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I
^
p
RAILROAD ENGINEERING
By WILLIAM G. RAYMOND
Railroad Field Geometry (Without Tables)
i6mo. 242 + iz pages, 138 figures, leather,
gilt edges. Price $1.80 net.
Elements of Railroad Engineering
Second Edition, Revised. 8vo. 408 + xvi
pages, 1 10 figures, 1 8 plates. Cloth, $3.25 net.
Railroad Field Manual for Civil Engineers
vii -h 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
COPYRIGHT, 1915,
BY
WILLIAM G. RAYMOND
Stanbope press
F. H.GILSON COMPANY
BOSTON, U.S.A.
1
200190
DEC 31 !9i5
SSFF
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
• • •
m
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:
"1. It is useless to make linear measurements with a precision of
more than 1 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.
" 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,
1915-
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
vu
1
RAILROAD FIELD MANUAL
CHAPTER I
SIMPLE, COMPOUND, AND VERTICAL CURVES
Simple Curves
Fundamental notations and equations. — The curve running from
A to Bin Fig. I,
A — T.C. (tangent-curve).
B = C.T. (curve- tangent).
V = P. J. (point of intersection of tangents).
I = central angle or deflection angle of tangents.
T = tangent distance =* R tan i J.
C =* long chord = 2 R sin \ I.
M = middle ordinate = R vers J J.
E = external distance = R exsec J J.
E = rtan}/.
C = 2Mcot\I.
C = 2 7*cosi/.
64 « — seci/.
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 ioo 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,
R =
50
Fig. 2.
siniZ>
Table I gives R and its logarithm for various degree
curves.
O
Tangent offset / = R vers I = — ^ . The tangent offset for one
station is tabulated in Table I. For a subchord c the tangent offset is
h ~ *ioo
10,000
Approximate Fundamental Relations. — Approximately, radii are in-
versely as the degrees or
R D' ,
Radius of a i° curve is 5729.65 = 5730 (approx.).
Rd = ^^ (approx.).
Tangent distance for a D° curve of central angle I is
Td — -j~ (approx.).
Table II gives tangent distances for a i° curve and various values of
J, and Table III gives corrections to Td =-yr- for more precise re-
sults. Values of C, M , and E, for D° curves
are also found approximately by dividing the
values for a i° curve for a given I by D.
Table IV gives values of £ for a i° curve.
A curve departs from a tangent approxi-
mately thus: = J n 2 D f n being the number
of stations from the tangent point and D the
degree of curve. Two curves of degrees D
and D\ depart from each other by the same approximate law, sub-
stituting the difference of degrees D — A or A — D of the D of the
foregoing formula.
T.C.
Fig. 3.
1
SIMPLE, COMPOUND, AND VERTICAL CURVES 3
The deflection angle A, for a chord of 100 feet is \D\ for a sub-
chord, c, it is given by
c
sin 5 —
2R
or, with sufficient exactness for all curves under about 8°,
6
100
or 5 = 0.3 cD, in minutes.
— X — =0.5 cD, in hundredths of degrees,
100 2
Fig. 5.
The ordinate from a chord at any point given by its distance from
the center is
or, approximately,
Whence, for
4
(approx.).
If the point is given by the distance from one extremity,
K = -j-£- (approx.)
872QXSXD , .
— — i — - (approx.).
10,000,000 ^
Approximately
M =
C 2
8£'
w =
Jlf
RAILROAD FIELD MANUAL
Location by Offsets from Long Chord —
ft
AG = 2 Rain- D, » = number stations A to G,
2
5/7 = 2* sin ^-^A •
2
Ab=fG= AG ~ BF ; CE = ioo or 2 Rain?— ±D,
2 2
bc = ef=Bc' = c'F = BF ~ CE , etc.
2
Bb = Ff = Hh- HW* - R (verainD - vers i (» - 2) Z>), etc.,
c h e
or
Ab =*fG = 100 cos D.
J 2
Be' = e'F = 100 cos n ~~ 3 Z>, etc.
2
Bb = Ff= 100 sin^- 1 I>.
Cc' = Ee' « 100 sin
2
«-3
I>.
Use natural functions and
move decimal.
In the particular figure HW = 50 tan \ D because n is odd.
If the number\>f 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.
A h __ fl~_. long chord of n stations —long chord of (»— 2) 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 J D for natural
functions.
Bb — tangent offset of Table I.
A V or XA gives line of tangent.
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 56, etc. If the curve begins with
T.0.1A
Fig. 7.
SIMPLE, COMPOUND, AND VERTICAL CURVES 5
a subchord, /, swing / feet above A to B (Fig. 8) until Bb = / sin •
100 2
Swing 100 - /about A to F until Ff - (100 - /) sin I00 ""* ?. p ro _
100 2
duce Ff to c', 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. 8.
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 CT 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
RAILROAD FIELD MANUAL
desired result; change in CT or TC = change in tangent distances,
or m = T 2 - 7\ = (R 2 - RJ tan i 7.
Fig. ii.
Fig. 12.
^4 ewrw ewds in VB or V'B' '; required to end in VB' or VB.
does not change. It is necessary to find change in A.
K
Degree
AA' = W = BB' = 00' =
sin /
Fig. 13.
Fig. 14.
A curve ends in VB; required to end in V'B' or V'B" without changing
A . Degree changes.
R' = R
» that is, change in R =
vers I
vers I
A curve ends in VB; required to end in V'B', A and D change.
XT
R' *- R zk f , according as B' is inward or outward from B.
exsec 1
SIMPLE, COMPOUND, AND VERTICAL CURVES
A A 1 = (R — R') tan J, or K cot £ J", 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 AA' = R (tan | / - tan } /'),
and A is moved forward or back accord-
ing as /' is less or greater than I.
Changing degree and keeping A fixed,
T
T is unchanged. /. R f = =-77
tan £ I
or Tcot i /'.
A curve ends in VB; required to end
in VB. Fig. 16. D and A change.
vers/
R' =R
vers V
A A = (R sin / — R' sin /') and A is moved forward or back accord-
ing as V is greater or less than /.
A curve ends in VB; required to end in VB'. Fig. 17.
1. Assume new R' < R according as B' is inside or outside VB,
Then
K K
cos a = I —
R-R'
or
1 —
R' -R
n stations = -=z •
Begin at n stations from B and run in curve of D f for n f stations =
a
Fig. 16.
Fig. 17.
2. Or, retaining A and letting n be total number of stations A to B,
D' = D ± ■=—«* approx.
t n*
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 f for n' stations = k> *
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 1 by
n °V 7 (g-i?') ' approx -
(D — D') is to be taken as the difference of degrees, subtracting the
smaller from the larger, a = nD, n' = jr, to make the tangents par-
allel. Run curve of degree D' from C.C.
Fig. 18.
Fig. 19.
Compound Curves
Given Ri, Ri, A u and A 2 ; required T\ and T%. Fig. 18.
VlV* =/!+/,.
Solve triangle WiV x for ViV and VtV.
Tt-VVx + h; T t =VV* + h.
Given I, T it T%, and Ri; required R it Ai, and A 2 . Fig. 19.
T* + I\ cos I - Ri sin /
cot JAj
7i sin / — Ri vers J
Ai - I - A,,
7i sin / — R% vers /
Ri** Ri +
vers Ai
(a)
SIMPLE, COMPOUND, AND VERTICAL CURVES 9
Given I, T lt T% t and Rt; required Ai, Ai, and As.
. . Rt sin / — T\ — Tt cos /
cot J Ai = — = = — = — : — = — »
As vers /—7s sin /
A, - J - Ai,
z> « -R* vers / — T 9 sin / - x
Ai = As . (d)
vers Ai x '
Given I, 7i, Ai, and Ai; required As, As, and r s .
As = / — Ai.
Rt is given by equation (a) above.
Tt - (As - Ri) sin A 2 + Ai sin / - Ti cos /.
Given Iu T%, As, and A 2 ; required Ai f Ai, and 7\.
Ai = / - A*
2? is given by equation (b) above.
7i = Rt sin / — Ti cos J — {Rt — R x ) 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' = -r
sin/
(6) Retaining the first branch, changing
only the degree of the second,
K
Rt' « a, db
Ri' = Ri±
vers A 2
K
Fig. 20.
vers Ai
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
K
vers As' = vers As d:
versAi' = versAi ±
Rt-Ri
K
IO
RAILROAD FIELD MANUAL
To change the direction of the final tangent by a given amount.
Solve the triangle W'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 1. 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 0.1 per station. The total change in rate is 0.8 + 0.6 = 1.4,
giving 14 stations =1.4 + 0.1, 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 -j-
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
457
458
459
460
461
462
463
464
728.6
elev. 456 —
elev. 457 -
elev. 458 —
elev. 459 -
elev. 460 —
elev. 461 —
elev. 462 —
elev. 463 —
- (0.8
(0.8-
(0.8-
(0.8-
(0.8-
(0.8-
(0.8
(0.8
(0.8
- 0.05)
0.15]
0.25]
- 0.35]
- 0.45;
- 0.55)
- 0.65)
- 0.75!
- 0.85)
= 728.6 -
727.85 -
727.20 —
726.65 -
726.20 —
725.85 -
725.60 -
72545 ~
725.40 +
0.75
0.65
o.55
0.45
o.35
0.25
0.15
0.05
0.05
- 727.85
= 727.20
= 726.65
■ 726.20
- 725.85
= 72560
== 72545
- 72540
- 72545
SIMPLE, COMPOUND, AND VERTICAL CURVES II
465 = elev. 464 — (0.8 — 0.95)
466 = elev. 465 — (0.8 — 1.05)
467 = elev. 466 — (0.8 — 1. 1 5)
468 = elev. 467 — (0.8 — 1.25)
469 = elev. 468 — (0.8 — 1.35)
725.45 + 0.15
725.60 + 0.25
725.85 + 0.35
726.20 + 0.45
726.65 + 0.55
725.60
725.85
726.20
726.65
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 — 1.0, meet at station 867.0,
where the elevation is 466.0. To connect the grades with a vertical
curve changing at the rate of 0.1 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 -4X1.0 = 462.0
Elev. sta. 864 = elev. 863 — (0.2 + 0.05
Elev. sta. 865 = elev. 864 — (0.2 +0.15
Elev. sta. 866 = elev. 865 — (0.2 + 0.25
Elev. sta. 867 = elev. 866 — (0.2 + 0.35
Elev. sta. 868 = elev. 867 — (0.2 + 0.45
Elev. sta. 869 = elev. 868 — (0.2 + 0.55
Elev. sta. 870 = elev. 869 — (0.2 + 0.65
= 466.8 — 0.25 = 466.55
= 466.55 - 0.35 = 466.20
= 466.20 - 0.45 = 46575
= 46575 - 0.55 - 465.20
= 465.20 - 0.65 = 464.55
= 464.55 - o-75 = 463.80
= 463.80 — 0.85 « 462.95
= 462.95 — 0.95 = 462.00
Elev. sta. 871 = elev. 870 — (0.2 + 0.75]
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 ordinate*, for
100 feet chords of curves of degrees given. The degrees are given in de-
grees and decimals of a degree.
hf.
D
R.diui
R
UfUtUMi
UgR
Tin.
Off.
t
Mid.
Ord. u
*-
Rutin
R
Lcprithm
ug n
Tin.
Off.
t
Mid.
Ord.
o.oo
w
00
000
.000 1
OO
5729.65
3.75813
.873
.218
286478.90
5-45709
.OIJ
.004
5617-31
.890
04
143139.45
.15606
■035
.009
5509.39
.74110
■9=8
.06
■OS 2
■013
06
540534
-73283
.925
■231
.08
7l6ig.73
.85503
.017
OS
-942
.336
57a93.79
4.75813
.087
3.71674
.960
.240
47746-49
.67894
.105
.70891
4Q9>S.S7
.61109
5036.03
.70113
^348
.16
35809.87
•035
16
4939-38
.69367
.18
.50385
■039
iS
485S-66
.68625
.030
• 357
38647.00
4-45709
.175
3-67895
1.047
.163
30043-55
.41570
ojs
4696.46
.67177
.065
.366
23B73.16
■37791
.309
4610.71
.66471
.083
,j6
31036.86
-057
16
4547.38
.65776
■275
.38
■31097
18
4476.33
.65003
-JO
19098.61
.363
^5
4407.46
3.64419
1.134
:X
-31
i70tM.95
.15197
.279
Ji
4340.69
.63736
.388
16851.73
.33664
4275-00
.63103
.169
.36
15915.51
.30)83
.314
.079
36
4313.02
■62459
.187
.207
.38
15077-86
.17834
■333
.083
38
.61835
.204
.301
I4313 97
4.15606
■ 349
.087
40
3- 61 101
■30S
-4*
13641-88
.13487
.367
4035-01
.60585
.310
•44
...467
.384
3978.98
-59977
-1S7
■314
-46
«455-°4
■09537
46
3914-47
■59378
■27+
■319
■4»
.07688
48
3871-44
-58787
.292
■Jij
■SO
405915
.436
.109
SO
3819-83
3.58104
51
S3
3769.57
.57639
-326
■332
■s*
.118
3710.61
-5J06l
.336
.36
10131.43
.OO994
■ 489
56
3672.92
-S65OI
.361
■340
.38
□878.61
3-99470
■ 5=6
.137
S8
3616.43
-55948
•379
.60
954934
3-97997
60
3581.10
3-55403
1-396
■349
.61
-96573
■541
■135
61
3536.89
.54863
-353
M
■95194
64
3493.76
■54319
■431
■358
.66
8681.1b
-93858
'S7&
■144
66
345L67
.53803
.363
.68
8415.90
.148
68
■53183
.466
-367
■70
8185.16
3-91303
!6ii
■ 153
3370.46
3.53769
1.483
■37i
79S7.8o
■9O079
.628
■157
3331-28
.51161
7741-73
.asasg
.646
-iSi
3192-99
■51759
■ 518
.380
■76
.87731
.663
76
325557
■5«63
■536
.384
.78
7345-67
.86603
.681
78
3218.09
.50J72
■553
.388
.80
3-85504
.60S
So
3183.23
350187
1-571
■393
.81
6987 '35
.84431
.716
.179
Si
3148-25
.49807
.588
■397
84
683O.Q0
.83385
.183
84
■49332
.606
.86
6661.36
.83363
■750
86
3080.55
.48863
.623
.88
6510.95
.81364
.768
.11)2
88
3047.78
-48398
.641
.410
«o
6366.16
3.80388
.785
.196
V
3015.71
1.658
■415
01
-79434
.803
V
2984.29
■47484
■67S
.419
94
6095.36
.78500
.305
94
3953-53
-47034
.693
•423
06
5968.38
.77586
!s 3 8
.109
96
■46589
.710
.428
■98
5846.58
.76690
■855
98
389387
.46148
-738
.432
1.00
5719-65
3.75813
■873
lilS 3
°°
3864.93
3.457"
I.74S
«6
TABLE I. — (Continued)
D
fitdlui
LaprHhn
Lo|R
Off."
Drd!
Du-
ll
Ridlui U
■
flrittim
LhR
IM.
Off.
Hid.
Ord.
j.»
2864.93
3-457"
1. 745
■436
3.00
1910.08 3
28.05
3.618
.655
1836.57
-45279
■763
1807.43
27817
■635
■650
3808.76
.44853
.780
■44 s
1SS4.95
27530
.653
s
.06
.444,8
.798
■449
.06
1872.63
27245
.670
I7S4.76
.44008
.8.s
■454
.08
1860.48
26962
.6S7
.6-/3
2728.32
3-43593
1-832
-458
1848.48 3
26681
X
3 70 J .70
.43181
.850
-463
1836.63
.722
2677-53
■41773
.867
.467
1824.93
26r! =
■740
Ms
.16
"6sa.74
■42369
.885
■47'
.16
18.3.39
2584O
.689
1618.40
.41969
.476
.18
1801.99
25575
■775
a
3604.51
3.41573
.480
1700.73 3
25303
2581.05
.41180
•037
.484
1779.61
35032
■Bio
14
■40790
■489
1768.63
34764
.827
.707
2535-38
.40404
.072
1757.78
34406
■845
.Ifl
1513-14
■497
.28
1747-06
34231
.862
.30
3-39643
■ 502
■30
1736.48 3
2.i9<>?
2.879
3469.81
.39366
.024
.506
.736.03
23705
.897
■M
3448.7.
.38894
.042
■ 511
171S-69
.914
.36
1437.06
.38524
.050
■5'5
.36
.705.48
33185
-J8
3407.56
.38158
.077
■ 519
.38
1605-39
-049
■737
3387.50
3-37794
1685,43 3
326? I
3,967
.742
-4*
2367.77
.5*8
1675-56
.984
■746
■44
3348.36
.129
■532
1665.83
22.63
3.002
■751
.46
3339.38
.36712
-147
■46
1656.19
21911
-48
3310.49
■36370
.164
-541
-48
1646.(18
31661
.036
• 759
■So
■50
1637.38 3
3 054
■7(>4
■51
"73-83
■35676
■100
■ 550
.52
1637.98
3i.6 S
.071
.768
H
315593
■35332
.216
1618.78
20919
.089
.36
■34991
.234
■ 559
.56
1609.60
30674
• 777
58
3J30.9S
.34654
.563
.58
1600.70
.134
.78.
.60
3303.87
334319
3.360
.567
.60
1591.81 3
30.89
3-141
.785
.62
3187.05
.33086
.386
.61
1583.02
■SO
790
.64
3170.40
■33656
■ 304
.576
.64
157431
19709
.176
• 794
.66
3 1 5*. '7
33328
.580
.66
1565-72
.68
3138.10
■330=3
•330
-5*5
.68
1557-32
'0335
J03
TO
3113.36
332681
3.356
589
1548.80 3
3- 2 28
.807
2106.66
-32359
-5M
1540.48
1S766
.346
.812
209I.I9
.32041
18533
.263
.8.6
.76
.31716
.408
.602
.76
1524.10
1S30.
.381
.820
•78
.607
.78
18071
.298
825
.80
3.31101
.611
.So
I5o8!o6 3
17842
3.316
.82g
2031.07
.460
.615
.83
17614
■333
.S33
.84
1017.66
.30485
.478
.84
1493.36
17387
■351
.838
.30180
■496
.86
1484.63
.368
.842
.88
1989.65
.29878
.513
.628
.88
1476.08
16037
-385
,847
1975-93
3-29577
.633
■00
.469.4. 3
16714
3-403
.85.
0»
■20279
■ 548
-<>;! 7
.02
1461.91
16492
.420
-855
■94
.28982
■565
.641
■04
1454.49
■438
.860
*6
1035-88
.28688
■ S83
.646
■96
■455
.864
.98
.28396
■08
1439-88
15833
-473
.868
3.00
1910.08
3.28105
2.618
■655
■43269 3
IS6.5
3-490
.873
TABLE I. — (Continued)
D
Kidlut
Logarithm
LofR
Tin.
Off.
■id.
Drd.
D
Ridlm
Logarithm
UgR
IU.
Off.
id.
ti.
4.00
i43»-6»
3.15615
3-49°
■873
500
1146.28
3-05929
4-362 .
091
M=S.56
■ 5o7
.877
1141.7s
•05756
■379
095
■04
1418.51
.15183
-525
-05583
■397
.06
141 1.5 2
.14969
'.W>
.06
1132.70
.05411
104
.08
1404.60
-14755
.560
.890
.08
1128.24
.05340
■432
108
1307-76
314543
.895
1133-83
3.05070
4-449 I
113
1390.98
■ 595
.899
1119.43
.04000
.467
117
1384.26
.612
.903
UIS-08
.04731
.484
.16
i37J-6i
.908
.16
-04571
126
I37I.oa
.13704
.647
.iS
.519
130
1364.49
3-13497
3.664
304327
4.536 1
135
1358.03
■13291
.682
1098.00
•554
139
.13076
-699
-925
1093.81
-03894
:s
■345-a8
.128B1
.717
.929
.26
1089.66
.03729
'5S9
.18
.12678
■ 734
•934
jB
1085.53
■03564
.606
152
■io
'33>.77
3.12476
.938
1081.44
156
1326.61
-12274
.769
1077.38
.641
•34
1320.49
.786
■34
1073.34
.658
16s
.30
1314-44
.11874
.804
■951
.36
1069.34
.676
■38
1308.44
-11675
.821
■956
■38
1065.37
■02750
.693
1302.50
3.1 1478
3839
.960
1061.43
302589
178
■43
1296.61
.11281
.856
.964
1057-51
.02429
.728
I S3
■44
1290.77
.11085
.874
.969
-44
IOS3-63
.02269
.746
,87
46
.10800
.891
-973
.46
1049.77
.03 log
-763
191
.48
127925
.10696
.978
■48
-01951
.780
196
■So
"73-57
3.926
.982
1042.14
3.01793
4.708 I
1267,93
.10310
-943
.52
1038.37
-01635
.8.5
205
■ 54
.10118
.96.
1034.6 a
.01478
■831
209
■S6
1256.8a
.978
■995
.56
.01333
.850
.58
1251-33
*9737
.996
■999
■ S8
218
1245.89
3.09548
4-013
1.004
.60
4885 1
.61
.09360
.031
.008
.62
.00856
.64
1235.16
.09172
.048
64
1016.29
231
.66
.08986
.065
.66
1012.70
.00548
■937
33s
.68
.083
.68
1009.14
■0039S
-955
I2L9-40
1.026
3.00243
4972 I
344
1214-24
,118
1002.09
248
1209.12
.08347
.135
.034
998.60
2-99939
S-OO?
