.14673 9.91069 5 50 Si 9.76406 9-85343 T /r 0.14657 9-91063. 49 1ft 52 9.76417 ii 9.85359 IO 0.14641 9-91058 5 48 53 9.76427 10 ii 9-85375 16 16 0.14625 9-91052 6 5 47 I 2 i .6 32 54 9.76438 9.85391 0.14609 9.91047 46 3 4.8 55 9.76448 10 9-85407 16 0.14593 9-91042 5 45 4 6.4 56 9-76459 II II 9-85423 16 16 0.14577 9-91036 6 5 44 5 6 8.0 9-6 57 9.76470 9-85439 0.14561 9.91031 43 7 II. 2 58 9.76480 10 9-85455 16 0.14545 9-91025 6 42 8 12.8 59 9.76491 II 9-8547I 16 if. 0.14529 9.91020 5 41 9 14.4 60 9-76501 JJ 9-85487 10 16 O.I45I3 9.91014 40 61 9-76512 9-85503 T 0.14497 9-91009 5 39 15 62 9-76523 II 9-85519 IO _/- 0.14481 9.91004 5 38 i i.S 63 9.76533 IO II 9-85535 IO 16 0.14465 9.90998 6 5 37 2 3 3-0 4 5 64 9.76544 9.8555I T 0.14449 9.90993 36 4 6^0 65 9-76554 10 9-85567 IO t-A 0.14433 9.90987 35 5 7-5 66 9.76565 II 9-85583 10 -e 0.14417 9-90982 5 34 6 9-0 67 9-76575 IO 9-85599 10 0.14401 9.90976 6 33 I 10.5 12 O 68 9-76586 II 9-85615 16 T/? 0-14385 9.90971 5 32 9 13.4 69 9-76597 II IO 9-85631 ID 16 0.14369 9.90966 5 6 31 70 9.76607 JJ 9-85647 T 0.14353 9.90960 30 71 9.76618 9-85663 IO 0.14337 9.90955 5 29 11 72 9.76628 10 9-85679 16 0.14321 9.90949 28 i I j 73 9.76639 II IO 9.85695 16 16 0.14305 9.90944 5 6 27 2 2.2 74 9.76649 9.857H _/ 0.14289 9.90938 26 3 33 75 9.76660 II 9-85727 10 T 0.14273 9.90933 5 25 4 4-4 76 9.76670 10 II 9-85743 IO 16 0.14257 9.90927 5 24 6 5-5 6.6 77 9.76681 9.85759 T / 0.14241 9-90922 23 7 7-7 78 9.76691 10 9-85775 IO -f- 0.14225 9.90916 22 8 8.8 79 9-76702 II IO 9.85791 IO 16 0.14209 9.90911 5 21 9 9-9 80 9.76712 JJ 9-85807 16 0.14193 9.90906 5 g 20 81 9-76723 9-85823 _/ O.I4I77 9.90900 19 10 82 9.76733 IO 9-85839 IO 0.14161 9.90895 5 18 i I.O 83 9-76744 II IO 9-85855 16 16 O.I4I45 9.90889 6 5 17 2 3 2.O 3-O 84 9.76754 9.85871 _ 0.14129 9.90884 16 4 4.0 85 9-76765 II 9-85887 10 0.14113 9-90878 15 5 S.o 86 9.76775 10 II 9-85903 16 16 0.14097 9-90873 5 6 14 6 7 6.0 7 o 87 9.76786 9.85919 _/; 0.14081 9.90867 13 8 l.o 88 9.76796 IO 9-85935 IO 0.14065 9.90862 5 12 9 9-0 89 9.76807 II 9-85951 16 16 0.14049 9-90856 II 90 9.76817 JJ 9-85967 16 0.14033 9-90851 g 10 9i 9.76828 9.85983 -r 0.14017 9.90845 09 6 92 9-76838 10 9-85999 IO 0.14001 9.90840 5 08 i 0.6 93 9-76849 II IO 9 . 86014 15 16 0.13986 9-90834 6 5 07 2 1.2 94 9-76859 9.86030 -r 0.13970 9.90829 06 3 1.8 95 9.76870 II 9.86046 IO 0.13954 9-90823 05 4 2.4 96 9.76880 IO II 9.86062 16 16 0.13938 9.90818 5 6 04 i 3-0 3-6 97 9-76891 9.86078 T. 0.13922 9.90812 03 7 4-2 98 9.76901 IO 9.86094 IO -e 0.13906 9.90807 5 02 8 4-8 99 9.76911 IO JJ 9.86110 10 16 0.13890 9.90801 OI 9 5-4 100 9.76922 9.86126 0.13874 9.90796 00 Cos d. Cot d. c. Tan Sin d. P. P. 125 54< (170) 143' Sin d. Tan d. c. Cot Cos d. I '. P. 00 9.76922 9.86126 16 0.13874 9.90796 g 100 01 02 03 04 05 o6 07 08 09 9-76932 9-76943 9-769S3 9-76964 9-76974 9-76984 9-7699S 9-77005 9.77016 ii 10 ii 10 10 II 10 II 9.86142 9.86158 9.86174 9.86190 9.86206 9.86222 9.86238 9-86254 9.86269 16 16 16 16 16 16 16 15 16 0.13858 0.13842 0.13826 0.13810 0.13794 0.13778 0.13762 0.13746 O.I373I 9.90790 9-90785 9.90779 9-90774 9.90768 9.90763 9-90757 9-90752 9.90746 5 6 5 6 5 6 5 6 99 98 97 96 95 94 93 92 9i I 2 3 4 5 6 7 8 9 16 1.6 is U 9-6 II. 2 12.8 14-4 10 9.77026 9.86285 16 O.I37I5 9.90741 6 90 ii 12 13 14 IS 16 17 18 19 9.77036 9-77047 9-77057 9.77068 9.77078 9.77088 9.77099 9.77109 9.77119 II IO II 10 IO II 10 IO 9.86301 9-86317 9-86333 9-86349 9-86365 9.86381 9-86397 9-86413 9.86429 16 16 16 16 16 16 16 16 16 0.13699 0.13683 0.13667 0.13651 0.13635 0.13619 0.13603 0.13587 O.I357I 9-90735 9.90730 9.90724 9.90718 9-907I3 9.90707 9.90702 9-90696 9-90691 5 6 6 5 6 5 6 5 5 89 88 8? 86 8s 84 83 82 81 I 2 3 4 5 6 7 8 9 16 1. 5 3-0 4-5 6.0 7-5 9-0 10.5 12.0 13 5 , 20 9.77130 9-86445 TC 0.13555 9-90685 80 21 22 33 24 35 26 27 28 29 9.77140 9-77150 9.77161 9.77171 9.77181 9.77192 9.77202 9.77212 9-77223 10 II 10 10 II 10 10 II 9.86460 9.86476 9.86492 9.86508 9-86524 9.86540 9-86556 9.86572 9.86588 16 16 16 16 16 16 16 16 0.13540 0.13524 0.13508 0.13492 0.13476 0.13460 0.13444 0.13428 0.13412 9.90680 9-90674 9-90669 9.90663 9-90657 9.90652 9.90646 9.90641 9.90635 6 5 6 6 5 6 5 6 79 78 77 76 75 74 73 73 71 I 2 3 4 5 6 7 8 9 11 I.I 2.2 3-3 4-4 1:1 7-7 8.8 9-9 30 9.77233 9.86603 16 0.13397 9 90630 5 70 31 33 33 34 35 36 37 38 39 9.77243 9-77254 9.77264 9.77274 9.77285 9-77295 9-77305 9.77316 9.77326 II 10 10 II 10 10 II IO 9.86619 9-86635 9.86651 9.86667 9.86683 9-86699 9-86715 9-86731 9.86746 16 16 16 16 16 16 16 15 16 0.13381 0.13365 0.13349 0.13333 O.I33I7 0.13301 0.13285 0.13269 0.13254 9.90624 9.90618 9-90613 9.90607 9.90602 9-90596 9.90591 9.90585 9-90579 6 5 6 5 6 5 6 6 69 68 67 66 65 64 63 62 61 I 2 3 4 I 7 8 9 10 I.O 2.0 3-0 4-0 S.o 6.0 7-0 8.0 9-0 40 9.77336 9.86762 16 0.13238 9-90574 g 60 41 43 43 44 45 46 47 48 49 9-77346 9-77357 9-77367 9-77377 9-77387 9-77398 9.77408 9.77418 9.77429 II IO 10 10 II 10 10 II 9.86778 9-86794 9.86810 9.86826 9.86842 9-86857 9.86873 9.86889 9.86905 16 16 16 16 15 16 16 16 16 0.13222 0.13206 0.13190 O.I3I74 0.13158 O.I3I43 0.13127 0.13111 0.13095 9-90568 9-90563 9-90557 9-9055I 9.90546 9.90540 9-90535 9.90529 9-90523 5 6 6 5 6 5 6 6 59 58 57 56 55 54 53 52 51 I 2 3 4 S 6 7 8 9 5 0.5 I.O 1-5 2.O 2.5 3-0 3-5 4-0 4-5 50 9-77439 9.86921 0.13079 9-905I8 50 Cos d. Cot d. c. Tan Sin d. I >. P. 126' 53' (171) 143' Sin d. Tan d. c. Cot Cos d. p. p. 60 9-77439 9.86921 16 0.13079 9-90518 50 51 9-77449 10 9-86937 Tfi 0.13063 9-90512' 49 53 9-77459 10 9-86953 IO 0.13047 9.90507 5 48 53 9-77469 IO II 9.86968 15 16 0.13032 9-90501 6 6 47 2 1.0 3 2 54 9.77480 9-86984 T 0.13016 9.90495 46 3 4-8 55 9-77490 IO 9.87000 10 _/- 0.13000 9.90490 5 45 4 6.4 56 9.77500 IO 10 9.87016 10 16 0.12984 9.90484 6 44 s 8.0 o 6 57 9.77510 9-87032 Tfi 0.12968 9.90479 5 43 7 y.o II. 2 58 9.77521 11 9.87048 IO 0.12952 9.90473 42 8 12.8 59 9-77531 IO IO 9-87063 15 16 0.12937 9.90467 6 41 9 14.4 60 9-77541 10 9 87079 16 0.12921 9.90462 5 g 40 6i 9-77551 9-87095 Tfi 0.12905 9.90456 39 15 62 9-7756I IO 9.87111 ID -rf\ 0.12889 9.90450 38 I 1-5 63 9-77572 II 9.87127 10 T 0.12873 9-90445 5 37 2 3-0 64 9-77582 10 9.87143 10 0.12857 9-90439 6 36 3 4-5 6 o 65 9-77592 10 9-87158 15 f A 0.12842 9-90434 5 35 4 s 7-5 66 9.77602 IO IO 9.87174 10 16 0.12826 9.90428 6 34 6 9-0 67 9.77612 9.87190 -f? 0.12810 9.90422 33 7 10.5 68 69 9-77623 9-77633 II IO IO 9.87206 9.87222 10 16 16 0.12794 0.12778 9.90417 9.90411 5 6 g 32 31 9 12. 13-5 70 9-77643 IO 9-87238 rr 0.12762 9-90405 30 71 9-77653 9-87253 5 T 0.12747 9.90400 5 29 11 73 9-77663 10 9-87269 10 Tfi 0.12731 9.90394 6 28 I I.I 73 9-77673 IO II 9-87285 IO 16 0.12715 9.90388 5 27 a 2.2 74 9-77684 9.87301 T 0.12699 9-90383 26 3 3-3 75 9.77694 IO 9-873I7 10 0.12683 9.90377 25 4 4-4 76 9-77704 IO IO 9-87332 15 16 0.12668 9.90371 6 5 24 I 11 77 78 9-77714 9.77724 IO 9-87348 9-87364 16 - o . 12652 0.12636 9.90366 9.90360 6 23 22 I 1:1 79 9-77734 IO IO 9-87380 IO 16 0.12620 9.90354 6 21 9 9-9 80 9-77744 9-87396 ifi 0.12604 9.90349 20 81 9-77755 II 9.87412 J.Q 0.12588 9.90343 19 10 82 9-77765 IO 9.87427 IS T r 0.12573 9.90337 6 18 I I.O 83 9-77775 IO IO 9-87443 10 16 0.12557 9-90332 5 6 17 2 3 2.O 3-0 84 9-77785 9-87459 tFi 0.12541 9.90326 6" 16 4 4-o 85 9-77795 10 9-87475 IO 0.12525 9.90320 15 5 i 86 9-77805 IO 10 9-87490 15 16 0.12510 9-903I5 5 6 14 6 7 o.o 7 o 87 9-778I5 9-87506 T& 0.12494 9-90309 13 8 8.0 88 9-77825 10 9.87522 IO _/ 0.12478 9-90303 12 9 9.0 89 9.77835 IO 9.87538 IO 16 0.12462 9.90298 5 g II 90 9.77846 10 9-87554 Tf 0.12446, 9.90292 g 10 91 9.77856 9.87569 *o Tfi 0.12431 9.90286 09 6 92 9-77866 IO 9.87585 10 rA 0.12415 9.90280 08 I 0.6 93 9.77876 10 IO 9.87601 10 16 0.12399 9-90275 5 6 07 2 1.2 1.8 94 9.77886 9.87617 rA 0.12383 9.90269 06 4 2.4 95 9.77896 IO 9-87633 10 0.12367 9.90263 05 3-0 96 9.77906 10 10 9.87648 15 16 0.12352 9-90258 5 6 04 6 3-6 97 9.77916 9.87664 T 0.12336 9-90252 03 7 g 4-2 4-8 98 9.77926 10 9.87680 IO _/r 0.12320 9.90246 02 n 1.1 99 9-77936 IO 9.87696 IO Tff 0.12304 9.90241 5 g OX 100 9.77946 9.877H *J 0.12289 9-90235 00 Cos d. Cot d. C. Tan Sin d. P.P. 126' 53 (172) 37' 142 < Sin d. Tan d. c. Cot Cos d. P. P. 00 9.77946 9.87711 16 0.12289 9-90235 g 100 01 9-77956 9.87727 16 0.12273 9-90229 g 99 16 02 9.77966 9.87743 0.12257 9.90223 98 03 9.77976 II 9-87759 16 0.12241 9.90218 6 97 I 2 I.O 3-2 04 9.77987 9-87775 0.12225 9.90212 g 9b 3 4-8 05 9-77997 9.87790 IS O.I22IO 9 . 90206 95 4 6.4 o6 9.78007 9.87806 16 O.I2I94 9.90201 6 94 5 8.0 9f. 07 9.78017 9.87822 O.I2I78 9.90195 93 7 II. 2 08 9.78027 9.87838 10 O.I2I62 9-90189 92 8 12.8 09 9-78037 9.87853 15 16 0.12147 9-90183 9i 9 14-4 10 9-78047 9.87869 16 0.12131 9.90178 6 90 ii 9-78057 9-87885 0.12115 9.90172 89 15 12 9.78067 10 9.87901 0.12099 9.90166 88 I 1-5 13 9-78077 9.87916 15 0.12084 9.90160 87 2 30 14 9.78087 9-87932 0.12068 9.90155 g 86 3 45 6 o 15 9-78097 9-87948 0.12052 9-90149 g 85 5 7-5 16 9.78107 10 9.87964 15 0.12036 9.90143 6 84 6 9-0 17 18 9-78II7 9.78127 9.78I37 IO 10 9-87979 9-87995 9.88011 16 16 Ih O.I202I O.I2OO5 0.11989 9-90137 9.90132 9.90126 5 6 83 82 81 7 8 9 10.5 12.0 13 5 20 9.78147 9.88027 O.II973 9.90120 6 80 21 9-78157 9.88042 O.II958 9.90114 79 11 22 9.78167 9.88058 O.II942 9-90109 5 78 23 9.78177 10 9.88074 IS O.II926 9-90103 6 77 2 2.2 24 9.78187 9.88089 tfi O.II9II 9-90097 g 7b 3 33 25 9.78197 9.88105 O.II895 9.90091 75 4 4-4 26 9.78207 10 9.88121 16 O.II879 9.90086 6 74 i o\6 27 9.78217 9.88137 O.II863 9.90080 73 7 7-7 28 9.78227 9.88152 IS O.II848 9.90074 72 8 8.8 29 9.78236 9 9.88168 16 O.II832 9.90068 71 9 9-9 30 9.78246 9.88184 rfi o. 11816 9.90063 g 70 31 9.78256 9.88200 O.IISOO 9.90057 g 69 9 32 9.78266 9.88215 15 O.II785 9-90051 68 I 0.9 33 9.78276 10 9.88231 16 O.II769 9.90045 6 67 2 3 1.8 2.7 34 9.78286 9.88247 O.H753 9.90039 66 4 3-6 35 9.78296 9.88262 15 O.II738 9-90034 65 5 4-5 36 9.78306 10 9.88278 16 O.II722 9.90028 6 64 6 ol 37 9-78316 9.88294 O.II706 9.90022 63 8 7.2 38 9-78326 9.88310 O.II690 9.90016 62 9 8.1 39 9.78336 9-88325 15 16 O.II675 9.90011 5 g 61 40 9-78346 9-88341 16 O.II659 9.90005 g 60 41 9 78356 9-88357 O.II643 9-89999 59 5 42 9-78366 9.88372 15 O.II628 9 89993 58 I 0-5 43 9-78375 9 IO 9.88388 16 o. 11612 9.89987 57 2 I.O 44 9-78385 9.88404 0.11596 9.89982 56 3 1-5 45 9-78395 9.88420 0.11580 9.89976 55 2 5 46 9.78405 10 9.88435 15 16 0.11565 9.89970 6 54 6 3-0 47 9-784I5 9-88451 o.i 1549 9.89964 53 7 3-5 48 9.78425 IO 9.88467 16 O.H533 9-89958 52 8 4.0 49 9-78435 9.88482 15 16 0.11518 9-89952 51 50 9-78445 9.88498 0.11502 9-89947 50 Cos d. Cot d. c. Tan Sin d. P.P. 127' 62 d73) 37< 142 Sin d. Tan d. c. Cot Cos d. P. P. 50 9-78445 10 9.88498 16 0.11502 9.89947 g 50 51 9-78455 9-88514 0.11486 9.89941 49 16 52 9.78464 9 9-88529 15 T-A 0.11471 9.89935 48 i 6 53 9.78474 10 IO 9-88545 IO 16 o II455 9.89929 6 47 2 3-2 54 9.78484 9.88561 -r 0.11439 9.89923 46 3 4-8 55 9.78494 IO 9-88577 10 0.11423 9.89918 5 45 4 6.4 56 9.78504 10 10 9-88592 15 16 0.11408 9.89912 6 44 5 6 8.0 9.6 57 9-785I4 9.88608 Tfi 0.11392 9.89906 43 7 II. 2 58 9-78524 10 9.88624 10 0.11376 9.89900 42 8 12.8 59 9-78533 9 9-88639 15 ifi o. 11361 9.89894 g 41 9 14.4 60 9.78543 10 9-88655 10 16 O.I 1345 9.89888 40 61 9-78553 9.88671 0.11329 9-89883 39 15 62 9.78563 10 9.88686 15 0.11314 9.89877 38 I 1-5 63 9-78573 10 10 9.88702 16 16 0.11298 9.89871 6 6 37 2 3 3-0 4 5 64 9.78583 9.88718 0.11282 9.89865 36 4 6.0 65 9-78592 9 9-88733 IS T-A 0.11267 9-89859 35 5 7-5 66 9.78602 10 IO 9.88749 10 16 0.11251 9-89853 6 34 6 7 9.0 10.5 67 9.78612 9.88765 0.11235 9.89847 33 8 12. O 68 9 . 78622 IO 9.88780 IS T r O.II22O 9-89842 5 32 9 13.5 69 9-78632 10 9.88796 IO 16 O.II204 9-89836 6 31 70 9.78642 10 9.88812 TC 0.11188 9-89830 5 30 71 9-786si 9 9.88827 15 rfi 0.11173 9.89824 g 29 10 72 9.78661 10 9.88843 10 rf\ 0.11157 9.89818 28 I I.O 73 9.78671 IO IO 9.88859 IO TC 0.11141 9.89812 6 27 2 2.0 74 9.78681 9.88874 A O T 0.11126 9.89806 26 3 3-0 75 9.78691 IO 9.88890 IO Tfi O.IIIIO 9.89801 5 g 25 4 5 4.0 5 o 76 9.78700 9 10 9.88906 IO 15 0.11094 9-89795 6 24 6 6^0 77 9.78710 9.88921 *A 0.11079 9.89789 23 7 7.0 78 9.78720 10 9-88937 10 ifi 0.11063 9-89783 g 22 8 8.0 79 9-78730 IO 9-88953 IO T 0.11047 9-89777 g 21 9 9.0 80 9-78739 9 IO 9.88968 AO 16 0.11032 9.89771 6 20 81 9-78749 9.88984 tfi 0.11016 9-8976S g 19 9 82 9 78759 10 9.89000 10 O.IIOOO 9-89759 18 i 83 9.78769 10 10 9-89015 15 16 0.10985 9-89754 5 6 17 2 3 18 I . 2.7 84 9.78779 9-89031 0.10969 9.89748 g 16 4 3.6 85 9.78788 9 9.89046 IS 0.10954 9.89742 15 5 4-5 86 9.78798 10 10 9.89062 16 16 0.10938 9-89736 6 14 7 5-4 6.3 87 9.78808 9.89078 0.10922 9.89730 g 13 8 7-2 88 9.78818 10 9-89093 15 Tfi 0.10907 9.89724 g 12 9 8.1 89 9-78827 9 9-89109 10 rfi 0.10891 9.89718 5 II 90 9-78837 xo 9-89125 xo TC 0.10875 9.89712 g 10 9i 9-78847 10 9.89140 *5 T 0.10860 9-89706 09 6 92 9.78856 9 9-89156 10 TJ\ 0.10844 9.89701 5 08 I 0.6 93 9.78866 10 9-89172 10 0.10828 9.89695 07 2 1.2 94 9.78876 10 9.89187 15 ,/? 0.10813 9-89689 g 06 3 1.8 95 9.78886 10 9-89203 ID 0.10797 9.89683 g 05 4 5 2. 4 3 96 9-78895 9 10 9.89218 15 16 0.10782 9.89677 6 04 6 3-'6 97 9-78905 9.89234 _ 0.10766 9.89671 03 7 4-2 98 9.78915 10 9-89250 IO 0.10750 9-89665 02 8 4-8 99 9.78924 9 9.89265 15 16 0.10735 9-89659 g OI 9 5-4 100 9-78934 IO 9.89281 0.10719 9.89653 00 Cos d. Cot d. c. Tan Sin d. P.P. 127 62 < (174) 38' 141' Sin d. Tan d. c. Cot Cos d. P. P. 00 9.78934 9.89281 16 0.10719 9 89653 6 100 01 9 78944 9.89297 0.10703 9.89647 99 Ifi 02 9-78954 10 9-89312 15 if> 0.10688 9.89641 98 j 1.6 03 9-78963 9 10 9-89328 10 IS 0.10672 9.89635 5 97 2 32 04 9.78973 9-89343 rfi 0.10657 9 89630 g 96 3 4-8 05 o6 9-78983 10 9 IO 9-89359 9-89375 10 16 15 0.10641 0.10625 9-89624 9.89618 6 6 95 94 4 5 6 6.4 8.0 9-6 07 9-79002 9.89390 ifi o. 10610 9.89612 g 93 7 II. 2 08 9.79012 10 9.89406 IO T y- 0.10594 9.89606 92 8 12.8 09 9.79021 9 9.89422 10 jr 0.10578 9.89600 6 9i 9 14-4 10 9.79031 10 9-89437 A O 16 o . 10563 9-89594 6 90 ii 9.79041 10 9-89453 0.10547 9-89588 89 15 12 9.79050 9 9.89468 IS T /r 0.10532 9-89582 88 5 i- 1-5 13 9.79060 10 IO 9.89484 It) 16 0.10516 9.89576 6 87 2 3-0 A r 14 9.79070 9-89500 0.10500 9.89570 86 4 6.1 15 9-79079 9 9-895I5 IS ifi 0.10485 9-89564 85 5 7-5 16 9-79089 IO 9-8953I IO 0.10469 9.89558 84 6 9-0 17 18 9.79099 9.79108 10 9 9.89546 9.89562 IS 16 Tfi 0.10454 0.10438 9.89552 9.89546 6 83 82 7 8 9 10.5 12. ti.if 19 9.79118 IO 10 9.89578 IO je 0.10422 9-89540 O 6 81 20 9.79128 9-89593 A O 16 0.10407 9-89534 80 21 9-79137 9 9-89609 0.10391 9-89528 79 in 22 9-79147 10 9.89624 IS rfi o . 10376 9.89522 78 j I 33 9.79156 9 10 9.89640 IO 16 0.10360 9-89516 6 77 2 2.0 24 9.79166 9-89656 0.10344 9-89510 76 3 3-0 25 9.79176, 10 9.89671 IS rfi 0.10329 9.89504 75 4 4-0 26 9.79185 9 IO 9-89687 10 IS 0.10313 9.89499 5 6 74 5 6 5-0 6.0 27 9.79195 9-89702 jf. 0.10298 9-89493 73 7 7-0 28 9-79204 9 9-89718 IO ffi o . 10282 9.89487 72 8 8.0 29 9.79214 IO 10 9.89734 10 TC 0.10266 9-89481 g 71 9 9-0 30 9-79224 9-89749 A O 16 0.10251 9.89475 g 70 3i 9-79233 9 9.89765 0.10235 9-89469 69 9 32 9-79243 IO 9.89780 IS _ O.I022O 9-89463 68 i 0-9 33 9 IO 9.89796 10 IS o . 10204 9.89457 6 67 2 -i 1.8 2 7 34 9.79262 9.89811 rfi O.IOI89 9-89451 66 o 4 3^6 35 9.79272 IO 9-89827 10 rf\ O.IOI73 9.89445 65 5 4-5 36 9.79281 9 10 9.89843 10 15 O.IOI57 9.89439 6 64 6 ii 37 9.79291 9-89858 rfi O.IOI42 9-89433 63 7 8 0.3 7 2 38 9-79300 9 9.89874 IO O.IOI26 9.89427 62 g 8.1 39 9-79310 10 9.89889 IS 16 O.IOIII 9.89421 61 \ iO 9.79319 9 IO 9.89905 T^ 0.10095 9-89415 5 60 4i 9-79329 9-89920 ID T 0.10080 9.89409 59 6 42 9-79339 IO 9.89936 IO tfi 0.10064 9.89403 58 I 0.6 43 9.79348 9 IO 9.89952 IO IS 0.10048 9.89397 6 57 2 1.2 44 9-79358 9.89967 T /? 0.10033 9 89391 56 3 1.8 45 9.79367 9 9-89983 10 0.10017 9-89385 55 4 2.4 46 9-79377 IO 9 9.89998 IS 16 O.IOO02 9.89379 6 54 S 6 3-0 3-6 47 9.79386 9 90014 0.09986 9.89373 53 7 4.2 48 9.79396 IO 9.90029 IS T /r 0.09971 9-89366 7 52 8 4.8 49 9-79405 9 IO 9-90045 IO 16 0.09955 9-89360 5 51 9 5-4 50 9-79415 9.90061 0.09939 9-89354 50 Cos d. Cot d. c. Tan Sin d. P. P. 128* (175) Sin d. Tan d. c. Cot Cos d. F .P. 50 9.79415 9.90061 0.09939 9-89354 g 50 51 52 53 54 55 56 57 58 59 9-79424 9-79434 9-79444 9-79453 9-79463 9-79472 9.79482 9-79491 9-79501 10 10 9 10 9 10 9 10 9.90076 9-90092 9.90107 9.90123 9-90138 9.90154 9.90169 9.90185 9.90200 16 15 16 15 16 15 16 15 16 0.09924 0.09908 0.09893 0.09877 0.09862 0.09846 0.09831 0.09815 0.09800 9.89348 9.89342 9.89336 9.89330 9.89324 9-89318 9-89312 9-89306 9-89300 6 6 6 6 6 6 6 6 g 49 48 47 46 45 44 43 42 41 I 2 3 4 I 7 8 9 16 1.6 J:l 6-4 8.0 9-6 II. 2 12.8 14-4 60 9-79510 9.90216 16 0.09784 9.89294 40 61 62 63 64 65 66 67 68 69 9-79520 9-79529 9-79539 9-79548 9-79558 9-79567 9-79576 9-79586 9-79595 9 10 9 10 9 9 10 9 9.90232 9.90247 9-90263 9.90278 9.90294 9.90309 9-90325 9.90340 9-90356 IS 16 15 16 15 16 15 16 0.09768 0.09753 0.09737 0.09722 0.09706 0.09691 0.09675 0.09660 0.09644 9.89288 9.89282 9.89276 9.89270 9.89264 9-89258 9-89252 9.89246 9.89239 6 6 6 6 6 6 6 7 g 39 38 37 36 35 34 33 32 31 I 2 3 4 I 7 8 9 15 l-S 3-0 4-5 6.0 7-5 9-0 10.5 12.0 13 5 70 9-79605 9 90371 16 0.09629 9-89233 6 30 71 72 73 74 75 76 77 78 79 9-796I4 9.79624 9.79633 9.79643 9-79652 9.79662 9.79671 9.7968o 9.79690 10 9 10 9 10 9 9 10 9.90387 9-90403 9.90418 9.90434 9.90449 9.90465 9.90480 9.90496 9.90511 16 15 16 15 16 15 16 IS l6 0.09613 0.09597 0.09582 0.09566 0.09551 0-09535 0.09520 0.09504 0.09489 9.89227 9.89221 9-892I5 9.89209 9.89203 9.89197 9.89191 9-89185 9.89179 6 6 6 6 6 6 6 6 6 29 28 27 26 25 24 23 22 21 I 2 3 4 5 6 7 8 9 10 I.O 2.0 3-0 4.0 5-0 6.0 7-0 8.0 9-o 80 9.79699 9 9.90527 0.09473 9.89173 20 81 82 83 84 85 86 87 88 89 9-79709 9-79718 9.79728 9-79737 9-79746 9-79756 9-79765 9-79775 9.79784 9 10 9 9 10 9 10 9 9.90542 9.90558 9-90573 9-90589 9.90604 9.90620 9-90635 9.90651 9.90666 16 IS 16 15 16 IS 16 IS 16 0.09458 0.09442 0.09427 0.09411 0.09396 0.09380 0.09365 0.09349 0.09334 9.89166 9.89160 9-89I54 9.89148 9-89142 9-89136 9.89130 9.89124 9.89118 6 6 6 6 6 6 6 6 g 19 18 17 16 IS 14 13 12 ii I 2 3 4 5 6 7 8 9 9 0.9 1.8 1:1 4-5 5-4 6-3 7.2 8.1 90 9-79793 9 9.90682 0.09318 9.89112 10 9i 92 93 94 95 96 97 98 99 9.79803 9.79812 9.79822 9-7983I 9.79840 9.79850 9.798S9 9.79868 9.79878 9 10 9 9 10 9 9 10 9.90697 9-90713 9.90728 9.90744 9-90759 9-90775 9.90790 9.90806 9.90821 16 IS 16 IS 16 IS 16 IS 16 0.09303 0.09287 0.09272 0.09256 0.09241 0.09225 0.09210 0.09194 0.09179 9-89105 9.89099 9-89093 9.89087 9.89081 9-89075 9.89069 9-89063 9-89056 6 6 6 6 6 6 6 7 5 09 08 07 06 05 04 03 02 OI I 2 3 4 5 6 7 8 9 7 0-7 i-4 2.1 2.8 3-5 4-2 4-9 5-6 6-3 100 9.79887 9-90837 0.09163 9.89050 00 Cos d. Cot d. c. Tan Sin d. I 5 . P. 128* 61 39' 140' Sin d. Tan d. c. Cot Cos d. P. P. 00 9.79887 9.90837 0.09163 9-89050 g 100 ox 9.79897 IO 9.90852 15 Tfi 0.09148 9.89044 g 99 16 02 9.79906 9 9.90868 IO 0.09132 9.89038 98 T 1.6 03 9.799IS 9 IO 9.90883 15 16 0.09117 9.89032 6 97 2 3-2 04 9-79925 9.90899 0.09101 9.89026 96 3 4.8 05 9-79934 9 9-90914 15 -* 0.09086 9.89020 95 4 6.4 8_ 06 9-79943 9 IO 9-90930 IO IS 0.09070 9.89013 7 6 94 5 6 .O 9-6 07 9-79953 9-90945 _/- 0.09055 9.89007 93 7 II. 2 08 9.79962 9 9.90961 10 0.09039 9.89001 92 8 12.8 09 9-79971 9 9.90976 IS rfi 0.09024 9-88995 5 9i 9 14.4 10 9.79981 IO 9.90992 IO yr 0.09008 9-88989 5 90 ii 9-79990 9 9.91007 o T 0.08993 9-88983 89 15 12 9-79999 9 9.91023 10 0.08977 9.88976 7 88 I I- 5 13 9.80009 10 y 9.91038 15 16 0.08962 9.88970 6 87 2 3 3-0 4-5 14 9.80018 9-91054 0.08946 9-88964 86 4 6.0 IS 9.80027 9 9.91069 15 0.08931 9.88958 85 5 7-5 16 9-80037 10 9.91085 16 0.08915 9.88952 84 6 9.0 17 9.80046 9 9.91100 15 1- 0.08900 9.88946 83 7 8 10. 5 12. If 9.80055 9 9.91116 IO 0.08884 9-88939 7 g 82 9 I3-S 19 9.80064 9 9.9H3I 15 16 0.08869 9-88933 g 81 20 9.80074 9-9"47 Tg 0.08853 9.88927 g 80 21 9.80083 9.91162 1-5 T 0.08838 9.88921 79 10 22 9.80092 9 9.91178 10 0.08822 9-88915 78 1 1.0 23 9.80102 IO 9-9II93 15 16 0.08807 9-88909 1 77 2 2.0 24 9.80111 9-91209 0.08791 9.88902 76 3 3-0 25 9.80120 9 9.91224 IS 0.08776 9.88896 g 75 4 5 4.0 5 o 26 9.80129 9 10 9.91239 15 16 0.08761 9.88890 6 74 6 6.0 27 9-80139 9-91255 0.08745 9.88884 73 7 7.0 28 9.80148 9 9.91270 15 ,c 0.08730 9.88878 72 8 8.0 29 9.80157 9 9.91286 10 0.08714 9.88871 7 g 71 9 9.0 30 9.80166 9 IO 9.91301 15 16 0.08699 9.88865 g 70 3i 9.80176 9.9I3I7 0.08683 9-88859 69 9 32 9.80185 9 9-9I332 15 _/ 0.08668 9.88853 68 I 33 9.80194 9 IO 9.91348 10 15 0.08652 9.88847 7 67 2 3 18 I.o 2.7 34 9.80204 9-9I363 T 0.08637 9.88840 66 4 3-6 35 9.80213 9 9.91379 IO 0.08621 9-88834 65 5 4-5 36 9 . 80222 9 9 9.9I394 15 16 0.08606 9.88828 6 64 6 7 5-4 6.3 37 9.80231 9.91410 0.08590 9.88822 63 8 7-2 38 9 . 80240 9 9.91425 15 tfi 0.08575 9.88815 7 62 9 8.1 39 9.80250 IO 9.91441 IO 0.08559 9.88809 g 61 40 9-80259 9.9I456 IS T(- 0.08544 9.88803 g 60 41 9.80268 9.91471 10 0.08529 9-88797 59 6 42 9.80277 9 9.91487 16 0.08513 9-88791 6 58 I 0.6 43 9.80287 IO 9 9-91502 15 16 0.08498 9.88784 7 6 57 2 1.2 44 9.80296 9-9I5I8 0.08482 9-88778 56 3 1.8 45 46 9-80305 9-80314 9 9 9 9.9I533 9-91549 15 16 IS 0.08467 0.08451 9.88772 9.88766 6 7 55 54 4 i 2-4 3-0 3-6 47 9.80323 9.91564 T 0.08436 9-88759 53 7 4.2 48 9-80333 IO 9.91580 10 0.08420 9.88753 52 8 4-8 49 9-80342 9 9-91595 15 TC 0.08405 9-88747 g Si 9 5-4 50 9-80351 9.91610 A O 0.08390 9.88741 50 Cos d. Cot d. c. Tan Sin d. P.P. 129' 50 C (177) 140' Sin d. Tan d. c. Cot Cos d. P. P. 50 Si 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 9i 92 93 94 95 96 97 98 99 100 9-80351 9 9 10 9 9 9 9 10 9 9 9 9 9 9 10 9 9 9 9 9 9 10 9 9 9 9 9 9 9 9 10 9 9 9 9 9 9 9 9 9 9 9 9 9 10 9 9 9 9 9 9.91610 16 15 16 15 16 IS 16 15 15 16 IS 16 IS 16 IS IS 16 IS 16 IS 16 IS IS 16 IS 16 IS IS 16 IS 16 15 16 IS IS 16 IS 16 15 IS 16 IS 16 IS IS 16 IS 16 IS IS 0.08390 9-88741 7 6 6 6 7 6 6 6 7 6 6 7 6 6 6 7 6 6 7 6 6 6 7 6 6 7 6 6 7 6 6 6 7 6 6 7 6 6 7 6 6 7 6 6 7 6 7 6 6 7 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 II 10 09 08 07 06 05 04 03 02 01 00 I 2 3 4 7 8 9 I 2 3 4 5 6 7 8 9 I 2 3 4 I I 9 I a 3 4 I 7 8 9 I 2 3 4 I 7 8 9 16 1.6 3-2 4.8 6-4 8.0 9-6 II. 2 12.8 14-4 15 1-5 3-0 4-5 6.0 7-5 9.0 10.5 12. 13.5 10 I.O 2.O 3-0 4.0 5-0 6.0 7-0 8.0 9.0 9 SI 1:1 4-5 5-4 6.3 7.2 8.1 7 o-7 1.4 2.1 2.8 3-5 4-2 4.9 5.6 6.3 9.80360 9-80369 9.80379 9.80388 9.80397 9.80406 9.80415 9.80425 9.80434 9.91626 9.91641 9.9I657 9.91672 9.91688 9.91703 9.91719 9.9I734 9.91749 0.08374 0.08359 0.08343 0.08328 0.08312 0.08297 0.08281 0.08266 0.08251 9-88734, 9.88728 9.88722 9.88716 9.88709 9-88703 9.88697 9.88691 9.88684 9.80443 9-9I765 0.08235 9.88678 9.80452 9.80461 9.80470 9.80479 9.80489 9.80498 9.80507 9.80516 9-80525 9.91780 9.91796 9.91811 9.91827 9.91842 9-9I857 9-9I873 9.91888 9.91904 0.08220 o . 08204 0.08189 0.08173 0.08158 0.08143 0.08127 o. 08112 0.08096 9.88672 9.88665 9.88659 9-88653 9.88647 9.88640 9-88634 9.88628 9.88621 9.80534 9.91919 0.08081 9.88615 9-80543 9-80553 9.80562 9-8057I 9.80580 9.80589 9.80598 9.80607 9.80616 9-91935 9.91950 9-9I965 9.91981 9.91996 9.92012 9.92027 9-92042 9-92058 0.08065 o . 08050 0.08035 0.08019 o . 08004 0.07988 0.07973 0.07958 0.07942 9.88609 9.88603 9-88596 9-88590 9.88584 9-88577 9.88571 9-88565 9-88558 9.80625 9.92073 0.07927 9-88552 9.80635 9.80644 9.8o653 9.80662 9.80671 9.80680 9.80689 9.80698 9.80707 9-92089 9.92104 9.92120 9-92135 9.92150 9.92166 9.92181 9.92197 9.92212 0.07911 0.07896 0.07880 0.07865 0.07850 0.07834 0.07819 0.07803 0.07788 9.88546 9-88540 9-88533 9.88527 9.88521 9.88514 9.88508 9.88502 9-88495 9.80716 9.92227 0.07773 9-88489 9.80725 9-80734 9.80743 9.80752 9.80762 9.80771 9.80790 9-80789 9.80798 9-92243 9.92258 9.92274 9.92289 9.92304 9.92320 9.92335 9.9235I 9.92366 0.07757 0.07742 0.07726 0.07711 o . 07696 0.07680 0.07665 0.07649 0.07634 9.88483 9.88476 9.88470 9.88464 9.88457 9.88451 9.88444 9-88438 9.88432 9.80807 9.92381 0.07619 9.88425 Cos d. Cot d. c. Tan Sin d. P. P. 129 6. P. 132* (184) 43' 136' Sin d. Tan d. c. Cot Cos d. I \ P. 00 9-83378 g 9.96966 0.03034 9-86413 100 01 02 03 04 OS 06 07 08 09 9-83386 9-83395 9-83403 9.83411 9-834I9 9.83427 9-83435 9-83443 9-83451 9 8 8 8 8 8 8 8 g 9.96981 9.96996 9.97011 9.97026 9.97042 9-97057 9.97072 9.97087 9-97102 15 15 15 16 IS 15 15 15 16 0.03019 0.03004 0.02989 0.02974 0.02958 0.02943 0.02928 0.02913 0.02898 9-86406 9-86399 9-86392 9-86384 9.86377 9-86370 9-86363 9-86356 9-86349 7 7 8 7 7 7 7 7 99 98 97 96 95 94 93 92 9i i 2 3 4 1 9 16 1.6 ii a 9-6 II. 2 12.8 14.4 10 9-83459 9.97118 0.02882 9-86342 90 ii 12 13 U 15 16 17 18 19 9-83468 9.83476 9.83484 9.83492 9-83500 9-83508 9-83516 9.83524 9.83532 8 8 8 8 8 8 8 8 g 9-97I33 9.97148 9.97163 9.97178 9-97193 9.97209 9.97224 9.97239 9-97254 IS 15 IS IS 16 IS IS IS 0.02867 0.02852 0.02837 0.02822 0.02807 0.02791 0.02776 0.02761 0.02746 9-86335 9.86328 9.86321 9-86314 9.86306 9.86299 9.86292 9.86285 9.86278 7 7 7 8 7 7 7 7 89 88 87 86 85 84 83 82 81 i 2 3 4 5 6 7 8 9 15 i. 5 3-0 4-5 6.0 7-5 9.0 10. 5 12. 13-5 20 9.83540 g 9.97269 16 0.02731 9.86271 80 21 22 33 24 25 26 27 28 29 9-83548 9.83556 9.83565 9-83573 9-83581 9.83589 9-83597 9-83605 9-83613 8 9 8 8 8 8 8 8 g 9-97285 9.97300 9-97315 9-97330 9-97345 9-9736I 9-97376 9-97391 9.97406 IS IS IS IS 16 IS IS IS 0.02715 0.02700 0.02685 0.02670 0.02655 0.02639 o . 02624 0.02609 0.02594 9.86264 9-86257 9.86250 9.86242 9-86235 9.86228 9.86221 9.86214 9.86207 7 7 8 7 7 7 7 7 79 78 77 76 75 74 73 72 7i i 2 3 4 5 6 7 8 9 9 0-9 1.8 2.7 3-6 4-5 11 7-2 8.1 30 9.83621 g 9.97421 16 0.02579 9.86200 g 70 31 32 33 34 35 36 37 38 39 9-83629 9-83637 9-83645 9.83653 9.83661 9-83669 9.83677 9-83685 9-83693 8 8 8 8 8 8 8 8 g 9-97437 9-97452 9.97467 9.97482 9-97497 9-97512 9-97528 9-97543 9-97558 IS IS IS IS IS 16 IS 15 IS 0.02563 0.02548 0.02533 0.02518 0.02503 0.02488 0.02472 0.02457 0.02442 9.86192 9.86185 9.86178 9.86171 9.86164 9-86157 9.86150 9.86142 9-86135 7 7 7 7 7 7 8 7 69 68 67 66 65 64 63 62 61 i 2 3 4 7 8 9 8 0.8 1.6 2.4 3-2 4-0 4-8 5-6 6.4 7-2 40 9.83701 g 9-97573 0.02427 9.86128 60 4i 42 43 44 45 46 47 48 49 9-83709 9-837I7 9-83725 9-83733 9-83741 9-83749 9-83757 9.83765 9-83773 8 8 8 8 8 8 8 8 8 9.97588 9.97604 9.97619 9-97634 9.97649 9.97664 9.97679 9-97695 9.97710 16 IS IS IS IS IS 16 IS 15 0.02412 0.02396 0.02381 0.02366 0.02351 0.02336 0.02321 0.02305 0.02290 9.86121 9.86114 9.86107 9-86099 9.86092 9.86085 9.86078 9.86071 9.86063 7 7 8 7 7 7 7 8 7 59 58 57 56 55 54 53 52 Si I 2 3 4 7 8 9 7 0-7 1-4 2.1 2.8 3-5 4-2 4-9 5-6 6-3 50 9-83781 9-97725 0.02275 9.86056 60 Cos d. Cot d. c. Tan Sin d. P. P. 133< 46' (185) 43 136 < Sin d. Tan d. c. Cot Cos d. p. p. 50 Si 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 9.83781 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 7 8 8 8 8 8 8 8 8 8 8 8 7 8 8 8 8 8 8 8 8 7 8 8 8 8 8 8 8 7 8 8 9-97725 IS 15 16 15 15 IS IS IS 16 IS IS IS IS 16 IS IS IS IS IS 16 IS IS IS IS IS 16 IS IS IS IS IS 16 IS IS IS IS 16 IS IS IS IS IS 16 IS IS IS IS IS 16 IS 0.02275 9.86056 7 7 7 8 7 7 7 7 8 7 7 7 8 7 7 7 7 8 7 7 7 8 7 7 7 8 7 7 7 8 7 7 8 7 7 7 8 7 7 8 7 7 7 8 7 7 8 7 7 8 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 II 10 09 08 07 06 05 04 03 02 01 00 I 2 3 4 5 6 7 8 9 i 2 3 4 5 6 7 8 9 I 2 3 4 5 6 7 8 9 i 2 3 4 5 6 7 8 9 16 1.6 3-2 4.8 6.4 8.0 9-6 II. 2 12.8 14-4 15 i.S 3-0 4-5 6.0 7-5 9-0 10.5 12.0 13-5 8 0.8 1.6 2-4 3-2 4.0 4-8 5.6 6.4 7.2 7 0-7 i-4 2.1 2.8 3-S 4-2 4.9 5-6 6.3 9.83789 9.83797 9-83805 9-83813 9.83821 9.83829 9.83837 9.83845 9.83853 9.97740 9-97755 9-97771 9-97786 9.978oi 9.97816 9.97831 9.97846 9.97862 o . 02260 0.02245 0.02229 0.02214 0.02199 0.02184 0.02169 0.02154 0.02138 9-86049, 9.86042 9-86035 9.86027 9.86020 9.86013 9.86006 9-85999 9.85991 9.83861 9.97877 0.02123 9.85984 9-83869 9-83877 9-83885 9-83898 9.83901 9.83909 9.839I7 9.83925 9-83932 9.97892 9.97907 9-97922 9-97938 9-97953 9.97968 9.97983 9.97998 9.98013 0.02108 0.02093 0.02078 0.02062 0.02047 0.02032 0.02017 0.02002 0.01987 9-8S977 9.85970 9-85962 9-85955 9.85948 9-85941 9-85934 9.85926 9.85919 9.83940 9.98029 O.OI97I 9-85912 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 9i 92 93 94 95 96 97 98 99 100 9.83948 9.83956 9.83964 9.83972 9.83980 9.83988 9.83996 9.84004 9.84012 9-98044 9-98059 9-98074 9.98089 9.98104 9.98120 9.98135 9.98150 9-98165 O.OI956 O.OI94I O.OI926 O.OI9II 0.01896 0.01880 0.01865 0.01850 0.01835 9-85905 9.85897 9.85890 9-85883 9.85876 9.85868 9.85861 9-85854 9-85847 9.84020 9.98180 O.OI82O 9.85839 9.84027 9-84035 9-84043 9.84051 9.84059 9.84067 9.84075 9-84083 9.84091 9.98195 9.98211 9.98226 9.98241 9.98256 9.98271 9.98287 9.98302 9-983I7 O.OI805 0.01789 0.01774 0.01759 0.01744 O.OI729 O.OI7I3 0.01698 0.01683 9-85832 9-85825 9-85817 9.85810 9.85803 9.85796 9.85788 9-85781 9.85774 9-84098 9-98332 0.01668 9-85766 9.84106 9.84114 9.84122 9-84130 9-84138 9.84146 9-84154 9.84161 9.84169 9-98347 9-98362 9.98378 9.98393 9-98408 9-98423 9.98438 9.98453 9.98469 O.OI653 0.01638 0.01622 O.OI607 O.OI592 0.01577 0.01562 0.01547 0.01531 9-85759 9-85752 9-85745 9-85737 9.85730 9-85723 9.857I5 9-85708 9-85701 9.84177 9.98484 0.01516 9.85693 Cos d. Cot d. c. Tan Sin d. P. P. 133 46 < (i 86) 44' 135< Sin d. Tan d. c. Cot Cos d. P. P. 00 OI 02 03 04 05 06 07 08 09 10 ii 12 13 14 IS 16 17 18 i9 20 21 22 23 24 25 26 27 28 29 30 3i 32 33 34 35 36 37 38 39 40 4i 42 43 44 45 46 47 48 49 50 9.84177 8 8 8 8 7 8 8 8 8 7 8 8 8 8 8 7 8 8 8 8 7 8 8 8 8 7 8 8 8 7 8 8 8 7 8 8 8 7 8 8 8 7 8 8 8 7 8 8 7 g 9.98484 IS IS 15 IS 16 IS IS 15 15 15 16 IS IS IS IS IS 16 IS IS IS IS IS 16 IS IS IS IS IS 16 15 IS IS IS IS 16 IS IS IS IS IS 16 IS IS IS IS IS IS 16 IS IS 0.01516 9-85693 7 7 8 7 7 8 7 7 8 7 7 8 7 7 8 7 7 8 7 7 8 7 8 7 7 8 7 8 7 7 8 7 8 7 7 8 7 8 7 7 8 7 8 7 8 7 7 8 7 8 100 99 98 97 96 95 94 93 92 9i 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 7i 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 5i 50 I 2 3 4 5 6 7 8 9 i 2 3 4 5 6 7 8 9 I 2 3 4 5 6 7 8 9 i 2 3 4 S 6 7 8 9 16 1.6 3-2 4-8 6.4 8.0 9.6 II. 2 12.8 14-4 15 1-5 3-0 4-5 6.0 7-5 9-0 10.5 12.0 13-5 8 0.8 1.6 2.4 3-2 4.0 4-8 5-6 6.4 7-2 7 o.7 4 .8 5 .2 9 5-6 6.3 9-84185 9.84193 9.84201 9.84209 9.84216 9.84224 9.84232 9.84240 9 84248 9.98499 9.98514 9.98529 9.98S44 9-98560 9.9857S 9.98590 9.98605 9.98620 0.01501 0.01486 0.01471 0.01456 0.01440 0.01425 0.01410 0.01395 0.01380 9.85686 9-85679 9-85671 9.85664 9-85657 9.85649 9.85642 9-85635 9.85627 9.84255 9.98635 0.01365 9.85620 9.84263 9.84271 9-84279 9.84287 9-84295 9.84302 9-84310 9.84318 9.84326 9-98651 9.98666 9.98681 9.98696 9.98711 9.98726 9.98742 9.98757 9.98772 0.01349 0.01334 0.01319 0.01304 0.01289 0.01274 0.01258 0.01243 0.01228 9-85613 9.85605 9.85598 9-85591 9-85583 9.85576 9-85569 9 85561 9-85554 9-84334 9.98787 0.01213 9-85547 9-8434I 9-84349 9-84357 9-8436S 9-84373 9.84380 9-84388 9.84396 9.84404 9.98802 9.98817 9.98833 9.98848 9-98863 9.98878 9.98893 9-98908 9.98924 0.01198 0.01183 0.01167 0.01152 0.01137 O.OII22 O.OII07 O.OIO92 O.OIO76 9-85539 9.85532 9-85524 9.855I7 9-85510 9-85502 9-85495 9-85487 9-85480 9.84411 9.98939 o. 01061 9.85473 9.84419 9-84427 9-84435 9.84442 9.84450 9.84458 9.84466 9.84473 9.84481 9-98954 9.98969 9.98984 9.98999 9.990I5 9.99030 9.99045 9.99060 9.99075 0.01046 O.OI03I O.OIOI6 O.OIOOI 0.00985 0.00970 0.00955 0.00940 0.00925 9-85465 9.85458 9-85450 9-85443 9.85436 9.85428 9.85421 9.854I3 9-85406 9.84489 9.99090 0.00910 9-85399 9.84497 9.84504 9.84512 9.84520 9-84528 9.8453S 9.84543 9.8455I 9.84558 9.99106 9.99121 9.99136 9.99I5I 9.99166 9.99181 9.99196 9-99212 9.99227 0.00894 0.00879 0.00864 0.00849 0.00834 0.00819 0.00804 0.00788 0.00773 9.85391 9-85384 9.85376 9-85369 9-85361 9-85354 9-85347 9-85339 9-85332 9-84566 9.99242 0.00758 9.85324 Cos d. Cot 1 d. c. Tan Sin d. P. P. 134 45 < (187) 44 Sin d. Tan d. c. Cot Cos d. 1 >. P. 50 9.84566 g 9.99242 0.00758 9.85324 50 5i 52 53 54 55 56 57 58 59 9-84574 9.84582 9.84589 9-84597 9.84605 9.84612 9.84620 9.84628 9-84635 8 7 8 8 7 8 8 7 g 9-99257 9.99272 9.99287 9-99303 9.99318 9-99333 9.99348 9.99363 9 99378 IS 15 16 IS IS IS IS IS 16 0.00743 0.00728 0.00713 0.00697 0.00682 0.00667 0.00652 0.00637 0.00622 9-853I7' 9-85309 9-85302 9-85294 9-85287 9.85279 9-85272 9.85265 9.85257 8 7 8 7 8 7 7 8 49 48 47 46 45 44 43 42 41 I 2 3 4 5 6 16 1.6 32 4-8 6.4 8.0 9.6 60 9.84643 g 9-99394 0.00606 9-85250 g 40 8 12.8 6i 62 63 64 65 66 67 68 69 9.84651 9-84659 9.84666 9.84674 9.84682 9.84689 9.84697 9.84705 9-84712 8 7 8 8 7 8 8 7 g 9.99409 9.99424 9-99439 9-99454 9.99469 9.99485 9-99500 9-99515 9-99530 IS IS IS IS 16 IS IS IS 0.00591 0.00576 0.00561 0.00546 0.00531 0.00515 0.00500 0.00485 0.00470 9.85242 9.85235 9-85227 9.85220 9.85212 9-85205 9.85197 9-85190 9.85182 7 8 7 8 7 8 7 8 39 38 37 36 35 34 33 32 31 9 I 2 3 4 5 14.4 !5 1-5 3-0 4-5 6.0 7-5 70 9.84720 9-99545 0.00455 9.85175 g 30 6 9.0 71 72 73 74 75 76 77 78 79 9.84728 9.84735 9.84743 9.8475I 9.84758 9.84766 9-84773 9.84781 9.84789 7 8 8 7 8 7 8 8 9.99560 9-99576 9-99591 9.99606 9.99621 9.99636 9.99651 9-99666 9.99682 16 IS IS IS IS IS IS 16 0.00440 0.00424 0.00409 0.00394 0.00379 0.00364 0.00349 0.00334 0.00318 9.85167 9.85160 9-85I52 9-85I4S 9.85137 9-85130 9.85122 9.85H5 9.85107 7 8 7 8 7 8 7 8 29 28 27 26 25 24 23 22 21 8 9 I 12.0 13-5 8 0.8 80 9.84796 g 9.99697 0.00303 9.85100 g 20 2 1.6 8s 82 83 84 85 86 87 88 89 9.84804 9.84812 9.84819 9.84827 9-84835 9.84842 9.84850 9-84857 9.84865 8 7 8 8 7 8 7 8 9.99712 9.99727 9.99742 9-99757 9-99773 9-99788 9-99803 9.99818 9.99833 IS IS IS 16 IS IS IS IS 0.00288 0.00273 0.00258 0.00243 0.00227 O.OO2I2 0.00197 0.00182 O.OOI67 9-85092 9-85085 9.85077 9.85069 9.85062 9.85054 9.8S047 9-8S039 9-85032 7 8 8 7 8 7 8 7 g 19 18 17 16 15 14 13 12 II 3 4 S 6 7 8 9 3-2 4.0 4-8 5.6 6.4 7-2 7 90 9.84873 9.99848 16 0.00152 9-85024 10 i 0.7 9i 92 93 94 95 96 97 98 99 9.84880 9.84888 9.84895 9.84903 9.84911 9.84918 9.84926 9.84933 9.84941 8" 7 8 8 7 8 7 8 9.99864 9.99879 9.99894 9-99909 9-99924 9-99939 9-99955 9.99970 9-99985 IS IS IS IS IS 16 IS IS O.OOI36 O.OOI2I 0.00106 0.00091 O.OOO76 0.0006I O.OOO45 O.OO03O O.OOOI5 9-85017 9-85009 9.85001 9.84994 9.84986 9-84979 9.84971 9.84964 9.84956 8 8 7 8 7 8 7 8 7 09 08 07 06 05 04 03 02 OI 2 3 4 5 6 7 8 9 i-4 2.1 2.8 3-5 4-2 4-9 5.6 6.3 100 9.84949 IO.OOOOO o.ooooo 9.84949 00 Cos d. Cot d. c. Tan Sin d. P P. 134 C 45' (188) TABLE XXVI. LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS Log vs = 2 log a + V Log exsec = 2 log a -f- E Hun- Hun- 1 dredths Vers v E Exsec dredths Vers V E Exsec 00 [nf. neg. 6. 725 182 725 Inf. neg. 00 6.18271 6. 714 182 780 6.18278 02 04 06 08 IO 12 14 16 18 2.78478 3-38684 .73903 .98890 4.18272 .34109 .47498 .59096 .69327 725 725 725 725 725 725 725 724 724 725 725 725 725 725 726 726 726 727 2.78478 3.38684 .73903 .98890 4.18272 .34109 .47498 .59096 .69327 02 04 06 08 10 12 14 16 18 .19991 .21678 23332 .24956 6.26549 .28115 .29652 .31163 .33-461 713 713 712 712 711 711 711 710 709 782 785 787 789 791 794 797 799 801 .19998 .21685 .23339 .24964 6.26557 .28123 .29661 .31172 33470 20 4.78478 724 727 4.78479 20 6.34107 709 804 6.34116 22 24 26 28 30 32 34 36 38 .86757 .94315 5.01267 .07704 5.13697 .19302 .24568 .29533 .34229 724 724 724 724 724 724 723 723 723 727 728 729 729 730 731 731 732 733 .86757 94315 5.01268 .07705 5.13697 .19303 .24569 .29534 .34230 22 24 26 28 30 32 34 36 38 -35543 .36955 .38345 .39713 6.41059 .42385 .43691 44977 .46246 708 708 707 707 706 706 70S 704 704 807 810 812 815 818 821 824 827 830 35553 .36965 .38356 .39724 6.41070 42397 .43703 .44989 46259 40 5.3868 4 723 734 5.38685 40 6.47496 703 833 6.47509 42 44 46 48 50 52 54 56 58 .42922 .46962 .50824 54521 5.58066 .61473 .64751 .67910 .70958 723 722 722 722 722 722 721 721 721 735 735 736 737 739 740 740 742 743 .42923 .46964 .50825 54522 5.58068 .6x475 .64753 .67912 .70960 42 44 46 48 50 52 54 56 58 .48728 49943 .5U4I .52322 6.53488 .54639 55774 .56895 .58001 703 702 701 700 700 699 699 698 697 836 839 842 845 849 852 855 859 863 .48741 49957 SUSS 52336 6.53503 .54654 55790 .56911 .58018 60 5.73902 721 745 5.73904 60 6.59093 697 866 6.59110 62 64 66 68 70 72 74 76 78 .76750 .79508 .82181 .84774 5.87292 .89738 .92118 94435 .96691 720 720 720 720 719 719 719 718 718 746 747 749 750 752 753 755 756 758 .76753 795II .82184 .84777 5-87295 .89742 .92122 .94438 -96695 62 64 66 68 70 72 74 76 78 .60173 .61238 .62291 .63331 6.64359 .65375 .66379 .67372 .68353 696 695 694 694 693 692 692 691 690 870 873 877 88l 884 888 892 896 900 .60190 .61256 .62309 .63350 6.64378 65395 .66399 .67392 .68374 80 5-98890 718 760 5.98894 80 6.69323 689 903 6.69345 82 84 86 88 90 92 94 96 98 6.01034 .03128 .05171 .07168 6.09120 .11029 .12897 .14726 -16517 717 717 717 716 716 715 715 715 714 762 764 766 767 769 772 774 776 778 6 . 01039 .03132 .05176 07173 6.09125 11035 .12903 . 14732 16523 82 84 86 88 90 92 94 96 98 .70383 .71232 .72171 73100 6.74019 .74929 .75829 .76718 .77601 688 68? 686 686 685 684 683 682 681 907 911 915 919 924 928 932 936 941 .70405 71254 .72194 .73123 6.74043 74953 .75854 .76743 .77627 100 6.18271 714 780 6.18278 100 6.78474 681 945 6.78500 (189) TABLE XXVI. LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS Logvs = 2 logo; + V Logexsec = 2logo: + Hun- dredths 2 Hun- dredths 3 x Vers V E Exsec Vers V \ E Exsec 00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 So 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 94 96 98 100 6.78474 6. 681 680 679 678 677 676 675 674 673 672 671 670 669 668 667 666 665 664 663 662 661 660 659 658 657 656 655 654 652 651 650 649 648 646 645 644 643 642 640 639 638 637 636 634 633 632 631 630 628 627 626 182 945 950 954 958 963 968 973 978 982 987 992 997 *O02 006 Oil 016 022 027 032 037 042 048 053 058 063 069 075 081 086 092 098 103 109 5 121 127 133 139 145 151 157 163 169 176 182 189 195 201 208 215 221 .. 6.78501 00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 TO 72 74 76 78 ] 80 82 84 86 88 90 92 94 96 98 100 7-13687 6.182 626 624 623 621 620 619 618 616 614 613 612 610 609 607 606 605 603 602 600 599 597 596 595 593 591 590 588 587 585 583 582 58l 579 577 575 574 572 571 569 567 566 564 562 56o 558 557 555 553 552 550 548 6.183 221 228 234 241 248 255 261 268 275 283 290 296 303 3ii 318 325 333 340 348 355 362 370 377 385 393 400 408 416 424 432 440 448 456 464 472 480 488 496 504 512 521 529 538 547 556 564 572 581 590 599 608 7.13746 .79338 .80194 .81041 .81880 6.82712 .83535 .84350 .85158 85959 .79365 .80221 .81069 .81909 6.82741 -83565 .84381 .85189 .85990 .14264 14837 15406 .15972 7.I6S34 .17093 .17648 .18199 .18747 14324 .14898 .16468 16035 7.16598 I7I57 .17713 . 