UC-NRLF
GIFT OF
MICHAEL REESE
THE
RAILROAD SPIRAL.
THE THEORY OF THE
COMPOUND TRANSITION CURVE
REDUCED TO
PRACTICAL FORMUL/E AND RULES FOR
APPLICATION IN FIELD WORK;
COMPLETE TABLES OF DEFLECTIONS AND ORDINATES
FOR FIVE HUNDRED SPIRALS.
KY
WILLIAM H. SEARLES, C.E.,
MEMBER AMERICAN SOCIETY OF CIVIL KNG1NKERS,
AUTHOR "FIELD ENGINEERING."
UNIVERSITY!
NEW YORK :
JOHN WILEY & SONS.
1882.
COPYRIGHT,
1882,
BY JOHN WILEY & SONS.
PRE FACE.
THE object of this work is to reduce the well-known
theory of the cubic parabola or multiform compound
curve, used as a transition curve, to a practical and con-
venient form for ordinary field work.
The applicability of this curve to the purpose in-
tended has been fully demonstrated in theory and prac-
tice by others, but the method of locating the curve on
the ground has been left too much in the mazes of
algebra, or else has been described as a system of off-
sets, or fudging. Where a system of deflection angles
has been given,- the range of spirals furnished has been
much too limited for ge^r&^nictice. In consequence
the great majority of engineers have contented them-
selves with locating circular curves only, leaving to the
trackman the task of adjusting the track, not to the
centres given near the tangent points, but to such an
approximation to the spiral as he could give "by eye."
The method here described is that of transit and
chain, analogous to the method of running circular
curves ; it is quite as simple in practice, and as accu-
rate in result. No offsets need be measured, and the
curve thus staked out is willingly followed by the track-
men because it " looks right," and is right.
The preliminary labor of selecting a proper spiral for
a given case, and of calculating the necessary distances
to locate it at the proper place on the line, is here ex-
plained, and reduced to the simplest method. Many of
iii
IV PREFACE.
the quantities required have been worked out and tabu-
lated once for all, leaving only those values to be found
which are peculiar to the individual case in hand. A
large number of spirals are thus prepared, and their
essential parts are given in Table III.
In section 22 is developed the method of applying
spirals to existing circular curves, without altering the
length of line, or throwing the track off of the road bed,
an important item to roads already completed. Table
V. contains samples of this kind of work arranged in
order, so that, by a simple interpolation, the proper se-
lection can be made in a given case.
The series of spirals given in Table III. are obtained
by a simple variation of the chord-length, while the de-
flections and central angles remain constant. This is
the converse of our series of circular curves, in which
the chord is constantly 100 feet, while the deflections
and central angles take a series of values.
The multiform compound curve has been chosen as
the basis of the system, rather than the cubic parabola,
because, while there is no practical difference in the
two, the former is more in keeping with ordinary field
methods, and is far more convenient for the calculation
and tabulation of values in terms of the chord-unit, or of
measurement along the curve. While the several com-
ponent arcs of the spiral are thus assumed to be circu-
lar, yet the chord-points are points of a true spiral, to
which the track naturally conforms when laid according
to the chord-points given as centres.
The " Railroad Spiral " is in the nature of a sequel to
" Field Engineering ; " the same system of notation is
adopted, and any tables referred to, but not given here,
will be found in that work.
WM. H. SEARLES, C. E.
NEW YORK, July i, 1882.
CONTENTS.
CHAPTER I.
INTRODUCTION.
SECTION PAGE
1. Objections to simple circular curves I
2. Office of the spiral 2
CHAPTER II.
THEORY OF THE SPIRAL.
3. Description of the spiral 3
4. Co-ordinates of the spiral 3
5. Deflection angles from the main tangent . . . . 5
6. Deflection angles from an auxiliary tangent 6
7. The chord-length as a variable g
Construction of Table of Co-ordinates 10
9. Elements of the spiral 10
10. Selection of a spiral 1 1
CHAPTER III.
ELEMENTARY PROBLEMS.