253
-rt
.08064
■153
.76
995-14
.99788
»57
.78
.07883
■043
-78
991.68
■09638
.042
261
.So
3.0770I
4-188
1.047
.So
2.99488
5-059 1
266
1189.06
.205
.OS 2
.82
08+89
270
.84
1184.15
•07341
.84
981.53
.99100
■094
374
.86
.07162
.86
978.17
■99041
.88
1171-45
.06983
• 257
.065
.88
974-66
.9888s
383
1169.66
3.06806
4-275
1.069
.90
971-54
2.9B746
5.146 1
287
■sn
1164.91
.074
.92
068.36
-98599
.164
9*
.06453
.310
■94
965-01
■98453
.181
.96
ioRs
■06
961.77
.98307
■199
301
98
1150.88
■ 345
.087
■98
958.56
.98162
305
5.00
1146.28
3-05929
4-362
1.091
6.00
955-37
2.980.7
5-234 I
309
TABLE II. — (Continued)
.00
.03
.04
.06
.08
.10
.13
14
l6
.18
.33
24
.26
.28
.30
.33
■34
36
38
.40
•4*
•44
46
■48
SO
53
■54
56
58
.€0
.62
64
.66
68
70
73
74
76
78
.80
.82
84
.86
.88
9o
93
94
.96
98
11.00
24 1
21787
218.92
219.07
221.01
222.06
223.10
324.14
225.19
226.24
227.28
228.33
229.38
230.42
231.47
232.52
233 56
23461
23565
236.70
23775
238.79
239 84
240.89
24193
242.98
24403
245-08
246.12
247.17
248.22
249-27
250.31
251.36
252.41
25346
254. So
255. 55
256.60
25765
258.70
25975
260.79
261.84
262.89
26394
26499
266.04
267.09
268.14
269.18
270.23
25'
270.23
271.28
27233
27338
27443
27548
276.53
27758
278.63
279.68
280.73
281.78
282.83
28388
284.93
285.98
287.03
288.08
289.13
290.18
29123
292.28
29333
29439
295.44
296.49
29754
298.59
299-64
300.69
301.74
302.80
30385
30490
30595
30700
308.06
309.11
310.16
311. 21
312.27
31332
31437
31542
316.48
31753
318.58
31963
320.69
321.74
322.79
26*
133333
322.79
32385
32490
32595
327.01
328.06
32912
330.17
33122
332.28
334-39
33544
33650
33755
338.60
33966
340.71
34177
342.82
343.88
34493
34599
347.04
348.10
349- 15
350.21
35126
352.32
35338
35443
35549
356.54
35760
358.66
35971
360.77
361.83
362.88
36394
36500
366.05
367.11
368.17
36922
370.28
371-34
372.39
37345
37451
37557
27°
1375-57
1376.62
1377.68
1378.74
1379.80
1380.86
138191
1382.97
138403
138509
1386.15
1387.21
1388.27
138932
139038
1391.44
I392.50
139356
139462
1395.68
1396.74
139780
139886
1399 92
1400.98
1402.04
140310
1404.16
1405.22
1406.28
140734
1408.40
1409.46
1410.52
1411.S8
1412.64
I4I370
I4I476
141582
1416.88
141794
1418.01
1420.07
1421.13
1422.19
142325
I4243I
142538
1426.44
I42750
1428.56
28*
1428.56
1429.62
1430.69
1431.75
1432.81
1433.87
1434.94
1436.00
1437.06
1438.13
1439.19
1440.25
1441.32
1442.38
1443.44
1444.51
1445.57
1446.63
1447.70
1448.76
1449.82
1450.89
1451.95
1453.02
1454.08
1455.15
1456.21
1457.27
1458.34
1459.40
1460.47
1461.53
1462.60
1463.66
1464.73
1465.80
1466.86
1467.93
1468.99
1470.06
1471.12
1472.19
1473.26
1474.32
1475.39
1476.45
1477.52
1478.59
1479.66
1480.72
1481.79
39*
1481
1482
1483
1484
i486
1487
1488
1489
1490
1491
1492
1493
1494
1495
1496
1497
1498
1499
1501
1502
1503
1504
150s
1506
1507
1508
1509
1510
1511
1512
1513
1514
1515
1517
1518
1519
1520
1521
1522
1523
1524
1525
1526
1527
1528
1529
1530
1532
1533
1534
1535
79
86
92
99
06
12
19
26
33
39
46
53
60
67
73
80
87
94
ox
08
15
21
28
35
42
49
56
63
70
77
84
91
98
05
12
19
26
33
40
47
54
61
68
76
83
90
97
04
11
18
25
30 #
535.25
536.33
537.40
538.47
53954
540.62
54169
542.76
54383
544 91
54598
547.05
548.12
549-20
550.27
55134
553.42
553.49
55456
555.64
556.71
55779
558.86
55993
561.01
562.08
563.16
56423
56531
566.38
567.46
568.53
56960
570.68
571.76
572.83
57391
57498
576.06
577.13
578.21
57928
580.36
581.44
582.51
58359
584.67
585.74
586.82
587-90
588.97
31'
588.97
590.05
59113
592.20
59328
59436
59544
596 51
59759
598.67
59975
600.82
601.90
602.98
604.06
605.14
606.22
607.30
608.37
609.45
610.53
.611.61
612.69
613.77
614.85
61593
617.01
618.09
619.17
620.25
621.33
622.41
623.49
624.57
625.65
626.73
627.81
628.89
629.97
631.05
632.14
63321
63430
63538
636.46
637-54
638.62
63970
640.79
641.87
642.95 1
00
02
04
06
08
10
12
14
16
18
22
34
36
38
30
33
34
36
38
40
46
48
5o
53
54
56
58
60
63
64
66
68
70
73
74
76
78
83
84
86
88
90
93
94
96
98
00
(23)
TABLE L — (Continual)
hf.
D
Ftidl w L
Eiftttn
L.,R
iff.'
t
to. D
1-
D
Hidlui
Logirftlim
UfR
iff.'
t
rd.'
a.oo
716.78 3
85539 6
976
1.746 9
00
63M7
a. 80433 7
846 1
965
714-99
85430 ,
093
•750
635.86
.80336
713-ai
853" 7
06
711.45
8S«S
06
633.06
^145
898
978
08
709.69
85107
045
.764
08
631.67
.80049
982
707.94 1
06.,
1.768
1-79954 7
987
706.10
84893
080
■771
7<H-47
84786
09S
■777
617.53
-79704
968
16
84680
.781
16
6a6.. 7
9S5
84574
'33
.785
18
-79574 i
609,33 a
84468 7
ISO
613.4S
008
697.63
84361
167
.794
■79386
695.04
84157
'85
.798
610:76
j6
694.25
84151
.803
16
.79198
071
18
691.58
84047
.807
iS
090
026
30
83943 7
137
1.811
616.76
1.79011 8
689.36
83B38
.816
615.44
.789.9
687.61
83734
173
.830
.78816
36
685.96
83630
189
.835
36
tn.Bi
■ 33
'S9
38
684.33
83517
306
38
176
048
40
683.70 1
834a3 7
3=4
'■833
610,31
i. 48 8
194 I
681.09
833ao
.838
608.91
- 56
056
670.47
83H7
359
.841
44
607.63
- 61
061
.46
677.87
83 "5
376
■847
46
606.35
146
065
-48
676.37
83013
393
.851
48
605.08
: B.
069
674.69' a
83910 7
I-85S
So
603.80
sSl 1
074
67311
83808
31
298
67153
83707
446
.864
601.38
.7790S
316
rt}
36
669.97
83605
463
36
600.0 a
.77817
oSj
38
668.41
81504
480
.873
38
598.77
.77716
3 SO
091
60
666.86 3
81403 7
498
1.877
597-53
1.77636 8
368 a
096
665.33
83303
515
.881
596.19
■77546
385
64
663.77
.886
64
595.06
■77456
66
550
.890
66
S93.83
.77366
68
567
■895
68
591.60
■77276
TO
6jo.ii 1
6S7.70
3.8o 3
S8 S
1899
003
71
591.38
3.77187 8
■77098
4S5 1
471
74
656.19
81703
.908
388:96
490
126
76
654.70
81604
637
.913
76
587-76
.76910
507
78
653.11
81505
<>5S
78
586.56
.76831
514
.So
651.73 >
81407 7
67 >.
80
585.36
3.76743 8
'39
.81
650.35
81308
68g
■915
81
58417
■76654
559
.84
648.79
81110
707
84
581.99
.76566
576
I48
So
647.33
714
.934
86
581.81
.76478
594
as
645.87
741
.938
88
580.64
.76390
611
'57
644-*3 a
80917 7
1*43
579^6
1-76303 8
.6.
91
641.98
80810
776
91
578.30
.761.5
646
.66
641.54
80713
794
.951
94
S77-I4
663
170
-9*
80616
811
■956
96
575-99
.76041
68.
.98
638.69
80519
.960
98
574.83
698
179
9
00
637.37 1
80433 7
846
1.965 10
573-69
liJsB^T 8
7.6 3
183
TABLE t— (Coniinutf)
D
R*dlM Lo
girlttim
UfH
Off.
ri.
D H .
Ridlw U
B
ofH
IU.
Off.
t
: :
IO.0
573-69 a
75867
87.6 3
183
'SO
383*6 1
58327
13.053
3.377
368.0a
7S436
.803
sos
380.54
58040
■ '39
.399
S«Sa-47
.889
378*5
5775S
.331
74587
.97 6
37S.6o
57473
.312
.343
■4
351*8
74169
9.063
-365
546.44 1
a 03
370.78 3
569"
13-485
3-387
*
5+1.30
73344
.237
314
6
368.42
56634
.409
536.2 s
72937
.314
336
56358
!6 S 8
J
531-30
7 '534
358
a
363.78
56084
9
516.44
72"35
■498
J 80
9
361-51
558"
.83.
.474
521.67 a
71740
9.58s 2
16
339.26 3
55541
3-496
S16.99
7'348
.671
433
357*5
55=73
14*04
.518
515-38
70960
.758
445
354-86
S5°o6
■090
.540
.3
507.86
70575
■845
467
3S1-70
.177
.S63
503 -42
JOIB3
.932
489
A
350.56
54476
.263
■584
4OO.06 a
69815
10.019 1
348.45 1
14-349
3.606
;J
404-77
490.56
69441
69069
:!g1
555
6
346-37
344-3"
53953
436
.638
!g
486.43
.279
577
.8
342.17
53437
*o8
* 7 2
481.34
68336
.366
598
*
340-36
■695
-694
478.34 a
67974
10.453 2
17.0
338.17 2
52927
14.781
3.716
474.40
67614
■540
336-3I
52674
.S67
.738
470.53
6 7 a 5 8
.626
664
334-37
52433
466.71
66905
.713
686
332.45
.78'
•4
66555
.800
4
330.56
51934
.803
-s
450- "8 3
IO.887 2
730
328.69 2
51*77
3.825
.6
455-65
65863
■973
6
316.83
51432
-847
.T
453*7
6S3"
11.060
32S*o
51188
-385
.869
.a
448.56
65182
.147
8
313.18
50945
-891
.9
44S-09
64S45
■ 334
3i7
32139
.913
IS*
441.68 3
64511
839
iS
319-61 2
S0464
15.643
3.935
438.33
64180
861
317.87
50225
■730
■957
435.02
63851
883
49988
■979
431.76
63525
.580
9°5
3M-43
.902
428.56
63=01
.667
937
495';
.98a
•3
62873
949
5
3"*6 1
492B4
16.074
.6
422.28
.840
6
49052
.067
419.1a
6a 244
992
307-76
48S21
^46
,o8g
.8
416.19
12.014 3
S
306.14
48591
-333
■»
61617
9
304-53
48363
-4'9
■133
410.18 a
6130S
laiJs? 3
058
19
4S136
16.505
4-1S5
407 38
61000
080
301.37
4-gir)
404.53
60695
!3&o
199.82
47686
.677
.199
■3
401.71
60391
■447
198.18
47462
-763
-4
398-94
396-20 a
5979?
ia.620 3
146
168
4
S
296.;5
47240
.849
16.935
.343
4.365
1 *
393-50
.706
190
6
293.76
46799
.387
390.84
SQ30O
-793
46580
.107
.308
1 .8
388.2.
58907
.880
8
4636!
.193
-330
385.62
58616
.966
355
9
289-37
46145
1 'SO
383*6 2
58327
13*53 3
377
10
°
387.94 1
45930
17.363
4-374
TABLE I. — {Continued)
D
flidlm U
;lrithm
LogR
Off.'
t
Ord.
hf.
D
Rid in 1 La
1
prlthm
Lojfl
Off.'
t
Ord.
10.0
187.94 a
45930
17.36s
4-374
2 S °0
331.01 1
36363
11.644
S-476
186.52
4571b
•396
36193
.719
-498
185.11
45502
-537
319.31
36033
.8.4
■J
283.73
.613
•440
HS.32
35854
.899
■543
-4
181.3s
45079
.708
.462
337-43
35685
-985
.564
-s
3S0.99 5
44869
4.484
226.55 '
3S5'7
S-586
6
179.64
.S80
.506
2 25.68
3535°
■153
.608
37B.30
44453
.066
.518
s5
.140
8
176.08
44345
8
113.96
■325
.6«
9
37s .67
.138
9
213.II
34853
-675
274-37 '
43834
18.324
4-594
36
34688
22.405
5-697
=73.o8
221.43
34534
• 58c
-719
271.81
434'7
395
-638
210.6o
3436'
.665
■74'
-3
3TO-55
.481
219.78
34199
.75o
.763
•4
269.30
43034
S67
.681
218.96
34037
.835
-785
168.36 1
,1824
.8. 1.5 2
33875
33.910
5.807
.6
266.84
43634
.738
.,16
317-34
33715
.829
165.62
42426
.814
.748
116.54
33555
.090
-8S2
.8
264.42
43339
8
-874
■9
163.32
■702
9
214.96
>3^J7
-8g6
itetr '
41837
41643
19.081
.167
4.814
-836
17
114.18 2
113.41
33078
33931
33-345
s t$
1S8-7I
41449
.3 S 3
2
112.64
)»;'>■!
■514
■1
258.56
41356
,338
m.87
3260s
.084
■4
157-43
41064
.903
31452
.684
6.006
156.19 1
4087.;
I9.5O9
4-925
HO.36 3
33-769
.6
155-17
40(1*3
■595
■947
■yn-y
.853
■051
154.06
40404
.680
.969
308.87
3198S
■938
.8
151-06
40306
.766
.991
31835
34.033
■095
■g
251.87
401 18
.85.
9
.108
JJ.O
2S0.79 3
39931
19937
31539
6.139
307*6
■057
31378
.277
148.66
39561
.108
.079
31337
.362
.184
147.61
39376
■i
104.53
3 10 J 6
■446
4
346.56
iQ'0.5
.133
4
203.83
iO()2<>
-531
.218
345-53 3
39010
3O.304
SMS
103-13 3
3077'>
34.61S
6.250
244.50
■4SO
.167
6
202.43
30627
.273
3+3-49
38647
•53S
.189
IOI.74
.784
-394
.8
141.48
(846;
.630
8
30 i. u
.869
.316
I4I.48
38188
.706
•
30184
.954
140.49 3
38109
5-15S
39
199.70 3
30037
25.03B
6.361
37931
-877
.378
20891
.383
338.53
37754
.963
198.36
3974s
■405
237-S&
11.047
•331
37402
343
197-04
=9455
-376
■4SO
335-65 3
17*1)
31.218
5-366
196.38 =
94311
15-460
134.71
37053
303
6
195-74
19 1(1 J
-S4S
1J3 -77
36870
.388
19509
2tw:j
.516
233.84
36707
-474
S
19445
jMSKi
.713
.538
9
33193
36535
■454
■9
193. 83
28730
■79«
.560
1S
'
131.0. 3
36363
iJw
5476
JO
193.19 3
28557
25.881
6-583
TABLE I. — (Conclude®
n*g.
D
Radlu* Lo
R
ogH
Tin. Hid.
Oft. Ori.
Dag.
D
Ridlui
R
Logarithm
Ld ( H
Tin.
Off.
t
Kid.
Ord.
30.0
193. ig 2
28597
25-882 6
583
3S-o
166.28
a.22083
30.071
7.696
192.56
28456
.966
605
.65.82
.21963
154
.718
191-94
283x5
627
.65-36
.2.843
191.3a
28175
-135
649
16491
.21724
.320
■763
190.70
2*o.j6
.219
67.
164.46
.2.605
-403
■78s
190.09 2
27896
>94
5
164.01
2.21486
30.486
7.807
.6
189.49
.367
6
163.56
.21368
.830
27*19
73S
163.12
.21250
■653
.852
.8
188.28
J74Sl'
'ss6
760
.8
162.68
-21133
.736
-874
9
187.69
782
■0
162.24
.21016
.897
31.0
187. 2
2~-'o7
2672,, 6
log
36.0
161.80
2.20899
30.901
7919
186.51
27071
.808
.20782
■985
.942
■85.03
26935
.892
8.19
160.94'
.20666
31068
-964
•J
.85.35
26709
-976
871
3
.986
184.77
26664
894
.60.08
-20435
■»M
8.009
5
.84.2c 2
26530
916
15966
JI.3I6
8.03.
.6
183.63
183.07
36395
2626;
.228
.311
938
6
159.24
158.82
.20205
:X
OS3
.076
'.a
182.51
HSlaB
5R3
1
.58.40
-10977
-56s
.098
9
.81.95
J 5996
■480 7
00;
9
.19863
.048
32 .o
181.40 2
1S0.85
25*'>3
27.S64 7
.648
027
"
\
.5758
157..7
"9636
31-730
.813
8.143
.165
180.30
■73"
072
.56.76
-.9523
.896
.188
179.76
2S469
094
156.35
179-22
25338
.899
'55-95
■19299
233
17B.68 2
25208
27-983 7
.38
2.19187
32.144
8.255
IJ8.I5
25078
6
'55.15
.19076
.137
-7
177-62
24049
.150
183
.18964
.300
.8
177-09
205
8
154.36
.18854
..192
.322
4
176.57
24691
.318
9
.■8743
-474
■345
33-0
176.05 2
24563
24435
28.402 j
.48s
2,2
as
°
15358
2.18633
.18523
32-557
8.367
■390
"7S-02
24308
.569
2y-t
.■8413
.732
■J
241S1
.652
316
152.42
.18304
.804
•434
174.00
■736
339
152-04
.18195
.8S7
■457
'73-49 2
2302 S
28.820 7
■M
15. .66
2.18086
32.969
8.479
.6
172.99
;;,So2
383
6
.17978
2.S677
.9R7
150.00
.17870
134
2J5S2
29.070
428
150.53
.17762
■569
■9
171-50
23428
9
■17655
.298
34-0
171.02 1
17053
23303
23180
39-237 7
.321
495
39
J
M9-79
149.42
2.17547
.1744'
33-381
.463
8.592
.614
170.04
7. jo ; 6
517
149-05
.636
169-56
2 20.13
■487
-3
.48.69
.659
169.09
1 1811
-S7.
S62
148.33
.17111
.710
.6S1
2z68U
29.654 7
584
1.17016
33.792
8.704
.6
168.14
.67.67
2256;
■737
(io6
.6
147-61
147.25
.16911
.16805
-874
.956
Tn
J
167.10
22324
■904
6S?
