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TRANSITION CURVES.
A FIELD BOOK FOR ENGINEERS,
CONTAINING
RULES AND TABLES FOR LAYING
OUT TRANSITION CUEVES.
BY
WALTER G. FOX,
CIVIL ENGINEEK.
NEW YORK:
D. VAN NOSTRAND COMPANY, PUBLISHERS,
23 MURRAY AND 27 WARREN STREETS.
1893.
COPYRIGHT, 1803.
Do VAN NOSTRAND COMPANY.
PREFACE.
In this work the endeavor has been to
condense the essential facts and principles
constituting the theory of the transition
curve and to demonstrate how the curve
can be conveniently laid out in field prac-
tice.
The same methotl nas'been adopted as is
usually employed in running circular curves
so that the field operations may be more
readily comprehended and performed.
WALTER G. Fox.
NEW YOKK, October 3d, 1893.
CHAPTER I.
INTRODUCTION.
Near the ends of a railway curve the
curvature should be gradually diminished
to enable trains at high speed to deviate
gradually from the tangent and avoid any
shock caused by changing too suddenly
from a straight line to a sharp curve.
A transition curve is generally used to
connect the circular curve with the tangent
as it makes easy transition between them.
Moreover, as the outer rail of a curved
track is elevated above the inner rail, a
transition curve will unite better with the
tangent ; for the super elevation of the
outer rail is proportional to the degree of
curve which is least at the initial point.
CHAPTER II.
A transition curve is a compound curve
with many changes of radius. It has nearly
the form of the cubic parabola.
The curve is laid out on chords with the
transit in the same manner as circular
curves. A uniform chord length of 10
feet has been arbitrarily assumed to facili-
tate calculation, and the degree of curve
changed at the end of every chord. The
deflection angles have been computed and
will be found in the tables.
To compute the deflection angles it was
necessary to have the co-ordinates of every
point of compound curve. The sines and
cosines of the different angles between the
tangent and the chords of the curves mul-
tiplied by 10 will give the required co-or-
dinates.
In Fig. I. T is the beginning of the
curve. The distance measured along the
tangent in the direction of L is the latitude
and in the direction of D at a right angle
ITig. 1..
with the tangent is the departure of the
various points of compound curves.
For example let us refer to Table L,
which begins with a 10' curve and
compounds into a sharper curve at the end
of each chord, the degree of curve chang-
ing to 20', 30', 40', etc. See Fig. 2.
1
ft
Degree of
curve.
Angle of
chord.
Cosine
xlO.
Sine
xlO.
Co-ordinates
Lati-
tude.
Depar-
ture.
10
10'
O 7 30"
10.000000
0.001454
10.000000
0.001454
20
20'
2' 00"
9.999998
0.005818
19.999998
0.007272
30
30'
4' 30"
9.999991
0.013090
29.999989
0.020362
40
40'
8' 00"
9.999973
0.023271
39.999962
0.043633
50
50'
12' 30"
9.999933
0.036361
49.999895
0.079994
Having calculated the co-ordinates of
the different points the deflection angles
were determined by
- = tan. T.
Example: To find the deflection angle
for the first point in Table I.
C. 20'
C. 10
10
Dep., 0.001454 log. 3.1625644
Lat., 10.000000 log. 1.0000000
Ans. Del angle, 0' 30" log. tan. 6.1625644
The difference in the degree of curve
between the different arcs of the transition
curve should be maintained as nearly as
possible between the terminating arc and
the circular curve.
In the field the transit should be placed
at T and the angles deflected from the
tangent. In case there should be an ob-
stacle making a point invisible at T, the
next point should be measured on the lung-
chord. The intermediate point can then
be easily set.
To find the length of the long chord to
any point on a curve. Fig. 3.
Let C be the length of chord, L the
latitude of the point, and A the deflection
angle.
c= L
cos A
Example :
To find the length of the long chord to
the 8th point of the curve in Table III.
11
12
Lat. 79.991 log. 1.9030411
Def. angle, 38' 15" log. cos. 9.9999731
Ans. Long chord, 79.996 log. 1.9030680
CHAPTER III.
EXPLANATION OF TABLES.
The curvature increases more rapidly in
each succeeding table.
Table I. begins with a 10' curve which
is the difference in curvature between the
first arc of each table, the degrees of curve
being 10', 20', 30', etc. The de-
gree of curve of the first arc is equal to-
the difference between all the other arcs in
the same table, so that the sixth arc in
Table I. is a 1 curve and the sixth arc in
Table VI. is a 6 curve.
Column I. gives the length of curve ac-
cording to the central angle consumed
which will be found on the same line in a
different column, and also the other corres-
ponding parts of the tables under their
respective headings.
Column II. has the degree of curve of
each separate arc.
14
Column III. contains the deflection an-
gles. It will be observed that the value
of the angles has been calculated to the
nearest second. Such accuracy is not ex-
pected in practice but was necessary in or-
der to correctly determine the long chords.
Column IV. gives the central angles of
the curves.
Columns V. and VI. have the co-ordi-
nates of every point of compound curve,
By using the co-ordinates in place of the
deflection angles the curve may be laid out
with offsets from the tangent.
Column VII. has the long chords to ev-
ery point on the curves.
Column VIII. contains numbers by which
the tangents of half the intersection angles
of the curves should be multiplied and the
product added to the tangent distances as
a correction.
CHAPTER IV.
PROBLEMS.
Given, a 5 degree curve, to run a transi-
tion curve connecting it with a tangent
having an intersection angle of 20 degrees,
If we select the curve from Table VL
its length will be 40 feet, consisting of 4
arcs, and the central angle will be 1 de-
gree. As this must be repeated at the
other end of the curve, the central angle
should be multiplied by 2 and the sum sub-
tracted from the intersection angle. The
remainder will be the central angle of the
circular curve. To find its length divide
the central angle by the degree of curve.
Example :
Intersection angle, 20
Central angle of trans, curve X 2, 2
5/ 18
Length of circular curve, 360 ft.
16
To find the tangent distance V.