18265 .18814 6.86752 6.86784 7.19291 7.19359 .87538 .88317 .89089 -89854 6.90612 91364 .92110 .92849 .93582 .87570 .88350 .89122 .89888 6.90647 .91400 .92146 .92886 .93619 .18832 .20370 .20904 21435 7-21963 .22488 .23010 .23528 .24043 .19901 .20439 .20975 .21507 7.22035 .22561 .23083 .23603 .24119 6.94308 6.94347 7.24556 7.24632 .95029 95744 .96453 .97156 6.97854 .98546 .99232 .99913 .00589 .95068 .95783 .96493 .97197 6.97895 .98588 .99275 99957 .00633 .25065 .25571 .26075 26575 7-27073 .27567 .28059 .28549 .29035 .251.42 .25649 .26154 .26655 7-27154 .27649 .28142 .28632 .29120 7.01260 7.01304 7.29519 7 29605 .01925 .02586 .03241 .03891 7.04537 .05178 .05814 .06446 .07073 .01971 .02632 .03288 .03939 7.04585 .05227 .05864 .06496 .07124 .30000 30478 30954 31427 7.31898 .32366 .32831 .33294 33755 .30087 .30566 .31043 .31517 7.31988 .32457 32924 .33388 .33850 7.07695 7-07747 7-34213 7.34309 .08314 .08927 .09537 .10142 7-10743 .11340 .11932 . 12521 .13106 .08366 .08981 .09591 .10197 7.10799 .11396 .11790 .12579 .13165 .34669 .35122 .35574 .36022 7.36469 36913 37355 37794 .38232 .34766 .35220 .35672 .36122 7-36569 37014 37457 .37898 .38337 7.13687 7.13746 7-38667 7.38773 (190) TABLE XXVI. (Continued) Hun- dredths 4 Hun- dredths 5 Vers Diff. .001 Exsec Diff. .001 Vers. Diff. .001 Exsec Diff. .001 00 7-38667 21 6s 7.38773 00 7.58039 7.58204 02 04 06 .39100 39531 .39960 21.55 21.45 21 30 .39207 .39639 .40069 21.60 .21.50 21 35 02 04 06 .58385 .58731 .59074 17-30 17.15 17.15 .58552 .58899 .59244 17-35 17-25 17.20 08 IO -4P386 7.40811 21.25 .40496 7.40922 21.30 08 10 .59417 7-59758 17-05 .59588 7-59930 17.10 12 14 16 18 .41233 .41654 .42072 .42488 21. IO 2I.O5 20.90 20.80 .41346 .41767 .42187 .42604 21. 2O 21.05 21.00 20.85 12 14 16 18 .60098 .60436 .60773 .61109 16.90 16.85 16.80 .60271 .60611 .60949 .61287 17.05 17.00 16.90 16.90 20 7.42903 7-43020 20 7.61443 16 70 7.61622 22 24 26 28 30 32 .43315 .43726 .44134 44541 7.44946 45349 20.55 20.40 20.35 20.25 20.15 20.05 43433 .43845 .44255 .44662 7.45068 45472 2O.6O 20.50 20.35 20.30 20.20 20.10 22 24 26 28 30 32 .61777 .62108 .62439 .62769 7.63097 .63424 16.55 16.55 16.50 16.40 16.35 16.25 .61957 .62290 .62622 .62953 7-63283 .63611 16.65 16.60 16.55 16.50 16.40 16.35 34 36 38 45750 .46149 .46546 19-95 19.85 .45874 .46275 .46673 20.O5 19.90 34 36 38 .63749 .64074 .64397 16.25 16.15 .63938 .64264 .64589 16.30 16.25 40 7.46942 7.47070 40 7.64719 7.64912 42 44 46 47335 .47727 .48118 19.60 19-55 19.40 .47465 .47858 .48249 19.65 19-55 19.50 42 44 46 .65040 .65360 .65678 16.00 15.90 15.85 .65235 .65556 .65876 16.05 16.00 15.90 48 50 52 .48506 7.48893 .49278 19-35 19-25 19.15 .48639 7.49027 .49413 19.40 19.30 19.20 48 50 52 .65995 7.66312 .66627 15.85 15-75 15.70 .66194 7.66512 .66829 15.90 15.85 15-75 54 56 58 .49661 .50043 .50422 19.10 18.95 18 95 .49797 .50180 .50561 I9-I5 19.05 IQ OO 54 56 58 .66941 .67253 .67565 15.60 15.60 15 55 .67144 .67458 .67771 15.70 15.65 15 60 60 7.5o8oi 18 80 7.50941 60 7.67876 7.68083 62 64 66 .5H77 .51552 .51926 18.75 18.70 18.55 .51319 .51695 .52070 18.80 18.75 18.65 62 64 66 .68185 68493 .68800 15.40 15-35 15-35 .68394 .68704 .69013 15.50 15 45 15-35 68 70 72 .52297 7.52668 .53036 18.55 18.40 18.35 .52443 7.52814 .53184 18-55 18.50 18.40 68 70 72 .69107 7.69412 .69716 15-25 15.20 15.15 .69320 7.69627 .69932 15-35 15.25 15.25 74 76 78 53403 .53769 54133 18.30 18.20 18 10 53552 -539I9 .54284 18.35 18.25 18 20 74 76 78 .70019 .70320 .70621 15-05 15-05 .70237 .70540 .70843 15.15 15.15 80 7-54495 18 05 7.54648 18 10 80 7.70921 14 Q^ 7.7"44 82 84 86 .54856 .55216 55574 17.90 17.90 17.80 .55010 .55371 55730 18.05 17-95 17.90 82 84 86 .71220 .71517 .71814 14.85 14.85 14.80 .71444 .71743 .72042 14-95 14-95 14-85 88 90 92 55930 7.56285 .56639 17-75 17.70 17.65 .56088 7.56444 .56799 17.80 17-75 17.70 88 90 92 .72110 7.72405 .72698 14.75 14.65 14.65 .72339 7.72635 .72930 14.80 14-75 14-75 94 96 98 .56991 57342 .57691 17-55 17.45 17 40 .57153 57505 .57855 17.60 17.50 17 45 94 96 98 .72991 .73283 73573 14.60 14.50 .73225 .73518 .738io 14.65 14.60 14 ^ 100 7.58039 7.58204 100 7.73863 J. 74102 (191) TABLE XXVI. (Continued) Hun- dredths 6 Hun- dredths 7 Vers Diff. .001 Exsec Diff. .001 Vers Diff. .001 Exsec Diff. .001 00 7.73863 7.74102 00 7.87238 7-87563 02 .74152 74392 02 .87486 .87812 04 74440 14.40 .74681 14-45 04 .87732 12.30 .88061 12.45 06 .74727 14-35 14.25 74970 14-45 14.35 06 .87978 12.30 12.30 .88309 12.40 12.35 08 .75012 .75257 08 .88224 .88556 10 ia 7-75297 .75581 14.25 14.20 14.20 7-75544 -75830 14-35 14-30 14.20 IO ia 7-88469 .88713 12.25 12.20 12.15 7.88803 .89049 12.35 12.30 12.25 14 .75865 .76114 14 .88956 .89294 16 .76147 14.10 .76398 14.20 16 .89199 12.15 .89539 12.25 18 .76428 14.05 .76681 14.15 18 .89441 .89782 12.15 20 7.76708 7-76963 20 7.89682 7 . ooo jo 22 .76988 4 ' 77244 22 .89922 ' .90268 ' 24 .77266 13.9 -77525 14-05 24 .90162 12. OO .90510 12. IO 26 77544 13.9 13-85 .77804 13-95 13.90 26 .90402 11.90 .90751 12.05 28 30 33 .77821 7.78097 .78372 13.80 13-75 I3-70 .78082 7.78360 .78637 13-90 13.85 13-75 28 30 32 .90640 7.90878 .91116 11.90 II.9O 11.80 .90992 7 .in _.<_ .91471 12. OO 11-95 II 95 34 36 38 40 .78646 .78919 .79192 13-65 13 65 13-55 .78912 .79187 .79462 13-75 13-75 13-65 TI fin 34 36 38 40 91352 .91588 .91824 II.80 II.80 11.70 .91710 .91948 .92185 11.90 11.85 11.85 7.79463 7-79735 7.92058 7.92422 43 44 46 .79734 .80004 .80273 13.50 13-45 13.40 .80007 .80279 .80550 13.60 13 55 13 50 43 44 46 -92293 .92526 .92759 11.65 11.65 II. 60 .02658 .92893 .93128 11.75 11.75 11.70 48 50 53 54 56 58 .80541 7.80809 .81075 .81341 .81606 .81870 13-40 13.30 13.30 13.25 13-20 .80820 7.81089 .81357 .81624 .81891 .82157 13-45 13-40 13-35 13-35 13.30 48 50 53 54 56 58 .92991 7.93223 -93454 .93684 .93914 -94I43 II. 60 11-55 11.50 11.50 H-45 .93362 7.93596 .93829 .94061 .94293 .94524 11.70 11.65 II. 60 II. 60 11-55 60 62 7-82133 13-15 7.82422 13-20 60 62 7-94372 11.35 7.94755 11.50 .82396 .82686 94599 .94985 64 66 68 70 73 .82657 .82918 .83179 7.83438 -83697 13.05 13-05 13.05 12.95 12.95 12.85 .82950 .83212 .83474 7.83736 .83996 13.20 13.10 13.10 13.10 13.00 13 oo 64 66 68 70 73 .94827 .95054 .95280 7.95505 95730 11.40 H.35 11.30 11.25 11.25 11.25 .95214 95443 .95671 7-95899 .96126 11-45 11-45 11.40 11.40 11-35 11.30 74 76 78 .83954 .84211 .84468 12.85 12.85 .84256 .845M .84773 12.90 12.95 74 76 78 95955 .96178 .96402 II. IS II. 20 .96352 .96578 .96803 11.30 11.25 80 7.84723 7.85030 80 7. 966-M 7.97028 82 84 86 .84978 .85232 .85485 12.70 12.65 12.65 .85286 .85542 .85797 12.80 12.75 12.75 82 84 86 .96846 .97068 .97289 II. IO 11.05 II. OO .97252 .97476 .97699 11.20 11.15 II. 10 88 90 93 94 96 98 .85738 7.85990 .86241 .86491 .86741 .86990 12.60 12.55 12.50 12.50 12.45 .86052 7.86306 .86558 .86811 .87062 .87313 12.70 12.60 12.65 12.55 12.55 88 90 93 94 96 98 -97509 7.97729 .97948 .98167 .98385 .98603 11.00 10.95 10.95 10.90 10.90 10 85 -97921 7.98143 .98365 .98585 .98806 .99025 II. 10 II. IO II. OO 11.05 10.95 100 100 7.99245 (192) TABLE XXVI. (Continued) Hun- dredths 8 Hun- dredths 9 Diff. Vers .001 Exsec Diff. .001 Diff. Vers ! .001 Exsec Diff. .001 00 02 04 06 08 10 12 7.98820 I0.8o 10.80 10.80 10.75 10.70 10.70 10.65 7-99245 10.90 10.90 10.90 10.85 10.85 10.80 10.75 00 02 04 06 08 10 12 8.09032 9.60 9.60 9.60 9-55 9-50 9-55 9-50 8.09570 9-70 9 65 9.60 9-65 9-70 9.60 965 .99036 .99252 .99468 .99683 7.99897 8.00III .99463 .99681 .99899 8.00116 8.00333 .00549 .09224 .09416 .09608 .09799 8.09989 .10180 -09764 09959 .10153 .10346 8.10540 10732 14 16 18 .00324 .00537 .00749 10.65 10.60 .00764 .00979 .01193 10.75 10.70 14 16 18 . 10370 10559 .10748 9-45 9-45 Q 4^ .10925 .11116 .11308 9-55 9.60 9CC 20 8.00961 8 - OI 47 "I, 20 8.10937 8.II499 22 24 26 .01172 .01383 .01593 10.55 10.50 10.50 .01621 .01834 .02046 10.65 10. 60 10. 60 22 24 26 .11125 .11313 .11500 9-40 9-35 9-35 .11690 .11880 .12070 9-50 9-50 9-45 28 30 32 .01803 8.02012 .02221 10. 45 10.45 10.40 .02258 8.02469 .02680 10.55 10.55 10-55 28 30 32 .11687 8.11874 .12060 9-35 9-30 9-30 .12259 8.12448 .12637 9-45 9-45 9-40 34 36 38 .02429 .02637 .02844 10.40 10.35 .02891 .03101 .03310 10.50 10.45 34 36 38 .12246 .12431 .12616 9-25 9-25 .12825 .13013 .13201 9-40 9-40 9-7C 40 8.03051 8.03519 40 8.12801 9 2O 8.13388 9 -JO 42 44 46 .03257 .03462 .03668 10.25 10.30 IO.2O .03727 03935 .04143 10.40 10.40 10.35 42 44 46 .12985 .13169 .13352 9-20 9-15 9-15 .13574 .13761 13947 9.35 9.30 9.25 48 50 52 .03872 8.04077 .04280 10.25 IO.I5 IO.2O .04350 8.04556 .04762 10.30 10.30 10. 2O 48 So 53 .13535 8.13718 .13900 9-15 9.10 9.10 .14132 8.I43I7 .14502 9-25 9-25 9-25 54 56 58 .04484 .04686 .04889 10.10 10.15 .04968 .05173 .05378 10.25 10.25 54 56 58 .14082 .14263 -14444 9-05 9-05 .14687 .14871 .15054 9.20 9-15 9 2O 60 8.05091 10 05 8.05582 10 15 60 8.14625 8.15238 910 62 64 66 .05292 .05493 05693 10.05 IO.OO 10.00 .05785 .05989 .06191 IO.2C 10.10 10.15 62 64 66 .14805 .14985 .15165 9.00 9.00 8-95 .15420 .15603 .15785 9-iS 9.10 9.10 68 70 73 .05893 8.06093 .06292 IO.OO 9-95 9-95 06394 8.06595 .06797 10.05 10.10 10.05 68 70 72 15344 .8.15523 .15702 8-95 8-95 8.90 15967 8.16148 .16330 9-05 9.10 9.00 74 76 78 .06491 .06689 .06886 9.90 9-85 9 GO .06998 .07198 .07398 IO.OO 10.00 74 76 78 .15880 .16057 .16235 8.85 8.90 8 85 . 16510 .16691 .16871 9-05 9.00 8 62 64 66 .71886 .71978 .72071 4.60 4.65 4.60 .74221 .74318 .74416 4-85 4.90 4-85 62 64 66 .76387 .76475 .76563 4.40 4.40 4-35 .78985 .79078 .79171 4.65 4.65 4.65 68 70 72 .72163 8.72255 .72347 4.60 4.60 4.60 74513 8.74610 .74707 4-85 4-85 4-85 68 70 72 .76650 8.76737 .76825 4-35 4-40 4-35 79264 8.79357 79449 4.65 4.60 4-65 74 76 78 .72439 .72531 .72622 4.60 4-55 A fio .74804 .74901 .74998 4-85 4-85 A R go .91010 .91093 4-15 4-15 36 38 .91365 .91448 3.65 3-65 j ge 95079 .95169 4.00 4.00 40 8.87768 -y 0- 8.9H75 40 8.91511 7 fiC 8.95238 42 44 46 48 .87845 .87921 .87998 .88074 3-80 3.85 3-80 .91258 .91341 91423 .91506 4-15 4.10 4-15 42 44 46 48 .91584 .91657 .91730 .91803 3.65 3-65 3-65 .95318 95398 95478 95557 4.00 4.00 3-95 So 52 8.88150 .88226 3.80 3.8o 3-80 8.91589 .91671 4-15 4.10 4.10 50 52 8.91876 .91949 3.05 3-65 3.6S 8.95637 .95716 4.00 3-95 3-95 54 56 58 60 .88302 .88378 .88454 3-8o 3.80 3-80 7 80 91753 .91836 .91918 4-15 4.10 4.10 54 56 58 60 .92022 .92095 .92167 3.65 3-60 3-65 -, gc 95795 .95875 95954 4.00 3-95 3-95 3ne 8.88530 8.92000 8.92240 8.96033 62 .88606 . 92082 62 92313 .96112 64 .88682 3.80 .92165 4-15 64 .92385 3.6o .96192 4.OO 66 68 70 72 .88758 .88833 8.88909 .88984 3-8o 3-75 3-80 3-75 3.80 .92247 .92328 8.92410 .92492 4.10 4-05 4.10 4.10 4.10 66 68 70 72 .92457 .92530 8 . 92602 .92674 3-6o 3-65 3-60 3.6o 3.6o .96271 .96350 8.96428 .96507 3-95 3-95 3-90 3-95 3-95 74 76 78 .89060 .89135 .89210 3 75 3-75 .92574 .92656 .92737 4.10 4-05 74 76 78 .92746 .92818 .92891 3.6o 3-65 .96586 .96665 .96744 3-95 3-95 80 8.89286 8.92819 80 8.92962 <3 6O 8.96822 2 QtT 82 84 86 .89361 .89436 .89511 3-75 3-75 3-75 .92900 .92982 93063 4.10 4-05 4.10 82 84 86 .93034 .93io6 .93178 3.6o 3-6o 3-60 .96901 .96979 .97058 3-9 3-95 3-90 88 90 92 .89586 8.89661 .89735 3-75 3-70 3 75 .93145 8.93226 .93307 4-05 4-05 4.05 88 9o 92 .93250 8.93321 -93393 3-55 3-60 3.60 .97136 8.97215 .97293 3-95 3-90 3-9 94 96 98 .89810 .89885 .89960 3-75 3-75 93388 .93469 93550 4-05 4-05 94 96 98 .93465 .93536 .93607 3-55 3-55 t 60 97371 97450 .97528 3- 95 3.90 3 9 100 8.90034 8.93631 100 8.93679 8.97606 (200) TABLE XXVI. (Continued) Hun- dredths 24 Hun- dredths 25 Vers Diflf. .001 Exsec Diflf. .001 Vers Diflf. .001 Exsec Diflf. .001 00 8.93679 8.97606 00 8.97170 9-0*443 02 93750 .97684 02 97239 .01518 04 .93821 3-55 .97762 3-9 04 97307 3-40 01594 3 80 06 .93892 3-55 .97840 3-9 06 97375 3.40 .01669 3-75 08 .93964 97918 08 97443 .01744 10 12 8.94035 .94106 3-55 3-55 8-97995 98073 3 85 3-90 10 12 8.97512 .97580 3-45 3-40 9.01819 .01895 3-75 3-80 3-55 3-9 3-40 3-75 14 16 18 94177 94247 .94318 3-50 3-55 .98151 .98229 .98306 3.90 3.85 3 go 14 16 18 .97648 .97716 .97784 3-40 3-40 .01970 .02045 .02120 3-75 3-75 20 8.94389 8.98384 i fie 20 8.97851 9-02195 22 .94460 .98461 22 .97919 .02270 24 26 94530 .94601 3-50 3.55 .98539 .98616 3.90 3.85 7 Re 24 26 97987 .98055 3-40 3-40 02345 .02419 3-75 3-70 28 30 .94671 8.94742 3-55 .98693 8.98771 3-90 28 30 .98122 8.98190 3-40 .02494 9.02569 3-75 3-75 32 .94812 3-50 .98848 3 85 32 98257 3-35 .02644 3-75 34 36 .94882 94953 3-55 .98925 .99002 3-85 34 36 .98325 .98392 3-35 .02718 .02793 3-75 38 40 .95023 3-50 3-50 .99079 3-85 3.85 j Or 38 40 .98459 3-35 3-40 .02867 3-70 3-75 8.95093 8.99156 8.98527 9.02942 42 44 46 48 50 52 54 56 .95163 95233 .95303 95373 8.95443 95513 .95582 .95652 3.50 3-50 3-50 3-50 3-50 3-45 3.50 .99233 99310 .99387 .99464 8.99541 .99617 .99694 99771 3-85 3-85 3-85 3-85 3.8o 3-85 3-85 42 44 46 48 50 52 54 56 .98594 .98661 .98728 .98795 8.98862 .98929 .98996 99063 3-35 3-35 3-35 3-35 3-35 3 35 3-35 .03016 .03091 .03165 03239 9.03314 .03388 .03462 .03536 3-70 3-75 3-70 3-70 3-75 3-70 3-70 3-70 58 95722 3-5O .99847 1 Or 58 . 99130 3-35 . 03610 3-70 60 62 64 66 8-95791 3-50 3-45 3-50 8.99924 3-80 3-80 3-85 60 62 64 66 8.99197 3.30 3-35 3 35 9-03684 3-70 3-70 3-70 .95861 95930 .96000 9.00000 ' . 00076 00153 .99263 99330 99397 .03758 .03832 .03906 68 70 72 74 .96069 8.96138 .96207 .96277 3-45 3-45 3-45 3-50 .00229 9.00305 .00382 .00458 3.8o 3.8o 3-85 3-80 68 70 72 74 .99463 8.99530 99596 .99663 3-30 3-35 3 30 3-35 .03980 9.04054 .04127 .04201 3-70 3-70 3-65 3-70 76 78 80 82 .96346 .96415 3-45 3-45 3-45 3-45 00534 .00610 3.8o 3-80 3-80 3.8o 76 78 80 82 .99729 99795 3-30 3-30 3-30 3-35 .04275 .04348 3.70 3.65 3-70 3-65 8.96484 9.00686 8.99861 9. 04422 .96553 .00762 99928 . 04495 84 86 88 .96621 .96690 .96759 3-40 3-45 3-45 .00838 .00913 00989 3-8o 3-75 3-80 84 86 88 99994 9.00060 .00126 3-30 3-30 3-30 .04569 .04642 .04715 3-70 3.65 3-65 90 92 94 96 8.96828 .96896 .96965 .97033 3-45 3.40 3-45 3-40 9.01065 .01141 .01216 .01292 3-8o 3.8o 3-75 3.8o 90 92 94 96 9.00192 .00258 .00323 .00389 3-30 3-30 3-25 3-30 9.04789 .04862 04935 .05008 3-70 3-65 3-65 3-65 98 100 .97102 3-45 3-40 .01367 3-75 3-80 98 100 00455 3-30 3-30 .05082 3-70 3-65 8.97170 9-01443 9.00521 9.05155 (201) TABLE XXVI. (Continued) Hun- dredths 26 Htm- dredths 27 Vers Dift. .001 Exsec Diff. .001 Vers Diff. .001 Exsec Diff. .001 00 02 04 06 08 10 J 12 9.00521 3-25 3-30 3-25 3-30 3-25 3-30 3-25 9.05155 3.65 3-65 3-65 3-60 3-65 3-65 3.65 00 02 04 06 08 10 12 9.03740 3-15 3-15 3-15 3-15 3-15 3-15 3-15 9-08752 3-55 3-55 3-50 3-55 3-55 3-50 3-55 .00586 .00652 .00717 .00783 9.00848 .00914 . 05228 .05301 .05374 .05446 9.05519 .05592 .03803 .03866 03929 .03992 9.04055 .04118 .08823 .08894 .08964 .09035 9.09106 .09176 14 16 18 20 .00979 .01044 .01109 3-25 3-25 3-30 .05665 .05738 .05810 3-65 3.6o 3-65 i fio 14 16 18 20 .04181 .04244 .04306 3-15 3.10 3-15 09247 .09318 .09388 3-55 3-50 3-55 9-OII75 9.05883 9-04369 9-09459 22 24 .01240 .01305 3-25 .05955 .06028 3.65 22 24 .04432 .04494 3.10 .09529 09599 3-50 26 .01370 3-25 .06101 3.65 26 04557 3-15 .09670 3-55 28 30 32 .01435 9 . 01500 .01565 3-25 3-25 3-25 3.20 .06173 9.06245 .06318 3-6o 3-60 3-65 3.60 28 30 32 ..04619 9.04682 .04744 3-10 3-15 3-10 3-15 .09740 9.09810 .09881 3-50 3-50 3-55 3-50 34 36 38 .01629 .01694 .01759 3-25 3-25 .06390 .06462 .06535 3.6o 3-65 i go 34 36 38 .04807 .04869 .04931 3-10 3-10 .09951 .IO02I .lOOgi 3-50 3-50 40 9.01824 9.06607 3 fin 40 9-04993 9.IOI6I 42 .01888 .06679 42 .05055 .10231 44 46 .01953 .02017 3-25 3-20 3.25 .06751 .06823 3.60 3.60 3.60 44 46 .05118 .05180 3-15 3-10 3-10 . I030I .10371 3-50 3-50 3-50 48 .02082 .06895 48 .05242 .10441 50 52 9.02146 .02210 3-20 3-20 3.25 9.06968 .07039 3.65 3.55 3.60 50 52 9.05304 .05366 3-io 3-10 3-05 9-I05II . I058I 3-50 3-50 3-45 54 56 58 60 .02275 02339 .02403 3.20 3-20 3-20 .07111 .07183 .07254 3.60 3.55 3.60 -, 60 54 56 58 60 .05427 .05489 .05551 3-10 3-10 3.10 3 IO .IO650 . 10720 .10790 3-50 3-50 3-50 i 415 9-02467 9.07326 9-05613 9.10860 62 64 66 68 .02531 .02596 .02660 .02723 3.25 3-20 3.15 .07398 .07469 .07541 .07613 3-55 3-60 3-60 62 64 66 68 .05675 .05736 .05798 .05859 3-05 3-10 3-05 .10929 .10999 .11068 .11138 3-50 3-45 3-50 70 72 74 76 78 9.02787 .02851 .02915 .02979 .03043 3-20 3.20 3-20 3-20 3-20 9.07684 .07756 .07827 .07898 .07970 3-55 3.6o 3-55 3-55 3-60 70 72 74 76 78 9-05921 .05982 .06044 .06105 .06166 3-io 3-05 3-10 3-05 3-05 9.II207 .11277 .11346 .11415 .11485 3-45 3-50 3-45 3-45 3-50 80 9.03106 9 . 08041 80 9.06228 2 O^ 9-II554 3 45 82 84 86 .03170 -03233 03297 3.15 3-20 3.15 .08112 .08184 .08255 3-60 3-55 3-55 82 84 86 .06289 .06350 .06411 3-05 3-05 3-05 .11623 .11692 .11762 3 45 3-50 3-45 88 90 92 .03360 9.03424 .03487 3-20 3.15 3.15 .08326 9.08397 .08468 3-55 3-55 3-55 88 .90 92 .06472 9.06533 .06594 3-05 3-05 3-05 .11831 9.II900 .11969 3-45 3-45 3-45 94 96 98 .03550 .03614 .03677 3-20 3.15 .08539 .08610 .08681 3-55 3-55 94 96 98 .06655 .06716 .06777 3-05 3-05 7 O 1 ? .12038 .12107 .12176 3-45 3-45 3 45 100 9-03740 9.08752 100 9.06838 9-12245 (202) TABLE XXVI. (Continued) Hun- dredths 28 Hun- dredths 29 Vers Diff. .001 Exsec Diff. .001 Vers Diff. .001 Exsec Diff. .001 00 9.06838 3 05 9-12245 3.40 00 9-09823 2-95 9-15641 3 35 02 04 06 .06899 .06960 . 07020 3-05 3-00 3-05 .12313 .12382 .12451 3-45 3-45 3-45 02 04 .06 .09882 .09940 .09999 2.90 2.95 2.90 .15708 15775 .15842 3-35 3-35 3-35 08 IO 12 .07081 9.07141 07202 3.00 3-05 3-05 . 12520 9.12588 12657 3-40 3 45 3-45 08 10 12 .10057 9 10115 .10174 2.90 2 95 2.90 .15909 9.15976 . 16043 3-35 3 35 3-30 14 16 18 .07263 .07323 -07383 3.00 3.00 3 05 .12726 .12794 .12863 3-40 3-45 3 4 14 16 18 . 10232 . 10291 .10349 2.95 2.90 2 90 .16109 .16176 . 16243 3-35 3-35 3 3 20 9.07444 3 oo 9.I293I 3 45 20 9.10407 2 90 9-16309 3 35 22 24 26 -07504 .07564 .07625 3.00 3-05 3.00 .13000 .13068 .13137 3-40 3-45 3-40 22 24 26 .10465 .10523 .10581 2.90 2.90 2.90 .16376 .16443 .16509 3-35 3-30 3-35 28 30 32 .07685 9-07745 .07805 3.00 3.00 3.00 .13205 9.13273 .13341 3.40 3-40 3-45 28 30 32 .10639 9.10697 10755 2.90 2.90 2.90 .16576 9.16642 .16709 3-30 3-35- 3-30 34 36 38 .07865 -07925 .07985 3-00 3.00 3 oo .13410 .13478 .13546 3-40 3-40 3 4 34 36 38 .10813 .10871 .10929 2.90 2.90 2 QO 16775 .16842 .16908 3-35 3-30 40 9.08045 3 oo 9-13614 3 4 40 9-10987 2 90 9.16974 3 35 42 44 46 .08105 .08165 .08225 3.00 3-00 2-95 .13682 .13750 .13818 3-40 3-40 3-40 42 44 46 .11045 .11102 .IIl6o 2.85 2.90 2.90 .17041 .17107 .17173 3-30 3-30 3-30 48 50 52 .08284 9-08344 .08404 3.00 3.00 2.95 .13886 9-13954 . 14022 3-40 3-40 3-40 48 So 52 .11218 9.II275 II333 2.85 2.90 2.85 .17239 9-I7306 .17372 3-35 3-30 3-30 54 56 58 .08463 .08523 .08583 3.00 3.00 2 Hun- fi 7 dredths Vers Diff. .001 Exsec Diff. .001 dredths Vers Diff. .001 Exsec Diff. .001 00 9.64425 9.89669 00 9-65836 9.92225 02 .64453 .89720 02 .65863 .92276 04 .64482 -45 .89771 2.55 04 .65891 .40 .92327 2.55 06 .64510 40 .89822 2.55 06 .65919 .40 .92379 08 64539 .89873 08 .65947 .92430 10 9-64567 40 9.89924 2.55 10 9-65975 .40 9.92481 2.55 12 -64596 45 .89975 2.55 12 .66003 .40 .92532 2.55 14 .64624 .90026 14 .66031 .92584 16 .64653 45 .90077 2.55 16 .66059 4 .92635 18 .64681 .40 .90128 2.55 18 .66086 35 .92686 2.55 2 60 20 9.64709 9.90179 20 9.66114 9.92738 22 .64738 .90230 22 .66142 .92789 34 26 .64766 .64794 40 .40 45 .90281 .90332 2-55 2.55 2.55 24 26 .66170 .66198 .40 .40 35 .92840 .92892 2.55 2.60 2.55 28 30 32 .64823 9.64851 .64880 40 .40 .90383 9-90434 90485 2.55 2.55 28 30 32 .66225 9.66253 .66281 .40 .40 .92943 9.92994 .93046 2.55 2.60 2 55 34 36 38 .64908 64936 .64964 .40 .40 .90536 .90587 .90638 2.55 2.55 34 36 38 .66309 .66336 .66364 35 .40 .93097 -93I49 .93200 2.60 2.55 2 55 40 9-64993 9.90689 2 60 40 9-66392 9-93251 2 60 42 44 46 .65021 .65049 .65077 .40 .40 45 .90741 .90792 .90843 2.55 2.55 2.55 42 44 46 .66419 .66447 .66475 .40 .40 35 .93303 93354 .93405 2.55 2.55 2.60 48 So 52 .65106 9.65134 .65162 .40 .40 4 .90894 9.90945 .90996 2.55 2.55 2.55 48 50 52 .66502 9-66530 .66558 .40 .40 35 93457 9-93508 .93560 2.55 2.6o 2.55 54 56 .65190 .65218 .40 .91047 .91098 2.55 54 56 .66585 .66613 .40 .936ii .93663 2.60 58 .65247 .91149 2.55 2 60 58 .66640 35 .93714 2 60 60 9-65275 40 9.91201 60 9.66668 9-93766 2 55 62 64 66 65303 65331 .65359 -40 .40 .40 .91252 91303 91354 2.55 2.55 2.55 62 64 66 .66696 66723 .66751 35 .40 35 .93817 .93869 .93920 2.6o 2.55 2.55 68 70 72 .65387 9.65415 .65444 .40 45 .40 .91405 9.91456 .91508 2.55 2.60 2.55 68 70 72 .66778 9.66806 .66833 .40 35 .40 93971 9.94023 .94074 2.60 2.55 2.60 74 76 65472 .65500 .40 91559 .91610 2.55 74 76 .66861 .66888 -35 .94126 94177 2.55 78 .65528 .91661 2.55 78 .66916 .40 .94229 2 60 80 9.65556 40 9.91712 2 60 80 9.66943 9.94281 2 55 82 .65584 .91764 82 .66971 94332 2.60 84 .65612 91815 84 .66998 .94384 2.55 86 .65640 .40 .91866 2-55 86 .67026 35 94435 2.60 88 90 .65668 9-65696 . .40 .91917 9.91968 2-55 2 60 88 90 .67053 9.67080 35 .94487 9-94538 2.55 2 6O 92 .65724 40 . 92020 2-55 92 .67108 35 94590 2.55 94 96 98 .65752 .65780 .65808 .40 .40 40 .92071 .92122 .92173 2.55 2.55 2 60 94 96 98 .67135 .67163 .67190 40 35 .94641 .94693 94745 2.60 2.60* 2 55 100 9-65836 9 . 92225 100 9.67217 9.94796 (217) TABLE XXVI. (Continued) Hun- 5 8 Hun- 5 9 dredths Vers Diff. .001 Exsec Diff. .001 dredths Vers Diff. .001 Exsec Diff. .001 00 9.67217 9.94796 2 60 00 9-68571 9.97387 2 fin 02 04 67245 .67272 35 .94848 .94900 2.60 02 04 68598 .68624 30 -97439 -97491 2.60 06 .67299 35 40 94951 2.55 2 60 06 .68651 35 35 97543 2 60 08 .67327 95003 08 .68678 97595 10 9-67354 35 9-95054 2.55 10 9-68705 35 9.97647 2.6o 12 .67381 35 .95106 2 60 12 .68731 -30 .97699 2 fin 14 .67408 .95158 14 68758 97751 16 18 67436 .67463 40 35 95209 .95261 2-55 2.60 2 60 16 18 .68785 .68811 -35 -30 .97803 .97855 2.60 2.60 2 fir 20 9.67490 9-95313 20 9.68838 9.97908 2 60 22 24 67517 .67545 40 .95364 .95416 2.60 22 24 .68865 .68892 -35 .97960 .98012 2.60 26 .67572 35 35 95468 2.60 2.60 26 .68918 -30 35 ..98064 2.6O 2 60 28 67599 95520 28 .68945 .98116 30 32 9.67626 67653 35 35 4 9-95571 95623 2.55 2.60 2.60 30 32 9.68971 .68998 30 35 35 9.98168 .98220 2.60 2.60 2 65 34 .67681 .95675 34 .69025 .98273 36 .67708 35 .95726 2.55 36 .69051 -30 .98325 2.60 38 67735 35 95778 2 fin 38 .69078 35 .98377 2.60 40 9.67762 9-95830 2 60 40 9-69104 3O 9-98429 42 .67789 .95882 42 .69131 .98481 2 65 44 .67816 35 95934 44 .69158 35 .98534 2 60 46 67843 35 35 95985 2.55 2 60 46 .69184 30 35 .98586 2.60 48 .67870 96037 48 .69211 .98638 2 60 So 52 9.67897 67925 35 40 35 9.96089 .96141 2.60 2 60 50 52 9.69237 .69264 3O -35 30 9.98690 .98743 2.65 2.60 54 .67952 .96193 54 .69290 .98795 2 60 56 58 67979 .68006 35 35 .96244 .96296 2-55 2.60 2 60 56 58 -69317 .69343 35 -30 .98847 .98899 2.60 2.65 60 9.68033 9.96348 2 fin 60 9.69370 9.98952 2 60 62 64 66 .68060 .68087 .68114 35 35 35 .96400 -96452 96504 2.60 2.60 2 60 62 64 66 -69396 .69423 .69449 -35 -30 35 .99004 .99056 .99109 2.60 2.6 5 2.60 68 .68141 .96556 68 .69476 .99161 2 60 70 72 9.68168 .68195 35 35 35 9.96607 .96659 2.55 2.60 2 60 70 72 9-69502 .69528 30 -30 35 9.99213 .99266 2.6 5 2.60 74 76 78 .68222 .68248 .68275 30 35 .96711 96763 .96815 2.60 2.60 2 60 74 76 78 .69555 .69581 .69608 -30 35 99318 99371 99423 2.65 2.60 2.60 80 9.68302 9.96867 2 60 80 9-69634 9-99475 2.65 82 .68329 .96919 82 .69660 .99528 2 60 84 86 .68356 .68383 35 35 35 .96971 97023 2.60 2 60 84 86 .69687 .69713 35 30 30 .99580 .99633 2.65 2.60 88 90 92 .68410 9-68437 .68463 35 30 35 97075 9.97127 97179 2.60 2.60 2 60 88 90 92 -69739 9.69766 .69792 35 30 30 .99685 9-99738 .99790 2.6 S 2.60 2.65 94 96 98 .68490 .68517 68544 35 35 .97231 97283 .97335 2.60 2.60 2 60 94 96 98 .69818 .69844 .69871 30 35 .99843 .99895 99947 2.60 2.60 2 65 100 9-68571 9.97387 100 9.69897 o.ooooo (218) TABLE XXVI. (Continued) Hun- dredths 60 Hun- dredths 61 Vers Diff. .001 Exsec Diff. .001 Vers Diff. .001 Exsec Diff. .001 00 9.69897 o.ooooo % .- 00 9.71197 0.02640 2 6s 02 04 06 .69923 .69949 .69976 1.30 1-35 1.30 .00053 .00105 .00158 2.60 2.65 2.60 02 04 06 .71223 .71248 .71274 1.25 1.30 1.30 .02693 .02746 .02799 2.65 2.65 2.65 08 .70002 .00210 08 .71300 .02852 10 12 9.70028 .70054 1.30 1-30 1-35 0.00263 .00315 2.65 2.60 2.65 IO 12 9.71325 .71351 1.25 1.30 1.30 0.02905 .02958 2.65 2.65 2.70 14 16 18 .70081 .70107 .70133 1.30 1.30 .00368 .00420 .00473 2.60 2.65 2 6? 14 16 18 .71377 .71402 .71428 1.25 1-30 .03012 .03065 .03118 2.65 2.65 2 6"? 20 9.70159 0.00526 2 fin 20 9.71454 0.03171 2 6^ 22 24 26 28 30 32 .70185 .70211 . -70237 .70264 9.70290 .70316 1.30 1.30 1-35 1.30 1.30 1.30 .00578 .00631 .00684 .00736 0.00789 .00842 2.65 2.65 2.60 2.65 2.65 2.65 22 24 26 28 30 32 -71479 .71505 .71530 .71556 9-71582 .71607 1.30 1.25 1.30 1.30 I 25 1.30 .03224 .03278 .03331 .03384 0.03437 .03491 2.70 2.65 2.65 2.65 2.70 2.65 34 36 38 70342 .70368 .70394 1.30 1.30 .00894 .00947 .01000 2.65 2.65 2 60 34 36 38 .71633 .71658 .71684 1.25 1.30 .03544 .03597 .03651 2.65 2.65 2 6 1 ; 40 9.70420 0.01052 "2 fix 40 9.71709 0.03704 2 fir 42 .70446 .01105 42 71735 -03757 44 46 .70472 .70498 1.30 1.30 .01158 .01211 2.65 2.65 44 46 .71760 .71786 1.25 1.30 .03811 .03864 2.70 2.65 48 So 52 .70524 9.70550 70576 1.30 1.30 .01263 O.OI3I6 .01369 2.65 2.65 48 So 52 .71811 9.7I837 .71862 1.30 1.25 .03917 0.03971 .04024 2.70 2.65 54 56 58 .70602 .70628 70654 1.30 1.30 . 01422 01475 .01528 2.65 2.65 fin 54 56 58 .71888 .71913 .71939 1.25 1.30 .04078 .04131 .04184 2.65 2.65 60 62 64 66 68 70 72 9.70680 1.30 1.30 1.30 1.30 1.30 1.25 1.30 0.01580 2.65 2.65 2.65 2.65. 2.65' 2.65 2.65 60 62 64 66 68 70 72 9.71964 1.25 1.30 1.25 1.30 1.25 1.25 1.30 1.25 0.04238 2.65 2.70 2.65 2.70 2.65 2.70 2.65 .70706 .70732 .70758 .70784 9.70810 .70835 .01633 .01686 01739 .01792 0.01845 .01898 .71990 .72015 .72041 .72066 9.72091 .72117 .04291 .04345 .04398 .04452 0.04505 .04559 74 76 .70861 .70887 1.30 .01951 .02004 2.65 74 76 .72142 .72167 1.25 .04612 .04666 2.70 78 .70913 1.30 .02056 2.60 9 fie 78 .72193 1.30 .04720 2.70 2 fit 80 82 84 9.70939 1.30 1.30 0.02109 2.65 2.65 80 82 84 9.72218 1.25 1.25 1.30 0.04773 2.70 2.65 -70965 .70991 .02162 .02215 .72243 .72269 .04827 .04880 86 .71016 1.25 1.30 .02268 2.65 2.65 86 72294 1.25 1.25 .04934 2.70 2.70 88 90 .71042 9.71068 1.30 .02321 0.02374 2.65 88 9o .72319 9.72345 1.30 .04988 0.05041 2.6S 92 71094 1.30 1.30 ,02427 2.65 2.65 92 .72370 1.25 1.25 -05095 2.70 2.70 94 96 98 .71120 7II45 .71171 1.25 1.30 .02480 .02534 .02587 2.70 2.65 2 6t 94 96 98 72395 .72420 72446 1.25 1.30 .05149 .05202 .05256 2.65 2.70 100 9.71197 o . 02640 100 9.72771 0.05310 (219) TABLE XXVI. (Continued) Hun- e: 1 Hun- I" dredths Vers Diff. .001 Exsec Diff. .001 dredths Vers Diff. .001 Exsec Diff. .001 00 9.72471 0.05310 00 9.73720 0.08015 02 04 .72496 .72521 -25 -05364 05417 2.65 02 04 73745 .73769 .20 .08070 .08124 2.75 2.70 06 08 72547 .72572 -30 25 05471 .05525 2.70 2.70 06 08 73794 73819 .25 25 .08179 .08233 2.75 2.70 10 9.72597 .25 0.05579 2.70 IO 9-73844 .25 0.08288 2-75 12 14 .72622 .72647 .25 25 .05633 .05686 2.70 2.65 12 14 .73868 .73893 .20 -25 08343 08397 2.75 2.70 16 .72672 25 .05740 2.70 16 .73918 .25 .08452 2.75 18 .72698 30 05794 2.70 18 73942 .20 .08506 2.70 20 9.72723 0.05848 20 9.73967 0.08561 22 .72748 .05902 22 73992 .08616 2.75 24 .72773 25 .05956 2.70 24 .74016 .20 .08670 2.70 26 .72798 25 .06010 2.70 26 74031 25 .08715 2.75 28 .72823 .06064 28 .74065 .08780 2-75 30 9.72848 .25 o. 06118 2.70 30 9.74090 -25 0.08835 2.75 32 72873 .25 .06172 2.70 32 74II5 .25 .08889 2.70 34 36 .72898 72923 25 .06226 .06280 2.70 34 36 74139 .74164 .25 .08944 .08999 2.75 38 .72948 .25 06334 2.70 38 .74188 .09054 2-75 40 9-72973 0.06388 40 9-74213 0.09108 42 44 .72999 73024 .25 .06442 .06496 2.70 42 44 74237 .74262 .25 .09163 .09218 2.75 46 48 73049 73074 .25 25 .06550 .06604 2.70 2.70 46 48 .74286 743II .20 25 09273 .09328 2-75 2.75 So 9-73099 .25 0.06658 2.70 50 9-74335 0.09383 2.75 52 .73123 .06712 2.70 52 .74360 25 .09438 2.75 54 .73148 .06766 54 .74384 09493 56 73173 .25 .06820 2.70 56 .74409 .25 -09548 2.75 58 .73198 25 .06874 2.70 58 74433 .09602 2.70 60 9 73223 0.06929 60 9.74458 0.09657 62 .73248 .06983 62 .74482 .09712 64 66 68 73273 .73298 .73323 .25 .25 .25 07037 .07091 .07146 2.70 2.70 2.75 64 66 68 74507 -74531 74556 .25 .20 25 .09767 .09823 .09878 2.75 2.80 2-75 70 9-73348 .25 0.07200 2.70 70 9-74580 0.09933 2.75 72 73373 .25 07254 2.70 72 .74604 .20 .09988 2.75 74 73398 .07308 74 .74629 .10043 76 73422 .07363 2.70 76 .74653 .10098 2-75 78 73447 .25 .07417 2.70 78 .74678 25 . IOI53 2.75 80 9-73472 0.07471 80 9.74702 0.10208 82 73497 .07526 82 .74726 . 10263 84 73522 .25 .07580 2.70 84 -74751 -25 . I03I9 2.80 86 73547 .25 .07634 2.70 86 74775 10374 2.75 .20 2-75 .20 2.75 88 73571 .07689 88 74799 .10429 90 9.73596 .25 0.07743 2.70 9o 9.74824 -25 o . 10484 2.75 92 .73621 .25 .07798 2-75 92 .74848 .10540 2.80 94 .73646 .07852 94 .74872 .10595 96 .73671 .25 .07906 2.70 96 .74896 .10650 2.75 98 .73695 .07961 2.75 98 74921 -25 . 10705 2.75 2 80 100 9-73720 0.08015 100 9-74945 0.10761 (220) TABLE XXVI. (Continued) Hun- 64 Hun- 6 dredths Vers Diff. .001 Exsec Diff. .001 dredths Vers Diff. .001 Exsec Diff. .001 00 9-74945 0.10761 00 9.76146 0.13551 2 8s 02 .74969 .10816 02 .76170 .13608 04 06 74993 .75018 -25 .20 . 10871 . 10927 2.75 2.80 2.75 04 06 76194 .76218 .20 .15 .13664 .13720 2.80 2.80 2.85 08 .75042 .10982 08 .76241 13777 zo 9.75066 0.11038 10 9-76265 0.13833 2.80 12 .75090 .25 .11093 2.75 2.80 12 .76289 .20 .20 .13890 2.85 2.80 14 16 18 75II5 75139 .75163 .20 .20 .11149 .11204 .11260 2.75 2.80 14 16 18 76313 .76336 .76360 -15 .20 .13946 .14003 .14059 2.85 2.80 2 Of 20 9-75I87 0.11315 2 80 20 9.76384 0.14116 2 80 22 7521 I .11371 22 .76408 . 14172 24 26 75235 .75260 .25 .11426 .11482 2-75 2.80 24 26 .76431 .76455 .15 .20 . 14229 .14285 2.85 2.80 2 8s 28 30 32 .75284 9.75308 75332 .20 .20 .20 .11537 O.H593 .11649 2.80 2.80 2.75 28 30 32 .76479 9.76502 .76526 -IS .20 .20 .14342 0.14398 .14455 2.80 2.85 2.85 34 36 38 75356 .75380 .75404 .20 .20 2O .11704 .11760 .11816 2.80 2.80 34 36 38 76550 .76573 .76597 .15 .20 .14512 .14568 .14625 2.80 2.85 2 8S 40 9.75428 20 0.11871 2 80 40 9.76620 0.14682 2 80 42 44 46 75452 75476 75500 .20 .20 .20 .11927 .11983 .12039 2.80 2.80 2.75 42 44 46 .76644 .76668 .76691 .20 .15 .20 .14738 .14795 .14852 2.85 2.85 2.85 48 50 .75524 9-75549 .25 .12094 0.12150 2.80 2 80 48 50 76715 9.76738 IS .14909 0.14966 2.85 20 n 52 75573 .20 .12206 2.80 52 .76762 .15 . 15022 2.85 54 56 58 75597 .75621 .75645 .20 .20 2O . 12262 .12318 - 12373 2.80 2.75 2 80 54 56 58 76785 .76809 -76833 .20 .20 .15079 .15136 .15193 2.85 2.85 60 9.75669 15 0.12429 2 80 60 9-76856 0.15250 2 8s 62 64 66 75692 757i6 75740 .20 .20 .20 . 12485 . 12541 12597 2.80 2.80 2.80 62 64 66 .76880 .76903 .76927 15 .20 .15 .15307 .15364 .15421 2.85 2.85 2.85 68 70 .75764 9-75788 .20 .12653 0.12709 2.80 2 80 68 70 -76950 9.76974 .20 . 15478 0.15535 2.85 i Re 72 .75812 .20 .12765 2.80 72 .76997 .20 .15592 2.85 2.85 74 76 78 .75836 .75860 -75884 .20 .20 .20 . 12821 .12877 12933 2.80 2.80 2 80 74 76 78 .77021 .77044 .77068 IS .20 15 .15649 .15706 .15764 2.85 2.90 2 85 80 9.75908 .20 0.12989 2 85 80 9.77091 0.15821 82 84 86 75932 75956 .75980 .20 .20 -IS . 13046 . 13102 .13158 2.80 2.80 2.80 82 84 86 77II4 .77138 .77161 .20 .15 .20 .15878 .15935 15992 2.85 2.85 2.85 88 90 92 .76003 9.76027 .76051 .20 .20 .20 .13214 0.13270 .13326 2.80 2.80 2.85 88 9o 92 77185 9.77208 77231 IS .15 .20 .16049 0.16107 .16164 2.90 2.85 2.85 94 96 98 -76075 .76099 .76123 .20 .20 .15 .13383 13439 13495 2.80 2.80 2.80 94 96 98 77255 77278 77301 .15 15 . 16221 .16279 .16336 2.90 2.85 2 8* 100 9.76146 o. 13551 100 9 77325 o 16393 (221) TABLE XXVI. (Continued) Hun- | i Hun- 6' r 1 dredths Vers Diff. .001 Exsec Diff. .001 dredths Vers Diff. .001 Exsec Diff. 1 .001 1 00 9-77325 0.16393 00 9.78481 0.19293 oa 04 77348 77371 i.iS . 16451 . 16508 2.85 02 04 .78504 -78527 I.iS 19352 .19410 2-90 1 06 77395 i 15 .16566 2.90 2 85 06 .78550 I.IO .19469 2-95 1 2.95 1 08 .77418 .16623 08 .78572 . 19528 10 9-77441 I.iS 0.16681 2.90 10 9-78595 i.iS 0.19587 95 1 xa .77465 .16738 2.85 12 .78618 i -15 .19645 .90 1 14 z6 18 .77488 77SI i 77534 I.IS I.IS . 16796 .16853 .16911 2.85 2.90 14 16 18 .78641 .78664 .78687 I.iS I.iS 19704 19763 . 19822 95 1 .95 1 20 9-77558 0.16969 2 8? 20 9.78710 0.19881 2 00 1 aa 24 26 .77581 .77604 .77627 I.IS I.IS 1. 20 .17026 .17084 .17142 2.90 2.90 2.85 22 24 26 .78732 .78755 .78778 I.iS I.iS 1.15 19939 .19998 .20057 2-95 1 2-95 1 2-95 1 28 30 32 .77651 9.77674 .77697 I.IO I.iS 1.15 .17199 0.17257 .17315 2.90 2.90 2.85 28 30 32 .78801 9-78823 .78846 I.IO i-iS I.iS .20116 0.20175 .20234 2-95 1 2-95 1 2-95 1 34 36 38 .77720 77744 .77767 1.20 1.15 17372 17430 .17488 2.90 2.90 34 36 38 .78869 .78892 .78915 I.iS I.IS .20293 .20352 . 20412 295 3-00 1 2 Q5 1 40 9-77790 O.I7S46 40 9-78937 I 15 0.20471 2 05 1 42 44 .77813 .77836 I.iS .17604 .17662 2.90 42 44 .78960 .78983 I.IS .20530 .20589 s 46 77859 1.15 .17720 2.9O 46 .79005 I 15 .20648 yo i 48 50 .77882 9.77906 1.20 .17778 0.17836 2.90 48 50 .79028 9.7905I I.IS .20708 0.20767 52 .77929 I.IS .17894 2 90 52 79073 I 15 .20826 i 54 56 58 77952 77975 .77998 1 C/l C/l t 17952 . 18010 .18068 2.90 2.90 54 56 58 .79096 79II9 .79141 i.iS I.IO i IS .20885 .20945 .21004 300 3- 60 9.78021 0.18126 60 9.79164 I IS 0.21064 2 95 1 62 64 66 .78044 .78067 .78090 I.IS I.IS I.iS .18184 .18242 .18300 2.90 2.90 2.90 62 64 66 .79187 .79209 .79232 I.IS I.IS I.IS .21123 .21183 .21242 3-00 1 2-95 1 3 oo 1 68 70 72 .78113 9.78I36 .78159 I.IS I.IS I.IS .18358 0.18417 .18475 2.95 2.90 2.90 68 70 72 .79255 9.79277 .79300 I.IO I.iS I.IO .21302 0.21361 .21421 2-95 3-00 1 2-95 1 74 76 78 .78182 .78205 .78228 I.iS I.IS .18533 .18592 .18650 2.95 2.90 74 76 78 79322 79345 .79368 I.iS 1.15 I.IO .21480 .21540 .21600 3.00 1 3-00 1 2-95 1 80 9.78251 I 15 0.18708 80 9-79390 i.iS 0.21659 3.00 1 82 .78274 .18767 82 79413 .21719 84 86 .78297 .78320 I.IS I.IS 1.15 .18825 .18883 2.90 2.90 2.95 84 86 79435 79458 I.iS I.IO .21779 .21839 3-00 1 2-95 1 88 90 92 78343 9-78366 .78389 I.IS I.IS 1.15 .18942 0.19000 .19059 2.90 2.95 2.90 88 90 92 .79480 9-79503 .79525 I.iS I.IO 1.15 .21898 0.21958 .22018 3-00 1 3-00 1 3-00 1 94 96 98 .78412 .78435 .78458 i IS I.IS .19117 .19176 .19234 2.95 2.90 2 95 94 96 98 .79548 79570 79593 I.IO I.iS I.IO . 22078 .22138 . 22198 il 100 9 78481 o 19293 100 9.79615 o . 22258 (222) TABLE XXVI. (Continued) Hun- dredths 68 Hun- dredths 69 Vers Diff. .001 Exsec Diff. .001 Vers Diff. .001 Exsec Diff. .001 00 9-796I5 0.22258 00 9-80729 0.25296 02 .79638 .22318 02 .80751 25357 04 06 .79660 .79683 i. IS .22378 .22438 3.00 3-00 04 06 .80773 .80795 I.IO 25419 .25480 3.10 3-OS 08 79705 .22498 08 .80817 25542 IO 9.79728 i. IS 0.22558 3-00 IO 9-80839 I.IO 0.25604 3.10 12 .79750 I.IO .22618 3-00 12 .80861 I.IO .25666 3.10 I.IO 3.00 I.IO 3.05 14 79772 .22678 14 .80883 .25727 16 79795 i. IS .22739 3-05 16 .80905 .25789 3.10 18 .79817 .22799 3-oo 18 .80927 .25851 3-10 20 9.79840 0.22859 20 9.80949 0.25913 22 24 26 .79862 .79884 79907 I.IO i. IS . 22920 .22980 .23041 3-00 3-OS 22 24 26 .80971 .80993 .81015 I.IO I.IO -25975 .26037 .26099 3.10 3-10 28 .79929 .23101 28 .81037 .26161 30 32 9-79952 79974 i. IS I.IO I.IO 0.23161 .23222 3-oo 3-OS 3 oo 30 32 9-81059 .81080 i. 05 I.IO 0.26223 .26285 3.10 3.10 3-io 34 36 .79996 .80019 i. IS .23282 .23343 3-OS 34 36 .81102 .81124 I.IO 26347 .26409 3.10 38 .80041 I IO .23403 3-oo 38 .81146 .26471 3.10 40 9.80063 0.23464 40 9.81168 0.26533 42 44 .80085 .80108 i. IS .23524 .23585 3-05 42 44 .81190 .81212 I.IO .26596 .26658 3-10 46 .80130 I.IO .23646 3 05 3.00 46 -81234 I.IO . 26720 3-io 3.15 48 So 52 .80152 9-80175 .80197 i. IS I.IO I.IO .23706 0.23767 .23828 3-05 3-05 3-OS 48 So 52 .81256 9.81277 .81299 1.05 I.IO I.IO -26783 0.26845 .26907 3.10 3.10 3-iS 54 56 58 .80219 .80241 .80264 I.IO 1. 15 .23889 23949 .24010 3-00 3-05 54 56 58 .81321 -81343 .81365 I.IO I.IO .26970 .27032 -27095 3-io 3-15 60 9.80286 I IO 0.24071 60 9.81387 0.27157 62 64 66 .80308 .80330 .80352 I.IO I.IO 1. 15 .24132 24193 .24254 3-05 3-05 3-05 62 64 66 .81408 .81430 .81452 I.IO I.IO I.IO . 