!■
146.89
.16701
34-038
■771
■9
■987
673
9
■46.54
.16596
35°
166.18 1
22083
30.071 7
696
■"
2.16492
34.101
8.S16
TABLE II. — Tangent Distances for a i° Curve for Varying I's
T d - T x o/D + C of Table III
/
0*
l a
2 #
3°
4°
5 #
6 #
7*
I
.00
.02
.04
.06
0.00
50.00
100.01
150.04
200.08
250.16
300.28
350.44
.00
.02
04
.06
1. 00
2.00
. 3.00
51.00
52.00
5300
10 1. 01
102.01
103.01
15104
152.04
15304
201.08
202.09
203.09
251 . 16
252.17
25317
301.28
302.28
30329
35144
352.45
35345
.08
.10
.12
4.00
5. 00
6.00
54.00
55.00
56.00
104.01
105.01
I06.OI
154.04
15504
156.04
204.09
205.09
206.09
254.17
25517
256.17
30429
305.29
306.30
35446
35546
356.46
.08
.10
.12
14
l6
.18
.20
.22
•34
.26
7.00
8.00
9.00
57.00
58.00
59.00
107.01
I08.0I
109.01
157.04
158.04
159-04
207.09
208.09
209.10
25718
258.18
259-18
30730
308.30
30930
357.47
358.47
35948
14
.16
.18
.SO
.22
•24
.26
10.00
60.00
110. 01
160.04
210.10
260.18
310.31
360.48
11.00
12.00
13.00
61.00
62.00
63.00
III. 02
112.02
113.02
161.04
162.05
163.05
211. 10
212.10
213.10
261.18
262.19
263.19
311. 31
312.31
31332
361.48
362.49
363.49
.38
.30
.32
14.00
15.00
16.00
64.00
65.00
66.00
114.02
115.02
II6.02
164.05
165.05
166.05
214.10
215.10
216. 11
264.19
265.19
266.19
31432
31532
316.32
36450
365.50
366.50
.28
.30
33
•34
36
38
.40
•43
44
46
17.00
18.00
19.00
67.00
68.00
69.00
117.02
II8.02
119.02
167.05
168.05
169.05
217. 11
218. 11
219. 11
267.20
268.20
269.20
31733
318.33
31933
367.51
368.51
36952
34
36
38
.40
43
44
46
20.00
70.00
120.02
170.05
220.11
270.20
320.34
370.52
21.00
22.00
23.00
71.00
72.00
73.00
121.02
122.02
123.02
17105
172.05
17305
221. 11
222.11
223.12
271.21
272.21
273.21
321.34
322.34
32335
371.52
372.53
37353
48
.50
•53
24.00
25.00
26.00
74-01
75.01
76.01
I24.02
125.02
126.02
174.06
175.06
176.06
224.12
225.12
226.12
27421
275.21
276.22
32435
32535
326.36
37454
37554
376.55
48
SO
.52
■54
56
•58
.60
.62
64
.66
27.00
28.00
29.00
77-01
78.01
79.01
127.02
128.02
129.02
17706
178.06
17906
227.12
228.12
229.12
277.22
278.22
279.22
32736
328.36
32937
37755
378.55
37956
54
.56
.58
.60
.62
64
.66
30.00
80.01
130.02
180.06
230.13
280.23
330.37
380.56
31.00
32.00
33.00
81.01
82.01
83.01
I3I.02
132.03
133.03
181.06
182.06
183.06
231.13
232.13
23313
281.23
282.23
283.23
331.37
332.38
33338
381.57
382.57
38358
.68
.70
72
34.00
35.00
36.00
84.01
85.01
86.01
134.03
135.03
136.03
184.07
185.07
186.07
23413
23513
236.14
284.24
285.24
286.24
33438
33539
336.39
384.58
38559
386.59
.68
.70
73
74
76
78
.80
.82
84
.86
37.00
38.00
39.00
87.01
88.01
89.01
137.03
138.03
I3903
187.07
188.07
189.07
237.14
238.14
23914
287.24
288.25
289.25
33739
338.40
33940
387.59
388.60
38960
74
.76
78
.80
.82
84
.86
40.00
90.01
140.03
190.07
240.14
290.25
340.40
39061
41.00
42.00
43.00
91.01
92.01
93.01
141.03
142.03
143.03
191.07
192.07
19307
241.15
242.15
24315
291.25
292.26
29326
34141
342.41
34342
39161
392.62
393.62
.88
.90
.92
44.00
4500
46.00
94.01
95. 01
96.01
14403
I45.O3
I46.O3
194.08
19508
196.08
244.15
24515
246.15
294.26
29526
296.27
344.42
345.42
346.43
394.63
395.63
396.64
.88
90
.93
•94
96
98
LOO
47.00
48.00
4900
97.01
98.01
9901
147.03
I48.O3
I4904
19708
198.08
19908
247.16
248.16
249.16
29727
298.27
29928
34743
348.43
34944
39764
398.65
39965
94
96
98
1.00
50.00
100. 01
I50.04
200.08
250.16
300.28
350.44
400.66
(20)
1
TABLE II. — (Continued)
I
&•
9 #
IO #
xx°
13°
I3 #
14°
15*
J
.00
.02
.06
400.66
450.93
501.38
551.70
602.2I
652.81
703.51
75432
.00
.03
.04
.06
401.66
403.67
403.67
451.94
452.95
45395
503. 29
503.29
504.30
552.71
553.72
554. 73
603.22
604.23
605.34
653.82
654.84
655.85
704.53
705.54
706.56
755-34
756.36
75738
.08
.10
.13
404.68
495.68
406.69
45496
455.96
456.97
505.31
506.32
507.33
555.74
556.75
55776
606.36
607.27
608.28
656.86
657.88
658.89
707.57
708.59
709.60
758.39
759.41
760.43
.08
.10
.13
14
.16
.18
.10
.33
34
.36
407.69
408.70
40970
45798
458.98
459 99
508.33
509.34
510.35
558.77
55978
560.79
609.29
610.30
6ll. 31
65990
660.92
661.93
710.62
711.63
712.65
761.45
762.46
76348
14
.16
.18
.10
.33
•34
.36
410.71
461.00
511.36
561.80
6l2.32
662.94
71367
76450
411. 71
413.72
413.72
462.00
463.01
464.03
512.37
513.37
514.38
562.81
563.82
56483
61333
61435
615.36
663.96
664.97
665.98
714.68
71570
716.71
76552
766.53
767.55
.38
•3©
.33
414.73
415.73
416.74
465.03
466.03
467.04
515.39
516.40
517.41
565.84
566.85
567.86
616.37
6l738
618.39
667.00
668.01
669.02
71773
718.74
71976
768.57
76959
770.61
.38
.30
.33
•34
36
38
.40
43
44
46
417.74
418.75
4197s
468.04
469.05
470.06
518.41
519.42
520.43
568.87
56988
S70.89
619.40
620.42
621.43
670.04
671.05
672.07
720.78
721.79
722.81
771.62
772.64
773-66
34
36
38
.40
43
44
46
430.76
471.06
521.44
571.90
622.44
67308
723.83
774-68
431.76
422.77
423.78
472.07
473.08
474.08
522.45
533.46
524.46
572.91
57392
57493
623.45
624.46
625.47
674.09
67511
676.12
72484
725.86
726.87
77570
776.72
77773
48
So
4P
43478
425.79
426.79
475.09
476.10
477-10
525.47
526.48
527.49
575-94
576.95
57796
626.49
627.50
638.51
677.13
678.15
679.16
727.89
728.90
72992
778.75
779-77
780.79
48
50
53
54
56
58
.00
.63
64
.66
437.80
428.80
439.81
478.H
47912
480.13
528.50
529.51
530.51
578.97
57998
580.99
629.52
630.54
631.55
680.18
681.19
682.21
730.94
731.95
732.97
781.81
782.83
7838s
54
56
58
.00
03
.64
.66
430.81
481.13
531.52
582.00
632.56
683.22
73399
784.86
431.83
432.83
43383
482.14
483.15
484.15
532.53
533.54
534.55
58301
584.02
58503
63357
63458
635.60
684.23
685.35
686.26
73500
736.03
737.03
785.88
786.90
78792
.68
•70
.7a
434.84
43584
436.85
485.16
486.17
487.18
535.56
536.57
537.57
586.04
58705
588.06
636.61
637.62
638.63
687.28
688.29
689.31
738.05
739.07
740.08
788.94
78996
790.98
.68
.70
73
74
76
78
80
.83
84
.86
437.8s
438.86
43987
488.18
489.19
490.20
538.58
539.59
540.60
589.07
590.08
591.09
63965
64O.66
64I.67
690.32
69133
692.35
741.10
743.13
743.13
792.00
793.03
794.04
74
76
78
.80
.83
84
.86
440.87
491.20
541.61
592.10
642.68
69336
744. IS
795.05
441.88
442.88
443.89
492.21
493.22
494.23
542.62
543.63
544.64
593. 11
594.12
595.13
64370
64471
64572
69438
695.39
696.41
745.17
746.19
747.20
796.07
797.09
798.11
.88
90
.93
444.90
445.90
446.91
495.23
49624
497.25
545.65
546.66
547.67
596.14
597.16
598.17
646.73
64775
648.76
697.42
698.44
69945
748.33
74934
750.35
79913
800.15
801.17
.88
90
.93
94
1 96
L 98
447.91
448.92
44993
498.36
499.36
500.37
548.67
549.68
550.69
59918
600.19
601.20
649-77
650.79
651.80
700.47
701.48
702.50
751.27
752.29
75331
803.19
803.31
804.23
94
96
98
1.00
Iloo
450.93
501.28
551.70
602.21
652.81
703.51
754.32
805.35
(21)
TABLE II. — (Continued)
.00
.03
.04
.06
.08
.10
.12
14
.16
.18
.80
.22
•»4
36
.28
30
.33
•34
36
38
.40
46
i6°
.84
.86
90
93
94
96
98
1.00
805.35
806.37
807.39
808.31
809.33
810.35
811.37
813.39
81341
81443
815.45
816.47
81749
818.51
819.53
830.55
831.57
833.59
833.61
834.63
835.66
^836.68
837.70
838.73
48 83974
830.76
831.78
833.80
833.83
83484
835-87
So
53
54
56
.58
.00
.63 1 836.89
64 83791
.66 838.93
.68 8399s
70 840.97
73 841.99
.74 84303
.76 844.04
.78 _845.o6
.80
.83
17*
846.08
847.10
848.13
849. IS
850.17
85119
853.31
85334
854.36
855.38
856.30
856.30
85733
858.35
859-37
860.39
861.41
863.44
863.46
864.48
865.51
866.53
867.5S
868.57
869.60
870.63
871.64
873.67
873.69
87471
875-74
876.76
877.78
878.81
87983
880.85
881.88
883.90
883.92
884.95
885.97
887.00
888.03
889.04
890.07
891.09
893.13
893.14
894.17
89519
896.31
z8 (
897.34
898.36
89929
900.31
90134
903.36
90339
904.41
90544
906.46
907.49
907.49
908.51
909.54
910.56
911.59
913.61
91364
914.66
91569
916.73
917.74
918.77
919.79
930.83
931.84
933.87
93390
924.92
925.95
926.97
938.00
929-03
930.05
931.08
932.11
933.13
934.i6
93519
936.31
93724
19°
938.37
93929
94032
941.35
942.37
94340
94443
94546
946.48
94751
948.54
949- 57
950.59
951.63
953.65
953-68
954- 70
95573
956.76
957-79
95881
958.81
95984
960.87
961.90
963.93
963.96
96498
966.01
96704
968.07
969.10
970.13
971.16
973.19
97331
974.34
97537
976.30
97733
978.36
97939
980.43
981.45
983.48
98350
98453
985.56
986.59
987.63
988.65
20'
98968
990.71
99174
993.77
993.80
994 83
995-86
99689
99792
998.95
99998
IOOI.OI
1003.04
1003.08
1004. II
1005.14
1006.17
1007.20
1008.33
1009.36
1010
*
OIO.39
OII.33
013.35
013.39
014.43
015.45
016.48
017.51
018.54
019.58
030.61
021.64
033.67
033.70
034.73
035.77
036.80
037.83
038.86
039.89
030.93
031.96
033.99
034.03
035.06
036.09
037.12
038.16
039.19
040.33
041.35
21 •
043.39
04333
044.35
045.39
046.43
047.45
048.49
04953
050.55
051.59
053.63
053.66
054.69
055.73
056.76
05779
O58.83
05986
060.89
06l.93
IO61.93
IO63.96
1064.00
IO65.O3
IO66.O7
IO67.IO
I068.I4
IO69.I7
1070.31
1071 . 34
1073.37
1073.31
1074.35
1075.38
1076.43
1077.45
1078.49
1079.53
1080.56
1081.59
1083.63
1083.66
1084.70
1085.74
1086.77
1087.81
1088.84
1089.88
1090.93
1091.95
22'
1093.99
I094.03
IO95.06
IO96.IO
1097.14
1098.17
1099.31
1100.35
I IOI.38
II03.33
1103.36
1104.39
HO5.43
IIO6.47
IIO7.5I
II08.54
IIO9.58
II 10. 62
mi. 66
1112.69
111373
III373
111477
1115.81
1116.85
1117.88
1118.93
1119.96
1131.00
1133.04
1133.07
II34.II
1135.15
II36.I9
1137.33
1138.37
1129.31'
II3O.3S
II3I38
1133.43
II3346
"34.50
"35. 54
1136.58
113763
1138.66
1139.70
1140.74
1141.78
1143.83
114386
114490
1145-94
1146.98
1148.02
1149.06
1 150. 10
I 151. 14
I 153. 18
1153.33
I 154 26
II55.30
1156.34
II5738
1158.42
115946
1160.50
1 161. 55
1163.59
1163.63
1164.67
1165.71
23 #
16571
166.75
16779
168.83
169.88
170.92
171.96
17300
174.04
17509
176.13
177.17
178.31
17935
180.30
181.34
183.38
18343
18447
185.51
186.55
187.60
188.64
189.68
190.73
191.77
193.81
19386
19490
195.94
196.99
198.03
19907
300.13
30I.I6
303.31
303.35
304.39
205.34
306.38
307.43
208.47
209.52
310.56
311. 60
212.65
313.69
314.74
31578
316.83
317.87
.00
.03
.04
.06
.08
.10
.23
14
16
l8
.SO
.23
24
.26
.28
30
32
34
36
38
.40
46
48
5o
52
54
56
.58
.00
.62
64
.66
.68
70
72
74
76
78
80
.82
84
.86
.88
90
.92
94
.96
98
.00
(22)
^
TABLE II. — (Continued)
.00
.02
.04
.06
.08
.10
.12
.14
.16
.18
.22
34
.26
.28
.30
.32
34
36
3«
.40
46
48
50
53
•54
56
58
.60
.62
64
.66
.68
70
72
74
76
78
.80
.82
84
.86
.88
90
92
•94
96
98
Hoo
24°
217.87
218.92
219.97
221.01
222.06
223.10
224.14
225.19
226.24
227.28
228.33
229 38
230.42
231.47
232.52
233 56
23461
235.65
236.70
23775
238.79
239.84
240.89
241.93
242.98
244.03
24508
246.12
247.17
248.22
24927
250.31
251.36
252.41
25346
254.50
255-55
256.60
25765
258.70
259-75
260.79
261.84
262.89
26394
264.99
266.04
267.09
268.14
269.18
270.23
25"
270.23
271.28
272.33
273.38
274.43
275.48
276.53
277.58
278.63
279.68
280.73
281.78
282.83
283.88
284.93
285.98
287.03
288.08
289.13
290.18
291.23
292.28
293.33
294.39
295.44
296.49
297.54
298.59
299.64
300.69
301.74
302.80
303.85
304.90
305.95
307.00
308.06
309.11
310.16
1311.21
312.27
313.32
314.37
315.42
316.48
317.53
318.58
319.63
320.69
321.74
322.79
26°
1322.79
1323.85
1324.90
1325.95
1327.01
1328.06
1329.12
1330.17
1331.22
1332.28
1333.33
1334.39
1335.44
1336.50
1337.55
1338.60
1339.66
1340.71
1341.77
1342.82
1343.88
1344.93
1345.99
1347.04
1348.10
1349.15
1350.21
1351.26
1352.32
1353.38
1354.43
1355.49
1356.54
1357.60
1358.66
1359.71
1360.77
1361.83
1362.88
1363.94
1365.00
1366.05
1367. II
1368.17
1369.22
1370.28
1371.34
1372.39
1373.45
1374 51
1375. 57
27°
137557
1376.62
1377.68
1378.74
1379.80
1380.86
1381.91
1382.97
138403
138509
1386.15
138721
1388.27
138932
1390.38
1391.44
139250
139356
1394.62
1395.68
1396.74
139780
139886
139992
1400.98
1402.04
1403.10
1404.16
1405.22
1406.28
1407.34
1408.40
1409.46
1410.52
1411-58
1412.64
I4I370
I4I476
141582
1416.88
I4I794
1418.01
1420.07
1421.13
1422.19
142325
I4243I
142538
1426.44
1427.50
1428.56
28°
1428.56
1429.62
1430.69
1431.75
1432.81
1433.87
1434.94
1436.00
1437.06
1438.13
1439 19
1440.25
1441.32
1442.38
1443.44
1444.51
1445.57
1446.63
1447.70
1448.76
1449.82
1450.89
1451.95
1453.02
1454.08
1455.15
1456.21
1457.27
1458.34
1459 .40
1460.47
1461.53
1462.60
1463.66
1464.73
1465.80
1466.86
1467.93
1468.99
1470.06
1471.12
1472.19
1473.26
1474.32
1475.39
1476.45
1477.52
1478.59
1479-66
1480.72
1481.79
29 s
1481.79
1482.86
148392
1484.99
1486.06
148712
1488.19
1489.26
149033
1491.39
149246
149353
1494.60
149567
1496.73
1497.80
1498.87
149994
1501.01
1502.08
150315
1504.21
1505.28
1506.35
150742
1508.49
150956
1510.63
151 1. 70
1512.77
151384
I5I49I
I5I598
15170s
1518.12
I5I9.I9
1520.26
1521.33
1522.40
152347
152454
152561
1526.68
152776
1528.83
152990
1530.97
1532.04
1533. 11
1534 18
1535.25
30 #
535.25
536.33
537.40
538.47
53954
540.62
541.69
542.76
54383
544-91
54598
547.05
548.12
549-20
550.27
55134
552.42
55349
55456
55564
556.71
55779
558.86
55993
561.01
562.08
563.16
564.23
56531
566.38
56746
568.53
569.60
570.68
571.76
572.83
57391
57498
576.06
577.13
578.21
579-28
580.36
581.44
582.51
58359
584.67
585.74
586.82
58790
588.97
31*
1588.97
1590.05
I59I.I3
1592.20
1593.28
159436
159544
1596.51
159759
1598.67
159975
1600.82
1601.90
1602.98
1604.06
1605.14
1606.22
1607.30
1608.37
l6o9.4S
1610.53
161 1. 61
1612.69
l6i377
1614.85
i6l593
1617.01
1618.09
1619.17
1620.25
1621.33
1622.41
1623.49
1624.57
1625.65
1626.73
1627.81
1628.89
1629.97
1631.05
1632.14
1633.21
163430
163538
1636.46
163754
1638.62
(23)
163970
1640.79
1641.87
94
96
98
1642.95 11.00
.00
.02
04
.06
.08
.10
.12
14
.16
.18
.22
34
.26
.28
.30
.32
34
36
38
.40
46
48
5o
52
54
56
58
.60
.62
64
.66
.68
70
•72
74
.76
.78
.80
.82
84
.86
•90
.92
r
TABLE II. — (Continued)
.00
.03
3
.08
.xo
.12
•14
.16
.18
.22
2(
30
3a
34
•3«
38
•4*
44
46
.48
50
.52
•54
.56
58
.00
.62
64
.66
70
7a
74
76
.78
.82
.84
90
.92
•94
96
98
H.00
3a #
642.95
644.04
64512
646.20
647-28
648.36
64945
650.53
651.61
652.70
65378
654.86
65595
65703
658. 11
659.20
660.28
661.36
662.45
663.53
664.62
665.70
666.79
667.87
668.96
670.04
671.13
672.21
67330
67438
67547
676.55
677.64
678.72
679.81
680.90
681.98
683.07
684.15
685.24
686.33
687.41
688.50
68959
690.67
691.76
692.85
69394
69502
696.11
697.20
33'
69720
698.29
69937
700.46
701.55
702.64
70373
704.82
705.91
706.99
708.08
709.17
710.26
711.35
712.44
713 -53
714.62
71571
716.80
717.89
718.98
720.07
721.16
722.25
72334
72443
725.52
726.61
727.70
728.79
72988
730.97
732.07
733l6
73425
73534
736.43
73752
738.62
73971
740.80
741.89
742.99
744.08
745.17
746.26
74736
748.45
749- 54
75064
751.73
34 #
751.73
752.82
75392
755.01
756.10
75720
758.29
75939
760.48
761.58
762.67
76377
764.86
7659S
76705
768.14
76924
770.34
771.43
772.53
773.62
774.72
775.81
776.91
778.00
779 10
780.20
781.30
782.39
78349
78459
785.68
786.78
787.88
788.08
790.07
791-17
792.27
79337
79446
795.56
79666
79776
79886
79996
801.06
802.15
803.25
804.35
805.45
806.55
35*
806.55
807.65
808.75
809.85
810.95
812.05
813.15
814.25
81535
816.45
817.55
818.65
81975
820.85
821.96
823.06
824.16
825.26
826.36
827.46
828.56
829.67
830.77
831.87
832.97
83408
835.18
836.28
837.38
838.49
83959
840.69
84180
842.90
844.00
84511
846.21
847.31
848.42
84952
850.63
851.73
852.84
85394
85504
856.15
85725
858.36
85946
860.57
861.68
36 #
861.68
862.78
863.89
864.99
866.10
867.21
868.31
869.42
870.53
871.63
872.74
873.85
874.95
876.06
877.17
878.27
879.38
880.49
881.60
882.71
883.81
884.92
886.03
887.14
888.25
889.36
890.46
891.57
892.68
893.79
894.90
896.01
897.12
898.23
89934
900.45
901.56
902.67
90378
90489
906.00
907.11
908.22
909.33
910.44
911.56
912.67
91378
91489
916.00
917. II
(24)
37*
1917
1918
1919
1920
1921
1922
1923
1924
1926
1927
1928
1929
1930
I93I
1932
1933
1934
1936
1937
1938
1939
1940
1941
1942
1943
1944
1946
1947
1948
1949
1950
1951
1952
1953
1955
1956
1957
1958
1959
i960
1961
1962
1963
1965
1966
1967
1968
1969
1970
I97I
1972
11
23
34
45
56
68
79
90
01
13
24
35
47
58
69
81
92
04
15
26
38
49
61
72
84
95
07
18
30
42
53
65
76
88
00
12
23
35
47
58
70
82
93
05
17
28
40
52
64
1L
88
38 #
972.88
974-00
975- 11
976.23
97735
978.47
97959
980.71
081.83
982.95
98407
98519
986.31
987.43
988.55
98967
990.79
99191
993.03
99415
995-28
996.40
997.52
998.64
99976
2000.88
2002.01
2003.13
2004.25
2005.37
2006.49
2007.62
2008.74
2009.86
2010.99
2012. 11
2013.23
2014.36
2015.48
2016.60
2017.73
2018.85
2019.98
2021.10
2022.22
2023.35
2024.47
2025.60
2026.72
2027.85
2028.98
39*
2028.98
2030.10
2031.23
2032.35
2033.48
2034.60
203573
2036.86
203798
2039- 11
2040.24
2041.36
2042.49
2043.62
2044.74
204587
2047.00
2048.13
2049.26
2050.38
2051.51
2052.64
2053.77
2054.90
2056.03
2057.15
2058.38
2059.41
2060.54
2061.67
2062.80
2063.93
2065.06
2066.19
2067.32
2068.45
2069.58
2070.71
2071.84
2072.97
2074.10
2075.24
2076.37
2077.50
2078.63
2079.76
2080.89
2082.03
2083.16
2084.29
2085.42
.00
.02
•04
.06
.08
.xo
.12
14
l6
l8
.23
24
.26
.28
30
32
•34
36
38
.40
46
48
50
5a
•54
56
.58
.00
.62
64
.66
.68
.70
7a
74
76
78
.80
.82
84
.86
.88
90
.92
•94
96
98
.00
TABLE II. — (Continued)
.00
.02
04
.06
.08
.10
.13
14
.16
.18
.20
.33
•*4
.26
.28
.30
.32
•34
36
38
.40
46
48
50
-53
54
56
•58
60
.62
64
.66
.68
.70
73
74
76
78
.80
.82
84
.86
.88
90
93
94
96
98
H.00
40°
2085.42
2086.55
2087.69
2088.82
20899s
2091.09
2092.22
2093.35
2094.49
2095.62
2096.75
2097.89
2099.02
2100.16
2101.29
2102.43
2103.56
2104.70
2105.83
2106.97
2108.10
2109.24
21 10. 37
2111.51
2112.64
21 13. 78
2114.92
2116.05
2117.19
2118.32
2119.46
2120.60
2121.74
2122.87
2124.01
2125.15
2x26.29
2x27.42
2128.56
2129.70
2130.84
2131.98
2133. IX
213425
2135-39
2136.53
2137.67
2138.81
2139-95
214109
2142.23
41'
2142.23
214337
2I445I
214565
2146.79
214793
214907
2150.21
2151.35
2152.49
2153.63
215478
215592
2157.06
2158.20
215934
2160.48
2x61.63
2162.77
2163.91
2x65.05
2166.20
2167.34
2168.48
2x69.63
2170.77
2171.91
2173.06
2174.20
217534
2176.49
217763
2178.78
2179.92
2181.07
2182.21
2x8336
2x84.50
2185.65
2186.79
2187.94
2x89.09
2x90.23
2191.38
2192.52
219367
2194.82
219597
2197. XI
2198.26
2199.41
43*
2199. 41
22O0.55
22OI.70
2202.85
2204.OO
2205.I5
2206.29
2207.44
2208.59
220974
2210.89
22X2.04
2213.19
2214.33
2215.48
2216.63
2217.78
22X8.93
2220.08
2221.23
2222.38
2223.53
2224.68
2225.84
2226.99
2228.14
2229.29
2230.44
2231.59
2232.74
223390
223505
2236.20
223735
2238.51
2239.66
2240.81
2241.97
2243.12
2244.27
224543
2246.58
224773
2248.89
2250.04
2251.20
2252.35
225350
225466
2255- 81
2256.97
43'
2256.97
2258.12
2259.28
2260.44
2261.59
2262.75
2263.90
2265.06
2266.22
2267.37
2268.53
2269.69
2270.84
2272.00
2273.16
2274. 3X
2275.47
2276.63
227779
2278.95
2280.10
2281.26
2282.42
2283.58
2284.74
2285.90
2287.06
2288.22
2289.38
2290.54
2291.70
2292.86
2294.02
2295.18
2296.34
2297.50
2298.66
2299.82
2300.98
2302.14
2303.30
230447
230563
2306.79
230795
2309.12
2310.28
2311.44
2312.60
23 J 377
231493
23X493
2316.09
23x7-26
23x8.42
231958
2320.75
2321.91
2323.08
232424
232540
2326.57
232773
2328.90
2330.06
233123
2332.40
233356
233473
2335.89
233706
2338.23
2339-39
2340.56
2341.73
234289
234406
234523
2346.40
234756
2348.73
2349- 90
2351.07
2352.24
235341
235458
235574
2356.91
2358.08
2359.25
2360.42
2361.59
2362.76
236393
2365.10
2366.27
2367.44
2368.61
236978
2370.96
2372.13
237330
45°
2373 30
237447
2375.64
2376.81
237799
2379- 16
2380.33
2381.50
2382.68
238385
2385.02
2386.20
238737
2388.54
2389.72
2390.89
2392.07
2393.24
239441
239559
2396.76
2397.94
2399.11
2400.29
2401.47
2402.64
2403.82
2404.99
2406. 17
240735
2408.52
2409.70
2410.88
2412.05
241323
2414.41
241559
24x6.76
2417.94
2419.12
2420.30
2421.48
2422.66
2423.84
2425.01
2426.19
242737
2428.55
242973
2430-9'
2432.09
46*
47*
243327
243445
243563
2436.71
2438.00
2439.18
2440.36
244154
2442.72
2432.09 I 2491-32
2492.51
249370
2494-89
244390
244508
2446.27
244745
2448.63
2449.82
2451.00
2452.18
245336
245455
245573
2456.91
2458.10
2459-28
2460.47
2461.65
2462.84
2464.02
2465. 21
2466.39
246758
2468.76
246995
2471.13
2472.32
2473- 5X
2474.69
2475.88
2477.07
2478.25
2479-44
2480.63
2481.82
2483.00
2484.19
248538
2486.57
2487.76
2488.95
2490- 13
2491.32
2496.08
249727
2498.46
249965
2500.84
2502.03
2503.22
250441
2505.60
2506.80
250799
2509.18
2510.37
25x1.56
2512.75
25X395
2515.14
2516.33
251753
25x8.72
25I9-9I
2521. 11
2522.30
2523 49
252469
252588
3527.08
2528.27
2529.47
2530.66
253186
2533.05
253425
253544
2536.64
253783
2539 03
2540.23
254143
2542.62
254382
254502
2546.21
254741
2548.61
254981
2551.00
Ha
.03
04
.06
.08
.10
.12
.14
l6
l8
.80
.33
34
.36
.38
.30
33
34
36
38
.40
46
48
.50
53
54
56
.58
.80
.63
64
.66
.68
70
73
74
76
78
.80
.83
-84
.86
.88
9o
93
94
96
98
.00
(25)
r
TABLE II. — (Continued)
48 s
00
.02
04
.06
.08
.10
.12
14
.16
.18
.10
.33
•*4
.36
.38
.30
.33
■34
36
38
.40
43
44
46
48
.50
•53
54
56
58
.60
.63
64
.66
.68
.70
73
74
76
78
.80
.83
84
.86
88
90
93
94
96
98
.00
2551.00
2553.20
3553.40
355460
3555.80
3557.00
2558.20
3559- 40
2560.60
2561.80
2563.00
2564.20
356540
2566.60
2567.80
2569.00
2570.20
3571.40
2572.60
257380
2575.01
2576.21
257741
2578.61
3579.82
2581.02
2582.22
258343
358463
258583
258704
2588.24
2589.44
2590.65
25918s
2593.o6
2594.36
359547
359667
3597.88
359908
2600.29
2601.50
2602.70
2603.91
2605.12
2606.32
2607.53
2608.74
260994
2611.15
49'
261 1. 15
2612.36
2613.57
2614.78
2615.98
2617.19
2618.40
2619.61
2620.82
2622.03
2623.24
262445
262566
2626.87
2628.08
2629.29
2630.50
2631.71
2632.92
263413
263534
2636.56
2637.77
2638.99
2640.19
2641.40
2642.62
2643.83
264504
2646.26
2647.47
2648.68
2649.90
2651. 11
2652.33
265354
2654.76
265597
2657.18
2658.40
2659.62
2660.83
2662.05
2663.26
2664.48
2665.69
2666.91
2668.13
2669.34
2670.56
2671.78
50°
2671.78
2672.99
2674.21
2675.43
2676.65
2677.86
2679.09
2680.31
2681.53
2682.74
268396
2685.18
2686.40
2687.62
2688.84
2690.06
2691.28
2692.50
269373
26949s
2696.17
3697.39.