Let M. be the length of circular curve
that the central angle of transition curve
consumes, a d the apex distance of cir-
cular curve and L the latitude of the ter-
minal point of the transition curve.
M,
Example :
Apex. dist. of 5 curve for 20, 202 . 120
Lat. of ter'l point of trans, curve, 39 . 998
242.118
Length of 5 curve to consume 1, 20.000
Ans. V = Tan. distance, 222.118
A correction should be added to the pro-
duct which will not materially change the
result unless the length of curve exceeds
70 feet, or the intersection angle contains
more than 50 degrees.
Rule: Multiply the tangent of half the
intersection angle by the distance in column
8 corresponding to the terminal point of
the curve. '
OF THE
UNIVERSITY
17
Example :
1/2 intersection angle 10
Tan. = .1763 X .086 ~ .015
then Y = - 222.118
Correction .015
Tan. dist. - 222.133
With curves of short radius there is a
slight error in the preceding formula, as M
does not always have exactly its true value.
The error, however, is too small to take
into account in field practice. In the above
example it is only one-thousandth of a
foot. In a ten degree curve the error is
about three-tenths.
The true value of M can be determined
by multiplying the cosine of half the cen-
tral angle of transition curve by the length
of circular curve it has consumed.
Example :
1/2 central angle of trans.
curve = 30' log. cos. 9.9999835
Length of circular curve con-
sumed = 20 feet log. 1.3010300
Ans. M, = 19.999 ft. loff. 1.3010135
18
Having measured the tangent distances,
the transition curves should be first laid
out and then connected by running the cir-
cular curve. See Fig. 4.
Fig.
If there are two circular curves of dif-
ferent radii, their ends will be a short dis-
tance apart, the sharper curve being on the
inner side, which can then be eased off to
unite with the other curve. See Fig. 5.
On railway location it may be desirable
to first lay out the circular curve, neglect-
ing the transition curves until the time of
construction.
19
In this case the N two terminal points of
the transition curves should be located with
their co-ordinates, the transit set over
either point and a foresight taken on the
other and the curve run in.
Compound curves should not be located
in this manner, as a backsight is then re-
quired.
TRANSITION CURVE TABLES.
23
TABLE I.
1 Length of 1
1 Curve.
j Degree of
1 Curve.
I ">
1.1.
ii
w
Co-ordinates
Long
Chord.
| Corr'n for
f Tang.Dist.
Lati-
tude.
Depar
ture.
10
10'
0' 30"
1' 0"
10.000
0.001
10.000
.000
20
20'
I' 11"
3' 0"
20.000
0.007
20.000
.004
30
30'
2' 17"
6' 0"
30.000
0.020
30.000
.007
40
40'
3' 41"
10' 0"
40.000
0.043
40.000
.014
50
50'
5' 30"
15' 0"
50.000
0.080
50.000
.029
60
1 00'
V 33"
21' 0"
60.000
0.132
60.000
.045
70
1 10'
9' 58"
28' 0"
69.999
0.203
69.999
.061
80
1 20'
12' 43"
36' 0"
79.999
0.296
79.999
.086
90
1 30'
15' 49"
45' 0"
89.998
0.414
89.999
.126
100
1 40'
O c 19' 15"
55' 0"
99.997
0.560
99.998
.168
110
1 50'
23' 0"
1 6' 0"
109.995
0.736
109.997
.208
120
2 00'
37' 4"
1 18' 0"
119.993
0.945
119.996
.265
130
2 10'
31' 30"
1 31' 0"
129.990
1.191
129.995
.341
140
2 20'
36' 15"
145' 0"
139.986
1.476
139.994
.417
150
2 30'
41' 19"
2 0' 0"
149.981
1.803
149.992
.494
160
2 40'
46' 44"
2 16' 0"
159.974
2.175
159.989
.593
170
2 50'
52' 30"
2 33' 0"
169.965
2.596
169.985
.717
180
3 00'
58' 35"
2 51' 0"
179.954
3.067
179.980
.843
190
3 10'
1 4' 59"
3 10' 0"
189.940
3.591
189.974
.967
200
3 20'
1 11' 45"
3 30' 0"
199.923
4.173
199. 966
1.119
210
3 30'
1 18' 50"
3 51' 0"
209.903
4.814
209.958
1.302
220
3 40'
1 26' 14",
4 13' 0"
219.878
6.517
219.947
1.487
230
3 50'
1 34' 0"
4 36' 0"
229.848
6.286
229.934
1.6T1
240
4 00'
1 42' 5"
5 0' 0"
239.814
7.123
239.920
1.889
24
TABLE II.
Length of 1
Curve. |
!|
g>3
ft
Deflection
Angle.
Central
Angle.
Co-ordinates
Long
Chord.
Corr'tJ for
Tang.Dist. (
Lati-
tude.
Depar
ture.
10
20'
o i' o"
2' 0"
10.000
0.003
10.000
.002
20
40'
2' 24"
6' 0"
20.000
0.014
20.000
.005
30
1 00'
4' 35"
012' 0"
30.000
0.040
30.000
.014
40
1 20'
7' 28"
020' 0"
40.000
0.087
40.000
.029
50
1 40'
11' 0"
030' 0"
50.000
0.160
50.000
.051
60
2 00'
15' 7"
042' 0"
59.999
0.264
59.999
.081
70
2 20'
19' 59"
056' 0"
69.998
0.407
69.999
.122
80
2 40'
25' 29"
112' 0"
79.996
0.593
79.998
.174
90
3 00'
31' 40"
130' 0"
89.993
0.829
89.996
.240
100
3 20'
38' 30"
150' 0"
99.989
1.120
99.995
.320
110
3 40'
45' 58"
212' 0"
109.983
1.471
109.993
.415
120
4 00'
54' 9"
236' 0"
119.974
1.890
119.989
.529
130
4 20'
1 3' 0"
302' 0"
129.962
2.382
129.983
.662
140
4 40'
1 12' 29"
330' 0"
139.946
2.951
139.977
.813
150
5 00'
1 22' 39"
4 0' 0"
149.924
3.605
149.967
.988
160
5 20'
1 33' 29"
432' 0"
159.897
4.349
159.956
1.185
170
5 40'
1 44' 59"
5 6' 0"
169.861
5.189
169.940
1.407
180
6 00'
1 57' 9"
542' 0"
179.817
6.130
179.921
1.655
190
6 20'
2 9' 58"
620' 0"
189.762
7.178
189.897
1.930
200
6 40'
2 23' 28"
7 0' 0"
199.694
8.339
199.868
2.234
2K 1 -
7 00'
2 37 39"
742' 0"
209.612
9.619
209.832
2.569
220
7 20'
2 52' 28"
826' 0"
219.513
11.022
219.789
2.934
230
7 40'
3 1' 58"
9<>i2/ 0"
229.395
12.555
229.738
3.332
240
8 00'
3 24' 6"
10 0' 0"
239.255
14.222
239.677
3.763
25
TABLE III.