27220 .27273 27345 3- IS 3-10 3-iS 68 70 72 80375 9-80397 80419 I.IO I.IO I.IO .24315 0.24376 24437 3-05 3-05 3-05 68 70 72 .81474 9-81496 81517 I.IO 1.05 I.IO .27408 0.27471 27533 3-iS 3-10 3-lS 74 76 78 .80441 .80463 .80485 I.IO I.IO I 15 .24498 24559 .24621 3-05 3-10 3 5 74 76 78 .81539 .81561 -81583 I.IO I.IO I 05 .27596 27659 .27722 3-IS 3-iS 80 9.80508 I IO 0.24682 3 05 80 9.81604 i 10 0.27785 3 IS 82 84 86 .80530 .80552 .80574 I.IO I.IO I.IO 24743 .24804 .24866 3-05 3-io 3-05 82 84 86 .81626 .81648 .81670 I.IO I.IO 1.05 .27848 .27911 27974 3-15 3-15 3-lS 88 9o 92 .80596 9.80618 .80640 I.IO I.IO I.IO .24927 0.24988 .25050 3-05 3-10 3-OS 88 9o 92 .81691 9-81713 81735 I.IO I.IO i-OS .28037 0.28100 .28163 3-15 3-15 3-15 94 96 98 .80662 .80684 .80707 I.IO I. IS I. IO .25111 .25173 .25234 3-10 3-05 3 10 94 96 98 .81756 .81778 .81800 I.IO I.IO i 05 .28226 .28290 .28353 3-20 3.15 100 9.80729 9.25296 100 9.81821 0.28416 (223) TABLE XXVI. (Continued) Hun- 7 D Hun- 7 L dredths Vers Diff. .001 Exsec Diff. .001 dredths Vers Diff. .001 Exsec Diff. .001 00 9.81821 0.28416 00 9-82894 0.31630 02 04 06 08 10 13 14 16 18 .81843 .81865 .81886 .81908 9.81929 81951 81973 .81994 .82016 .10 .05 .10 05 .10 .10 05 .10 .28479 .28543 .28606 .28670 0.28733 .28797 .28860 .28924 .28987 3-20 3 15 3-20 3-15 3.20 3-15 3.20 3-15 02 04 06 08 10 12 14 16 18 .82915 .82936 .82958 .82979 9.83000 .83021 .83042 .83064 .83085 .05 .10 05 .05 .05 05 .10 .05 31695 .31760 .31826 .31891 0.31957 .32022 .32088 32153 .32219 3-25 3-30 3-25 3-30 3.25 3-30 3-25 3.30 20 9.82037 0.29051 20 9.83106 0.32285 22 34 26 28 30 32 34 36 38 .82058 .82080 .82102 .82124 9-82145 .82167 .82188 .82210 .82231 .10 .10 .10 .05 .10 05 .10 .05 .29114 .29178 .29242 .29306 0.29370 29434 .29498 .29562 .29626 3.20 3-20 3-20 3.20 3.20 3-20 3.20 3-20 22 34 26 28 30 33 34 36 38 .83127 .83148 .83169 .83191 9.83212 83233 .83254 .83275 .83296 .05 .05 .10 .05 .05 .05 .05 05 32350 .32416 .32482 .32548 0.32614 .32680 32746 .32812 .32878 3-30 3.30 3-30 3-30 3-30 3-30 3.30 3-30 40 9-82253 0.29690 40 9-83317 0.32944 43 44 46 48 50 53 54 56 58 .82274 .82296 82317 82339 9.82360 .82381 .82403 .82424 .82446 .10 05 .10 05 .05 .10 05 .10 .29754 .29818 .29882 .29946 0.30010 30075 30139 .30203 .30268 3-20 3.20 3-20 3-20 3.25 3-20 3-20 3 25 42 44 46 48 50 53 54 56 58 .83338 .83359 .83381 .83402 9-83423 .83444 .83465 .83486 .83507 .05 .10 .05 .05 .05 05 .05 05 .33010 33076 33142 33209 0.33285 33341 33408 33474 33541 3-30 3-30 3-35 3-30 3-30 3-35 3 30 3-35 60 9.82467 0.30332 60 9-83528 0.33607 62 64 66 68 ?o 73 74 76 78 .82489 .82510 .82531 .82553 9-82574 82595 .82617 .82638 .82660 .05 05 .10 .05 05 .10 05 .10 30397 .30461 .30526 30590 0.30655 .30720 30784 .30849 30914 3-20 3-25 3.20 3-25 3-25 3-20 3-25 3-25 62 64 66 68 70 73 74 76 78 83549 83570 83591 .83612 9-83633 .83654 .83675 .83696 .83717 05 .05 05 05 .05 05 05 05 33674 33741 33807 33874 0-33941 .34008 .34075 .34142 34209 3-35 3-30 3-35 3-35 3-35 3-35 3-35 3-35 80 9.82681 0.30979 80 9.83738 0.34276 82 84 86 88 90 93 94 96 98 .82702 .82724 .82745 .82766 9.82787 .82809 .82830 .82851 .82873 .10 .05 .05 05 .10 05 .05 .10 31044 .31109 3II74 .31239 0.31304 .31369 .31434 31499 .31564 3-25 3-25 3-25 3-25 3-25 3-25 3-25 3-25 82 84 86 88 9o 93 94 96 98 .83759 .83780 .83800 .83821 9.83842 .83863 .83884 -83905 .83926 .05 .00 .05 .05 05 .05 05 .05 34343 34410 34477 34544 0.34611 .34679 .34746 .34814 .34881 3 35 3 35 3-35 3 35 3-40 3 35 3-40 3-35 100 9.82894 0.31630 100 9-83947 0.34948 (224) TABLE XXVI. (Continued) Hun- 7 2 Hun- 73 dredths Vers Diff. .001 Exsec Diff. .001 dredths Vers Diff. .001 Exsec Diflf. .001 00 9-83947 0.34948 00 9-84981 0.38387 02 04 06 08 xo 12 14 16 18 .83968 .83988 .84009 -84030 9.84051 .84072 .84093 .84113 .84134 .00 .05 .05 .05 .05 .05 .00 .05 .35016 .35084 .35151 .35219 0.35287 35354 35422 35490 35558 3-40 3-35 3-40 3-40 3-35 3-40 3-40 3-40 02 04 06 08 IO 12 14 16 18 .85001 .85021 .85042 .85062 9.85083 .85104 .85124 .85144 .85165 .00 .05 .00 .05 .05 .00 .00 .05 .38457 38527 .38597 .38668 0.38738 38809 38879 .38949 .39020 3.50 3-50 3-55 3-50 3-55 3-50 3-50 3 55 20 9.84155 0.35626 20 9-85I85 0.39090 22 24 26 28 30 3* 34 36 38 .84176 .84197 .84217 .84238 9 84259 .84280 .84300 .84321 .84342 .05 .00 .05 .05 05 .00 -05 .05 .35694 .35762 .35830 .35899 0.35967 .36035 .36103 .36172 .36240 3-40 3-40 3-45 3-40 3.40 3-40 3-45 3-40 22 24 26 28 30 32 34 36 38 .85205 .85226 .85246 .85267 9.85287 .85307 .85328 85348 85368 -05 .00 .05 .00 .00 05 .00 .00 .39161 .39232 39303 39373 0.39444 39515 .39586 .39667 39728 3-55 3-55 3-50 3-55 3-55 3-55 3 55 3-55 > fa 40 9-84363 0.36309 40 9-85389 0.39800 42 44 46 48 50 5* 54 56 58 .84383 .84404 .84425 -84445 9.84466 .84487 .84507 .84528 .84549 .05 .05 .00 .05 .05 .00 .05 .05 .36377 .36446 .36514 .36583 0.36652 .36721 .36789 .36858 36927 3-45 3-40 3 45 3-45 3-45 3-40 3-45 3-45 42 44 46 48 So 52 54 56 58 85409 .85429 .85450 .85470 9.85490 .85511 .85531 .85551 .85572 .00 .05 .00 .00 .05 .00 .00 .05 .39871 39942 .40013 .40085 0.40156 .40228 .40299 .40371 40443 3-55 3-55 3.6o 3 55 3-60 3-55 3-Co 3-60 3rr 60 9-84569 QC 0.36996 60 9.85592 0.40514 > go 62 64 66 68 70 72 74 76 78 .84590 .84611 .84631 .84652 9.84672 84693 .84714 84734 .84755 .05 .00 05 .00 .05 05 .00 .05 37065 .37134 .37204 37273 0.37342 374II .37481 37550 .37619 3-45 3-50 3-45 3-45 3-45 3-50 3-45 3-45 62 64 66 68 70 72 74 76 78 .85612 .85632 .85653 .85673 9-85693 .85713 .85733 .85754 .85774 .00 .05 .00 .00 .00 .00 .05 .00 .40586 .40658 .40730 .40802 0.40874 .40946 .41018 .41090 .41163 3.60 3.60 3.60 3.60 3.60 3.60 3.60 3.65 -j 60 80 9.84775 0.37689 80 9-85794 0.41235 go 82 84 86 88 90 92 94 96 98 .84796 .84816 .84837 .84857 9.84878 .84899 .84919 .84940 .84960 .00 05 .00 .05 .05 .00 .05 .00 37759 .37828 .37898 37968 0.38037 38107 .38177 38247 .38317 3-45 3-50 3-50 3-45 3-50 3-50 3-50 3-50 82 84 86 88 90 92 94 96 98 .85814 .85834 85855 .85875 9.85895 .85915 .85935 .85955 .85975 .00 .05 .00 .00 .00 .00 .00 .00 .41307 .41380 .41452 41525 0.41598 .41670 .41743 .41816 .41889 3-65 3.60 3-65 3-65 3.60 3.6s 3-65 3-65 -> fir 100 9-84981 0.38387 100 9.85996 0.41962 (225) TABLE XXVI. (Continued} Hun- 74 Hun- 75 dredths Vers Diff. Exsec Diff. dredths Vers Diff. Exsec Diff. .001 .001 00 9.85996 0.41962 T. 6* 00 9-86992 0.45693 7 80 02 .86016 42035 02 .87012 .45769 04 .86036 I.OO .42108 3-65 04 .87032 I.OO .45846 3-85 06 .86056 .42181 3.65 06 .87052 45922 3-80 08 .86076 ' 42254 3- 5 08 .87071 45999 3-5 10 9.86096 I.OO 0.42328 3-70 10 9-87091 I.OO 0.46075 3-80 12 .86116 I.OO .42401 3-65 3.65 12 .87111 I.OO .46152 3.8o 14 .86136 .42474 14 .87131 . 46228 16 .86156 I.OO 42548 3-70 16 .87150 95 .46306 3-90 18 .86176 I.OO .42621 3.65 18 .87170 I.OO 46383 3-85 30c 20 9.86196 0.42695 7 6=; 20 9-87190 0.46460 22 24 26 .86216 .86236 .86257 I.OO 1.05 .42768 .42842 .42916 3-70 3-70 22 24 26 .87209 .87229 .87249 I.OO I.OO .46537 .46614 .46691 3.85 3-85 28 30 32 .86277 9.86297 .86317 I.OO I.OO I.OO .42990 0.43064 .43138 3-70 3-70 3-70 28 30 32 .87268 9.87288 87308 I.OO I.OO 95 .46769 0.46846 .46923 3-85 3-85 3.90 34 .86337 .43212 34 87327 .47001 36 38 .86357 .86377 I.OO .43286 .4336o 3-70 3-70 36 38 87347 87367 I.OO .47079 .47156 3-9 3-85 -1 QQ 40 9-86397 0.43434 40 9.87386 0.47234 42 44 46 .86417 .86436 .86456 95 I.OO 43509 .43583 .43657 3-70 3-70 42 44 46 .87406 87425 .87445 95 I.OO 47312 47390 .47468 3-90 3-90 48 .86476 ' 43732 48 87465 ' .47546 So 52 9.86496 .86516 I.OO I.OO 0.43806 .43881 3-70 3-75 50 52 9.87484 87504 95 I.OO 0.47624 .47702 3.90 3-90 } QC 54 56 58 .86536 .86556 .86576 I.OO I.OO 43956 44030 .44105 3-70 3-75 54 56 58 .87523 87543 .87562 I.OO 95 .47781 .47859 47938 3-90 3-95 7 QO 60 9.86596 0.44180 60 9-87582 Qf 0.48016 3 95 62 .86616 .44255 62 .87601 .48095 64 66 .86636 .86656 I.OO I.OO 95 44330 .44405 3-75 3-75 3 80 64 66 .87621 .87641 I.OO I.OO 95 .48173 .48252 3-9 3-95 3-95 68 .86675 .44481 68 .87660 48331 70 72 9-86695 .86715 I.OO I.OO I.OO 0.44556 .44631 3-75 3-75 3-75 70 72 9.87680 .87699 95 I.OO 0.48410 .48489 3-95 3-95 3-95 74 76 .86735 .86755 I.OO .44706 .44782 3-80 74 76 .87719 .87738 95 .48568 .48647 3-95 78 .86775 .44857 3-75 78 87758 .48727 3 95 80 9.86795 0-44933 7 8O 80 9-87777 0.48806 3 95 82 84 86 .86814 .86834 .86854 I.OO I.OO I.OO .45009 45084 .45160 3-75 3-80 3.8o 82 84 86 .87796 .87716 .87835 I.OO 95 I.OO .48885 48965 .49044 4.00 3-95 4.00 88 90 92 .86874 9.86894 .86913 I.OO 95 .45236 0.45312 45388 3-80 3-80 3 80 88 90 92 .87855 9.87874 .87894 95 I.OO 95 .49124 0.49204 .49284 4.00 4.00 4.00 94 96 98 .86933 .86953 .86973 I.OO I.OO .45464 45540 .45617 3.8o 3-85 380 94 96 98 .87913 .87933 .87952 I.OO 95 .49364 .49444 .49525 4.00 4-05 3-95 100 9.86992 0.45693 100 9.87971 0.49604 (226) TABLE XXVI. (Continued) Hun- dredths 76 Hun- dredths 77 Vers Diff. .001 Exsec Diff. .001 Vers Diff. .001 Exsec Diff. .001 00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 94 96 98 100 9-87971 1. 00 95 1. 00 95 95 I.OO 95 -95 I.OO 95 95 I.OO 95 95 I.OO 95 95 I.OO 95 95 95 I.OO 95 95 -95 I.OO 95 95 95 95 I.OO 95 95 95 95 95 I.OO 95 95 95 95 95 95 95 I.OO 95 95 95 95 95 0.49604 4.00 4.00 4-05 4.00 4-05 4-05 4.00 4-05 4-05 4-05 4-05 4-05 4.10 4-05 4.10 4-05 4.10 4-05 4.10 4.10 4.10 4-10 4.10 4.10 4-iS 4.10 4-15 4.10 4.15 4-15 4-15 4.15 4.15 4-15 4-iS 4-15 4.20 4-15 4.20 4.20 4.20 4-15 4.20 4-25 4.20 4.20 4.20 4-25 4.20 4-25 00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 So 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 94 96 98 100 9-88933 95 95 95 95 95 95 -95 -95 95 95 95 95 95 95 95 95 95 95 95 95 95 95 90 95 95 95 95 95 95 95 90 95 95 95 95 95 .90 95 95 95 95 .90 95 95 95 .90 95 95 95 .90 0.53724 4-25 4-25 4-25 4-25 4-25 4-25 4.30 4-25 4-30 4- 2S 4-30 4-30 4-30 4-30 4-30 4-30 4-35 4-30 4-35 4-35 4-30 4-35 4-35 4-35 4.40 4-35 4-35 4.40 4.40 4-35 4.40 4-40 4.40 4.40 4-45 4-40 4-45 4-40 4-45 4-45 4-45 4-45 4-45 4-45 4-50 4-45 4-50 4.50 4-45 4-50 .87991 .88010 .88030 .88049 9.88068 .88088 .88107 .88126 .88146 .49684 .49764 .49845 .49925 0.50006 .50087 .50167 . 50248 .50329 .88952 .88971 .88990 .89009 9.89028 .89047 .89066 .89085 .89104 53809 .53894 53979 .54064 0.54149 54234 54320 .54405 54491 9.88165 0.50410 9-89123 0.54576 .88184 .88204 .88223 .88242 9.88262 .88281 .88300 .88320 88339 50491 .50572 .50654 .50735 0.50817 .50898 .50980 .51061 5II43 .89142 .89161 .89180 .89199 9.89218 .89237 .89256 .89275 .89294 .54662 .54748 .54834 54920 0.55006 .55092 55179 55265 55352 9.88358 0.51225 9.89313 0.55439 .88377 .88397 .88416 -88435 9.88454 .88474 .88493 .88512 .88531 51307 .51389 .51471 .51553 0.51636 51718 .51801 .51883 .51966 89332 .89351 89369 .89388 9.89407 .89426 89445 .89464 -89483 55525 556i2 55699 55786 0.55874 .55961 .56048 .56136 .56224 9-88550 0.52049 9.89502 0.56311 .88570 .88589 .88608 .88627 9.88646 .88665 .88685 .88704 .88723 .52132 .52215 .52298 .52381 0.52464 52547 .52631 .52714 .52798 .89520 89539 .89558 89577 9.89596 .89615 .89633 .89652 .89671 .56399 .56487 .56575 .56663 0.56752 .56840 .56929 57017 .57io6 9.88742 0.52882 9.89690 0.57195 .88761 .88780 .88799 .88818 9.88838 .88857 .88876 .88895 .88914 .52966 53049 .53133 53218 0.53302 -53386 53470 53555 .53639 .89709 .89727 89746 .89765 9.89784 .89802 .89821 .89840 .89859 .57284 57373 .57462 57551 0.57641 57730 .57820 57910 57999 9-88933 0.53724 9.89877 o . 58089 (227) TABLE XXVI. (Continued') Hun- dredths 78 Hun- dredths 79 Vers Diff. .001 Exsec Diff. .001 Vers Diff. .001 Exsec Diff. .001 00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 5* 54 56 58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 94 96 98 100 9.89877 95 95 .90 95 95 95 .90 95 95 90 95 95 .90 95 .90 95 95 .90 95 90 95 95 90 .90 95 95 .90 95 90 95 95 90 95 .90 95 .90 .90 95 90 95 90 95 90 95 .90 .90 95 .90 95 90 0.58089 4-55 4-50 4-50 4-55 4-50 4-55 4-55 4-55 4-55 4-55 4-55 4-55 4.60 4.60 4-55 4.60 4.60 4.60 4-65 4.60 4-65 4.60 4-65 4.60 4.70 4-65 4-65 4-70 4.65 4-70 4-65 4.70 4.70 4-75 4-70 4.70 4-75 4-75 4-70 4-75 4-75 4.80 4-75 4-75 4-80 4.80 4.80 4.80 4.80 4.80 00 02 04 06 08 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 94 96 98 9-90805 90 95 90 95 .90 .90 95 90 90 95 90 90 95 .90 90 95 .90 90 90 95 90 .90 .90 95 .90 .90 .90 .90 95 .90 .90 .90 90 95 .90 90 90 .90 .90 95 .90 .90 90 .90 90 90 .90 90 .90 .90 0.62745 4.85 4.80 4.80 4.80 4.85 4.85 4.85 4.90 4.85 4.90 4.90 4-90 4-90 4.90 4-95 4.90 4-95 4-95 4-95 4-95 5.00 5-05 S-oo 5.00 S.oo S.oo S.oo 5.00 5-05 5-05 S.oo 5-05 5.05 5-05 5-05 5.10 5-10 S.io 5.10 S.io 5-15 S.io 5-iS 5-iS S.iS 5.20 5-iS 5.20 5-20 5.15 .89896 .89915 .89933 .89952 9.89971 .89990 .90008 .90027 .90046 .58180 .58270 58360 .58451 0.58541 .58632 .58723 .58814 .58905 .90823 .90842 .90860 .90879 9.90897 .90915 .90934 .90952 .90970 .62842 .62938 .63036 .63132 0.63229 .63326 .63423 .63521 .63618 9.90064 0.58996 9.90989 0.63716 .90083 . 90102 .90120 -90I39 9.90157 .90176 .90195 .90213 .90232 .59087 .59178 .59270 .59362 0-59453 59545 .59637 .59729 .59822 .91007 .91025 .91044 .91062 9.91080 .91099 .91117 .9"35 9II53 .63814 .63912 .64010 .64108 0.64207 64305 .64404 .64503 . 64602 9.90250 0.59914 9.91172 o . 64701 .90269 .90288 .90306 .90324 9.90343 .90362 .90380 90399 .90417 .60007 .60099 .60192 .60284 0.60378 .60471 .60564 .60658 .60751 .91190 .91218 .91226 .91245 9.91263 .91281 .91299 .91317 .91336 .64801 .64900 .65000 .65100 0.65200 .65300 .65400 .65500 .65601 9-90436 0.60845 9-91354 0.65702 90455 .90473 .90492 .90510 9.90529 .90547 90565 .90584 .90602 .60938 .61032 .61126 .61221 0.61315 .61409 .61504 .61599 .61693 .91372 .91390 .91408 .91427 9.9I445 91463 .91481 91499 .91517 .65802 .65903 .66005 .66106 0.66207 .66309 .66411 .66513 .66615 9 . 90621 0.61788 9.9I536 0.66717 .90639 .90658 .90676 90695 9.90713 90731 90750 .90768 .90787 .61883 .61979 .62074 .62169 o . 62265 . 62361 .62457 .62543 .62649 .91554 .91572 .91590 .91608 9.91626 .91644 .91662 .91680 .91698 .66820 .66922 .67025 .67128 0.67231 .67335 -67438 .67542 .67646 9-90805 0.62745 100 9.91716 0.67749 (228) TABLE XXVI. - (Continued) Hun- dredths 80 Hun- dredths 81 Vers Diff. .001 Exsec Diff. .001 Vers Diff. .001 Exsec Diff. .001 00 9.91716 0.67749 00 9.92612 QO 0.73179 5 65 02 04 06 08 IO 12 14 16 18 9I73S 9!753 .91771 .91789 9.91807 .91825 .91843 .91861 .91879 95 .90 .90 .90 .90 .90 .90 .90 .90 .67854 .67958 .68062 .68167 0.68272 .68377 .68482 .68587 .68693 5.20 5-20 5.25 5.25 5.25 5.25 5.25 5.30 02 04 06 08 10 12 14 16 18 .92630 .92647 .92665 .92683 9.92701 .92718 .92736 .92754 .92771 .85 .90 .90 .90 .85 .90 .90 .85 .73292 .73406 73520 .73634 0.73749 .73863 73978 .74093 .74208 5-70 5-70 5-70 5-75 5-70 5-75 5-75 5-75 er go 20 9.91897 9 0.68799 20 9.92789 0.74324 5 80 22 24 26 .91915 .91933 .91951 9 .90 .90 9 .68904 .69010 .69117 5.30 5-35 5-30 22 24 26 .92807 .92824 .92842 .85 90 9 74440 .74556 .74671 5-80 5-85 5-85 28 30 32 .91969 9.91987 .92005 .90 .90 90 .69223 0.69330 69436 5-35 5-30 5-35 28 30 32 .92860 9.92877 .92895 .85 .90 -90 .74788 0.74905 .75022 5-85 5-85 5-85 34 36 38 .92023 .92041 .92059 .90 .90 .69543 .69650 .69758 5-35 5-40 34 36 38 .92913 .92930 .92948 -85 90 QO 75139 .75256 75373 5-85 5-85 5 9 40 9.92077 0.69865 40 9-92966 Qe 0.75491 42 44 46 48 SO 52 .92095 .92111 .92130 .92148 9.92166 .92184 9 .90 95 90 .90 .90 90 69973 .70080 .70189 .70297 0.70405 .70514 5-35 5-45 5-40 5-40 5-45 5 45 42 44 46 48 50 52 .92983 .93001 .93018 .93036 9-93054 .93071 .90 .85 90 .90 .85 9 .75609 .75727 .75846 .75965 0.76083 .76203 5.90 5.95 5.95 5.90 6.00 5-95 54 56 58 . 92202 .92220 .92238 .90 .90 .70623 .70732 .70841 5-45 5-45 54 56 58 .93089 .93106 .93124 .85 90 .76322 .76442 .76561 6.00 5-95 6 05 60 9.92256 .90 0.70950 60 9.93142 g^ 0.76682 6 oo 62 64 66 .92274 .92291 .92309 .85 .90 QO .71060 .71169 .71279 5-45 5-50 5-5O 62 64 66 93159 .93176 .93194 .85 90 .90 .76802 .76922 77043 6.00 6.05 6.10 68 70 .92327 9-92345 .90 71389 0.71500 5-55 68 70 .93212 9.93229 .85 .77165 0.77286 6.05 72 .92363 .90 9 .71610 5-5O 5-55 72 93247 .90 .85 .77407 6.10 74 76 .92381 .92398 85 .71721 .71832 5-55 74 76 .93264 93282 .90 .77529 .77651 6.10 78 .92416 .90 .71943 5-55 78 93299 o5 77774 6 10 80 9.92434 9 0.72054 <; 60 80 9.93317 gc 0.77896 6 15 82 84 86 .92452 .92470 .92487 .90 .85 .90 .72166 .72278 .72389 5.6o 5-55 5.65 82 84 86 93334 93352 .93369 .90 .85 .90 .78019 .78142 .78265 6.15 6.15 6.20 88 90 92 .92505 9.92523 .92541 90 .90 .90 .72502 0.72614 .72726 5.60 S.6o 5.65 88 90 92 .93387 9-93404 93422 .85 .90 85 .78389 0.78513 .78637 6.20 6.20 6.20 94 96 98 .92559 .92576 .92594 .85 .90 .72839 .72952 .73065 5-65 5-65 94 96 98 93439 93457 93474 90 .85 .78761 .78886 .79011 6.25 6.25 6 25 100 9.92612 0.73179 100 9-93492 0.79136 (229) TABLE XXVI. (Continued) Hun- dredths 82 Hun- dredths 83 Vers Diff. .001 Exsec Diff. .001 Yers Diff. .001 Exsec Diff. .001 00 9-93492 Or 0.79136 69C 00 9-94356 0.85766 02 04 .93509 .93526 85 .79261 .79387 6.30 02 04 -94373 -94390 .85 .85907 .86048 7.05 7-05 06 93544 90 .85 79513 6 35 06 94407 85 85 .86190 7.10 7 10 08 93561 .79640 08 .94424 .86332 IO 12 9-93579 93596 .90 .85 0.79766 .79893 6.30 6.35 6 IO 12 9-94442 94459 .90 .85 0- 0.86474 .86616 7.10 7.10 14 93614 .80020 14 .94476 .86759 16 .93631 5 .80147 35 16 94493 .85 .86903 18 .93648 85 .80275 18 94510 85 Of .87046 7-15 20 9.93666 0.80403 20 9-94527 Qp 0.87190 22 24 26 .93683 93700 93718 -85 .90 85 .80531 .80659 .80788 6.40 6.45 6.45 22 24 26 94544 .94561 94578 .85 -85 85 .87335 .87470 .87625 7-25 7-25 7.25 28 30 32 93735 9-93752 93770 -85 .90 .85 .80917 0.81046 .81176 6.45 6.50 6.50 28 30 32 94595 9.94612 .94629 .85 .85 85 .87770 0.87916 .88062 7-30 7-30 7-35 34 36 38 .93787 93804 .93822 .85 .90 fir .81306 .81436 .81567 6.50 6.55 34 36 38 .94646 .94663 .94680 .85 .85 g- .88209 .88356 .88504 7-35 7.40 40 9.93839 0- 0.81697 6 60 40 9-94697 8? 0.88651 42 .93856 .81829 42 94714 .88800 44 46 .93874 .93891 .90 .85 85 .81960 .82092 55 6.60 6.60 44 46 94731 94748 85 .85 85 .88948 .89097 7-4O 7-45 7.50 48 50 52 .93908 9-93926 93943 .90 .85 .85 .82224 0.82356 .82488 6.60 6.60 6.65 48 50 52 .94765 9-94782 94799 .85 .85 .85 .89247 0.89397 89547 7-50 7-50 7.50 54 56 58 .9396o 93977 93995 .85 .90 c- .82621 .82755 .82888 6.70 6.65 54 56 58 .94816 .94833 .94850 .85 .85 0- .89697 .89848 .90000 7-55 7.60 7 60 60 9.94012 Or 0.83022 60 9.94867 Of 0.90152 62 64 66 .94029 .94047 .94064 .90 .85 85 .83156 .83290 .83425 6.70 6-75 6.75 62 64 66 .94884 .94901 .94918 .85 .85 85 .90304 .90457 .90610 7-65 7-65 7.7O 68 70 72 .94081 9.94098 94II5 .85 .85 90 .83560 0.83696 .83831 6.80 6.85 6.85 68 70 72 94935 9-94952 .94969 .85 .85 .85 .90764 0.90918 .91072 7.70 7.70 7-75 74 76 78 .94133 94150 .94167 .85 .85 .83968 .84104 .84241 6.80 6.85 6 85 74 76 78 .94986 -95003 .95020 .85 .85 .91227 .91383 .91538 7.80 7-75 7 80 80 9.94184 0- 0.84378 6 85 80 9-95037 ' S 0.91694 7 8? 82 84 86 .94201 .94219 .94236 .90 -85 .85 .84515 .84653 .84791 6.90 6.90 6.90 82 84 86 .95053 .95070 .95087 .85 .85 .85 .91851 .92008 .92166 7-85 7-90 7.90 88 90 92 .94253 9.94270 .94287 .85 -85 .85 .84929 0.85068 .85207 6.95 6.95 6.95 88 90 92 95104 9-95I2I .95138 .85 .85 .92324 0.92482 .92641 7-90 7-95 8.00 94 .94304 .85346 94 .95155 .92801 96 94322 90 .85486 7.00 96 .95171 .92961 98 94339 .85 g- .85626 7.00 98 .95188 85 gr .93121 8 05 100 9-94356 0.85766 100 9-95205 0.93282 (230) TABLE XXVI. (Continued} Hun- dredths 84 Hun- dredths 85 Vers Diff. .001 Exsec Diff. .001 Vers Diff. .001 Exsec Diff. .001 00 9-95205 Oe 0.93282 8ne 00 9.9604P go 1. 02010 02 04 06 .95222 .95239 .95256 .85 .85 .80 93443 .93605 .93767 8.10 8.10 8.15 02 04 06 .96056 -96073 .96089 .85 .80 .85 .02200 .02391 .02583 9-55 9.60 9.60 08 10 12 .95272 9.95289 .95306 .85 .85 Oe -93930 0.94093 94257 8.15 8. 20 08 10 12 .96106 9.96122 .96139 .80 .85 go .02775 1.02968 .03162 9-65 9.70 14 16 18 .95323 95340 .95356 .85 .80 Or .94421 94586 94751 8.25 8.25 8 -j O 14 16 18 .96155 .96172 .96188 .85 .80 .03356 .03551 .03747 9-75 9.80 90- 20 9-95373 gr 0.94917 20 9 96205 go 1.03944 22 -95390 .95183 22 .96221 .04141 24 26 95407 .95424 .85 85 .80 .95250 95417 8.35 8.35 8.40 24 26 .96238 .96254 .85 .80 85 .04339 .04538 9.90 9-95 10 OO 28 95440 .95585 28 .96271 .04738 30 32 9-95457 95474 8S 85 85 0.95754 .95922 8.45 8.40 8.50 30 32 9.96287 .96304 .80 .85 .80 1.04938 .05140 10. OO 10.10 10.10 34 36 38 95491 .95507 .95524 .80 85 X- .96092 .96262 96432 8.50 8.50 See; 34 36 38 .96320 .96337 .96353 .85 .80 .05342 .05544 .05748 10.10 10.20 40 9-95541 go 0.96603 8 60 40 9-96369 Of 1.05953 42 44 46 95557 95574 95591 .85 .85 85 .96775 .96947 .97120 8.60 8.65 8.65 42 44 46 .96386 .96402 .96419 .80 .85 .80 .06158 .06364 .06571 10.30 10.35 10.40 4* .95608 .97293 48 .96435 .06779 50 52 9.95624 .95641 85 .85 0.97467 .97641 8.70 8.75 50 52 9.9645I .96468 .85 .80 1.06987 .07197 10.50 10.50 54 56 58 .95658 .95674 .95691 .80 .85 - .97816 97992 .98168 8.80 8.80 8 85 54 56 58 .96484 .96501 .96517 .85 .80 .07407 .07618 .07830 10.55 10. 60 rn fie 60 9.95708 go 0.98345 8 81 60 9.96533 0- 1.08043 62 64 66 68 70 72 95724 95741 95758 95774 9-95791 .95807 .85 .85 .80 .85 .80 .85 .98522 .98700 .98879 .99058 0.99237 .99418 8.90 8-95 8.95 8-95 9-05 9-05 62 64 66 68 70 72 .96550 .96566 .96582 .96599 9-96615 .96632 .80 .80 .85 .80 .85 .80 .08257 .08472 .08687 .08904 1.09121 .09340 10.75 10.75 10.85 10.85 10.95 10.95 74 76 78 80 82 84 86 88 .95824 .95841 .95857 .85 .80 .85 .85 .80 85 .80 99599 .99780 .99963 9-05 9-iS 9-iS 9-15 9.20 9-25 9-25 74 76 78 80 82 84 86 88 .96648 .96664 .96680 .80 .80 .85 .80 .80 .85 .80 .09559 .09779 .IOCOO 11.00 11.05 ii. 15 ii. IS II. 25 II. 25 11.30 9.95874 i . 00146 9.96697 I. 10223 .95891 .95907 .95924 95940 .00329 .00513 .00698 .00883 .96713 .96729 .96746 .96762 .10446 . 10671 .10896 .11122 90 92 9-95957 95973 .85 .80 .85 1.01069 .01256 9-30 9-35 9.40 90 92 9-96778 .96795 .80 .85 .80 I. I 1349 .11578 11-35 ii. 45 II 45 94 95990 .01444 94 .96811 11807 96 .96007 .85 .01633 9-45 96 .96827 .80 . 12038 il.SS 98 100 .96023 .85 .01821 9.40 9-45 98 100 .96843 .80 .85 .12269 11-55 ii. 60 9.96040 I.O2OIO 9.96860 1.12501 (231) TABLE XXVI. (Continued) Hun- dredths 86 Hun- dredths 87 Vers Diff. .001 Exsec Diff. .001 Vets Diff. .001 Exsec Diff. .001 00 9.96860 80 .1 . 12501 00 9.97665 80 1-25785 02 04 06 .96876 .96892 .96908 .80 .80 .85 ' .12735 .12969 .13205 11.70 11.80 11.85 02 04 06 .97681 .97697 97713 .80 .80 .80 .26092 . 26400 .26710 15.40 15-50 15.60 08 .96925 .13442 08 .97729 . 27022 10 12 9.96941 .96957 .80 .80 I . 13680 .13919 11-95 12.05 IO 12 9-97745 .97761 .80 .80 1.27336 .27652 15.70 15.80 15-95 14 16 18 .96973 .96990 .97006 .85 .80 80 .14160 .14401 .14644 12.05 12.15 14 16 18 97777 97793 .97809 .80 .80 80 .27971 .28291 .28614 16.00 16.15 20 9.97022 80 1.14888 20 9.97825 80 1.28939 22 24 26 .97038 97054 .97071 .80 .85 .80 .15133 .15379 . 15626 12.30 12.35 12.45 22 24 26 .97841 .97857 .97873 .80 .80 .80 .29266 29595 .29926 16.45 16.55 16 70 28 30 32 .97087 9.97103 .97"9 .80 .80 .80 .15875 1.16125 .16376 12.50 12.55 12.65 28 30 32 .97889 9-97904 .97920 75 .80 .80 .30260 1.30596 .30935 16.80 i6.95 17.05 34 .97135 .16619 34 .97936 .31276 36 38 .97151 .97168 .85 80 .16883 .17138 12.70 12.75 36 38 97952 .97968 .80 go .31619 31965 I7-I5 17-30 40 9.97184 80 I. 17394 40 9.97984 80 1.32314 42 44 46 .97200 .97216 .97232 .80 .80 .80 .17652 .17910 .18172 12.90 13.10 13.10 42 44 46 .98000 .98016 .98031 .80 75 .80 .32665 .33018 33375 17-65 17-85 1,7.90 48 .97248 .18434 48 .98047 33733 50 52 9.97264 .97280 .80 .85 1.18697 .18961 13.15 13-20 13.35 50 52 9-98063 .98079 .80 .80 1.34095 .34460 18.25 18.35 54 56 58 .97297 97313 .97329 .80 .80 80 . 19228 .19495 . 19764 13.35 13-45 54 56 58 .98095 .98111 .98126 .80 75 go .34827 35197 -35570 18.50 18.65 18 80 60 9-97345 i. 20035 60 9.98142 go 1.35946 62 64 66 .97361 97377 97393 .80 .80 .80 .20307 .20580 20855 13-65 13-75 13.85 62 64 66 .98158 .98174 .98190 .80 .80 75 .36325 .36707 .37092 19.10 19-25 19-45 68 70 72 .97409 9-97425 97441 .80 .80 .80 .21132 1.21410 .21690 13.90 14.00 14.05 68 70 72 .98205 9.98221 98237 .80 .80 .80 .37481 1.37872 .38267 19-55 19-75 19.90 74 76 78 97457 97473 .97489 .80 .80 80 .21971 .22254 .22539 14.15 14-25 74 76 78 .98253 .98268 .98284 75 .80 g o .38665 .39067 39472 20. 10 20.25 80 9-97505 80 1.22825 80 9.98300 go I.3988I 20 60 82 84 86 97521 97537 97553 .80 .80 .80 .23113 .23403 .23695 14.50 14.60 14.65 82 84 86 .98316 .98331 .98347 75 .80 .80 .40293 .40709 .41128 20.8O 20.95 21. 2O 88 90 92 .97569 9.97585 .97601 .80 .80 .80 .23988 1.24283 .24580 14-75 14.85 14.90 88 90 92 .98363 9.98379 .98394 .80 75 .80 .41552 I.4I979 .42410 21.35 21.55 21.75 94 96 98 .97617 .97633 .97649 .80 .80 80 .24878 .25179 .25481 14-95 15.10 94 96 98 .98410 .98426 .98442 .80 .80 .42845 .43284 .43728 21.95 22.20 22 35 100 9-97665 1.25785 100 9.98457 I-44I75 (232) TABLE XXVI. (Continued) Hun- dredths 88 Hun- dredths 89 Vers Difif. .001 Exsec Diff. .001 Vers Diff. .001 Exsec Diff. .001 00 9-98457 I-44I75 00 9-99235 80 1.75050 02 04 06 .98473 .98489 98504 .80 75 80 .44627 45084 45545 22.85 23.05 23 25 02 04 06 .99251 .99266 .99282 75 .80 75 75943 76853 .77783 45.50 46.50 47-45 08 10 12 .98520 9-98536 .98551 .80 75 .80 .46010 I . 46481 .46956 23-55 23-75 24.00 08 IO 12 .99297 9.99312 99328 75 .80 75 .78732 1.79702 .80694 48.50 49.60 50.65 14 16 .98567 .98583 .80 .47436 47921 24.25 14 16 99343 99359 .80 .81707 .82744 51.85 18 .98598 75 80 .48411 24.50 18 99374 75 .83806 53-10 20 9.98614 80 1.48906 2C o 1 ? 20 9-99389 80 1.84894 ctr 75 22 24 26 .98630 .98645 .98661 75 .80 75 . 49407 .49913 50425 25-30 25.60 25.85 22 24 26 .99405 .99420 99435 75 75 .80 .86009 .87152 .88326 57.15 58.70 60.15 28 30 32 .98676 9.98692 .98708 .80 .80 75 .50942 1.51466 .51995 26.20 26.45 26.80 28 30 32 99451 9.99466 .99481 75 75 .80 .89531 1.90770 .92044 61.95 63.70 65.60 34 36 38 .98723 .98739 .98755 .80 .80 .52531 53073 .53621 27.IO 27.40 34 36 38 99497 99512 99527 75 75 80 .93356 .94707 .96101 67.55 69.70 40 9.98770 80 1.54176 28 10 40 9-99543 I- 97541 42 44 46 .98786 .98801 .98817 75 .80 75 .54738 .55306 .55882 28.40 28.80 29.15 42 44 46 -99558 99573 .99589 75 .80 75 .99028 2.00568 .02162 77.00 79.70 82.75 48 50 52 .98832 9.98848 .98864 .80 .80 75 56465 1.57056 .57654 29-55 29.90 30.35 48 50 52 .99604 9-99619 .99635 -75 .80 -75 .03817 2.05535 .07323 85.90 89.40 93-20 54 56 58 .98879 .98895 .98910 .80 75 80 .58261 .58875 59498 30.70 31 15 54 56 58 .99650 .99665 .99680 .75 75 80 .09187 -III33 .13168 97-30 101.75 60 9.98926 7S 1.60130 60 9.99696 2.15302 62 64 66 .98941 .98957 .98972 .80 75 .80 .60770 .61419 .62078 32-45 32-95 33-45 62 64 66 997II .99726 99742 75 .80 75 .17545 .19909 .22406 118.20 124.85 132.40 68 70 72 .98988 9.99003 .99019 75 .80 75 .62747 1.63425 .64114 33-90 34-45 34-95 68 70 72 99757 9.99772 .99787 -75 75 75 .25054 2.27872 30884 140.90 150.60 161.70 74 76 78 .99034 .99050 .99065 .80 75 .So .64813 65523 66245 35-50 36.10 36 65 74 76 78 .99802 .998i8 99833 .80 -75 75 .34118 .37609 41403 174-55 189.70 80 9.99081 75 1.66978 "V7 2<\ 80 9.99848 7C 2.45557 82 84 86 .99096 .99112 .99127 .80 75 .80 67723 .68481 .69252 37-90 38.55 39-20 82 84 86 .99863 .99879 .99894 .80 -75 75 .50148 55279 .61093 256.55 290.70 335-50 88 90 92 99143 9.99158 .99174 75 .80 75 .70036 1.70834 .71646 39-90 40.60 41-35 88 90 92 .99909 9.99924 99939 75 -75 75 .67803 2.75736 .85443 396.65 485.45 625-45 94 96 98 .99189 .99204 .99220 75 .80 75 72473 73316 74174 42.15 42.90 43 80 94 96 98 99954 99970 99985 .80 75 97952 3.I5576 .45694 881.20 1505-90 100 9-99235 I 75050 100 o.ooooo 00 (233) TABLE XXVII. NATURAL SINES, TANGENTS, COTANGENTS AND COSINES Deg. Sin d. Tan d. Cotan d. Cosin d. P. P. 0.0 o.oooo o.oooo inf. I.OOOO 90.0 i 0.0017 T o 0.0017 T o 572-9572 I.OOOO 9 2 0.0035 0.0035 286.4777 I.OOOO 8 3 0.0052 17 18 0.0052 17 18 190.9842 I.OOOO 7 4 0.0070 0.0070 143.2371 I.OOOO 6 5 0.0087 17 18 0.0087 17 18 114-5887 I.OOOO 5 6 0.0105 17 0.0105 17 95.4895 0.9999 o 4 7 O.OI22 T o O.OI22 T o 81.8470 0.9999 3 8 O.OI40 O.OI4O 71-6151 0.9999 2 9 0.0157 18 0.0157 18 63.6567 0.9999 I 1.0 0.0175 0.0175 57.2900 0.9998 89.0 i 0.0192 0.0192 52.0807 0.9998 9 2 0.0209 18 0.0209 18 47-7395 .... 0.9998 8 18 3 0.0227 0.0227 44 . 0661 0.9997 7 I 1.8 4 0.0244 17 0.0244 40.9174 0.9997 6 2 3.6 5 0.0262 0.0262 38.1885 .... 0-9997 5 3 5-4 6 0.0279 17 18 0.0279 18 35-8006 0.9996 o 4 4 7-2 5 9 o 7 0.0297 0.0297 33.6935 0.9996 3 6 10.8 8 0.0314 18 0.0314 18 31-8205 0-9995 o 2 7 12.6 9 0.0332 0.0332 30.1446 0-9995 I 8 14.4 2.0 0.0349 0.0349 18 28.6363 0-9994 88.0 9 16.2 i o . 0366 18 0.0367 27.2715 0-9993 9 2 0.0384 17 0.0384 18 26.0307 0-9993 i 8 3 0.0401 18 0.0402 17 24.8978 0.9992 T 7 4 5 0.0419 0.0436 17 18 0.0419 0.0437 18 17 23.8593 22.9038 9555 8821 0.9991 0.9990 I o 6 5 6 0.0454 17 0.0454 18 22.0217 8168 0.9990 I 4 7 8 9 0.0471 0.0488 0.0506 17 18 0.0472 0.0489 0.0507 17 18 21 . 2049 20.4465 19.7403 7584 7062 0.9989 0.9988 0.9987 I I 3 2 I 3.0 0.0523 18 0.0524 18 I9.08ll 6166 0.9986 I 87.0 17 i 2 3 0.0541 0.0558 0.0576 17 18 17 0.0542 0-0559 0.0577 17 18 18.4645 17.8863 17.3432 5782 5431 5113 0.9985 0.9984 0.9983 I I I 9 8 7 i 1.7 2 3-4 3 5-1 4 5 6 0.0593 0.0610 0.0628 18 0.0594 0.0612 0.0629 18 17 18 16.8319 16.3499 15.8945 4820 4554 4307 0.9982 0.0981 0.9980 I I I 6 5 4 4 6.8 5 8.5 6 10.2 7 H-9 7 8 9 0.0645 0.0663 0.0680 18 17 18 0.0647 0.0664 0.0682 17 18 15.4638 15.0557 14.6685 4081 3872 3678 0.9979 0.9978 0.9977 I I 3 2 I 8 13.6 9 15-3 4.0 0.0698 17 0.0699 18 14.3007 35oo 0.9976 2 86.0 i 2 0.0715 0.0732 17 18 0.0717 0.0734 17 18 13.9507 13.6174 3333 3178 0.9974 0-9973 I I 9 8 3 0.0750 0.0752 17 13.2996 3034 0.9972 I 7 4 5 6 0.0767 0.0785 0.0802 18 17 17 0.0769 0.0787 0.0805 18 18 17 12.9962 I2.7O62 12.4288 2900 2774 2656 0.9971 0.9969 0.0968 2 I 2 6 5 4 7 8 9 0.0819 0.0837 0.0854 18 17 18 0.0822 0.0840 0.0857 18 17 18 12 . 1632 11.9087 11.6645 2545 2442 2344 0.9966 0.9965 0.9963 I 2 I 3 2 I 5.0 0.0872 0.0875 11.4301 0.9962 85 Cosin d. "otan d. Tan d. Sin d. Deg. (234) TABLE XXVII. (Continued) Deg. Sin d. Tan d. Cotan d. Cosin d. P.P. 50 o . 0872 0.0875 11.4301 0.9962 85 i 0.0889 0.0892 18 11.2048 2166 0.9960 9 2 3 0.0906 0.0924 18 17 0.0910 0.0928 18 17 10.9882 10.7797 2085 2008 0-9959 0-9957 2 I 8 7 4 5 6 0.0941 0.0958 0.0976 17 18 17 0.0945 0.0963 0.0981 18 18 17 10.5789 10.3854 10.1988 1935 1866 1801 0.9956 0.9954 0.9952 2 2 I 6 5 4 7 8 0.0993 O.IOII 18 0.0998 0.1016 18 10.0187 9-8448 1739 1680 0.9951 0.9949 2 3 2 9 o . 1028 0-1033 18 9.6768 0.9947 I 6.0 0.1045 18 0.1051 18 9-5144 0.9945 84.0 i 2 3 0.1063 0.1080 0.1097 17 17 18 0.1069 0.1086 0.1104 17 18 18 9-3572 9-2052 9-0579 1520 1473 1427 0-9943 0.9942 0.9940 I 2 2 9 8 7 18 I 1.8 2 3-6 4 5 6 0.1115 0.1132 0.1149 17 17 18 O.II22 0.1139 0.1157 17 18 18 8.9152 8.7769 8.6427 1383 1342 1301 0-9938 0-9936 0-9934 2 2 2 6 5 4 3 5-4 4 7-2 5 9-0 6 10.8 7 8 9 0.1167 0.1184 O.I2OI 17 17 18 0.1175 0.1192 O.I2IO 17 18 18 8.5126 8.3863 8.2636 1263 1227 IIQ-J 0.9932 0-9930 0.9928 2 2 3 2 I 7 12.6 8 14.4 9 16.2 7.0 0.1219 0.1228 18 8.1443 0.9925 83.0 i 0.1236 0.1246 8.0285 0.9923 9 2 3 0.1253 0.1271 18 0.1263 0.1281 18 18 7.9158 7.8062 1096 1066 0.9921 0.9919 2 2 8 7 4 0.1288 17 0.1299 18 7.6996 IO38 0.9917 3 6 5 0.1305 18 0.1317 17 7-5958 IOII 0.9914 2 5 6 0.1323 17 0.1334 18 7-4947 985 0.9912 2 4 7 0.1340 17 0.1352 18 7.3962 960 0.9910 3 8 9 0.1357 0.1374 17 18 0.1370 0.1388 18 7-3002 7.2066 936 012 0.9907 0.9905 2 2 I 8.0 0.1392 17 0.1405 18 7-II54 890 0.9903 82 17 i 0.1409 0.1423 18 7.0264 8fio 0.9900 9 i 1.7 2 3 0.1426 0.1444 18 17 0.1441 0.1459 18 18 6-9395 6.8548 847 828 0.9898 0.9895 3 2 8 7 2 3-4 3 5-1 4 0.1461 0.1477 18 6.7720 808 0.9893 6 4 6.8 5 6 0.1478 0.1495 17 18 0.1495 0.1512 17 18 6.6912 6.6122 790 772 0.9890 0.9888 2 3 5 4 5 8.5 6 IO.2 7 H-9 7 8 9 O.I5I3 0.1530 0.1547 17 17 0.1530 0.1548 0.1566 18 18 18 6-5350 6.4596 6.3859 754 737 0.9885 0.9882 0.9880 3 2 3 2 I 8 13.6 9 15-3 9.0 0.1564 18 0.1584 18 6.3138 706 0.9877 81.0 i 2 3 0.1582 0.1599 0.1616 17 17 17 0.1602 0.1620 0.1638 18 18 17 6.2432 6.1742 6.1066 690 676 661 0.9874 0.9871 0.9869 3 2 3 9 8 7 4 5 6 0.1633 0.1650 0.1668 i? 18 17 0-1655 0.1673 0.1691 18 18 18 6.0405 5-9758 5.9124 647 634 622 0.9866 0.9863 0.9860 3 3 3 6 5 4 7 0.1685 0.1709 18 5.8502 608 .0.9857 3 8 0.1702 17 0.1727 18 5.7894 597 0.9854 2 9 0.1719 17 0.1745 18 5.7297 1584 0.9851 I 10 0.1736 0.1763 5.6713 0.9848 80 Cosin d. Cotan d. Tan d. Sin d. Deg. (235) TABLE XXVII. (Continued) Deg. Sin d. Tan d. Cotan d. Cosin d. ] 3 . P. 10.0 0.1736 18 0.1763 18 5.6713 0.9848 80.0 i 2 0.1754 0.1771 17 0.1781 0.1799 18 5.6140 5.5578 562 0.9845 0.9842 3 9 8 3 0.1788 0.1817 18 5-5026 552 0.9839 3 7 4 0.1805 0.1835 5.4486 0.9836 6 5 0.1822 17 0.1853 18 5-3955 531 0.9833 3 5 6 0.1840 0.1871 5-3435 520 0.9829 4 4 19 7 8 0.1857 0.1874 17 0.1890 0.1908 *9 18 5.2924 5.2422 502 0.9826 0.9823 3 3 3 2 i 2 3-8 9 0.1891 17 0.1926 18 5.1929 493 .0, 0.9820 3 I 3 5-7 11.0 0.1908 0.1944 18 5.1446 /17fi 0.9816 79.0 5 7.0 9-5 i 2 0.1925 0.1942 0.1962 0.1980 18 5-0970 5.0504 466 0.9813 0.9810 3 9 8 6 7 n. 4 13-3 3 0.1959 18 0.1998 18 5-0045 459 0.9806 4 7 8 15.2 4 0.1977 0.2016 4-9594 0.9803 6 9 17.1 5 6 0.1994 0.20II 17 17 17 0.2035 0.2053 19 18 18 4-9152 4.8716 442 436 428 0.9799 0.9796 4 3 4 5 4 7 0.2028 0.2071 4.8288 0.9792 3 8 0.2045 17 0.2089 18 4.7867 421 0.9789 3 2 9 0.2,062 17 0.2107 18 4-7453 414 0.9785 4 I 12.0 0.2079 0.2126 18 4-7046 0.9781 78.0 i 0.2096 0.2144 4.6646 0.9778 9 18 2 0.2113 I? 0.2162 .6252 394 0-9774 4 8 i 1.8 3 0.2130 i? 0.2180 .5864 388 0.9770 4 7 2 3-6 4 5 6 0.2147 0.2164 0.2181 17 17 17 17 0.2199 0.2217 0.2235 19 18 18 19 .5483 .5107 4737 38i 376 370 364 0.9767 0.9763 0.9759 3 4 4 4 6 5 4 3 4 5 6 5-4 7-2 9-0 10.8 7 0.2198 0.2254 18 4373 li-g 0-9755 3 7 12.6 8 9 0.2215 0.2233 18 0.2272 0.2290 18 4015 .3662 353 0-9751 0.9748 3 2 I 8 q 14.4 16.2 13.0 0.2250 0.2309 18 4.3315 0-9744 77.0 i 0.2267 0.2327 4.2972 0.9740 9 2 0.2284 i? 0.2345 4-2635 337 0.9736 4 8 3 0.2300 0.2364 19 18 4-2303 332 0-9732 4 7 4 0.2317 0.2382 4.1976 0.9728 6 6 0.2334 0.2351 17 o . 2401 0.2419 19 18 19 4.1653 4 1335 323 3i8 313 0-9724 0.9720 4 4 5 4 17 7 0.2368 0.2438 4.1022 0-9715 3 i 1-7 8 0.2385 17 0.2456 18 4-0713 309 0.9711 4 2 2 3-4 9 0.2402 0.2475 19 18 4.0408 305 0.9707 4 I 3 5-1 14.0 0.2419 0.2493 4.0108 2Ofi 0.9703 76.0 4 6.8 i 2 3 4 0.2436 0.2453 0.2470 0.2487 17 17 0.2512 0.2530 0.2549 0.2568 18 19 19 3-9812 3-9520 3.9232 3-8947 292 288 285 0.9699 0.9694 0.9690 0.9686 5 4 4 9 8 7 6 6 7 8 9 8.5 IO.2 ii. 9 13-6 15.3 5 0.2504 17 0.2586 3-866? 0.9681 5 6 0.2521 17 0.2605 19 18 3.8391 276 0.9677 4 4 7 0.2538 0.2623 3.8n8 0.9673 3 8 0.2554 16 0.2642 19 3-7848 270 0.9668 5 2 9 0.2571 i? 0.2661 19 T o 3.7583 265 262 0.9664 4 I 15 0.2588 0.2679 3-7321 0.9659 75 Cosin d. Cotan d. Tan d. Sin d. Deg. (236) TABLE XXVII. (Continued") Deg. Sin d. Tan d. Cotan d. Cosin d. P.P. 18.0 0.2588 0.2679 3-7321 0.9659 78.0 i 0.2605 0.2698 3.7062 0.9655 9 2 0.2622 i? 0.2717 19 3.6806 250 0.9650 5 8 3 0.2639 17 0.2736 19 18 3.6554 252 249 0.9646 4 5 7 19 4 6 0.2656 0.2672 0.2689 16 17 0.2754 0.2773 0.2792 19 19 19 3-6305 3.6059 3.58i6 246 243 240 0.9641 0.9636 0.9632 5 4 5 6 5 4 i 2 3 1-9 3-8 5-7 7 0.2706 0.2811 3.5576 0.9627 3 4 7-6 0.2723 i? 0.2830 19 3-5339 237 0.9622 5 2 5 9-5 9 0.2740 16 0.2849 19 18 3-5105 234 0.9617 5 I 6 II. 4 16.0 0.2756 0.2867 3.4874 228 0.9613 74.0 7 13-3 i 2 0.2773 0.2790 17 0.2886 0.2905 19 3-4646 3-4420 226 0.9608 0.9603 5 9 8 9 15.2 17.1 3 0.2807 16 0.2924 19 19 3.4197 223 220 0.9598 5 7 4 0.2823 0.2943 3-3977 0-9593 6 5 0.2840 i? 0.2962 19 3-3759 0.9588 5 5 18 6 0.2857 i? 0.2981 19 19 3-3544 215 212 0.9583 5 4 i 1.8 7 0.2874 0.3000 3-3332 0.9578 3 2 3-6 8 9 0.2890 0.2907 17 0.3019 0.3038 19 19 3-3122 3.2914 208 0.9573 0.9568 5 5 2 I 3 4 5-4 7-2 17.0 0.2924 if) 0.3057 3-2709 0.9563 73 5 g 9-0 10 8 i 0.2940 0.3076 3-2506 0.9558 9 7 12.6 2 3 0.2957 0.2974 16 0.3096 0.3H5 19 19 3.2305 3.2106 199 196 0.9553 0-9548 5 6 8 7 8 9 14.4 16.2 4 0.2990 0-3134 3.I9IO 0-9542 6 5 0.3007 17 0.3153 19 3.1716 194 0-9537 5 5 6 0.3024 17 0.3172 19 3.1524 192 0-9532 5 4 8 9 0.3040 0.3057 0.3074 17 17 16 0.3191 0.3211 0.3230 20 19 3-1334 3.II46 3-0961 188 185 0.9527 0.9521 0.9516 6 5 3 2 I i 2 1-7 3-4 18.0 0.3090 0.3249 3-0777 182 0.95H fi 72.0 3 4 5-1 6 8 i 2 3 0.3107 0.3123 0.3140 16 17 16 0.3269 0.3288 0.3307 19 19 20 3-0595 3.0415 3-0237 180 178 176 0.9505 0.9500 0.9494 5 6 9 8 7 5 6 7 8-5 10.2 ii. 9 4 0.3156 0-3327 3.0061 0.9489 6 8 13.6 S 0.3173 17 0.3346 19 2.9887 174 0.9483 5 9 15.3 6 0.3190 17 Tfi 0.3365 19 20 2.9714 173 170 0.9478 5 6 4 8 9 0.3206 0.3223 0.3239 17 16 0.3385 0.3404 0.3424 19 20 2.9544 2.9375 2.9208 169 167 166 0.9472 0.9466 o . 9461 6 5 g 3 2 I 16 i 6 19.0 0.3256 16 0.3443 2.9042 164 0.9455 6 71.0 j 2 3-2 i 0.3272 0-3463 2.8878 0.9449 9 3 4-8 2 3 0.3289 0.3305 17 16 i? 0.3482 0.3502 19 2O 20 2.8716 2.8556 160 159 0.9444 0.9438 5 6 6 8 7 4 5 6.4 8.0 4 S 6 0.3322 0.3338 0.3355 16 17 16 0.3522 0.3541 o.356i 19 2O 20 2.8397 2.8239 2.8083 IS8 156 154 0.9432 0.9426 0.9421 6 5 6 6 5 4 b 7 8 9-6 II. 2 12.8 14 4 7 8 o.337i 0.3387 16 0.3581 0.3600 19 2.7929 2.7776 153 0.9415 0.9409 6 3 2 9 0.3404 17 0.3620 2.7625 151 0.9403 g I 20.0 0.3420 0.3640 2-7475 0-9397 70.0 Cosin d. Cotan d. Tan d. Sin d. Deg. (237) TABLE XXVII. (Continued) Deg. Sin d. Tan d. Cotanj d. Cosin d. P.P. 20.0 0.3420 0.3640 2.7475 0.9397 70.0 i 0.3437 0.3659 2.7326 0-9391 9 2 0.3453 0.3679 2.7179 147 0.9385 8 3 0.3469 0.3699 2.7034 145 0-9379 6 7 145 6 22 4 5 6 0.3486 0.3502 0.3518 16 16 17 0.3719 0-3739 0-3759 20 2O 20 2.6889 2.6746 2.6605 143 141 141 0.9373 0.9367 o . 9361 6 6 7 6 5 4 I 2 3 2.2 4-4 6.6 7 8 9 0.3535 0.3551 0.3567 16 16 0.3779 0.3799 0.3819 20 20 2 . 6464 2.6325 2.6l87 139 138 0.9354 0.9348 0.9342 6 6 g 3 2 I 4 5 6 8.8 II. 13-2 21.0 0.3584 16 0.3839 2 . 6O5I 3 0.9336 69.0 7 15-4 i 2 0.3600 0.3616 16 0.3859 0.3879 2O 2.5916 2.5782 134 0.9330 0.9323 7 9 8 8 9 17.6 19.8 3 0.3633 16 0.3899 20 2.5649 133 132 0.9317 6 6 7 4 0.3649 0.3919 2-5517 0.9311 6 5 6 0.3665 0.3681 16 16 16 0.3939 0.3959 20 20 2.5386 2.5257 131 129 128 0.9304 0.9298 7 6 7 5 4 i 21 2.1 7 0.3697 0.3979 2.5129 0.9291 3 2 4-2 8 9 0.3714 0.3730 16 16 0.4000 0.4020 20 2.5002 2.4876 127 126 0.9285 0.9278 6 7 2 I 3 4 6.3 8-4 22.0 0.3746 16 0.4040 2-4751 0.9272 68.0 5 10.5 2 0.3762 0.3778 16 0.4061 0.4081 2O 2.4627 2.4504 123 0.9265 0.9259 6 9 8 7 14-7 Tfi 3 0-3795 17 16 0.4101 21 2.4383 121 121 0.9252 7 7 7 9 18.9 4 5 0.3811 0.3827 16 0.4122 0.4142 20 2 . 4262 2.4142 120 0.9245 0.9239 6 6 5 6 0.3843 16 0.4163 2O 2.4023 119 117 0.9232 7 7 4 7 8 9 0.3859 0.3875 0.3891 16 16 16 0.4183 0.4204 0.4224 21 20 21 2.3906 2.3789 2.3673 117 116 0.9225 0.9219 0.9212 6 7 3 2 I i 2 17 3-4 23.0 0.3907 16 0.4245 2.3559 0.9205 67.0 3 S.I 6 8 i 0.3923 0.4265 2.3445 0.9198 9 5 8 5 2 3 0-3939 0.3955 16 16 0.4286 0.4307 21 20 2.3332 2.3220 112 III 0.9191 0.9184 7 7 6 8 7 6 7 10.3 11.9 4 0.3971 0.4327 2.3109 0.9178 6 8 13.6 5 0.3987 16 0.4348 2.2998 III 0.9171 7 5 9 15-3 6 0.4003 16 0.4369 21 2.2889 IO9 108 0.9164 7 7 4 7 8 0.4019 0.4035 16 0.4390 0.4411 21 2.2781 2.2673 108 0.9157 0.9150 7 3 2 9 0.4051 16 16 0.4431 2.2566 107 106 0.9143 7 g I 16 24.0 0.4067 16 0.4452 21 2.2460 0.9135 66.0 2 3-2 i 0.4083 0.4473 2.2355 0.9128 9 3 4-8 2 0.4099 0.4494 2.2251 0.9121 7 8 4 6.4 3 0.4H5 0.4515 2.2148 103 0.9114 V 7 5 8.0 4 0.4131 , t - 0.4536 2.2045 0.9107 6 6 9-6 5 6 0.4147 0.4163 16 0.4557 0.4578 21 2.1943 2.1842 101 0.9100 0.9092 7 8 5 4 7 8 II. 2 12.8 7 0.4179 0.4599 2.1742 0.9085 3 9 8 0.4195 0.4621 2.1642 0.9078 7 2 9 0.4210 15 16 0.4642 2.1543 99 08 0.9070 I 25 0.4226 0.4663 2.1445 0.9063 65 Cosin d. Cotan d. Tan d. Sin d. Deg. (238) TABLE XXVII. (Continued) Deg. Sin d. Tan d. Cotan d. Cosin d. P.P. 25 0.4226 16 0.4663 2.1445 0.9063 65.0 I 2 3 4 5 6 7 8 9 0.4242 0.4258 0.4274 0.4289 0.4305 0.4321 0-4337 0.4352 0.4368 16 16 IS 16 16 16 IS 16 ffi 0.4684 0.4706 0.4727 0.4748 0.4770 o.479i 0.4813 0.4834 0.4856 22 21 21 22 21 22 21 22 2.1348 2.1251 2.II55 2.1060 2.0965 2.0872 2.0778 2.0686 2.0594 97 96 95 95 93 94 92 92 0.9056 0.9048 0.9041 0.9033 0.9026 0.9018 0.9011 0.9003 0.8996 8 7 8 7 8 7 8 7 9 8 7 6 5 4 3 2 I i 2 3 4 5 6 23 2.3 4.6 6-9 9-2 II. 5 13-8 26.0 0.4384 0.4877 2.0503 91 0.8988 g 64.0 7 16.1 I 2 3 4 5 6 7 8 9 0-4399 0.44IS 0-4431 0.4446 0.4462 0.4478 0-4493 0.4509 0.4524 16 16 IS 16 16 IS 16 IS 16 0.4899 0.4921 0.4942 0.4964 o . 4986 0.5008 0.5029 0.5051 0.5073 22 21 22 22 22 21 22 22 2.0413 2.0323 2.0233 2. 0145 2.0057 1.9970 1.9883 1-9797 I-97II 90 90 90 88 88 87 87 86 86 0.8980 0.8973 0.8965 0.8957 0.8949 0.8942 0.8934 0.8926 0.8918 7 8 8 8 7 8 8 8 g 9 8 7 6 5 4 3 2 I 9 i 2 3 4 18.4 20.7 22 2.2 4-4 6.6 8.8 27.0 0.4540 0.5095 22 1.9626 5 0.8910 63.0 5 II. O i 2 3 4 5 6 7 8 9 0.4555 0-4571 0.4586 0.4602 0.4617 0.4633 0.4648 0.4664 0.4679 16 IS 16 IS 16 IS 16 IS 16 0.5117 O.SI39 0.5161 0.5184 0.5206 0.5228 0.5250 0.5272 0.5295 22 22 23 22 22 22 22 23 I-9S42 1-9458 1-9375 1.9292 1.9210 1.9128 1.9047 1.8967 1.8887 84 84 83 83 82 82 8l 80 80 0.8902 0.8894 0.8886 0.8878 0.8870 0.8862 0.8854 0.8846 0.8838 8 8 8 8 8 8 8 8 9 8 7 6 5 4 3 2 I 7 8 9 I 2 IS.4 17.6 19-8 16 1.6 3-2 28.0 0.4695 0-5317 1.8807 0.8829 g 62.0 3 4 4.8 6 4 I 2 3 4 5 6 7 8 9 0.4710 0.4726 0-4741 0.4756 0.4772 0.4787 0.4802 0.4818 0.4833 16 IS IS 16 15 IS 16 IS '0.5340 0.5362 0.5384 0.5407 0.5430 O.S452 0.5475 0.5498 0.5520 22 22 23 23 22 23 23 22 1.8728 1.8650 1.8572 I.849S 1.8418 1.8341 1.8265 1.8190 1.8115 79 78 78 77 77 77 76 75 75 0.8821 0.8813 0.8805 0.8796 0.8788 0.8780 0.8771 0.8763 0.8755 8 8 9 8 8 9 8 8 9 8 7 6 5 4 3 2 I 5 6 7 8 9 8.0 9-6 II. 2 12.8 14-4 15 29 0.4848 0.5543 i . 8040 75 0.8746 g 61.0 2 3-0 I 2 3 4 5 6 7 8 9 0.4863 0.4879 0.4894 0.4909 0.4924 0.4939 0.4955 0.4970 0.4985 16 IS IS IS IS 16 IS IS 0.5566 0.5589 0.5612 0.5635 0.5658 0.5681 0.5704 0.5727 0.57SO 23 23 23 23 23 23 23 23 1.7966 1.7893 1.7820 1.7747 1.7675 1.7603 1-7532 1.7461 I-739I 74 73 73 73 72 72 71 71 70 0.8738 0.8729 0.8721 0.8712 o . 