11. To find a long chord SL 13
12. To find the tangents SE and EL 13
13. To find a long chord QL. 14
14. To find the tangents QE' and EL 15
15. To find the tangent-distance T s SV 16
16. To find 7' s approximately 17
17. To find the radius R 1 in terms of T s and spiral 17
V
VI CONTENTS.
SECTION PAGE
18. To find diff. R' in terms of d'iff. Ts 19
19. To find the external distance E s 20
20. To find the radius R' in terms of E s and spiral 21
21. To find diff. R' in terms of diff. x for Es constant 23
CHAPTER IV.
SPECIAL PROBLEMS.
22. Given, a simple curve, to replace it by another with
spirals ; length of line unchanged 25
a. To find the radius R' 26
b. To find the offset h 26
c. To find the distance d A S. 27
d. To find lengths of old and new lines 27
e. To select a suitable spiral 28
f. To find diff. h in terms of diff. R' 29
23. Given, a simple curve, to apply spirals without change of
radius 32
24. Given, a simple curve, to compound it for spirals without
disturbing the middle portion 34
25. Given, a compound curve, to replace it by another, with
spirals ; length of line unchanged 36
26. Given, a compound curve, to apply spirals without change
of radii 40
27. Given, a compound curve, to introduce spirals without dis-
turbing the P. C. C 42
CHAPTER V.
FIELD WORK.
28. To locate a spiral from S to L 45
29. To locate a spiral from L to S 46
30. To interpolate the regular stations 47
31. Choice of method for locating spirals 47
32. To locate a spiral by ordinates 48
33. Use of spirals on location work 48
34. Description of line with spirals 48
35. Elevation of outer rail on spirals 49
36. Monuments 49
37. Keeping field-notes 49
CONTENTS.
TABLES.
PAGE
I. Elements .of the spiral of chord-length TOO 50
II. Deflection angles for the spiral 52
II. Co-ordinates and Degree of curve of the spiral 58
IV. Functions of the spiral angle s 77
V. Selected spirals for unchanged length of line. 22 78
UNIVERSITY;
A
^\^ Sr it C)Jd' ^*t V*w '/
THE RAILROAD SPIRAL.
CHAPTER I.
INTRODUCTION.
I. ON a straight line a railway track should be level
transversely ; on a curve the outer rail should be raised
an amount proportional to the degree of curve. At the
tangent point of a circular curve both of these condi-
tions cannot be realized, and some compromise is usually
adopted, by which the rail is gradually elevated for
some distance on the tangent, so as to gain at the tan-
gent point either the full elevation required for the
curve, or else three-quarters or a half of it, as the case
may be. The consequence of this, and of the abrupt
change of direction at the point of curve, is to give the
car a sudden shock and unsteadiness of motion, as it
passes from the tangent to the curve.
The railroad spiral obviates these difficulties entirely,
since it not only blends insensibly with the tangent on
the one side, and with the circle on the other, but also
affords sufficient space between the two for the proper
elevation of the outer rail. Moreover, since the curva-
ture of the spiral increases regularly from the tangent
to the circle, and the elevation of the outer rail does
the same, the one is everywhere exactly proportional to
the other, as it should be. The use of the spiral allows
i
2 THE RAILROAD SPIRAL.
the track to remain level transversely for the whole
length of the tangent, and yet to fie fully inclined for
the whole length of the circle, since the entire change
in inclination takes place on the spiral.
2. The office of the spiral is not to supersede the cir-
cular curve, but to afford an easy and gradual transition
from tangent to curve, or vice versa, in regard both to
alignment and to the elevation of the outer rail. A
spiral should not be so short as to cause too abrupt a
rise in the outer rail, nor yet so long as to render the
rise almost imperceptible, and therefore difficult of ac-
tual adjustment. Within these limits a spiral may be
of any length suited to the requirements of the curve
or the conditions of the locality. To suit every case in
practice an extensive list of spirals is required from
which to select.
THEORY OF THE SPIRAL.
CHAPTER II.
THEORY OF THE SPIRAL.
3. THE Railroad Spiral is a compound curve closely
resembling the cubic parabola ; it is very flat near the
tangent, but rapidly gains any desired degree of curva-
ture.