*
S
d
o
Co-ordinates
*
tl
>
"o T!
n
Long
Tft
So
O ?-i
j>
"3 g>
o 5
Lati-
Depar
Chord.
*f
|Sr
ft
A
Q ^
tude.
ture.
as
10
30
1' 30"
3' 0"
10.000
0.004
10.000
.003
20
1 00'
3' 47"
9' 0"
20.000
0.022
20.000
.010
4
30
1 30'
6' 59"
18' 0"
30.000
0.063
30.000
.022
40
2 00'
11' 15"
30' 0"
40.000
0.131
40.000
.044:
50
2 30'
16' 30"
45' 0"
49.999
0.240
49.999
.080
60
3 00'
22' 45"
1 3'0"
59.998
0.397
59.999
.126
70
3 30'
30' 0"
1 24' 0"
69.995
0.611
69.997
.183
80
4 00'
38' 15"
1 48' 0"
79.991
0.890
79.996
.262
90
4 30'
O o 477 29"
2 15' 0"
89.985
1.243
89.993
.366
100
5 00'
57' 44"
2 45' 0"
99.975
1.679
99.989
.487
110
5 30'
1 8' 59"
3 18' 0"
109.962
2.207
109.984
.624
120
6 00'
1 21' 14"
3 54' 0"
119.942
2.835
119.975
.793
130
6 30'
1 34' 30"
4 33' 0"
129.915
3.572
129.964
1.001
140
7 00'
1 48' 44"
5 15' 0"
139.878
4.426
139.948
1.231
150
7 30'
2 3' 59"
6 0' 0"
149.830
5.406
149.927
1.480
160
8 00'
2 20' 14"
6 48' 0"
159.768
6.521
159.901
1.776
170
8 30'
2 37' 28"
7 39' 0"
169.688
7.778
169.866
2.130
180
9 00'
2 55' 42"
8 33' 0"
179.589
9.187
179.824
2.491
190
9 30'
3 14' 57"
9 30' 0"
189.465
10.756
189.770
2.889
200
10 00'
3 35' 11"
10 30' 0"
199.313
12.492
199.704
3.342
210
10 30'
3 56' 25"
11 33' 0"
209.128
14.405
209.623
3.855
220
11 OO/
4 18' 39''
12 39' 0"
218.. 906
16.501
219.527
4.399
230
11 30'
4 41' 52"
13 48' 0"
228.641
18.V89
229.411
4.973
240
12 OO'
5 6' 6"
15 O'O"
238.327
21.276
239.275
5.612
26
TABLE IV.
Length of
Curve.
Degree of
Curve.
Deflection
Angle.
Central
Angle.
Co-ordinates
Long
Chord.
Corr'n for
Tang.Dist.
Lati-
tude.
Depar
ture.
10
40'
2' 0"
4' 0"
10.000
0.006
10.000
.003
20
1 20'
4' 59"
12' 0"
20.000
0.029
20.000
.012
30
2 00'
9' 17"
24' 0"
30.000
0.081
30.000
.029
40
2 40'
14' 57"
40' 0"
39.999
0.174
39.999
.058
50
3 20'
22' 0"
1 0' 0"
49.998
0.320
49.999
.102
60
4 00'
30' 19"
1 24' 0"
59.996
0.529
59.998
.162
70
4 40'
39' 59"
1 52' 0"
69.992
0.814
69.996
.244
FO
5 20'
50' 58"
2 24' 0"
79.985
1.186
79.993
.348
90
6 00'
1 3' 18"
3 0' 0"
89.974
1.657
89.989
.479
100
6 40'
1 16' 59"
3 40' 0"
99.957
2.239
99.982
.639
110
7 20'
1 31' 58"
4 24' 0"
109.932
2.942
109.971
.831
120
8 00'
1 48' 19"
5 12' 0"
119.897
3.779
119.956
1.057
130
8 40'
2 5' 59"
6 04' 0"
129.849
4.761
129.936
1.321
140
9 20'
2 24' 58"
7 O 7 0"
139.784
5.898
139.908
1.625
150
10 00'
2 45' 18"
8 0' 0"
149.698
7.204
149.871
1.972
160
10 40'
3 6' 58"
9 4' 0"
159.588
8.688
159.824
2.365
170
11 20'
3 29' 57"
10 12' 0"
169.446
10.361
169.762
2 805
180
12 00'
3 54' 15"
11 24' 0"
179.269
12.235
179.686
3.296
190
12 40'
4 19' 54"
12 40' 0"
189.050
14.320
189.591
3.841
200
13 20'
4 46' 52"
14 0' 0"
198.780
16.626
199.474
4.439
210
14 00'
5 15' 9"
15 24' 0"
208.453
19.163
209.332
5.094
220
14 40'
5 44' 45"
16 52' 0"
218.059
21.942
219.160
5.809
230
15 20'
6 15' 41"
18 24' 0"
227.589
24.971
228.955
6.585
240 16 00'
6 47' 56"
20 0' 0"
237.033
28.260
238.711
7.422
Y v* Or i *a r. \
UNIVERSITY ]
27
TABLE V.
I Length of
1 Curve.
Degree of
Curve.