8704 0.8695 0.8686 0.8678 0.8669 9 8 9 8 9 9 8 9 9 8 7 6 5 4 3 2 I 3 4 5 6 7 8 9 4-5 6.0 7-5 9.0 10.5 12. I3-S 30 0.5000 0.5774 i 7321 70 0.8660 9 60.0 Cosin d. Cotan d. Tan d. Sin d. Deg. (239) TABLE XXVII. (Continued} Deg. Sin d. Tan d. Cotan d. Cosin d. P.P. 30.0 0.5000 0.5774 1.7321 0^8660 60.0 I 0.5015 0.5797 7251 0.8652 9 a 0.5030 15 0.5820 23 .7182 69 0.8643 9 8 3 0.5045 IS 15 0.5844 24 23 7113 69 68 0.8634 9 7 24 4 0.5060 0.5867 7045 0.8625 6 I a. 4 5 0.507S 15 0.5890 23 .6977 68 0.8616 y 5 a 4-8 6 0.5090 15 O.S9I4 24 .6909 68 0.8607 9 4 3 7-2 7 0.5105 0.5938 .6842 0.8599 3 | 12. 8 0.5120 15 0.5961 23 6775 67 0.8590 9 a 6 14-4 9 0.5135 15 0.5985 24 .6709 66 0.8581 y i 7 16.8 31.0 0.5150 0.6009 .6643 66 0.8572 9 59.0 Q 19.2 21.6 i 0.5165 0.6032 6577 0.8563 9 a 3 0.5180 0.5I9S 15 IS 0.6056 0.6080 24 24 .6512 6447 65 65 ftA 0.8554 0.8545 y 9 8 7 I 25 2.5 4 0.5210 0.6104 .6383 0.8536 6 a S.o 5 0.5225 IS 0.6128 24 6319 64 0.8526 5 3 7-5 6 0.5240 IS 15 0.6152 24 24 6255 64 64 0.8517 y 9 4 4 5 10. 13.5 7 0.5255 0.6176 .6191 0.8508 3 6 15.0 8 0.5270 15 0.6200 24 .6128 63 0.8499 y a 7 17.5 9 0.5284 14 0.6224 24 .6066 62 fi-* 0.8490 y i 8 20. 32.0 0.5299 0.6249 .6003 62 0.8480 58.0 i 3 3 0.5314 0.5329 0.5344 IS IS 14 0.6273 0.6297 0.6322 24 25 24 5941 .5880 .5818 61 62 61 0.8471 0.8462 0.8453 9 9 10 9 8 7 I 2 26 2.6 5-2 4 0.5358 0.6346 5757 0.8443 6 3 7-8 5 6 0.5373 0.5388 IS IS 14 0.6371 0-6395 25 24 25 .5697 .5637 60 60 60 0.8434 0.8425 y 9 10 5 4 4 I 10.4 13-0 15-6 7 0.5402 0.6420 .5577 0.8415 3 7 18.2 8 0.5417 IS 0.6445 25 5517 60 0.8406 y a 8 20.8 9 0.5432 IS 0.6469 24 5458 59 0.8396 i y 23.4 33.0 0.5446 0.6494 5399 0.8387 57.0 i 0.5461 0.6519 5340 0.8377 9 15 a 3 0.5476 0.5490 IS 14 IS 0.6544 0.6569 25 25 25 .5282 5224 58 58 58 0.8368 0.8358 y 10 10 8 7 a 3 IS 3-0 4-5 4 0.5505 0.6594 .5166 0.8348 6 4 6.0 5 6 0.5519 0.5534 14 IS 14 0.6619 0.6644 25 25 25 .5108 SCSI 58 57 57 0.8339 0.8329 y 10 9 5 4 s 6 7 7-5 9-o 10. s 7 0.5548 0.6669 4994 0.8320 3 8 za.o 8 0.5563 15 0.6694 25 4938 56 0.8310 a 9 13-5 9 0.5577 14 0.6720 .4882 So -6 0.8300 i 34.0 0.5592 0.6745 26 .4826 rfi 0.8290 56.0 14 i a 3 0.5606 0.5621 0.5635 IS 14 15 0.6771 0.6796 0.6822 25 26 25 .4770 -47IS .4659 55 56 54 0.8281 0.8271 0.8261 10 10 10 9 8 7 i 2 3 4 1.4 2.8 4-2 5.6 4 5 0.5650 0.5664 14 0.6847 0.6873 26 .4605 4550 55 0.8251 0.8241 10 6 5 s 7-o 8.4 o 8 6 0.5678 14 0.6899 26 .4496 54 0.8231 4 7 0.5693 0.6924 4442 0.8221 3 9 12.6 8 0-5707 14 0.6950 .4388 54 0.8211 3 9 0.5721 14 0.6976 26 -4335 53 0.8202 y I 35 0.5736 0.7002 1.4281 0.8192 55 Cosin d. Cotan d. Tan d. Sin d. Deg. (240) TABLE XXVII. (Continued) Deg. Sin d. Tan d. Cotan d. Cosin d. P.P. 35.0 0-5736 0.7002 26 1.4281 0.8192 55.0 i 0.5750 0.7028 1.4229 0.8181 9 a 0.5764 14 0.7054 26 1.4176 53 0.8171 8 3 0-5779 15 0.7080 26 27 1.4124 52 53 0.8161 10 7 27 4 0.5793 0.7107 1.4071 0.8151 6 i 2.7 5 6 0.5807 0.5821 14 14 0-7133 0.7159 26 26 27 1.4019 1.3968 52 51 52 0.8141 0.8131 10 10 5 4 2 3 4 5-4 8.1 10.8 7 0.5835 0.7186 1.3916 0.8121 3 s 13-5 8 0.5850 15 0.7212 26 1.3865 51 0.8111 2 6 16.2 9 0.5864 14 0.7239 27 o(\ 1.3814 51 0.8100 I 7 g 18.9 21 6 36.0 0.5878 14 0.7265 1.3764 0.8090 54.0 9 24.3 i 0.5892 14 0.7292 I.37I3 0.8080 9 a 3 0.5906 0.5920 14 14 0.7319 0.7346 27 27 1.3663 1.3613 So 50 40 0.8070 0.8059 II 8 7 i 28 2.8 4 0.5934 0-7373 1.3564 0.8049 6 2 5-6 S 6 0.5948 0.5962 14 14 0.7400 0.7427 2? 2? i 35U 1.3465 SO 49 AQ 0.8039 0.8028 II 5 4 3 4 5 8.4 II. 2 I4.O 7 0.5976 0.7454 1.3416 0.8018 3 6 16.8 8 0.5990 14 0.7481 2? 1.3367 49 0.8007 2 7 19.6 9 37.0 0.6004 14 14 0.7508 27 28 I.33I9 49 48 0.7997 II I 53.0 8 9 22.4 25.2 0.6018 0.7536 1.3270 0.7986 z 0.6032 14 0.7563 1.3222 0.7976 9 2 3 0.6046 0.6060 14 14 0.7590 0.7618 27 28 28 I.3I7S 1.3127 47 48 48 0.7965 0.79S5 IO II 8 7 i 2 2.9 5-8 4 0.6074 0.7646 1.3079 0.7944 6 3 8.7 S 0.6088 14 0.7673 27 1.3032 47 0-7934 5 4 II. 6 6 0.6101 13 14 0.7701 28 28 1.2985 47 47 0.7923 II 4 i 17-4 7 0.6115 0.7729 1.2938 0.7912 3 7 20.3 8 9 0.6129 0.6143 14 14 0.7757 0.7785 28 28 1.2892 1.2846 40 46 0.7902 0.7891 II 2 I 8 9 3:? 38.0 0.6157 14 0.7813 1.2799 0.7880 52.0 i 0.6170 13 0.7841 1.2753 4 0.7869 9 14 2 0.6184 14 0.7869 28 1.2708 45 0.7859 8 2 8 3 0.6198 14 0.7898 29 1.2662 46 0.7848 7 3 4.2 4 s 0.6211 0.6225 13 14 0.7926 0.7954 28 28 1.2617 1.2572 45 45 0.7837 0.7826 II 6 5 4 5 5.6 7-0 6 0.6239 14 0.7983 29 1.2527 45 0.7815 4 7 9.8 7 0.6252 0.8012 1.2482 0.7804 3 8 II. 2 8 0.6266 14 0.8040 28 1-2437 45 0.7793 2 9 12.6 9 0.6280 14 0.8069 29 1.2393 44 0.7782 I 39.0 0.6293 13 0.8098 29 1-2349 0.7771 51.0 13 I 0.6307 0.8127 1.2305 0.7760 9 i 1:1 2 3 o . 6320 0.6334 13 14 J 3 0.8156 0.8185 29 29 29 1.2261 1.2218 44 43 44 0.7749 0.7738 II II 8 7 3 3.9 r! 4 0.6347 0.8214 I. 2174 0.7727 6 5 6.5 5 6 0.6361 0.6374 14 13 0.8243 0.8273 29 30 1.2131 1.2088 43 43 0.7716 0.7705 II S 4 6 7 8 7-8 9-1 10.4 7 0.6388 0.8302 1.2045 0.7694 3 9 II. 7 8 o . 6401 13 0.8332 30 1.2002 43 0.7683 2 9 0.6414 13 0.8361 29 I . I960 42 0.7672 I 40 0.6428 0.8391 I.I9I8 0.7660 5.0 Cosin d. Cotan d. Tan d. Sin d. Deg (241) TABLE XXVII. (Continued) Deg. Sin d. Tan d. Cotan d. Cosin d. F .P. 40.0 0.6428 0.8391 1.1918 0.7660 50.0 i 2 3 4 5 6 7 8 9 0.6441 0.6455 0.6468 0.6481 0.6494 0.6508 0.6521 0.6534 0.6547 14 13 13 13 14 13 13 13 0.8421 0.8451 0.8481 0.8511 0.8541 0.8571 0.8601 0.8632 0.8662 30 30 30 30 30 30 30 1.1875 1-1833 1.1792 I.I750 1.1708 1.1667 i . 1626 1.1585 i. 1544 42 41 42 42 41 41 0.7649 0.7638 0.7627 0.7615 0.7604 0.7593 0.7581 0.7570 0.7559 ii ii 12 II II 12 II II 9 8 7 6 5 4 3 2 I i 2 3 4 5 6 7 8 31 9.3 12.4 15.5 18.6 21.7 24.8 41.0 0.6561 0.8693 1.1504 0.7547 49.0 9 i 2 3 4 5 6 7 8 9 0.6574 0.6587 0.6600 0.6613 0.6626 0.6639 0.6652 0.6665 0.6678 13 13 13 13 13 13 13 13 0.8724 0.8754 0.8785 0.8816 0.8847 0.8878 0.8910 0.8941 0.8972 30 31 31 32 1.1463 1.1423 1.1383 i -1343 1.1303 i . 1263 i . 1224 1.1184 i. 1145 40 40 40 40 40 39 40 39 0.7536 0.7524 0.7513 0.7501 0.7490 0.7478 0.7466 0.7455 0-7443 12 II 12 II 12 12 II 12 9 8 7 6 5 4 3 2 I i 2 3 4 1 7 8 9 32 3-2 6.4 9.6 12.8 16.0 19.2 22.4 25.6 28.8 42.0 0.6691 0.9004 1.1106 0.7431 48.0 i 2 3 4 5 6 7 8 9 0.6704 0.6717 0.6730 0.6743 0.6756 0.6769 0.6782 0.6794 0.6807 13 13 13 13 13 13 12 13 0.9036 0.9067 0.9099 0.9131 0.9163 0.9195 0.9228 0.9260 0.9293 32 32 32 32 33 32 33 1.1067 i . 1028 1.0990 1.0951 1.0913 1.0875 1.0837 1.0799 i . 0761 39 38 39 38 38 38 38 38 0.7420 0.7408 0.7396 0.7385 0.7373 0.7361 0.7349 0.7337 0.7325 12 12 II 12 12 12 12 12 9 8 7 6 5 4 3 2 I i 2 3 4 5 6 I 9 33 3-3 6.6 9-9 13.2 16.5 19.8 26.4 29-7 43.0 0.6820 0.0325 1.0724 0.7314 12 47.0 i 2 3 4 5 6 7 8 9 0.6833 0.6845 0.6858 0.6871 0.6884 0.6896 0.6909 0.6921 0.6934 12 13 13 13 12 13 12 13 0.9358 0-9391 0.9424 0.9457 0.9490 0.9523 0.9556 0.9590 0.9623 33 33 33 33 33 33 34 33 1.0686 1.0649 i. 0612 1.0575 1.0538 i . 0501 1.0464 1.0428 1.0392 37 37 37 37 37 37 36 36 0.7302 0.7290 0.7278 0.7266 0.7254 0.7242 0.7230 0.7218 0.7206 12 12 12 12 12 12 12 12 9 8 7 6 5 4 3 2 I i 2 3 4 5 6 7 8 9 34 3-4 6.8 IO.2 13.6 17.0 20.4 23.8 27.2 30.6 44.0 0.6947 0.9657 1.0355 0.7193 12 46.0 13 i 2 3 4 6 7 8 9 0.6959 0.6972 0.6984 0.6997 0.7009 0.7022 0.7034 0.7046 0.7059 13 12 13 12 13 12 12 13 0.9691 0.9725 0-9759 0.9793 0.9827 0.9861 0.9896 0.9930 0.9965 34 34 34 34 34 35 34 35 1.0319 i . 0283 1.0247 I. 0212 I.OI76 I.OI4I 1. 01 05 I.OO70 1.0035 36 36 35 36 35 36 35 35 0.7181 0.7169 0.7157 0.7145 0.7133 0.7120 0.7108 0.7096 0.7083 12 12 12 12 13 12 12 13 12 9 8 7 6 S 4 3 2 I i a 3 4 5 6 I 9 1.3 2.6 3-9 5-2 6.5 7-8 9-1 10.4 n. 7 45.0 0.7071 I. 0000 I . 0000 0.7071 45.0 Cosin d. Cotan d. Tan d. Sin d. Deg. (242) TABLE XXVIII. NATURAL VERSED SINES AND EXTERNAL SECANTS Angle Vers Exsec Angle Vers Exsec Angle Vers Exsec Angle Vers Exsec 0.0 .0000 .0000 5.0 .0038 .0038 10.0 .0152 .0154 15.0 0341 0353 .1 .0000 .0000 .1 .0040 .0040 .j .0155 .0157 .1 0345 0358 .2 .0000 .0000 .2 .0041 .0041 .2 .0158 .0161 .2 0350 .0363 3 .0000 .0000 3 .0043 .0043 3 .0161 .0164 3 0354 .0367 .4 .0000 oooo 4 .0044 .0045 .4 .0164 .0167 4 0359 .0372 5 .0000 .0000 5 .0046 .0046 S .0167 .0170 5 .0364 0377 .6 .0001 .0001 .6 .0048 .0048 .6 .0171 .0174 .6 .0368 .0382 .7 .0001 .0001 .7 .0049 .0050 .7 0174 .0177 .7 0373 .0388 .8 .0001 .0001 .8 .0051 .0051 .8 0177 .0180 .8 0378 0393 .9 .0001 .0001 9 0053 .0053 9 .0180 .0184 9 .0383 .0398 1.0 .0002 .0002 6.0 0055 .0055 11.0 .0184 .0187 16.0 .0387 .0403 .1 .0002 .0002 .1 .0057 .0057 .1 .0187 .0191 .1 0392 .0408 .2 .0002 .0002 .2 .0058 .0059 .2 .0190 .0194 .2 0397 0413 3 .0003 .0003 3 .0060 .0061 3 .0194 .0198 3 .0402 .0419 4 .0003 .0003 4 .0062 .0063 4 .0197 .0201 4 .0407 .0424 5 .0003 .0003 5 .0064 .0065 S .0201 .0205 5 .0412 .0429 .6 .0004 .0004 .6 .0066 .0067 .6 .0204 .0209 .6 .0417 0435 .7 .0004 .0004 7 .0068 .0069 7 .0208 .0212 -7 .0422 .0440 .8 .0005 .0005 .8 .0070 .0071 .8 .0211 .O2l6 .8 .0427 .0446 9 .0005 .0005 9 .0072 .0073 9 .0215 .0220 9 .0432 .0451 2.0 .0006 .0006 7.0 .0075 -0075 12.0 .0219 .0223 17.0 0437 .0457 .1 .0007 .0007 ,i .0077 .0077 .1 .0222 .O227 .1 .0442 .0463 .2 .0007 .0007 .2 .0079 .0079 .2 .0226 .0231 .2 .0447 .0468 3 .0008 .0008 3 .0081 .0082 3 .0230 0235 3 .0452 .0474 4 .0009 .0009 4 .0083 .0084 4 .0233 .0239 4 .0458 .0480 .5 .0010 .0010 5 .0086 .0086 5 .0237 .0243 5 .0463 .0485 .6 .0010 .0010 .6 .0088 .0089 .6 .0241 .0247 .6 .0468 .0491 .7 .OOII .0011 7 .0090 .0091 7 .0245 .0251 .7 0473 0497 .8 .0012 .0012 8 .0093 .0093 .8 .0249 .0255 .8 0479 .0503 9 .0013 .0013 9 .0095 .0096 9 .0252 .0259 9 .0484 .0509 30 .0014 .0014 8.0 .0097 .0098 13.0 .0256 .0263 18.0 .0489 .0515 .1 .0015 .0015 .1 .0100 .OIOI .1 .0260 .0267 .1 0495 .0521 .2 .0016 .0016 .2 .0102 .0103 .2 .0264 .0271 .2 .0500 0527 3 .0017 .0017 3 .OIO5 .0106 3 .0268 .0276 3 .0506 .0533 4 .0018 .0018 .4 .010? .0108 4 .0272 .0280 4 .0511 .0539 5 .0019 .0019 5 .OIIO .0111 5 .0276 .0284 5 .051? .0545 .6 .0020 .0020 .6 .OI 12 .0114 .6 .0280 .0288 .6 .0522 0551 .7 .0021 .0021 7 .OII5 .0116 7 .0285 .0293 .7 .0528 .0557 .8 .0022 .O022 .8 .0118 .0119 .8 .0289 .0297 .8 0534 .0564 9 .0023 .O023 9 .OI20 .0122 9 .0293 .0302 9 0539 0570 40 .0024 .OO24 9.0 .0123 .0125 14.0 .0297 .0306 19.0 .0545 .0576 .1 .0026 .0026 .1 .0126 .0127 .1 .0301 .0311 .1 0551 .0583 .2 .0027 .OO27 .2 .0129 .0130 .2 .0306 .0315 .2 .0556 0589 3 .0028 .0028 3 .0131 -0133 3 .0310 .0320 3 .0562 0595 .4 .0030 .0030 .4 0134 .0136 4 0314 .0324 4 .0568 .0602 5 .0031 .0031 .5 0137 0139 5 .0319 0329 5 .0574 .0608 .6 .0032 .0032 .6 .0140 .0142 .6 0323 0334 .6 .0579 .0615 .7 .0034 .0034 .7 .0143 .0145 7 .0327 .0338 .7 .0585 .0622 .8 .0035 .0035 .8 .0146 .0148 .8 03,32 0343 .8 .0591 .0628 9 .0037 .0037 9 .0149 .0151 9 .0336 .0348 .9 .0597 .0635 5.0 .0038 .0038 10.0 .0152 .0154 15.0 .0341 -0353 20.0 .0603 .0642 (243) TABLE XXVIII. NATURAL VERSED SINES AND EXTERNAL SECANTS Angle Vers Exsec Angle Vers Exsec Angle Vers Exsec Angle Vers Exsec 20.0 .0603 .0642 25.0 0937 .1034 30.0 .1340 1547 35.0 .1808 .2208 .1 .0609 .0649 .1 .0944 -1043 .1 .1348 -1559 .1 .1819 .2223 .2 .0615 0655 .2 .0952 .1052 .2 1357 .1570 .2 .1829 .2238 3 .0621 .0662 3 0959 .1061 3 .1366 .1582 3 .1839 2253 4 .0627 .0669 4 .0967 .1070 4 .1375 1594 .4 .1849 .2268 5 -0633 .0676 5 0974 .1079 5 -1384 .1606 5 .1859 .2283 .6 .0639 .0683 .6 .0982 .1089 .6 1393 .1618 .6 .1869 .2299 .7 .0646 .0690 7 .0989 .1098 .7 .1401 .1630 7 .1879 .2314 .8 .0651 .0697 .8 .0997 .1107 .8 .1410 .1642 .8 .1889 .2329 -9 .0658 0704 9 .1004 .1117 9 .1419 .1654 9 .1900 2345 21.0 .0664 .0711 26.0 .1012 .1126 31.0 .1428 .1666 36.0 .1910 .2361 .1 .0670 .0719 .1 .IO2O .1135 .1 1437 .1679 .1 .1920 .2376 .2 .0677 .0726 .2 .1027 .1145 .2 .1446 .1691 .2 .1930 .2392 3 .0683 .0733 3 1035 .1155 3 1455 .1703 3 .1941 .2408 4 .0689 .0740 4 1043 .1164 4 .1464 .1716 4 .1951 .2424 5 .0696 .0748 5 .IO5I .1174 5 .1474 .1728 5 .1961 .2440 .6 .0702 0755 .6 .1058 .1184 .6 .1483 .1741 .6 .1972 .2456 .7 .0709 .0763 .7 .1066 .1194 7 .1492 1753 .7 .1982 .2472 .8 .0715 .0770 .8 1074 .1203 .8 .1501 1766 .8 1993 .2489 9 .0722 .0778 .9 .1082 .1213 9 .1510 .1779 9 .2003 .2505 22.0 .0728 .0785 27.0 .1090 .1223 32.0 .1520 .1792 37.0 .2014 .2521 .1 0735 0793 .1 .1098 1233 .1 .1529 .1805 .1 .2024 .2538 .2 .0741 .0801 .2 .1106 1243 .2 .1538 .1818 .2 .2035 2554 3 .0748 .0808 3 .1114 .1253 3 .1547 .1831 3 2045 .2571 .4 .0755 .0816 4 .1122 .1264 4 1557 .1844 4 .2056 .2588 5 .0761 .0824 5 .1130 1274 5 .1566 .1857 5 .2066 .2605 .6 .0768 .0832 .6 .1138 .1284 .6 .1575 .1870 .6 .2077 .2622 .7 0775 .0840 .7 .1146 .1294 7 .1585 .1883 .7 .2088 .2639 .8 .0781 .0848 .8 .1154 .1305 .8 1594 .1897 .8 .2098 .2656 9 .0788 .0856 9 .1162 .1315 .9 .1604 .1910 9 .2109 .2673 23.0 0795 .0864 28.0 .1171 .1326 33.0 .1613 .1924 38.0 .2120 .2690 .1 .0802 .0872 .1 .1179 .1336 .1 .1623 .1937 .1 .2131 .2708 .2 .0809 .0880 .2 .1187 1347 .2 .1632 .1951 .2 .2141 .2725 3 .0816 .0888 3 .1195 1357 3 .1642 .1964 3 .2152 2742 .4 .0822 .0896 4 .1204 .1368 4 .1652 .1978 4 .2163 .2760 5 .0829 .0904 5 .1212 1379 5 .1661 .1992 5 2174 .2778 .6 .0836 .0913 .6 .I22O 1390 .6 .1671 .2006 .6 .2185 .2796 .7 .0843 .0921 .7 .1229 .1401 .7 .1680 .2020 7 .2196 .2813 .8 .0850 .0929 .8 1237 .1412 .8 .1690 2034 .8 .2207 .2831 9 .0857 .0938 9 1245 .1423 .9 .1700 .2048 9 .2218 .2849 24.0 .0865 .0946 29.0 .1254 .1434 34.0 .1710 .2062 39.0 .2229 .2868 .1 .0872 .0955 .1 .1262 .1445 .1 .1719 .2076 .1 .2240 .2886 .2 .0879 0963 .2 .1271 -1456 .2 .1729 .2091 .2 .2251 .2904 3 .0886 .0972 3 .1279 .1467 3 1739 .2IO5 3 .2262 .2923 .4 .0893 .0981 4 .1288 .1478 4 1749 .2120 4 2273 .2941 5 .0900 .0989 5 .1296 .1490 5 1759 2134 5 .2284 .2960 .6 .0908 .0998 .6 .1305 .1501 .6 .1769 2149 .6 .2295 2978 .7 .0915 .1007 .7 -I3I4 .1512 .7 .1779 .2163 7 .2306 .2997 .8 .0922 .1016 .8 .1322 -1524 .8 .1789 .2178 .8 .2317 .3016 9 .0930 .1025 .9 .1331 -1535 9 .1798 .2193 9 .2328 3035 25 0937 1034 SO -1340 .1547 35.0 .1808 .2208 40 2340 .3054 (244) TABLE XXVIII. NATURAL VERSED SINES AND EXTERNAL SECANTS Angle Vers Exsec Angle Vers Exsec Angle Vers Exsec Angle Vers Exsec 40.0 .1 .2 3 4 5 .6 .7 .8 9 41.0 .1 .2 3 4 !e .7 .8 .9 42.0 .1 .2 3 4 !e .7 .8 9 43.0 .1 .2 3 4 !e .7 .8 9 44.0 .1 .2 3 4 5 .6 .7 .8 9 46 .2340 3054 45 .1 .2 3 4 5 .6 7 .8 9 46.0 .1 .2 3 4 5 .6 '.8 9 47.0 .1 .2 3 4 5 .6 .7 .8 9 48.0 .1 .2 3 4 .6 .7 .8 9 49.0 .1 .2 3 4 5 .6 .7 .8 9 50.0 .2929 .4142 50.0 .1 .2 3 4 5 .6 7 .8 9 51.0 .1 .2 3 4 5 .6 7 .8 9 52.0 .1 .2 3 4 5 .6 .7 .8 9 53.0 .1 .2 3 4 .6 .7 .8 9 54.0 .1 .2 3 4 '.6 .7 .8 9 55.0 3572 5557 55.0 .1 .2 3 4 5 .6 7 .8 9 56.0 .1 .2 3 4 5 .6 7 .8 9 57.0 .1 .2 3 4 5 .6 7 .8 9 58.0 .1 .2 3 4 .6 .7 .8 9 59.0 .1 .2 3 4 5 .6 .7 .8 9 60.0 .4264 7434 2351 .2362 2373 2385 .2396 .2407 .2419 .2430 .2441 3073 3093 .3102 3131 3151 3171 3190 .3210 3230 .2941 2954 .2966 .2978 .2991 3003 .3016 .3028 3041 .4167 .4192 .4217 .4242 .4267 .4293 4318 4344 4370 3586 3599 .3612 .3626 3639 .3653 .3666 .3680 3693 5590 .5622 .5655 .5688 5721 5755 5788 .5822 -5856 .4279 4293 4307 4322 .4336 4350 .4365 4379 4394 7478 .7522 7566 .7610 .7655 .7700 7745 7791 .7837 .2453 3251 3053 4396 3707 .5890 .4408 .7883 .2464 .2476 .2487 .2499 .2510 .2522 2534 .2545 .2557 .3270 3291 3311 3331 3352 3373 3393 -34I4 3435 .3066 3079 3091 .3104 .3116 3129 .3142 3155 3167 .4422 4448 4474 4501 4527 4554 4S8i .4608 4635 3720 3734 3748 .376l 3775 .3789 .3802 .3816 3830 5925 5959 5994 .6029 .6064 .6099 .6i35 .6171 .6207 4423 4437 4452 .4466 .4481 4495 4510 4524 4539 .7929 .7976 .8023 .8070 .8118 .8166 .8214 .8263 .8312 .2569 -3456 .3180 .4663 .3843 .6243 4554 .8361 .2580 .2592 .2604 .2615 .2627 2639 .2651 .2663 .2675 3478 3499 3520 3542 .3563 .3585 3607 .3629 3651 .3193 .3206 .3218 3231 .3244 3257 .3270 3283 -3296 .4690 .4718 .4746 4774 .4802 .4830 .4859 .4887 4916 .3857 3871 .3885 .3899 .3912 .3926 3940 3954 -3968 .6279 .6316 .6353 6390 .6427 .6464 .6502 .6540 -6578 4568 4583 4598 .4612 .4627 .4642 4656 .4671 .4686 .8410 .8460 .8510 .8561 .8612 .8663 .8714 .8766 .8818 .2686 .3673 .3309 4945 .3982 .6616 4701 .8871 .2698 .2710 .2722 2734 .2746 .2758 .2770 .2782 2794 .3696 3718 3741 .3763 3786 3809 3832 .3855 .3878 3322 .3335 .3348 .336l 3374 .3387 3400 3413 .3426 4974 5003 .5032 .5062 .5092 .5121 .5151 .5182 .5212 3996 .4010 .4024 .4038 4052 .4066 .4080 .4094 .4108 6655 .6694 .6733 .6772 .6812 .6852 .6892 .6942 .6972 47i6 4730 4745 .4760 4775 4790 4805 .4820 .4835 .8924 8977 -903I .9084 9139 .9194 .9249 9304 .936o .2807 .3902 3439 5243 .4122 7013 4850 .9416 .2819 .2831 2843 .2855 .2868 .2880 .2892 .2904 .2917 .3925 3949 3972 3996 .4020 .4044 .4069 .4093 .4118 3453 3466 3479 3492 .3506 .3519 3532 3545 3559 5273 5304 5335 5366 5398 .5429 .5461 5493 5525 .4136 .4150 .4165 4179 .4193 .4207 .4221 .4236 .4250 7054 .7095 7137 .7179 .7221 .7263 7305 7348 7391 .4865 .4880 .4895 .4910 4925 .4940 .4955 4970 .4985 9473 9530 .9587 .9645 9703 .9762 .9821 .9880 -9940 .2929 .4142 3572 5557 .4264 7434 .5000 I.OOOO (245) TABLE XXVIII. NATURAL VERSED SINES AND EXTERNAL SECANTS Angle Vers Exsec Angle Vers Exsec Angle Vers Exsec 60.0 .1 .2 3 4 5 .6 7 .8 9 61.0 .1 .2 3 4 5 .6 .7 .8 9 62.0 .1 .2 3 4 !e .7 .8 9 63.0 .1 .2 3 4 5 .6 7 .8 9 64.0 .1 .2 3 4 5 .6 .7 .8 9 65.0 .5000 I. 0000 65.0 .1 .2 3 4 '.6 7 .8 9 66.0 .1 .2 3 4 5 .6 .7 .8 9 67.0 .1 .2 3 4 '.6 .7 .8 9 68.0 .1 .2 3 4 '.6 .7 .8 9 69.0 .1 .2 3 4 5 .6 .7 .8 9 70.0 5774 1.3662 70.0 .1 .2 3 4 5 .6 7 .8 9 71.0 .1 .2 .3 4 5 .6 .7 .8 9 72.0 .1 .2 3 4 5 .6 .7 .8 9 73.0 .1 .2 3 4 5 .6 7 .8 9 74.0 .1 .2 3 4 5 .6 .7 .8 9 75 .6580 1.9238 5015 .5030 .5045 .5061 5076 .5091 .5106 .5121 5137 .0061 .0122 .0183 .0245 1.0308 .0371 .0434 .0498 .0562 5790 5805 .5821 .583? .5853 .5869 .5885 5901 .5917 3751 .3841 3931 .4022 1.4114 .4207 4300 4395 .449 .6596 .6613 .6629 .6645 .6662 .6678 .6695 .6711 .6728 9379 9521 .9665 .9811 1.9957 2.0106 .0256 .0406 .0561 .5152 1.0627 5933 4586 .6744 2.0716 5167 .5182 5198 .5213 .5228 .5244 .5259 5274 .5290 .0692 .0757 .0824 .0890 1.0957 .1025 .1093 .1162 .1231 5949 .5965 .5981 5997 .6013 .6029 .6045 .6061 .6077 .4683 .4780 .4879 .4978 1.5078 .5180 .5282 .5384 .5488 .6761 6777 .6794 .6810 .6827 .6844 .6860 .6877 .6893 .0872 .1030 .1190 .1352 2.1515 .1681 .1848 .2017 .2188 .5305 .1301 .6093 1-5593 .6910 2 . 2361 .5321 .5336 5352 .5367 .5383 5398 .5414 5429 5445 .1371 .1441 .1513 .1584 1.1657 .1730 .1803 .1877 .1952 .6109 .6125 .6141 .6157 .6173 .6189 .6205 .6222 .6238 .5699 5805 5913 .6022 1.6131 .6242 .6354 .6466 .6580 .6926 .6943 .6960 .6976 .6993 .7010 .7026 .7043 .7060 2535 .2712 .2891 .3072 2.3255 3440 .3628 3817 .4009 .5460 i . 2027 .6254 1-6695 .7076 2.4203 .5476 .5491 5507 5522 5538 5554 .5569 .5585 .5601 .2103 .2179 .2256 2333 1.2412 .2490 .2570 .2650 .2730 .6270 .6286 .6303 .6319 .6335 .6351 .6367 .6384 .6400 .6811 .6927 .7046 .7165 1.7285 .7407 .7529 .7653 .7778 .7093 .7110 .7126 7143 .7160 7177 .7193 .7210 .7227 4399 4598 4789 .5003 2.5209 5418 .5629 .5843 .6060 .5616 1.2812 .6416 .7904 .7244 2.6280 .5632 .5648 .5663 .5679 .5695 5711 .5726 -5742 5758 .2894 .2976 .3060 .3144 1.3228 3314 .3400 .3486 3574 .6433 .6449 .6465 .6482 .6498 6514 .6531 .6547 .6563 .8032 .8161 .8291 .8422 1.8554 .8688 .8824 .8960 .9099 .7260 .7277 .7294 .7311 .7328 7344 .7361 -7378 7395 .6502 .6728 .6955 .7186 2.7420 .7657 .7897 .8140 .8387 5774 1.3662 .6580 1.9238 7412 2.8637 (246) TABLE XXVIII. NATURAL VERSED SINES AND EXTERNAL SECANTS Angle Vers Exsec Angle Vers Exsec Angle Vers Exsec 75.0 .1 .2 3 .4 5 .6 .7 .8 9 76.0 .1 .2 3 4 .5 .6 .7 .8 9 77.0 .1 .2 3 .4 5 .6 .7 .8 -9 78.0 .1 .2 3 4 .5 .6 .7 .8 9 79.0 .1 .2 .3 4 5 .6 .7 .8 9 80 .7412 2.8637 80.0 .1 .2 3 4 !e j 9 81.0 .1 .2 3 4 5 .6 .7 .8 9 82.0 .1 .2 3 M 5 .6 .8 9 83.0 .1 .2 3 4 5 .6 7 .8 9 84.0 .1 .2 3 4 .6 .8 9 85.0 .8264 4-7588 85.0 .1 .2 3 4 5 .6 7 .8 9 86.0 .1 .2 3 4 5 .6 .7 .8 9 87.0 .1 .2 3 4 .5 .6 .7 .8 9 88.0 .1 .2 3 4 !e .7 .8 9 89.0 .1 .2 3 4 3 .6 7 .8 9 90.0 .9128 10.4737 .7429 .7446 .7462 7479 .7496 .7513 7530 7547 .7564 .8890 .9147 .9408 .9672 2.9939 3.0211 .0486 .0765 .1048 .8281 .8298 .8315 .8332 .8350 .8367 .8384 .8401 .8418 .8164 8751 9351 .9963 5.0589 .1227 .1880 .2546 .3228 .9146 9i63 .9181 .9198 9215 .9233 9250 .9268 .9285 7073 .9506 11.2043 .4690 H.7455 12.0346 3371 .6541 9865 .7581 3.1336 .8436 5.3925 9302 13.3356 .7598 .76iS .7632 .7649 .7666 .7683 .7700 .7716 7733 .1627 .1923 .2223 .2527 3.2837 .3150 .3469 3792 .4121 .8453 .8470 .8487 .8505 .8522 .8539 .8556 .8574 .8591 .4637 .5366 .6111 .6874 5.7655 .8454 .9273 6.0II2 .0972 .9320 9337 9355 9372 9390 .9407 .9424 9442 9459 .7026 14.0889 .496i .9260 15.3804 .8616 16.3720 .9142 17.4915 7750 3-4454 .8608 6.1853 9477 18.1073 .7767 .7785 .7802 .7819 .7836 .7853 .7870 .7887 .7904 4793 .5137 .5486 .5841 3.6202 .6569 .6942 7320 .7706 .8626 .8643 .8660 .8677 .8695 .8712 .8729 8747 .8764 2757 .3684 .4635 .5611 6.6613 .7642 .8700 .9787 7-0905 9494 9512 .9529 .9546 .9564 .9581 9599 .9616 .9634 .7656 19.4709 20.2285 21.0444 21 . 9256 22.8802 23.9179 25.0499 26.2898 7921 3-8097 .8781 7-2055 .9651 27.6537 .7938 7955 7972 .7989 .8006 .8023 .8041 .8058 .8075 .8496 .8901 9313 9732 4-0159 0593 .1034 .1484 .1942 .8799 .8816 .8833 .8851 .8868 .8885 .8903 .8920 8937 .3238 4457 5711 .7004 7.8337 97ii 8.1129 2593 .4105 .9668 .9686 .9703 .9721 .9738 9756 9773 9791 .9808 29 . 1612 30.8362 32.7083 34.8145 37.2016 39.9296 43.0775 46.7500 51.0903 .8092 4.2408 .8955 8.5668 .9825 56.2987 .8109 .8126 .8143 .8160 .8178 .8195 .8212 .8229 .8246 .2883 .3367 .3860 .4362 4-4874 .5396 .5928 .6470 .7023 .8972 .8989 .9007 .9024 .9042 9059 .9076 .9094 .9111 7283 .8955 9.0685 .2477 9-4334 .6261 .8260 10.0336 2493 .9843 .9860 .9878 9895 9913 9930 .9948 .9965 .9983 62.6646 70.6221 80.8532 94-4947 H3.5930 142.2406 189.9868 285.4795 571.9581 .8264 4-7588 .9128 10.4737 I. 0000 00 (247) 248 RAILROAD FIELD MANUAL TABLE XXIX Seconds in decimals of a degree Sec. Degree Sec. Degree Sec. Degree Sec. Degree Sec. Degree Sec. Degree i 0.00028 II 0.00306 21 0.00584 3i 0.00862 4i 0.01139 5i 0.01417 2 0.00056 12 0.00334 22 0.00612 32 0.00890 42 0.01167 52 0.01445 i 0.00083 13 0.00361 23 0.00639 33 0.00917 43 0.01195 53 0.01473 4 O.OOIII 14 0.00389 24 0.00667 34 0.00945 44 O.OI222 54 0.01500 5 0.00139 IS 0.00417 25 0.00695 35 0.00973 45 0.01250 55 0.01528 6 0.00167 16 0.00445 26 o . 00723 36 O.OIOOO 46 0.01278 56 0.01556 7 0.00195 17 0.00473 27 0.00751 37 0.01028 47 0.01306 57 0.01584 8 0.00222 18 0.00500 28 0.00778 38 0.01056 48 0-01334 58 0.01612 9 o . 00250 19 0.00528 29 0.00806 39 0.01083 49 0.01361 59 0.01639 10 O.OO278 20 0.00556 30 0.00834 40 O.OIIII So 0.01389 60 0.01667 TABLE XXX Minutes in decimals of a degree Min. Degree Min. Degree Min. Degree Min. Degree Min. Degree Min. Degree i 0.01667 II 0.18333 21 0.35000 31 0.51667 4i 0.68333 5i 0.85000 2 0.03333 12 0.2OOOO 22 0.36667 32 0.53333 42 0.70000 52 0.86667 3 0.05000 13 .21667 23 0.38333 33 0.55000 43 0.71667 53 0.88333 4 0.06667 14 23333 24 0.40000 34 0.56667 44 0.73333 54 0.90000 5 0.08333 IS .25000 25 0.41667 35 0.58333 45 0.75000 55 0.91667 6 O.IOOOO 16 .26667 26 0.43333 36 0.60000 46 0.76667 56 0.93333 7 0.11667 17 .28333 27 0.45000 37 0.61667 47 0.78333 57 0.95000 8 0.13333 18 .30000 28 0.46667 38 0.63333 48 0.80000 58 0.96667 9 0.15000 19 .31667 29 0.48333 39 0.65000 49 0.81667 59 0.98333 10 0.16667 20 33333 30 0.50000 40 0.66667 50 0.83333 60 I.OOOOC From Roberts' " Track Formulae and Tables." CHAPTER IV LOCATION THEORIES AND TABLES FOR problems relating to improvements of existing lines exact data and volumes of discussion of methods will be available. The methods here suggested are for use in locating new lines for which precise data respecting motive power, business, or expense are indeterminate. Most of the problems will relate to saving in operation of freight trains. A single locomotive may be assumed, since in the solution of problems the results will be practically the same relatively for any probable differ- ences in locomotives. It must be remembered that not all of the trains will be affected by probable changes in ruling grades; the engineer must use his judgment in determining what trains will be affected. All trains must be considered in estimating the cost of distance, rise and fall, and curvature. The formulas given in what follows are based on the full discussion of the subjects in the author's "Elements of Railroad Engineering." Tractive Effort. For approximate computations the tractive effort of a locomotive may be assumed to be ~ 146 # Tb = - 43,100 24,350 16,950 35 09 41,500 23,850 16,700 4-0 1 40,000 23,350 16,500 45 g 38,650 22,850 16,250 5-0 Q 37,350 22,4OO 16,000 55 8 36,150 21,950 15,800 6.0 v> 35,000 21,550 15,550 6.5 33,950 21,150 15,350 7.0 32,950 20,750 15,150 7.5 32,000 20,350 14,950 8.0 31,100 20,000 14,750 8.5 30,300 19,650 14,550 9.0 29,500 19,300 14,350 9-5 28,750 19,000 14,200 Consolidation locomotive. Weight on drivers: 210,000 pounds. Weight of engine and tender: 400,000 pounds. Cylinders: 28" diam. by 32" stroke. Heating surface: 3837 sq. ft. Diameter drive wheels: 63". Boiler working pressure: 163 pounds. For this locomotive: Te = 55,174 pounds (say 55,000 pounds); To, = 49,412 pounds (say 50,000 pounds); Tt = 50,000 pounds at 1 1. 2 mi./hr., and as To limits the effort, this will be taken as constant and equal to 50,000 pounds at speeds below IX mi./hr. LOCATION THEORIES AND TABLES 259 TABLE XXXIV NET LOADS IN TONS OF 2000 POUNDS BEHIND THE TENDER OF THE CONSOLIDATION LOCOMOTIVE OF TABLE XXXIII, ON GIVEN GRADES AT GIVEN SPEEDS *\" Speed Grade" Under II mi 15 30 35 30 35 40 o.oo 11430 79i8 5400 3873 2908 2224 I?l8 .05 9234 6468 446? 3246 2464 1905 1487 .10 7736 5458 3800 2787 2131 1660 1305 .15 6650 4714 3300 2435 1872 1467 "59 .30 5825 4142 2911 2158 1665 1310 1039 .35 5177 3690 2600 1934 1495 "79 938 30 46SS 3323 2346 1748 1354 1070 853 35 4225 3019 2134 1592 1235 977 779 40 3866 2764 1954 1459 1132 896 715 45 3S6o 2546 1800 1345 1043 826 659 So 3297 2358 1667 1245 966 764 609 55 3068 2194 1550 1158 897 709 565 .60 2868 2050 1447 1080 836 660 525 .65 2691 1922 1356 IOII 782 616 490 .70 2533 1808 1274 949 733 577 457 75 2391 1706 1200 893 688 541 428 .80 2263 1613 1 133 842 648 508 401 .85 2148 1529 1073 795 6" 478 376 -90 2042 1453 1017 753 577 450 353 95 1946 1382 967 714 546 425 332 1. 00 1858 1318 920 678 517 401 3X3 05 1776 1259 877 645 491 380 295 .10 1702 1204 837 6i5 466 359 278 IS 1632 1153 800 586 443 341 262 .30 1567 1106 765 559 422 323 247 .35 1507 1062 733 534 402 306 233 30 1450 1020 703 5ii 383 291 220 35 1398 982 675 489 365 276 208 .40 1348 946 649 469 348 262 197 45 1302 911 624 449 333 249 186 50 1258 879 600 431 3i8 237 175 55 1217 849 578 414 304 226 1 66 .60 1178 820 557 397 291 214 156 .65 1141 793 537 382 278 204 147 .70 1106 768 5i8 367 266 194 139 75 1072 743 500 353 255 185 131 .80 1041 720 483 340 244 176 123 .85 IOII 698 467 327 234 167 "6 90 982 677 451 315 224 159 109 95 955 657 436 303 214 151 102 3.00 929 637 422 292 205 143 96 The average car weight for this table Is 45 tons. TABLE XXXV. RELATIVE NUMBER OF TRAINS TO DO THE SAME BUSINESS TONNAGE ON DIFFERENT GRADES, FOR DIFFERENT SPEEDS. COMPUTED FROM TABLE XXXIV ^-v^Speed Grade^^ Under ii mi. 15 20 25 30 35 40 0.00 0.34 0.49 0.72 .00 1-33 74 2.25 0.05 0.42 0.60 .87 19 1-57 03 2.60 O.IO 0.50 0.71 .02 39 1.81 -33 2.96 0.15 0.58 0.82 17 59 2.07 .64 3 34 O.2O 0.66 0.93 33 79 2.32 95 3.72 0.25 0.75 1.05 49 .00 2-59 3.29 4.12 0.30 0.83 1.16 -65 .21 2.86 3-6i 4-53 0-35 0.92 1.28 .81 43 3.13 3.96 4-96 0.40 .00 1.40 98 65 3-42 4-31 5-41 0-45 .09 1.52 -15 .87 3-71 4.68 5.87 0.50 -17 1.64 32 3- II 4.00 5.o6 6.35 0.55 .26 1.76 49 3-34 4-31 5-45 6.84 0.60 35 1.89 2.67 3-58 4.62 5.86 7.36 0.65 44 2.01 2.85 3.82 4-94 6.28 7-89 0.70 53 2.14 3 03 4.07 5.27 6.70 8.46 0.75 .62 2.27 3-22 4-33 5.62 7.15 9 03 0.80 71 2.40 3-41 4-59 5-97 7-61 9.64 0.85 .80 2.53 3.6o 4.86 6.33 8.09 10.28 0.90 89 2.66 3.8o S.I3 6.70 8.59 10.95 95 99 2.80 4.00 5-41 7-o8 9.10 11.64 .00 2.08 2-93 4.20 5-70 7.48 9.64 12.35 05 2.18 3.07 4-41 5-99 7-- 87 10.17 13.10 .10 2.27 3-21 4.62 6.29 8.30 10.77 13.91 IS 37 3-35 4-83 6.60 8-73 11-34 14.76 .20 47 3-50 5-05 6.92 9.16 11.97 15.65 25 57 3-64 5-27 7.24 9.62 12.63 16-59 30 67 3-79 S-So 7-57 10.09 13-29 17-57 35 77 3-94 5-73 7-91 10.59 14.01 18.59 40 2.87 4-09 5.96 8.24 n. ii 14.76 19.63 45 2.97 4-24 6.20 8.61 ii. 61 15-53 20.79 50 3-07 4-40 6.44 8.97 12. 16 16.31 22.09 -55 3.18 4-55 6.69 9-34 12.72 17.11 23.29 .60 3-28 4-71 6-94 9-74 13-29 18.07 24-78 .65 3-39 4-87 7.20 IO.I2 13.91 18-95 26.30 70 3-SO 5-03 7.46. 10.53 14-53 19-93 27.82 75 3-6l 5-20 7-73 10-95 15.16 20.90 29.51 .80 3-71 5.37 8.00 11-37 15.84 21.97 31-43 .85 3-82 5-54 8.28 11.82 16.52 23.15 33-33 90 3-94 S-7I 8.57 12.27 17.26 24-32 35-47 95 4-05 5-88 8.87 12.76 18.07 25.60 37.90 .00 4.16 6.07 9.16 13-24 18.86 27.04 40.27 TABLE XXXVI. PUSHER GRADES The tabular quantities are pusher grades for the through grades indicated at the side and top of table. Computed for the Consolida- tion Engine of Table XXXIII. Through grade 0.0 O.I 0.2 0-3 0.4 o.S 0.6 0.7 0.8 0.9 0.0 I.O 2.O 0.187 !.88 9 3-359 0-369 2.045 0.548 2.2OO 0.725 2-353 0.899 2.502 1.070 2.650 1.238 2.796 1.404 2-939 1-569 3.082 1.730 3-222 (260) TABLE XXXVII. GRADES AND GRADE ANGLES Feet Feet Incli- Feet Feet Incli- Feet Feet Incli- Feet Feet Incli- per sta- tion per mile nation, itfeg. per sta- tion per mile nation, deg. per sta- tion per mile nation, deg. per sta- tion per mile nation, deg. 00 .000 .000 50 26.400 .286 .00 52.800 573 So 79.200 "~859~ OI .528 .006 Si 26.928 .292 .01 53.328 579 Si 79.728 .865 02 1.056 .Oil 52 27.456 .298 .02 53.856 .584 52 80.256 .871 03 1.584 .017 53 27.984 304 03 54.384 590 53 80.784 .877 04 2,112 .023 54 28.512 .309 04 54-912 -596 -54 81.312 .882 05 2640 .029 55 29.040 .315 05 55-440 .601 -55 81.840 .888 06 3:168 034 56 29.568 .321 .06 55-968 .607 -56 82.368 .894 07 3696 .040 57 30.096 327 .07 56.496 .613 57 82.896 -899 08 4.224 .046 58 30.624 332 .08 57-024 .619 -58 83.424 90S 09 4752 .052 59 31-152 .338 .09 57-552 .624 -59 83.952 .9" 10 5,280 .057 60 31.680 344 .10 58.080 .630 .60 84.480 .917 II 5,8o8 .063 61 32.208 349 .11 58.608 .636 .61 85.008 .922 12 6.336 .069 62 32.736 355 .12 59-136 .642 .62 85.536 .928 13 6864 .074 63 33.264 .361 13 59-664 .648 -63 86.064 934 14 7392 .080 64 33-792 .367 14 60.192 .653 -64 86.592 939 IS 7.920 .086 65 34-320 372 IS 60.720 .659 -65 87.120 945 16 8.448 .092 66 34.848 378 .16 61.248 .665 .66 87.648 951 17 8.976 .098 67 35.376 .384 .17 61.776 .670 .67 88.176 957 18 9-504 .103 68 35.904 390 .18 62.304 .676 .68 88.704 .962 19 10.032 .109 69 36.432 395 .19 62.832 .682 .69 89.232 .968 20 10.560 .115 70 36.960 .401 .20 63.360 .688 .70 89.760 974 21 ii. 088 .120 7i 37-488 .407 .21 63.888 .693 71 90.288 .980 22 11.616 .126 72 38.016 .412 .22 64.416 .699 -72 90.816 .985 23 12.144 .132 73 38.544 .418 23 64.944 70S 73 91-344 .991 24 12.672 .138 74 39-072 424 24 65.472 .710 -74 91.872 ,997 25 13.200 143 75 39-600 430 25 66.000 .716 -75 92.400 .002 26 13.728 .149 76 40.128 .436 .26 66.528 .722 .76 92.928 .008 27 14.256 .155 77 40.656 .441 27 67.056 .728 77 93.456 .014 28 14-784 .160 78 41 . 184 447 .28 67.584 733 .78 93.984 .020 29 15.312 .166 79 41.712 452 .29 68.112 739 79 94-512 .026 30 15.840 .172 80 42.240 -458 30 68.640 745 .80 95.040 .031 31 16.368 .178 81 42.768 .464 '31 69.168 751 .81 95.568 .037 32 16.896 .183 82 43-296 .470 32 69.696 756 .82 96.096 043 33 17.424 .189 83 43-824 .476 33 70.224 .762 -83 96.624 .048 34 17-952 195 84 44-352 .481 34 70.752 .768 .84 97-152 .054 35 18.480 .200 85 44.880 .487 35 71.280 773 .85 97.68o .060 36 19.008 .206 86 45.408 493 .36 71.808 779 .86 98.208 .066 37 19.536 .212 87 45.936 498 37 72.336 .785 .87 98.736 .071 38 20.064 .218 88 46.464 .504 .38 72.864 791 .88 99.264 .077 39 20.592 .223 89 46.992 510 39 73-392 .796 -89 99.792 .083 40 21 . 120 .229 90 47-520 .516 40 73-920 .802 90 100.320 .089 4i 21.648 .235 9i 48.048 .521 41 74.448 .808 9i 100.848 .094 42 22.176 .240 92 48.576 527 42 74.976 .814 92 101 . 376 .100 43 22 . 704 .246 93 49-104 533 43 75.504 .8 I9 93 101.904 .106 44 23.232 .252 94 49.632 539 44 76.032 .825 94 102.432 .ill 45 23.760 .258 95 50.160 .544 45 76.560 .831 95 IO2 . 960 .117 46 24.288 .264 96 50.688 .550 .46 77.088 .836 -96 103.488 .123 47 24.816 .269 97 51.216 .556 47 77.616 .842 -97 IO4.OI6 .129 48 25.344 -275 98 51-744 .561 48 78.144 .848 98 104.544 .134 49 25.872 .281 99 52.272 .567 49 78.672 .854 99 105.072 .140 5 26.400 .286 1. 00 52.800 573 So 79-200 .859 .00 105.600 .146 (261) CHAPTER V ESTIMATING AND CONSTRUCTION TABLES TABLE XXXVIII Relative prices in place that can be paid for articles for the same purpose lasting N and N' years. The article lasting N years is assumed to cost $1.00 or one unit. Interest 4 per cent. Example. If a lasts 10 years and costs 50 cents, there may be paid for b lasting 7 years 50 X 0.74 = 37 cents. \ N N / Amount \^ 5 6 7 8 9 IO II 12 13 14 / $i.oo at N' / N' comp'd int.. 5 I.OO ~s 0-74 0.66 0.60 0.55 0-51 0.47 o.45 0.42 5 .21665 6 1.18 I.OO 0.87 0.78 0.70 0.65 0.60 0.56 0.52 0.50 6 .26532 7 1.35 1. 14 I.OO 0.89 0.81 0.74 0.69 0.64 0.60 0.57 7 31593 8 1.51 1.28 1. 12 I.OO 0.91 0.83 0.77 0.72 0.67 0.64 8 .36857 9 1.67 1.42 1.24 1. 10 I.OO 0.92 0.85 0.79 0.74 0.70 9 .42331 10 1.82 1-55 1.35 1.20 1.09 I.OO 0.93 0.86 0.81 0-77 10 .48024 ii 1.97 1.67 1.46 1.30 1.18 1. 08 I.OO 0.93 0.88 0.83 ii 53945 12 2. II 1.79 1.56 1-39 1.26 1.16 1.07 I.OO 0.94 0.89 12 .60103 13 2.43 1.90 1.66 1.48 1-34 1.23 1. 14 1. 06 I.OO 0.95 13 .66507 14 2.37 2.01 1.76 1-57 1.42 1.30 1. 21 1. 13 1. 06 I.OO 14 .73167 IS 2.50 2.12 1.85 1.65 1.50 1.37 1.27 1.18 I. II 1.05 15 .80094 16 2.62 2.22 1.94 1-73 1.57 1.44 1.33 1.24 1. 17 1. 10 16 87297 17 2.73 2.32 2.03 1.81 1.64 1.50 1.39 1.30 1.22 I. IS 17 .94790 18 2.84 2.41 2. II 1.88 1.70 .56 1.44 35 1.27 1.20 18 2.02581 19 2.95 2.50 2.19 1.95 1.77 .62 1.50 .40 1.32 1.24 19 2.10685 20 3-05 2.59 2.26 2. 02 1.83 .68 1.55 45 1.36 1.29 20 2.19112 21 3-15 2.68 2.34 2.08 1.89 73 1. 60 49 1.40 1-33 21 2.27876 22 3-25 2.76 2.41 2.15 1-94 .78 1.65 54 1.45 1-37 22 2.36991 23 3-34 2.83 2.48 2.21 2.0O .83 1.70 .58 1.49 1.41 23 2.46471 24 3-42 2.91 2.54 2.26 2.05 .88 1.74 .62 1.53 1.44 24 2.56330 25 3 Si 2.C)8 2.60 2.32 2.10 -93 1.78 .66 1.56 1.48 25 2.66583 262 ESTIMATING AND CONSTRUCTION TABLES 263 TABLE XXXVIII. (Continued) >\ IS 16 17 18 19 20 21 22 33 24 35 !% 5~~ 0.40 0.38 0.37 0.35 0.34 0.33 0.32 0.31 0.30 0.29 0.28 5 6 0-47 0.45 0-43 0.41 0.40 0.39 0-37 0.36 0.35 0.34 0.34 6 7 0.54 0.52 0.49 0.47 0.46 0.44 0.43 0.42 0.40 0.39 0.38 7 8 0.61 0.58 0.55 0.53 0.51 0.50 0.48 0.47 0-45 0.44 0.43 8 9 0.67 0.64 0.61 0.59 0.57 0.55 0.53 0.51 0.50 0.49 0.48 9 10 c.73 0.70 0.67 0.64 0.62 0.60 0.58 0.56 0.55 0.53 0.52 10 ii Q.79 0.75 0.72 0.69 0.67 0.64 0.62 0.61 0.59 0.57 0.56 ii 12 0.84 0.81 0.77 0.74 0.71 0.69 0.67 0.65 0.63 0.62 0.60 12 13 0.90 0.86 0.82 0.79 0.76 0.73 0.71 0.69 0.67 0.65 0.64 13 14 0-95 0.91 0.87 0.83 0.80 0.78 0.75 0.73 0.71 0.69 0.68 14 IS .00 0-95 91 0.88 0.85 0.82 0.79 0-77 0.75 0.73 0.71 IS 16 .05 .00 .96 0.92 0.89 0.86 0.83 0.81 0.78 0.76 0.75 16 17 09 04 00 0.96 0.93 0.90 0.87 0.84 0.82 0.80 0.78 17 18 .14 09 .04 I.OO 0.96 0-93 0.90 0.88 0.85 0.83 0.81 18 19 .18 .13 .08 1.04 I.OO 0-97 0.94 0.91 0.88 0.86 0.84 19 20 .22 17 .12 1.07 1.03 I.OO 0.97 0.94 0.91 0.89 0.87 20 21 .26 .20 IS .11 .07 .03 .00 0.97 0.94 0.92 0.90 21 22 30 .24 19 14 .10 .06 03 I.OO 0.97 0.95 0.92 22 33 34 .27 .22 17 13 .09 .06 1.03 I.OO 0.97 0.95 23 24 37 31 .25 .20 .16 .12 .08 1. 06 1.03 I.OO 0.98 24 35 41 34 .28 .23 19 15 .11 1. 08 1.05 1.02 I.OO 35 TABLE XXXIX Relative prices in place that can be paid for articles for the same purpose lasting N and N' years. The article lasting N years is as- sumed to cost $1.00 or one unit. Interest 5 per cent. Example. If a lasts 10 years and costs 60 cents, there may be paid for b lasting 12 years, 60 X 1.15 = 79 cents. \ N N / Amount N. 5 6 7 8 9 10 ii 12 13 14 / fi.oo at N' \ / N' comp'd int. 5 I.OO ~ 0.75 0.67 0.61 0.56 0.52 0.49 0.46 o.44 5 .27628 6 1. 17 I.OO 0.88 0-79 0.71 0.66 0.61 0.57 0.54 0.51 6 .34009 7 1-34 1. 14 I.OO 0.90 0.81 0.75 0.70 0.65 0.62 0.58 7 .40710 8 1.49 .27 1. 12 I.OO 0.91 0.84 0.78 0.73 0.69 .65 8 47745 9 1.64 .40 1.23 I.IO I.OO 0.92 0.86 0.80 0.76 72 9 .55132 10 1.78 .52 1-33 1.19 1.09 I.OO 0-93 0.87 0.82 78 10 .62889 ii 1.92 .64 1.44 1.29 1. 17 1. 08 I.OO o.94 0.88 .84 ii .71034 12 2.05 75 1-53 1-37 1.25 I. IS 1.07 I.OO 0.94 .90 12 79585 13 2.17 1.85 1.62 1-45 1.32 1.22 1. 13 1. 06 I.OO 95 13 .88564 14 2.29 1-95 I.7I 1.53 1.39 1.28 1. 19 1. 12 1.05 00 14 97993 IS 2.40 2.04 1.79 1.61 1.46 1.34 1.25 1. 17 I.IO .05 IS .07892 16 2.50 2.14 1.87 1.68 1.52 1.40 1.30 1.22 I. IS 09 16 .18287 17 2.60 2.22 1-95 1-74 1-59 1.46 1.36 1.27 1.20 14 17 2.29202 18 2.70 2.30 2.02 1.81 1.64 I. Si 1.41 1.32 1.24 .18 18 2.40662 19 2.79 2.38 2.09 1.87 1.70 1.56 1.45 1.36 1.29 .22 19 2.52695 20 2.88 2. 4 6 2.15 1-93 1-75 1.61 I. SO I.4I 1.33 .25 20 2.65330 21 2.96 2.53 2.22 1.98 1. 80 1.66 1.54 1-45 1.36 29 21 2.78596 22 3-04 2.59 2.2? 2.04 1.85 1.70 1.58 1-49 1.40 33 22 2.92526 23 3-12 2.66 2.33 2.09 1.90 1-75 1.62 1-52 1-44 .36 23 3.07152 24 3.19 2.72 2.38 2.13 1.94 1-79 1.66 1.56 1-47 -39 24 3-22510 25 3.26 2.78 2.44 2.18 1.98 1.83 1.70 1.59 1.50 .42 25 3-38635 TABLE XXXIX. (Continued) \ N AT'\ IS 16 17 18 19 20 21 22 23 24 25 N / /W 5 0.42 0.40 0.38 0.37 0.36 0.35 0.34 0.33 0.32 0.31 0.31 5 6 o.49 0.47 0.45 o.43 0.42 0.41 0.40 0.39 0.38 0.37 0.36 6 7 0.56 0.53 0.51 0.49 0.48 0.46 0.45 0.44 0.43 0.42 0.41 7 8 0.62 0.60 o.57 0-55 0.53 0.52 0.50 0.49 0.48 0.47 0.46 8 9 0.68 0.66 0.63 0.61 0.59 0-57 0.55 0-54 o.53 0.52 0.50 9 10 0.74 0.71 0.68 0.66 0.64 0.62 0.60 0.59 0.57 0.56 o.SS 10 ii 0.80 0.77 0.74 0.71 0.69 0.67 0.65 0.63 0.62 0.60 0.59 ii 12 0.85 0.82 0.79 0.76 o.73 0.71 0.69 0.67 0.66 0.64 0.63 12 13 0.90 0.87 0.83 0.80 0.78 0.75 0-73 0.71 0.70 0.68 0.67 13 14 0.95 0.91 0.88 0.85 0.82 0-79 0.77 0.75 0.73 0.72 0.70 14 IS I.OO 0.96 0.92 0.89 0.86 0.83 0.81 0.79 0.77 0.75 0.74 IS 16 1.04 I.OO 0.96 0.93 0.90 0.87 0.85 0.82 0.8o 0.79 0.77 16 17 1.09 1.04 .00 0.96 0.93 0.90 0.88 0.86 0.84 0.82 0.80 17 18 1.13 1. 08 .04 I.OO 0.97 0-94 0.91 0.89 0.87 0.85 0.83 18 19 1.16 1. 12 .07 1.03 I.OO 0.97 0.94 0.92 0.90 0.88 0.86 19 20 1.20 I. IS .11 1.07 1.03 I.OO 0-97 0.95 0.92 0.90 0.88 20 21 1.24 1.18 14 I.IO 1. 06 1.03 I.OO 0-97 0.95 0.93 0.91 21 22 1.27 1. 21 17 1. 13 1.09 1. 06 1.03 I.OO 0.98 0.95 0.93 22 23 1.30 1.24 1. 20 I. IS 1. 12 1. 08 1.05 1.02 I.OO 0.98 0.96 23 24 1-33 1.27 1.22 1.18 1. 14 I. II i. 08 i. 05 1. 02 I.OO 0.98 24 25 1-36 1-30 1.25 1. 21 1. 17 1. 13 I.IO 1.07 1.04 1. 