The spiral is constructed upon a series of chords of
equal length, and the curve is compounded at the end
of each chord. The chords subtend circular arcs, and
the degree of curve of the first arc is made the com-
mon difference for the degrees of curve of the suc-
ceeding arcs. Thus, if the degree of curve of the first
arc be o ic/j that of the second will be o 20', of the
third, o 30', &c.
The spiral is assumed to leave the tangent at the be-
ginning of the first chord, at a tangent point known as
the Point of Spiral, and designated by the initials P. S. r
or on the diagrams by the letter S.
4. To determine the co-ordinates of the sev-
eral chord extremities, let the point S be taken as
the origin of co-ordinates, the tangent through S as the
axis of Y, and a perpendicular through S as the axis of
X. Then x, y, will represent the co-ordinates of any
point of compound curvature in the spiral, x being the
perpendicular offset from the point to the tangent, and
y the distance on the tangent from the origin to that
offset.
For the purpose of calculation let us assume 100 feet
as the chord-length, and o 10' as the degree of curve of
THE RAILROAD SPIRAL.
the first arc of a given spiral. Then, since the degree
of curve is an angle at the centre of a circle subtended
by a chord of 100 feet, the central angle of the first
chord is 10', of the second 20', of the third 30', &c., and
the angles which the chords make with the tangent are :
For ist chord, Y x 10' = 5'
" 2d - u 10' + 1 X 20 = 20'
" 3 d " 10' + 20' 4- i x 30' = 45'
" 4th " 10' + 20 + 30 + i x 40 = 80'
&c., &c., &c.,
or in general the inclination of any chord to the tan-
gent at S is equal to half the central angle subtended
by that chord added to the central angles of all the
preceding chords. If now we consider the tangent as
a meridian, the latitude of a chord will be the product of
the chord by the cosine of its inclination, and its depart-
ure will be the product of the chord by the sine of its
inclination to the tangent. A summation of the several
latitudes for a series of chords will give us the required
values of _>', and a summation of the several departures
will give us the required values of x. By the aid of a
table of sines and cosines, we may therefore readily pre-
pare the following statement :
Chord.
Inclin.
to tang.
Dep.
100 sine.
x.
Lat. =
ioo cosine.
y-
I
o 05'
0.145
145
100.000
TOO.OOO
2
20'
0.582
.727
99.998
199.998
3
o 45 '
1.309
2.036
99.991
299.989
4
1 20'
2.327
4-3 6 3
99.979
399.968
&c.
&c..
&c.
In this manner Table I. has been constructed.
THEORY OF THE SPIRAL.
5. To calculate the deflection angles of the
Spiral ; Inst. at S. If in the diagram, Fig. i, we
draw the long chords 82, 83, 84, &c., g
we may easily determine the angle /,
which any long chord makes with the
tangent by means of the co-ordinates
of the further extremity of the chord,
for
x
tan / = .
y
Having calculated a series of values
of the angle /, we may lay out the
spiral on the ground by transit deflec-
tions from the tangent, the transit t>
ing at the point S.
The statement of the calculation is
as follows : FIG
Point.
X
/
tan / = - .
i
y
I
.145
too.ooo
.00145
o 05' oo"
2
.727
199.998
.00364
12' 30"
3
2.036
299.989
.00679
23' 20"
4
4-3 6 3
399.968
.01091
37' 30"
&c.
&c.
The values of / are more readily found by logarithms
however, since
log tan / = log x logy.
By this formula the first part of Table II. (Inst. at S)
THE RAILROAD SPIRAL.
FIG. 2.
has been calculated, and these are
the only deflections needed for field
use when the entire spiral is visible
from S.
6. To calculate the deflection
angles when the transit is at any
other chord-point than S : Sup-
pose the transit at point I, Fig. 2.
In the diagram draw through the
point i a line parallel to the tangent
at S, and also the long chords 1-3,
1-4, &c., and let a { represent the
angle between any one of these long
chords and the parallel. Then, from
the right-angled triangles of the dia-
gram we have the following expres-
sions :
For point 2, tan #, = - ^ = Q = .00582.
y* y\ 99-99 8
x z .\\ 1.891
" 3, tan a, = - -~~ = .00945.
y, -yi I99*9 8 9
4, tan #! =
&c.,
4.218
299.968
&c.