I &
"
<P "^
14
5
^
Co-ordinates
Long
Chord.
!!
Corr'n for
Tang.Dist,]
Lati-
tude.
Depar
ture.
10
-60'
2' 30"
5' 0"
10.000
0.007
10.000
.004
20
1 40'
6' 11"
15' 0"
20.000
0.036
20.000
.016
30
2 30'
11' 41"
30' 0"
29.999
0.102
29.999
.037
40
3 20'
18' 44"
50' 0"
39.999
0.218
39.999
.073
50
4 10'
27' 30"
1 15' 0"
49.997
0.400
49.998
.131
60
5 00'
37' 52"
1 45' 0"
59.994
0.661
59.997
.207
70
5 60'
50' 0"
2 20' 0"
69.987
1.018
69.994
.305
80
6 40'
1 3' 44"
3 0' 0"
79.977
1 .483
79.990
.436
90
7 30'
1 19' 10"
3 45' 0"
89.959
2.072
89.982
.606
100
8 20'
1 36' 14"
4 35' 0"
99.933
2.798
99.972
.806
110
9 10'
1 54' 59"
5 30' 0"
109.894
3.677
109.955
1.038
120
10 00'
2 15' 23"
6 30' 0"
119.839
4.722
119.932
1.321
130
10 50'
2 37' 28"
7 35' 0"
129.764
5.948
129.900
1.659
140
11 40'
3 V 13"
8 45' 0"
139.662
7.369
139.856
2.039
150
12 30'
3 26' 37"
10 0' 0"
149.529
8.998
149.799
2.461
160
13 20'
3 53' 41"
11 20' 0"
159.356
10.849
159.725
2.950
170
14 10'
4 22' 24"
12 45' 0"
169.136
12.935
169.630
3.509
180
15 00'
4 52' 46"
14 15' 0"
178.860
15.269
179.510
4.119
190
15 50'
5 24' 48"
15 50' 0"
188.517
17.865
189.361
4.781
200
16 40'
5 58' 29"
17 30' 0"
198.097
20.733
199.179
5.521
210
17 30'
6 33' 49"
19 15' 0"
207.587
23.885
208.95*
6.344
220
18 20'
7 10' $1"
21 5' 0"
216.974
27.332
218.689
7.223
230
19 10'
7 49' 23"
23 0' 0"
226.243
31.085
228.368
8.157
240
20 00'
8 29' 39"
25 0' 0"
235.379
35.153
237.989
9.180
28
TABLE VI.
ii
F
10
<
>
s
Be
p
Deflection
Angle.
1
M
p rt
Co-ordinates
Long
Chord.
Corr'n for
Tang.Dist.
Lati-
tude.
Depar
turs.
1 oo'
3' 0"
G' 0"
10.000
0.008
10.000
.004
20
2 00'
7' 23"
18' 0"
20.000
0.043
20.000
.017
30
3 00'
13' 58"
36' 0"
29.999
0.122
29.999
.043
40
4 00'
22' 26"
1 0' 0"
39.998
0.261
39.999
.086
50
5 00'
33' 0"
1 30' 0"
49.996
0.480
49.998
.153
60
6 00'
43' 30"
2 6' 0"
59.991
0.794
59.996
.244
70
7 00'
59' 58"
2 48' 0"
69.982
1.221
69.992
.366
80
8 00'
1 16' 28"
3 36' 0"
79.966
1.779
79.985
.523
90
9 00'
1 35' 0"
4 30' 0"
89.941
2.486
89.975
.719
100
10 00'
1 55' 28"
5 30' 0"
99.903
3.357
99.959
.958
110
11 00'
2 17' 58"
6 36' 0"
109.848
4.411
109.936
1.245
120
12 00'
2 42' 29"
7 48' 0"
119.769
5.665
119.903
1 584
130
13 00'
3 8' 57"
9 06' 0"
129.660
7.134
129.856
1.978
140
14 00'
3 37' 26"
10 30' 0"
139.514
8.836
139.793
2.431
150
15 00'
4 V 55"
12 00' 0"
149.322
10.787
149.711
2.947
160
16 00'
4 40' 23"
13 36' 0"
159.073
13.003
159.603
3.531
170
17 00'
5 14' 49"
15 18' 0"
168.757
15.498
169.467
4.183
180
18 00'
5 51' 15"
17 06' 0"
178.360
18.288
179.295
4.908
190
19 00'
6 29' 39"
19 0' 0"
187.868
21.386
189.081
5.706
200
20 00'
7 10' 2"
21 0' 0"
197.265
24.806
198.818
6.582
210
21 00'
7 52' 24"
23 6'0"
206.534
28.561
208.499
7.538
220
22 00'
8 36' 42"
25 18' 0"
215.655
32.660
218.114
8.571
230
23 00'
9 22' 58"
27 36' 0"
224.608
37.114
227.654
9.683
240/24 00'
10 11' 10"
30 0' 0"
233.371
41.932
237.108
10.874
TABLE OF TANGENT DISTANCES
FOB
CIRCULAR RAILROAD CURVES.
This table contains the tangent distances
to a 1 degree curve for every two minutes
of intersection angle to 90 degrees.
The tangent distance to any other de-
gree of curve may be determined by divid-
ing the number corresponding to the inter-
section angle by the degree of curve.