02 I.OO 25 (264) ESTIMATING AND CONSTRUCTION TABLES 265 Volumes of Triangular Prisms. Table XL. In railroad earth-work the volume of a given length of cut or fill is given by the average end area method as / X - in which / is the length, usually 100 feet, and AI and A 2 are the cross-section areas at the two ends. This is equivalent to - AI + - A 2 and Table XL is made on this basis with 100 for /. Hence the volumes are for 5O-foot lengths and are to be used twice, once on one side of a station section for the part of the station FIG. 25. volume on that side, and once on the other side as part of the station volume on that side. The commonest formula for the cross-section area of a 3-level section is A = -D +- (hi + fh). This is equivalent to the area of two triangles of bases c and - and altitudes D and hi + h 2 or it may be taken as three triangles of bases D and and altitudes c, hi and hz. Whence the volume in cubic yards for a 5O-foot length is ^=50/7 + 50 ( - 5? *+ <* + *>. D is the sum of the slope stake distances. The table is made to give the quantity f f x y, in which x is the base or altitude and y is the altitude or base of the triangular base or end of a triangular prism. In using the table for such work the x numbers at the side of the page should be used for heights c and and the y numbers at the top for D and hi + h%. The quantities are given for only whole numbers of y 266 RAILROAD FIELD MANUAL from i to 9 inclusive, but are given to hundredths so that for y = 50 or 70, etc., ten times the tabular values for 5 or 7, etc., are taken and for y = 0.5 and 0.7, etc., one-tenth the tabular values for 5 and 7 are taken. Example. The cross-section notes of two stations show Sta. 762 763 L -3.6 C R -5-3 13-6 -4.8 4.2 -5-7 15-3 16.4 14.8 the minus sign indicating a cut. The volume between stations 762 and 763 is had from the tables as follows: For 762. D = 13.6 + 15.3 = 28.9, done mentally, c = 4.2, w w = 20, = 10, At + fa = 8.9. From Table XL opposite x = 4.2 under 2 take 10 X 7.78 = 77.8 under 8 1X31.11 =31.11 under 9 ^ X 35-OO = 3.5 j = 10 .'. opposite x = 10.0 under 8 take i X 74.00 = 74.00 under 9 ^ X 83.33 = 8.33 Total 194.8 = 194.8 For 763. D = 14.8 + 16.4 = 31.2. c = 5-7, w W =20, - = 10, hi -f fa = n.2. Opposite x = 5.7 take 10 X 15.83 = 158.3 i X 5-28 = 5.3 & X 10.56 = I.I ! = " .'. opposite * = 10 take 10 X 9.72 = 97.2 i X 9-72 = 9.7 & X 1944 = 1-9 Total 273.5 273.5 Total for sta. 468.3 If the length between two sections is 50 feet, take | the result from the tables; if 40 feet, take & the result, and so on. For a single right ESTIMATING AND CONSTRUCTION TABLES 267 prism of any length take that portion of the tabular quantity that the length is of 50. Prismoidal Correction. Table XLI. To get the volume of a station of earthwork by the prismoidal formula, get it by the average end area method and subtract the prismoidal correction. That is, get the vol- ume from Table XL, and subtract the correction of Table XLI. The correction is for full stations of 100 feet; for shorter lengths use pro- portional parts of the tabular quantities. The arguments of the table Ci Co and D\ D are the differences in center heights and total widths respectively of the two end sections. Level Section Volumes. Table XLI I. This table is used only in preliminary estimates and gives the volumes in cubic yards for 100- foot lengths for varying center heights. Its use will be evident. TABLE XL. VOLUMES OF TRIANGULAR PRISMS 50 FEET IN LENGTH Y / N. I 2 3 4 S 6 7 8 9 X ' X .1 .09 19 .28 37 .46 56 .65 74 .83 .1 .2 19 37 .56 74 93 i. ii 1.30 1.48 1.67 .2 3 .28 56 .83 I. II 1-39 1.67 1-94 2.22 2.50 3 4 37 74 .11 1.48 1.85 2.22 2.59 2.96 333 4 5 .46 93 39 1.85 2.31 2.78 3.24 3-70 4.17 -5 .6 56 I. ii .67 2.22 2.78 3-33 3-89 4-44 5-00 .6 7 .65 30 94 2.59 3-24 3-89 4-54 5-19 5.8 3 -7 .8 74 .48 .22 2.96 3-70 4-44 5-19 5-93 6.67 .8 9 .83 .67 50 3 33 4-17 5.00 5-83 6.67 7-50 .9 1.0 93 .85 2.78 3 70 4.63 5.56 6.48 7-41 8-33 1.0 .1 1. 02 .04 3-06 4-07 5-09 6. ii 7-13 8. is 9-17 .1 .2 I. ii .22 3 33 4-44 5.56 6.67 7-78 8.89 IO.OO .2 3 1.20 41 3.6l 4.81 6.02 7.22 8.43 9-63 10.83 3 .4 1.30 59 3-89 S-I9 6.48 7-78 9-07 10.37 11.67 -4 5 1-39 78 4.17 5-56 6.94 8-33 9-72 ii. ii 12.50 -5 .6 1.48 2.96 4-44 5-93 7.41 8.89 10.37 11.85 13.33 .6 .7 1-57 3-15 4-72 6.30 7.87 9-44 11.02 12.59 14.17 -7 .8 1.67 3-33 5-00 6.6? 8.33 IO.OO 11.67 13-33 15.00 .8 9 1.76 3 52 5-28 7.04 8.80 10.56 12.31 14.07 15.83 9 2.0 1.85 3-70 5-56 7-41 9-26 II. II 12.96 14.81 16.67 2.0 .1 1.94 3-89 5-83 7-78 9-72 11.67 I3.6l 15.56 17.50 .1 .2 2.04 4.07 6.H 8.15 10.19 12.22 14.26 16.30 18.33 .2 3 2.13 4.26 6.39 8.52 10.65 12.78 14.91 17.04 19.17 3 4 2.22 4-44 6.67 8.89 II. II 13-33 15.56 17.78 20.00 .4 5 2.31 4-63 6-94 9-26 H-57 13.89 16.20 18.52 20.83 -5 .6 2.41 4.81 7.22 9.63 12.04 14.44 16.85 19.26 21.67 .6 7 2.50 5.00 7.50 IO.OO 12.50 15-00 17.50 20.00 22.50 7 .8 2.59 5-19 7-78 10.37 12.96 15-56 18.15 20.74 23-33 .8 9 2.69 5-37 8.06 10.74 13-43 16.11 18.80 21.48 24.17 9 3.0 2.78 5-56 8.33 II. II 13-89 16.67 19.44 22.22 25.OO 3.0 .1 2.87 5-74 8.61 11.48 14-35 17.22 20.09 22.96 25.83 .1 .2 2.96 5-93 8.89 11.85 14-81 17.78 20.74 23.70 26.67 .2 3 3-06 6. ii 9-17 12.22 15.28 18.33 21.39 24.44 27.50 3 .4 3-iS 6.30 9-44 12.59 15-74 18.89 22.04 25.19 28.33 4 5 3-24 6.48 9-72 12.96 16.20 19-44 22.69 25-93 29.17 5 .6 3-33 6.67 10.00 13-33 16.67 20.00 23-33 26.67 30.00 .6 7 3-43 6.85 10.28 13.70 17.13 20.56 23.98 27.41 30.83 .7 .8 3-52 7.04 10.56 14.07 17-59 21. II 24-63 28.15 31.67 .8 9 3-61 7.22 10.83 14-44 18.06 21.67 25.28 28.89 32.50 .9 4.0 3.70 7-41 II. II I4.8I 18.52 22.22 25-93 29.63 33 33 4.0 .1 3-80 7-59 11-39 15-19 18.98 22.78 26.57 30.37 34-17 .1 .2 3-89 7-78 11.67 15.56 19-44 23-33 27.22 31- II 35-00 .2 3 3-98 7-96 11.94 15-93 19.91 23-89 27.87 31.85 35-83 3 4 4.07 8.15 12.22 16.30 20.37 24-44 28.52 32.59 36.67 4 5 4-17 8.33 12.50 16.67 20.83 25.00 29-17 33 33 37.50 5 .6 4.26 8.52 12.78 17.04 21.30 25.56 29.81 34-07 38.33 .6 7 4-35 8.70 13.06 I7-4I 21.76 26.11 30.46 34-81 39-17 7 .8 4-44 8.89 13 33 17.78 22.22 26.67 31.11 35.56 40.00 .8 .9 4-54 9-07 13.61 18.15 22.69 27.22 3I.76 36.30 40.83 9 50 4-63 9.26 13.89 18.52 23.15 27.78 32.41 37-04 41-67 5.0 (268) TABLE XL. (Continued] \ Y F / \ I 2 3 4 5 6 7 8 9 / x\ / x 5.1 4-72 9-44 14.17 18.89 23.61 28.33 33.o6 37.78 42.50 51 .2 4.81 9-63 14-44 19.26 24.07 28.89 33-70 38.52 43-33 .2 3 4-91 9-8i 14.72 19.63 24-54 29-44 34-35 39-26 44-17 3 4 S.oo 10. OO 15.00 20.00 25.00 30.00 35.00 40.00 45-00 4 5 5-09 10.19 15.28 20.37 25.46 30.56 35.65 40.74 45.83 5 .6 5.19 10.37 15.56 20.74 25-93 31.11 36.30 41-48 46.67 .6 7 5.28 10.56 15.83 21. II 26.39 31.67 36.94 42.22 47-50 7 .8 5-37 10.74 i6.ii 21.48 26.85 32.22 37-59 42.96 48.33 .8 9 5.46 10.93 16.39 21.85 27.31 32.78 38.24 43-70 49-17 9 6.0 5.56 ii. n 16.67 22.22 27.78 33-33 38.89 44-44 50.00 6.0 .1 5.65 11.30 16.94 22.59 28.24 33.89 39-54 45.19 50.83 .1 .2 5-74 11.48 17.22 22.96 28.70 34-44 40.19 45-93 51.67 .2 3 5.83 11.67 17.50 23-33 29.17' 35-00 40.83 46.67 52.50 3 4 5-93 11.85 17.78 23.70 29.63 35.56 4L48 47-41 53-33 4 5 6.02 12.04 18.06 24.07 30.09 36.11 42.13 48.15 54-17 5 .6 6. II 12.22 18.33 24.44 30.56 36.67 42.78 48.89 55-00 .6 7 6. 20 12.41 18.61 24.81 31.02 37-22 43-43 49.63 55.83 7 .8 6.30 12.59 18.89 25.19 31.48 37.78 44.07 50.37 56.67 .8 9 6.39 12.78 19-17 25.56 31-94 38.33 44-72 51. ii 57-50 9 70 6.48 12.96 19.44 25-93 32.41 38.89 45.37 51.85 58.33 7.0 ! 6. 57 13-15 19.72 26.30 32.87 39.44 46.02 52.59 59-17 .1 .2 6.67 13 33 20.00 26.67 33 33 40.00 46.67 53 33 60.00 .2 3 6.76 13-52 20.28 27.04 33-80 40.56 47-31 54-07 60.83 3 -4 6.85 13-70 20.56 27.41 34-26 41.11 47.96 54-81 61.67 4 5 6.94 13.89 20.83 27.78 34.72 41.67 48.61 55.56 62.50 -5 .6 7-04 14.07 21. II 28.15 35-19 42.22 49-26 56.30 63.33 .6 -7 7-13 14.26 21-39 28.52 35.65 42.78 49-91 57-04 64.17 .7 .8 7.22 14.44 21.67 28.89 36.li 43.33 50.56 57-78 65.00 .8 9 7-31 14-63 21.94 29.26 36-57 43.89 51.20 58.52 65.83 9 8.0 7-41 14.81 22.22 29.63 37-04 44-44 51.85 59-26 66.67 8.0 .1 7-50 15-00 22.50 30.00 37-50 45-00 52.50 60.00 67-50 .1 .2 7-59 15.19 22.78 30.37 37.96 45.56 53.15 60.74 68.33 .2 3 7.69 15-37 23.06 30.74 38.43 46.11 53.80 61.48 69.17 3 4 7.78 15.56 23-33 31. II 38.89 46.67 54.44 62.22 70.00 4 5 7-87 15.74 23.6l 31.48 39-35 47-22 55.09 62.96 70.83 5 .6 7.96 15-93 23.89 31.85 39-81 47.78 55.74 63.70 71.67 .6 7 8.06 16.11 24.17 32.22 40.28 48.33 56.39 64.44 72.50 -7 .8 8.15 16.30 24.44 32.59 40-74 48.89 57.04 65.19 73-33 .8 9 8.24 16.48 24.72 32.96 41.20 49.44 57.69 65.93 74-17 -9 9.0 8.33 16.67 25.OO 33-33 41-67 50.00 58.33 66.67 75-00 9.0 .1 8.43 16.85 25.28 33-70 42.13 50.56 58.98 67.41 75.83 .1 .2 8.52 17.04 25.56 34-07 42.59 Si. ii 59.63 68.15 76.67 .2 3 8.6l 17.22 25.83 34-44 43-06 5I-67 60.28 68.89 77-50 3 4 8.70 17.41 26.11 34-81 43-52 52.22 60.93 69-63 78-33 4 5 8.80 17-59 26.39 35-19 43-98 52.78 6l.57 70.37 79-17 5 .6 8.89 17.78 26.67 35.56 44-44 53-33 62.22 71. n 80.00 .6 7 8.98 17.96 26.94 35-93 44-91 53.89 62.87 71.85 80.83 -7 .8 9.07 18.15 27.22 36.30 45-37 54-44 63.52 72.59 81.67 .8 -9 9.17 18.33 27.50 36.67 45.83 55-00 64.17 73-33 82.50 -9 10.0 9.26 18.52 27.78 37-04 46.30 55-56 64.81 74-07 83.33 10.0 (269) TABLE XL. (Continued) \ Y Y / \ i 2 3 4 S 6 7 8 9 / x\ ' X 10.1 9 35 18.70 28.06 37-41 46.76 56.11 65.46 74.81 84.17 10.1 .2 9-44 18.89 28.33 37.78 47-22 56.67 66.11 75-56 85.00 .2 3 9-54 I9-07 28.61 38.15 47-69 57-22 66.76 76.30 85-83 3 .4 9.63 19.26 28.89 38.52 48.15 57.78 67.41 77.04 86.67 -4 5 9-72 19-44 29.17 38.89 48.61 58.33 68.06 77-78 87-50 -5 .6 9-81 I9.63 29.44 39-26 49-07 58.89 68.70 78-52 88.33 .6 .7 9.91 I9.8I 29.72 39-63 49-54 59.44 69.35 79 26 89-17 7 .8 10.00 20.00 30.00 40.00 50.00 60.00 70.00 80.00 90.00 .8 9 10.09 2O.I9 30.28 40-37 50.46 60.56 70.65 80.74 90.83 9 11.0 10.19 20.37 30.56 40-74 50.93 6i.n 71-30 81.48 91.67 11.0 .1 10.28 20.56 30.83 41.11 51-39 61.67 71-94 82.22 92-50 .1 .2 10.37 20.74 31. II 41.48 51.85 62.22 72-59 82.96 93-33 .2 3 10.46 20.93 31-39 41.85 52.31 62.78 73-24 83-70 94-17 3 4 10.56 21. II 31.67 42.22 52.78 63.33 73-89 84-44 95.00 -4 5 10.65 21.30 31-94 42-59 53-24 63-89 74-54 85-19 95-83 5 .6 10.74 21. 4 8 32.22 42.96 53-70 64.44 75-19 85.93 96.67 .6 7 10.83 21.6? 32.50 43-33 54-17 65.00 75.83 86.67 97-50 -7 .8 10.93 21.85 32.78 43-70 54.63 65.56 76.48 87-41 98-33 .8 9 11.02 22.04 33.06 44-07 55-09 66.11 77-13 88.15 99-17 -9 12.0 II. II 22.22 33 33 44-44 55.56 66.67 77-78 88.89 100.00 12.0 .1 11.20 22.41 33.6i 44-81 56.02 67.22 78-43 89-63 100.83 .1 .2 11.30 22.59 33.89 45.19 56.48 67.78 79-07 90-37 101.67 .2 3 ".39 22.78 34-17 45.56 56.94 68.33 79-72 9I.II 102.50 3 4 11.48 22.96 34-44 45-93 57-41 68.89 80.37 91.85 103-33 4 5 11-57 23.15 34-72 46.30 57.87 69.44 81.02 92.59 104.17 -5 .6 11.67 23-33 35-00 46.67 58.33 70.00 81.67 93 33 105.00 .6 .7 11.76 23.52 35-28 47-04 58.80 70.56 82.31 94.07 105.83 .7 .8 11.85 23.70 35-56 47-41 59-26 71.11 82.96 94-81 106.67 .8 9 11.94 23.89 35-83 47-78 59-72 71.67 83-61 95-56 107.50 9 13.0 12.04 24-07 36.11 48.15 60.19 72.22 84.26 96.30 108.33 13.0 .1 12.13 24.26 36-39 48.52 60.65 72.78 84.91 97-04 109.17 .1 .2 12.22 24.44 36.6 7 48.89 61.11 73-33 85-56 97.78 IIO.OO .2 , -3 12.31 24.63 36.94 49-26 6i.57 73.89 86.20 98.52 110.83 3 4 12.41 24.81 37-22 49.63 62.04 74-44 86.85 99-26 111.67 4 .5 12.50 25.OO 37.50 50.00 62.50 75-00 87-50 100.00 112.50 5 .6 12.59 25.19 37.78 50.37 62.96 75-56 88.15 100.74 113.33 .6 7 12.69 25-37 38.06 50.74 63.43 76.11 88.80 101.48 114.17 7 .8 !2. 7 8 25.56 38.33 Si- ii 63.89 76.67 89-44 102.22 115.00 .8 -9 12.87 25-74 38.61 51.48 64-35 77-22 90.09 102.96 US 83 9 14 12.96 25-93 38.89 51.85 64.81 77.78 90-74 103.70 116.67 14.0 .1 13.06 26.11 39.17 52.22 65.28 78.32 91-39 104.44 117-50 .1 .2 13-15 26.30 39.44 52.59 65.74 78.89 92.04 105.19 "8.33 .2 3 13.24 26.48 39.72 52.96 66.20 79-44 92.69 105-93 119.17 3 4 13-33 26.67 40.00 53-33 66.67 80.00 93-33 I06.67 120.00 4 5 13-43 26.85 40.28 53-70 67-13 80.56 93-98 107.41 120.83 5 .6 13.52 27-04 40.56 54-07 67.59 81.11 94.63 108.15 121.67 .6 7 I3.6I 27.22 40.83 54-44 68.06 81.67 95.28 108.89 122.50 7 .8 13.70 27.41 41.11 54-81 68.52 82.22 95-93 109.63 123 33 .8 9 13-80 27-59 41.39 55-19 68.98 82.78 96.57 H0.37 124.17 -9 15.0 13.89 27-78 41.67 55-56 69.44 83.33 97.22 III. II 125.00 15.0 (270) TABLE XL. (Continued) \ Y X" \ X 2 3 4 5 6 7 8 9 x\ 15.1 13.98 27.96 41-94 55-93 69.91 83-89 97-87 111.85 125.83 15.1 .2 14-07 28.15 42.22 56.30 70.37 84.44 98.52 112.59 126.67 .2 3 14.17 28.33 42.50 56.67 70.83 85.00 99-17 "3-33 127.50 3 .4 14.26 28.52 42.78 57-04 71-30 85.56 99-81 114.07 128.33 4 5 14.35 28.70 43.o6 57-41 71.76 86.11 100.46 114.81 129.17 5 .6 14.44 28.89 43-33 57.78 72.22 86.67 IOI.II 115.56 130.00 .6 -7 14.54 29.07 43.61 58.15 72.69 87.22 101.76 116.30 130.83 7 .8 14.63 29.26 43-89 58.52 73-15 87.78 102.41 117.04 131.67 .8 9 14.72 29.44 44-17 58.89 73.6i 88.33 103.06 117.78 132.50 9 16.0 14.81 29.63 44-44 59-26 74-07 88.89 103.70 118.52 133-33 16.0 .1 14.91 29.81 44.72 59.63 74-54 89.44 104.35 119.26 134-17 .1 .2 15.00 3O.OO 45-00 60.00 75-00 90.00 105.00 120.00 135-00 .2 3 15.09 30.19 45-28 60.37 75.46 90.56 105.65 120.74 135-83 3 4 15.19 30.37 45.56 60.74 75-93 91.11 106.30 121.48 136.67 4 5 15.28 30.56 45.83 61.11 76.39 91.67 106.94 122.22 137-50 5 .6 15.37 30.74 46.11 61.48 76.85 92.22 107.59 122.96 138.33 .6 7 15.46 30.93 46.39 61.85 77-31 92.78 108.24 123.70 139-17 7 .8 15.56 31.11 46.67 62.22 77-78 93 33 108.89 124-44 140.00 .8 9 15.65 31.30 46.94 62.59 78.24 93.89 109.54 125.19 140.83 9 17.0 15.74 31-48 47-22 62.96 78.70 94-44 110.19 125-93 141-67 17.0 .1 15.83 31.67 47.50 63-33 79-17 95-00 110.83 126.67 142.50 .1 .2 15.93 31.85 47.78 63.70 79.63 95.56 111.48 127-41 143-33 .2 3 16.02 32.04 48.06 64.07 80.09 96.11 112.13 128.15 144.17 3 4 i6.n 32.22 48.33 64.44 80.56 96.67 112.78 128.89 145.00 4 S 16.20 32.41 48.61 64.81 81.02 97-22 113-43 129.63 145.83 5 .6 16.30 32-59 48.89 65-19 81.48 97.78 114.07 130.37 146.67 .6 7 16.39 32-78 49.17 65.56 81.94 98.33 114.72 I3I.II 147-50 7 .8 16.48 32.96 49.44 65.93 82.41 98.89 115.37 131.85 148.33 .8 9 16.57 33-15 49.72 66.30 82.87 99-44 116.02 132.59 149-17 9 18.0 16.67 33-33 50.00 66.67 83.33 IOO.OO 116.67 133 33 150.00 18.0 .1 16.76 33-52 50.28 67.04 83.80 100.56 117.31 134.07 150.83 .1 .2 16.85 33-70 50.56 67-41 84.26 IOI.II 117.96 I34.8I 151-67 .2 3 16.94 33-89 50.83 67.78 84.72 101.67 118.61 135.56 152.50 3 4 17.04 34-07 51- II 68.15 85.19 IO2.22 119.26 136.30 153-33 4 5 17-13 34.26 51-39 68.52 85.65 102.78 119.91 137-04 154-17 5 .6 17.22 34-44 51-67 68.89 86.ii 103-33 120.56 137.78 155.00 .6 7 I7.3I 34.63 51-94 69.26 86.57 103.89 121.20 138.52 155.83 .7 .8 17.41 34-81 52.22 69-63 87.04 104.44 121.85 139.26 156.67 .8 9 17-50 35-00 52-50 70.00 87.50 IO5.OO 122.50 140.00 157.50 9 19.0 17-59 35-19 52.78 70.37 87-96 105.56 123.15 140.74 158.33 19.0 .1 17.69 35-37 53-o6 70.74 88.43 106.11 123.80 141.48 159-17 .1 .2 17.78 35-56 53-33 71. II 88.89 106.67 124-44 142.22 160.00 .2 3 17.87 35-74 53.6l 71.48 89-35 107.22 125.09 142.96 160.83 3 4 17.96 35-93 53.89 71.85 89.81 107.78 125-74 143.70 161.67 4 S 18.06 36.11 54-17 72.22 90.28 108.33 126.39 144-44 162 . 50 5 .6 18.15 36.30 54-44 72.59 90-74 108.89 127.04 145.19 163-33 .6 .7 18.24 36.48 54-72 72.96 91.20 109.44 127.69 145-93 164.17 7 .8 18.33 36.67 55-00 73-33 91.67 110.00 128.33 146.67 165.00 .8 9 18.43 36.85 55-28 73-70 92.13 110.56 128.98 147.41 165.83 9 20.0 18.52 37-04 55-56 74.07 92.59 III. II 129.63 148.15 166.67 20.0 (271) TABLE XL. (Continued) \ F Y / A I 2 3 4 5 6 7 8 9 / x 20.1 18.61 37-22 55.83 74-44 93-06 111.67 130.28 148.89 167.50 20 1 .2 18.70 37.41 56.11 74-81 93-52 112.22 130.93 149-63 168.33 .2 3 18.80 37.59 56.39 75-19 93-98 112.78 I3L57 150.37 169.17 3 4 18.89 37.78 56.67 75.56 94-44 113-33 132.22 151.11 170.00 4 -5 18.98 37.96 56.94 75-93 94-91 113.89 132.87 151.85 170.83 5 .6 19.07 38.15 57-22 76.30 95-37 114.44 133.52 152.59 171.67 .6 .7 19-17 38.33 57.50 76.67 95.83 II5.OO 134.17 153-33 172.50 .7 .8 19.26 38.52 57.78 77-04 96.30 115.56 134-81 154-07 173-33 .8 9 19-35 38.70 58.06 77-41 96.76 116.11 135.46 154-81 174-17 9 21.0 19-44 38.89 58.33 77-78 97-22 116.67 136.11 155.56 175.00 21.0 .1 19-54 39.07 58.61 78.15 97.69 117.22 136.76 156.30 175.83 .1 .2 19-63 39.26 58.89 78.52 98.15 II7-78 I37-4I 157-04 176.67 .2 3 19.72 39-44 59.17 78-89 98.61 118.33 138.06 157.78 177-50 3 4 19.81 39.63 59-44 79-26 99-07 118.89 138.70 158.52 178.33 4 5 19-91 39-Si 59-72 79.63 99-54 119-44 139-35 159-26 I79-I7 5 .6 20.00 40.00 60.00 80.00 IOO.OO 120. OO 140.00 160.00 180.00 .6 .7 20.09 40.19 60.28 80.37 100.46 I2O.56 140.65 160.74 180.83 7 .8 20.19 40.37 60.56 80.74 100.93 121. II 141.30 161.48 181.67 .8 9 20.28 40.56 60.83 81.11 101.39 121.67 141.94 162.22 182.50 9 22.0 20.37 40.74 6i.n 81.48 101.85 122.22 142.59 162.96 183-33 22.0 .1 20.46 40-93 6i.39 81.85 102.31 122.78 143-24 163.70 184.17 .1 .2 20.56 41.11 61.67 82.22 102.78 123.33 143-89 164.44 185.00 .2 3 20.65 41-30 6i.94 82.59 103.24 123.89 144-54 165.19 185.83 3 4 20.74 41.48 62.22 82.96 103.70 124.44 145.19 165.93 186.67 4 5 20.83 41.67 62.50 83.33 104.17 125.00 145.83 166.67 187.50 -5 .6 20.93 41.85 62.78 83-70 104.63 125.56 146.48 167.41 188.33 .6 .7 21. 02 42.04 63.06 84-07 105.09 126.11 147.13 168.15 189.17 7 .8 21. II 42.22 63.33 84-44 105.56 126.67 147.78 168.89 190.00 .8 9 21.20 42.41 63-61 84.81 106.02 127.22 148.43 169.63 190.83 9 23.0 2I.3O 42.59 63.89 85.19 106.48 127.78 149.07 170.37 191-67 23.0 .1 21.39 42.78 64.17 85.56 106.94 128.33 149.72 171.11 192.50 .1 .2 21.48 42.96 64.44 85-93 107.41 128.89 150.37 171.85 193-33 .2 3 21.57 43-15 64.72 86.30 107.87 129.44 151.02 172-59 194-17 3 4 21.67 43-33 65.00 86.67 108.33 I3O.OO 151.67 173.33 195.00 4 5 21.76 43-52 65-28 87.04 108.80 130.56 152.31 174.07 195.83 5 .6 21.85 43-70 65-56 87.41 109.26 I3I.II 152.96 174.81 196.67 .6 .7 21-94 43-89 65.83 87.78 109.72 131.67 I53.6i 175.56 197.50 .7 .8 22.04 44.07 66.11 88.15 110.19 132.22 154-26 176.30 198.33 .8 9 22.13 44.26 66.39 88.52 110.65 132.78 I54.9I 177.04 199-17 9 24.0 22.22 44-44 66.67 88.89 in. II 133-33 155.56 177.78 200.00 24.0 .1 22.31 44.63 66.94 89.26 in- 57 133.89 156.20 178.52 200.83 .1 .2 22.41 44-81 67.22 89.63 112.04 134.44 156.85 179-26 201.67 .2 3 22.50 45.00 67.50 90.00 112.50 135-00 157.50 180.00 202 . SO 3 4 22.59 45-19 67-78 90.37 112.96 135.56 158.15 180.74 203.33 -4 S 22.69 45-37 68.06 90.74 113-43 136.11 158.80 181 . 48 204.17 5 .6 22.78 45.56 68.33 91.11 113-89 136.67 159-44 182.22 2O5.OO .6 7 22.87 45-74 68. 61 91.48 114-35 137-22 160.09 182.96 205.83 -7 .8 22.96 45-93 68.89 91-85 114.81 137.78 160.74 183-70 206.67 .8 9 23.06 46.11 69.17 92.22 115.28 138.33 161.39 184.44 207.50 9 25.0 23.15 46.30 69.44 92-59 115-74 138.89 162 . 04 185.19 208.33 25 (272) TABLE XL. (Continued) X I 2 3 4 5 6 7 8 9 A 25.1 23.24 46.48 69.72 92.96 116.20 139-44 162.69 185.93 209.17 25.1 .2 23 33 46.67 70.00 93-33 116.67 140.00 163.33 186.67 210.00 .2 3 23-43 46.85 70.28 93-70 117.13 140.56 163.98 187.41 210.83 3 .4 23.52 47.04 70.56 94-07 117-59 141.11 164.63 188.15 211.67 4 5 23.61 47-22 70.83 94-44 118.06 141.67 165.28 188.89 212.50 -5 .6 23.70 47.41 7I.II 94.81 118.52 142.22 165.93 189.63 213-33 .6 7 23.80 47-59 71-39 95.19 118.98 142.78 166.57 190.37 214.17 .7 .8 23-89 47-78 71.67 95.56 119-44 143-33 Il6?.22 191.11 2I5.OO .8 9 23.98 47.96 71-99 95-93 119.91 143-89 167.87 191-85 215.83 9 26.0 24.07 48.15 72.22 96.30 120.37 144.44 168.52 192.59 216.67 26.0 .1 24.17 48-33 72.50 96.67 120.83 145.00 169.17 193.33 217.50 .1 .2 24.26 48.52 72.78 97-04 121.30 145.56 169.81 194.07 218.33 .2 3 24-35 48.70 73.06 97-41 121.76 146.11 170.46 194-81 219.17 3 .4 24-44 48.89 73-33 97-78 122.22 146.67 I7I.II 195.56 220.00 4 S 24-54 49-07 73-6i 98.15 122.69 147-22 171.76 196.30 220.83 5 .6 24-63 49-26 73-89 98.52 123.15 147-78 172.41 197.04 221.67 .6 .7 24.72 49-44 74-17 98.89 I23.6I 148.33 173.06 197.78 222.50 7 .8 24.81 49-63 74-44 99-26 124.07 148.89 173.70 198.52 223-33 .8 9 24.91 49-Si 74.72 99.63 124-54 149-44 174.35 199-26 224.17 9 27.0 25.00 50.00 75.oo 100.00 125.00 150.00 175.00 200.00 225.00 27.0 j 25.09 50-19 75-28 100.37 125.46 150.56 175.65 200.74 225.83 .1 .2 25.19 50.37 75.56 100.74 125-93 151.11 176.30 201.48 226.67 .2 3 25.28 50.56 75.83 IOI.II 126.39 151.67 176.94 202.22 227.50 3 4 25-37 50.74 76.11 101.48 126.85 152.22 177-59 202 . 96 228.33 4 5 25-46 50-93 76.39 101.85 127.31 152.78 178.24 203.70 229.17 5 .6 25.56 Si-il 76.67 102.22 127.78 153-33 178.89 204.44 230.00 .6 7 25.65 51.30 76.94 102.59 128.24 153-89 179-54 205.19 230.83 -7 .8 25.74 51.48 77.22 102.96 128.70 154-44 180.19 205-93 231.67 .8 9 25-83 51-67 77-50 103.33 129.17 155-00 180.83 206.67 232.50 -9 28.0 25-93 51-85 77.78 103.70 129.63 155.56 181.48 207.41 233-33 28.0 .1 26.02 52.04 78.06 104.07 130.09 156.11 182.13 208.15 234.17 .1 .2 26.11 52.22 78.33 104.44 130.56 156.67 182 . 78 208.89 235-00 .2 3 26.20 52.41 78.61 I04.8I I3I.O2 157-22 183.43 209.63 235.83 3 4 26.30 52-59 78.89 105.19 131.48 157.78 184-07 210.37 236.67 -4 5 26.39 52.78 79-17 105.56 131-94 158.33 184.72 211. II 237.50 -5 .6 26.48 52.96 79-44 105-93 132.41 158.89 185-37 211.85 238-33 .6 .7 26.57 53-15 79.72 I06.30 132.87 159-44 186.02 212.59 239.17 -7 .8 26.67 53-33 80.00 106.67 133 33 160.00 186.67 213.33 240.00 .8 9 26.76 53-52 80.28 107.04 I33.8o 160.56 187.31 214.07 240.83 -9 29.0 26.85 53-70 80.56 107.41 I34.26 i6i.n 187.96 214.81 241.67 29.0 .1 26.94 53.89 80.83 107.78 134.72 161.67 188.61 215.56 242.50 .1 .2 27.04 54.07 81.11 108.15 135.19 162.22 189.26 216.30 243-33 .2 3 27-13 54-26 81.39 108.52 135.65 162.78 189.91 217.04 244.17 -3 4 27.22 54-44 81.67 108.89 136.11 163.33 190.56 217.78 245.00 -4 5 27-31 54-63 81.94 109.26 136.57 163.89 I9I.2O 218.52 245.83 5 .6 27.41 54-81 82.22 109.63 137.04 164.44 191.85 219.26 246.67 .6 7 27.50 55-00 82.50 IIO.OO 137.50 165.00 192.50 220.00 247.50 .7 .8 27-59 55-19 82.78 110.37 137.96 165.56 193.15 220.74 248.33 .8 9 27.69 55-37 83.06 110.74 138.43 i66.ii 193.80 221 . 48 249.17 9 30.0 27-78 55-56 83.33 III. II 138.89 166.67 194-44 222 . 22 25O.OO 30 (273) TABLE XL. (Continued) \ Y Y / \ I 2 3 4 5 6 7 8 9 / x \ ' X 30.1 27.87 55-74 83-61 111.48 139-35 167.22 195.09 222.96 250.83 30.1 .2 27.96 55-93 83.89 111.85 I39.8i 167.78 195-74 223.70 251.67 .2 3 28.06 56.11 84.17 112.22 140.28 168.33 196.39 224.44 252.50 3 .4 28.15 56.30 84.44 112.59 140.74 168.89 197.04 225.19 253-33 -4 5 28.24 56.48 84.72 112.96 141.20 169.44 197.69 225.93 254.17 .5 .6 28.33 56.67 85.00 113-33 141.67 170.00 198.33 226.67 255.00 .6 .7 28.43 56.85 85.28 113.70 142.13 170.56 198.98 227.41 255.83 -7 .8 28.52 57-04 85-56 114.07 142.59 171.11 199.63 228.15 256.67 .8 9 28.61 57-22 85.83 114-44 143.06 171.67 200.28 228.89 257.50 9 31.0 28.70 57.41 86.11 114.81 143.52 172.22 200.93 229.63 258.33 31.0 .1 28.80 57.59 86.39 115.19 143.98 172.78 201.57 230.37 259-17 .! .2 28.89 57.78 86.67 115.56 144-44 173-33 202 . 22 231.11 260.00 .2 3 28.98 57.96 86.94 115-93 144.91 173-89 202.87 231.85 260.83 3 .4 29.07 58.15 87.22 II6.30 145-37 174-44 203-52 232.59 261.67 -4 5 29.17 58.33 87.50 116.67 145.83 175.00 204.17 233-33 262 . 50 5 .6 29.26 58.52 87.78 117.04 146.30 175.56 204.81 234-07 263.33 .6 -7 29.35 58.70 88.06 II7-4I 146.76 176.11 205.46 234-81 264.17 -7 .8 29-44 58.89 88.33 117.78 147.22 176.67 206. II 235.56 265.00 .8 9 29-54 59.07 88.61 II8.I5 147.69 177.22 206.76 236.30 265.83 9 32.0 29.63 59.26 88.89 118.52 148.15 177.78 207.41 237.04 266.67 32.0 .1 29.72 59-44 89.17 118.89 148.61 178.33 208.06 237.78 267.50 .1 .2 29.81 59.63 89-44 119.26 149-07 178.89 2O8.70 238.52 268.33 .2 3 29.91 59-81 89.72 119.63 149-54 179-44 209.35 239-26 269.17 3 4 30.00 60.00 90.00 120.00 150.00 180.00 210.00 240.00 270.00 4 5 30.09 60.19 90.28 I2O.37 150.46 180.56 210.65 240.74 270.83 5 .6 30.19 60.37 90.56 I2O.74 150.93 i8l.il 211.30 241-48 271-67 .6 .7 30.28 60.56 90.83 121. II 151-39 181 . 67 211.94 242 . 22 272.50 .7 .8 30.37 60.74 9I.II 121.48 151 85 182.22 212.59 242.96 273-33 .8 9 30.46 60.93 91-39 121.85 152.31 182.78 213.24 243-70 274.17 9 33.0 30.56 6i.n 91.67 122 . 22 152.78 183.33 213.89 244-44 275-00 33 .1 30.65 61.30 91.94 122.59 153.24 183.89 214.54 245.19 275-83 .1 .2 30.74 61.48 92.22 122.96 153.70 184.44 215.19 245-93 276.67 .2 3 30.83 61.67 92.50 123-33 I54-I7 185.00 215.83 246.67 277-50 3 4 30.93 61.85 92.78 123.70 154.63 185.56 216.48 247.41 278.33 4 -5 31.02 62.04 93.o6 124 . 07 155.09 i86.il 217.13 248.15 279-17 -5 .6 31.11 62.22 93-33 124-44 155.56 186.67 217.78 248.89 280.00 .6 7 31.20 62.41 93-61 124.81 156.02 187.22 218.43 249.63 280.83 .7 .8 31.30 62.59 93.89 125.19 156.48 187.78 219.07 250.37 281.67 .8 .9 31-39 62.78 94.17 125.56 156.94 188.33 219.72 251.11 282 . 50 9 34.0 31.48 62.96 94-44 125-93 I57.4I 188.89 220.37 251.85 283.33 34.0 .1 31-57 63.15 94-72 126.30 157.87 189.44 221.02 252.59 284.17 .1 .2 31-67 63.33 95-00 126.67 158.33 190.00 221.67 253-33 285.00 .2 3 31.76 63-52 95.28 127.04 158.80 190.56 222.31 254.07 285.83 3 4 31.85 63.70 95.56 127.41 159-26 191.11 222.96 254.81 286.67 4 5 31-94 63.89 95.83 127.78 159-72 191-67 223.61 255.56 287.50 5 .6 32.04 64.07 96.11 128.15 160.19 192 . 22 224.26 256.30 288.33 .6 7 32.13 64.26 96.39 128.52 160.65 192.78 224.91 257-04 289.17 7 .8 32.22 64.44 96.67 128.89 i6i.n 193-33 225.56 257.78 290.00 .8 .9 32.31 64.63 96.94 129.26 i6i.57 193.89 226.20 258.52 290.83 .9 35.0 32.41 64.81 97.22 129.63 162 . 04 194.44 226.85 259.26 291.67 35.0 (274) TABLE XL. (Continued) \ F / \ A. V i 2 3 4 5 6 7 8 9 / x 35.1 32.50 65.00 97-50 130.00 162.50 195.00 227.50 260.00 292.50 35.1 .2 32.59 65.19 97-78 130.37 162.96 195.56 228.15 260.74 293 33 .2 3 32.69 65.37 98.06 130.74 163.43 196.11 228.80 261.48 294.17 3 4 32.78 65.56 98.33 131.11 163.89 196.67 229.44 262.22 295.00 4 5 32.87 65.74 98.61 131.48 164.35 197-22 230.09 262.96 295.83 5 .6 32.96 65.93 98.89 131-85 164.81 197.78 230.74 263.70 296.67 .6 7 33-06 66.11 99-17 132 . 22 165.28 198.33 231.39 264.44 297.50 .7 .8 33-15 66.30 99-44 132.59 165.74 198.89 232.04 265.19 298.33 .8 -9 33 24 66.48 99-72 132.96 166.20 199.44 232.69 265.93 299.17 9 36.0 33 33 66.67 IOO.OO 133-33 166.67 200.00 233 33 266.67 300.00 36.0 .1 33-43 66.85 100.28 133.70 167.13 200.56 233.98 267.41 300.83 .x .2 33.52 67.04 100.56 134-07 167.59 2OI.II 234.63 268.15 301.67 . .a 3 33.61 67.22 100.83 134-44 168.06 201.67 235.28 268.89 302.50 3 4 33-70 67.41 IOI.II 134-81 168.52 202.22 235-93 269.63 303.33 .4 .5 33-80 67-59 101.39 135-19 168.98 202.78 236.57 270.37 304.17 5 .6 33.89 67-78 101.67 135.56 169-44 203-33 237-22 271.11 305.00 .6 .7 33-98 67.96 101.94 135-93 169.91 203.89 237.87 271.85 305.83 7 .8 34-07 68.15 102 . 22 136.30 170.37 204-44 238.52 272.59 306.67 .8 9 34-17 68.33 IO2 . 50 136.67 170.83 2O5.OO 239 I? 273 33 307.50 9 37.0 34-26 68.52 102.78 I37.04 171.30 205.56 239-81 274-07 308.33 37.0 .x 34-35 68.70 103.06 137-41 171.76 206. II 240.46 274-81 309.17 .1 .2 34-44 68.89 103-33 137.78 172.22 206.67 241.11 275 56 310.00 .2 3 34-54 69.07 IO3.6I 138.15 172.69 207 . 22 241.76 276.30 310.83 3 4 34.63 69.26 103.89 138.52 173.15 207.78 242.41 277.04 311.67 4 5 34-72 69.44 104.17 138.89 173-61 208.33 243 06 277.78 312.50 5 .6 34-81 69.63 104.44 139.26 174-07 208.89 243.70 278.52 313.33 .6 7 34-91 69.81 104.72 139.63 174-54 209.44 244-35 279.26 314.17 .7 .8 35-00 70.00 I05.OO 140.00 175.00 210.00 245-00 280.00 315-00 .8 .9 35-09 70.19 105.28 140.37 175.46 210.56 245.65 280.74 315.83 .9 38.0 35-19 70.37 105.56 140.74 175-93 211. II 246.30 281.48 316.67 38.0 .1 35-28 70.56 105.83 I4I.II 176.39 211.67 246.94 282.22 3I7.50 .1 .2 35-37 70.74 io6.ii 141.48 176.85 212.22 247-59 282.96 318.33 .2 3 35.46 70.93 106.39 141.85 177.31 212.78 248.24 283.70 319.17 3 4 35.56 71.11 106.67 142.22 177.78 213-33 248.89 284.44 320.00 .4 5 35.65 71-30 106.94 142.59 178.24 213.89 249-54 285.19 320.83 -5 .6 35-74 71.48 107 . 22 142.96 178.70 214.44 250.19 285.93 321.67 .6 7 35.83 71.67 107.50 143-33 179-17 2I5.OO 250.83 286.67 322.50 .7 .8 35-93 71-85 107.78 143.70 179.63 215.56 251.48 287.41 323.33 .8 .9 36.02 72.04 108.06 144-07 180.09 2l6.II 252.13 288.15 324.17 9 39.0 36.11 72.22 108.33 144-44 180.56 216.67 252.78 288.89 325.00 39.0 .1 36.20 72.41 108.61 144-81 181.02 217.22 253-43 289.63 325.83 .1 .2 36.30 72.59 108.89 145.19 181.48 217.78 254-07 290.37 326.67 .2 3 36.39 72.78 109.17 145.56 181.94 218.33 254-72 291.11 327.50 3 4 36.48 72.96 109.44 145-93 182.41 218.89 255-37 291.85 328.33 4 .5 36.57 73-15 109.72 146-30 182.87 219.44 256.02 292.59 329.17 5 .6 36.67 73-33 IIO.OO 146.67 183.33 220.00 256.67 293-33 330.00 .6 7 36.76 73-52 110.28 147.04 183.80 220.56 257-31 294.07 330.83 .7 .8 36.85 73.70 110.56 147-41 184.26 221. II 257.96 294.81 331 67 .8 9 36.94 73.89 110.83 147.78 184.72 221 . 67 258.61 295.56 332.50 9 40.0 37-04 74-07 III. II 148.15 185.19 222 . 22 259.26 296.30 333-33 40.0 (275) TABLE XLI Prismoidal Corrections to be subtracted from average end area volumes 100 feet long. D DI = difference in total width; C C\ = difference in center height. Corrections in cubic yards. \D->, D-DJ Ny, I 2 3 4 5 6 V 8 9 c-cX /C-C, .2 0.06 O.I2 0.19 0.25 0.31 0-37 0.43 0-49 0.56 .2 4 O.I2 0.25 0.37 o.49 0.62 0.74 0.86 0-99 i. ii 4 .6 O.I9 0.37 0.56 0.74 0.93 I. ii 1.30 1.48 1.67 .6 .8 0.25 0.49 0-74 0-99 1.23 1.48 1-73 1.98 2.22 .8 I.O 0.31 0.62 0,93 23 1-54 1.85 2.16 2-47 2.78 I.O .2 0-37 0.74 I. II -48 1.85 2.22 2.59 2.96 3-33 .2 -4 0.43 0.86 1-30 73 2.16 2-59 3-02 3.46 3.89 4 .6 0.49 99 1.48 98 2.47 2.96 3-46 3-95 4-44 .6 .8 0.56 .11 1.67 .22 2.78 3-33 3.89 4-44 S.oo .8 2.0 0.62 23 1.85 2-47 3-09 3.70 4.32 4-94 5-56 2.0 .2 0.68 .36 2.04 2.72 3-40 4-07 4-75 5-43 6. ii .2 4 0.74 .48 2.22 2.96 3-70 4-44 5-19 5-93 6.67 4 .6 0.80 .60 2.41 3-21 4.01 4.81 5-62 6.42 7-22 .6 .8 0.86 73 2.59 3.46 4-32 5-19 6.05 6.91 7-78 .8 3-0 0.93 .85 2. 7 8 3.70 4-63 5-56 6.48 7-41 8.33 30 .2 0.99 .98 2. 9 6 3.95 4-94 5-93 6.91 7-90 8.89 .2 4 1.05 .10 3-15 4.20 5-25 6.30 7-35 8.40 9-44 4 .6 i. ii .22 3-33 4-44 5-56 6.67 7-78 8.89 10.00 .6 .8 1. 17 35 3.52 4.69 5-86 7-04 8.21 9-38 10.56 .8 4.0 1.23 2.47 3-70 4-94 6. 17 7-41 8.64 9.88 ii. ii 4.0 .2 1.30 2.59 3-89 5-19 6.48 7.78 9-07 10.37 11.67 .2 4 1.36 2.72 4.07 5-43 6-79 8.15 9-Si 10.86 12.22 4 .6 1.42 2.84 4-26 5-68 7.10 8.52 9-94 11.36 12.78 .6 .8 1.48 2.96 4-44 5-93 7-41 8.89 10-37 11.85 13-33 .8 S.o I -54 3-09 4-63 6.17 7-72 9.26 10.80 12-35 13.89 So .2 1. 60 3.21 4.81 6.42 8.02 9-63 11.23 12.84 14.44 .2 4 1.67 3-33 S.oo 6.67 8.33 IO.OO 11.67 13-33 15-00 4 .6 1-73 3.46 5-19 6.91 8.64 10-37 12. IO 13-83 15.56 .6 .8 1-79 3-58 5-37 7.16 8.95 10.74 12.53 14-32 16.11 .8 6.0 1.85 3-70 5-56 7-41 9.26 ii. ii 12.96 14.81 16.67 6.0 .2 1.91 3-83 5-74 7-65 9-57 11.48 13.40 15.31 17.22 .2 4 1.98 3-95 5-93 7-90 9.88 11.85 13.83 I5.8o 17.78 4 .6 2.04 4-07 6. ii 8.15 10.19 12.22 14.26 16.30 18-33 .6 .8 2.10 4.20 6.30 8.40 10.49 12.59 14.69 16.79 18.89 .8 70 2.16 4-32 6.48 8.64 10.80 12.96 15.12 17.28 19-44 70 .2 2.22 4-44 6.67 '8.89 ii. ii 1333 15.56 17.78 20.00 .2 4 2.28 4-57 6.85 9-14 11.42 13.7 15.99 18.27 20.56 4 .6 2.35 4-69 7.04 9-38 11-73 14.07 16.42 18.77 21. II .6 .8 2.41 4.81 7.22 9-63 12.04 14.44 16.85 19.26 21.67 .8 8.0 2.47 4-94 7-41 9-88 12.35 14.81 17.28 19-75 22.22 8.0 .2 3-53 5.o6 7-59 10.12 12.65 15.19 17.72 20.25 22.78 .2 4 2.59 5-19 7-78 10.37 12.96 15.56 18.15 20.74 23-33 -4 .6 2.65 5-31 7.96 10.62 13-27 15-93 18.58 21.23 23-89 .6 .8 2.72 5-43 8.15 10.86 13-58 16.30 19.01 21.73 24.44 .8 9.0 2.78 5.56 8.33 ii. ii 13-89 16.67 19.44 22.22 25-00 9.0 .2 2.84 5.68 8.52 11.36 14.20 17.04 19.88 22.72 25.56 .2 4 2.90 5-8o 8.70 ii. 60 14-51 17.41 20.31 23.21 26.11 .4 .6 2.96 5-93 8.89 11.85 14.81 17.78 20.74 23.70 26.67 .6 .8 3.02 6.05 9.07 12.10 15.12 18.15 2T.I7 24.20 27.22 .8 10.0 3-09 6.17 9.26 12.35 15-43 18.52 21. 60 24.69 27.78 10 (276) TABLE XLI. (Continued} \D-Di DDi/ I 2 3 4 5 6 7 8 9 ^/ c-c^\ /c-c, 10.2 3-15 6.30 9-44 12.59 15-74 18.89 22.04 25.19 28.33 IO.2 .4 3-21 6.42 9-63 12.84 16.05 19.26 22.47 25.68 28.89 4 .6 3-27 6-54 9-81 13-09 16.36 19-63 22.90 26.17 29.44 .6 .8 3-33 6.67 IO.OO 13 33 16.67 20.00 23-33 26.67 30.00 .8 II. O 3.40 6.79 10.19 13.58 16.98 20.37 23-77 27.16 30.56 II. O .2 3.46 6.91 10.37 13-83 17.28 20.74 24.20 27-65 31.11 .2 4 3.52 7.04 10.56 14.07 17-59 21. II 24-63 28.15 31-67 4 .6 3-58 7.16 10.74 14-32 17.90 21.48 25.06 28.64 32.22 .6 .8 3.64 7.28 10.93 14-57 18.21 21.85 25-49 29.14 32.78 .8 12.0 3.70 7.41 ii. ii 14.81 18.52 22.22 25-93 29-63 33-33 12.0 .2 3-77 7-53 11.30 15.06 18.83 22.59 26.36 30.12 33.89 .2 4 3.83 7-65 11.48 15.31 19.14 22.96 26:79 30.62 34-44 4 .6 3-89 7-78 11.67 15.56 19.44 23-33 27.22 31.11 35.00 .6 .8 3-95 7-9 11.85 15.80 19.75 23.70 27.65 31.60 35.56 .8 13 o 4.01 8.02 12.04 16.05 20.06 24.07 28.09 32.10 36.11 13 o .2 4.07 8.15 12.22 16.30 20.37 24.44 28.52 32.59 36.67 .2 4 4.14 8.27 12.41 16.54 20.68 24.8l 28.95 33-09 37-22 4 .6 4.20 8.40 12.59 16.79 20.99 25.19 29.38 33-58 37.78 .6 .8 4.26 8.52 12.78 17.04 21.30 25.56 29.81 34-07 38.33 .8 14.0 4-32 8.64 12.96 17.28 21.60 25-93 30.25 34-57 38.89 14.0 .2 38 8.77 13.15 17-53 21.91 26.30 30.68 35.o6 39-44 .2 4 44 8.89 13.33 17-78 22.22 26.67 31- II 35.56 40.00 4 .6 51 9.01 13.52 18.02 22.53 27.04 31-54 36.05 40.56 .6 .8 .57 9-14 13.70 18.27 22.84 27.41 31.98 36.54 41. II .8 IS o .63 9.26 13.89 18.52 23.15 27.78 32.41 37-04 41-67 15 o .2 69 9-38 14.07 18.77 23.46 28.15 32.84 37-53 42.22 .2 .4 4-75 9-51 14.26 19.01 23-77 28.52 33-27 38.02 42.78 4 .6 4.81 9.63 14.44 19.26 24.07 28.89 33-70 38.52 43-33 .6 .8 4.88 9-75 14.63 19.51 24.38 29.26 34.14 39-01 43-89 .8 16.0 4-94 9.88 14-81 19-75 24.69 29.63 34-57 39-51 44-44 16.0 .2 S.oo 10.00 15.00 20.00 25.OO 30.00 35-00 40.00 45.00 .2 -4 S.o6 10.12 15-19 20.25 25.31 30.37 35-43 40.49 45.56 4 .6 5.12 10.25 15-37 20.49 25.62 30-74 35-86 40.99 46.11 .6 .8 S-I9 10.37 15 56 20.74 25 93 31.11 36.30 41.48 46.67 .8 17.0 5-25 10.49 15-74 20.99 26.23 31.48 36.73 41.98 47.22 17.0 .2 5-31 10.62 15-93 21.23 26.54 31.85 37-l6 42.47 47-78 .2 4 5-37 10.74 16.11 21.48 26.85 32.22 37-59 42.96 48.33 4 .6 5-43 10.86 16.30 21-73 27.16 32.59 38.02 43-46 48.89 .6 .8 5-49 10.99 16.48 21.98 27.47 32.96 38.46 43-95 49-44 .8 18.0 5-56 ii. ii 16.67 22.22 27-78 33-33 38.89 44-44 50.00 18.0 .2 5-62 11.23 16.85 22.47 28.09 33-70 39-32 44-94 50.56 .2 4 5-68 11.36 17.04 22.72 28.40 34-07 39-75 45-43 51. ii 4 .6 5-74 11.48 17.22 22.96 28.70 34-44 40.19 45-93 5I.67 .6 .8 5.8o ii. 60 17.41 23.21 29.01 34.81 40.62 46.42 52.22 .8 19.0 5-86 11-73 17-59 23.46 29.32 35-19 41-05 46.91 52.78 19.0 .2 5-93 11.85 17.78 23.70 29.63 35.56 41.48 47-41 53-33 .2 4 5-99 11.98 17.96 23-95 29.94 35-93 4I-9I 47-90 53-89 4 .6 6.05 12. IO 18.15 24.20 30.25 36.30 42.35 48.40 54-44 .6 .8 6. ii 12.22 18.33 24-44 30.56 36.67 42.78 48.89 55-00 .8 20.0 6.17 12.35 18.52 24.69 30.86 37-04 43-21 49.38 55.56 20.0 (277) TABLE XLII. LEVEL SECTION VOLUMES 100 feet long. Roadbed 14 feet. Slope i| : i. Center height 0.0 O.I 0.2 03 0.4 0.5 0.6 0.7 0.8 0.9 Center height 5 II 16 22 27 33 39 45 51 i "57 64 70 77 83 90 97 104 III 119 i 2 126 133 141 149 156 164 172 181 189 197 2 3 206 214 223 232 241 250 259 268 277 287 3 4 296 306 316 326 336 346 356 366 377 387 4 5 398 409 420 431 442 453 465 476 488 499 5 6 523 535 547 559 572 584 597 609 622 6 7 635 648 661 675 688 701 715 729 742 756 7 8 770 785 799 813 828 842 857 872 887 902 8 9 917 932 947 963 978 994 IOIO 1026 1042 1058 9 10 1074 1090 1107 1123 1140 H57 1174 1191 1208 1225 10 ii 1243 1260 1278 1295 1313 I33i 1349 1367 1385 1404 ii 12 1422 1441 1459 1478 1497 1516 1535 1555 1574 1593 12 13 1613 1633 1652 1672 1692 1713 1733 1753 1774 1794 13 14 1815 1836 1857 1878 1899 1920 1941 1963 1984 2006 14 2028 2050 2072 2094 2116 2138 2161 2183 2206 2229 IS SO 2252 2275 2298 2321 2345 2368 2392 2415 2439 2463 16 3 2487 2733 2511 2759 2535 2784 2560 2809 2584 2835 2861 2633 2886 2658 2912 2683 2938 2708 2965 a 19 2991 3017 3044 3070 3097 3124 3151 3178 3205 3232 19 Center o n 9 A Center height O.O I .O o.o O height 20 3259 3539 3830 4131 4444 4769 5104 5450 5807 6176 20 30 6556 6946 7348 776l 8185 8620 9067 9524 9993 10472 30 40 10963 11465 11978 12502 13037 13583 14141 14709 15289 25880 40 TABLE XLIII. LEVEL SECTION VOLUMES 100 feet long. Roadbed 16 feet. Slope i : i. Center height o.o O.I O.2 0.3 0-4 o.S 0.6 0.7 0.8 09 Center height o ~ 12 is" 25 31" 38 44 Si 58 O i "65 72 79 86 94 101 109 117 125 133 I 2 141 149 157 166 174 183 192 2OI 209 219 2 3 228 237 247 256 266 275 285 295 305 3i6 3 4 326 336 347 358 368 379 390 401 412 424 4 I 435 556 447 568 458 58l 470 594 482 607 494 620 506 633 518 646 531 660 673 I I 68? 