= .01411.
But these are better worked by logarithms, and the
values of a l found directly from the logarithmic tan-
gent.
Let s the spiral angle = the angle subtended by
any number of spiral chords, beginning at S. Then
s = the sum of the central angles of the several chords
considered ; and it therefore equals the angle between
OO
10'
10'
20'
3;
3'
1 00-
40'
o t
I 40
&c.
THEORY OF THE SPIRAL. 7
the tangent at S and a tangent at the last point consid-
ered. The series of values of the angle s is as follows :
Point. Angle under single chord. Angle f.
S
I
2
3
4
&c.
Since the values of a\ found above are deflections
at point i from a parallel to the main tangent, it is evi-
dent that if we subtract from each the value of s for
point i, or 10', we shall have the deflections, /, from an
auxiliary tangent through the point i, which we require
for use in the field. The statement is as follows :
Instrument at point i ; (s = 10').
Point. Angle ,. Angle/.
2 20" I0 f
3 32' 30" 22' 30"
4 48' 20" 38' 20"
&c., &c., &c.
The instrument will read zero on the auxiliary tan-
gent through point i where it stands, and of course the
back deflection over the circular arc Si is 05'. Hence
we have the complete table of deflections when the
instrument is at point i.
Similarly, if we suppose the instrument to be at
point 2, we shall have the statement :
Point. _
3 tan a> 2 = = = .01018. -
y* _f* 99-99 1
4 tan a* -3^3 _ OI r- 2 7.
y y* 199-97
&c., &c.,
8
THE RAILROAD SPIRAL.
and since for point 2, s = 20', we have :
Point. Angle a^. Angle i.
3 55' ' . o 15'
30
&c.,
32 30
&c.
The instrument will read zero on the auxiliary tangent
through the point 2, the back deflection to the point i
is half the central angle under the second chord, or 10',
and the back deflection to S is the difference between
S? and the deflection at S for point 2, or 30' 12' 30"
17' 30". We thus may complete the table of deflections
for the instrument at point 2.
By a similar process the deflections required at any
other chord-point may be deduced. It should be noted,
however, in forming the table, that the back deflection
5 \ , y to any point is equal to the value
of s for the place of the instru-
ment, less the value of s for the
back-point, less the forward de-
flection at the back-point for the
place of the instrument. This is
obvious from an inspection of the
triangle formed by the two auxil-
iary tangents and the chord join-
ing the two points in question.
Thus, Fig. 3, when the instru-
ment is at point 4, the back de-
flection for point 2 is equal to
100' 30' 32' 30" = 37' 30."
In the manner above described
has been calculated the complete
table of deflections from auxiliary
tangents at chord-points, for every chord-point of the
spiral up to point 20, Table II. It is evident, that by
FIG.
THEORY OF THE SPIRAL. 9
means of this table the entire spiral may be located, the
transit being set over any chord-point desired, while the
chain is carried around the curve in the usual manner ;
also, that the curve may be laid out in the reverse direc-
tion from any chord-point not above the 2oth, since all
the back deflections are also given.
7. Variation in the chord-length.
We have thus far assumed the spiral to be constructed
upon chords of 100 feet, but it is evident that such a
spiral would be entirely too long for practical use ; it
would be 1700 feet long before reaching a 3 curve.
We must, therefore, assume a shorter chord ; but in
so doing it will not be necessary to recalculate the
angles and deflections, for these remain the same whatever
be the chord-length. By shortening the chord-length we
merely construct the spiral on a smaller scale. The
values of x and y and of the radii of the arcs at corre-
sponding points are proportional to the chord-lengths,
and the degrees of curve for corresponding chords are
(nearly) inversely proportional to the same.
Thus for any chord-length c we have :
c
x : x 1QO :: c : 100, or x = ^ 100 -
100
c
y : Vioo :: c : 100, or y y 100 .
100
c
R s : jR iM :: c : 100, or R 3 ^? 100 .