. OF THF
UNIVERSITY )
31
t
1
2
3
/
0.00
50.00
100.00
150.07
2
1.67
51.67
101.67
151.74
2
4
3.33
53.33
103 34
153.41
4
6
5.00
55.00
105.01
155.08
6
8
6.67
56.67
106.68
156 . 75
8
10
8.33
58.33
108.35
158.42
10
12
10.00
60.00
110.02
160.09
12
14
11.67
61.67
111.69
161.76
14
16
13.33
63.33
113.36
163.43
16
18
15.00
65.00
115.02
165.09
18
20
16.67
66.67
116.69
166.76
20
22
18.33
68.33
118.36
168.43
22
24
20.00
70.00
120.03
170.10
24
26
21.67
71.67
121.70
171.77
26
28
23.33
73.33
123.37
173.44
28
30
25.00
75.00
125.03
175.10
30
32
26.67
76.67
126.70
176.72
32
34
28.33
78.33
128.37
178.39
34
36
30.00
80.00
130.04
180.06
36
38
31.67
81.67
131.71
181.73
38
40
33.33
83.33
133.38
183.40
40
42
35.00
85.00
135.05
185.07
42
44
36.67
86.6^
136.72
186.74
44
46
38.33
88.33
138.38
188.40
46
48
40.00
90.00
140.05
190.07
48
50
41.67
91.67
141.72
191.74
50
52
43.33
93.33
143.39
193.41
52
54
45.00
95.00
145.06
195.08
54
56
46.67
96.67
146 . 73
196.75
56
58
48.33
98.33
148.40
198.42
58
60
50.00
100.00
150.07
200.09
60
32
/
4
5
6 7
/
200.09
250.17
300.30
350.44
2
201 . 76
251.84
301.97
352.11
2
4
203.43
253.51
303 . 64
353.79
4
6
205.10
255.18
305.31
355.46
6
8
206.77
256 . 85
306.98
357.13
8
10
208.44
258.52
308.65
358.81
10
12
210.11
260.20
310.32
360.48
12
14
211.77
261.86
311.99
362.15
14
16
213.45
263.54
313.66
363.83
16
18
215.11
265.20
315.33
365.50
18
20
216.78
266.87
317.00
367.17
20
22
218.45
268.54
318.67
368.85
22
24
220.12
270.21
320.34
370.52
24
26
221.79
271.88
322.01
372.19
26
28
223.46
273.54
323 . 68
373.86
28
30
225.13
275.21
325.35
375.54
30
32
226.80
276.88
327.02
377.22
32
34
228.47
278.55
328.69
378.89
34
36
230.14
280.23
330.37
380.57
36
38
231.81
281.90
332.04
382.24
38
40
233.48
283.57
333.71
383.92
40
42
235.15
285.24
335.38
385.60
42
44
236.82
286.91
337.05
387.27
44
46
238.48
288.59
338.73
388.95
46
48
240.15
290.26
340.40
390.62
48.
50
241.82
291.93
342.07
392.30
50
52
243.49
293.60
343.74
393.98
52
54
245.16
295.27
345.41
395.65
54
56
246.83
296.95
347.08
397.33
56
58
248.50
298.62
348.76
399.01
58
60
250.17
300.30
350.44
400.70
60
33
t
8
9
10
11
t
~0
400.70
450.95
501.32
551.74
2
402.37
452.63
503.00
553.42
2
4
404.05 454.31
504.68
555.10
4
6
405 . 72
455.98
506.36
556.78
6
8
407.39 457.66
508.04
558.46
8
10
409.06 459.34
509.72
560.14
10
12
410.74 461.02
511.40
561.82
12
14
412. 41 j 462. 70
513.08
563.50
14
16
414.08 464.37
514.76
565.18
16
18
415.75
466.05
516.44
566.86
18
20
417.43
467.73
518.12
568.54
20
22
419.10
469.41
519.80
570.22
22
24
420.77
471.08
521.48
571.90
24
26
422.45
472.76
523.16
573.58
26
28
424.12
474.43
524.85
575.27
28
30
425.79
476.10
526.53
576.95
30
32
427.47
477.78
528.21
578.63
32
34
429.15
479.46
529.89
580.32
34
36
430.82
481.14
531.57
582.00
36
38
432.50
482.83
533.25
583.69
38
40
434.18
484.51
534.93
585.37
40
42
435.86
486.19
536.61
587.05
42
44
437.54
487.87
538.29
588.74
44
46
439.21
489.56
539.97
590.42
46
48
440.89
491.24
541.65
592.11
48
50
442.57
492.92
543.33
593.79
50
52
444.25
494.60
545.01
595.47
52
54
445.93
496.28
546.69
597.16
54
56
447.60
497.96
548.37
598.84
56
58
449.28
499.65
550.06
600.53
58
60
450.95
501.32
551.74
602.22
60
34
/
12
13
14
15
/
602.22
652.87
703.53
754.35
2
603.91
654.56
705.23
756.05
2
4
605.60
656.25
706.92
757.74
4
6
607.28
657.93
708.62
759.44
6
8
608.97
659.62
710.31
761.13
8
10
610.66
661.31
712.01
762.83
10
12
612.35
663.00
713.71
764.53
12
14
614.04
664.69
715.40
766.22
14
16
615.72
666.37
717.10
767.92
16
18
617.41
668.06
718.79
769.61
18
20
619.10
669.75
720.49
771.31
20
22
620.79
671.44
722.20
773.01
22
24
622.48
673.13
723.89
774.70
24
26
624.16
674.81
725.59
776.40
26
28
625.85
676.51
727.28
778.09
28
30
627.55
678.20
728.97
779.79
30
32
629.24
679.89
730.66
781.49
32
34
630-93
681.58
732.35