830 701 845 715 859 729 875 743 890 757 90S 771 921 786 936 800 952 815 967 I 9 983 999 1015 1032 1048 1064 1081 1098 1114 1131 9 10 1148 1165 1182 I2OO 1217 1235 1252 1270 1288 1306 10 ii 1324 1342 1361 1379 1398 1416 1435 1454 1473 1492 ii 12 I5II 1530 1550 1569 1589 1609 1629 1649 1669 1689 12 13 1709 1730 1750 1771 1792 1813 1833 1855 1876 1897 13 14 1919 1940 1962 1983 2005 2027 2049 2072 2094 2116 14 2139 2162 2184 2207 2230 2253 2276 2300 2323 2347 IS 16 2370 2394 2418 2442 2466 2490 2515 2539 2564 2588 16 3 2613 2867 2638 2893 2663 2919 2688 2945 2713 2971 2738 2998 2764 3024 2789 3051 2815 3078 2841 3105 3 i9 3131 3159 3186 3213 3241 3268 3296 3323 3351 3379 19 Center height O.O I.O 2.0 30 4.0 So 6.0 7.0 8.0 90 Center height 20 3407 3694 3993 4302 4622 4954 5296 5650 6015 6391 20 30 6778 7176 7585 8006 8437 8880 9333 9798 10274 10761 30 40 H259 11769 12289 12820 13363 I39I7 14481 15057 15644 16243 40 (278) TABLE XLIV. LEVEL SECTION VOLUMES 100 feet long. Roadbed 20 feet. Slope ij : i. Center height 0.0 O.I O.2 03 0.4 o.S 0.6 07 0.8 09 Center height o 7 15 23 31 38 46 55 63 7i o i 'So' 88 97 106 US 124 133 142 161 i 2 3 170 272 180 283 190 294 200 305 210 316 220 327 230 339 240 350 I 262 373 2 3 4 385 397 409 421 446 458 471 484 496 4 I IS 522 659 535 673 549 687 702 575 716 589 731 603 746 617 761 630 776 I 7 791 806 821 837 852 868 884 900 916 932 7 8 948 964 981 998 IOI4 1031 1048 1065 1082 8 9 1117 1 134 1152 1169 IIS? 1205 1223 1241 1259 1278 9 10 1296 1315 1334 1352 1371 1390 1409 1429 1448 1461 10 ii 1487 1507 1527 1546 1566 1587 1607 1627 1648 1668 ii 12 1689 1710 1731 1752 1773 1794 1815 1837 1858 1880 12 13 1902 1924 1946 1968 1990 2012 2035 2058 2080 2103 13 14 2126 2149 2172 2195 2219 2242 2266 2289 2313 2337 14 2361 2607 2385 2633 2409 2658 2458 2709 2483 2735 2508 2761 2532 2786 2557 2812 2582 2839 11 z 2865 3133 2891 3161 2918 3188 2944 3216 2971 3244 2998 3272 3025 3300 3052 3328 3356 3106 3384 17 18 19 3413 3442 3470 3499 3528 3557 3586 3615 3645 3674 19 Center height 0.0 I.O 2.0 30 4.0 So 6.0 7.0 8.0 9-0 Center height 20 3704 4006 4319 4643 4979 5324 568i 6050 6430 6820 20 30 7222 7635 8059 8494 8941 9398 9867 10346 10837 "339 30 40 II852 12376 I29II 13457 14015 14583 15163 15754 16356 16969 40 TABLE XLV. LEVEL SECTION VOLUMES 100 feet long. Roadbed 20 feet. Slope i : i. Center OH o 6 n T OQ ft A Center height 3 0.4 5 U.U O-7 .O o.y height 7 15 23 30 38 46 54 62 70 i 78 86 94 103 in 114 128 137 145 154 i 2 163 172 181 190 199 208 218 227 236 246 2 3 256 265 275 285 295 305 315 325 335 345 3 4 356 366 376 387 398 408 419 430 441 452 4 I 463 578 474 590 485 602 497 614 508 626 638 531 lal 554 675 566 687 I 7 700 713 725 738 751 764 777 790 803 816 7 8 9 830 967 843 981 856 995 870 1009 884 1024 897 1038 9" 1052 925 1067 939 1082 1096 8 9 10 IIII 1126 1141 1156 1171 1186 1 201 1217 1232 1247 IO ii 1263 1279 1294 1310 1326 1342 1358 1374 1390 1406 ii 12 1422 1439 1455 1471 1488 1505 1521 1538 1555 1572 12 13 1589 1606 1623 1640 1658 1675 1692 1710 1728 1745 13 14 1763 1781 1799 1817 1835 1853 1871 1889 1908 1926 14 15 1944 1963 1982 2OOO 2019 2038 2057 2076 2095 2114 IS IO 2133 2153 2172 2191 2211 2231 2250 2270 2290 2310 16 11 2330 2533 2350 2554 2370 2575 2390 2596 24IO 2617 2431 2638 2451 2659 2471 2680 2492 2702 2513 2723 17 18 19 2744 2766 2788 2809 2831 2853 2875 2897 2919 2941 19 Center height 0.0 I.O 2.0 30 4.0 So 6.0 7.0 8.0 9-0 Center height 20 2963 3189 3422 3663 3991 4167 4430 4700 4978 5263 20 30 5556 5856 6163 6478 68OO 7130 7467 7811 8163 8522 3O 40 8889 9263 9644 10033 10430 10833 11244 11663 12089 12522 40 (279) TABLE XLVI. LEVEL SECTION VOLUMES 100 feet long. Roadbed 24 feet. Slope i| : i. Center height 0.0 O.I 0.2 0-3 0.4 o.S 0.6 0.7 0.8 0.9 Center height o 9 18 27 36 46 55 65 75 85 z 94 105 US 125 135 146 156 167 178 189 I 2 200 211 222 234 245 257 269 280 292 304 2 3 317 329 341 354 366 379 392 405 418 431 3 4 444 458 471 485 499 512 526 540 555 569 4 5 583 598 612 627 642 657 672 687 702 7i8 6 733 749 765 780 812 829 845 861 878 7 894 fcn 928 945 962 979 996 1014 1031 1049 7 8 1067 1085 1 102 II2I "39 1157 1175 1194 1212 1231 8 9 1250 1269 1288 1307 1326 1346 1365 1385 1405 9 10 1444 1465 1485 1505 1525 1546 1566 1587 1608 1629 10 ii 1650 1671 1692 1714 1735 1757 1779 1800 1822 1845 ii 12 1867 1889 I9II 1934 1956 1979 2OO2 2025 2048 2071 12 13 2094 2118 2141 2165 2189 2213 2236 2261 2285 2309 13 14 2333 2358 2382 2407 2432 2457 2482 2507 2532 2558 14 IS 2609 2635 2661 2686 2713 2739 2765 2791 2818 15 16 2844 2871 2898 2925 2952 2979 3006 3034 3061 3089 16 11 3H7 3400 3145 3429 3172 3458 3201 3487 3229 35i6 3257 3546 3285 3575 3314 3605 3342 3635 3371 3665 3 19 3694 3725 3755 3785 38i5 3846 3876 3907 3938 3969 19 Center height o.o I.O 2.0 3-0 40 S.o 6.0 7.0 8.0 90 Center height 20 4000 4317 4644 4983 5333 5694 6067 6450 6844 7250 20 30 7667 8004 8533 8983 9444 9917 10400 10894 11400 11917 30 40 12444 12983 13533 14094 14667 15250 15844 16450 17067 17694 40 TABLE XLVII. LEVEL SECTION VOLUMES 100 feet long. Roadbed 24 feet. Slope i : i. Center height 0.0 O.I 0.2 0-3 o-4 o.S 0.6 0.7 0.8 o.9 Center height 9 18 27 36 45 55 64 74 83 i 93 102 112 122 132 142 152 162 172 182 i 2 193 203 214 224 235 245 267 278 289 2 3 300 3" 322 334 345 356 368 380 391 403 3 4 415 427 439 451 463 475 487 500 5io 524 4 5 537 550 562 575 588 601 614 627 640 653 6 667 680 694 707 721 734 748 762 756 790 7 804 818 832 846 861 875 890 904 919 933 7 8 948 963 978 993 1008 1023 1038 1054 1069 1084 8 9 IIOO 1116 1131 1147 1163 H79 1195 I2II 1226 1243 9 xo 1259 1276 1292 1308 1325 1342 1358 1375 1392 1409 10 ii 12 1426 1600 1443 1618 1460 1636 1477 1654 1495 1672 1512 1690 1530 1708 1547 1726 1745 1582 1763 ii 12 13 1782 1800 1819 1837 1856 1875 1894 1913 1932 1951 13 14 1970 1990 2009 2028 2048 2068 2087 2107 2127 2147 14 11 2167 2370 2187 2391 2207 2412 2227 2433 2247 2454 2268 2475 2288 2496 2308 2517 2329 2539 2560" 11 17 2582 2603 2625 2646 2668 2690 2712 2734 2756 2778 17 18 2800 2822 2845 2867 2889 2912 2935 2957 2980 3003 18 19 3026 3049 3072 3095 3"8 3142 3165 3188 3212 3236 19 Center height o.o I.O 2.0 3-0 4.0 So 6.0 7.0 8.0 9.0 Center height 20 3259 3500 3748 4004 4267 4537 4815 5100 5393 5692 20 30 6000 6315 6637 6967 7304 7648 8000 8359 8726 9100 30 40 9481 9870 10267 10670 11081 11500 11926 12359 12800 13248 40 (280) TABLE XLVIII. LEVEL SECTION VOLUMES 100 feet long. Roadbed 26 feet. Slope ij : i. Center height 0.0 O.I 0.2 0.3 04 o.S 0.6 0.7 0.8 0.9 Center height 10 19 29 39 50 6o~ 70 81" 9I i 102 113 124 135 146 157 168 180 191 203 i 2 215 227 239 251 263 275 288 301 313 326 2 3 339 352 365 378 392 405 419 432 446 460 3 4 474 488 517 531 546 56i 575 590 605 4 620 636 651 666 682 698 713 729 745 762 5 778 794 811 827 844 861 878 895 912 929 6 7 946 964 981 999 1017 1035 1053 1071 1089 7 8 1126 1144 1163 1182 1 201 1220 1258 1278 1297 8 9 1317 1333 1356 1376 1396 I4l6 1436 1457 1477 1498 9 10 1519 1539 1560 1581 I6O2 1624 1645 1666 1688 1710 10 ii 1731 1753 1775 1798 1820 1842 1865 1887 1910 1933 ii 12 1956 1979 2002 2025 2048 2072 2095 2119 2143 2167 12 13 2191 2215 2239 2263 2288 2312 2337 2362 2387 2412 13 14 2437 2462 2488 2513 2539 2564 2590 2616 2642 2668 14 13 2694 2963 2721 2990 2747 3018 2774 3046 2800 3073 2827 3101 2854 3129 2881 3158 2908 3186 2936 3214 3 X 3243 3533 3300 3593 3329 3623 3358 3653 3387 3683 3713 3445 3743 3474 3774 3504 3804 a i9 3835 3866 3897 3928 3959 3990 4022 4053 4085 4116 19 Center height 0.0 I.O 2.O 30 4.0 So 6.0 7-0 8.0 90 Center height 20 4148 4472 4807 5154 5511 5880 6259 6650 7052 7465 20 30 7889 8324 8770 9228 9696 10176 10667 11168 11681 12205 30 40 12741 13287 13844 I44I3 14992 15583 16185 16798 17422 18057 40 TABLE XLIX. LEVEL SECTION VOLUMES 100 feet long. Roadbed 26 feet. Slope i : i. Center height 0.0 O.I 0.2 0-3 0.4 o.S 0.6 0.7 0.8 0-9 Center height o IO 19 29 39 49 59 69 79 90 i 100 no 121 131 142 153 164 174 185 196 i 2 207 219 230 241 252 264 275 287 299 3io 2 3 322 334 346 358 370 382 395 407 419 432 3 4 444 457 470 483 495 508 521 534 548 56i 4 5 574 587 601 614 628 642 655 669 683 697 5 6 711 725- 739 754 768 782 797 811 826 841 6 7 856 870 885 900 915 931 946 961 976 992 7 8 1007 1023 1039 1054 1070 1086 1 102 1118 "34 1150 8 9 1167 1183 1199 1216 1232 1249 1266 1283 1299 1316 9 10 1333 1350 1368 1385 1402 1419 1437 1454 1472 1490 10 ii 1507 1525 1543 1561 1579 1597 1615 1634 1652 1670 ii 12 1689 1707 1726 1745 1764 1782 1801 1820 1839 1859 12 13 1878 1897 1916 1936 1955 1975 1995 2014 2034 2054 13 14 2074 2094 2114 2134 2155 2175 2195 2216 2236 2257 14 IS 2278 2299 2319 2340 2361 2382 2404 2425 2467 IS 16 2489 2510 2532 2554 2575 2597 2619 2641 2663 2685 16 11 2707 2933 2730 2956 2752 2979 2774 3003 2797 3026 2819 2842 3072 2865 3096 3H9 2910 3143 3 19 3l67 3190 3214 3238 3262 3286 33io 3334 3359 3383 19 Center height o.o I.O 2.0 30 4.0 S.o 6.0 7.0 8.0 9.0 Center height 20 3407 3656 3911 4174 4444 4722 5007 53oo 5600. 5907 20 30 6222 6544 6874 7211 7556 7907 8267 8633 9007 9389 30 40 9778 10174 10578 10989 11407 11833 12267 12707 13156 13611 40 (281) TABLE L. LEVEL SECTION VOLUMES 100 feet long. Roadbed 28 feet. Slope i : Center height 0.0 O.I O.2 03 0.4 o.S 0.6 0-7 0.8 0.9 Center height .... 10 21 32 42 53~ 64 75 sT 98 i '109 121 132 144 156 168 180 192 205 217 i a 230 242 255 268 281 294 307 320 334 348 2 3 361 375 389 403 417 431 445 460 474 489 3 4 504 519 534 549 564 595 610 626 fie 4 I 656 822 673 839 690 856 706 874 722 891 909 755 926 772 944 788 962 805 980 I 7 998 1016 1035 1053 1072 1090 1109 1128 1147 1166 7 8 1185 1204 1224 1243 1263 1283 1303 1322 1343 1363 8 9 1383 1403 1424 1445 1465 1486 ISO? 1528 1549 1571 9 10 1592 1614 1635 1657 1679 1701 1723 1745 1767 1790 10 ii 1812 1835 1858 1881 1904 1927 1950 1973 1997 2020 ii 12 2044 2068 2092 2116 2140 2164 2189 2213 2238 2262 12 13 2287 2312 2337 2362 2387 2413 2438 2464 2489 2515 13 14 2541 2567 2619 2645 2672 2698 2725 2752 2779 14 11 11 2806 3081 3368 3667 2833 3109 3397 3697 3138 3427 3728 2887 3166 3456 3758 2915 3195 3486 3789 2942 3223 35i6 3820 2970 3252 3546 3851 2997 3281 3576 3882 3025 33 39i3 3053 3339 3636 3944 3 11 19 3976 4007 4039 4070 4102 4134 4166 4198 4231 4263 i9 Center height 0.0 I.O 2.0 30 4.0 So 6.0 7.0 8.0 90 Center height 20 4296 4627 4970 5324 5689 6065 6451 6850 7259 7680 20 30 40 8111 13037 8554 I3S9I 9007 14156 9472 I473I 9948 I53I8 10435 I59I7 10933 16526 1 1443 17146 11963 17778 12494 18420 30 40 TABLE LI. LEVEL SECTION VOLUMES loo feet long. Roadbed 28 feet. Slope i : i Center height 0.0 O.I 0.2 0-3 0.4 0-5 0.6 0-7 0.8 o.9 Center height o IO 21 31 42 53 64 74 85 96 i 107 119 130 141 152 164 175 167 199 210 i 2 222 234 2 4 6 258 270 282 295 307 319 332 2 3 344 357 370 383 395 408 421 434 448 461 3 4 474 487 501 514 528 542 555 569 583 597 4 5 6ll 625 639 654 668 682 697 711 726 74i 5 6 656 770 785 800 816 831 846 861 876 892 6 7 908 923 939 954 970 986 1002 1018 1034 1050 7 8 1067 1083 1099 1116 1132 1149 1166 1182 "99 1216 8 9 1233 1250 1267 1285 1302 1319 1337 1354 1372 1390 9 10 1407 1425 1443 1461 1479 1497 1515 1534 1552 1570 10 iz 12 1589 1778 1607 1797 1626 1816 1645 1836 1664 1855 1682 1875 1701 1895 1720 1914 1739 1934 1759 1954 ii 12 13 1974 1994 2014 2034 2055 2075 2095 2116 2136 2157 13 14 2178 2199 2219 2240 2261 2282 2304 2325 2346 2367 14 3 2389 2607 2410 2630 2432 2652 2454 2674 2475 2697 2497 2719 2519 2742 2541 2765 2563 2788 2585 2810 51 3 2833 3067 2856 3090 2879 3H4 2903 3138 2926 3162 2949 3186 2972 3210 2996 3234 3019 3259 3043 3283 3 19 3307 3332 3356 3381 3406 3431 3455 348o 3505 3530 i9 Center height 0.0 I.O 2.O 30 4.0 S.o 6.0 7-0 8.0 90 Center height 20 3556 3811 4074 4344 4622 4907 5200 5500 5807 6122 20 30 6444 6774 7111 7456 7807 8167 8533 8907 9289 9678 30 40 10074 10478 10889 11307 II733 12167 12607 13056 I35H 13974 40 (282) TABLE LII. LEVEL SECTION VOLUMES ioo feet long. Roadbed 30 feet. Slope i$ : i. Center O O 4 o.S 0.6 O-7 0.8 O Q Center height ** y height o II 22 34 45 57 69 80 92 104 i ii? 129 141 154 166 179 192 205 218 231 i 2 244 258 271 285 299 312 326 340 355 369 2 3 4 383 533 398 549 412 565 42? 580 442 596 457 612 472 487 645 502 661 678 3 4 5 694 711 728 745 762 779 796 814 831 849 5 6 7 867 1050 884 1069 902 1088 920 1107 939 1126 957 1146 975 1165 994 1185 1012 1205 1031 1224 6 7 8 9 1244 1450 1264 1471 1285 1492 1305 1514 1325 1535 1346 1557 1366 1579 1387 1600 1408 1622 1644 8 9 10 1667 1689 1711 1756 1779 1802 1825 1848 1871 10 ii 1894 1918 1941 1965 1981 2012 2036 2060 2085 2109 ii 12 2133 2158 2182 2207 2232 2257 2282 2307 2332 2358 12 13 2383 2409 2435 2460 2486 2512 2539 2565 2591 2618 13 14 2644 2671 2698 2725 2752 2779 2806 2834 286l 2889 14 15 16 2917 3200 2944 3229 2972 3258 3000 3287 3029 33i6 3057 3346 3085 3375 3405 3142 3435 3464 3 3 3494 3800 3525 3831 3555 3862 3615 3925 3646 3957 3676 3989 3707 4020 3738 4052 3769 4084 3 19 4117 4149 4181 4214 4246 4279 4312 4345 4378 4411 19 Center height 0.0 I.O 2.0 30 4.0 So 6.0 7.0 8.0 9-0 Center height 20 4444 4783 5133 5494 5867 6250 6644 7050 7467 7894 20 30 8333 8783 9244 9717 10200 10694 1 1200 11717 12244 12783 30 40 13333 13894 14467 15050 15644 16250 16867 17494 18133 18783 40 TABLE LI 1 1. LEVEL SECTION VOLUMES ioo feet long. Roadbed 30 feet. Slope i : i. Center Om 0.6 Okl o 8 f\ a Center height O.3 0.4 -5 7 u.o o.y height O II 22 34 45 56 68 80 91 103 I 115 127 139 151 163 175 187 200 212 224 i 2 237 250 262 275 288 301 314 327 340 353 2 3 4 367 504 380 518 393 532 407 546 421 56l 434 575 448 589 462 604 476 619 490 633 3 4 1 648 800 663 816 678 831 693 847 708 863 723 879 738 895 754 911 769 927 784 943 I 7 959 976 992 1008 1025 1042 1058 1075 1092 1109 7 8 1126 H43 1160 1177 1195 1212 1229 1247 1265 1282 8 9 1300 1318 1336 1354 1372 1390 1408 1426 1445 1463 9 10 1481 1500 1519 1537 1556 1575 1594 1613 1632 1651 10 ii 1670 1690 1709 1728 1748 1768 1787 1807 1827 1847 ii 12 1867 1887 1907 1927 1947 1968 1988 2008 2029 2050 12 13 2070 2091 2112 2133 2154 2175 2196 2217 2239 2260 13 14 2281 2303 2325 2346 2368 2390 2412 2434 2456 2478 14 15 2500 2522 2545 2567 2589 2612 2635 2657 2680 2703 16 2726 2749 2772 2795 2818 2842 2865 2888 2912 2936 16 3 2959 3200 2983 3224 3007 3249 3031 3274 3298 3079 3323 3103 3127 3373 3151 3399 3176 3423 3 19 3448 3473 3499 3524 3549 3575 3601 3626 3652 3678 19 Center height 0.0 I.O 2.0 30 4.0 So 6.0 70 8.0 9.0 Center height 20 3704 3967 4237 4515 4800 5093 5393 5700 6015 6337 20 30 6667 7004 7348 7700 8059 8426 8800 9181 9570 9967 30 40 10370 10781 II2OO 11626 12059 12500 12948 13404 13867 14337 40 (283) TABLE LIV. LEVEL SECTION VOLUMES ioo feet long. Roadbed 32 feet. Slope i| : i. Center height 0.0 O.I 0.2 0-3 0.4 o.S 0.6 0.7 0.8 0.9 Center height o 12 24 36~ 48 eT 73 sT 98 in i 124 137 ISO 163 177 190 204 218 231 245 i 2 259 273 288 302 316 331 346 360 375 390 2 3 406 421 436 452 467 483 499 515 547 3 4 563 579 596 612 629 646 680 697 714 4 5 731 749 767 784 802 820 838 856 874 6 911 930 948 967 986 1005 1024 1043 1063 1082 6 7 IIO2 1122 1141 1161 1181 1201 1222 1242 1262 1283 7 8 1304 1324 1345 1366 1388 1409 1430 1452 1473 1495 8 9 1517 1539 1561 1583 1605 1627 1650 1672 1695 1718 9 10 1741 1764 1787 1810 1833 1857 1881 1904 1928 1952 10 ii 1976 2000 2024 2049 2073 2099 2122 2147 2172 2197 ii 12 2222 2247 2273 2298 2324 2350 2375 2401 2427 2453 12 13 2480 2506 2532 2559 2586 26l2 2639 2666 2694 2721 13 14 2748 2776 2803 2831 2859 2887 2915 2943 2971 2999 14 3 3028 3319 3056 3348 3085 3378 3H4 3408 3143 3438 3172 3468 3201 3498 3230 3259 3559 3289 3590 3 ; 8 7 362O 3933 3651 3965 3682 3997 3713 4029 3744 4062 3775 4094 3807 4126 4159 3870 4192 3902 4224 11 19 4257 4290 4324 4357 4390 4424 4457 4491 4525 4559 19 Center height o.o I.O 2.O 30 4-0 5-0 6.0 7.o 8.0 9-0 Center height 20 4593 4939 5296 5665 6044 6435 6837 7250 7674 8109 20 30 8556 9013 9482 9962 10452 10954 11467 11991 12526 13072 30 40 13630 14198 14778 15369 15970 16583 17207 17843 18489 19146 40 TABLE LV. LEVEL SECTION VOLUMES ioo feet long. Roadbed 32 feet. Slope i : i. Center height O.O O.I O.2 03 0.4 0-5 0.6 0.7 0.8 0.9 Center height 12 24 36 48 60 72 85 97 no o i 122 135 148 160 173 186 199 212 225 239 i 2 252 265 279 292 306 319 333 347 36i 375 2 3 389 403 417 431 446 460 475 489 504 519 3 4 31 548 701 563 716 578 732 593 748 608 764 624 780 654 812 670 828 4 5 6 844 861 877 894 910 927 944 960 977 994 6 7 ion 1028 1045 1063 1080 1097 HIS 1132 1150 1167 7 8 1185 1203 1221 1239 1257 1275 1293 13" 1330 1348 8 9 1367 1385 1404 1423 1441 1460 1479 1498 1517 1536 9 10 1556 1575 1594 1614 1633 1653 1672 1692 1712 1732 10 ii 1752 1772 1792 1812 1832 1853 1873 1894 1914 1935 ii 12 1956 1976 1997 2018 2039 2060 2081 2103 2124 2145 12 13 2167 2188 22IO 2231 2253 2275 2297 2319 2341 2363 13 14 2385 2407 2430 2452 2475 2497 2520 2543 2565 2588 14 15 2611 2634 2657 2680 2704 2727 2750 2774 2797 2831 15 16 2844 2868 2892 2916 2940 2964 2988 3012 3036 3061 16 17 3085 3110 3134 3159 3184 3208 3233 3258 3283 3308 17 18 3333 3359 3384 3409 3435 346o 3486 35ii 3537 3563 18 i9 3589 3615 3641 3667 3693 3719 3746 3772 3799 3825 19 Center height 0.0 I.O 2.0 30 4.0 So 6.0 7.0 8.0 9-0 Center height 20 "3852" 4122 4400 4685 4978 5278 5585 5900 6222 6552 20 30 6889 7233 7585 7944 8311 8685 9067 9456 9852 10256 30 40 10667 11085 II5II 11944 12385 12833 13289 13752 14222 14700 40 (284) TABLE LVI. LEVEL SECTION VOLUMES TOO feet long. Roadbed 20 feet. Slope | : i Center height 0.0 O.I O.2 0-3 0.4 o.S 0.6 0-7 0.8 0.9 Center height 7 IS 22 30 37 45 52 60 67 i 75 83 90 98 106 121 129 136 144 i 2 152 160 167 175 183 191 199 207 214 223 2 3 231 238 246 254 263 271 279 287 295 303 3 4 319 327 . 336 344 352 360 369 377 385 4 5 394 402 410 419 427 435 444 452 461 469 5 6 478 486 495 503 512 521 529 538 546 555 6 7 564 573 581 590 599 608 616 625 634 643 7 8 652 661 670 679 688 696 706 714 724 733 8 9 742 75i 760 769 778 787 796 806 8i5 824 9 10 ii 833 927 936 852 946 861 955 870 965 880 974 889 984 899 993 908 1003 917 1013 10 ii 12 1022 1032 1042 1051 1061 1071 1080 1090 IIOO IIIO 12 13 III9 1129 H39 1149 1 159 1169 1179 1189 1199 1209 13 14 1219 1229 1239 1249 1269 1279 1289 1299 1309 14 11 1319 1422 1330 1433 1340 1443 1350 1453 1464 I37i 1474 1381 1485 1495 1402 1506 1412 1516 ;i 17 1527 1537 1548 1559 1569 1580 1591 1601 1612 1623 17 18 1633 1644 1655 1666 1676 1687 1698 1709 1720 I73i 18 19 1742 1753 1764 1774 1785 1796 1808 1819 1830 1841 19 Center height 0.0 1.0 2.0 30 4.0 5-0 6.0 7.0 8.0 9-0 Center height 20 1852 1964 2078 2194 2311 2431 2552 2675 2800 2927 20 30 3056 3186 3319 3453 3589 3727 3867 4008 4152 4297 30 40 4444 4593 4744 4897 5052 5208 5367 5527 5689 5853 40 TABLE LVII. LEVEL SECTION VOLUMES ioo feet long. Roadbed 24 feet. Slope : i. Center OA Og 0.6 Om 0.8 o O Center height 3 4 5 7 u.y height o 9 18 27 36 45 54 63 72 sT i 90 99 108 117 126 145 154 172 i 2 181 191 200 209 219 228 237 247 256 266 2 3 275 284 294 303 313 322 332 342 351 361 3 4 370 380 39 399 409 419 428 438 448 458 4 5 468 477 487 497 507 517 527 537 547 557 5 6 567 577 587 597 607 617 627 637 647 657 6 7 668 678 688 698 708 719 729 739 750 760 7 8 770 78i 791 802 812 822 833 843 854 864 8 9 875 886 896 907 917 928 939 949 960 971 9 10 981 992 1003 1014 1025 1035 1046 1057 1068 1079 10 ii 1000 IIOI III2 1123 1 134 H45 1156 1167 1178 1189 ii 12 I2OO I2II 1222 1233 1245 1256 1267 1278 1289 1301 12 13 1312 1323 1335 1346 1357 1369 1380 1392 1403 1414 13 14 1426 1437 1449 1460 1472 1484 1495 1507 1518 1530 14 3 1542 1659 1553 1671 1565 1683 1577 1695 1588 1707 1600 1719 1612 1731 1624 1743 1636 1755 1647 1767 3 17 1779 1791 1803 1815 1827 1839 1851 1863 1876 1888 17 18 1900 1912 1924 1937 1949 1961 1974 1986 1998 201 1 18 i9 2023 2036 2048 2060 2073 2085 2098 2110 2123 2I 3 6 19 Center height 0.0 1.0 2.0 30 4-0 5-0 6.0 7.0 8.0 9-0 Center height 20 2148 2275 2404 2534 2667 2801 2937 3O75 3215 3356 20 30 35oo 3645 3793 3942 4093 4245 4400 455 4715 4875 30 40 5037 5201 5367 5534 5704 5875 6048 6223 6400 6579 40 (285) 286 RAILROAD FIELD MANUAL TABLE LVIIL MIDDLE ORDINATES FOR CURVING RAILS Ordinate given in inches From Roberts' " Track Formulae and Tables." Degree of curve o ; S i.5 2 2.5 3 3 ; 5 4-5 5 5-5 6 6.5 7 7-5 8 8.5 9 9-5 10 10.5 11 ii.5 12 12.5 13 13-5 14 U-S e 15 15-5 16 Length of rails in feet H H 14 16 18 H H i iH xH M 24 ifi 2*4 2% 26 28 2H 214 2% 2% Hi 3% 2% 2% 2% 3 3M 30 2 2% 2% 2% 2% 2% 3 3 3W 3*4 33 ~ 2% 23/4 3 3*6 3H 3H 3% 3*4 3% 4 4H 4*6 TABLE LVIIIa. GAGE ON CURVES Degree of curve Gage Degree of curve Gage 8 and under 4' 8H" i5-i6 4' 9" 9-io 4' 8H" i7-i8 4' 9*6" II-I2 4' 8%" I9-20 4' 9H" I3-I4 4' 8%" ESTIMATING AND CONSTRUCTION TABLES 287 CANTING THE TRACK Using the gage for the base, if R = radius, 5 = speed in miles per hour and e is the difference in level of the two rails in feet, 4-708 + 223.5 -^ Using gage plus one rail head which the author recommends for use with the common track level, The corresponding approximate formulas are e = and e = . In inches E = 0.00066 &D and E = 0.000686 S*D. K The formula of the American Railway Engineering Association for difference of level measured at the gage lines is E = 0.00066 S*D, but the author recommends E = 0.000686 S 2 D. Both formulas are tabulated in Tables LIX and LIXa. TABLE LIX Difference of level in inches of the two rails of a standard gage track on curves of various degrees for various speeds. Formula: = 0.00066 S 2 D (Nearest | inch) Degree nf Speed in miles per hour Degree nf curve 10 IS 2O as 30 35 40 45 So 55 60 65 70 curve i _ H H % H % Itt i% i 5 /i 2 2% 2% 3H i 2 H % Vn. ^i iH I $ 2^ 2^fj 3H 4 4 3 /4 5H 6^i 2 3 M- y> H iH i?4 2% 3H 4 4% 6 8% 9% 3 4 H H i 1% 2% 3/4 4/4 5% 6% 8 9^ 4 5 % % i/4 2 3 4 sH 65/^ 8H 5 6 % i iH 2^ 3/i 4 7 /^ 6H 8 6 7 y* iH i% 27^ 4H 5 5 4 7 3/ 6 7 8 v/ yi/ ^ ^T/ 4% 6^-4 Q3Z 8 9 5 iH ->*<<* 3% 7H 9 10 % iV 5% 10 ii M T 3/j 2 7 A .1^ 8% ii 12 % _i^ .7/ 7H 12 13 7Z 2 3?4 5% 7M- 13 14 I ? 5 >* 5% 83-8 14 15 I 2H 3% 8% IS 16 l}4 2K* 4H 6H 16 17 jl/ ^ 1W 7 17 18 i/4 *3Kj l*d 18 i 3 i ^ 5 7% 19 20 2 3 5H 20 TABLE LIXa From Formula E = 0.000686 S 2 D (Nearest T \ inch) Degree of curve Speed in miles per hour Degree of curve 10 IS 20 25 30 35 40 45 So 55 60 65 70 o.S i 2 3 4 5 6 7 8 9 10 ii 12 14 16 18 20 He He M Me H M He H Me H 'He % i*i 'Me iHe iH ift He M Me Me % % 'Me iHe tJ4 i% i?i i'He i% 2Me 2H 2% 3He H H Me 'Me iHe iH i% i'Me 2Me 2^ 2% 3.o 3Me 3' Me 4% 4'Me SHe Me He % iMe I'He 2^ 2Me 3-0 3% 3'Me 4H 4' He SH 5' Me 6'Me 7H Me H iH iji 2^ 3He 3'He 4Me 4^6 5^ 6Vi 6% 7Me He 'Me I' He afc 3% 4Me 5-0 5'Me 6Me 7H Me iHe 2Me 3Me 4% SHe 6Me 7H 'He iH 2H 4H 5Me 6H % i' He 3He SH 6' Me iHe 2He 4H 6Me i'/4 2Yz 4' Me 7 3 /i iHe 2% 5'Me 8' He I' He 3% 6% o.S i 2 3 4 5 6 7 8 9 10 ii 12 14 16 18 20 ... (288) Z" xxxxxxxxxxxxxxxxxxxxxxxx NX NX NX NX NX NX NX NX coX cox MX MX cox MX MX MX cox cox cox cox MX cox .a-s-s O v V > V V V V V V S S ^sl it i O lO W *C ^t ' \\o N. to o ** ^ ) PO CN o*o rj-^t'^Tj-oooooooooooo M o o o^o cnoooooooo 00 MOO MOO MOO MOO MOO MOOOOOOOOOOOOOOOOOOOOOOOO o llglSlitglRl voo ^rffOfOr-'r^M M^O^O ^fO o^w ^^t'O^'^'O^o w S^Mw^Tr^io^^^XMOc^i>HiONioqoor- r^r^ioiO'^f'^f< v 5r / 5Cs d M MM M O O O O* CT* Oi 00 GO CO ^H OOOOOOOOOOOOOOOOOOOOOOOO 80 IS 155 co gj n '%" d *f 3% 2 fc 31 II sa SI O. SR ss ^ bo jo-d as I JTl- 5 " I ^' g n I*S ^ T3 O 5 (289) TABLE LXI Conversion of linear feet of 100 ft. wide right-of-way into acres From Roberts' " Track Formulae and Tables." Acres Lin. ft. R. of W. Acres Lin. ft. R. of W. Acres Lin. ft. R.-of W. Acres Lin. ft. R. of W. I.O 435-6 II. 4791.6 21. O 9147.6 31.0 13503.6 2.O 871.2 12.0 5227 .2 22 . 9583-2 32.0 13939-2 3-o 1306.8 13.0 5662.8 23.0 10018.8 33-0 14374 -8 4.0 1742.4 14.0 6098.4 24.0 10454-4 34-0 14810.4 S.o 2178.0 15.0 6534.0 25.0 10890.0 35-0 15246.0 6.0 2613.6 16.0 6969.6 26.0 11325.6 36.0 15681.6 7.0 3049.2 17.0 7405.2 27.0 11761.2 37-0 16117.2 8.0 3484.8 18.0 7840.8 28.0 12196.8 38.0 16552.8 9.0 3920.4 19.0 8276.4 29.0 12632.4 39-0 16988.4 10. 4356.0 20.0 87I2.O 30.0 13068.0 40.0 17424.0 Acres Lin. ft. Acres Lin. ft. Acres Lin. ft. Acres Lin. ft. Acres Lin. ft. 2.2 89-3 176.4 263.5 350.7 O.OI 0.21 0.41 0.61 0.81 6.5 93- 7 180.8 267.9 355-0 O.O2 O.22 0.42 0.62 0.82 10.9 98.0 185.1 272.3 359-4 0.03 0.23 0-43 0.63 0.83 15.2 102. 4 189.5 276.6 363-7 0.04 0.24 0.44 0.64 O.I 54 19.6 IO6.7 193-8 281 .0 368.1 0.05 0.25 0.45 0.65 0.85 24.0 III. I 198.2 285-3 372-4 0.06 0.26 0.46 0.66 o.l 56 28.3 IIS- 4 202.6 289.7 376.8 0.07 0.27 0.47 0.67 0.87 32.7 119- 8 206.9 294 .0 381.2 0.08 0.2 8 0.48 0.68 o.i 58 37-0 124. I 211. 3 298 4 385.5 0.09 O.29 0-49 0.69 0.89 41.4 128.5 215.6 302.7 389.9 O.IO 0.30 0.50 0.70 0.90 45-7 132.9 220.0 307 .1 394-2 O.II 0.31 0.51 0.71 0.91 50.1 137- 2 224-3 311 5 398-6 0.12 0.32 0.52 0.72 0.92 54-5 MX. 6 228.7 315 .8 402.9 0.13 0.33 0.53 0.73 0-93 58.8 145.9 233-0 320 .2 407.3 0.14 0.34 0.54 0.74 0-94 63.2 150.3 237-4 324 5 411.6 0.15 0-35 0.55 0.75 0.95 67-5 154. 6 241.8 328.9 416.0 0.16 0.36 0.56 0.76 0.96 71-9 159. 246.1 333-2 420.4 0.17 0.37 0.57 0.77 0.9? 76.2 163. 4 250.5 337 .6 424-7 0.18 0.38 0.58 0.78 0.98 80.6 167.7 254-8 341 -9 429-1 0.19 0.39 0.59 0.79 0.99 84.9 172. I 259-2 346 -3 433 4 0.20 0.40 0.60 0.80 1. 00 89-3 176. 4 263.5 350.7 437 8 (290) ESTIMATING AND CONSTRUCTION TABLES 291 TABLE LXII. DRAINAGE AREAS Sq. ft. opening = C $ (drainage area, in acres) 3 From Roberts' " Track Formulae and Tables." Acres drained Steep slopes C = i Moderately steep slopes C = % Rolling land C*M Flat farm lands C = H Sq. ft. opening required 10 5-6 2.7 1.9 i.i 20 9-5 6.3 32 1-9 30 12.8 8.5 4-3 2.6 40 iS-9 10.6 5-3 3-2 50 18.8 12.5 6.3 3-8 60 21.5 14-3 7-2 4-3 70 24.2 16.1 8.1 4-8 80 26.7 17.8 8.9 5-3 90 29.2 19-5 9-7 5-8 100 31-6 21. 1 10.5 6-3 ISO 42-9 28.6 14-3 8.6 160 44-9 29.9 15-0 9-0 200 53-2 35.5 17.7 10.6 240 60.9 40.6 20.3 12.2 300 72.1 48.1 24.0 14.4 320 75-9 50.6 25-3 15. a 400 89-4 59-6 29.8 17-9 480 102. S 68.3 34-2 20.5 500 105-7 70-S 35.2 21. 1 560 II5-I 76-7 38.4 23.0 600 121. 2 80.8 40.4 24.2 640 127.2 84.8 42.4 25-4 800 150.4 100.3 50.1 30.1 1000 177.8 118.5 59-3 35.6 2000 299.0 199-3 99-7 59-8 2500 353-5 235-7 117.8 70.7 3600 464.8 309.9 154-9 93-0 5000 594-6 396.4 198.2 118.9 6000 681.7 454-5 227.2 136.3 7000 765.3 510.2 255-1 I53-I 8000 845.9 563.9 282.0 169.2 9000 924.4 616.3 308.1 184.9 10 OOO IOOO.O 666.7 333-3 200.0 I. C. R.R. Co., 1906. Extended, 1909, by S. S. R. 292 RAILROAD FIELD MANUAL ' 30 O 3 | 20 r 10 TABLE LXllI From Roberts' " Track Formulae and Tables." CURVES OF CONTENTS OF ABUTMENTS APPROXIMATE CONTENTS OF"T" ABUTMENTS SINGLE TRACK 50 100 150 200 250 300 350 400 450 500 Contents of Two Abutments in Cu.Yds. Illinois Central R-R.Cc. Distance from Base of Rail to Ground o . S S g , *~ ^-*~ ,^ ^ -^ .X x*^ X X^^ x x / X / / APPROXIMATE CONTENTS STRAIGHT ABUTMENTS SINGLE TRACK, / 100 200 300 400 500 600 700 Contents of Two Abutments in Cu.Yds. 800 ESTIMATING AND CONSTRUCTION TABLES 293 TABLE LXIV. COST CURVES TIMBER TRESTLES From Roberts' " Track Formulae and Tables." Unit Prices. Material in Place. Creosoted Material: Caps $55-oo per M. Ft. B.M. Stringers 55.00 per M. Ft. B.M. Other material 43.00 per M. Ft. B.M. Piles 0.37 per linear foot. Untreated Material: Caps $42.00 per M. Ft. B.M. Stringers 42.00 per M. Ft. B.M. Other Material 30.00 per M. Ft. B.M. Piles 0.24 per M. Ft. B.M. Iron 0.05 per pound. To the cost of the trestle obtained from the following curves by multiplying the cost per foot by the length, add for single track: $25.00 for two bulkheads in all cases. 0.70 per linear foot for ties and ballast for ballast floors only. For double track add: $50.00 for two bulkheads in all cases. 1.40 per linear foot for ties and ballast for ballast floors only. Creosoted open deck trestles have piles, caps and braces creosoted. Ballast floor trestles have all material creosoted. The cost per linear foot given by the following curves does not include longitudinal bracing. This should be figured separately for each trestle as the conditions at the opening may require. 28 825 24 23 022 21 2d ^19 11? P fa S 1S K s s / s /' s pfr s _<5 w ag ^ *^ ^,- \e , _, % ^*- *** ^^ 2f ^ ^ r ^- ^-* ^~* ^ "** \ r ^ ^ ,- ,- -^\ ^-* ^^ ^ ^~~ SINGLE TRACK, UNTREATED, OPEN DECK TRESTLE. .--" ^ **^~ ! 1 1 12 4M6 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 Height-Base of Rail to Ground Line IlUnoii Central R.B.C*. f 18.00 1 16 gl5 I 14 fill 12 pll i: ^-^ x^ ,*-* x^" ^^*^ ^ >V\e ^ ^^"*" i^ (^ -r ^. , ' *-* - ^ -^^* Alt ^-* c)^2 \^ . ^^ ^^* ^x ^ **^ - ^- * .X" ^ ' ,^ g X-- ^* SINGLE TRACK, CREOSOTED, OPEN DECK TRESTLE. ' ^"" ii * 1 1 1 1 1 1 1 10 15 20 25 30 35 40 Height-Base of. Rail to Ground Line 45 5( IlUoola Central R.R.CO. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 Height- Base of Bail to Ground Line niiaoii cotna B.R.O*. ESTIMATING AND CONSTRUCTION TABLES COST CURVES TIMBER TRESTLES (Continued) 295 f 23.00 2 21 320 g 19 5 18 1 17 a 16 |14 13 a ^ ^ ..^ cA ^- ^ r > x^ ^ ** / . " ^\V V - --- x YO^ ^-*" / ^ X SINGLE TRACK, BALLAST FLOOR TRESTLES. "CREOSOTED." *" 15 20 25 30 35 40 45 Wl Height -Base of Bail to Ground.Line UliuoU CentnlJLR.Co. 42 * 41 o 40 h 39 S f~ . S^ .. ***^ ^* oW 5 ^^ f\\ *^Je*Z. Cost per Linei ofTrest e8852.8gS2SS?J ^v.^ ^^ ^\ 'V^ ^^, [ ' DOUBLE TRACK. BALLAST FLOOR TRESTLE "CREOSOTED" ^s ^^x x" ^^x* *-*^" \ 12 14 .16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 44 Height- Base of Bail to Ground Line ^ TABLE LXV. PRELIMINARY CULVERT ESTIMATES AND BRIDGE WEIGHTS Iron pipe Concrete arch eter, inches Area, sq. ft. Weight, Ibs. per lin. ft. Cu. yds. masonry, two ends Dimen- sions, feet Area, sq. ft. Cu. yds. per ft. length Cu. yds., two ends 12 0.79 72.5 4.0 2X 2 3-5 0.57 8.0* 16 1.40 107.8 S.o 3X 4 10.9 1. 06 26.1* 18 1.77 127.7 5-7 4X 6 22. 1.63 60.9* 20 2.18 165.7 6.5 6X 6 30.4 3.10 42. 5t 24 3-14 200. 6 7.8 SX 8 54-4 3-99 76. at 30 4-91 290.2 10.8 10X10 85-4 5-57 109. ot 36 7.07 391-6 13.2 12X12 123-3 6.89 i68.4t 42 9.62 512.2 17.8 15X15 193-2 8.81 278. 5t 48 12.57 665.2 22.3 20X18 306.2 12.08 499-ot Square end walls. t Thirty-degree wing walls. 296 RAILROAD FIELD MANUAL BRIDGE WEIGHTS* W = weight of steel in pounds. / = span in feet for truss bridges and length overall for girder bridges. Plate Girders: Deck plate girder W = 12 P + 150 / Through plate girder, iron floor system . . W = 12 P + 500 / Through plate girder, large ties on shelf or flange angles W = 9^ I 2 + 150 / Through plate girder, solid iron floor W = 12 P- + 800 / Riveted Lattice Bridges: Deck bridge, cross-ties on top chord W = 7 P + 200 / Through bridge, iron floor system W = 7 1 2 + 300 / Pin Connected Bridges: Deck span, cross-ties on top chord W = 5 / 2 + 250 / Deck span, iron floor system . . , JF = 5 1 2 + 475 / Through span, iron floor system W = 7 P + 650 / TABLE LXVI. PRELIMINARY BALLAST ESTIMATES Ties 6" X 8" X 8'. 18 ties to a 33-ft. rail. Tabular quantities are in cubic yards per mile. Depth Gravel 3" above tie at center slope 2-1 Broken stone level with top of ties I ft. outside slopes iH-i j j.' in inches Single Double track Single Double track track 14' C. to C. track 14' C. to C. 6 1400 4iS9 1965 4418 8 1737 4952 2396 5306 10 2095 5767 2845 6210 12 2476 6603 3309 7i3i From "Modern Framed Structures" by Johnson, Bryan and Turneaure. CHAPTER VI TURNOUTS AND CROSSOVERS Let F = frog angle. S switch angle. / = toe length of frog from theoretic point. T = heel spread of switch. / = point thickness of switch rail. /' = point thickness of actual frog point. N = number of frog. Frog Distance I FIG. 26. / = length of switch rail. R = radius of turnout curve. G = gage of track. T t N = 3 cot 5 F , sin S = i Frog distance = I + [G - (T +/sin F)] cot | (F + S) + /cos F + ?N p . i^_ G-(r+/sin/Q 297 2Q8 RAILROAD FIELD MANUAL Values for F, R, frog distance, S, and other quantities for varying values of N and / are given in Tables XLVII and XLVIII, and for spring rail frogs in Table XLIX. ^^"""^ A . ' JG W cH&W ^^ > ^- P D ^^ > ' 1 G FIG. 27. BD = (W -G)cotF- - ED = y^-GcotF. sm F Total length = BD + 2 X frog distance to theoretic point. FIG. 28. = r-CR-f-|G)versS, R TURNOUTS AND CROSSOVERS e = (F + e) - F, p = / - (R + i G) sin 5, L = length of crossover 2 \R sin (Jf 7 + 6) 299 Distance between frogs = L 2 X frog distance. rt Length of connecting rail = 2R 57*3 FIG. 29. W-G-fsin/? versF Curve length is R 57-3 300 RAILROAD FIELD MANUAL FIG. 30 K and the elements for both turnouts with equal frog angles F are known. To find F" and the crotch frog distance. O = T-(R + %G) vers 5, AC = 2(12 AC 'cos/3, 1?" = /8 a. Crotch frog distance from first point is 07' = -4- or -*., sin a. cos a F" = 2fi, xr_ 32 TURNOUTS AND CROSSOVERS 3 OI B* 3 8 < JA CO C4NC) (S 33 S SS s a t- 00 t- * O>OiOOOO 4 id pdodt--tofo ^ 1 a 2 H 8 8 88oo8io n- 10 vot~t^ooo>iH u,^ 2^ 1 t-4 s '8. .2 o 3 a ^ 3 5 S U 4^ 31 jplh 11 til 11 1 ^ II t! IK ll M { 1 c. oooo oooooo oooo oo oooo oooo obob oboo 5 'S 1 i . II III IS 1 11 It t$ X5 H ex 1 1i do a 1 is the latitude, h is the altitude, and s' = % (90 5 + + h), 5 being the declin- ation taken with its proper sign, + when north and when south. The difference between Z and the observed azimuth from the fixed line gives the azimuth of that line from which the meridian may be run out. To make the observation, set the transit over a point at least one and one-half hours before or after noon and as long after sunrise or before sunset, and use any fixed distant point for zero azimuth. Use a piece of colored or smoked glass in the eyepiece cap or before the object glass, and bring the cross wires tangent to the sun 's disc, approximately by hand motion of the alidade and telescope and clamps and precisely by both slow motions together; read the vertical circle and the azimuth. Subtract the refraction correction of Table LXX for the measured alti- tude from the vertical circle reading. If the horizontal wire has been made tangent to the lower limb (edge) of the sun, add the sun's semi-diameter, i6 m or 0.26667, to the corrected vertical circle reading; if the upper limb has been observed subtract the semi-diameter. The result is the correct altitude of the sun at the instant of observation. The sun's semi-diameter is not exactly i6 m but varies during the year and may be had from the Nautical Almanac mentioned later if greater precision is desired. The measured azimuth from the reference line is increased or dimin- ished according as the edge nearest or farthest from the reference line is observed, by 16' X sec h or 0.27 sec h. If the observer is satisfied to'quarter the sun's image with the cross wires no semi-diameter correc- tion is required either for altitude or azimuth. Explanation of Astronomical Terms. Celestial bodies are located on the celestial sphere by coordinates corresponding to latitude and longitude of the terrestrial sphere. The celestial equator is a circle cut from the celestial sphere by the terrestrial equatorial plane ex- tended, and angular distances north or south of the celestial equator are called declinations, corresponding to terrestrial latitude. An arbi- trary meridian of the celestial sphere is chosen as the reference for what is called right ascension, corresponding to longitude. Right ascension is not used in the methods of this chapter. The declination of the sun changes constantly. The Nautical Almanac or American Ephemeris, published by the Government annu- ally, gives the hourly change. Several instrument makers distribute gratis in pocket form reprints of that part of the Nautical Almanac relating to the sun and useful to surveyors. Every surveyor should have one of these reprints. 308 RAILROAD FIELD MANUAL To determine the declination at a given place and hour on a given day the approximate Greenwich time of the observation must be known. In North America if standard time is carried' by the observer, he will know what meridian time he carries and hence how many hours slow he is of Greenwich time; Eastern time is 5 hours, Central time 6 hours, Mountain time 7 hours, and Pacific time 8 hours slow of Greenwich time. The Almanac gives the coordinates of the sun for Greenwich apparent and mean noon, and some reprints give the coordinates for one and some for the other. Noon. The sun does not appear to move at a uniform rate around the earth. Apparent noon is the instant the sun appears to cross the meridian. Mean noon is the instant that an imaginary sun, moving at a uniform rate and making the same number of revolutions in a year as the real sun, appears to cross the meridian. Mean time is time accord- ing to the mean sun and is what is carried by clocks and watches. The difference between apparent and mean time is called the equation of time and is found in the Nautical Almanac and reprints. It is some- times to be added and sometimes subtracted to convert one time into the other. The sign to be used is given with the equation. Assuming standard time at any place where central time is used, 9 o'clock in the morning, being 3 hours before noon, would be (6 3) hours = 3 hours after noon at Greenwich, and the declination of the table for a given day must be corrected for 3 hours change. In the table means south and + north. If the change is marked the sun is going south, and north or + declination is decreasing, while south or declination is increasing. If local mean time is carried by the observer he must know his approx- imate longitude and must convert this into time, 15 to the hour, to find the difference between Greenwich and local time. In either case if his reprint gives the position of the sun for apparent noon, he must apply the equation of time to his mean time to find the local or stand- ard apparent time. The positions of heavenly bodies are figured from the center of the earth. With a body as near as the sun this gives rise to a correction to altitudes measured at the surface known as the correction for paral- lax. It is but a few seconds and is neglected in this discussion. The methods of this chapter are such as are suited to field instruments read- ing to minutes or o.oi of a degree and many refinements necessary in astronomical work are omitted. Latitude. I. By Polaris. The altitude of the north pole equals the latitude of the place of observation. Measure the altitude of Polaris AZIMUTH, LATITUDE, AND TIME 309 at upper or lower culmination, subtract the refraction correction found in Table LXX. Add or subtract the pole distance of the star as found in Table LXX, according as lower or upper culmination is observed. The result is the latitude. TABLE LXX. POLAR DISTANCE OF POLARIS For January i of years named 1915 1918 1921 1924 1927 1930 1933 1936 1939 1942 i.i49 1.133 1.118 1.102 1.087 1.071 1.056 1.041 1.026 1.011 . .. sin polar distance Sm of azimuth at elongation = -. : : -= cosine latitude Latitude = altitude of Polaris at culmination polar distance refraction correction given below. Latitude or altitude Correction Latitude or altitude Correction 20 30 40 0.043 0.027 50 60 0.013 0.009 To observe, set up a little before the time of culmination found in Table LXX I; set the horizontal wire on the star and follow with the slow motions till the star's motion seems to be wholly in azimuth and not at all in altitude. Read the vertical angle. 2. By a Noon Observation of the Sun. Measure the altitude of the sun when at its highest point; subtract the refraction correction of Table LXX for the altitude found; subtract the sun's declination if north or add if south; the result is the co-latitude. To observe set up the transit a little before noon; set the horizontal wire on the upper or lower limb (edge) of the sun's disc and keep it there as the sun rises and until it has ceased to rise and its motion seems to be wholly in azimuth; read the vertical circle very carefully and subtract or add the sun's semi-diameter according as the upper or lower limb was observed. Colored or smoked glass must be used, prefer- ably in the eyepiece cap rather than before the object glass, but either may answer. The sun's semi-diameter is an average of 16 minutes or 3io RAILROAD FIELD MANUAL 0.26667. It varies during the year and may be had from the Nautical Almanac or makers' reprint (see previous article) if desired to greater precision. TABLE LXXI* Approximate local mean times (counting from noon 24 hours) of the elongations and culminations of polaris in the year 1915 for lati- tude 40 N.; longitude 6 H W. from Greenwich. Date East elongation West elongation Upper culmination Lower culmination Jan. i h. m. o 52 i h. m. 12 42 I h. m. h. m. IS 23 53 o II 46 8 Feb. i 22 45 9 10 39 7 16 42 8 *5 21 50 6 944 2 Mar. i 15 20 55.4 20 CO 2 8 49-2 7 54 2 54.2 I V) 14 52-3 Apr. i is May i 18 53-3 17 58.1 16 55 3 6 47.1 5 51-9 4 49 I o 52.1 23 43-i 12 50.2 ii 55-0 IS 16 oo 4 7 CA 2 June i . 14 S3 8 2 47 6 8 So 7 15 13 58 9 I 52 7 7 55 8 July i 15 12 S6.3 12 OI 5 o 50.1 23 51-5 18 51.3 6 53-2 5 58 4 Aug. i 10 55 . o 22 45 . o 16 50 o 4 51-9 IS Sept. i 10 OO.I 8 53.5 21 50.1 2O 43-5 IS 55.1 14 48 <> 3 57-0 2 50.4 je 7 58 6 19 48 6 I 55 5 Oct. i 6 55 8 18 45 8 12 50 8 o 52.7 IS 6 oo 8 i? So 8 II 55 8 23 53 9 Nov. i 4 54.o 16 44-O 10 49.0 22 47.1 15. .. 