100
Let D s = the degree of curve due to radius JR S , and
.Z} 100 = the degree of curve due to radius ^ 10 o? then,
100 100
A ~ : TTv and -*MOO
2 sin
whence
IOO
sin 1 D, = sin 4- Z> JO o,
T* C
TO THE RAILROAD SPIRAL.
in which D s is the degree of curve upon any chord in a
spiral of chord-length c, and Z> 100 is the degree of curve
upon the corresponding chord in the spiral of chord-
length 100.
Accordingly, if we assume a chord-length of 10 feet
the values of x and y will be of those calculated for
100
a chord-length of 100 feet, while the degree of curve
on each chord will be (nearly) 10 times as great as be-
fore.
8. In the construction of Table III., we have as-
sumed the chord to have every length successively from
10 feet to 50 feet, varying by a single foot, and have
calculated the corresponding values of x, y and Z) 8 .
The logarithm of x is also added, and the, length of
spiral nc.
We are thus furnished with 41 distinct spirals, but
since the same spiral may be taken with a different
number of chords (not less than three) to suit different
cases, the variations which the tables furnish amount to
no less than 500 spirals, some one or more of which
will be adapted to any case that can arise. The maxi-
mum length of spiral has been taken at 400 feet ; the
shortest spiral given is 3x10 feet = 30 feet. Be-
tween these limits may be found spirals of various
lengths.
9. The elements of a spiral are :
D^ The degree of curve on the last chord,
, The number of chords used,
c y The chord-length,
n x c y The length of spiral,
s, The central angle of the spiral,
x y y, The coordinates of the terminal point.
Every spiral must terminate, or join the circular curve
THEORY OF THE SPIRAL. II
at a regular chord-point of which the coordinates are
known.
10. To select a spiral.
The terminal chord of a spiral must subtend a degree
of curve less than that of the circular curve which fol-
lows, but the next chord beyond (were the spiral pro-
duced) must subtend a degree of curve equal to or
differing but a little from that of the circular curve.
Thus, if the circle were a 10 degree curve, the spiral
may consist of 5 chords 10 feet long (the degree of
curve on the 6th chord being 10 oo' 45"), or of 15
chords 26 feet long (the degree of curve on the i6th
chord being 10 16' 09"), the length of 'spiral is 50 feet
in one case and 390 in the other ; between these limits
the tables furnish 15 other spirals of intermediate length,
all adapted to join a 10 degree curve.
We may therefore introduce one more condition which
will fix definitely the proper spiral to employ. If the
length of spiral be assumed, we seek in the tables those
values of n and c which are consistent with the required
value of D s for (;/ -h i), at the same time that their
product, nc, equals as nearly as may be the assumed
length of spiral. Thus, if with a 10 degree curve a
length of about 130 feet were desirable, we should select
either
n = 8, f= 15, D 9 10 oo' 45"; nc 120 ft;
or ;z =-9, c= 16, D s 10 25' 51"; nc 144 ft.
D s is always taken for (n.+ i). When circumstances
permit, a chord-length of about 30 feet will give the
best proportioned spirals. With a 30 foot chord-length
the length of spiral will be about 770 times the super-
elevation of the outer rail at a velocity of 35 miles per
hour.
12 THE RAILROAD SPIRAL.
The value of s depends on the number of chords (n)
and is independent of the chord-length. If the angle s
were selected from the table, this would fix the number
n, and we must then choose the chord-length c so as to
give the proper value of D s . Thus, if s were assumed
= 9 10' then n = 10, and = 18 ft. or 19 ft., giving
D s 10 ii r 54" or 9 39' 36" to suit a 10 degree curve,
and making the length (nc) of the spiral either 170 or
1 80 ft., according to the spiral selected.
The coordinates (<x,y) depend on the values of both
n and c. They are used in solving the problems of
the spiral, being taken directly from Table III. for this
purpose, under the value of c and opposite the value
of n.
ELEMENTARY PROBLEMS.
CHAPTER III.
ELEMENTARY PROBLEMS.
ii. To find the length C of any long chord
beginning at the point of spiral S. Fig. 4. Let
L be the other extremity of the long
chord, x, y the coordinates of L, and
/ the deflection angle YSL at S for
the point L.