2698.61
2699.83
2701.06
2702.28
2703.50
2704.72
270595
2707.17
2708.39
2709.62
2710.84
2712.06
2713.29
2714.51
271574
2716.96
2718.19
271941
2720.64
2721.86
2733.09
2724.32
373554
2726.77
2727.99
2729.22
3730.45
273168
3732.90
51'
3732.90
3734.13
373536
3736.59
3737.81
3739- 04
2740.27
374150
3743.73
374396
374519
2746.43
3747.65
274888
2750.11
3751.34
3753.57
3753.80
3755.03
2756.26
3757.49
2758.73
3759.96
2761 . 19
2762.42
3763.65
2764.89
2766.12
27673s
2768.59
2769.82
2771.05
2772.29
277353
377476
377599
2777.33
2778.46
277970
2780.93
2782.17
2783.41
2784.64
2785.88
2787.11
2788.35
278959
2790.82
2792.06
279330
3794.54
53'
3794.54
3795.77
3797.01
2798.25
2799.49
2800.73
2801.97
2803.21
3804.45
2805.69
2806.93
2808.17
2809.41
2810.65
2811.89
2813.13
2814.37
2815.61
2816.85
2818.10
2819.34
2820.58
2821.82
2823.06
2824.31
2825.55
2826.79
2828.04
2829.29
2830.53
3831.77
2833.01
2834.36
3835.50
3836.75
3837.99
3839.34
2840.48
2841.73
2842.98
2844.22
2845.47
2846.72
2847.96
2849.21
2850.46
2851.71
2852.95
2854.20
2855.45
2856.70
53 c
2856.70
2857.95
2859.20
2860.44
2861.69
2862.94
2864.19
3865.44
2866.69
286794
2869.20
2870.45
2871.70
2872.95
287420
2875.45
2876.70
287796
2879.21
2880.46
2881.71
2882.97
2884.22
3885.47
2886.73
2887.98
2889.24
2890.49
2891.74
2893.00
2894.26
289551
2896.77
2898.02
2899.28
2900.53
290179
3903.05
3904.30
3905.56
2906.82
2908.07
290933
2910.59
2911.85
2913. 11
391437
391563
2916.88
2918.14
2919.40
54°
2919.40
2920.66
2921.92
2923.18
292444
2925.70
2926.96
2928.23
292949
2930.75
2932.01
293327
2934.53
293580
293706
3938.33
393959
2940.85
2942.11
294338
294464
294590
2947.17
2948.43
294970
2950.96
2952.23
295349
295476
2956.03
295729
2958.56
295983
2961.09
2962.36
296363
296490
2966.16
2967.43
2968.70
2969.97
297124
2972.51
297378
397505
3976.31
397758
2978.86
2980.13
2981.48
2982.67
55 f
2982.67
398394
2985.21
2986.48
398775
2989.02
3990.30
2991.57
2992.84
299412
2 99539
2996.66
2997.94
3999.21
3000.48
3001.76
3003.03
3004.31
3005.58
3006.86
3008.13
3009.41
3010.68
3011.96
301324
3014.51
3015.79
301707
3018.35
3019.62
3020.90
3022.18
3023.46
3024.74
3026.02
3027.29
3028.57
3029.85
3031 . 13
3032/41
3033-69
303497
3036.25
3037.54
3038.82
3040.10
304138
3042.66
3043.94
304523
3046.51
.00
.02
04
.06
.08
.10
.13
■14
l6
l8
.33
34
.36
.38
.30
33
■34
.36
.38
.40
•43
44
46
48
50
53
•54
56
58
.60
.63
64
.66
.68
.70
73
74
.76
78
.80
.83
84
.86
.88
9o
.93
94
96
98
.00
(26)
TABLE II. — (Continued)
.00
.03
04
.06
.08
.10
.12
.14
l6
.18
.10
.22
•34
.26
.28
.30
.32
•34
36
38
.40
•43
44
46
48
50
53
54
56
58
.60
.62
64
.66
.68
To
72
74
76
78
.80
.82
■84
.86
.88
90
93
94
.96
98
H.00
56 #
3046.SI
3047-79
304907
3050.36
3051.64
3052.92
3054.31
305549
3056.78
3058.06
305935
3060.63
3061.92
3063.20
3064.49
3065.78
3067.06
3068.35
3069.64
3070.92
3072.21
307350
307479
3076.08
307736
3078.65
307994
3081.23
3082.52
3083.81
3085.10
3086.39
3087.68
3088.97
3090.26
309155
3092.84
3094.13
3095.43
3096.72
3098.01
309930
3100.60
3x01.89
3103.18
3104.48
310577
310706
3108.36
3109.65
3H0.95
57*
3H0.95
3112.24
3II354
311483
3x16.13
311742
3118.73
3120.02
3x21.31
3122.61
3I339I
3125.20
3126.50
3x27.80
3129.10
3130.40
3131.69
3133.99
313439
313559
3136.89
3138.19
3X39-49
3140.79
3143.09
314339
314469
3146.00
314730
3148.60
314990
315120
3X53.5X
3153.81
3155. xi
3156.43
315773
315902
3160.33
3161.64
3162.94
3164.24
316555
3166.85
3168.16
3169.46
3170.77
3x73.08
317338
3x7469
3176.00
58*
3176.00
317730
3178.61
3X7993
3181.23
3182.54
3x8385
318515
3186.46
3187.77
3189.08
3190.39
319170
319301
319433
319563
3196.95
3198.26
319957
3200.88
3202.19
3303.51
3204.82
3206.13
3307.44
3208.76
3210.07
32H.39
3212.70
321401
3215.33
3216.64
321796
3219.27
3220.59
3221.91
3323.22
3324.54
3225.86
3227.17
3228.49
3229.81
323X.13
3232.45
323376
333508
3336.40
333773
333904
3340 36
3241.68
59°
3241-68
3343.00
334433
334564
3346.96
3348.28
3349- 61
3350.93
3352.25
335357
3354.89
3256.22
335754
3358.86
3260-19
3261.51
3262.83
3264.16
336548
3366.81
3368.14
3269.46
3370.79
3373.11
337344
3374.76
3376.09
337743
3378.75
3280.07
3281.40
3282.73
3284.06
3285.39
3386.72
3388.04
3389.37
3390.70
3393.03
3393.36
329469
3396.03
339736
3398.69
3300.02
330X.3S
3302.68
330402
3305.35
3306.68
3308. ox
6o'
3308.01
330935
3310.38
3312.02
331335
33x469
33x6.02
331736
331869
3320.03
3321.36
3333.70
333403
333537
3336.71
3338.05
333938
3330.73
3332.06
333340
3334.74
3336.08
333741
3338.75
3340.09
3341.43
3342.77
334411
334546
3346.80
3348.14
334948
335082
3353.16
335351
3354-85
3356.19
335753
3358.88
3360.22
3361.57
3362.91
336436
336560
3366.94
3368.29
336964
3370.98
3373.33
337367
3375.02
61
3375
3376
3377
3379
3380
338i
3383
3384
3385
3387
3388
3389
3391
3393
3393
3395
3396
3397
3399
34O0
3403
3403
3404
3406
3407
3408
34io
341 1
3413
3414
3415
34i6
3418
3419
3430
3432
3423
3435
3436
3427
3439
3430
3431
3433
3434
3435
3437
3438
3440
3441
1442
02
37
72
06
41
76
11
46
81
16
51
85
21
56
91
26
61
97
3i
66
02
37
73
08
43
78
14
49
85
20
56
91
37
62
98
34
69
05
41
77
12
48
84
20
57
92
28
64
00
36
72
(27)
62°
3443.73
3444.08
3445-44
3446.81
3448.17
344953
3450.89
3453.26
345362
345498
3456.35
3457 71
345907
3460.44
3461.80
3463.17
346453
3465.90
346727
3468.63
3470.00
347137
3472-73
3474.10
3475-47
3476.84
3478.20
347957
3480.94
3482.31
348368
3485.05
3486.42
3487.79
348916
3490.53
349190
3493.37
3494.65
3496.02
3497.39
3498.77
3500.14
3501.51
3502.89
3504.26
3505.63
350701
3508.38
3509-76
35II.I4
63'
35II.I4
3513.51
3513.88
34X536
3516.64
3518.02
351939
3530.77
3522.15
352353
3534-90
3536.28
3537.66
353904
3530.43
353180
3533.18
353456
353594
353733
3538.70
3540.09
3541.47
3543.85
354433
3545-62
3547.00
3548.38
354976
3551.15
3552.53
355392
355530
3556.69
3558.07
355946
3560.84
3562.23
3563.62
3565-Qi
3566.39
3S6778
356917
3570.56
357X.95
3573.33
357472
3576.11
3577.50
3578.89
3580.28
00
02
04
06
08
10
12
14
16
18
22
34
26
28
30
33
34
36
38
40
46
48
5o
53
54
56
58
60
62
64
66
68
70
73
74
76
78
80
82
84
86
88
90
93
94
96
98
00
TABLE II. — (Continued)
64"
.00
.02
.04
.06
.08
.10
.13
14
.16
18
.20
.22
.34
.26
.28
.30
.33
■34
36
38
.40
.4*
44
46
48
SO
53
■54
56
58
.60
.62
64
.66
.68
70
7a
74
76
78
.80
.82
84
.86
.88
90
9a
94
96
98
1.00
3580.28
3S81.67
358306
3584.46
358585
3587.24
3588.63
3590.02
3591.43
359*.8i
359420
359560
3596.99
359839
359978
3601.17
3602.57
3603.96
3605.36
3606.76
3608.15
3609.55
3610.95
3612.34
361374
3615.14
3616.54
361794
361934
3620.74
3622.14
3623.54
3624.93
3626.34
3627.74
3629.14
3630.54
363194
363334
363474
3636. IS
3637.55
3638.95
3640.36
3641.76
3643.16
364457
364597
364738
3648.78
3650.19
65*
3650.19
3651.60
365300
3654.41
3655.82
3657.22
3658.63
3660.04
3661.45
366286
3664.26
3665.67
3667.08
3668.49
366988
367131
3672.7a
367413
367555
3676.96
3678.37
367978
3681.20
3682.61
3684.02
3685.43
3686.85
3688.26
3689.68
369109
3692.51
369393
369534
3696.76
3698.17
3699 59
370X.01
3703.43
3703.84
3705.36
3706.68
3708.10
3709S3
37W.93
3713.35
37X377
3715.19
3716.61
3718.04
371946
3720.88
66°
3730.88
3732.30
373373
373515
3736.57
373799
373941
3730.84
373336
3733.69
373511
3736.54
373796
3739 39
3740.81
3743.34
3743.67
374509
3746.53
374795
374938
3750.81
3753.34
3753.66
3755.09
3756.53
3757- 95
3759-38
3760.81
3763.24
376368
3765.11
3766.54
3767.97
3769,.4I
3770.84
3773.37
3773.71
3775.14
3776.57
3778.01
377944
3780.88
3783.31
378375
378519
3786.62
3788.06
378950
3790.93
3793.37
67*
3793.37
379381
379535
379669
379813
3799 57
3801.01
3802.45
3803.89
3805.33
3806.77
3808.21
3809.65
3811.10
3812.54
381398
381543
3816.87
3818.31
381976
3821.20
3822.65
3824.09
3825.54
3836.98
3828.43
3839.88
383132
3833.77
3834.32
383567
383713
3838.56
3840.01
384146
3843.91
384436
384581
3847.36
3848.71
3850.17
3851.62
3853.07
385453
385597
385743
3858.88
3860.33
3861.79
3863.34
68*
3864.70
3866.15
3867.61
3869.06
3870.53
387198
387343
3874.89
3876.35
387781
387937
3880.73
3882.18
3883.67
3885.10
3886.56
3888.02
3889.48
3890.94
3893.40
3893.87
389533
389679
3898.35
389973
3901.18
3903.64
390411
390557
390704
3908.50
390997
3911.43
3912.90
3914.37
391583
391730
3918.77
393034
3921.70
392317
3924.64
3926. 11
392758
3929.05
3930.68
3931.99
393346
393494
3936.4X
3864.70 I 3937-88
(28)
69*
393788
3939-35
3940.82
3942.30
394377
3945.25
3946.72
3948.19
3949.67
3951.15
3952.62
3954.10
3955. 57
395705
3958.53
3960.00
3961.48
3962.96
396444
396592
396740
3968.88
3970.36
3971.84
3973.33
397480
3976.38
397776
397934
3980.73
3983.21
3983.69
3985.18
3986.66
3988.15
398963
3991.13
3993.60
3994.09
399557
399706
3998.55
4000.03
4001.52
4003.01
4004.50
4005.99
4007.48
4008.96
4010.45
4011.94
7o #
4011.94
401343
4014.93
4016.42
4017.91
4019.40
4020.89
4022.39
4023.88
4025.37
4026.86
4028.36
4029.85
4031.35
4032.84
4034.34
403584
403733
4038.83
4040.33
4041.82
4043,33
4044.82
4046.33
404783
404933
4050.82
4053.33
405382
405533
4056.82
4058.33
405983
4061.32
4062.83
4064.33
4065.83
4067.34
4068.84
4070.34
4071.85
407336
4074.86
4076.37
407787
407938
4080.89
4082.39
4083.90
4085.41
4086.92
71*
4086.9a
4088.43
4089.94
409I.45
4093.96
4094.47
409598
409749
409900
4100.51
4103.03
410354
4105.05
4106.57
4108.08
4109.59
41n.1i
4112.62
4II4-I4
411565
4II7.I7
4118.69
4120.20
4x21.73
413334
413476
4X36.28
4127.80
4I293I
4X30.83
413235
4133.87
4135.40
4136.92
4138.44
413996
4141.48
414300
414453
4146.05
4148.57
4149.10
4150.62
4152. xs
415367
4I5530
4156.73
41S8.35
4159.78
4161.31
4163.83
.00
.02
•04
.06
.08
.10
.12
14
.16
.18
.10
.22
34
.26
.28
.30
.33
.34
.36
.38
.40
43
44
46
48
SO
.53
.54
56
58
.60
.62
64
.66
.68
70
.73
74
76
78
.80
.82
84
.86
.88
90
•93
94
96
98
.00
TABLE II. — (Continued)
I
73*
73*
74*
75°
76°
77 #
78°
I
.00
.03
04
.06
4162.83
423972
43x760
439652
4476.50
455756
463978
.00
.03
.04
.06
4x64.36
416S89
416743
424127
4242.81
424436
4319.17
4320.74
4322.31
4398.11
439969
4401.28
4478.10
447972
4481.33
4559.20
4560.84
4562.47
4641.44
4643.09
4644.7S
.08
.10
.13
4168.9s
4170.48
4x73.01
4245 91
424746
4249.01
4323.88
432545
4327.02
4402.87
4404.46
4406.06
4482.94
448455
4486.16
4564.10
4565.74
4S67.37
4646.41
4648.06
464972
.08
.10
.13
• 14
.16
.18
.SO
.33
•*4
.36
417354
417507
4x76.60
4250.56
4252. 11
4253.66
4328.59
4330.I6
433173
4407.65
440924
4410.83
4487.78
448939
449101
456901
4570.65
4572.28
4651.38
465304
465470
.14
l6
.18
.20
.33
•*4
.36
4x78.13
425521
433330
4412.42
4492.62
457392
4656.36
4X79-66
4x81.20
4182.73
4256.77
425832
425987
4334.87
4336.45
4338.02
44x402
441563
441721
449423
4495.85
4497-47
4575.56
4577.19
4578.83
4658.02
465968
4661.34
.38
.30
.33
418426
4x85.80
4x87.33
4261.42
4262.98
4264.S3
433959
4340.97
4342.74
4418.80
4420.39
4421.99
449909
4500.70
4502.32
4580.47
4582.11
4583.75
4663.01
4664.67
4666.33
.38
.30
32
34
36
38
.40
.43
.44
46
4x88.87
4x90.40
4191.93
4266.09
4267.64
4269.19
434432
434589
4347.47
442359
442518
4426.78
4503.94
4505.55
4507.17
458539
458703
4S88.67
4668.00
4669.66
467132
.34
.36
.38
.40
42
44
46
4X9347
4270.85
434904
4428.38
4508.79
4590.31
4672.99
4195.01
419654
4198.08
4272.31
' 427386
4275.42
4350.62
4352.19
435377
442997
443157
443317
4510.41
4S12.03
4513.65
459196
4593.60
459524
467465
4676.32
467799
48
.50
53
419962
4201.15
4202.69
4276.98
4278.53
4280.09
435535
4356.93
4358.51
443477
4436.37
443797
4515.27
4516.89
4518. 51
4596.89
4598.53
4600.18
4679.65
4681.32
4682.99
48
50
53
54
.56
58
.00
.63
.64
.66
4204.23
4205.77
4207.31
4281.65
4283.21
4284.77
4360.08
4361.66
4363.34
443957
4441.17
4442.77
4520.14
4521.76
452338
4601.82
4603.47
4605.19
4684.66
4686.32
4687.99
54
,.56
58
.80
.63
64
.66
4308.85
4286.33
436481
444437
452501
4606.76
4689.67
4210.39
43H.93
4213.47
4287.89
4289.45
4291.01
4366.40
436798
436956
444597
444758
4449 18
4526.63
4528.25-
4529.88
4608.40
4610.05
4611.70
4691.33
469300
469468
.68
70
73
4315.01
4216.55
4218.09
4292.57
429413
429569
4371.15
4373.73
4374.31
4450.78
4452.39
445399
4531.50
4533-13
453476
461335
461499
4616.64
4696.35
4698.02
469969
.68
70
73
74
76
78
.80
.83
84
.86
42x963
4221.18
4222.72
429725
4298.82
4300.38
437590
437748
437906
445559
445720
4458.81
4536.38
4538.01
453964
46x8.29
461994
4621.59
4701.37
470304
4704.71
74
76
.78
.80
.83
84
.86
4224.26
4301.94
4380.65
4460.41
4541.27
4623.24
4706.39
4225.81
4227.35
4228.90
430351
430507
4306.64
4382.23
438382
438540
4462.02
4463.62
446523
4542.89
454452
4S46.IS
4624.90
4626.55
4628.20
4708.06
470974
47II.4I
.88
90
.93
4230.44
4231.98
423353
4308.20
430977
4311.33
4386.99
4388.58
4390.16
4466.84
4468.45
4470.0S
4547.78
4549-41
4551.04
4629.85
463151
463316
4713.09
471477
47I6.4S
.88
90
.93
94
96
■98
1.00
423508
4236.62
4238.17
4312.90
431446
431603
439175
439334
439493
4471.66
447327
447488
455267
455431
455594
4634.81
4636.47
4638.12
4718.12
4719-80
4721.48
94
96
98
1.00
4239.72
431760
4396.52
447650
45S7.56
463978
4723. x6
(29)
TABLE II. — (Continued)
l
79°
80°
8i°
8a»
83*
84°
85"
I
.00
.02
.04
.06
472316
4807.85
489358
4980.71
5069.17
5I59-0O
5250.26
.00
.02
04
.06
4724.84
4726.53
4728.20
4809.45
4811.16
4812.86
489531
489704
4898.78
4982.47
498422
498598
5070.95
5072.73
5074.52
5160.80
5162.62
5164.43
5252.10
525394
525578
.08
.10
. ia
472988
4731.56
4733.25
4814.57
4816.27
4817.98
4900.51
4902.24
490397
498774
498949
4991.25
5076.30
5078.09
5079-87
5166.25
5168-06
516988
525762
525946
5261.30
.08
.10
.12
■14
.16
.18
.10
.33
24
.36
473493
4736.6i
4738.29
4819.69
4821.40
4823.10
490570
4907.44
490917
4993-01
499477
4996.53
5081.66
5083.45
5085.23
5171.69
5173.50
517532
5263.15
5264.99
5266.84
14
.16
.18
.80
.22
24
.36
473998
4824.81
4910.90
4998.29
5087.02
5177.14
5268.68
4741.66
474335
4745.03
4826.52
4828.23
482994
4912.64
491437
4916. IX
. 5000.05
5001.82
5003.58
5088.81
5090.60
5092.39
5178.95
5180.77
5182.59
5270.53
5272.38
5274.22
.38
.30
.33
4746.72
4748.41
4750.09
483165
483336
4835.08
491783
491958
4921.32
5005.34
5007.10
5008.87
5094.18
509597
509776
5184.41
5186.23
5188.05
5276.07
527792
527977
.38
.30
32
34
36
38
.40
•43
■44
46
475178
475347
4755.16
4836.79
4838.50
4840.22
492306
4924.80
4926.54
5010.63
5012.40
5014.16
509955
5101.34
5103.14
518987
519169
5I935I
5281.62
5283.47
5285.32
34
36
38
.40
•42
•44
.46
4756.85
4841.93
4928.28
501593
5104.93
519533
5287.17
4758.54
4760.23
4761.92
484364
484536
4847.08
4930.02
4931.76
493350
5017.70
5019.46
5021.23
5106.73
5108.52
5110.32
5197.15
5198.98
5200.80
5289.02
5290.87
5292.73
■ 48
.50
• 52
476361
4765.30
4766.99
4848.79
4850.51
4852.22
493524
4936.98
4938.73
5023.00
5024.77
5026.54
5x12. 11
5II39I
5115.70
5202.62
5204.45
5206.27
5294.58
5296.42
5298.29
.48
50
52
•54
56
■58
.60
.63
64
.66
4768.69
4770.38
4772.07
485394
4855.66
485738
4940.47
4942.21
494396
5028.31
5030.08
5031.85
5"7.5o
5II930
5121.10
5208.10
5209.93
5211.75
5300.13
5302.00
5303.86
54
56
58
.60
.63
64
.66
477376
4859.10
494570
503362
5122.90
5213.58
5305.71
477546
477715
4778.85
4860.82
4862.54
4864.26
4947-45
4949.19
4950.94
503539
5037.17
5038.94
5124.70
5126.50
5128.30
52I54I
521724
5219.07
5307.57
530943
531129
.68
70
•7*
4780.55
4782.24
478394
4865.98
4867.70
4869.42
4952.68
495443
4956.18
5040.71
5042.49
5044.26
513010
513190
513370
5220.90
5222.73
522456
5313.15
5315.01
5316.87
.68
70
72
74
.76
.78
.80
.83
84
.86
478564
4787.33
478903
4871.14
4872.87
4874.59
495793
4959-68
4961.63
5046.04
5047.81
5049-59
5I355I
5I373I
5139- 12
5226.39
5228.22
5230.06
5318.73
5320.59
5322.46
74
.76
.78
.80
.83
84
.86
4790.73
4876.31
496318
5051.37
5140.92
523189
532432
4792.43
479413
4795.83
4878.04
487976
4881.49
496493
4966.68
4968.43
50S3.I4
505492
5056.70
5142.73
514453
5146.34
523372
523556
523739
5326.18
5328.05
532991
.88
•90
9a
479753
4799-23
4800.93
4883.22
4884.94
4886.67
4970.18
497194
497369
5058.48
5060.26
5062.04
5148.15
5149-95
5151.76
523923
5241.07
5242.90
5331.78
5333.65
533551
.88
.90
.92
94
96
98
1.00
4802.64
4804.34
4806.04
4888.40
4890.13
48918s
4975-44
4977.20
4978.95
5063.82
5065.60
5067.38
515357
515538
5157- 19
524474
5246.58
5248.42
533738
533925
534112
94
96
98
1.00
4807.85
489358
4980.71
5069.17
5159-00
5250.26
5342.99
(30)
TABLE II. — (Continued)
I
86'
87*
88'
89"
90°
91
.00
■04
06
5341.99
5437.14
5533.06
5630.51
57196s
5830
5344-85
5346.73
5348.60
5439.14
553499
5538.86
5631-48
5636-41
5731.65
5733.65
5735.6S
5831
5834
5836
.08
5350.46
5351.34
5354-"
5444.84
5445-65
5540.79
5544-67
5638.38
5640.35
5643.33
S737.6S
5739*6
5741.66
5838
5840
584=
.16
.iS
M
.36
5356.09
5357.96
5359.84
5453 46
5548.54
5644.3V
5646.16
5648.13
574367
5745.67
5747 68
5S44
S846
5848
5361.71
5456.17
5SS1.41
5650.30
5749.69
S8S0
5363.59
5365.46
5367.34
5458.18
5460.09
5554-36
5558^4
5653.18
5654. IS
5656.I.