3 58 9 15 48 . 9 9 53-9 21 52.O Dec. i IS 2 55. 8 2 00.6 14 45-8 13 50.6 8 50.8 7 55-6 20 48.9 19 53-7 * From data furnished by the U. S. Coast and Geodetic Survey. Approximate Determination of Time. i. To Find the Error of a Watch. The observation for azimuth on the sun may be utilized. The instant of the observation should be noted on the watch. Then sin/ sin Z cos h cos 6 AZIMUTH, LATITUDE, AND TIME 311 in which / is the hour angle in degrees before or after apparent local noon. Reduce t to hours by dividing by 15 and find the apparent local time; apply the equation of time for the day from the Nautical Almanac, and the result is mean local time. The longitude of the place must in general be had from a map, if possible to the nearest minute. The difference between local longitude and the standard meridian whose time is carried reduced to time is applied to the determined mean local time to get the mean standard time. The difference between this and the observed time of the observation is the error of the watch. If the watch carries local time, longitude is needed only to compute change in declination and need not be so precisely determined. To refer to any calendar day other than the first and fifteenth of each month, subtract 3.94 for every day between it and the preced- ing tabular day, or add 3.94"* for every day between it and the succeed- ing tabular day. To find the times for the tabular dates after 1915, to the tabular value add 1.36 for each year after 1915 less 3.9 for each leap year. In any leap year deduction for that year is not made until March i. To find the time of western elongation for Jan. 18, 1919: For January 15, 1915, western elongation, tabular time is II A 46.8"*. 1919 1915 =4; 4 X 1.36 = 5.4. One leap year has intervened, and, therefore, 5.4 3.9 = 1.5 to be added. II A 46.8 + 1.5 = 11*48.3 for January 15, 1919. For January 18 subtract 38 15 = 3 X 3 X 3-94 m = n-8 m getting n h 36.5. To refer to any other than the tabular latitude between the limits of 25 and 50 north add to the time of west elongation 0.13*" for every degree south of latitude 40, and subtract from the time of west elonga- tion o.i8 m for every degree north of 40. Reverse these signs for cor- rections to the times of east elongation. For latitudes as high as 60 diminish the times of west elongation and increase the times of east elongation by 0.23 for every degree north of latitude 40. To refer to other longitudes, add o.i6 TO for each hour east of 6 hours and subtract o.i6 m for each hour west of 6 hours. 312 RAILROAD FIELD MANUAL TABLE LXXII Length of 0.01 of latitude and 0.01 of longitude to the nearest whole foot Latitude Length 0.01 latitude Length 0.01 longitude Latitude Length 0.01 latitude Length 0.01 longitude I 3628 3652 31 3637 3133 2 3628 3650 32 3638 3100 3 3628 3647 33 3638 3066 4 3628 3643 34 3639 3031 5 3628 3638 35 3640 2995 6 3628 3632 36 3641 2958 7 3628 3625 37 3641 2920 8 3628 36i7 38 3641 2882 9 3628 3608 39 3642 2842 10 3629 3597 40 3643 2802 ii 3629 3586 4i 3643 .2761 12 3629 3573 42 3644 2719 13 3629 3559 43 3645 2675 14 3630 3545 44 3646 2632 IS 3630 3529 45 3646 2587 16 3631 3512 46 3647 2542 17 3631 3494 47 3647 2495 18 3631 3475 48 3648 2449 19 3631 3455 49 3649 2401 20 3632 3433 50 3649 2353 21 3632 3411 51 3650 2303 22 3633 3388 52 3650 2254 23 3633 3364 53 3651 2203 24 3634 3338 54 3652 2152 25 3634 3312 55 3652 2099 26 3635 3285 56 3653 2047 27 3635 3256 57 3653 1994 28 363S 3227 58 3654 1940 29 3636 3197 59 3655 1886 30 3637 3166 60 3655 1831 AZIMUTH, LATITUDE, AND TIME TABLE LXXIII. CONVERGENCE OF MERIDIANS , Latitude Angular convergence per mile, degrees Distance for convergence of 0.01, feet Latitude Angular convergence per mile, degrees Distance for convergence of 0.01, feet i o.ooo 209,240 31 0.009 6084 2 .001 104,588 32 .009 5851 3 .001 69,690 33 .009 5629 4 .001 52,231 34 .010 5420 5 .001 4L747 35 .010 5222 6 .002 34,750 36 .010 5032 7 .OO2 29,747 37 .Oil 4852 8 .002 26,002 38 .Oil 4681 9 .002 23,062 39 .012 45i6 10 .003 20,715 40 .012 4359 ii .003 18,792 41 .013 4208 12 .003 17,185 42 .013 4062 13 .003 15,823 43 .013 3922 14 .004 14,651 44 .014 3788 IS .004 13,634 45 .014 3658 16 .004 12,740 46 .015 3533 i? .004 ii,9So 47 .015 3412 18 .005 11,244 48 .016 3295 19 .005 10,611 49 .017 3181 20 .005 10,039 50 .017 3071 21 .006 9,5i8 Si .018 2964 22 .006 9,044 52 .018 2860 23 .006 8,609 53 .019 2758 24 .006 8,208 54 .020 2660 25 .007 7,837 55 .021 2563 26 .007 7,493 56 .021 2469 27 .007 7,173 57 .022 2377 28 .008 6.874 58 .023 2288 29 .008 6,594 59 .024 22OO 30 .008 6,331 60 .025 2114 CHAPTER VIII TABLES FOR METRIC CURVES METRIC curves are used in Latin-American countries. The "degree" is the angle subtended by a chord of 20 meters. Practically all usually tabulated curve functions may be converted from feet values in tables for curves used in the United States to meter values for metric curves by dividing by five. The more commonly used functions have been tabulated in the three following tables. The tabular values are in meters. 314 TABLE LXXIV. RADII, TANGENT OFFSETS AND MIDDLE ORDINATES FOR METRIC CURVES Degree = angle subtended by chord of 20 meters Deg., D Radius, R Logarithm, log/? Tan. off., t Mid. ord., m Deg., D Radius, R Logarithm, log/? Tan. off.,< Mid. ord., m 0.0 00 00 .000 .000 10.0 114.74 2.05970 1-743 437 .2 5729.57 3.75812 035 .009 .2 112.49 2.05113 1.778 445 4 2864.80 3.45709 .070 .017 4 110.34 2.04272 1.813 454 .6 1909.87 3.28100 .105 .026 .6 108.26 2.03447 1.847 .463 .8 1432 '.41 3-15607 .140 .035 .8 106.26 2.02637 1.882 472 I.O 1145-93 3.059J6 .175 .044 II. 104-33 2 . 01843 I.9I7 .480 .2 954-95 2.97998 .209 .052 .2 102.48 2 . 01063 1-952 .489 -4 8i8.53 2.91304 .244 .061 4 100.68 2.00296 1.986 498 .6 716.22 2.85505 .279 .070 .6 98.95 1-99544 2.021 507 .8 636.65 2.80390 .314 .079 .8 97.28 I . 98804 2.056 SIS 20 572.99 2.75814 349 .087 12.0 95.67 1.98077 2.O9I 524 .2 520.90 2.71676 .384 .096 .2 94-11 I.9736I 2.125 533 -4 477-50 2.67897 .419 .105 4 92.59 1.96658 2.160 542 .6 440-77 2.64422 454 .113 .6 91.13 1.95966 2.195 550 .8 409.30 2.61204 .489 .122 .8 89.71 I.9528S 2.229 559 30 382.02 2.58208 .524 .131 13 o 88.34 I.946I4 2.264 .568 .2 358-15 2.55406 .558 .140 .2 87.00 1-93954 2.299 577 4 337-08 2.52774 .593 .I 4 8 4 8S-7I 1-93304 2.333 .585 .6 318-36 2.50292 .628 .157 .6 84.46 1.92663 2.368 594 .8 301.61 2.47945 .663 .166 .8 83.24 1.92032 2.403 .603 4.0 286.54 2.45718 .698 .175 14.0 82.06 1.91411 2.437 .612 .2 272.90 2.43600 733 .183 .2 80.91 1.90798 2.472 .620 4 260 . 50 2.41581 .768 .192 4 79-79 1.90193 2.507 .629 .6 249-18 2.39651 .803 .201 .6 78.70 1.89598 2.541 .638 .8 238.80 2.37804 .838 .209 . 8 77.64 1.89010 2.576 .647 So 229.26 2.36032 .872 .218 15.0 76.61 1.88430 2.611 .655 .2 220.44 2.34330 .907 .22? .2 75-61 1.87858 2.645 .664 4 212.29 2.32692 942 .236 4 74.63 1.87294 2.680 .673 .6 204.71 2.31114 977 .244 .6 73.68 1.86737 2.714 .682 .8 197-66 2.29591 I.OI2 253 .8 72.76 1.86187 2.749 .690 6.0 191.07 2.28120 1.047 .262 16.0 71.85 1.85644 2.783 .699 .2 184.92 2.26697 .082 .271 .2 70-97 1.85109 2.818 .708 4 179.14 2.25320 .116 .279 4 70.11 1.84579 2.853 .717 .6 173.72 2.23985 .151 .288 .6 69.27 1.84056 2.887 .726 .8 168.62 2.22690 .186 .297 .8 68.45 1.83540 2.922 -734 7.0 163.80 2.21432 .221 .306 17.0 67.65 I . 83030 2.956 743 .2 159-26 2.2O2II .256 .314 .2 66.87 1.82526 2.991 752 4 154.96 2.19022 .291 .323 4 66.11 1.82027 3-025 .761 .6 150.89 2.17866 .325 332 .6 65.37 I.8I535 3-o6o .769 .8 147.03 2.16739 .360 340 .8 64.64 1.81048 3.094 -778 8.0 143.36 2.15642 395 349 18.0 63.92 1.80567 3.129 .787 .2 139.87 2.I457I .430 .358 .2 63 23 1.80091 3-163 -796 4 136.54 2.13526 .465 .367 4 62.55 1.79620 3.198 .805 .6 133-37 2.I25O6 .500 375 .6 61.88 I.79I55 3-232 .813 .8 130.35 2.II5IO 534 .384 .8 61.23 1.78694 3.267 .822 9.o 127-45 2.10536 .569 -393 19.0 60.59 1.78239 3-301 .831 .2 124.69 2.09583 .604 .402 .2 59.96 1.77789 3-335 .840 4 122.04 2.08651 .639 .410 4 59-35 1.77343 3-370 .849 .6 II9-5I 2.07739 .674 .419 .6 58.75 1.76902 3.404 .857 .8 117.07 2.06846 .708 .428 .8 58.16 1.76465 3-439 .866 10 "4.74 2.05970 1-743 437 20.0 57-59 1.76033 3-473 .875 (315) TABLE LXXV. METRIC CURVES Degree Actual arc, Long chords Degree of curve one station 2Sta. 3Sta. 4Sta. .5 Sta. 6 Sta. 7 Sta. 8 Sta. of curve 02 ao.ooom. 40.00 60.00 80.00 IOO.OO 120.00 139-99 159-99 02 4 .000 o.oo o.oo o.oo o.oo H9-99 9-98 9-98 4 .6 .000 o.oo o.oo 79.99 99-99 9-98 9-97 9-95 .6 .8 .000 o.oo o.oo 9-99 9-98 9-97 9-95 9.92 .8 I.O 20.000 40.00 59-99 79-99 99-97 119-95 139.92 159-87 I.O .2 .000 o.oo 9-99 9.98 9.96 9-92 9.88 9.82 .2 4 .000 o.oo 9-99 9-97 9-94 9-90 9-83 9-75 4 .6 .001 o.oo 9.98 9.96 9.92 9.86 9.78 9.67 .6 .8 .001 o.oo 9-98 9-95 9-90 9-83 9.72 9-59 .8 2.0 2O.OOI 39.99 59-97 79-94 99-88 119.78 139.66 159-49 2.0 .2 .001 9-99 9-97 9-93 9-85 9-74 9-59 9-38 .2 4 .001 9-99 9.96 9-91 9.82 9.69 9-51 9.26 4 .6 .002 9-99 9.96 9.90 9.80 9-64 9.42 9-14 .6 .8 .OO2 9-99 9-95 9.88 9.76 9-58 9-33 9.00 .8 30 20.002 39-99 59-94 79.86 99-73 II9.52 I39.23 158.85 30 .2 .003 9.98 9-94 9-84 9.69 9-45 9-13 8.69 .2 4 .003 9-98 9-93 9.82 9-65 9-39 9.02 8.53 4 .6 .003 9-98 9.92 9.80 9.60 9-31 8.90 8.35 .6 .8 .004 9.98 9.91 9-78 9 56 9-23 8-77 8.16 .8 4.0 20.004 39-98 59-90 79-76 99-51 "9-15 138.64 157-96 4.0 .2 .004 9-97 9.89 9-73 9-46 9.06 8.50 7-75 .2 -4 .005 9-97 9.88 9-71 9-41 8.97 8.36 7-54 4 .6 .005 9-9? 9-87 9.68 9.36 8.87 8.20 7-30 .6 .8 .006 9-97 9.86 9-65 9-30 8.78 8.04 7-07 .8 S.o 20.006 39.96 59-85 79-62 99-24 118.67 137-88 156.82 S.o .2 .007 9.96 9-84 9-59 9.18 8.56 7.71 6.56 .2 4 .007 9.96 9.82 9-56 9.12 8.45 7-52 6.29 4 .6 .008 9-95 9.81 9-52 9-05 8-34 7-34 6.02 .6 .8 .009 9-95 9.80 9-49 8.98 8.21 7-iS 5-73 .8 6.0 20.009 39-94 59-78 79-45 98.91 118.09 136.95 155-43 6.0 .2 .010 9-94 9-77 9-41 8.83 7-96 6.74 5-12 .2 4 .010 9-94 9-75 9-38 8.76 7-83 6.53 4.81 4 .6 .Oil 9-93 9-74 9-34 8.68 7-69 6.31 4.48 .6 .8 .012 9-93 9-72 9-30 8.60 7-55 6.09 4-15 .8 7.0 20.012 39-92 59-70 79.26 98.51 117.40 135.86 I53.8o 7.0 .2 .013 9.92 9.69 9-21 8-43 7-25 5-62 3-45 .2 -4 .014 9.92 9-67 9.17 8-34 7.10 5-37 3-o8 4 .6 CIS 9-91 9-65 9.12 8.25 6-94 5.12 2.71 .6 .8 .015 9.91 9-63 9.08 8.16 6.78 4.87 2.32 .8 8.0 20.016 39-90 59-6i 79-03 98.06 116.62 134-60 151-94 8.0 .2 .017 9-90 9-59 8.98 7-96 6-45 4-33 I. S3 .2 4 .018 9.89 9-57 8.93 7.86 6.27 4.06 1. 12 4 .6 .019 9-89 9-55 8.88 7.76 6.09 3-77 0.70 .6 .8 .020 9.88 9-53 8.83 7.66 5-91 3-49 0.27 .8 9-0 20.O2I 39-88 59-51 78.77 97-55 H5-73 I33-I9 149.83 9-o .2 .022 9-87 9-49 8.72 7-44 5-54 2.89 9.38 .2 4 .022 9.87 9.46 8.66 7-33 5-34 2.58 8.93 4 6 .023 9.86 9-44 8.60 7.21 5-15 2.27 8.46 .6 .8 .024 9-85 9-42 8.55 7.10 4-94 1-95 7-99 .8 10 2O . O25 39.85 59-39 78.48 96.98 114-74 131.62 147-50 10.0 Degree Actual arc 2Sta. 3 Sta. 4 Sta. 5 Sta. 6 Sta. 7 Sta. 8 Sta. Degree (316) TABLE LXXV. (Continued) Degree of Actual arc, one Long chords Degree of curve station 2Sta. 3Sta. 4 Sta. 5 Sta. 6 Sta. 7 Sta. 8 Sta. curve 10.0 20.025m. 39-85 59-39 78.48 96.98 H4-74 131.62 147-50 10.0 .2 .026 9-84 9-37 8.42 6.86 4-53 1.29 7.01 .2 .4 .027 9-84 9-34 8.36 6.74 4-31 0.95 6.51 -4 .6 .029 9-83 9-32 8.30 6.61 4.10 0.61 6.00 .6 .8 .030 9-82 9-29 8.23 6.48 3-87 0.25 5-48 .8 II. 20.031 39-82 59-27 78.17 96.35 113-65 129.90 144-95 II. .2 .032 9.81 9-24 8.10 6.22 3-42 9-54 4.42 .2 4 033 9.80 9.21 8.03 6.08 3-19 9-17 3-87 4 .6 034 9.80 9.18 7-97 5-95 2.95 8.80 3-32 .6 .8 .035 9-79 9-15 7-90 5-8i 2.71 8.42 2.76 .8 12.0 20.037 39-78 59-13 77-82 95-6? 112.47 128.03 142.19 12.0 .2 .038 9-77 9.10 7-75 5-52 2.21 7.64 1.61 .2 4 039 9-77 9-07 7-68 5.38 1.96 7.24 1.03 4 .6 .040 9-76 9-04 7.60 5-23 I.7I 6.84 0-43 .6 .8 .042 9-75 9.01 7-53 5-o8 1-45 6.43 139-83 .8 13 o 20.043 39-74 58.98 77-45 94-93 111.18 126.01 139-22 13 o .2 044 9-73 8.94 7-37 4-77 0.92 5-59 8.60 .2 4 .046 9-73 8.91 7.29 4-61 0.65 5-17 7-98 4 .6 .047 9-72 8.88 7.21 4-45 0.37 4-73 7-34 .6 .8 .048 9-71 8.85 7-13 4-29 0.09 4-30 6.70 .8 14.0 20.050 39-70 58.81 77-05 94-13 109.81 123.86 136.05 14.0 .2 .051 9-69 8.78 6.96 3-96 9-53 3-41 5-40 .2 -4 053 9.68 8.74 6.88 3.8o 9-23 2-95 4-73 4 .6 .054 9-68 8.71 6-79 3-63 8.95 2.50 4.06 .6 .8 .056 9-67 8.67 6.70 3 45 8.65 2.03 3-38 .8 IS o 20.057 39-66 58.64 76.61 93-28 108.35 121.56 132.70 15 o .2 059 9-65 8.60 6.52 3-10 8.04 1.09 2.0O .2 4 .060 9.64 8.56 6.43 2.92 7-74 0.61 1.30 4 .6 .062 9-63 8.53 6.34 2.74 7-43 0.12 O.6O .6 .8 .064 9.62 8.49 6.25 2.56 7. II 119.63 129-88 .8 16.0 20.065 39 -61 58.45 76.15 92-37 106.79 119.14 I29.I6 16.0 .2 .067 9.60 8.41 6.06 2.18 6.48 8.6 4 8.44 .2 4 .068 9-59 8.37 5-96 1.99 6.15 8.13 7-70 4 .6 .070 9-58 8.33 5-86 1. 80 5-82 7.62 6.96 .6 .8 .072 9-57 8.29 5.76 1.61 5-49 7.11 6.21 .8 17.0 20.074 39-56 58.25 75-66 91.41 105.16 116.59 125.46 17.0 .2 .075 9-55 8.21 5-56 1.22 4.82 6.06 4-70 .2 4 077 9-54 8.17 5-46 1. 01 4-47 5-53 3-93 4 .6 079 9-53 8.13 5-36 0.8l 4-13 S.oo 3-16 .6 .8 .081 9 52 8.08 5-25 0.61 3-78 4-46 2.38 .8 18.0 20.082 39-51 58.04 75-15 90.40 103-43 H3-9I 121.59 18.0 .2 .084 9-50 8.00 5-04 0.19 3-o8 3-36 0.80 .2 4 .086 9-48 7-95 4-93 89-98 2.72 2.81 o.oo 4 .6 .088 9-47 7-91 4-83 9-77 2.36 2.26 119.20 .6 .8 .090 9-46 7-87 4-71 9-56 i-99 1.69 8.39 .8 19.0 20.092 39-45 57-82 74-6o 89-34 101 . 63 111.13 117-58 19.0 .2 .094 9-44 7-78 4-49 9-12 1.26 0.56 6.76 .2 4 .096 9-43 7-73 4-38 8.90 0.88 109.98 5-93 4 .6 .098 9-42 7-68 4-27 8.68 0.51 9.40 5.10 .6 .8 .100 9-40 7.64 4-15 8.46 0.13 8.82 4.27 .8 20.0 20.102 39-39 57-59 74.03 88.23 99-74 108.23 113-43 20.0 Degree Actual arc 2Sta. 3 Sta. 4 Sta. 5 Sta. 6 Sta.^ 7 Sta. 8 Sta. Degree (317) TABLE LXXVI. METRIC CURVES Degree Middle ordinates Degree of curve i Sta. 2 Sta. 3 Sta. 4 Sta. 5 Sta. 6 Sta. 7 Sta. 8 Sta. of curve 0.2 .009 -035 .079 .140 .218 '314 .428 558 0.2 4 .017 .070 .157 .279 436 .628 .855 1. 117 -4 .6 .026 .105 .236 419 .654 942 1.283 1.675 .6 .8 .035 .140 314 558 .873 .256 1.710 2.233 .8 I.O .044 175 393 .698 .091 570 2.137 2.791 I.O .2 .052 .209 471 .838 .309 .884 2.565 3-349 .2 4 .061 .244 550 977 .527 .198 2.992 3.907 4 .6 .070 .279 .624 1.117 745 512 3.418 4.464 .6 .8 .079 314 .707 1.256 .963 .825 3-845 5.020 .8 2.0 .087 349 .785 1-396 2.180 3-139 4-271 5.576 2.0 .2 .096 .384 .864 1-535 2.398 3 452 4.697 6.132 .2 4 .105 .419 942 1.675 2.616 3.765 5-122 6.687 -4 .6 .113 454 1. 021 1.814 2.833 4-078 5.548 7-241 .6 .8 .122 .489 .099 1-953 3-051 4-391 5-973 7-795 .8 30 .131 .524 .178 2.093 3-268 4.703 6-397 8-348 30 .2 .140 558 .256 2.232 3.485 5-015 6.821 8.900 .2 4 .148 593 335 2.371 3.703 5-327 7.244 9-451 -4 .6 .157 .628 413 2.510 3-920 5.639 7.667 10. OO2 .6 .8 .166 .663 491 2.650 4-136 5-950 8.090 10.551 .8 4.0 .175 .698 570 2.789 4-353 6.262 8.511 II.IOO 4.0 .2 .183 733 .648 2.928 4-570 6.572 8.933 11.647 .2 4 .192 .768 .726 3.067 4-786 6.883 9-353 12.194 4 .6 .201 .803 .805 3-205 5.002 7-193 9-773 12.739 .6 .8 .209 .838 .883 3-344 5.218 7-502 10.192 13.283 .8 50 .218 .872 .961 3.483 5-434 7.812 10.611 13.826 50 .2 .227 .907 2.040 3-622 5-650 8. 121 11.028 14.367 .2 4 .236 942 2.118 3.76o 5-865 8.429 11-445 14.907 -4 .6 .244 977 2.196 3-899 6.081 8.737 11.862 15.446 .6 .8 .253 I.OI2 2.274 4-037 6.296 9-045 12.277 15.983 .8 6.0 .262 1.047 2.352 4.175 6.511 9-352 12.691 16.519 6.0 .2 .271 .082 .431 4-314 6.725 9.658 13-104 17.053 .2 -4 .279 .116 .509 4-452 6.940 9.965 13.517 17.586 4 .6 .288 .151 .587 4-590 7-154 10.270 13.929 18.117 .6 .8 .297 .186 .665 4.728 7.368 10-575 14-339 18.646 .8 70 .306 .221 743 4.866 7.581 10.880 14-749 19.174 7.0 .2 .314 .256 .821 5-003 7-795 11.184 I5-I57 19.700 .2 4 323 .291 2.899 5.I4I 8.008 11.487 15.565 20.223 4 .6 332 .325 2.977 5-279 8.221 11.790 15.971 20.745 .6 .8 340 .360 3.055 5.4i6 8.433 12.092 16.376 21.265 .8 8.0 349 395 3 133 5-553 8.645 12.394 16.780 21.783 8.0 .2 358 430 3-2II 5-691 8.857 12.695 17.183 22.299 .2 4 .367 .465 3-288 5-828 9.069 12.995 17.584 22.813 .4 .6 375 .500 3.366 5-965 9.280 13-294 17.985 23.325 .6 .8 .384 534 3-444 6.101 9.491 13-594 18.384 23-834 .8 9.0 393 .569 3-522 6.238 9.702 13.892 18 . 782 24.342 90 .2 .402 .604 3-599 6.375 9.912 14.189 19.178 24.847 .2 4 .410 -639 3.677 6.511 IO.I22 14.486 19-573 25.350 4 6 .419 .674 3-754 6.647 10.332 14.782 19.967 25.850 .6 .8 .428 .708 3-832 6.784 10-541 15.078 20.359 26.348 .8 10.0 437 1-743 3.9io 6.919 10.750 15-372 20.750 26.843 10 Degree I Sta. 2 Sta. 3 Sta. 4 Sta. 6 Sta. 7 Sta. 8 Sta. Degree (318) TABLE LXXVI. (Continued) Degree Middle ordinates Degree of curve I Sta. 2 Sta. 3 Sta. 4 Sta. 5 Sta. 6 Sta. 7 Sta. 8 Sta. of curve 10.0 437 1-743 3-910 6.919 10.750 15-372 20.750 26.843 10.0 .2 445 1.778 3.987 7-055 10.958 15-666 21.139 27.336 .2 4 454 1-813 4.065 7.191 11.167 15-959 21.527 27.827 4 .6 .463 1.847 4.142 7-327 H-374 16.251 21.913 28.315 .6 .8 .472 1.882 4.219 7-462 11.582 16.542 22 . 298 28.800 .8 XI. O .480 1.917 4-297. 7-597 11.789 16.832 22.681 29.282 II. .2 .489 1-952 4-374 7-732 H-995 17.122 23.063 29.762 .2 4 .498 1.986 4-451 7.867 12.201 17.410 23-443 30.239 4 .6 SO? 2.O2I 4-539 8. 002 12.407 17.698 23.821 30.714 .6 .8 SIS 2.O56 4-605 8.137 I2.6I2 17-985 24.198 31 185 .8 12.0 .524 2.O9I 4.682 8.271 12.817 18.271 24-573 31 654 12.0 .2 .533 2.125 4-759 8.405 13-021 18.556 24.946 32.119 .2 4 542 2.160 4-836 8.539 13.225 18.840 25.317 32-582 4 .6 550 2.195 4.913 8.673 13.429 19-123 25-687 33-041 .6 .8 559 2.229 4.990 8.807 13.632 19-405 26.055 33.498 .8 13.0 .568 2.264 5.067 8.940 13-834 19.686 26.421 33.951 13 o .2 577 2.299 5-144 9.074 14.036 19.966 26.785 34-402 .2 4 .585 2.333 5-220 9.207 14.238 20 . 245 27-147 34-849 4 .6 594 2.368 5-297 9-339 14.439 20.523 27.507 35.292 .6 .8 .603 2.403 5-374 9-472 14.639 20.800 27.866 35-733 .8 14.0 .612 2.437 5-450 9-6os 14.839 21.076 28.222 36.171 14.0 .2 .620 2.472 5.527 9-737 15.039 21-351 28.576 36.604 .2 4 .629 2.507 5.603 9.870 15.238 21.625 28.929 37-035 4 .6 .638 2.541 5.679 IO.OOI 15.437 21.897 29.279 37.462 .6 .8 .647 2.576 5.756 10.133 15.634 22.169 29.628 37-886 .8 15.0 .655 2.611 5.832 10.264 15.832 22.439 29.974 38.306 15 o .2 .664 2.645 5.908 10.395 16.028 22.709 30.318 38.723 .2 4 .673 2.680 5.984 10.526 16.225 22.977 30.660 39-137 4 .6 .682 2.704 6.060 10.657 16.421 23.244 31.000 39-546 .6 .8 .690 2.749 6.136 10.788 16.616 23-509 31.338 39-952 .8 16.0 699 2.783 6.212 10.918 16.810 23.774 31.673 40.355 16.0 .2 .708 2.818 6.288 11.048 17.005 24-037 32.007 40-753 .2 4 .717 2.853 6.364 11.178 17.198 24.299 32.338 41.148 4 .6 .726 2.887 6.439 11.308 17-390 24-560 32.667 41-540 .6 .8 734 2.922 6.515 H-437 17-583 24.820 32-993 41.927 .8 17.0 743 2-956 6.591 11.566 17-774 25.078 33-317 42.311 17.0 .2 752 2.991 6.666 11.696 17.966 25-335 33.639 42.691 .2 4 .761 3-025 6.741 11.824 18.156 25-591 33-959 43-067 4 .6 .769 3.060 6.817 11-953 18.346 25.846 34.276 43-439 .6 .8 .778 3-094 6.892 12.081 18-534 26.099 34-591 43-807 .8 18.0 .787 3.129 6.967 12.208 18.723 26.351 34.903 44.171 18.0 .2 796 3-163 7-043 12.336 18.911 26 . 601 35-213 44-531 .2 4 .805 3.198 7.117 12.464 19.098 26.850 35-521 44.887 4 .6 .813 3.232 7-193 12.593 19-285 27.098 35.826 45-239 .6 .8 .822 3.266 7.267 12.717 19.470 27-345 36.129 45.587 .8 19.0 .831 3-301 7-342 12.844 19.656 27.590 36.429 45.931 19.0 .2 .840 3 335 7.417 12.971 19.840 27-833 36.727 46.271 .2 4 .849 3-370 7-492 13.097 20.024 28.076 37-022 46.606 4 .6 .857 3.404 7.566 13.222 20 . 207 28.316 37.314 46.938 .6 .8 .866 3-439 7-641 13.348 20.389 28.556 37.604 47-265 .8 20 .875 3-473 7.715 13 473 20.571 28.794 37.892 47-588 20.0 Degree i Sta. 2 Sta. 3 Sta. 4 Sta. 5 Sta. 6 Sta. 7 Sta. 8 Sta. Degree (319) CHAPTER IX MISCELLANEOUS TABLES TABLE LXXVII. TRIGONOMETRIC FORMULAS, CIRCULAR MEASURE, ETC. Trigonometric Functions and Formulas. Solution of Triangles By definition, if R = I , ED = sine a. OD = cosine a. DA versed sine a. HF = coversed sine a. BA = tangent a. FC = cotangent a. OB = secant a. OC = cosecant a. If R is other than I, it follows from the above definitions and the proportionality of similar figures, that 1. ED = Rsina. 5. BA =R tan a. 2. OD = R cos a. 6. FC = R cot a. 3. DA '= R versin a. 7. OB = R sec a. 4. HF = R coversin a. 8. OC = R cosec a. from which also in any right triangle of angles a and ft, if o be the side opposite the angle a, o the side adjacent thereto, and h the hypotenuse, sin a IO. cos a cos ft. 13- 14. sec a = - = cosec ft. 1 1 . tan a = - = cot ft. a 12. cot a = - = cosec a versa - = sec 0. 1 6. covers a = fr-a h h-o covers /3. vers 0. Hence, 17- o = h sin a = h cos 0. cos/3 18. 320 h cos a = h sin /3. a a cos a sin /3 MISCELLANEOUS TABLES o = a tan a = a cot 0. 3 21 I9 . 21. tan a cot /3 a = o cot a = o tan /3. 22. 24. h = o cosec a = o sec 0. o = cosec a sec /3 a) (ha). = acosec/3. 2 5- *= 322 RAILROAD FIELD MANUAL Oblique triangles may be solved by some one of the following formulas: GIVEN SOUGHT FORMULAS 27. A,B, a, 28. A,a,b, 29. C, a, b, 30. C, a, b, 31. C, a, b, 32. C, a, b, 33- C, o, b, 34. a, 6, c, 35- a,b,c t 36. 4,B, C, a, B, C, c, % (A + 5), 4(4- 5), 4,5, c, Area, Area, Area, c-^tai'-jg^ ~: n R_ sln 4T ^ _ q n o / 4 I R \ sin .4 01 4 (-4 + 5) = 90 - 4 C. 7 tan l ( A Tt\ ~ tan * ( A -H- R\ | 5 = | (A + B} - | (A - B)'. Area \ab sin C. If ^ = i (a + b + c), 4 /( 5 _ &) ( s c ) sin 2 A V/ , 1 A- V / 5 ^ ~ ^ ^ 7( 5 _ ft) ( 5 - c ) uu 2 4 - y 5 ( 5 _ a ) j _ 2 V(^ a ) ( 5 6 ^ ( 5 ~~ C ) 2(5-6)(f-c) V^lo^ ^ Area = Vs(s a) (s 6) (5 c). A a 2 sin 5 sin C 2 sin A MISCELLANEOUS TABLES 323 From the definitions of the trigonometric functions, the geometrical properties of right triangles and in some cases algebraic transformations, it may be shown that if A is any angle and B any other angle, 37. sin 2 A + cos 2 A = I . -58. sin A = 7 = VI cos 2 A = tan A cos A cosec A = 2 sin A cos A = vers A cot 5 .4 39- cos 40. tan 41. cot V (i cos 2 A). A A/T sin 2 A = cot A sin A > cos 2 \A i = i 2 sin 2 j /I sec A i vers A = : cos 2 \ A sin 2 sin A i j /I = V + % COS 2 .4 . = Vsec 2 A - i cos A cot A -\/ ' -!- Vi cos 2 A sin 2 A ' VcosM I COS 2 A cos A i + cos 2 A vers 2 A sin 2 A cos A i sin2A sin A tan A sin 2 A sin 2 A I + cos 2 A tan | /I I COS 2 A vers 2 -4 sin 2 /I sec /4 i 42. vers A = i cos A = sin A tan \A = 2 sin 2 % A = cosA (sec A i). 43. sin (/I 5) = sin -4 cos B sin 5 cos -4. 44. cos 04 B) = cos /4 cos B =F sin -4 sin B. . , . 4 /i cos ^4 4 /vers -4 45. sm^==y =V-T-' 46. sin 2/1 = 2 sin .4 cos A . 'i 47. cos* ,4 48. cos 2 ^4 = 2 cos 2 A i = cos 2 -4 sin 2 A = i 2 sin 2 A. tan A I cos ^4 4/1 cos A 49. tan $ .4 = : -; = cosec A cot A = ; -. = V ; T i +sec A sin A T i + cos A 2 tan A si s{nA I+cosA vers /4 sin ^4 cosec A cot -4 * S2 . MiA- **-. 1 2 COt A 324 RAILROAD FIELD MANUAL 5 vers A 53. vers^4= I + V i 5 vers .4 54. vers 2 A = 2 sin 2 -4 . 55. sin A + sin B = 2 sin 56. sin -4 sin B = 2 cos 57. cos A -f cos 5 = 2 cos 58. cos 5 cos A = 2 sin cos A 2 + V 2 (i + cos A) (A + B) cos%(A - B). (A + B} sin \(A - B}. (&'&) cosi (4 - 5).. (A + B) sin J(4 - B). B = cos 2 5 - cos 2 ,4 = sin (A + B} sin (A - B). 59. sin 2 4 - sin 60. cos 2 A - sin 2 B = cos (-4 + B) cos C4 - ' sin (A + 5) 61. tanA+tanB = - . ' cos. -4 cos 5 62. tan 4 tan B sin (A ff) cos A cos B LENGTH OF CIRCULAR ARCS TO RADIUS i Deg. Length Deg. Length Deg. Length Deg. Length Deg. Length 10 0.1745329 50 0.8726646 90 1.5707963 130 2.2689280 170 2.967059? 20 0.3490659 60 1.0471976 IOO i - 7453293 140 ; 2. 4434610 180 3.1415927 30 0.5235988 70 i . 2217305 no 1.9198622 150 2.6179939 190 3.3161256 40 0.6981317 80 1.3962634 1 20 2.0943951 160 2.7925268 200 3.4906585 IT = 3.I4I59 Degrees in arc of length equal to radius, Degrees in arc of length equal to TT, Circumference = 2 irr = Area = irr 2 . If / = length of circular arc d = number of degrees in same r = radius of same c = chord of same m middle ordinate J7T = 0.497150 5, 57.295780. 180. 360. I T 1 80 7T -^57- 3 approx. I d 1 80 *S 3 approx. d ' -d I = r- irr = o r approx. 1 80 57.3 Area of sector \ Ir. :.f d Area of sector = ^- irr 2 . Approx. area of segment = \cm. - = 0.31831. Volume of sphere = f irr 3 . Square feet in i acre = 43,560. Cubic feet in i cubic meter = 35.3145. MISCELLANEOUS TABLES 325 TABLE LXXVIII. STADIA FUNCTIONS Differences of elevation for 100 feet rod reading Deg. o I 2 3 4 5 6' 7 8 9 Deg. .00 o.oo 1-74 3-49 5-23 6.96 8.68 10.40 12.10 13-78 15-45 .00 05 0.09 1.83 3-57 5-31 7-05 8.77 10.48 12.18 13-87 15.53 05 .1 0.17 1.92 3.66 5-40 7-13 8.85 10.57 12.26 13-95 15.62 .1 IS 0.26 2.01 3-75 5-49 7.22 8-94 10.65 12.35 14-03 15-70 .15 .2 0.35 2.09 3-84 5-57 7-3 9-03 10.74 12.43 14.12 15.78 .2 .25 0.42 2.18 3-92 S-66 7-39 9 .11 10.82 12.52 14.20 15-87 25 3 0.52 2.27 4.01 5-75 7.48 9.20 10.91 12.60 14.28 15.95 3 35 0.61 2. 3 6 4.10 5.83 7.56 9.28 10.99 12.69 14-37 16.03 35 4 0.70 2.44 4.18 5-92 7.65 9-37 11.08 12.77 14-45 16.11 4 45 0.79 2. S3 4.27 6.01 7-73 9.46 ii. 16 12.86 14-54 16.20 45 .5 0.87 2.62 4-36 6.09 7.82 9-54 11.25 12.94 14.62 16.28 5 55 0.96 2.70 4-44 6.18 7.91 9-63 H-33 13-03 14.70 16.36 55 .6 i. 05 2.79 53 6.27 7-99 9-71 11.42 13.11 14-79 16.44 .6 -65 1. 13 2.88 .62 6.35 8.08 9.80 11.50 13-19 14.87 16.52 65 7 1.22 2.97 71 6-44 8.17 9-88 11-59 13.28 14-95 16.61 .7 75 I.3I 3-OS 79 6.53 8.25 9-97 11.67 13.36 15.04 16.69 75 .8 1.40 3-14 .88 6.61 8.34 10.05 11.76 13.45 15.12 16.77 8 .85 1.48 3-23 4-97 6.70 8.42 10.14 11.84 13 53 15.20 16.86 .85 9 1.57 3-31 5-05 6.79 8.51 IO.22 H-93 13.61 15 28 16.94 9 95 1.66 3-40 5-14 6.87 8.60 10.31 12. OI 13.70 15-37 17.02 95 I.OO 1-74 3-49 5-23 6.96 8.68 10.40 12.10 13.78 15-45 17.10 I.OO Corrections to rod readings for horizontal distance Rod o 1 2 3 4 5 6 7 8 9 Rod 100 0.0 0.0 O.I 0.3 o.S 0.8 i.i 1-5 i-9 2.5 100 200 0.0 O.I O.2 o.S I.O 1-5 2.2 3-0 3-9 4-9 2OO 300 o.o O.I 0.4 0.8 1-5 2.3 3-3 4-5 5-8 7-4 300 400 0.0 O.I o.S i.i 2.O 3-0 4-4 6.0 7-8 9-8 400 500 o.o 0.2 0.6 1-4 2.5 3-8 5-5 7-5 9-7 12.3 500 600 0.0 0.2 0-7 1.6 2-9 4.6 6-5 8-9 ii. 6 14-7 600 700 0.0 0.2 0.8 ii-9 3-4 5-3 7-6 10.4 13.6 17.2 700 800 o.o O.2 I.O 2.2 3-9 6.1 8-7 11. 9 I5-S 19.6 800 900 o.o 0-3 I.I 2.4 4-4 6.8 9-8 13-4 17-5 22.1 900 IOOO o.o 0.3 1.2 2.7 4-9 7-6 10.9 14-9 19.4 24:5 IOOO 326 RAILROAD FIELD MANUAL TABLE LXXVIII. (Continued) Deg. 10 ii 12 13 14 15 16 17 18 19 Deg. .00 17.10 18.73 20.34 21.92 23.47 25.00 26.50 27.96 29-39 30.78 .00 .05 17.18 18.81 20.42 22.00 23.55 25.08 26.57 28.03 29-46 30.85 .05 .z 17.26 18.89 20.50 22.08 23.63 25.15 26.64 28.10 29.53 30.92 .1 .15 17.35 18.97 20.58 22.15 23.70 25 23 26.72 28.18 29.60 30.99 15 .2 17.43 19-05 20.66 22.23 23.78 25.30 26.79 28.25 29.67 31.06 .2 25 17.51 19.13 20.73 22.31 23.86 25.38 26.87 28.32 29-74 3I-I3 25 3 17.59 19.21 20.81 22.39 23.93 25.45 26.94 28.39 29.81 3I.I9 3 35 17.67 19.30 20.89 22.47 24.01 25.53 27.01 28.46 29.88 31.26 35 4 17.76 I9.38 20.97 22.54 24.09 25.60 27.09 28.54 29.95 31-33 4 45 17.84 19.46 21.05 22.62 24.16 25.68 27.16 28.61 30.02 31.40 45 5 17.92 19-54 21.13 22.70 24.24 25-75 27.23 28.68 30.09 31-47 5 .55 18.00 19.62 21.21 22.78 24.32 25-83 27-31 28.75 30.16 31-53 .55 .6 18.08 19.70 21.29 22.85 24.39 25.90 27-38 28.82 30.23 31.60 .6 .65 18.16 19.78 21.37 22.93 24.47 25-98 27-45 28.89 30.30 31 67 .65 .7 18.24 19.86 21.45 23.01 24.55 26.05 27.52 28.96 30-37 31-74 .7 75 18.33 19-94 21.53 23.09 24.62 26.13 27.60 29.04 30.44 31-80 75 .8 18.41 20.02 21. 60 23.16 24.70 26.20 27.67 29.11 30.51 31-87 .8 .85 18.49 20. 10 21.68 23.24 24.77 26.27 27-74 29.18 30.58 31-94 .85 9 18.57 20.18 21.76 23.32 24-85 26.35 27.81 29.25 30.65 32.01 9 95 18.65 20.26 21.84 23.40 24.92 26.42 27-89 29.32 30.71 32.07 95 x.oo 18.73 20.34 21.92 23.47 25.00 26.50 27.96 29-39 30.78 32.14 1. 00 Corrections to rod readings for horizontal distance Rod 10 11 12 13 14 15 16 17 18 19 Rod _____ 100 3-0 3-6 4-3 S.I 5-9 6.7 7-6 8.5 9-5 10.6 100 200 6.0 7-3 8.6 IO.I II. 7 13-4 15.2 17.1 19.1 21.2 2OO 300 9-1 10.9 13-0 15-2 17.6 20.1 22.8 25.6 28.6 31-8 300 400 12. 1 14.6 17-3 20. 2 23-4 26.8 30.4 34-2 38.2 42-4 400 500 I5-I 18.2 21.6 25-3 29-3 33-5 38.0 42.7 47-7 53-0 500 600 18.1 21.8 25-9 30-4 35-1 40.2 45-6 51-3 57-3 63.6 600 700 21. 1 25-5 30.2 35-4 41.0 46.9 53-2 59-8 66.8 74.2 700 800 24.2 29.1 34-6 40.5 46.8 53-6 60.8 68.4 76.4 84.8 800 900 27.2 32.8 38.9 45.5 52.7 60.3 68.4 76-9 85-9 95-4 900 1000 30.2 36.4 43-2 50.6 ** 67.0 76.0 85-5 95-5 106.0 IOOO MISCELLANEOUS TABLES 327 TABLE LXXVIII. (Continued) Deg. 20 21 22 23 24 25 26 2? 28 29 Deg. .00 32.14 33.46 34 73 35-97 37-i6 38-30 39-40 40.45 41-45 42.40 .00 .05 32.21 33.52 34-80 36.03 37-22 38.36 39-45 40.50 41-50 42-45 .05 .1 32.27 33-59 34-86 36.09 37-27 38-41 39-51 40.55 41-55 42-49 .x IS 32.34 33.65 34.92 36.15 37-33 38-47 39.56 40.60 41.60 42.54 -IS .2 32.41 33-72 34-98 36.21 37-39 38.53 39-6i 40.66 41-65 42.59 .2 25 32.47 33.78 35-05 36-27 37-45 38-58 39-67 40.71 41.69 42.63 .25 3 32-54 33.84 35-11 36.33 37-51 38.64 39-72 40.76 41-74 42.68 3 35 32.61 33-91 35-17 36.39 37-56 38.69 39-77 40.81 41-79 42.72 .35 4 32.67 33-97 35-23 36.45 37.62 38.75 39-83 40.86 41.84 42.77 4 45 32.74 34-04 35-29 36.51 37-68 38.80 39-88 40.91 41.89 42.81 45 5 32.80 34-10 35-36 36.57 37-74 38.86 39-93 40.96 41-93 42.86 5 55 32.87 34-i6 35-42 36.63 37-79 38-91 39-98 41.01 41-98 42.90 55 .6 32.93 34-23 35-48 36.69 37-85 38.97 40.04 41.06 42.03 42.95 .6 .65 33-00 34.29 35-54 36.75 37-91 39-02 40.09 41.11 42.08 42.99 .65 7 33-07 34-35 35-6o 36.80 37.96 39-08 40.14 41.16 42.12 43-04 7 .75 33 13 34-42 35-66 36.86 38.02 39-13 40.19 41.21 42.17 43-08 75 .8 33 20 34.48 35-72 36.92 38.08 39-18 40.24 41.26 42.22 43-13 .8 .85 33.26 34-54 35-78 36.98 38.13 39-24 40.30 41-31 42.26 43-17 .85 9 33-33 34-6i 35-85 37-04 38.19 39-29 40-35 41-35 42.31 43-21 .9 95 33-39 34.67 35-91 37-10 38.25 39 35 40.40 41.40 42.36 43-26 95 1. 00 33.46 34-73 35-97 37-16 38.30 39-40 40.45 41-45 42.40 43-30 1. 00 Corrections to rod readings for horizontal distance Rod 20 21 22 23 24 25 26 27 28 29 Rod 100 II. 7 12.8 14-0 15-3 16.5 17.9 19.2 20.6 22.0 23.5 100 200 23.4 25.7 28.1 30.5 33-1 35-7 38-4 41.2 44-1 47.0 200 300 35- i 38.5 42.1 45-8 49-6 53-6 57-7 61.8 66.1 70.5 300 400 46.8 51.4 56.1 61.1 66.2 7i-4 76.9 82.4 88.2 94-0 400 500 58.5 6 4 .2 70.2 76.4 82.7 89-3 96.1 103.1 IIO. 2 II7-5 500 600 70.2 77-0 84.2 91.6 99-2 107.2 II5-3 123.7 132.2 141.0 600 700 81.9 89.9 98.2 106.9 115.8 125.0 134-5 144.3 154-3 164-5 700 800 93-6 102.7 112. 2 122.2 132-3 142.9 153-8 164.9 176.3 188.0 800 900 105-3 115.6 126.3 137-4 148.9 160.7 173-0 185.5 198.4 211.5 900 1000 117.0 128.4 140-3 152.7 165.4 178.6 192.2 206.1 22O.4 235.0 1000 328 RAILROAD FIELD MANUAL TABLE LXXIX. BAROMETRIC ELEVATIONS Giving altitudes above arbitrary sea level (barometer reading 30 inches) for various barometer readings B. To determine difference of elevation of two points having barometer readings B and Bi, take from the table the altitudes corresponding to B and BI, and correct their difference by Table LXXX. The corrected difference is the quantity required. B A Diff. for o . 01 B A Diff. for c . 01 B A Diff. for o . 01 Inches Feet Feet Inches Feet Feet Inches Feet Feet II. ii. i 27,336 27,090 -24.6 14.0 14.1 20,765 20,570 -19-5 17.0 17.1 15,476 I5,3i6 ' 16.0 II. 2 ii. 3 11. 4 ii. 5 26,846 26,604 26,364 26,126 24.4 24.2 24.0 23.8 23.6 14.2 14-3 14.4 14-5 20,377 20,186 19,997 19,809 19-3 19.1 18.9 18.8 18.6 17.2 17-3 17-4 17-5 15,157 14,999 14,842 14,686 15-9 15.8 15-7 15.6 15.5 II. 6 ii. 7 25,890 25,656 23.4 14.6 14-7 19,623 19.437 18.6 ' I7 .6 17-7 14,531 14,377 15.4 II. 8 25,424 23.2 14.8 19,252 18.5 17.8 14,223 15-4 23.0 18.4 15-3 11.9 12.0 12. 1 25,194 24,966 24,740 22.8 22.6 22.4 14.9 iS.o I5-I 19,068 18,886 18,705 18.2 18.1 18.0 17-9 18.0 18.1 14,070 I3,9i8 13,767 15-2 15.1 15.0 12.2 24,516 15.2 18,525 18.2 13,617 12.3 24,294 22.2 15-3 18,346 17.9 18.3 13,468 14-9 12.4 24,073 22.1 21.9 15-4 18,168 17.8 17.6 18.4 13,319 14.9 14-7 12.5 23,854 15-5 17,992 18.5 13,172 12.6 12.7 23,637 23,421 21.7 21.6 15.6 15-7 17,817 17,643 17-5 17-4 18.6 18.7 13,025 12,879 14-7 14.6 21.4 17.3 14.6 12.8 23,207 15.8 17,470 18.8 12,733 12.9 22,995 21.2 15,9 17,298 17.2 18.9 12,589 14.4 13.0 22,785 21. 2O.9 16.0 17,127 17.1 16.9 19.0 12,445 14.4 14.3 13. 1 13,2 13 3 22,576 22,368 22,162 20.8 20.6 16.1 16.2 16.3 16,958 16,789 16,621 16.9 16.8 I9-I 19.2 19-3 12,302 12,160 12,018 14-2 14.2 20.4 16.7 14. 1 13.4 13.5 13.6 21,958 21,757 ' 21,557 20.1 20. o 19.9 16.4 16.5 16.6 16,454 16,288 16,124 16.6 16.4 16.3 19-4 19-5 19.6 11,877 11,737 n,598 14.0 13 9 13-9 13-7 13-8 13-9 21,358 21,160 20,962 19.8- 19.8 19.7 16.7 ' 16.8 16.9 I5,96i 15,798. 15,636 16.3 16.2 16.0 19.7 19.8 19-9 11,459 11,321 11,184 13-8 13-7 13-7 14.0 20,765 17.0 15,476 20.0 11,047 Taken from Appendix 10, " U. S. Coast and Geodetic Survey Report " for 1881. MISCELLANEOUS TABLES 3 2 9 TABLE LXXIX. (Continued) B A Diff. for o.oi B A Diff. for o.oi B A Diff. for o.oi Inches Feet Feet Inches Feet Feet Inches Feet Feet 20.0 2O. I 11,047 10,911 -13-6 23-7 23.8 6,423 6,308 -ii. 5 27.4 27-5 2,470 2,371 -9-9 20.2 10,776 13 5 ' 23-9 6,194 11.4 11.4 27.6 2,272 9-9 9-9 20.3 10,642 24-0 6,080 27.7 2,173 20.4 10,508 13-4 24-1 5,967 II. 3 27.8 2,075 9.8 20.5 10,375 13 3 24-2 5,854 II. 3 27.9 1,977 9.8 13-3 11.3 9.7 20.6 10,242 24-3 5,741 28.0 1,880 20.7 10,110 13-2 24-4 5,629 II. 2 28.1 1,783 9-7 20.8 9,979 13- 1 T1 T 24-5 5,518 II. I II I 28.2 1,686 9-7 20.9 21.0 9,848 9-7i8 J-,5- ! 13-0 24-6 24-7 5,407 5,296 II. I 28.3 28.4 1,589 1,493 9-7 9-6 21. 1 9,589 12.9 12.9 24.8 5,i86 II. 10.9 28.5 1,397 9-6 9.5 21.2 9,46o 24-9 5,077 28.6 1,302 21.3 21.4 9,332 9,204 12.8 12.8 25.0 25.1 4,968 4,859 IO.9 10.9 28.7 28.8 1,207 1,112 9-5 9-5 12.7 10.8 9.4 21-5 21.6 9,077 8,951 12.6 __ 25-2 25-3 4,751 4,643 10.8 28.9 29.0 1,018 924 9-4 21.7 8,825 12. 12.5 25-4 4,535 10.8 10.7 29.1 830 9-4 9.4 21.8 8,700 25-5 4,428 29.2 736 21.9 8,575 12.5 25.6 4,321 10.7 29-3 643 93 22.0 8,451 12.4 12.4 25-7 4,215 10.6 10.6 29.4 550 9-3 9.2 22.1 8,327 25-8 4,109 29-5 458 22.2 8,204 12.3 25-9 4,004 10.5 29.6 366 9.2 22.3 8,082 12.2 26.0 3,899 10.5 29.7 274 9.2 12.2 10.5 92 * 22.4 7,960 26.1 3,794 29.8 182 22.5 7,838 12.2 26.2 3,690 10.4 29-9 91 9-1 22.6 7,717 12. 1 12 O 26.3 3,586 10.4 30.0 00 9-1 22.7 7,597 26.4 3,483 10.3 30.1 -91 9- 1 22.8 7,477 12.0 26.5 3,38o 10.3 30.2 181 9-0 22.9 7,358 II-9 26.6 3,277 10.3 30-3 271 9-0 II.9 10.2 9.0 23-0 7,239 26.7 3,175 30.4 36i 23-1 7,121 ii. 8 26.8 3,073 IO.2 30-5 45i 9-0 23-2 7,004 11.7 TT 7 26.9 2,972 IO.I 30-6 540 8.9 8 n 23 3 6,887 il . / 27.0 2,871 ' 30-7 629 9 23.4 23.5 6,770 6,654 II.7 ii. 6 ii. 6 27.1 27.2 2,770 2,670 IO.I IO.O IO.O 30.8 30-9 717 805 8.8 8.8 8.8 23.6 6,538 27-3 2,570 31-0 -893 23.7 6,423 11.5 27-4 2,470 IO.O 330 RAILROAD FIELD MANUAL TABLE LXXX. CORRECTION COEFFICIENT FOR TEMPERATURE AND HYGROMETRIC CONDITIONS This correction is used when no hygrometric observations have been made. To the difference in altitude found in Table LXXIX for the given barometer readings is added algebraically the product of that difference and the correction below given, according to the formula, diff. alt. = (diff. by Table LXXIX) (i + c). Sum O. T.i Corr. coeff. Sum O. T. Corr. Coeff. Sum O. T. Corr. coeff. 10 20 30 40 So 60 0.1024 -0.0915 0.0806 0.0698 -0.0592 0.0486 0.0380 70 80 90 IOO no 120 130 0.0273 0.0166 0.0058 +0.0049 +0.0156 +0.0262 +0.0368 140 150 160 170 180 +0.0471 +0.0575 +0.0677 +0.0779 +0.0879 Computed from Tables I and IV, Appendix 10, " U. S. Coast Survey Report " for 1881. TABLE LXXXI. 1 Volume in cubic feet per second discharging over a thin plate weir one foot in length without end contractions 2 according to Francis' Formula Q = 3.33 L,VH 3 . Head, H , in ft. Cu. ft. per sec. Head, H, in ft. Cu. ft. per sec. Head, H, in ft. Cu. ft. per sec. Head, H, in ft. Cu. ft. per sec. Head, H, in ft. Cu.ft. per sec. .01 0.003 Si .213 .01 3-380 5i 6.179 2.01 9.489 .02 0.009 52 -249 .02 3-430 52 6.240 2.02 9.56o .03 0.017 53 .285 03 3-481 53 6.302 2.03 9 631 .04 0.027 54 .321 .04 3-532 54 6.364 2.04 9.703 .05 0.037 55 358 05 3.583 55 6.426 2.05 9-774 .06 0.049 56 395 .06 3.634 -56 6.488 2.06 9-846 .07 0.062 57 433 .07 3.686 57 6-551 2.07 9.917 .08 0.075 -58 471 .08 3-737 58 6.613 2.08 9.989 .09 0.090 59 509 .09 3-790 59 6.676 2.09 10.062 .10 0.105 .60 548 .10 3-842 .60 6.739 2.10 10.134 .11 O.I2I .61 .586 .11 3.894 .61 6.803 2. II 10.206 .12 0.138 .62 .626 .12 3-947 .62 6.866 2.12 10.279 .13 0.156 63 .665 13 4.000 .63 6-930 2.13 10.352 .14 0.174 .64 70S .14 4-053 .64 6.994 2.14 10.425 IS 0.193 .65 745 IS 4-107 .65 7-058 2.15 10.498 .16 0.213 .66 786 .16 4-160 .66 7.122 2.16 10.571 .17 0.233 .67 .826 17 4-214 .67 7-187 2.17 10.645 .18 0.254 .68 .867 .18 4.268 .68 7-251 2.18 10.718 19 0.276 .69 .909 19 4.323 -69 7.316 2.19 10.792 .20 0.298 70 950 .20 4-377 .70 7.381 2. 2O 10.866 .21 0.320 71 992 .21 4-432 71 7.446 2.21 10.940 .22 0.344 72 2.034 .22 4.487 72 7-512 2.22 11.015 .23 0.367 73 2.077 23 4-543 73 7-577 2.23 11.089 24 0-392 74 2.I2O .24 4-598 74 7-643 2.24 11.164 .25 0.416 75 2.163 25 4.654 75 7.709 2.25 11.239 .26 0.441 76 2.206 .26 4-710 76 7-775 2.26 11.314 .27 0.467 77 2.250 .27 4.766 77 7-842 2.27 11.389 .28 0-493 78 2.2 94 .28 4.822 -78 7.908 2.28 11.464 .29 0.520 79 2.338 .29 4.879 79 7-975 2.29 11.540 .30 0-547 .80 2.383 30 4.936 .80 8.042 2.30 11.615 .31 0-575 .81 2.428 31 4-993 .81 8.109 2.31 11.691 .32 0.603 .82 2-473 32 5.050 .82 8.176 2.32 11.767 33 0.631 83 2.518 33 5-108 .83 8.244 2-33 11.843 34 0.660 .84 2.564 34 5.165 .84 8.311 2-34 11.920 35 0.690 85 2.610 35 5.223 .85 8-379 2.35 11.996 .36 0.719 .86 2.656 36 5.281 .86 8-447 2.36 12.073 37 0.749 87 2.702 37 5-340 .8? 8-515 2-37 12.150 .38 0.780 .88 2-749 38 5-398 .88 8.584 2.38 12.227 39 0.811 .89 2.796 39 5-457 .89 8.652 2.39 12.304 .40 0.842 .90 2.843 .40 5-516 .90 8.721 2.40 12.381 41 0.874 91 2.891 .41 5-575 91 8.790 2.41 12.459 42 0.906 92 2-939 .42 5.635 92 8-859 2.42 12.536 43 0-939 93 2.987 43 5.694 93 8-929 2-43 12.614 44 0.972 94 3-035 -44 5-754 94 8.998 2.44 12.692 45 1.005 95 3-083 45 5.814 95 9.068 2.45 12.770 .46 1-039 96 3.132 .46 5.875 96 9.138 2.46 12.848 47 1.073 97 3.181 47 5-935 97 9.208 2-47 12.927 .48 1.107 98 3-231 .48 5.996 98 9.278 2.48 13 005 49 1.142 99 3-280 49 6.057 99 9-348 2.49 13-084 50 1. 177 I.OO 3-330 50 6.118 .00 9-419 2.50 13-163 1 From Trautwine's Engineers' Pocketbook. 1 The table values are not seriously in error when there are end contractions provided L is at least 10 H. The original formula was limited to heads, H, between H foot and 2 feet. The tabular values are probably not seriously in error for the range given. For any weir of length L feet multiply the tabular values by L. (331) 332 RAILROAD FIELD MANUAL TABLE LXXXII Feet to Meters i 2 3 4 5 6 7 8 9 Meters Meters Meters Meters Meters Meters Meters Meters Meters Meters o.ooo 0.305 0.610 0.914 1.219 1.524 1.829 2.134 2-438 2.743 IO 3.048 3-353 3.658 3-962 4.267 4-572 4.877 5.182 5-486 5-791 20 6.036 6.401 6.706 7.010 7-315 7.620 7.925 8.229 8-534 8.839 30 9.144 9-449 9-753 10.058 10.363 10.668 10.972 11.277 11.582 11.887 40 12.192 12.496 12.801 13-106 13.411 13.716 14.020 14.325 14-630 14-935 So 15.239 15-544 15.849 16.154 16.459 16.763 17.068 17-373 17-678 17.983 60 18.287 18.592 18.897 19 . 