783.19
34
36
632.61
683.26
734.05
784.89
36
38
634.30
684.95
735.74
786.59
38
40
635.99
686.64
737.43
788.29
40
42
637.68
688.33
739.12
789.99
42
44
639.37
690.02
740.81
791.69
44
46
641. C5
691.70
742.51
793.39
46
48
642.74
693.39
744.20
795.09
48
50
644.43
695.08
745.89
796.79
50
52
646.12
696.77
747.58
798.49
52
54
647.81
698.46
749.27
800.19
54
56
649.49
700.14
750.97
801.89
56
58
651.18
701.83
752.66
803.59
58
60
652.87
703.53
754.35
805.29
60
35
/
16
17
18
19
/
9
805.29
856.35
907.52
958.86
2
806.99
858.05
909.23
960.57
2
4
808.64
859.76
910.94
962.30
4
6
810.39
861.46
912.65
964.00
6
8
812.09
863 . 16
914.36
965.72
8
10
813.79
864.87
916.07
967.43
10
12
815.49
866.57
917.78
969.15
12
14
817.19
868.27
919.49
970.86
14
16
818.89
869.98
921.20
972.58
16
18
820.59
871.68
922.91
974.29
18
20
822.29
873.38
924.63
976.01
20
22
823 . 99
875.09
926.34
977.72
22
24
825.69
876.79
928.05
979.44
24
26
827.39
878.49
929.76
981.15
26
28
829.09
880.20
931.47
982.86
28
30
830.79
881.90
933.18
984.58
30
32
832.49
883.61
934.89
986.30
32
34
834.20
885.32
936.60
988.02
34
36
835.90
887.02
938.32
989.74
36
38
837.61
888.73
940.03
991.46
38
40
839.31
890.44
941 . 74
993.18
40
42
841.01
892.15
943.45
994.90
42
44
842.72
893.86
945.16
996.62
44
46
844.42
895.56
946.88
998.34
46
48
846.13
897.27
948.59
1000.0
48
50
847.83
898.98
950.30
1001.8
50
52
849.53
900.69
952.01
1003.5
52
54
851.24
902.40
953.72
1005.2
54
56
852.94
904.10
955.44
1006.9
56
58
854.65
905.81
957.15
1008.6
58
60
856.35
907.52
958.86
1010.4
60
36
/
20
21
22
23
/
1010.4
1062.0
1113.8
1165.8
~cT
2
1012.1
1063.7
1115.5
1167.5
2
4
1013.8
1065.4
1117.3
1169.2
4
6
1015.5
1067.2
1119.0
1171.0
6
8
1017.2
1068.9
1120.7
1172.7
8
10
1019.0
1070.6
1122.4
1174.4
10
12
1020.7
1072.4
1124.2
1176.2
12
14
1022.4
1074.1
1125.9
1177.9
14
16
1024.1
1075.8
1127.6
1179.7
16
18
1025.8
1077.5
1129.4
1181.4
18
20
1027.6
1079.3
1131.1
1183.1
20
22
1029.3
1081.0
1132.8
1184.9
22
24
1031.0
1082.7
1134.6
1186.6
24
26
1032.7
1084.4
1136.3
1188.4
26
28
1034.4
1086.2
1138.0
1190.1
28
30
1036.1
1087.9
1139.7
1191.8
30
32
1037.9
1089.6
1141.5
1193.6
32
34
1039.6
1091.3
1143.2
1195.3
34
36
1041.3
1093.1
1144.9
1197.1
36
38
1043.0
1094.8
1146.7
1198.8
38
40
1044.8
1096.5
1148.4
1200.5
40
42
1046.5
1098.3
1150.1
1202.3
42
44
1048.2
1100.0
1151.9
1204.0
44
46
1049.9
1101.7
1153.6
1205.8
46
48
1051.7
1103.4
1155.4
1207.5
48
50
1053.4
1105.2
1157.1
1209.2
50
52
1055.1
1106.9
1158.8
1211.0
52
54
1056.8
1108.6
1160.6
1212.7
54
56
1058.6
1110.3
1162.3
1214.5
56
58
1060.3
1112.1
1164.0
1216.2
58
60
1062,OJ1113.8
1165,8
1218.0
60
37
'
24
25
26
27
'
1218.0
1270.3
1322.9
1375.6
2
1219.7
1272.0
1324.6
1377.4
2
4
1221.4
1273.8
1326.4
1379.2
4
6
1223.2
1275.5
1328.1
1380.9
6
8
1224.9
1277.3
1329.9
1382.7
8
10
1226.7
1279.0
1331.6
1384.5
10
12
1228.4
1280.8
1333.4
1386.2
12
14
1230.2
1282.5
1335.2
1388.0
14
16
1231.9
1284.3
1336.9
1389.8
16
18
1233.6
1286.1
1338.7
1391.5
18
20
1235.4
1287.8
1340.4
1393.3
20
22
1237.1
1289.6
1342.2
1395.0
22
24
1238.9
1291.3
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6
8
3317.1
3384.2
3452.0
3520.5
8
10
3319.3
3386.4
3454.3
3522.8
10
12
3321.6
3388.7
3456.6
3525.1
12
14
3323.8
3390.9
3458.8
3527.4
14
16
3326.0
3393.2
3461.1
3529.7
16
18
3328.3
3395.4
3463.4
3532.0
18
20
3330.5
3397.7
3465.7
3534.3
20
22
3332.7
3399.9
3467.9
3536.6
22
24
3334.9
3402.2
3470.2
3538.9
24
26
3337.2
3404.4
3472.5
3541.2
26
28
3339.4
3406.7
3474.7
3543.5
28
30
3341.6
3408.9
3477.0
3545.8
30
32
3343 . 9
3411.2
3479.3
3548.1
32
34
3346.1
3413.5
3481.6
3550.4
34
36
3348.3
3415.7
3483 . 9
3552.7
36
38
3350.6
3418.0
3486.2
3555.0
38
40
3352.8
3420.3
3488.5
3557.3
40
42
3355.0
3422.5
3490.7
3559.6
42
44
3357.3
3424.8
3493.0
3562.0
44
46
3359.5
3427.1
3495.3
3564.3