5751.69
5753-TO
58S3
5855
5857
.38
30
5369.11
5371-10
5371.9s
546581
5467.7a
5560.18
5561.U
S564.a6
5660] 07
575973
5761.74
5859
S861
5863
*6
-38
.10
44
*6
5376.74
5469.64
547346
S566.0I
556990
5664.03
5666.00
5667.98
S763.75
5767.78
586s
5867
S869
5380.50
u -. .■
5571.84
5669.96
5769.79
5871
5381.3a
5384.16
5474-39
5481.11
557379
5577^68
5673.91
5771-80
5773-83
577S.84
5873
5875
5S77
■4S
W
s>
5388.03
5389.91
5391.80
548303
5484.95
5486.86
5581.58
5583.53
5677.88
5679-87
5681.85
5777-85
5779-87
5781.89
5879
58Si
.588,3
■8*
58
.63
64
66
539369
539746
5488.78
5490-70
55S5.48
5587-43
5683.83
5685.83
568780
3783.91
5785.93
5787.95
S885
5887
S889
539535
549454
5591.33
5089-79
5789.97
5891
Es
5495.38
5593-19
5597-19
5691.78
5693.76
5695-73
5791-99
5796.03
5893
5896
5S98
68
.70
5*6.91
5408. Bo
5410.69
SI
559915
5601.10
5697-74
5699.73
5798.06
5800.08
5904
.19
7*
SO
.81
8,
.86
541158
5507.99
5509.91
5511.84
5605.01
5606.97
5608. 93
5707^69
5804.13
5806.16
580819
SSof
59'°
M78.J6
5513.77
5610.89
S709 68
S8I0.1I
59"
5430.16
5515-69
5517.61
56ia.85
5616^77
5711.68
5713.67
5813.14
5814.37
5816.30
5916
9"
.91
5415.84
5419-64
553148
5513-40
5618.73
5717-66
5719-66
5711.65
5818.33
5830.37
5813.40
59=4
■94
96
.98
1.00
543344
5517.16
5539- 19
5616! 58
5618. 55
5723-65
5735.65
5717-65
5814.43
5836.47
5818.50
SOI?
5919
5931
5437.14
5533.06
S630.S.
5739.65
5830.53
5933
TABLE II. — (Continued)
I
93°
94°
95°
96°
97°
98°
99°
/
.00
.03
.04
.06
6037.79
6144.30
6252.82
6363.42
6476.19
659121
6708.56
.00
.02
04
.06
6039.90
6042.01
6044.12
6146.45
6148.60
6150.7S
6255.01
6257.20
625939
6365.66
6367.89
6370.13
6478.47
6480.75
648303
659353
6595.86
659818
6710.93
671330
6715.67
.08
.10
6046.25
6048.3s
6050.47
6152.90
6155.06
6157.21
6261.59
6263.78
6265.98
6372.36
637460
6376.84
648531
648759
6489.87
6600.51
6602.84
6605.16
6718.05
6720.42
6722.80
.08
.10
.12
•14
.16
.18
.SO
.22
•*4
.36
6052.58
605470
6056.82
615937
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
672993
.14
.16
.18
.10
.22
24
.26
6058.96
6165.84
627477
638580
649901
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
650359
6505.88
6616.82
6619.17
6621.49
6734.70
673708
673946
.28
• 30
.32
6067.42
6069.55
6071.66
6174-48
6176.64
6178.80
6283.57
6285.78
6287.98
639477
6397.02
639927
6508.17
6510.46
6512.7s
6623.83
6626.16
6628.50
674185
674423
6746.62
.28
.30
.32
34
3«
38
.40
•4*
44
46
607378
607591
6078.03
6180.96
6183.13
6185.29
6290.18
6292.39
6294.60
6401.52
6403.76
6406.01
651504
651733
651963
6630.84
6633.18
663552
6749.00
675139
6753.78
34
36
38
.40
•4*
•44
46
6080.16
6187.46
6296.80
6408.26
6521.92
663786
6756.17
6082.28
6064.41
6086.54
6189.63
6191.79
619396
6299.01
6301.22
6303.43
6410.52
6412.77
641502
6524.22
6526.62
6528.81
6640.21
6642.5s
6644.90
6758.56
6760.96
6763.35
48
SO
53
6088.67
6090.80
6092.93
6196.13
6198.30
6200.47
630564
630785
6310.07
64x7.28
6419 53
6421.79
653x.11
6533.41
6535.71
664724
6649 59
6651.93
676574
6768.14
6770.53
48
.50
53
•54
56
58
.60
.62
64
.66
6095.06
609719
6099.32
6202.64
6204.81
6206.99
6312.28
631449
6316.71
6424.04
6426.30
6428.56
6538.02
6540.32
6542.62
665428
6656.63
6658.98
6772.93
677533
677773
54
56
.58
.60
.62
64
.66
6101.46
6209.16
6318.92
6430.82
654493
6661.33
6780.12
6103.59
6105.73
6107.86
6211.34
6213.51
6215.69
6321.14
6323.36
6325.58
643308
643534
643760
654723
6549.54
6551.85
6663.69
6666.04
6668.39
6782.53
678493
6787.33
.68
70
.73
6110.00
61 12. 14
6114.28
6217.87
6220.04
6222.22
6327.80
6330.02
6332.24
6439.86
6442.13
644439
6554.15
6556.46
6558.77
6670.75
6673.10
667546
678973
6792.14
6794.55
.68
.70
•73
74
76
78
.80
.82
•84
.86
61x6.42
6118.55
6120.70
6224.40
6226.58
6228.77
633446
6336.68
633890
6446.66
6448.93
6451.19
6561.08
656339
6565.71
6677.82
6680.18
6682.54
679695
679936
6801.77
74
.76
.78
.80
.82
84
.86
6122.84
6230.95
6341.13
6453.46
6568.02
6684.90
6804.18
6124.98
6127.12
6129.27
623313
623531
6237.50
634335
6345.58
6347.81
6455.73
6458.00
6460.27
6570.34
6572.65
657497
6687.26
6689.62
6691.99
6806.59
6809.00
6811.41
.88
.90
9»
6131.43
6133.56
6135.70
6239.69
6241.87
6244.06
6350.04
6352.26
6354.49
6462.54
6464.81
6467.09
6577.29
6579.60
6581.92
669435
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
646936
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
Td = Tio/D + tabular correction
I \
a°
4°
6°
8°
xo°
xa°
14°
16
x8°
30°
D y
o
1
2
0.00
0.00
0.00
0.01
0.01
0.01
0.01
0.01
0.01
O.OI
1
a
.00
.00
.01
.01
.01
.02
.02
.02
.02
.03
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
•OS
.06
.07
.08
09
.10
8
IO
O.OI
0.02
O.04
o.os
0.06
0.08
0.09
0.10
O.II
0.13
XO
xa
.01
.03
.04
.06
.08
.09
.11
.12
.14
• IS
xa
X4
.01
.03
•os
.07
.09
.11
.13
14
.16
.18
14
16
.02
.04
.06
.08
.10
.12
.14
.16
.18
.20
16
18
SO
22
.02
.04
.07
.09
.11
• 14
.16
.18
.21
.33
18
SO
aa
0.02
o.os
0.07
0.10
0.13
0.15
0.18
0.2I
0.23
0.26
.02
.os
.08
.11
.14
17
.20
33
35
.39
24
.02
.06
09
.12
• IS
.18
.22
.35
.28
.31
34
a6
.03
.06
.10
.13
.17
.20
.34
.37
.30
.34
36
a8
.03
.07
.11
.14
.18
.22
.35
.39
• 33
.36
a8
30
0.03
0.07
O.II
0.15
0.19
0.23
0.27
0.31
0.35
0.39
30
33
.03
.08
.12
.16
.21
.35
.39
.33
.38
.43
33
34
.03
.08
.13
.18
.22
.37
.31
.36
.40
.45
34
36
.04
09
.14
19
.23
.28
.33
.38
.43
.47
36
38
40
43
.04
.09
• IS
.30
• 25
• 30
.35
.40
.45
•So
38
40
43
O.O4
o.xo
0.15
0.2X
0.26
0.32
0.37
0.43
0.47
0.53
.04
.10
.16
.22
.28
.33
.39
.45
•So
.56
44
.04
.11
.17
.33
.39
.35
.41
.47
•S3
.59
44
46
.05
.12
.18
.34
.31
•37
.43
.49
.56
.62
46
48
.05
.12
.19
.26
.33
.39
.45
.53
•59
.65
48
50
O.05
0.13
0.20
0.27
0.34
0.40
0.47
0.54
0.61
0.68
50
5a
.05
.13
.21
.38
.35
.43
•SO
57
.64
•71
53
54
.06
.14
.22
39
.37
.44
•S3
•59
.67
•74
54
56
.06
• IS
.33
.30
.38
.46
•54
.62
.70
.77
56
58
60
6a
.06
.IS
.34
33
.40
.48
.56
.64
•73
.80
58
60
6a
0.06
0.16
0.24
0.33
0.42
0.50
0.58
0.67
0.75
0.84
.07
.16
.35
• 34
.43
•53
.61
•70
•79
.87
64
.07
.17
.37
.36
.45
.54
.64
73
.82
91
64
66
.07
.18
.38
• 37
.47
.56
.66
•75
.85
•95
66
68
.07
.18
.39
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
1.02
70
73
.08
.20
.31
• 43
• S3
.63
•74
.84
.95
1.06
73
74
.08
.21
.33
• 43
.54
.65
•77
.87
99
1. 10
74
76
09
.21
.33
.45
.56
.68
.79
•91
1.02
1. 14
76
78
80
8a
.09
.22
.34
.46
.58
•70
.82
94
1.06
1. 18
78
80
8a
0.09
0.23
0.36
O.48
0.60
0.73
0.84
0.97
1. 10
1.22
.09
.34
.37
•50
.63
.75
.88
1. 01
1. 14
1.27
84
.10
35
.38
• S3
.65
.78
•91
1. os
1. 18
1.3X
84
86
.10
.35
.40
•53
.67
.81
•95
x.08
1.22
1.36
86
88
.zi
.36
.41
.55
.70
.84
.98
I.I2
1.26
1. 41
88
9o
O.IX
0.27
0.42
0.57
0.72
0.87
1.02
1. 16
1. 31
1.45
90
93
.11
.28
.44
•59
•75
90
1. 05
1.20
1.36
1.50
93
94
.12
.39
.46
.61
•77
93
1.09
1.24
1.40
1.56
94
96
.12
.30
.47
.64
.80
.96
1. 13
1.29
1. 45
1.62
96
98
100
.13
.31
.49
.66
.83
1. 00
1. 17
1.34
1. 51
1.68
98
100
0.13
0.33
0.51
0.68
0.86
1.03
1. 21
1 1.38
X.56
174
(33)
TABLE IV. — External Distances for a i
Ed = Eio/D (approx.)
Curve
I
o°
IO*
20 -
30"
40°
5o°
6o°
70*
8o*
*>• 1
.0
.2
.4
.6
.0
21.9
88.4
202.1
367.7
592.3
886.4
1265.0
1749 9
2373.3 1
.0
.0
.1
22.8
237
24.6
90.2
92.0
93.8
04.9
07.7
10.5
71.6
75.5
794
975
602.7
07.9
93.1
99.8
906.5
735
82.1
90.8
60.9
71.9
83.0
87. S
240X.8
16. 1
.8
1.0
.2
.1
.2
.3
255
26.5
27.5
95. 7
97.6
995
13.4
216.2
19. 1
83.4
387.4
91-4
13. 1
618.4
237
133
920.1
27.0
995
1308.2
17.0
94.1
1805.3
16.6
30.5
2444.9
59.5
•4
.6
.8
8.0
.2
4
.6
•4
.6
•7
28.5
295
30.5
101.4
1033
1053
22.0
25.0
27.9
954
995
4035
29.0
34.4
39.8
339
40.8
478
25.8
347
43-6
279
39 3
50.7
74. a
88.8
2503.6
9
31.6
107.2
230.9
4076
6452
954-8
1352.6
1862.2
2518.5
i.i
1.3
1.5
32.6
337
34-8
9-2
11. 2
13.3
339
36.9
399
11. 8
159
20.1
50. 6^
56.1*
61.6
61.8
68.8
759
61.6
70.6
797
73-8
85.4
97.0
33-4
48. s
63.6
.8
.2
17
2.0
2.2
359
37.1
38.2
153
117.4
19. 5
430
246.1
49.2
24-3
428.5
32.7
67.1
672.7
78.2
83.1
.990.2
974
88.9
1398.0
1407.3
1908.7
1920.5
32.4
78.7
2594.0
2609.4
•4
.6
.8
4.0
.2
•4
.6
25
2.8
32
394
40.6
41.8
21.6
237
25.8
52.3
554
58.6
37.0
41.3
45.6
839
89.5
952
1004.7
12.0
193
16.5
259
353
443
56.3
68.3
24.8
40.3
559
35
43.O
128.0
261.8
450.0
700.9
1026.6
1444.6
1980.4
2671.6
39
42
4.6
443
455
46.8
30.2
32.4
346
65.0
68.2
715
543
58.7
63.2
06.6
12.4
18.2
340
414
48.9
54.1
636
732
92.5
2004.7
17.0
874
27033
19.2
.8
5.6
.2
5.0
55
59
48.I
49-4
50.8
36.9
139. 1
41.4
74.8
278.1
81.4
67.6
472.1
76.6
24.0
7299
357
56.4
10639
71.5
82.8
14924
1502. 1
293
2041.7
542
352
2751.3
675
4
.6
.8
6.0
.2
4
.6
6.4
6.8
73
52.1
535
549
437
46.0
48.4
847
88.1
915
81. 1
85.6
90.2
41.6
47-6
53-6
79 1
86.8
945
11. 8
21.7
31.5
66.7
793
91.9
83.8
2800.1
16.6
79
56.3
150.7
2949
494.8
759-6
I 103. 2
I54I.4
2104.6
28332
8.4
89
95
577
59-2
60.6
53.1
55-5
579
98.3
301.7
05.2
994
5041
08.8
65.6
71.7
77-8
09.9
177
25.6
51.3
61.3
71.3
174
30.3
43.2
49-8
66.5
83.4
.8
7.o
.2
10. 1
10.7
II. 3
62.1
63.6
65.2
60.4
162.8
653
08.7
312.2
15.8
135
518.2
22.9
839
790.1
96.3
335
11414
493
81.4
*59i-6
1601.8
56.2
2169.2
82.4
2900.3
29173
344
•4
.6
.8
8.0
.2
•4
.6
12.0
12.6
133
66.7
68.3
69.8
67.8
70.3
72.8
193
22.9
26.5
27.7
325
374
802.5
08.8
152
573
654
73-4
12.0
22.3
32.6
955
2208.8
22.1
51.6
68.9
86.3
14.0
71.4
1754
330.1
542.2
821.4
1181.6
1643.0
22355
3003.8
147
154
16.2
730
747
76.3
78.0
80.6
83.2
338
375
41.2
47.1
52.0
57-0
277
34.1
40.5
89.7
979
1206. 1
635
64.0
745
48.9
62.5
76.1
21.4
391
56.8
.8
90
.2
16.9
17.7
18.5
78.0
797
8I.4
85.8
188.5
91.2
449
348.6
52.4
61.9
566.9
72.O
47.0
853.5
60.0
144
1222.7
31. 1
85.1
16958
1706.5
89.8
23035
173
74.7
3092.7
3110.8
4
.6
.8
10.0
193
20.2
21.0
83.I
84.8
86.6
939
96.6
994
56.2
60.0
639
77-0
82.1
87.2
66.5
73.1
797
395
48.0
56.4
173
28.1
390
31.2
451
592
28.9
472
65.6
21.9
88.4
202.1
367.7
592.3
886.4
1265.0
17499
23733
3184. I
I
o°
10*
20*
30°
40°
50°
60*
70°
8o°
»:.
(34)
TABLE V. — Long Chords and Actual Arcs
Degree
of
Actual
arc, one
aSta.
3Sta.
4Sta.
5Sta.
6Sta.
7Sta.
8Sta.
Degree
of
curve
station
curve
0.1
.4
.6
.8
100.000
200.0
300.0
400.0
500.0
600.0
700.0
800.0
O.S
4
.6
.8
.000
.000
.001
00.0
00.0
00.0
00.0
00.0
00.0
00.0
00.0
00.0
00.0
4999
99-9
00.0
5999
99-8
6999
6998
99-7
7999
7998
99-6
1.0
.2
4
100. 001
.002
.002
200.0
00.0
00.0
300.0
00.0
2999
399-9
99-9
999
499-8
99-8
997
5997
996
995
699.6
994
99.2
7994
99.1
98.8
1.0
.2
• 4
.6
.8
SO
.2
4
.6
.003
.004
00.0
00.0
99-9
999
998
99-8
996
995
993
99.1
98.9
98.6
98.4
979
.6
.8
S.O
.2
4
.6
100.005
200.0
299.9
3997
4994
598.9
698.3
7974
.006
.007
.009
00.0
00.0
00.0
999
998
99-8
99-6
996
99-5
993
99.1
990
987
98.5
98.2
979
975
971
96:9
96.3
957
.8
30
.2
.010
100. on
.013
1999
199-9
999
99-8
299.7
997
994
3993
992
98.8
498.6
98.4
979
597.6
973
96.7
696.2
956
950
7943
935
.8
30
.2
4
.6
.8
4.0
.2
•4
.6
.ois
.016
.018
999
999
999
996
99-6
99-6
99.1
990
98.9
98.2
98.0
97.8
96.9
96.6
96.2
951
945
939
92.6
91.7
90.8
4
.6
.8
4.0
.2
4
.6
100.020
1999
299.5
398.8
497-6
5957
693.2
789.8
.022
.025
.027
999
999
998
995
994
994
98.7
98.5
98.4
973
971
96.8
953
949
94.4
92.5
91.8
91.0
88.8
87.7
86.5
.8
.2
.029
100.032
.034
99.8
199.8
998
993
299.2
992
98.2
398.1
979
96.5
4962
959
93.9
5934
92.8
90.2
689.4
88.5
853
784.1
82.8
.8
.2
4
.6
.8
6.0
.2
.4
.6
•037
.040
.043
998
998
997
991
990
990
97-8
97.6
974
95.6
952
94 9
92.3
91.7
91. 1
87.6
86.7
857
81.5
80.1
78.7
4
.6
.8
6.0
.2
4
.6
100.046
1997
298.9
3973
4945
590.4
684.7
777-2
.049
.052
.055
997
997
997
98.8
• 98.8
987
97.1
96.9
967
94.2
93.8
934
89.8
89.1
88.5
837
82.7
81.6
75-6
74-0
72.4
.8
7.0
.2
059
100.062
.066
99-6
199.6
99.6
98.6
298.5
98.4
96.5
396.3
96.1
930
492.6
92.1
877
5870
86.3
80.4
6793
78.1
70.7
7690
67.2
.8
70
.2
■4
.6
.8
8.0
.2
4
.6
.070
.073
.077
99.6
996
995
98.3
98.2
98.1
958
95.6
954
91.7
91.2
90.8
855
84.7
839
76.9
75.6
74.3
65.4
635
61.6
.4
.6
.8
8.0
.2
.4
.6
100.081
199- 5
298.x
395.1
490.3
5831
6730
7597
.085
.090
.094
995
995
99-4
98.0
979
97.8
949
94-6
944
89.8
893
88.8
82.2
81.4
80.5
71.7
70.3
68.9
57-7
556
535
.8
90
.2
098
100.103
.108
994
1994
99-4
97.6
2975
97-4
94-1
3939
93-6
88.3
4877
87.2
796
578.6
77-7
67.4
666.0
644
51.4
749-2
46.9
.8
9.0
.2
•4
.6
.8
10.0
.112
.117
.122
99 3
993
99-3
973
972
97.1
93 3
930
92.7
86.6
86.1
85.5
76.7
75.7
74.7
62.9
61.3
597
44.6
42.3
399
•4
.6
.8
10.0
100. 127
199. a
297.0
392.4
4849
5737
658.1
737.5
Degree
Actual arc
aSta.