Then
or
C = -
cos i
. (I.)
.
sin / "
The values of x, y and / are found
in Tables III. and II.
Example. In the spiral of chord-
length = 30 ft. what is the length of
the long chord from S to the loth
point ?
FIG. 4:
From Table III.,
log x 1.224491
/ 3 12' 28" log sin 8.747853
C 299.66 Ans.
2.476638
12. To find the lengths of the tangents from
the points S and L to their intersection E.
Fig. 4. Let x, y be the coordinates of L, and s the
14 THE RAILROAD SPIRAL.
spiral angle for the point L. Then s the deflection
angle between the tangents at E, and
LE = - SE^y x cot s . . . . (2.)
sm s
The values of x y y and s are found in Tables III. and
IV.
Example. In the spiral of chord-length 40 extending
to the pth point, what are the tangents LE and SE ?
From Table III., log x 1.219075
" IV., s 7 30' log sin 9.115698
LE = 126.87 2.103377
log A* 1.219075
s 7 30' log cot 0.880571
125.790 2.099646
y 359-352
SE = 233.562
13. To find the length C of any long chord
KL. Fig. 4. Let x, y be the coordinates of L, and
x r , y the coordinates of K ; and let a be the angle LKN
which LK makes with the main tangent, and / the de-
flection angle KLE', and /" the deflection angle LKE'.
Then a = (s i) at the point L, = (s ! + /') at K.
KN
KL = - T -^r- T or
cos LKN
(3.)
cos a
Example. In the spiral of chord-length 18 what is the
ELEMENTARY PROBLEMS. 15
length of the long chord from point 12 to point 20?
Here K = 12 and L = 20 = n.
From Table III., y 34 6.47 T J fl IV \
y 214.? '
log 2.119352
From Table II., / 13
f I0-0 7 '2 3 "
. * . a 23 07' 23" log cos 9.963629
.'. C 143-13 2 - I 557 2 3
14. To find the lengths of the tangents from
any two points L and K to their intersection at
E'. Fig. 4. Let s, s f be the spiral angles for the points
L and K respectively. Then (s /) the deflection
angle between tangents at E'. Having first found C
LK by the last problem we have in the triangle LKE'
TTT' c sin *' W- ^ sm/ ' (A \
L.& = -. jr KJl. 7 - 7T (4-)
sin (s s) sin (s s )
Example. In the spiral of chord-length 18 what are
the tangents for the points 12 and 20 ?
By last example, C log 2.155723
From Table IV.,
(s - s) 35 n 13 = 22 log sin 9-573575
2.582148
From Table II., i 10 07' 23" log sin 9.244927
.', LE' = 67.15 1.827075
Again: 2.582148
Table II. , / 11 52' 37'' log sin 9.313468
,-. KE' = 78.635 1.895616
16
THE RAILROAD SPIRAL.
FlG
15. Given : A circular curve
and spirals joining two tangents,
to find the tangent dis-
tance T, = VS. Fig. 5.
Let S be the point of spiral,
V the intersection of the tan-
gents, SL the spiral, LH one
half the circular curve, and O
its centre. In the diagram
draw GLI parallel to the tan-
Is, gent VS, and GN, LM, and
OI perpendicular to VS. Join
OL and OV.
Then
IOL = s ; IOV = i A ; OL = R' ; SM = y ; LM = x.
Now SV = SM 4- NV -f MN.
But NV = GN . tan VGN = x tan \ A.
MN = GL = OL^ = R' Sin (i A ~ S] -
Hence
sin OGI cos ^A
, sin (| A s)
...(s.)
Example. Let the degree of the circular curve be
>' = 7 20', and the angle between tangents, A =42.