202 19.507 19.811 20.116 20.421 20.726 21.031 70 21.335 21 . 640 21.945 22 . 250 22.555 22.859 23.164 23.469 23-774 24.079 80 24-383 24.688 24-993 25.298 25.602 25-907 26.212 26.517 26.822 27 . 126 90 27.431 27.736 28.041 28.346 28.651 28.955 29.260 29-565 29.870 30.174 100 30.479 30.784 31-089 31-394 31-698 32.003 32.308 32-613 32.918 33-222 Meters to Feet vr_i__- o i 2 3 4 5 6 7 8 9 JvieLers Feet Feet Feet Feet Feet Feet Feet Feet Feet Feet o o.oo 3.28 6.56 9.84 13.12 16.40 19.69 22.97 26.25 29.53 IO 32.81 36.09 39-37 42.65 45-93 49-21 52-49 55.78 59-06 62.34 20 65.62 68.90 72.18 75.46 78.74 82.02 85.30 88.58 91-87 95.15 30 98.43 101.71 104-99 108.27 111.55 114.83 118.11 121.39 124.67 127.96 40 131.24 134.52 137.80 141.08 144.36 147.64 150.92 154-20 157.48 160.76 SO 164.04 167.33 170.61 173.89 177.17 180.45 183.73 187.01 190.29 193-57 60 196.85 200.13 203.42 206.70 209.98 213.26 216.54 219.82 223.10 226.38 70 229.66 232.94 236.22 239-51 242.79 246.07 249-35 252.63 255.91 259-19 80 262.47 265.75 269.03 272.31 275.60 278.88 282.16 285.44 288.72 292.00 90 295.28 298.56 391.84 305.12 308.40 3H.69 314.97 318.25 321-53 324-81 100 328.09 331-37 334-65 337-93 341-21 344-49 347.78 35i.o6 354-34 357.62 i statute mile = 1.6093 kilometers, i kilometer = 0.6214 statute mile. CHAPTER X ADJUSTMENT OF INSTRUMENTS The Transit. The adjustments in order are: I. Axis of plate bubbles perpendicular to vertical axis of instrument. II. Line of sight perpendicular to and presumably coincident with horizontal axis of telescope. 1. Parallax. 2. Vertical wire perpendicular to horizontal axis of telescope. 3. Intersection of wires to axis of telescope. 4. Objective slide coincident with axis of telescope. 5. Line of sight perpendicular to horizontal axis of telescope. 6. Eyepiece to center. III. Horizontal axis of telescope truly horizontal (Standard). IV. Axis of telescope bubble parallel to line of sight. V. Error of vernier of vertical circle. To test for and make the adjustments: 1. Set up; turn the alidade through 180; if the plate bubbles depart from the centers of their tubes, bring them halfway back with the adjusting pin, and relevel. Repeat test and adjustment until complete. II. i. Carefully focus the eyepiece on the wires until they are sharp and there is no apparent motion over an object sighted as the eye is moved a little sidewise or up and down. 2. Sight a minute distant point and note whether the vertical wire remains on the point as the telescope is slightly revolved on its hori- zontal axis. If not, loosen two adjacent capstan screws carrying the wire ring, and turn the ring slightly until by further trial the adjustment is complete. 3. Construct a pair of Y standards of wood, as two notches cut in opposite sides of a box, that will support the telescope near the two ends after it has been removed from the standards. Remove the telescope and place it in the Y's; direct the line of sight to any minute distant point and revolve the telescope in the Y's 180; notice if the intersection of the wires remains on the point; if not, bring the wires halfway back to the point by opposite capstan screws, moving first 333 334 RAILROAD FIELD MANUAL one set and then the other. Repeat the test and adjustment until complete. This adjustment is not often made but should be made if careful leveling is to be done with the transit. 4. Not all object slides are adjustable. If the slide may be ad- justed, its adjustment may be tested by making a test for 3 on a very near object; if an error appears, the slide may be moved by moving the ring carrying it. The ring is often concealed by a band around the telescope. After adjusting the slide a test should be made again for 3, and both tests and adjustments repeated until complete. 5. Set up; sight a minute distant point, or set a pin some 200 to 400 feet away; transit the telescope and find a distant point in the line of sight or set a pin some 200 to 400 feet away; revolve in azimuth and sight the first point; transit and see whether the line of sight falls on the second point; if not, move the vertical wire one-fourth the apparent distance toward the second point; repeat the test and adjust- ment until complete. The same first point may be retained but new second points must be found or set for each test. 6. If the wires seem to be to one side of the field of view, the eyepiece is not centered. This need not cause error in work. To correct the condition, if the eyepiece is adjustable, move the ring carrying it between the eye end and the wire ring until the wires appear to be in the center of the field. III. (a} Hang a plumb line from a high point not far from the transit, turn the line of sight on the line near its top and plunge the telescope downward, noting if the intersection of the wires follows the line; if not, raise or lower one end of the horizontal axis until the required condition is met. (6) Sight a nearby high point, say, on a building, plunge the telescope and set a point near the level of the instrument or lower; reverse in azimuth, transit, and sight the high point; plunge again and note whether the line of sight cuts the first lower point set; if not, raise or lower one end of the horizontal axis until the required condition is met. A new lower point will be needed for each test. IV. Adjust as described for the Y level, II I (6). V. After making IV, with the transit carefully leveled, bring the telescope bubble to the center of its tube and note the reading of the vertical arc vernier. If it is not zero, either move the vernier till it does read zero or note the error as an index error to be applied to all vertical angle readings. Care must be taken to note whether the error is such as to diminish or increase angles of elevation; angles of depres- sion will be affected in the opposite way. ADJUSTMENT OF INSTRUMENTS 335 The Y Level. The adjustments are: I. Line of sight coincident with axis of telescope. 1. Parallax. 2. Horizontal wire perpendicular to vertical axis of instrument. 3. Intersection of wires to axis of telescope. 4. Objective slide coincident with axis of telescope. 5. Eyepiece to center. II. Bubble axis parallel to line of sight. 1. Vertical adjustment. 2. Lateral adjustment. III. Y adjustment bubble axis perpendicular to vertical axis of level. To Make the Adjustment: . 1. i. Set up; focus the eyepiece on the wires until they appear sharp and there is no apparent motion over the field as the eye is moved a little sidewise or vertically. 2. Sight a distant point near one end of the horizontal wire; turn the telescope a little on the vertical axis and note if the point remains covered by the wire from one end to the other; if not, loosen two ad- jacent screws carrying the wire ring and turn the ring until by trial the required condition is met. 3. Loosen the clips over the Y's and sight a distant point, clamping the azimuth motion; revolve the telescope in the Y's upside down, and note whether the intersection of the wires remains on the point; if not, move the wire ring first sidewise and then vertically about half the apparent error until the intersection of the wires will remain on a point through a complete revolution of the telescope. 4. If the object slide is adjustable, perform the same test on a very near point and adjust the object slide. The ring carrying the object slide when it is adjustable is likely to be concealed beneath a band around the telescope between the object focusing screw and the wire adjusting screws. 5. When the instrument is adjusted the wires may appear to be out of the center of the field of view; if so, the eyepiece is out of center and may be centered by moving the ring between the eye end and the wire ring until the wires appear to center the field. II. i (a) Set up carefully, loosen the clips over the Y's, remove the telescope and replace it end for end; if the bubble does not return to the center of the tube bring it halfway back by the bubble adjusting screws, relevel, and test and adjust again, until the required condition is met. A better method is the following: 336 RAILROAD FIELD MANUAL (b) Set up midway between two stakes about 200 feet apart; read a rod on the two stakes and get their difference of elevation; set up beyond one of the stakes about ^ the distance between them; read the rod on the near stake, apply the difference in level and add a cor- rection for curvature, = o.ooi ft. for 220 feet, for a trial reading on the distant stake; read the rod on the distant stake and if the reading is not the same as the trial reading, move the target up or down {% of the apparent error according as the instrument is pointing low or high; set the line of sight on the target and bring the bubble to the center by its adjusting screws. Without moving the instrument make another complete test on near and far rod and continue until the required con- dition is met. If the level is set outside one stake one-fourth of the distance between the stakes the apparent error will be multiplied by instead of ^, and similarly for other proportions. 2. With the level set up and the clips loose, turn the telescope a little in the Y's and note whether the bubble remains centered; if not, adjust it by the lateral adjusting screws at one end until it will so remain when the telescope is turned from side to side. III. Set up and level with particular care over one set of leveling screws; turn on the vertical axis 180 and note whether the bubble remains in the center of the tube; if not, bring it halfway back by rais- ing or lowering one Y by the capstan nuts through which its stem passes; relevel and repeat until the required condition is met. The Dumpy Level. The adjustments are: I. Bubble axis perpendicular to vertical axis of level. II. Line of sight parallel to bubble axis. 1. Parallax. 2. Horizontal wire perpendicular to vertical axis of instrument. 3. Line of sight. The adjustments are made as follows: I. Set up with telescope leveled with particular care over one set of screws; reverse in azimuth and note whether the bubble remains in the center of its tube; if not, bring it halfway back by the bubble adjusting screws; relevel and repeat until the condition is met. II. i and 2 are made as for the Y level. 3 is made as for Y level II I (6), adjusting the wires to the bubble rather than the bubble to the wires; i.e., centering the bubble and moving the horizontal wire to the computed correct reading on the distant rod. CHAPTER XI SEXAGESIMAL TRIGONOMETRIC FUNCTIONS TABLE LXXXIII. LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS FOR EACH MINUTE OF ARC To interpolate when the angles are less than 3, the quantities 5 and T as found in the 5th and 6th columns of the first three pages are used as follows: Log sine a = Log a' S, Log tangent or = Log a' T, Log a' = Log sine a + 5 = Log tangent a + T. For cosine and cotangent of angles near 90 use the sine and tangent of the complements. 337 179 C / / LSin d S T LTan cd LCot LCos o ~I~ 2 3 4 I I 9 i 2 3 4 i J 9 10 ii 12 13 14 15 16 3 19 20 21 22 23 24 25 26 27 28 29 30 3i 32 33 34 s 8 39 40 41 42 43 44 8 47 48 49 50 51 52 53 54 II 11 59 60 30103 17609 12494 9691 7918 6694 5800 5"5 4576 4139 3779 3476 3218 2997 2802 2633 2483 2348 2227 2119 202 1 1930 1848 1773 1704 1639 1579 1524 1472 1424 1379 1336 1297 1259 1223 1190 1158 1128 IIOO 1072 1046 IO22 999 976 954 934 914 896 877 860 843 827 812 797 782 769 755 743 730 30103 17609 12494 9691 7918 6694 5800 5"5 4576 4139 3779 3476 3219 2996 2803 2633 2482 2348 2228 2119 2O20 1931 l8 4 8 1773 1704 1639 1579 1524 1473 1424 1379 1336 1297 1259 1223 II9O "59 1128 IIOO 1072 1047 1022 998 976 955 934 915 895 878 860 843 828 812 797 782 769 756 742 730 0.00 000 60 59 58 57 56 55 54 53 52 Si 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 IS 14 13 12 II 10 1 7 6 5 4 3 2 I 6.46 373 6.76476 6.94 085 7.06 579 7.16 270 7.24 188 7.30 882 7.36 682 7.41 797 3-53 627 3-53 627 3-53 627 3-53 627 3-53 627 3-53 627 3-53 627 3-53 627 3-53627 3-53 627 3-53 627 3-53 627 3-53 627 3-53627 3-53 627 3-53 627 3-53 627 3-53 627 6-46 373 6.76 476 6.94085- 7.06 579 7.16 270 7.24 188 7.30 882 7.36 682 7.41 797 3-53627 3-23 524 3-05 915 2.93 421 2.83 730 2.75812 2.69 118 2.63 318 2.58 203 0.00 000 0.00000 o.oo ooo o.oo ooo 0.00 000 0.00000 o.oo ooo 0.00000 0.00000 10 7-46 373 3-53 627 3-53 627 7-46 373 2.53 627 0.00000 ii 12 13 14 15 16 3 19 20 21 22 23 24 11 11 29 30 31 32 33 34 35 36 11 39 40 7-505I2 7-54 291 7-57 767 7-6o 985 7.63 982 7.66 784 7.69417 7.71 900 7-74 248 3-53 627 3-53 627 3-53 627 3-53 628 3-53 628 3-53 628 3-53 628 3-53 628 3-53 628 3-53 627 3-53 627 3-53 627 3-53627 3-53 627 3-53627 3-53 627 3-53627 3-53627 7-50512 7.5429I 7-57 767 7.60 986 7-63 982 7-66 785 7.69 418 7.71 900 7-74 248 2.49 488 2-45 709 2.42 233 2-39014 2.36 018 2.33 215 2.30 582 2.28 100 2.25 752 0.00 000 0.00000 o.oo ooo O.OO 000 O.OO 000 O.OO 000 9.99 999 9-99 999 9-99 999 7.76475 3-53 628 3-53 627 7.76476 2.23 524 9-99 999 7.78 594 7.80615 7-82 545 7-84 393 7.86 1 66 7.87 870 7.89 509 7.91 088 7.92 612 3-53 628 3-53 628 3-53 628 3-53 628 3-53 628 3-53 628 3-53 628 3-53 628 3-53 628 3-53 627 3-53 627 3-53 627 3-53627 3-53 627 3-53627 3-53 626 3-53 626 3-53626 7.78 595 7.80615 7.82 546 7-84 394 7.86 167 7-87 871 7.89 Sio 7.91 089 7.92 613 2.21 405 2.19385 2.17 454 2.15 606 2.13 833 2.12 129 2.IO 490 2.089II 2.07 387 9-99 999 9-99 999 9-99 999 9-99 999 9-99 999 9-99 999 9-99 999 9-99 999 9.99 998 7-94 084 3-53 628 3-53 626 7.94086 2.05 914 9-99 998 7-95 5o8 7.96 887 7-98 223 7-99 520 8.00 779 8.O2 002 8.03 192 8.04 350 8.05 478 3-53 628 3-53 628 3-53 628 3-53 628 3-53 628 3-53 628 3-53 628 3-53 628 3-53 628 3-53 626 3-53 626 3-53 626 3-53 626 3-53 626 3-53 626 3-53 626 3-53 626 3-53 626 7-95 Sio 7.96 889 7.98 225 7-99 522 8.00 781 8. 02 004 8.03 194 8.04 353 8.05 481 2.04 490 2.03 in 2.01 775 2.00478 1.99219 1.97 996 1.96806 i-95 647 1-94 519 9-99 998 9-99 998 9-99 998 9-99 998 9-99 998 9-99 998 9-99 997 9-99 997 9-99 997 8.06 578 3-53 628 3-53 625 8.06 581 1-93 419 9-99 997 4i 42 43 44 8 47 48 49 5 8.07 650 8.08 696 8.09 718 8.10717 8. 1 1 693 8.12 647 8.13 581 8.14 495 8.15 39i 3-53 628 3-53 628 3-53 629 3-53 629 3-53 629 3-53 629 3-53 629 3-53 629 3-53 629 3-53 625 3-53 625 3-53 625 3-53 625 3-53 625 3-53 625 3-53 625 3-53 625 3-53 624 8.07 653 8.08 700 8.09722 8.10720 8. 1 1 696 8.12651 8.13 585 8.14 500 8.15 395 1.92 347 1.91 300 1.90 278 1.89 280 1.88304 1-87 349 1.86415 1.85 500 1.84605 9-99 997 9-99 997 9-99 997 9-99 996 9.99 996 9-99 996 9.99 996 9-99 996 9-99 996 8.16 268 3-53 629 3-53 624 8.16 273 1.83 727 9-99 995 5i 52 53 54 II 11 59 60 8.17 128 8.17 971 8.18 798 8.19 610 8.20407 8.21 189 8.21 958 8.22 713 8.23 456 3-53 629 3-53 629 3-53 629 3-53 629 3-53 629 3-53 629 3-53 629 3-53 629 3-53 630 3-53 624 3-53 624 3-53 624 3-53 624 3-53 624 3-53 624 3-53 623 3-53 623 3-53 623 8.17 133 8.17 976 8.18 804 8.19616 8.20413 8.21 195 8.21 964 8.22 72O 8.23 462 1.82 867 1.82 024 1.81 196 1.80384 1-79 587 1.78 805 1.78 036 1.77 280 1.76 538 9-99 995 9-99 995 9-99 995 9-99 995 9-99 994 9-99 994 9-99 994 9-99 994 9.99 994 8.24 186 3-53 630 3-53 623 8.24 192 1.75 808 9-99 993 LCos d LCot cd LTan LSin t 90 (338) 89 178 C t / LSin d S T LTan cd LCot LCos 60 8.24186 3-53 630 3-53623 8.24 192 7lH 1.75 808 9-99 993 60 61 62 63 64 % 67 68 69 i 2 3 4 7 8 9 8.24903 8.25609 8.26 304 8.26988 8.27 661 8.28324 8.28977 8.29621 8.30 255 706 684 6p 66 3 653 644 634 3-53 630 3-53 630 3-53 630 3-53 630 3-53 630 3-53 630 3-53 630 3-53 630 3-53 630 3.53623 3-53623 3-53 623 3-53622 3.53622 3-53622 3-53622 3-53622 3.53622 8.24910 8.25616 8.26312 8.26996 8.27 669 8.28332 8.28986 8.29629 8.30 263 706 696 684 673 663 654 643 634 74 384 .73 688 .73004 72331 .71 668 .71014 -70371 69 737 9-99 993 9-99 993 9-99 993 9-99 992 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3-53632 3-53 632 3-53 632 3.53632 3-53632 3-53632 3-536I9 3.53619 3.53619 3.S36I9 3.536i9 3.536i8 3.536i8 3.536i8 3.53618 8.37 229 8.37 762 8.38 289 8.38 809 8.39323 8.39 832 8.40 334 8.40 830 8.41 321 533 527 520 514 509 502 496 49i 486 1.62 771 1.62 238 1.61 711 1.61 191 1.60677 i. 60 168 1.59666 1.59170 1.58679 9-99 988 9.99 988 9-99 987 9-99 987 9-99 987 9-99 986 9-99 986 9-99 986 9-99 985 39 38 37 36 35 34 33 32 31 90 30 8.41 792 A Q Q 3-53632 3.53617 8.41 807 A g Q 1.58193 9-99 985 30 9i 92 93 94 95 96 97 98 99 3i 32 33 34 35 36 s 39 8.42 272 8.42 746 8.43 216 8.43 680 8.44 139 8.44 594 8.45 044 8.45 489 8.45 930 474 470 464 459 455 450 445 441 3-53632 3-53 633 3-53 633 3-53 633 3-53 633 3.53633 3.53633 3-53 633 3-53 633 3.53617 3536i7 3.53617 3.53617 3.536i6 3.536i6 3-536I6 3.536i6 3-536I5 8.42 287 8.42 762 8.43 232 8.43696 8.44 156 8.44611 8.45061 8-45 507 8.45 948 475 470 464 460 455 450 446 441 I.577I3 1-57 238 1.56768 1-56304 1-55844 1.55389 1-54939 1-54493 1-54052 9-99 985 9-99 984 9-99 984 9-99 984 9-99 983 9-99 983 9-99 983 9-99 982 9-99 982 29 28 27 26 25 24 23 22 21 100 40 8.46 366 3-53 634 3.536I5 8.46 385 1.53615 9.99 982 20 IOI 102 103 104 105 106 107 108 109 4i 42 43 44 45 46 8 49 8.46 799 8.47 226 8.47 650 8.48 069 8.48 485 8.48 896 8.49 304 8.49 708 8.50 108 427 424 419 416 411 408 404 400 3-53 634 3-53 634 3-53 634 3-53 634 3-53 634 3-53 634 3-53 634 3-53 635 3.53635 3.53615 3.53615 3.53614 3.53614 3-53614 3-53614 3-53613 3-53613 3.53613 8.46817 8.47 245 8.47 669 8.48 089 8.48 505 8.48917 8.49325 8.49729 8.50130 428 424 420 416 412 408 404 401 1.53183 1.52755 1-52331 1.51911 I. Si 495 1.51 083 1.50675 1.50271 1.49 870 9.99981 9-99 98i 9-99 98i 9-99 98o 9-99 98o 9-99 979 9-99 979 9-99979 9.99 978 :i 17 16 15 14 13 12 II no 60 8.50 504 390 3-53 635 3.53613 8.50527 397 1.49473 9-99 978 10 III 112 "3 114 "5 116 117 118 119 Si 52 53 54 55 56 % 59 8.50 897 8.51 287 8.51 673 8.52055 8-52 434 8.52 810 8.53 183 8.53 552 8.53919 390 386 382 379 376 373 369 367 *6a 3.53635 3.53635 3.53635 3-53 635 3.53635 3-53636 3-53 636 3-53 636 3-53 636 3.53612 3-53612 3.53612 3.536II 3.536II 3.536n 3.536ii 3.536io 3.536io 8.50920 8.51310 8.51 696 8.52079 8.52459 8.52835 8.53 208 8.53 578 8.53 945 393 390 386 383 380 376 373 370 $ i .49 080 1.48690 1.48304 1.47921 I.4754I 1.47 165 1.46792 1.46422 1.46055 9-99977 9-99977 9-99977 9-99 976 9.99976 9-99 975 9-99975 9-99 974 9-99974 9 8 7 6 5 4 3 2 I 120 60 8.54 282 3-53636 3.53610 8.54 308 1.45692 9-99 974 LCos d LCot cd LTan LSin t (339) 88 C 2 177 C 1 / LSin d S T LTan cd LCot LCos 120 2 3 4 5 6 I 9 10 ii 12 13 14 IS 16 17 18 19 20 21 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688 9.9I7I3 9-9i 739 9-91 765 9.91 791 9.91 816 9.91 842 26 26 25 26 26 26 25 26 0.08 364 0.08338 0.08312 0.08 287 0.08 261 0.08 235 0.08 209 0.08 184 0.08 158 9-88 730 9.88720 9.88 709 9.88 699 9.88 688 9.88678 9.88 668 9.88657 9.88 647 IO II 10 II 10 IO II IO 29 28 27 26 25 24 23 22 21 6 7 8 9 9.6 9.0 II. 2 IO.5 12.8 I2.O 14-4 I3-S 40 9-8o 504 9.91 868 0.08 132 9.88636 20 41 42 43 44 45 46 47 48 49 9-80519 9-8o 534 9-80550 9-8o 565 9.80 580 9-8o 595 9.80610 9.80625 9.80641 15 16 IS IS IS IS IS' 16 9-91 893 9.91919 9-91 945 9.91 971 9.91 996 9.92 022 9.92 048 9.92073 9.92099 26 26 26 25 26 26 25 26 26 0.08 107 0.08 081 0.08055 0.08 029 0.08 004 0.07 978 0.07 952 0.07927 0.07 901 9.88626 9.88615 9.88 605 9-88 594 9-88 584 9.88573 9-88 563 9-88552 9-88 542 II IO II 10 II IO II IO iQ 18 17 16 15 14 13 12 II 2 3 4 | 11 10 I.I I.O 2.2 2.O 3-3 3-0 4-4 4-0 5-5 5-0 50 9.80656 9.92 125 0.07 875 9-88531 10 7 7-7 7-0 Si 52 53 54 II 11 59 9.80671 9.80 686 9.80 701 9.80 716 9-8o 731 9.80 746 9.80 762 9-80777 9-80792 IS IS IS IS ;i IS IS 9-92 ISO 9.92 176 9.92 2O2 9-92 227 9-92 253 9.92 279 9.92 304 9-92 330 9.92356 26 26 25 26 26 25 26 26 0.07 850 0.07 824 0.07 798 0.07 773 0.07 747 0.07 721 0.07 696 0.07 670 0.07 644 9.88521 9.88510 9-88 499 9.88 489 9.88478 9.88 468 9.88457 9-88 447 9.88436 II II 10 II IO II IO II 9 8 7 6 5 4 3 2 I 8 9 8.8 8.0 9-9 9-0 60 9.80 807 9-92 38l 0.07 619 9-88425 LCos d LCot cd LTan LSin d r P P 129 (377) 50 40 139 C / LSin d LTan cd LCot LCos d P P 9.80 807 9-92381 26 0.07 619 9-88425 60 I 2 3 4 s 7 8 9 9.80822 9-80837 9.80852 9.80867 9.80882 9.80 897 9.80912 9.80927 9.80 942 15 15 15 15 15 15 15 15 9.92 407 9-92 433 9.92 458 9.92 484 9-92 5io 9-92 535 9-92 561 9.92 587 9.92 612 26 25 26 26 ll 26 11 0-07 593 0.07 567 0.07 542 0.07 516 0.07 490 0.07 465 0.07 439 0.07413 0.07 388 9-88415 9.88 404 9.88394 9-88383 9-88372 9.88362 9-88351 9-88 340 9-88330 ii 10 II II IO II II 10 3 57 56 55 54 53 52 51 26 25 10 9.80957 15 9.92 638 0.07 362 9-88319 11 50 I 2.6 2.5 ii 12 13 14 15 16 17 18 19 9.80972 9.80987 9-8l 002 9.8l 017 9.8l 032 9.8l 047 9.81 061 9.81076 9.81 091 IS 15 15 15 15 15 14 IS 15 9-92 663 9.92 689 9.92715 9-92 740 9.92 766 9-92 792 9.92817 9.92 843 9.92 868 26 26 25 26 26 25 26 25 26 0.07 337 0.07 311 0.07 285 0.07 260 0.07 234 0.07 208 0.07 183 0.07 157 0.07 132 9.88 308 9.88 298 9.88 287 9.88276 9.88 266 9-88 255 9.88 244 9-88 234 9.88 223 10 II II 10 II II 10 II 49 48 47 46 45 44 43 42 4i 3 4 5 6 I 9 5-2 5-o 7-8 7-5 10.4 10.0 13.0 12.5 15.6 15.0 18.2 17.5 20.8 20.0 23.4 22.5 20 9.81 106 9.92 94 0.07 106 9.88212 40 21 22 23 2 4 25 26 27 28 29 9.8l 121 9-8l 136 9-8l 151 9.81 166 9.81 180 9.81 195 9.8l 210 9.8l 225 9.8l 240 IS 15 IS 15 14 IS 15 IS IS 9.92 920 9-92 945 9.92971 9.92 996 9.93022 9.93 048 9-93 073 9-93 099 9-93 124 25 26 25 26 26 25 26 25 0.07 080 0.07 055 0.07 029 0.07 004 0.06978 0.06952 0.06927 0.06 901 0.06876 9.88 201 9.88 191 9.88 180 9.88 169 9.88 158 9.88 148 9.88137 9.88 126 9.88115 IO II II II 10 II II II 39 33 37 36 35 34 33 32 31 I 2 15 14 i.S i-4 3.0 2.8 30 9.8l 254 14 9-93 ISO 0.06 850 9.88 105 30 3 4-5 4-2 6.0 5.6 3i 32 33 34 35 36 37 38 39 9.8l 269 9-8l 284 9.8l 299 9-8I3I4 9.8l 328 9-8i 343 9-8l 358 9-81372 9.81 387 15 IS 15 IS 14 IS 15 14 IS 9-93 175 9-93 201 9-93 227 9-93 252 9-93 278 9-93 303 9-93 329 9-93 354 9-93 38o 26 26 25 26 25 26 il 26 0.06825 0.06 799 0.06 773 0.06 748 0.06 722 0.06 697 0.06 671 0.06 646 0.06 620 9.88 094 9.88 083 9.88072 9.88 061 9.88051 9.88 040 9.88029 9.88018 9.88007 II II II 10 II II II II 2 9 28 27 26 25 24 23 22 21 5 6 7 8 9 7-5 7-0 9.0 8.4 10.5 9.8 12.0 II. 2 I3-S 12-6 40 9.81 402 9.93 406 0.06 594 9.87 996 20 4i 42 43 44 45 46 47 48 49 9.81 417 9.81 431 9.81 446 9.81 461 9-8i 475 9.81 490 9-8i 505 9.81 519 9-8i 534 IS 14 IS 15 14 15 IS 14 IS 9-93 43i 9-93 457 9-93 482 9-93 5o8 9-93 533 9-93 559 9-93 584 9-93 610 9-93 636 26 25 26 25 26 25 26 26 0.06 569 0.06 543 0.06 518 0.06 492 0.06 467 0.06 441 0.06 416 0.06 390 0.06 364 9.87 985 9-87975 9.87 964 9.87 953 9.87 942 9.8793I 9.87920 9.87 909 9.87 898 10 II II II II II II II 19 18 i/ 16 15 14 13 12 II I 2 3 4 5 11 10 I.I 1.0 2.2 2.0 3-3 3-0 4-4 4-0 5-5 5- 50 9.81 549 9-93 661 26 0.06339 9.87 887 10 6 6.6 6.0 5i 52 53 54 55 56 11 59 9-8i 563 9-81578 9-8i 592 9.81 607 9.81 622 9.81 636 9.81 651 9.81 665 9.81 680 IS 14 IS IS 14 15 14 15 9-93 687 9-93 712 9-93 738 9-93 763 9-93 789 9-938i4 9-93 840 9-93 865 9-93 891 25 26 25 26 25 26 25 26 0.06313 0.06 288 0.06 262 0.06 237 0.06 211 0.06 186 0.06 1 60 0.06 135 0.06 109 9-87877 9.87 866 9-87855 9.87 844 9-87833 9.87 822 9.87811 9.87 800 9.87 789 II II II II II II II II 7 6 5 4 3 2 I I 9 7-7 7-o 8.8 8.0 9-9 9-0 60 9.81 694 14 9.93916 25 0.06 084 9.87778 LCos d LCot cd LTan LSin d t P P 130 (378) 49 138 C / LSin d LTan cd LCot LCos d P P 9.81 694 9.93916 26 0.06 084 9.87 778 60 I 2 3 4 I 7 8 9 9.81 709 9.81 723 9.81 738 9.81 752 9.81 767 9.81 781 9.81 796 9.81810 9.81825 14 15 14 15 14 15 14 IS 9-93 942 9-93 967 9-93 993 9.94018 9.94 044 9-94 069 9-94 95 9.94120 9-94 146 25 26 25 26 25 26 25 26 0.06 058 0.06 033 0.06 007 0.05 982 0.05 956 0.05931 0.05 905 0.05 880 0.05 854 9.87 767 9-87756 9-87 745 9.87 734 9-87 723 9.87 712 9.87 701 9.87 690 9-87679 ii ii ii ii ii ii ii ii 59 58 57 56 55 54 53 52 5i 26 25 26 25 10 9-81 839 9-94 171 26 0.05 829 9.87 668 50 2 5-2 5-0 ii 12 13 14 11 17 18 19 9-81 854 9.81 868 9.81882 9.81 897 9.81 911 9.81 926 9.81 940 9-8i 955 9.81 969 14 14 IS 14 IS 14 IS 14 9.94 197 9.94222 9-94 248 9-94 273 9-94 299 9.94324 9-94350 9-94375 9-94 401 25 26 25 26 25 26 3 0.05 803 0.05778 0.05 752 0.05 727 0.05 701 0.05 676 0.05 650 0.05 625 0-05 599 9-87657 9.87 646 9-87 635 9.87624 9-87613 9.87 601 9.87 590 9-87 579 9-87 568 ii ii ii ii 12 II II II 49 48 47 46 45 44 43 42 41 3 4 s 6 7 8 9 7-8 7-5 10.4 10.0 13.0 12.5 15.6 15.0 18.2 17.5 20.8 20.O 23.4 22.5 20 9.8i 983 9-94 426 26 0.05 574 9-87SS7 40 21 22 23 24 25 26 27 28 29 9.81 998 9.82012 9.82 026 9.82 041 9.82055 9.82 069 9.82 084 9.82 098 9.82 112 14 14 IS 14 14 IS 14 14 9-94452 9-94477 9-94 503 9-94 528 9-94 554 9-94579 9.94 604 9-94 630 9-94655 25 26 25 26 25 25 26 25 26 0.05 548 0.05 523 0.05 497 0.05472 0.05 446 0.05 421 0.05 396 0.05 370 0.05 345 9.87 546 9-87 535 9.87 524 9.87513 9.87 501 9.87 490 9.87479 9.87 468 9-87457 II II II 12 II II II II 39 38 37 36 35 34 33 32 3i i 2 3 15 14 1.5 1.4 3.0 2.8 4-5 4-2 30 9.82 126 9.94 681 0.05 319 9.87 446 30 4 6.0 5.6 3i 32 33 34 35 36 3 39 9.82 141 9.82155 9.82 169 9.82 184 9.82 198 9.82 212 9.82 226 9.82 240 9-82 255 14 14 IS 14 14 14 14 IS 9-94 7o6 9-94732 9-94757 9-94 783 9-94 808 9-94 834 9.94859 9.94 884 9.94910 26 25 26 25 26 25 25 26 0.05 294 0.05 268 0.05 243 0.05 217 0.05 192 0.05 166 0.05 141 0.05 116 0.05 090 9.87434 9-87423 9.87412 9.87 401 9.87 390 9.87378 9-87367 9-87356 9.87 345 II II II II 12 II II II 29 28 27 26 25 24 23 22 21 6 1 9 9.0 8.4 10.5 9.8 I2.O II. 2 I3-S 12.6 40 9.82 269 9-94935 26 0.05 065 9-87334 20 4i 42 43 44 a 47 48 49 9.82 283 9.82 297 9.823II 9.82326 9.82 340 9.82354 9.82 368 9.82382 9-82 396 14 14 IS 14 14 14 14 14 9.9496i 9-94 986 9.95012 9-95 037 9-95 062 9-95 088 9-95 H3 9-95 139 9.95 164 25 26 25 3 25 26 25 26 0.05 039 0.05 014 0.04 988 0.04 963 0.04 938 0.04912 0.04 887 0.04 861 0.04 836 9-87322 9.873H 9-87 300 9.87 288 9.87 277 9.87 266 9.87 255 9.87 243 9.87 232 II II 12 II II II 12 II 19 18 17 16 15 14 13 12 II z 2 3 4 I 12 11 1.2 I.I 2.4 2.2 3-6 3-3 4.8 4-4 6.0 5.5 7.2 6.6 50 9.82 410 9.95 190 0.04 810 9.87 221 10 I 8.4 7-7 51 52 53 54 55 56 11 59 9.82 424 9.82 439 9-82453 9.82467 9.82 481 9.82 495 9-82 509 9-82 523 9-82537 IS 14 14 14 14 14 14 14 9-95 215 9-95 240 9-95 266 9-95 291 9-95317 9-95342 9-95368 9-95 393 9-95 4i8 25 26 25 26 A 25 25 0.04 785 0.04 760 0.04 734 0.04 709 0.04 683 0.04 658 0.04632 0.04 607 0.04 582 9.87 209 9.87 198 9.87 187 9.87 175 9.87 164 9.87153 9.87 141 9-8? 130 9.87 119 II II 12 II II 12 II II 9 8 7 6 5 4 3 2 I 8 9 9.6 8.8 10.8 9.9 60 9-82551 9.95 444 0.04 556 9.87 107 LCos d LCot cd LTan LSin d f P P 131 C (379) 42 137 C / LSin d LTan cd LCot LCos d P P 9-82551 9-95 444 0.04 556 9.87 107 60 I 2 3 4 7 8 9 9-82 565 9-82 579 9-82 593 9.82 607 9.82 621 9-82 635 9.82 649 9.82 663 9.82677 14 14 14 14 14 14 14 14 9-95 469 9-95 495 9-95 520 9-9S 545 9-95571 9-95 596 9-95622 9-95 647 9-95 672 26 25 25 26 25 26 25 25 26 0.04531 0.04 505 0.04 480 0.04 455 0.04 429 0.04 404 0.04378 0.04 353 0.04328 9.87 096 9-87 085 9.87073 9.87 062 9.87 050 9.87 039 9.87028 9.87016 9-87 005 u 12 II 12 II II 12 II 59 58 57 56 55 54 53 52 5i 26 25 10 9.82 691 9-95 698 0.04 302 9-86 993 50 I 2.6 2.5 ii 12 13 14 15 16 17 18 19 9-82 705 9.82719 9-82 733 9.82 747 9.82 761 9.82775 9.82 788 9.82 802 9.82816 14 14 14 14 14 13 14 14 9-95 723 9-95 748 9-95 774 9-95 799 9-95825 9-95 850 9-95 875 9-95 9oi 9.95926 25 26 25 26 25 25 26 *6 0.04 277 0.04 252 0.04 226 0.04 201 0.04 175 0.04 150 0.04125 0.04 099 0.04 074 9.86982 9.86 970 9-86 959 9-86 947 9-86 936 9.86924 9.86913 9.86 902 9.86 890 12 II 12 II 12 II II 12 40 48 47 46 45 44 43 42 4i 3 4 5 6 I 9 7-8 7-5 10.4 10. o 13.0 12.5 15.6 15.0 18.2 17.5 20.8 20.0 23-4 22.5 20 9.82 830 9-95952 0.04 048 9.86879 40 21 22 23 24 y 27 28 29 9.82 844 9.82 858 9.82872 9.82 885 9.82 899 9-82 913 9.82927 9.82 941 9-82 955 14 14 13 14 14 14 14 u 9-95 977 9.96 002 9.96028 9-96053 9.96078 9.96 104 9.96129 9.96155 9.96 180 25 26 25 25 26 25 26 25 0.04023 0.03 998 0.03 972 0.03 947 0.03 922 0.03 896 0.03 871 0.03 845 0.03 820 9.86 867 9-86855 9.86 844 9.86832 9.86821 9.86 809 9.86 798 9.86 786 9-86 775 12 II 12 II 12 II 12 II 39 38 37 36 35 34 33 32 3i i 2 3 14 13 1.4 1.3 2.8 2.6 4-2 3-9 30 9.82 968 9-96 205 26 0.03 795 9-86 763 12 30 4 5-6 5-2 3i 32 33 34 35 36 11 39 9.82982 9.82 996 9.83010 9-83 023 9-83 037 9-83051 9-83 065 9-83078 9-83 092 14 14 13 14 14 14 13 14 9-96 231 9-96 256 9.96 281 9.96 307 9.96332 9-96357 9-96 383 9.96 408 9-96 433 25 11 25 25 26 25 25 0.03 769 0.03 744 0.03 719 0.03 693 0.03 668 0.03 643 0.03 617 0.03 592 0.03 567 9-86752 9.86 740 9.86728 9.86717 9-86 705 9.86 694 9.86682 9.86670 9-86 659 12 12 II 12 II 12 12 II 29 28 27 26 25 24 23 22 21 6 7 8 9 8.4 7-8 9.8 9-1 1 1. 2 IO.4 12.6 11.7 40 9.83 106 *4 9.96 459 0.03 541 9.86 647 20 4i 42 43 44 45 46 47 48 49 9.83 120 9-83 133 9-83 147 9-83 161 9.83 174 9.83 188 9.83 202 9-83 215 9-83 229 13 14 14 13 14 14 13 14 9.96 484 9-96510 9-96535 9-96 56o 9-96 586 9.96611 9.96 636 9.96 662 9.96 687 2 5 26 25 25 26 25 25 26 25 0.03 516 0.03 490 0.03 465 0.03 440 0.03 414 0-03 389 0.03 364 0.03 338 0.03 313 0.86635 9.86624 9.86612 9.86 600 9-86 589 9.86577 9-86565 9-86 554 9-86 542 II 12 12 II 12 12 II 12 19 18 17 16 15 14 13 12 II i 2 3 4 5 6 12 11 1.2 I.I 2.4 2.2 3-6 3-3 4.8 4-4 6.0 5-5 7.2 6.6 50 9.83 242 9.96712 tf\ 0.03 288 9-86 530 12 10 7 g 8.4 7-7 Si 52 53 54 55 56 59 9-83 256 9-83 270 9.83 283 9.83 297 9-83 310 9-83 324 9.83 338 9-83 351 9-83 365 14 13 14 13 U 14 13 U 9-96 738 9.96 763 9.96 788 9.96814 9-96 839 9.96 864 9.96 890 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5-6 5-2 31 32 33 34 35 36 11 39 9-83 795 9.83 808 9.83821 9.83 834 9-83 848 9.83 861 9-83 874 9-83 887 9-83 9oi 14 13 13 13 14 13 13 13 14 9-97 750 9-97 776 9.97 801 9.97 826 9-9785I 9-97 877 9-97 902 9.97927 9-97 953 25 26 25 25 25 26 25 25 26 O.O2 250 0.02 224 O.O2 199 0.02 174 0.02 149 0.02 123 O.O2 098 0.02 073 O.02 047 9.86 044 9.86032 9.86020 9.86 008 9.85 996 9.85 984 9.85972 9-85 96o 9.85 948 12 12 12 12 12 12 12 12 29 28 27 26 25 24 23 22 21 6 7 8 9 8.4 7-8 9-8 9.1 II. 2 IO.4 12.6 11.7 40 9.83 914 *3 9.97978 0.02022 9-85 936 20 4i 42 43 44 45 46 47 48 49 9.83 927 9.83 940 9-83 954 9-83 967 9-83 98o 9-83 993 9.84 006 9.84 020 9-84033 ij> 13 14 13 13 13 13 14 13 9.98 003 9.98 029 9.98 054 9.98079 9.98 104 9.98 130 9-98i55 9.98 180 9.98 206 25 26 25 25 3 25 \\ o.oi 997 o.oi 971 o.oi 946 o.oi 921 o.oi 896 o.oi 870 o.oi 845 o.oi 820 o.oi 794 9.85 924 9-85912 9-85 9oo 9.85 888 9-85876 9-85 864 9-85851 9-85 839 9-85827 12 12 12 12 12 13 12 12 i9 18 i7 16 15 U 13 12 II i 2 3 4 I 12 11 1.2 I.I 2-4 2.2 3-6 3-3 4-8 4-4 6.0 5.5 7.2 6.6 50 9.84 046 9-98 231 2 5 o.oi 769 9-85815 10 7 g 8.4 7-7 96 8.8 Si 52 53 54 55 56 Si 59 9.84059 9.84072 9-84085 9.84098 9.84112 9-84125 9-84 138 9-84151 9.84 164 13 13 13 14 13 13 13 13 9-98256 9.98 281 9.98 307 9-98332 9-98357 9-98 383 9-98 408 9.98 433 9-98 458 25 26 25 25 26 25 25 25 o.oi 744 o.oi 719 o.oi 693 o.oi 668 o.oi 643 o.oi 617 o.oi 592 o.oi 567 o.oi 542 9-85 803 9-85 79i 9-85 779 9-85 766 9-85 754 9.85 742 9-85 730 9-85 7i8 9-85 7o6 12 12 13 12 12 12 12 12 1 7 6 5 4 3 I 9 10.8 9.9 60 9.84177 9.98484 o.oi 516 9-85 693 LCos d LCot cd LTan LSin d I P P (381) 46 135 t T i 2 3 4 5 6 7 8 9 10 ii 12 13 14 15 16 17 18 19 29 21 22 23 24 11 27 28 29 30 3i 32 33 34 51 S 39 40 4i 42 43 44 3 a 49 50 Si 52 53 54 II S 59 60 LSin d LTan cd LCot LCos d P P 9-84177 HHMMMMHHH H HHHHHHHHH M HHHHMMHHH H HHHHHMt-IHH H HHHHHHHHH H HHHHMHHHH H 9-98 484 25 25 26 25 25 II 25 11 25 25 25 26 25 25 25 26 25 25 26 25 25 25 26 25 25 25 26 25 25 26 25 25 25 26 25 25 25 26 25 25 25 26 25 11 25 25 25 26 25 25 25 26 25 25 25 26 25 o.oi 516 9.85 693 12 12 12 12 13 12 12 12 13 12 12 12 13 12 12 13 12 12 13 12 12 13 12 12 13 12 13 12 12 13 12 13 12 13 12 12 13 12 13 12 13 12 13 12 13 12 13 12 13 13 12 13 12 13 12 13 13 12 13 12 d 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 i7 16 15 14 13 12 II 10 9 8 7 6 5 4 3 2 I 26 25 14 i 2.6 2.5 1.4 2 5.2 5-0 2.8 3 7-8 7-5 4-2 4 10.4 10.0 5.6 5 13.0 12.5 7.0 6 15.6 15.0 8.4 7 18.2 17.5 9.8 8 2O.8 2O.O 1 1. 2 9 23.4 22.5 12.6 13 12 I 1-3 1-2 2 2.6 2.4 3 3-9 3-6 4 5-2 4-8 5 6.5 6.0 6 7-8 7-2 7 9.1 8.4 8 10.4 9.6 9 11.7 10.8 9.84 190 9-84 203 9.84 216 9.84229 9.84 242 9-84255 9.84269 9.84282 9.84 295 9.98 509 9.98 534 9-98 560 9-98 585 9.98 610 9-98635 9.98 661 9.98 686 9.98711 o.oi 491 o.oi 466 o.oi 440 o.oi 415 o.oi 390 o.oi 365 o.oi 339 o.oi 314 o.oi 289 9.85 681 9.85 669 9-85 657 9-85 645 9.85632 9.85 620 9.85 608 9-85 596 9-85 583 9-84308 9-98 737 o.oi 263 9.85571 9-84321 9-84334 9-84347 9.84 360 9-84373 9-84 385 9.84 398 9.84411 9.84424 9.98 762 9.98 787 9.98812 9.98 838 9.98 863 9.98 888 9.98913 9-98 939 9.98 964 o.oi 238 o.oi 213 o.oi 188 o.oi 162 o.oi 137 O.OI 112 o.oi 087 o.oi 061 o.oi 036 9-85 559 9-85 547 9-85 534 9-85522 9.85510 9-85 497 9-85 485 9-85473 9-85 460 9-84437 9.98 989 O.OI Oil 9-85 448 9.84450 9-84 463 9.84476 9-84 489 9.84 502 9-84515 9-84528 9.84 540 9-84 553 9.99015 9.99 040 9-99 065 9.99090 9.99116 9-99 141 9.99 166 9-99 I9i 9.99217 o.oo 985 o.oo 960 0.00935 o.oo 910 o.oo 884 0.00859 o.oo 834 o.oo 809 o.oo 783 9-85 436 9-85423 9.854" 9-85 399 9-85386 9-85374 9-85361 9-85 349 9-85337 9.84 566 9-99 242 0.00758 9-85324 9.84579 9-84 592 9-84 605 9.84618 9.84 630 9.84 643 9-84656 9.84669 9.84 682 9.99 267 9-99 293 9.99318 9-99 343 9-99 368 9-99394 9.99419 9-99 444 9-99 469 o.oo 733 o.oo 707 o.oo 682 0.00657 o.oo 632 o.oo 606 o.oo 581 o.oo 556 o.oo 531 9-853I2 9.85 299 9-85 287 9-85274 9.85 262 9-85 250 9-85 237 9.85225 9.85212 9.84 694 9-99 495 o.oo 505 9.85 200 9.84 707 9.84720 9-84 733 9-84 745 9.84758 9.84771 9.84 784 9.84 796 9.84 809 9-99 520 9-99 545 9-99 570 9-99 596 9.99621 9.99 646 9.99672 9-99697 9.99722 o.oo 480 0.00455 o.oo 430 o.oo 404 0.00379 0.00354 0.00328 o.oo 303 o.oo 278 9.85 187 9.85 175 9.85 162 9-85 150 9-85I37 9-85125 9.85 112 9.85 ioo 9.85 OS? 9.84822 9-99 747 o.oo 253 9.85074 9-84835 9.84847 9.84 860 9.84873 9-84885 9.84898 9.84911 9.84923 9.84 936 9-99773 9-99 798 9-99823 9-99 848 9.99874 9-99 899 9-99924 9-99 949 9-99975 o.oo 227 O.OO 202 0.00177 o.oo 152 0.00126 O.OO IOI 0.00076 0.00051 0.00025 9.85062 9.85 4 9 9.85037 9-85024 9.85012 9.84 999 9.84986 9.84974 9.84961 9-84 949 o.oo ooo 0.00000 9.84 949 LCos d LCot cd LTan LSin / P P 134 C (382) 45 C TOPICAL INDEX PAGE Adjustment of instruments 333 Azimuth, latitude, and time 305 Barometric elevations 328 Compound curves 8 Curvature, estimating 254 Curve resistance 250 Distance, estimating 252 Elevation of outer rail 287 Estimating elements for distance, curvature, etc 252 Gage on curves 286 Grade, ruling, estimating for changes in 254 Grades for unbalanced traffic 251 Latitude, azimuth, and time 305 Load on grade 250 Loads on grades, table 259 Location theories and tables 249 Logarithms, sines, etc., use of table 96 table 76 table, use of 70 versine and exsecants 189 Metric curves 314 Miscellaneous tables 320 Momentum grade "... 251 Prismoidal correction 267 table 276 Pusher grade 250 Pusher grades table 260 Pusher service, cost of 255 383 384 TOPICAL INDEX PAGE Right of way table 290 Rise and fall, estimating 253 Resistance, train and curve 250 Ruling grade, estimating for changes in 254 Simple curves, approximate fundamental relations 2 fundamental notations and equations I location by offsets from long chord 4 problems 5 table of radii and tangent offsets and middle ordi- nates for 100 foot chords 12 tangent distances for i curve 20 Sines, etc., natural 234 use of tables 96 Spiral, Chapter II 39 chord spiral, the 41 compound curve spiraling 44 deflection coefficients 46 general functions, table of 47 select and layout, to 43 tables, special 55 Spring frogs, leads 304 Switches and frogs, properties of 302 Tables, ballast estimates 296 barometric elevations 328 bridge masonry quantities in abutments 292 bridge weights 296 compound interest 262 convergence of meridians 313 conversion, English and metric units 332 minutes and seconds to decimals 248 cost curves, timber trestles 293 culvert estimates 295 drainage areas 291 elevation of outer rail 288 external distances for i curve 34 frogs and switches, properties of 302 gage on curves 286 grades and grade angles 261 latitude and longitude length 0.01 312 TOPICAL INDEX 385 PAGE Tables, level section volumes 278 loads hauled on grades 259 logarithmic versines and exsecants 189 sexagesimal system . . . 337 long chords and actual arcs 35 metric curves, long chords and actual arcs 316 middle ordinates of long chords 318 radii tangent offsets and middle ordi- nates, one sta 315 middle ordinates for long chords 37 for 100 feet 12 for rails 286 natural sines, etc 234 natural versines and exsecants 243 polaris, elongation and culmination 3 IQ pole distance 309 prices, relative, for articles of different lives 262 prismoidal correction 276 pusher grades 260 radii of simple curves 12 of metric curves 315 refraction correction to altitude 309 relative trains on different grades 260 right of way 290 sexagesimal logarithmic functions 337 spiral deflection coefficients 46 spiral functions, general 47 spiral, special 55 spring frog leads 304 stadia reduction 325 switch leads practical 303 tangent distance corrections 33 tangent distances for i curves 20 tangent offsets for 100 feet 12 track material 289 tractive effort of locomotive 258 train resistance 257 trigonometric formulas 320 velocity heads 256 volumes triangular prisms 268 386 TOPICAL INDEX Tables, weir volumes Time, azimuth, and latitude ............. , ................... 305 Track material ............................................ 289 Tractive effort ............................................ 249 of locomotive ................................. 25 Trains, relative number on different grades , .................. 26c Train resistance ......................... .................. 2 table ....................................... 25 Trestles, cost curves ........................................ 29, Trigonometric functions, logarithmic sines, etc ................. 96 sexagesimal ......................... 337 Turnouts and crossovers .................................... 297 i Unbalanced traffic, grades for '. 251 Velocity grade 256 heads, table 256 Versines and external secants, natural 243 and exsecants, table of logarithmic 189 Vertical curves IO Volumes, level section 267 tables 278 triangular prisms 265 tables 268 Weir volumes, table 33 1 Ul YB 53598 / ~i S08538 UNIVERSITY OF CALIFORNIA LIBRARY