3Sta.
4Sta.
5Sta.
6Sta.
7Sta.
8Sta.
Degree
(35)
TABLE V. — (Continued)
Degree
of curve
Actual arc,
Dne station
aSta.
3Sta.
4Sta.
5Sta.
6Sta.
7Sta.
8Sta.
Degree
of curve
10.0
100.127
1992
297.0
392.4
4849
573-7
658.1
737-5
10.0
XX.
.2
4
.6
.132
.137
.143
992
992
99.1
96.8
96.7
96.6
92.1
91.8
91.5
843
837
83.1
72.6
71.6
70.5
56.5
54.8
530
351
32.5
30.0
2
4
6
.8
.a
.148
100. 154
.159
991
199. 1
990
96.5
296.3
96.2
91.2
390.8
90.5
82.4
481.8
81. 1
694
568.2
67.1
51.3
6495
47-7
27.4
724.8
22.1
XX.
8
2
IS
.4
.6
.8
.0
.2
•4
6
.165
.171
• X77
990
99.0
98.9
961
959
958
90.2
89.8
895
80.4
797
790
659
64.8
635
45.8
440
42.1
194
16.6
13.8
18
4
6
8
.0
2
4
6
100.183
198.9
2956
389.1
478.3
562.3
640.1
710.9
.189
.195
.202
98.9
98.8
98.8
955
953
952
88.7
88.4
88.0
77.6
76.9
76.2
61. X
59-8
58.5
38.2
36.2
342
08.1
05.1
702.2
13
8
.2
.208
100.215
.222
98.8
1987
98.7
950
2949
947
87.6
387.2
86.8
754
474-6
739
572
5559
54.6
32.1
630.1
28.0
6992
696.1
93-0
13
8
.2
14
4
6
8
.2
4
.6
.228
235
.242
98.6
98.6
98.5
94.6
944
94-2
86.5
86.0
85.6
731
72.3
71.5
532
519
50.5
25.8
237
21.5
89.9
86.7
83.5
14.
• 4
6
8
.2
4
.6
100.249
198.5
2941
385.2
470.6
549.1
6193
680.3
.256
.264
.271
93.5
98.4
9M
939
937
935
84.8
84.4
839
69.8
69.0
68.1
47.6
46.2
447
17.0
14.8
12.5
77-0
737
70.3
15
8
.2
279
IOO.286
.294
98.3
198.3
98.2
934
2932
93.0
835
3831
82.6
673
466.4
655
43.2
541.7
40.2
10.2
607.8
05.4
66.9
663.5
60.0
IS
.8
.2
16
.4
6
8
.0
2
4
6
.302
.310
.318
98.2
98.2
98.1
92.8
92.6
924
82.2
81.7
81.2
64.6
63.7
62.8
38.7
37.1
35.6
03.0
600.6
5982
56.5
530
494
16
•4
.6
.8
.0
.2
4
.6
100.326
198. X
292.3
380.8
461.9
534.0
595-7
645.8
.334
•342
.351
98.0
98.0
979
92.1
91.9
91.7
80.3
79-8
793
60.9
60.0
59
32.4
30.7
29.1
932
907
88.1
42.2
38.5
34.8
17
8
2
•359
100.368
.376
979
197.8
97-8
91.5
2913
91. 1
78.8
378.3
778
58.0
457-1
56.1
27.5
5258
24.1
855
582.9
80.3
31. 1
627.3
235
17
.8
.2
18
4
6
8
.0
2
4
6
.385
394
.403
977
97.6
976
90.8
90.6
904
773
76.8
76.3
55.1
54.1
530
22.4
20.7
18.9
777
750
72.3
19.6
15.8
11. 9
IB
4
.6
.8
.2
.4
.6
100.412
1975
290.2
3757
452.0
517.2
5696
608.0
.422
.431
.440
975
974
974
90.0
89.8
89.6
752
747
74-1
51.0
499
48.9
154
136
xi. 8
66.8
64.1
61.3
04.0
600.0
596.0
19
8
2
.450
100.460
.469
973
1973
97.2
893
289.1
88.9
73.6
373.0
72.5
47.8
446.7
45.6
10.
508.1
06.3
58.5
555-6
52.8
92.0
587.9
83.8
19
.8
.2
■
4
6
8
.479
.489
•499
971
97.1
970
88.6
88.4
88.2
71.9
71.3
70.7
44*5
434
42.3
04.4
02.5
500.6
49-9
47-0
44.1
79-7
75-5
71.3
10.
■4
6
8
*0
100.510
197.0
287.9
370.2
441. 1
498.7
541. 1
567.x
I Degree
Actual arc
aSta.
3Sta.
4Sta.
5Sta.
6Sta.
7Sta.
8Sta.
Degree
(36)
TABLE VI. — Middle Ordinates
Degree
of curve
iSta.
aSta.
3Sta.
4Sta.
5Sta.
6Sta.
7Sta.
8Sta. ^
free
irve
o.a
.4
.6
.8
.04
17
39
•70
1.09
1.57
2.14
2.79 0.
S
4
6
8
•09
.13
.17
35
53
.70
.78
1. 18
1.57
x.40
3.09
3.79
3.18
327
436
3.14
4.71
6.38
4.28
6.41
8.55
558
8.38
11. 17
x.o
.2
• 4
.22
.36
.30
87
x.05
1.33
x.96
3.36
3.75
3 49
4.19
489
545
6.54
763
785
942
10.99
10.69
13.83
14.96
13.96 1.
16.75
1953
2
4
.6
.8
1.0
.2
■ 4
.6
.35
39
1.40
1.57
314
353
558
6.38
8.72
9.81
12.56
1413
I7.O9
19.33
23.33
35.10
6
8
2
4
6
•44
x.75
3 93
6.98
10.90
1569
21.36
27.88 t.
.48
.52
• 57
1.93
3.09
3.37
432
4 71
5.10
7.68
8.37
9.07
11.99
13.08
14.17
17.36
1883
30.39
2348
35.61
27.74
30.66
3343
36.21
.8
3o
.2
.61
65
.70
3.44
a. 62
3.79
5.50
5.89
6.28
977
xo.46
11. 16
15.25
16.34
1743
21.95
23.52
35.08
39.86
31.98
34.10
38.98
41. 74 3
44.SO
8
.0
2
4
.6
.8
4.0
.2
4
.6
•74
•78
.83
2.97
3.14
332
6.67
7.06
746
11.86
13.55
1325
18.51
19.60
30.68
36.64
38.30
2975
36.33
38.34
40.45
4726
50.01
52.76
4
6
8
2
4
6
87
3 49
7.85
13 94
ai.77
31 31
42 56
5550 4
.93
96
1. 00
366
384
4.01
8.34
8.63
9.03
14.64
1533
16.03
33.85
23.93
25.01
32.86
3441
3596
44.66
46.77
48.87
58.34
60.97
6369
.8
50
.2
x.05
1.09
X.14
419
4 36
4-54
9-43
9.81
10.30
16.73
1742
18. 11
36.09
2717
38.25
3751
3906
40.60
50.96
53.05
55.14
66.43.
69.13 5
71.84
8
2
■ 4
.6
.8
6.0
.2
■ 4
.6
1. 18
1.33
1.37
471
489
5.06
10.59
I0.O8
"37
18.80
19.49
30.19
2933
30.40
31.48
42.15
43.69
45.23
57.23
59-31
61.38
74-54
77 23
7992
4
6
8
2
4
6
X.3I
5.23
xx. 76
20.88
32.55
46.76
6346
82.60 6.
1.35
. 1.40
1.44
541
558
576
13.15
13.54
13.93
21.57
33.36
33.95
3363
3470
3577
48.39
49-83
5135
6552
6758
69.64
8527
8793
90.59
.8
70
.2
I.48
1 53
1. 57
593
6. 11
6.28
13.32
13. 71
14. 10
2364
24.33
25.03
36.84
3791
3897
53.88
5440
55.92
71.70
7374
7579
93-23
9587 7
9850
8
2
4
.6
.8
8.0
.2
.4
.6
1.63
1.66
1.70
6.45
6.63
6.80
14.49
14.88
1527
2571
26.39
37.08
40.04
41.10
43.17
5744
58.95
60.46
77.82
7985
81.88
101.12
103.73
106.33
4
6
8
2
4
6
x.75
6.98
15 66
3777
43 23
61.97
8390
X08.93 8
1.79
1.83
x.88
7.15
732
7.50
16.05
16.44
16.83
28.45
3914
39.83
44.29
4535
46.40
6347
64.97
66.47
8592
8792
89.93
in. 50
114.06
116.62
.8
9.0
.2
1.93
1.97
3.00
7.67
785
8.03
17.23
1761
18.00
30.51
31.19
31.87
4746
48.51
49-56
6797
6946
70.95
91.92
9391
9589
119. 17
121.71 9
124.33
.8
.2
.4
.6
.8
100
3.05
3.XO
3.14
8.19
8.37
8.54
18.38
18.77
19.16
33.56
33-34
3392
50.61
51.66
52.71
72-43
7391
7539
97.87
9983
101.80
126.75
13935
131.74
• 4
.6
.8
2.l8
8.73
1955
3460
5375
76.86
X0375
134.22 10
Degree
iSta.
aSta.
3Sta.
4Sta.
5Sta.
6Sta.
7Sta.
8 Sta. 1 De
gr~|
(37)
TABLE VI. — {Continued)
Degree
of curve
iSta.
2 Sta.
3Sta.
4Sta.
5Sta.
6Sta.
7Sta.
8 Sta.
Degree 1
of curve|
10.0
a.z8
8.7a
19 55
34.6o
53 75
76.86
10375
134.23
10.0 1
3
4
6
2.23
2.27
2.31
8.89
9.06
9.24
1994
20.32
20.71
35.28
3596
36.63
54-79
55-83
56.87
78.33
7979
81.25
105.70
107.64
109.57
136.68
139.14
141.58
.3
.4
.6
zi.
8
o
3
2.36
2 40
a. 44
941
959
9-76
21.10
31.48
21.87
3731
37 99
38.66
5791
58.94
59-98
82.71
8416
85.61
in. 49
X13.41
115.32
144.00
14641
148.81
.8
zz.o
.a
12
4
6
8
a
4
6
2.49
2.53
2.58
993
10. 11
10.28
22.25
22.64
23.03
3934
40.01
40.68
61.01
62.04
63.06
8705
88.49
8992
117. 21
119. 11
120.99
151.20
15357
15593
4
.6
.8
11.0
.3
4
.6
2.6a
10.45
2341
4136
6409
91 36
123.87
158.37
2.66
2.71
2.75
10.63
10.80
10.97
2380
24.18
2457
4203
42.70
43-37
65.11
66.13
6714
9278
9420
9562
124.73
126.59
128.43
160.59
162.91
165.21
13
8
o
a
2.80
2.84
2.88
11. 15
11.32
11.49
2495
2533
2572
44-03
4470
4537
68.16
69.17
70.18
97.03
98 43
99.83
130.27
132.10
133-92
167.49
169.75
172.01
.8
13.0
.2
14
4
6
8
.2
4
.6
293
2.97
301
11.67
11.84
12.01
26.10
26.48
26.87
46.03
46.70
4736
71.19
72.19
73-20
101.23
102.62
104.00
13573
13753
13933
17424
176.46
178.67
4
.6
.8
14.0
.3
4
.6
306
12.19
2725
48.03
74-20
105.38
x4x.11
180.85
310
3 15
319
12.36
12.53
12.71
2763
28.01
28.40
48.69
4935
50.01
75.20
76.19
77.18
106.76
108.12
109.49
142.88
14464
146.40
183.02
185.17
187.31
15
.8
o
.2
3.23
3.28
332
12.88
1305
1323
28.78
29.16
29.54
50.66
51.32
5198
78.17
79x6
80.14
110.85
1 13.30
II354
148.14
X49 87
151.59
18943
191.53
193.62
.8
150
.3
16
.4
6
.8
.0
.a
4
.6
336
341
345
13 40
13-57
1374
29.92
30.68
52.63
53 29
5394
81.12
82.10
8308
114.88
II6.22
II755
15330
155.00
156.69
19568
19773
199.76
.4
.6
.8
16.0
.3
4
.6
350
1392
31.06
5459
8405
IX8.87
158.37
201.77
354
358
363
1409
14.26
14.44
31-44
3182
32.20
5524
5589
56.54
85.02
8599
86.95
120.19
121.50
122.80
160.03
161.69
16333
203.77
205.74
207.70
17
8
a
367
3-72
376
14.61
14 78
14-95
32.57
32.95
3333
57.19
57.83
58.48
8791
88.87
89.83
124.10
135.39
126.68
164.96
166.59
168.20
209.64
311.55
21346
.8
170
.3
18
4
6
8
a
4
6
3.80
385
3.89
1513
1530
1547
3371
3408
3446
59.12
5976
60.40
90.78
91.73
92.67
I2796
129.23
130.49
16979
17138
172.95
21533
217.20
219.03
.4
.6
.8
18.0
.3
.4
.6
3 94
1564
3484
61.04
93.6a
131-75
17452
220.86
398
4.02
4.07
15.82
1599
16.16
35.21
3559
3596
61.68
62.32
62.96
94-55
9549
9642
133.01
I3425
13549
176.07
177.60
I79-I3
222.65
22443
226.20
19
8
a
4- 11
4x6
4.20
16.33
16.51
16.68
36.34
36.71
37.09
63.59
6432
6485
9735
98.a8
99-20
136.72
13795
139.17
180.64
182.15
18364
227.93
239.65
231.35
.8
19.0
.a
to
4
6
8
4.24
429
4 33
16.85
17.02
1719
3746
3783
38.20
6548
66.11
66.74
100.12
101.03
101.95
140.38
141.58
142.78
185. 11
186.57
188.02
23303
23469
236.33
•4
.6
.8
to.o-
437
1737
38.58
67 37
X02.86
H3 97
18946
23794
Degree
xSta.
2 Sta.
3Sta.
4Sta.
5Sta.
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 (7\ 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 i D . . . 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.
587
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 = } 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.
180
If the maximum allowable cant is 6 inches, L = ^-7= feet for curves
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.
LD
A =
200
39
40
RAILROAD FIELD MANUAL
Notation: Referring to Fig. 22
S.T.
Fig. 22.
T. 5. = Tangent-Spiral. 5. C. » Spiral-Curve.
C. S. — Curve-Spiral. 5. 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 7*. 5. as origin.
Y = ordinate with T, S. as origin.
U = spiral long tangent.
V = spiral short tangent.
— curve offset.
Z =» distance on tangent from T. S. to 7\ C of the offset curve.
T 9 = 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
1. 7, I9,37»6i,9i, 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 Us of any given cases are to be multi-
plied to find the functions C, X, Y t U, and V. The use of the table
to get and Z will be evident from the table headings.
C X Y
Referring to Fig. 22, the tabulated quantities are A, A, y* yt -= »
U V
-=-» y* coefficients for and Z, The following formulas give these
quantities:
A = , A *= \ A — 0.000000825 A 1 , B — A — A.
Q
j = cos 0.3 A + 0.004 exsec } A (approx.).
X = Ccos-4.
Y — Csin-4.
Z = *-£sinA.
_. ~sin 2?
U » C -; 7*
sin A
*
T . ^ sin A
sin A
= F — i? vers A.
42 RAILROAD FIELD MANUAL
61A
300
_ . v LI A . 7A , 19 A. 37 A.
Exactly, X = — I cos h ' h cos — h cos — h cos
J 10 \ 300 300 300 300
, 01 A , 127 A , 169 A , 217 A . 271 A\
+ cos h cos — - r cos — \- cos — - h cos — — 1 •
300 300 30° 300 300 /
Exactly, F = — (sin h sin h . . . etc. ) •
J 10 \ 300 300 /
Y
Exactly, tan A = -=s •
The following formulas give the tangent distance and external dis-
tance of the curve:
T. = CR + o)tan$/ + £
E 9 — (R + o) exsec J / + 0.
The central 8 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 -=-Z>. The
degree at the end of any chord p (p being the number of the chord)
P . P
is — D for the 10-chord spiral or jb^ for the JV-chord spiral.
For precise computation of positions of points on line, A should be
computed from A = } A — 0.000000825 A 8 or taken from Table VIII,
or Tables IX to XXIII. The deflection a x for the first chord is always
— for the 10-chord spiral.
300 ^
For field use the deflection from the T. S. to any chord point should
be taken as a\ times the square of the number of the chord point to be
located so long as 6 does not exceed 15 . A may be taken as 1/3 A
for A^ 1 5 . When A is more than 15 one or more intermediate transit
points should be used. Such points should be so chosen that the 8
from the T. 5. to the first intermediate point shall not exceed 15 , and
so that 8 from any occupied point to the next transit point less the 8
from the T. 5. 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 io°
and an error of 3 seconds or less than 0.00 1° if A is taken as J A.
THE SPIRAL
43
For A = 1 5 the error is 10 seconds or less than 0.003 . If it is desired
to find the deflection from the T. S. to any point to which the 5 exceeds
1 5° 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 I and D, determine
L and A; from Table VIII find coefficients for and determine and Z
and substitute in the equation for T a . Knowing the P. 7. 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. 5. 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, laid
off and the curve D run in for the full central angle J, using an offset
back sight for the direction of the offset tangent, locating the 5. 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 I — 2 A, and off-
setting 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 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 i 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 1.5-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
r
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 XIII 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%) 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
curve of degree (D x — Dt) 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 ACoi the spiral is
i A (A being the central angle for spiral of length L connecting a tangent
and (Di — Dt)° 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 Dt be the degree of the larger
radius curve and A of the shorter, + J A — C, when A «
zoo
— - — — and C is the correction to be used when necessary or when
200 J
A is more than 15 . Or the curve may be considered as part of a spiral
from a tangent to the curve of degree D% and the deflection coefficients
of Table VII may be used.
Fig. 23.
THE SPIRAL
45
Illustrative Example. — A 4 curve is to connect with an 8° curve
the offset coming at Sta. 464 of the 4
-^ -^ ~ 600
D x -D t = 4 ; L = _,_ _ =300. A =
i°.
Dx
curve. D% = 4
(Di+Dt) L
2 100
■8°;
18 .
VD t - Dt
For computing o and Z we use D = D\ — D% = 4 , and L = 300,
whence A = 6°, and from the tables for D = 4, L = 300, and A = 6°
we find - 2.62 and Z = 149.92. Therefore C. 5. -= 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
8 L
its length the whole spiral would be Z* = ~ =r ■= 2 L = 600 feet
i/i — u%
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
600-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 at, 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:
1.20 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
D\ curve, are the deflections for the chord points of a five-chord spiral
of 300 feet connecting a tangent and 4 = (D\ — D%) 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 <*i or io°. It is also — - — 2 or, stated otherwise,
** 200
the angle between the long chord of — stations of curve D% and the
6 * 100
chord of the spiral is 2°, which is the deflection angle of a 300-foot spiral
connecting a tangent and 4 = (Di — Z)») degree curve.
Considering the whole spiraled curve, and referring to Fig. 24, in
which Ri is the shorter radius and R% 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\ = 0\ sec Ji. The arc of R% must be thrown
PPt = 0% sec 1%. PP\ — PP% must equal 0', the proper offset for
the connecting spiral. Therefore determine Z* to find o% and PPt; to
4 6
RAILROAD FIELD MANUAL
PP -4- n f
PPt add o' previously found and find a trial value for 0\ = — = — ;
sec J- 1
if found too small, o% or o' or both may be increased, which will
necessitate increase in L% or L'
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% tan A2 and 0\ 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 Jo', the
proper offset for the chosen spiral ; find new radii for the curves such that
i o f cos 1% -|- 0%
R% = 2?2 —
Bi' = Ri-
vers I 2
01 — i 0' COS I\
vers h
Both tangent points will be moved toward the vertex by
/ 2 = (R t — .Ra' + ^ 0') sin h
h = (Ri -!&'-! o') sin 7i.
TABLE VII. — Coefficients of a,i for Deflection Angles to
Chord Points
Deflection
angle to
Transit at chord-point number
chord-point
number
T.S.
I
2
3
4
5
6
7
8
9
10
S.C.
or. S.
2
8
18
32
50
72
98
128
162
200
1
1
O
5
14
27
44
65
90
119
152
189
2
4
4
8
20
36
56
80
108
140
176
3
9
10
7
XI
26
45
68
95
126
161
4
16
18
16
10
14
32
54
80
no
144
5
25
28
27
22
13
O
17
38
63
92
125
6
36
40
40
36
28
16
20
44
72
104
7
49
54
55
52
45
34
19
23
50
81
8
64
70
72
70
64
54
40
22
26
56
9
81
88
91
90
85
76
63
46
25
29
10 5. C.