46
48
3361.8
3429.3
3497.6
3566.6
48
50
3364.0
3431.6
3499.9
3568.9
50
52
3366.2
3433.9
3502.2
3571.2 52
54
3368.5
3436.1
3504.5
3573.5 54
56
3370.7
3438.4
3506.8
3575.8 ! 56
58
3373.0
3440.7
3509.0
3578.1 58
60
3375.2
3442.9
3511.3
3580.4 60
47
/
64
65
66
67
/
3580. 4 | 3650. 4
3721.1
3792.6
2
3582.8
3652.8
3723.4
3795.0
2
4
3585.1
3655.1
3725.8
3797.4
4
6
3587.4
3657.5
3728.2
3799.8
6
8
3589.7
3659.8
3730.6
3802.2
8
10
3592.1
3662.2
3732.9
3804.6
10
12
3594.4
3664.5
3735.3
3807.0
12
14
3596.7
3666.9
3737.7
3809.4
14
16
3599.1
3669.2
3740.1
3811.8
16
18
3601.4
3671.6
3742.4
3814.2
18
20
3603.7
3673.9
3744.8
3816. G
20
22
3606.0
3676.2
3747.2
3819.0
22
24
3608.4
3678.6
3749.6
3821.4
24
26
3610.7
3680.9
3751.9
3823.8
26
28
3613.0
3683.3
3754.3
3826.2
28
30
3615.3
3685.6
3756.7
3828.6
30
32
3617.7
3688.0
3759.1
3831. C
32
34
3620.0
3690.4
3761.5
3833.4
34
36
3622.3
3692.7
3763.9
3835.9
36
38
3624.7
3695.1
3766.3
3838.3
38
40
3627.0
3697.4
3768.7
3840.7
40
42
3629.4
3699.8
3771.0
3843.1
42
44
3631.7
3702.2
3773.4
3845.5
44
46
3634.0
3704.5
3775.8
3847.9
46
48
3636.4
3706.9
3778.2
3850.4
48
50
3638.7
3709.3
3780.6
3852.8
50
52
3641.1
3711.6
3783.0
3855.2
52
54
3643.4
3714.0
3785.4
3857.6
54
56
3645.7
3716.3
3787.8
3860.0
56
58
3648.1
3718.7
3790.2
3862.5
58
60
3650.4
3721.1
3792.6
3864.9
60
48
/
08
69
70
71
/
3864.9
3938.1
4012.1
4087.1
2
3867.3
3940.6
4014.6
4089.7
2
4
3869.7
3943.0
4017.1
4092.2
4
6
3872. %
3945.5
4019.6
4094.7
6
8
3874.6
3947.9
4022.1
4097.2
8
10
3877.0
3950.4
4024.6
4099.8
10
12
3879.5
3952.9
4027.1
4102.3
12
14
3881.9
3955.3
4029.6
4104.8
14
16
3884.3
3957.8
4032.1
4107.3
16
18
3886.8
3960.2
4034.6
4109.8
18
20
3889.2
3962.7
4037.1
4112.4
20
22
3891.6
3965.2
4039.6
4114.9
22
24
3894.1
3967.6
4042.1
4117.4
24
26
3896.5
3970.1
4044.6
4119.9
26
28
3898.9
3972.5
4047.1
4122.4
28
30
3901.4
3975.0
4049.6
4125.0
30
32
3903.8
3977.5
4052.1
4127.5
32
34
3906.3
3980.0
4054.6
4130.0
34
36
3908.7
3982.4
4057.1
4132.6
36
38
3911.2
3984.9
4059.6
4135.1
38
40
3913.6
3987.4
4062.1
4137.7
40
42
3916.1
3989.9
4064.6
4140.2
42
44
3918.5
3992.3
4067.1
4142.7
44
46
3921.0
3994.8
4069.6
4145.3
46
48
3923.4
3997.3
4072.1
4147.8
48
50
3925.9
3999.8
4074.6
4150.4
50
52
3928.3
4002.2
4077.1
4152.9
52
54
3930.8
4004.7
4079.6
4155.4
54
56
3933.2
4007.2
4082.1
4158.0
56
58
3935.7
4009.7
4084.6
4160.5
58
60
3938.1
4012.1
4087.1
4163.1
60
49
72
73
74
75 '
4163.1
4240.0
4317.8
4396.7
4165.6
4242 . 6
4320.5
4399.4
2
4168.2
4245.1
4323.1
4402.1
4
4170.7.
4247.7
4325.7
4404.7
6
4173.3
4250.3
4328.3
4407.4
8
4175.8
4252.9
4330.9
4410.0
10
4178.4
4255.5
4333.6
4412.7
12
4181.0
4258.1
4336.2
4415.3
14
4183.5
4260.7
4338.8
4418.0
16
4186.1
4263.2
4341.4
4420.7
18
4188.6
4265.8
4344.0
4423.3
20
4191.2
4268.4
4346.7
4426.0
22
4193.7
4271.0
4349.3
4428.6
24
4196.3
4273.6
4351.9
4431.3
26
4198.8
4276.2
4354.5
4434.0
28
4201.4
4278.8
4357.1
4436.6
30
4204.0
4281.4
4359.8
4439.3
32
4206.5
4284.0
4362.4
4442.0
34
4209.1
4286.6
4365.1
4444.6
36
4211.7
4289.2
4367.7
4447.3
38
4214.3
4291.8
4370.3
4450.0
40
4216.8
4294.4
4373.0
4452.7
42
4219.4
4297.0
4375.6
4455.3
44
4222.0
4299.6
4378.3
4458.0
46
4224.5
4302.2
4380.9
4460.7
48
4227.1
4304.8
4383.5
4463.4
50
4229.7
4307.4
4386.2
4466.0
52
4232.3
4310.0
4388.8
4468.7
54
4234.8
4312.6
4391.5
4471.4
56
4237.4
4315.2
4394.1
4474.1
58
4240.0
431Y.8
4396.7
4476 . 7
60
50
t
76
77
78
79
/
4476.7
4557.8
4640.0
4723.4
2
4479.4
4560.5
4642.8
4726.2
2
4
4482.1
4563.3
4645 . 6
4729.0
4
6
4484.8
4566.0
4648.3
4731.8
6
8
4487.5
4568.7
4651.1
4734.7
8
10
4490.2
4571.5
4653.9
4737.5
10
12
4492.9
4574.2
4656.7
4740.3
12
14
4495 . 6
4576.9
4659.4
4743.1
14
16
4498.3
4579.7
4662.2
4745.9
16
18
4501.0
4582.4
4665.0
4748.7
18
20
4503.7
4585.1
4667.7
4751.5
20
22
4506.3
4587.9