100
108
1x2
112
xo8
100
88
72
52
28
TABLE VIII. — General Functions — Ten-chord Spiral
A
A
c
L
X
L
Y
u
L
V
L
o^mL
— nD
Z-mL
,—nD
A
m
n
m
n
e
0.0
.1
.2
3
•
0.000 1
:. 00000
1. 00000
.00000
.66667
.33333
.00000
.0000
.50000
.0000
0.0
.X
.3
3
033
.067
.100
.00000
.00000
.00000
•
.00000
.00000
.00000
.00058
.00116
.00175
.66667
.66667
.66667
.33333
.33333
.33333
.00015
.00029
.00044
.0000
.0000
.0000
.50000
.50000
.50000
.0001
.0003
.0004
•4
.5
.6
.133
.167
.300
.00000
.00000
.00000
.00000
0.99999
99999
.00233
.00391
.00349
.66667
.66667
.66667
.33333
.33334
33334
.00058
.00073
.00087
.0000
.0000
.0000
.50000
.50000
.50000
.0005
.0006
.0008
.4
.5
.6
.7
.8
9
1.0
.1
.2
•3
.233
.207
.300
99999
99999
99999
99999
99998
•99998
.00407
.00465
.00524
.66667
.66667
.66668
33334
.33334
.33334
.00102
.00116
.00131
.0000
.0000
.0000
.50000
.50000
.50000
.0009
.0010
.0011
7
.8
9
1.0
.1
.2
•3
.333
99999
99997
.00582
.66668
.33334
.00145
.0000
.50000
.0013
.367
.400
.433
99998
.99998
.99998
99996
99996
99995
.00640
.00698
.00756
.66668
.66668
.66668
•33335
.33335
33335
.00160
.00175
.00189
.0000
.0000
.0000
•49999
•49999
•49999
.0014
.0015
.0017
4
.5
.6
.467
.500
.533
•99997
.99997
99997
99994
99993
99992
.00814
.00873
.00931
.66669
.66669
.66669
•33335
.33336
.33336
.00204
.00218
.00233
.0000
.0000
.0000
49999
•49999
.49999
.0018
.0019
.0020
•4
5
.6
.7
.8
9
S.O
.z
.2
.3
.567
.600
.633
99996
99996
•99995
99991
•99990
99989
.00989
.01047
.01105
.66670
.66670
.66671
.33336
.33337
.33337
.00247
.00262
.00276
.0000
.0000
.0000
49999
•49998
•49998
.0032
.0023
.0024
•7
.8
9
s.o
.1
.2
3
.667
99995
•99988
.01163
.66671
33337
.00291
.0000
.49998
.0025
.700
.733
.767
.99994
99994
.99993
.99987
.99985
.99984
.01222
.01280
.01338
.66671
.66672
.66672
.33338
.33338
.33338
.00305
.00320
.00335
.0000
.0001
.0001
•49998
.49998
•49997
.0027
.0028
.0029
•4
5
.6
.800
.833
.867
99992
.99992
.99991
99983
.99981
.99980
.01396
.01454
.01512
.66673
.66673
.66674
.33339
.33339
33340
.00349
.00364
.00378
.0001
.0001
.0001
49997
•49997
•49997
.0031
.0032
•0033
4
5
.6
.7
.8
9
8.0
.z
.2
.3
.900
.933
.967
.99990
.99990
.99989
99978
.99976
99975
.01571
.01639
.01687
.66674
.66675
.66676
33340
.33341
.33342
.00393
.00407
.00422
.0001
.0001
.0001
•49996
.49996
.49996
•0034
.OO36
•OO37
7
.8
9
S.O
.X
.3
.3
1. 000
.99988
99973
•01745
.66676
33342
.00436
.0001
•49996
.0038
1.033
1.067
1. 100
.99987
.99986
.99985
99971
99969
.99967
.01803
.01861
.01919
.66677
.66678
.66678
33343
.33343
•33344
.00451
.00465
.00480
.0001
.0001
.0001
•49995
49995
•49995
0039
.0041
.0042
•4
• 5
.6
1.133
1. 167
1. 200
.99985
.99984
99983
99965
99963
.99961
.01978
.03036
.03094
.66679
.66680
.66681
•33345
33345
33346
.00494
.00509
.00524
.0001
.0001
.0001
•49994
49994
49994
OO43
.0045
.OO46
.4
5
.6
.7
.8
9
4.0
.X
.2
3
1.233
1.267
1.300
99982
99981
99980
99959
99956
99954
.03152
.022I0
.03368
.66681
.66682
.66683
33347
•33347
33348
00538
.00553
.00567
.0002
.0002
.0002
•49993
•49993
•49993
.0047
.OO48
.0050
7
.8
9
4.0
.X
.2
.3
1.333
•99979
99952
.02326
.66684
.33349
.00582
.0002
.49992
.O05I
1.367
1.400
1.433
•99978
.99976
•99975
99949
99947
99944
.02384
.02443
.02501
.66685
.66686
.66686
.33350
.33351
.33351
.00596
.00611
.00625
.0002
.0002
.0002
.49992
49991
.49991
.0052
O053
•0055
•4
5
.6
1.467
1.500
1.533
•99974
•99973
99972
•99941
99939
.99936
.02559
.02617
.02675
.66687
.66688
.66689
33352
33353
.33354
.00640
.00654
.00669
.0002
.0002
.0002
•49991
.49990
•49990
.OO56
.0057
.0059
.4
5
.6
.7
.8
9
5.0
1.567
1.600
1.633
99970
99969
99968
99933
•99930
■99927
.02733
.02791
.02849
.66690
.66691
.66692
33355
33356
33357
.00683
.00698
00713
.0002
.0003
.0003
.49989
.49989
.49988
.OO60
.0061
.0062
7
.8
9
5.0
1.667
99967
99924
.02907
.66693
33358
.00727
.0003
.49988
.OO64
(47)
TABLE VIII. — (Continued)
6.0
.x
.3
3
•4
5
.6
7
.8
9
6.0
.z
.3
3
•4
5
.6
7
.8
9
7.0
.1
.a
3
4
•5
.6
7
.8
9
8.0
.i
.a
•3
4
• 5
.6
7
.8
9
9.0
.z
.a
3
•4
5
.6
7
.8
9
10.0
,667
.700
733
.767
.800
833
.867
.900
■933
967
a. 000
a.033
2.067
a. 100
2. 133
2.167
2.200
2.233
2.267
2.300
2.333
2.367
2.400
2.433
2.467
2.500
2.533
2.567
2.600
2.633
2.666
2.699
2.733
2.766
2.799
2.833
2.866
2.899
2.933
2.966
2.999
3.Q33
3.066
3.099
3. 133
3.166
3.199
3232
3.266
3299
3332
C
L
99967
99965
99964
99962
99961
99960
99958
99957
99955
99953
99952
99950
99949
99947
99945
99944
99942
99940
99938
99936
99935
99933
99931
99929
99927
99925
99923
99921
99919
99917
999H
99912
99910
99908
99906
99903
99901
99899
99897
99894
99892
99889
99887
99884
99882
99879
99877
99874
99872
99869
99866
99924
99921
99918
99915
99912
99908
99905
99902
99898
99895
99891
99887
99884
99880
99876
99872
99868
99864
99860
99856
99852
99847
99843
99839
99834
99830
99825
99821
99816
998II
99806
99801
99797
99792
99786
99781
99776
99771
99766
99760
99755
99749
99744
99738
99733
99727
99721
99715
99709
99704
99698
02907
02965
03023
03082
03140
03198
03256
03314
03372
Q3430
03488
03546
03604
03662
03720
03778
03836
03894
03952
04010
04068
04126
04184
04242
04300
04358
04416
04474
04532
04590
U
04648
04706
04764
04822
04880
04937
04995
05053
05x11
05169
05227
05285
05343
05400
05458
05516
05574
05632
05690
Q5747
0S805
66693
66695
66696
66697
66698
66699
66700
66702
66703
66704
66705
66707
66708
66709
66711
66712
66713
6671s
66716
66718
66719
66721
66722
66724
66725
66727
66729
66730
66732
66734
66735
66737
66739
66741
66742
66744
66746
66748
66750
66752
66754
66756
66758
66760
66762
66764
66766
66768
66770
66772
66774
V
L
33358
33359
33360
33361
33362
33363
33364
33365
33366
33367
33369
33370
33371
33372
33373
33375
33376
33377
33379
33380
33381
33383
33384
33385
33387
33388
33390
33391
33393
33394
33396
33398
33399
33401
33402
33404
33406
33407
33409
334II
33413
33414
33416
33418
33420
33422
33424
33425
33427
33429
33431
(48)
o—mL—nD
m
00727
00742
00756
00771
00785
00800
00814
00829
00843
00858
00872
00887
00901
00916
00930
00945
00959
00974
00989
01003
01018
0x032
0x047
01061
0x076
01090
01105
01119
01134
01 148
01 163
01177
01192
01206
0x221
01235,
01250
01264
01279
01293
01308
01322
01337
01351
01366
01381
01395
01410
01424
OI439
01453
0003
0003
0003
0003
0003
0003
0003
0004
0004
0004
0004
0004
0004
0004
0005
0005
0005
0005
0005
0005
0005
0006
0006
0006
0006
0006
0006
0007
0007
0007
0007
0007
0007
0008
0008
0008
0008
0008
0009
0009
0009
0009
0009
OOIO
0010
0010
00X0
0010
0011
001 1*
OOII
Z-mL-nD
m
49988
49987
49987
49986
49986
49985
49985
49984
49984
49983
49982
49982
49981
4998i
49980
49979
49979
49978
49977
49977
49976
49975
49975
49974
49973
49972
49972
49971
49970
49969
49969
49968
49967
49966
49965
49965
49964
49963
49962
4996i
49960
49959
49959
49958
49957
49956
49955
49954
49953
49952
0064
0065
0066
0067
0069
0070
0071
0073
0074
0075
0076
0078
0079
0080
0081
0083
0084
0085
0086
0088
0069
0090
0091
0093
0094
0095
0097
0098
0099
0100
0x02
0103
0104
0105
0107
0108
0109
OIXO
0112
01X3
0114
0Il6
0X17
0Il8
0119
0I20
0122
QI23
0X24
OI26
0127
6
6.0
.x
.a
3
4
7
.8
9
1
.a
3
•4
■ 5
.6
7
.8
9
.0
1
.a
3
•4
5
.6
7
.8
9
.0
1
.a
3
4
5
.6
7
.8
9
.0
.1
.a
3
•4
5
6
9
10
1
TABLE ,
VIII.
— {Continued)
A
•
A
C
L
X
L
Y
L
u
L
V
L
o—mL—nD
Z^mL-nD
A
m
»
m
M
100
.1
.2
•3
•4
•5
.6
•7
.8
9
U.O
.1
.2
•3
4
•5
.6
.7
.8
9
U.O
.1
.2
•3
•4
.5
.6
.7
.8
•9
U.O
.i
.2
•3
.4
5
.6
7
.8
9
14.0
.1
.2
3
•4
5
.6
7
.8
•»
3.332
•99866
99698
.05805
.66774
33431
01453
.0011
49951
.OI27
10.0
.1
.2
3
4
5
.6
7
.8
9
U.O
.1
.2
3
4
5
.6
7
.8
9
U.O
.1
.2
3
4
5
.6
7
.8
9
U.O
.1
.2
•3
4
•5
.6
7
.8
•9
14.0
.1
.2
•3
4
5
.6
7
.8
9
15.0
3.366
3-399
3.432
3466
3499
3532
3.566
3599
3632
99864
99861
.99858
.99856
.99853
.99850
.99847
.99844
.99841
.99691
.99685
99679
99673
.99667
.99660
.99654
.99647
.99641
.05863
.05921
.05979
.06036
.06094
.06x52
.062x0
.06267
.06325
.66776
.66779
.66781
.66783
.66785
.66787
.66790
.66792
.66794
33433
33435
.33437
•33439
.33441
.33443
.33446
.33448
.33450
.01468
.01482
.01497
.0x5x1
.01526
.01540
.01555
.01569
.01584
.0011
.0012
.00x2
.0012
.0012
.00x2
.0013
.0013
.0013
49950
49949
49948
•49947
.49946
.49945
.49944
49943
49942
.0128
.0129
.0131
.0132
OI33
.0134
.0136
OI37
.0138
3.666
99838
99634
.06383
.66797
.33452
01598
.00x3
.49941
•OI39
.0141
.0142
.0143
.0144
.OI46
.OI47
.0148
.OI49
.0151
3.699
3.732
3.765
3.799
3832
3865
3«99
3.932
3965
99835
99832
.99829
.99826
.99823
.99820
.99817
.99814
.99811
99627
99621
.99614
.99607
99600
99593
99586
•99579
.99572
•06441
.06498
.06556
.06614
.06671
.06729
.06787
.06844
.06902
.66799
.66802
.66804
.66806
.66809
.668x1
.66814
.66816
.66819
33454
.33456
.33459
33461
.33463
.33465
.33468
.33470
•33472
.01612
.01627
.01641
.01656
.01670
.01685
01699
.01714
.01728
.0014
.0014
.0014
.00x4
.00x5
.0015
.0015
.0015
.0016
•49940
•49939
•49937
49936
•49935
49934
•49933
•49932
49931
3999
.99808
.99565
.06960
.66822
.33475
.OI743
.0016
•49929
.0152
4.032
4.065
4.098
4.132
4.165
4.198
4.232
4.265
4.298
.99804
99801
.99798
•99795
.99791
•99788
.99785
99781
.99778
.99557
.99550
•99543
.99535
.99528
•99520
99513
99505
99497
.07017
.07075
•07133
.07190
.07248
.07305
.07363
07421
.07478
.66824
.66827
.66830
.66832
.66835
.66838
.66840
.66843
.66846
•33477
•33479
.33482
.33484
.33487
.33489
33492
33494
33497
.01757
.01772
.01786
.01801
.01815
.01830
.0x844
.01859
.01873
.00x6
.00x6
.0017
.0017
.0017
.0018
.0018
.0018
.0018
.0019
.0019
.0019
.0020
.0020
.0020
.0020
.0021
.0021
.0021
49928
•49927
.49926
49925
.49923
49922
49921
.49920
.49918
.0153
0154
.OI56
.0157
.OI58
.0159
.OI60
.O162
.OI63
4.331
.99774
.99489
.07536
.66849
33499
.01888
49917
.OI64
4365
4.398
4.431
4.465
4.498
4.531
4.S64
4598
4.631
99771
99767
99764
.99760
99757
99753
99749
99746
99742
.99481
99474
.99467
99457
99449
99441
•99433
•99425
99416
.07593
.07651
.07708
.07766
.07823
.07881
.07938
.07996
.08053
.66852
.66854
.66857
.66860
.66863
.66866
.66869
.66872
.66875
33502
33504
33507
•33510
.33512
33515
335x8
33520
33523
.0x902
.01917
.01931
.01946
.0x960
•01974
.01989
.02003
.02018
.49916
.49915
•49913
.49912
•499"
.49909
.49908
49907
.49905
.OI65
.0167
.0168
.OI69
.OI70
.0172
0173
.0174
•0175
4.664
.99738
.99408
.08111
.66878
33526
.02032
.0022
49904
0177
4.698
4.731
4.764
4.797
4.831
4.864
4897
4930
4964
99735
99731
99727
99723
.99719
•99715
.99711
•99708
99704
.99399
99391
.99382
99374
99365
99356
99347
•99339
99330
.08x68
.08226
.08283
.08340
.08398
.08455
.08513
.08570
.08627
.66881
.66884
.66887
.66890
.66893
.66897
.66900
.66903
.66906
.33529
.33531
33534
33537
.33540
33543
.33546
33549
.33552
.02047
.02061
.02076
.02090
.02105
.02119
02134
.02148
.02162
.0022
.0022
.0023
.0023
.0023
.0024
.0024
.0024
.0025
•49903
.49901
.49900
49898
.49897
.49896
49894
.49893
40891
.0178
•0179
.0180
.0182
.0183
.OI84
.0185
.OI86
.0188
lu.o
4997
99700
.08685
.66909
33555
.02177
.0025
^0890
.0189
J
(49)
TABLE VIII. — (Continued)
li!o
.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
.z
.2
3
4
5
.6
7
.8
9
19.0
.z
.2
3
•4
• 5
.6
7
.8
9
SO.O
4-997
030
064
097
130
163
197
5.230
5.263
5. 296
5.330
5363
5396
5429
5463
5.496
5.529
5563
5596
5629
5.662
5.696
5729
5.762
5796
5829
5.862
5. 896
5.929
5962
5995
6.028
6.062
6.095
6.128
6. 161
6.194
6.228
6.261
6.294
6.328
6.361
6.394
6.427
6.461
6.494
6.527
6.560
6.594
6.627
6.660
C
L
99700
99696
99692
99687
99683
99679
99675
99671
99667
99662
99658
X
L
99654
99650
99645
99641
99637
99632
99628
99623
99619
99614
99610
99605
99601
99596
99591
99587
99582
99577
99572
99568
99563
99558
99553
99548
99543
99538
99533
99528
99523
99518
99513
99508
995Q3
99498
99493
99487
99482
99477
99472
99466
99321
99312
99302
99293
99284
99275
99265
99256
99246
99237
99227
99218
99208
99198
99188
99178
99169
99159
99148
99138
99128
99118
99108
99097
99087
99076
99066
99055
99044
99034
99023
99012
99001
98990
98979
98968
98957
98946
98935
98923
98912
98901
98889
98878
98866
98854
98843
98831
98819
98807
98795
08685
08742
08799
08857
08914
08971
09028
09086
09143
09200
09257
09315
09372
09429
09486
09543
09600
09658
09715
09772
09829
09886
09943
0000
0057
01 14
0171
0228
0285
0342
0399
0456
0513
0570
0627
0684
0741
0798
0855
0912
0968
025
082
139
196
252
309
366
423
479
536
U
L
66909
66913
66916
66919
66923
66926
66929
66933
66936
66940
66943
66947
66950
66954
66957
66961
66964
66968
66972
66975
V
L
66979
66983
66986
66990
66994
66998
67002
67005
67009
67013
67017
67021
67025
67029
67033
67037
67041
67045
67049
67053
67058
67062
67066
67070
67074
67079
67083
67087
67092
67096
67100
33555
33558
3356i
33564
33567
33570
33573
33576
33579
33582
33585
33588
33592
33595
33598
33601
33605
33608
336ii
33615
33618
33621
33625
33628
33632
33635
33639
33642
33646
33649
33653
33657
33660
33664
33667
33671
33675
33679
33682
33686
33690
33694
33697
33701
33705
33709
33713
33717
33721
33725
33729
(50)
o—mL—nD
m
02177
02191
02206
02220
02235
02249
02264
02278
02292
02307
02321
02336
02350
02365
02379
02393
02408
02422
02437
02451
02466
02480
02494
02509
02523
02538
02552
02567
02581
02595
02610
02624
02639
02653
02667
02682
02696
0271 1
02725
02739
02754
02768
02783
02797
0281 1
02826
02840
02854
02869
02883
02898
0025
0025
0026
0026
0026
0027
0027
0027
0028
0028
0028
0029
0029
0029
0030
0030
0030
0031
0031
0032
Z=*mL-nD
0032
0032
0033
0033
0033
0034
0034
0035
0035
0035
0036
0036
0037
0037
0037
0038
0038
0039
0039
0039
0040
0040
0041
0041
0041
0042
0042
0043
0043
0044
0044
m
49890
.49888
.49887
.49885
49884
49882
.49881
.49879
.49878
.49876
.49875
.49873
.49872
.49870
.49868
.49867
.49865
.49863
.49862
.49860
.49859
.49857
49855
.49853
.49852
.49850
49848
49847
.49845
.49843
■49841
.49840
.49838
.49836
.49834
.49833
.49831
.49829
.49827
■49825
.49823
.49821
.4982c
.49818
.49816
.49814
.49812
.49810
.49808
■49806
.49804
.0189
.0190
.0191
.0193
0194
.0195
.0196
.0198
.0199
.0200
.0201
.0202
.0204
.0205
.0206
.0207
.0209
.0210
.0211
.0212
.0213
.0215
.02l6
.0217
.0218
.0220
.022I
.0222
.0223
.0224
.0226
.0227
.0228
.0229
.O23O
.0232
.0233
0234
.0235
.0236
.0238
■ 0239
.0240
.O24I
.0242
.0244
• 0245
.O246
.0247
.0248
.0250
16
IT
1
.2
3
4
■ 5
.6
7
.8
9
.x
.2
.3
.4
5
.6
7
.8
9
.0
.1
.2
.3
•4
.5
.6
.7
.8
9
.z
.2
3
4
■5
.6
7
.8
9
.0
.x
.2
3
.4
5
.6
7
.8
9
.0
18
19
1
TABLE VIII. — (Continued)
A
A
£
X
Y
U
L
V
L
o=*mL—nD
Z=*mL-nD
A
L
L
L
tn
n
m
H
10*0
.1
.2
3
•4
• 5
.6
7
.8
9
U.O
.1
.2
•3
•4
•5
.6
.7
.8
9
11.0
.1
.2
3
•4
• 5
.6
.7
.8
9
18.0
.1
.2
3
•4
5
.6
•7
.8
9
24.0
.1
.2
3
•4
5
.6
7
.8
•9
6.66b
99466
98795
11536
.67100
.33729
.02898
.0044
.49804
.0250
io!o
.1
.2
3
4
5
.6
7
.8
•9
11.0
.1
.2
•3
4
5
.6
7
.8
9
11.0
.1
.2
3
.4
■ 5
.6
7
.8
9
18.0
.1
.2
3
4
5
.6
7
.8
9
14.0
.1
.2
3
4
5
.6
7
.8
9
16.0
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
"593
11649
11706
11763
11819
1 1876
11932
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
12102
.67145
.33770
.03041
.0048
.49784
.0262
7.026
7059
7.092
7125
7.158
7192
7.225
7.258
7.291
99406
99400
99395
99389
99383
99378
99372
99366
.99360
98660
98647
98634
98622
98609
98596
98583
98570
98557
12159
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
7324
99354
98544
12667
67193
33813
.03185
.0053
.49763
O273
7.358
7391
7.424
7.458
7491
7524
7-557
7590
7623
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
131 17
13173
.67197
.67202
.67207
.67212
.67217
.67222
.67227
.67232
67237
.33818
.33822
.33827
.33831
.33836
.3*840
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
7657
99295
98409
13230
.67242
33859
.03328
.0058
-49741
.0285
7.690
7723
7.756
7.789
7.822
7.856
7.889
7922
7955
.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
99i8o
99174
■
98255
.98240
.98226
.98211
98197
98182
98167
98153
98138
.13847
.13903
I39S9
.14015
.14071
14127
14183
14239
14295
.67300
.67305
.67310
.67316
.67321
.67327
67332
.67338
67343
339"
.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
.49711
.49709
.49707
.49704
.49702
.49699
.49697
.0298
0299
.0300
.0302
.0303
.0304
.0305
.0306
.0307
18.0 |
8.321
99167
98123
14350
.67349
.33956
.03614
.0068
.49695
.0309
(51)
TABLE VIII. — (Continued)
A
A
C
L
X
L
Y
L
U
L
V
L
o—mL—nD
Z~mL—*D
A
m
11
m
11
18*0
.1
.a
3
4
5
6
7
.8
9
16.0
.z
.2
•3
4
.5
.6
7
.8
9
17.0
.z
.2
3
4
• S
.6
7
.8
9
18.0
.1
.a
.3
.4
.5
.6
7
.8
.9
19.0
.1
.3
•3
• 4
• S
.6
• 7
.8
9
80.0
•
8.321
99167
.98133
.14350
67349
.33956
.03614
.0068
49695
.0309
S5.0
.1
.3)
.3
•4
•s
.6
.7
.8
9
16.0
.1
.a
•3
•4
•5
.6
•7
.8
9
17.0
.1
.a
•3
•4
5
.6
•7
.8
9
•8.0
.z
.a
•3
•4
5
.6
•7
.8
9
18.0
.1
.a
•3
••4
•5
.6
•7
.8
9
80.0
8.354
8.387
8.430
8.453
8.486
8.519
8.553
8.586
8.619
.99160
.99154
99147
.99140
99X33
99137
.99130
.99"3
99106
.98x08
98093
.98078
.98063
.98048
.98033
.98017
.98003
97987
.14406
.14463
.14518
.14574
.14639
.14685
.14741
.14797
.14853
.67354
.67360
.67365
.67371
.67377
67383
.67388
67394
.67400
.33961
.33966
33971
33976
.33983
.33987
.33993
33997
.34003
.03628
03643
.03657
.03671
.03685
.03700
.03714
.03728
.03742
.0069
.0069
.0070
.0071
.0071
.0073
.0073
0073
0073
49693
.49690
.49687
.49685
.49683
.49680
.49677
.49675
.49673
.0310
.0311
.0312
.0313
.0314
.0315
.0317
.03x8
.0319
8.653
9Q099
•97971
.14908
.67405
.34008
.03757
.0074
.49670
.0320
8.685
8.719
8.753
8.785
8.818
8.851
8.884
8.917
8.951
.99093
99085
99078
.99071
99064
99057
99050
99043
99036
97956
.97940
97935
97909
.97893
.97878
.97863
.97846
97830
.14964
.15019
.15075
.15131
.15186
.15343
.15397
.15353
.15408
.67411
.67417
.67433
.67439
.67435
67441
67447
.67453
.67458
.34013
.34018
.34024
.34039
.34035
.34040
.34045
.34051
.34056
.03771
.03785
.03800
.03814
.03838
.03843
.03857
.03871
.03885
.0074
.0075
.0076
.0076
.0077
.0077
.0078
.0078
.0079
.49667
.49665
49663
.49660
.49657
.49654
.49653
.49049
.49647
.0331
.0333
.0333
.0335
.0336
.0337
.0338
.0339
.0330
8.984
.99029
.97814
.15464
.67465
34062
.03899
.0080
.49644
.0331
9.017
9050
9.083
9.1x6
9.149
9.183
9.316
9.349
9.383
.99033
.99014
.99007
.99000
98993
.98985
.98978
98971
98963
.97798
97783
.97766
•97749
97733
.97717
•97700
.97684
.97667
.15519
.15575
.15630
.15686
.15741
.15796
.15853
.15907
.15963
.67471
.67477
.67483
.67489
.67495
.67501
67507
.67514
.67530
.34067
.34073
.34079
.34084
.34090
.34095
.34101
34107
.34H3
.03913
.03939
.03942
.03956
.03970
.03985
•03999
.04013
.04027
.0080
.0081
.0081
.0083
.0083
.0083
.0084
.0084
.0085
.49641
.49639
.49636
.49633
.49631
.49639
.49635
.49633
.49630
.0333
.0334
.0335
.0336
.0337
.0338
.0339
.0340
.0343
9315
.98956
97651
.16018
.67536
•341x8
.04041
.0085
.49617
.0343
9348
9381
9415
9448
9.481
9514
9547
9S8o
9613
.98948
98941
.98933
.98936
.98918
.98911
.98903
.98895
.98888
.97634
.97617
.97601
.97584
.97567
•97550
.97533
.97516
•97499
.16073
.16138
.16183
.16339
.16394
.16349
.16404
.16459
.16514
.67533
.67539
.67545
.67551
.67558
.67564
.67571
.07577
.67584
.34134
.34130
.34136
.34141
.34147
.34153
.34159
.34165
.34171
.04056
.04070
.04084
.04098
.04113
.04x27
.04141
.04155
.04169
.0086
.0087
.0067
.0088
.0088
.0089
.0090
.0090
.0091
.49615
.496ia
.49609
.49606
.49604
.49601
.49598
.49595
49593
.0344
.0345
.0346
.0347
.0348
.0349
.0351
.0353
.0353
9.647
.98880
.97483
.16569
.67590
.34177
.04184
.0093
.49590
.0354
9.680
9713
9.746
9779
9.81a
9845
9878
99"
9945
.98873
.98865
.98857
.98849
.98841
.98833
.98836
.98818
.98810
.97465
•97447
.97430
.97413
•97395
.97378
.97360
•97343
•97335
.16634
.16679
.16734
.16789
.16844
.16899
.16954
.17009
.17064
.67597
.67603
.67610
.67616
.67633
.67630
.67636
.67643
.67650
.34183
.34189
.34195
.34301
.34307
.34313
•34319
.34335
.34232
.04198
.04313
.04336
.04340
.04354
.04369
.04383
.04397
.04311
.0093
.0093
.0093
.0094
.0095
.0095
.0096
.0097
.0097
.49587
.49584
.49581
.49578
.49575
.49573
.49570
.49567
.49504
.0355
.0356
.0357
.0358
.0359
.0361
.036a
.0363
.0364
9.978
.98803
.97307
.17119
.67657
.34238
.04335
.0098
.49561
.0365
(52)
TABLE VU1. ~ (Continued)
r
TABLE VIII. — (Continued)
35.0
.1
.2
3
4
.5
.6
.7
.8
•9
36.0
.1
.2
3
4
•5
.6
•7
.8
9
87.0
.z
.a
•3
4
5
.6
7
.8
9
88.0
.z
.2
•3
.4
5
.6
7
.8
.8
89.0
.1
.2
3
•4
5
.6
7
.8
•9
40.0
.631
.664
698
731
764
797
.830
863
.896
.929
.961
994
2.027
2.061
2.094
2.127
a. 159
2.192
2.226
2.259
2.292
2.324
2.358
2.391
2.424
2.456
2.489
2.523
2.556
2.588
2.621
2.654
2.687
2.720
2.753
2.786
2.819
2.852
2.885
2.918
2.951
2.984
13.017
3050
3083
116
149
182
215
247
3.281
C
L
98371
98362
98353
98344
98334
98325
98315
98306
98297
98287
98278
98268
98259
98249
98239
98230
98220
98210
98201
98191
98181
98171
98162
98152
98142
98132
98122
98112
98102
98092
98082
98072
98062
98052
98042
98032
98022
98011
98001
97991
9798i
97970
97960
97950
97939
97929
97919
97908
97898
97887
97877
X
L
96351
96331
96310
96290
96269
96248
96227
96207
96186
96165
96144
96123
96102
96080
96059
96038
96017
95995
95974
95952
95931
95909
95887
95866
95844
95822
95800
95778
95756
95734
95712
95690
95668
95646
95623
95601
95578
95556
95533
955H
95488
95466
95443
95420
95397
95374
95351
95328
95305
95282
95259
L
19833
19887
19940
19994
20047
20101
20155
20208
20262
20315
20368
20422
20475
20529
20582
20635
20689
20742
20795
20848
20901
20955
21008
21061
21114
21167
21220
21273
21326
21379
21432
21485
21537
21590
21643
21696
21749
21801
21854
21907
21959
22012
22065
22117
22170
22222
22275
22327
22379
22432
22484
U
L
68027
68035
68043
68051
68059
68068
68076
68084
68092
68101
68109
68117
68126
68134
68142
68151
68159
68168
68176
68185
68194
68202
6821 1
68219
68228
68237
68246
68254
68263
68272
68281
68290
68299
68307
68316
68325
68334
68343
68353
68362
68371
68380
68389
68398
68408
68417
68426
68435
68445
68454
68464
V
L
34578
34585
34593
34600
346o8
34615
34623
34630
34638
34645
34653
34661
34668
34676
34684
34692
34699
34707
34715
34723
34731
34739
34747
34754
34762
34770
34779
34787
34795
34803
34811
34819
34827
34835
34844
34852
34860
34869
34877
34885
34894
34902
349"
34919
34928
34936
34945
34953
34962
3497Q
34979
(54)
o^mL—uD
in
05030
05044
05058
05072
05086
05100
05115
05129
05143
05157
05171
05185
05199
05213
05227
05241
05255
05269
05282
05296
05310
05324
05338
05352
05366
05380
05394
05408
05422
05436
05450
05464
05478
05492
05506
05520
05534
05548
05561
05575
05589
05603
05617
05631
05645
05659
05673
05687
05701
05714
05728
0132
0133
0133
0134
0135
0136
0136
0137
0138
0139
0139
0140
0141
0142
0142
0143
0144
0145
0145
0146
0147
0148
0149
0149
0150
0151
0152
0152
0153
0154
0155
0156
0156
0157
0158
0159
0159
0160
0161
0162
0163
0163
0164
0165
0166
0167
0168
0168
0169
0170
0171
Z=mL—nD
m
49404
49400
49397
49393
49390
49387
49383
4938o
49376
49373
49369
49366
49362
49359
49355
49352
49348
49345
49341
49338
49334
49330
49327
49323
49320
49316
49312
49309
49305
49302
49298
49294
49291
49287
49283
49279
49276
49272
49268
49265
49261
49257
49253
49250
49246
49242
49238
49234
49231
49227
49223
.0419
.0420
.0421
.0422
0423
.0424
.0425
.0426
.0427
.0428
.0429
.0430
.0431
0432
.0433
.0434
.0435
.0436
.0437
•0438
.0439
86
87
.0440
0441
.0442
.0443
.0444
.0445
.0446
.0447
.0448
.0449
.0450
.0451
.0452
.0453
.0454
.0455
0456
.0457
.0458
.0459
.0460
.0461
.0462
0463
.0464
.0465
.0466
.0467
■0468
.0469
88 JO
.z
.2
•3
•4
5
.6
7
.8
9
.z
.2
3
4
5
.6
7
.8
9
.z
.2
3
■4
5
.6
7
.8
9
.x
.2
3
4
>5
.6
7
.8
9
.x
.2
•3
■4
■5
.6
7
.8
9
88
88
40
TABLE IX. — Spiral Functions for Change of o.3°pebioo Feet
Suitable for Speeds of 104 Miles an Hour or Less and
Curves of 1° or Lbss
L
D
*
-
2
-
c
X
¥
V
V
30
40
SO
60
TO
to
90
100
30
40
jo
60
70
So
90
WO
JO
50
60
S3
90
soo
30
50
60
So
90
400
30
50
60
80
90
MM
06
08
16
16
38
13
36
38
46
48
S3
M
S5
58
60
6.
66
76
78
9°
'
98
009
036
0«
064
169
100
MS
3 S 6
m
361
441
484
519
576
fas
719
784
S4I
9«
061
089
156
SJS
396
369
600
681
764
849
936
025
lit
ao»
334
005
008
016
040
048
056
06s
085
096
108
133
161
176
19a
308
243
l6l
300
3»
363
38S
408
433
456
481
S60
588
618
64s
675
705
736
76S
Soo
833
65
8s
9°
95
105
115
135
150
160
■65
185
189
194
199
aos
jit
3U
34!
99
99
99
99
99
99
99
98
98
98
03
OS
°6
oS
15
18
16
39
33
35
»
48
62
68
Bo
86
93
oS
16
61
60.00
90.00
50. 00
80.00
90.00
30.00
80.00
3999
39 99
49.99
69-99
79.99
89-99
99.99
409-98
19.98
39.98
39.9s
49. 9»
69^97
79 97
89.96
49996
40
SO
60
TO
So
90
SO
60
80
90
50
79
89
99
309
19
39
39
49
SO
69
84
99
409
'9
39
39
49
69
79
89
499
99
99
99
99
99
99
98
98
98
98
97
97
96
96
96
95
94
94
92
91
9'
!l6
.38
.34
■ 54
.63
.80
■ 91
1.38
1. 91
3.09
3.39
2.49
3.19
3.45
4-63
4.96
5.3°
5.66
6.04
6.43
6. 84
6.67
36.67
46.67
66.67
73-33
80.00
86 67
93 33
06.67
36.67
46.67
66 67
73-33
80.00
86.67
9333
06. 6j
13.33
36.67
46.67
60.01
266.68
80 OJ
86.68
93 35
06.69
13-36
3.3.1
6.67
16.67
13.33
26.67
30.00
33.33
36.67
4333
46.67
50. 00
53.33
5«.67
60.00
63.33
66.67
73-33
76.67
So. 00
83.33
86. 67
93 34
96.67
106,67
116.67
123.34
136.67
136.68
146.68
153.35
156.69
163.36
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
C
X
F.
U
V
zo
•
0.03
•
O.OOI
O.OOI
5.00
0.00
10.00
10.00
0.00
6.67
3.33
20
.06
.006
.002
10.00
00.
20.00
20.00
.00
X333
6.67
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
.15
.037
.013
25.00
.00
50.00
50.00
.01
3333
16.67
60
.18
.054
.018
30.00
.00
60.00
60.00
.02
40.00
20.00
70
.21
073
.025
3500
.01
70.00
70.00
.03
46.67
33.33
80
24
.096
.032
40.00
.01
80.00
80.00
04
53.33
26.67
90
100
zo
27
.121
.041
45-00
.02
90.00
90.00
.06
60.00
30.00
.30
.150
.050
50.00
.02
100.00
100.00
09
66.67
33.33
33
.181
.061
55- 00
03
110.00
110.00
.12
7333
3667
20
.36
.216
.072
60.00
• 04
120.00
120.00
15
80.00
40.00
30
39
.253
.08s
65.OO
• 05
130.00
130.00
19
86.67
43.33
40
.43
.294
.098
70.00
.06
140.00
140.00
24
9333
4667
50
45
.337
.113
75.00
07
150.00
150.00
29
100.00
50.00
60
48
.384
.128
80.00
09
160.00
160.00
.36
106.67
53.33
70
•5Z
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
120.00
60.00
90
800
10
•57
•541
.181
95.00
15
190.00
190.00
.60
126.67
63.33
.60
.600
.200
100.00
17
200.00
200.00
.70
13333
66.67
.63
.661
.221
105.00
.20
210.00
210.00
.81
140.00
70.00
20
.66
.726
.242
Iio.oo
23
220.00
220.00
•93
I46.67
7333
30
69
.793
.265
IZ5.00
27
230.00
230.00
1.06
I5333
76.67
40
• 72
.864
.288
120.00
• 30
240.00
240.00
1. 21
160.OO
80.00
50
75
937
.313
125.00
.34
250.00
250.00
1.36
166.67
83.34
60
.78
1. 014
.338
I3O.OO
.38
260.00
25999
Z.53
17334
86.67
70
.81
1.093
.365
135.00
43
26999
26999
1.72
180.OO
90.00
80
.84
1.176
.393
140.00
• 48
27999
27999
1.92
I86.67
9334
90
800
zo
87
1. 261
• 421
145.00
.53
28999
28999
2.13
19334
9667
90
1.350
• 450
150.00
• 59
29999
29998
2.36
200:01
100.01
93
1. 441
.48I
155.00
•6S
30999
30998
2.60
206.67
IQ3.34
20
96
1.536
.512
160.00
71
3X999
31998
2.86
21334
106.67
30
99
Z.633
• 545
165.00
.79
32999
32997
3.14
220.01
IZO.OI
40
1.02
1.734
.578
170.00
.86
339 99
339 97
343
226.68
113.34
50
1. os
1.837
.613
175.00
93
34999
34996
3.74
23335
116.68
60
z.08
z.944
.648
180.00
1.02
35998
35996
4.07
240.02
120.01
70
1. 11
2.053
.685
184.99
1. 11
36998
36995
4.42
246.68
12335
80
1.14
2.166
.722
189.99
1.20
379 98
37994
479
25335
126.68
90
400
zo
117
2.281
.761
194.98
1.29
389 97
38994
518
260.02
I3O.O2
1.20
2.400
.800
199.98
1.40
399-97
39993
5. 58
266.69
Z33.36
1.23
2.521
.841
204.98
1. So
40996
40992
6.01
27336
136.69
20
1.26
2.646
.882
209.98
1.62
41996
4I99I
6.46
280.03
I4O.O3
30
Z.29
2.773
.9^5
214.98
1.74
42996
42990
6.94
286.70
I4337
40
13a
a. 904
•968
219.98
1.86
43995
439.89
743
29337
Z46.70
50
1. 35
3037
1. 013
224.98
1.98
44995
44987
795
300.04
ISO.04
60
1.38
3.174
Z.058
229.97
2.13
45994
459.86
8.49
306.72
I5338
70
Z.41
3.313
1. 105
23497
2.27
469.93
46984
9.06
31339
156.72
80
1.44
3456
z.152
23996
2.41
47992
47983
965
320.06
160.06
90
SOO
147
3.601
1. 201
24496
2.57
48992
489.81
10.27
326.74
163. 40 1
1. So
3.750
1.250
24996
2.72
49990
49979
10.91
33341
166.74 1
(56)
^
TABLE XI. — Spiral Functions for a Change of 0.4 ° per zoo
Feet. Suitable for Speeds of 83 Miles an Hour or
Less, or Curves of 2.0 or Less
L
D
A
A
Z
C
X
Y
U
V
zo
30
30.
•
0.04
.08
.12
•
0.003
.008
.018
•
0.001
.003
.006
500
10.00
15.00
0.00
.00
.00
10.00
20.00
30.00
10.00
20.00
30.00
0.00
.00
.00
6.67
13.33
30.00
3 33
6.67
10.00
40
50
60
.16
.20
•24
.033
.050
.073
.011
.016
.024
30.00
35.00
30.00
.00
.00
.01
40.00
50.00
60.00
40.00
50.00
60.00
.01
.02
.03
36.67
3333
40.00
13.33
16.67
20.00
To
80
90
100
zo
30
30
.38
.32
.36
.098
.138
.163
.033
.043
.054
35.00
40.00
45.00
.01
.01
.02
70.00
80.00
90.00
70.00
80.00
90.00
.04
.06
.08
46.67
53-33
60.OO
2333
26.67
30.00
.40
.300
.067
50.00
.03
100.00
100.00
.12
66.67
33.33
44
.48
•5*
.343
.388
.338
.081
.096
.113
55.O0
60.OO
65.00
.04
.05
.06
110.00
120.00
130.00
110.00
120.00
130.00
.16
.20
.26
7333
80.00
86.67
36.67
40.00
43.33
40
50
60
.56
.60
.64
.392
.450
.512
.131
.150
.171
70.O0
75.00
80.OO
.08
.10
.12
140.00
150.00
160.00
140.00
150.00
160.00
.32
39
.48
9333
100.00
106.67
46.67
50.O0
5333
To
80
90
SOO
zo
30
30
.68
.72
.76
•578
.648
.723
.193
.316
.341
85.OO
90.O0
9500
.14
.17
.20
170.00
180.00
190.00
170.00
18Q.00
190.00
.57
.68
.80
113.33
120.00
I26.67
56.67
60.OO
6333
.80
.800
.367
100.00
.23
200.00
200.00
93
I33.33
66.67
.84
.88
92
.883
.968
1.058
.294
.323
.353
105.00
110.00
1 15. 00
.27
.31
.35
210.00
220.00
230.00
210.00
21999
229.99
1.08
1.24
1. 41
140.00
I46.67
15334
70.00
7333
76.67
40
SO
60
.96
1. 00
z.04
1.Z53
1.350
I.352
.384
.417
.451
130.00
135.00
130.00
.40
.46
.51
240.00
250.00
25999
23999
249 99
25998
1. 61
I.82
2.04
160.00
166.67
17334
80.00
8334
86.67
To
80
90
800
zo
30
30
1.08
1.13
1. 16
I.458
1.568
1.683
.486
.523
.561
135.00
140.00
144.99
.57
.64
.71
26999
27999
289.99
269.98
27998
289.97
2.29
2.55
2.84
180.01
186.67
19334
90.01
9334
96.67
1.30
1.800
.600
149.99
.79
299-99
29997
314
200.01
100.01
1.34
1.38
1.33
1.933
3.048
3.Z78
.641
.683
.726
154.99
159.99
164.99
.86
.95
1.05
30998
31998
32998
30997
31996
32995
347
3.81
4.18
206.68
•213.35
220.02
103.34
106.68
110.03
40
50
60
1.36
1.40
1.44
3.313
2.450
2.592
.771
.817
.864
169.99
174.98
179-98
1. 14
1.25
1.36
33998
34997
35997
339 95
34994
35993
457
499
543
226.68
23336
24O.O3
113-35
XI6.69
130.03
To
80
90
400
zo
30
30
I.48
I. S3
1.56
2.738
3.888
3.043
.913
.963
1.0x4
184.98
189.98
194.98
1.48
1.6b
1.72
36996
379 96
389.95
36991
37990
38989
5.89
6.38
6.90
246.69
253.37
260.04
13336
136.70
130.03
1.60
3.300
1.067
199.97
1.86
39994
39988
7-44
266.71
13337
I.64
1.68
1.72
3.362
3.528
3.698
1. 121
1. 176
1.233
204.97
209.97
314.96
2.00
2.15
2.31
409.94
41993
429.92
409.86
41984
429.82
8.02
8.61
9.25
27338
280.06
286.73
136.71
140.05
I43.39
40
50
60
1.76
1.80
1.84
3.872
4.050
4.232
I.29I
1.350
1.4"
319.96
224.95
229.95
2.48
2.65
2.83
43991
449-90
45989
43980
44978
45975
990
10.60
11.32
29341
300.08
306.76
146.73
150.07
I534I
TO
80
9o
•00
1.88
1.92
1.96
4.418
4.608
4.802
1.473
1.533
1. 6oi
234.95
239.94
244.93
3.02
3.22
3.42
469.88
47987
48985
469.72
479 69
489.66
12.07
12.86
1368
31343
320.11
326.79
156.75
160.IO
163.44 1
2.00
1 5.000
1.667
249 93
1 3.63
499.84
499.62
1454
33347
166.79 1
(57)
TABLE XII. — Spiral Functions for a Change of 0.5 per 100
Feet. Suitable for Speeds of 77 Miles an Hour or
Less, or Curves of 2.0 or Less
L
D
A
A
Z
C
X
Y
U
V
zo
0.05
0.002
O.OOI
5.00
0.00
10.00
10.00
0.00
6.67
3.33
20
.zo
.0X0
.003
10.00
.00
20.00
20.00
.00
13.33
6.67
30
.15
.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
50
.25
.062
.021
25.00
.00
50.00
50.00
.02
33.33
16.67
60
.30
.090
.030
30.00
.ox
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
.160
• 053
40.00
.02
80.00
80.00
.07
53.33
26.67
90
100
zo
.45
.202
.067
4500
03
90.00
90.00
.11
60.00
3000
• SO
.250
.083
50.00
.04
100.00
100.00
15
66.67
33 33
.55
.302
.ior
55.00
.05
110.00
110.00
19
7333
36-67
20
.60
.360
.120
60.00
.06
120.00
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
.zo
140.00
140.00
.40
9333
4«.67
So
75
.562
.187
75.00
.12
150.00
150.00
49
zoo. 00
SO.OO
60
.80
.640
.213
80.00
.15
160.00
160.00
.60
Z06.67
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
j 80. 00
180.00
85
120.00
60.00
90
SOO
zo
95
.002
.301
95.00
25
190.00
189.99
1. 00
126.67
63.33
1. 00
1. 000
.333
100.00
29
200.00
19999
1. 16
I33.34
66.67
.05
1. 102
.367
105.00
• 34
210.00
209.99
135
X4O.OO;
70.00
20
.10
1. 210
.403
110.00
39
220.00
21999
1.55
I46.67
7334
30
15
X.322
441
115.00
•44
230.00
229.99
177
I5334
76.67
40
.20
1.440
.480
120.00
.50
239.99
239 99
2.01
160. oz
80.00
SO
.25
X.562
.521
125.00
57
249 99
24998
2.27
166.67
8334
60
•30
I.69O
.563
12999
.64
25999
259.98
2.56
173.34
86.67
70
.35
1.822
.607
13499
72
269.99
269.97
2.86
180.01
90.OI
80
.40
I.96O
653
13999
.80
27999
27997
3-19
186.68
93.34
90
SOO
10
45
2.102
.701
14499
.88
289.98
289.96
355
193.35
96.68
1.50
2.250
.750
149.99
98
299.98
29996
3 93
200.02
100.01
55
2.402
.801
15499
1.08
309.98
30995
4.33
206.69
10335
20
.60
2.560
853
15998
1. 19
31997
31994
476
213.36
106.69
30
•65
2.722
907
164.98
1. 31
32997
329-93
5.23
220.02
ZZ0.02
40
•70
2.890
963
16998
143
33996
339 92
572
226.70
ZZ3-36
50
• 75
3.062
X.021
17498
1.56
34996
34990
6.23
23337
II6.7O
60
.80
3.240
1.080
17997
1.70
35995
359.88
6.78
240.04
120.03
70
«5
3-422
1. 141
184.97
1.84
36994
36987
737
246.71
Z23.38
80
.90
3- 6l0
1.203
189.97
2.00
37994
379.85
798
25339
126.71
90
400
• 95
3.802
1.267
19496
2.16
38993
389.83
8.63
260.06
130.05
2.00
4.O00
1.333
19996
2.33
399.92
39981
9-30
266.74
133-40
(58)
TABLE XIII. — Spiral Functions for a Change of 0.75 ° per
100 Feet.
…[truncated]