4670.5
4754.3
22
24
4509.0
4590.6
4673.3
4757.1
24
26
4511.7
4593.3
4676.0
4760.0
26
28
4514.4
4596.0
4678.8
4762.8
28
30
4517.1
4598.8
4681.6
4765.6
30
32
4519.8
4601.5
4684.4
4768.4
32
34
4522.5
4604.3
4687.2
4771.2
34
36
4525.3
4607.0
4689.9
4774.1
36
38
4528.0
4609.8
4692.7
4776.9
38
40
4530.7
4612.5
4695.5
4779.7
40
42
4533.4
4615.3
4698.3
4782.6
42
44
4536.1
4618.0
4701.1
4785.4
44
46
4538.8
4620.8
4703.9
4788.2
46
48
4541.5
4623.5
4706.7
4791.0
48
50
4544.2
4626.3
4709.5
4793.9
50
52
4547.0
4629.0
4712.2
4796.7
52
54
4549 . 7
4631.8
4715.0
4799.5
54
56
4552.4
4634.5
4717.8
4802.4
56
58
4555.1
4637.3
4720.6
4805.2
58
60
4557.8
4640.0
4723.4
4808.0
60
51
'
80
81
82
83 '
4808.0
4893 . 9
4981.0
5069.4
2
4810.9
4896.8
4983.9
5072.4
2
4
4813.7
4899.7
4986.8
5075.4
4
6
4816.6
4902.6
4989.8
5078.4
6
8
4819.4
4905.4
4992.7
5081.4
8
10
4822.3
4908.3
4995.7
5084.4
10
12
4825.1
4911.2
4998.6
5087.3
12
14
4828.0
4914.1
5001.5
5090.3
14
16
4830.8
4917.0
5004.5
5093.3
16
18
4833 . 7
4919.9
5007.4
5096.3
18
20
4836.5
4922.8
5010.3
5099.3
20
22
4839.4
4925.7
5013.3
5102.3
22
24=
4842.2
4928.6
5016.2
5105.2
24
26
48-15.1
4931.5
5019.2
5108.2
26
2S
4847.9
4934.4
5022.1
5111.2
28
30
4850.8
4937.2
5025.0
5114.2
30
32
4853 . 7
4940.2
5028.0
5117.2
32
34
4856.5
4943.1
5031.0
5120.2
34
36
4859.4
4946.0
5033.9
5123.2
36
38
4862.3
4948.9
5036.9
5126.2
38
40
4865.1
4951.8
5039.8
5129.2
40
42
4868.0
4954.7
5042.8
5132.2
42
44
4870.9
4957.6
5045.8
5135.2
44
46
4873.8
4960.6
5048.7
5138.2
46
48
4876.6
4963.5
5051.7
5141.2
48
50
4879.5
4966.4
5054.6
5144.3
50
52
4882.4
4969.3
5057.6
5147.3
52
54
4885.3
4972.2
5060.6
5150.3
54
56
4888.1
4975.1
5063.5
5153.3
56
58
4891.0
49^8.0
5066.5
5156.3
58
60
4893.9
4981.0
5069.4
5159.3
60
/
84
85
86
87
'
5159.3
5250.6
5343.3
5437.5
2
5162.3
5253.6
5346.4
5440.7
2
4
5165.3
5256.7
5349.5
5443 . 9
4
6
5168.4
5259.8
5352.7
5447.1
6
8
5171.4
5262.9
5355.8
5450.3
8
10
5174.4
5266.0
5358.9
5453.4
10
12
51-77.5
5269.0
5362.0
5456.6
12
14
5180.5
5272.1
5365.2
5459.8
14
16
5183.5
5275.2
5368.3
5463.0
16
18
5186.6
5278.3
5371.4
5466.2
18
20
5189.6
5281.4
5374.6
5469.4
20
22
5192.6
5284.4
5377.7
5472.5
22
24
5195.6
5287.5
5380.8
5475 . 7
24
26
5198.7
5290.6
5383.9
5478.9
26
28
5201.7
5293.7
5387.1
5482.1
28 -
30
5204.7
5296.7
5390.2
5485.3
30
32
5207.8
5299.8
5393.4
5488.5
32
34
5210.8
5302.9
5396.5
5491.7
34
36
5213.9
5306.1
5399.7
5494.9
36
38
5216.9
5309.2
5402.8
5498 . 1
38
40
5220.0
5312.3
5406.0
5501.3
40
42
5223.1
5315.4
5409.1
5504.5
42
44
5226.1
5318.5
5412.3
5507.7
44
46
5229.2
5321.6
5415.4
5510.9
46
48
5232.2
5324.7
5418.6
5514.1
48
50
5235.3
5327.8
5421.8
5517.3
50
52
5238.3
5330.9
5424.9
5520.5
52
54
5241.4
5334.0
5428.1
5523.7
54
56
5244.5
5337.1
5431.2
5526.9
56
58
5247.5
5340.2
5434.4
5530.1
58
60
5250.6
5343.3
5437.5
5533.3
60
53
/
88
/
88
/
se
/
89
5533.3
30
5581.9
5630.8
30
5680.2
2
5536.6
32
5585.1
2
5634.1
32
5683.5
4
5539.8
34
5588.4
4
5637.4
34
5686.8
6
5543.1
36
5591.7
6
564C.7
36
5690.2
8
5546.3
38
5594.9
8
5644.0
38
5693.5
10
5549.5
40
5598.2
10
5647.3
40
5696.8
12
5552.8
42
5601.4
12
5650.6
42
5700.1
14[5556.0
44
5604.7
14
5653.9
44
5703.4
165559.2
46
5608.0
16
5657.1
46
5706.8
18
5562.5
48
5611.2
18
5660.4
48
5710.1
20
5565.7
50
5614.5
20
5663.7
50
5713.4
22
5568.9
52
5617.8
22
5667.0
52
5716.7
84
5572.2
54
5621.0
24
5670.3
54
5720.0
2(\
5575.4
56
5624.3
26
5673.6
56
5723.4
2*
5578.6
58
5627.5
28
5676.9
58
5726.7
30
5581.9
60
5630.8
30
5680.2
60
5730.0
TABLE
GIVING BADII OF DEGEEES
OF CURVE.
57
02
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00
8
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CO CO CO CO CO CO CO CO CO CO CO CO CO
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59
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