Let the spiral values be c = 2.3 ; n 9 . ' . s = 7,3o'
Then by the last equation and the tables,
y
x
21
206.627
log
log tan
0.978743
9-5 8 4i77
3,6,55
0.562920
ELEMENTARY PROBLEMS. 17
R' 7 20' C log 2.893118
iA s 13 30' log sin 9.368185
j-A 21 a. c. log cos 0.029848
195.502 2.291151
. ' . T a = 405-784
l6. When an approximate value of T, is only re-
quired we may employ a more convenient formula
derived from the fact that the line OI produced bisects
the spiral SL very nearly, and that the ordinate to the
spiral on the line OI, being only about -g- x, may be neg-
lected. Thus,
Approx. T, If tan |A + i nc. . . (6.)
Example. Same as above.
R' 7 20' C log 2.893118
I A 21 log tan 9.5,84177-'
300.1. 2.477295
i^ = ix9X23 103.5
. ' . 71 == approx. 403.6 >
Remark. This formula, eq. (6) when K is" taken equal
to the radius corresponding to the degree of curve
D s for (n -f i), gives practically correct results. But
as in practice, the value of R 1 will differ somewhat frcm
the radius of D# so the value of T 8 derived from this
formula will differ more or less from the true value, as
in the last example.
1/17. Given : the tangent distance T y SV, and the
angle A , and the length of spiral SL, to find the radius
K of the circular curve, LH, Fig. 5. The length
Ib- THE RAILROAD SPIRAL.
of spiral is expressed by nc, hence we have from the
last equation.
approx., ' = (T a \nc) cot -JA. . . .(7.)
After R' is thus found, the values of n and c are to be
determined, such that, while their product equals the
given length of spiral as nearly as may be, the value of
Z> 4 for (n -f i) shall correspond nearly with R '. The
values of n and c are quickly found by reference to
Table III.
Examble. Let T s = 406, A = 42, and nc = 170.
T 9 \nc 321 log 2.5065
lA 21 log COt. 0.4158
Jt' = say, 6 51' curve, 2.9223
By reference to Table III., we find that when n = 8
and c = 22, the product nc being 176, the value of D 3
for (n + i) is 6 49' 19", and this is the best spiral to
use in this case. But as this spiral is longer than our
assumed one, we should decrease the value of Jt' some-
what, if we would nearly preserve the given value of
T s . For instance, assume R' radius of 6 54' curve,
and using the same spiral, calculate by eq. (4) the re-
sulting value of T S9 and we shall find T = 408.646.
As this is an exact value of T* for the values of R\ n
and c last assumed, and is also a close approximation to
the value first given, it will probably answer the purpose
completely. If, however, for any * reason the precise
value of T s 406 is required, we may find the precise
radius which will give it by the following problem.
l8. Given: a curve, and spiral, and tangent-distance,
ELEMENTARY PROBLEMS. 19
T to find the difference in ' corresponding to
any small difference in the value of T s .
If in eq. (5) we assume a constant spiral, and give to
K two values in succession and subtract one resulting
value of T s from the other, we shall find for their dif-
ference,
diff. Ts = "h (**"'). diff. jf. . ( 8 .)
cos fa
Hence
d i ff . jf = . COS * A diff. T v . (9.)
sin (iA j)
Example. When J?' = rad. 6 54' curve, n = 8, c
22, 7^ = 408.646 ; what radius will make T s = 406 with
the same spiral ?
Eq. (9) diff. T,= 2.646 log 0.422590
3- A, 21 log COS 9.970152
(-3- A s), 15 a. c. log sin 0.587004
.'. diff. 1? 9-544 0.979746
*' 6 54' 830.876
. *. Required radius 821.332, or 6 58' 49" curve.
Remark. Care must be taken to observe whether in
thus changing the value of 7?', the value of Z>', the de-
gree of curve, is so far changed as to require a different
spiral according to the rule for the selection of spiral^
10. Should this be the case (which is not very likely),
we may adopt the new spiral, and proceed with a new
calculation as before.
1 9- Given : a circular curve with spirals joining two
tangents, to find the external distance . VH,
Fig- 5-
20 THE RAILROAD SPIRAL.
Let SL be the spiral, LH one-half the circular curve,
and O its centre.
Then VH = VG + GO - OH.
But VG = ^T^r - ~~TT > and in the tria ngle
cos VGN cos A
rni rn T n sin OLI -- & cos s
GOL, GO = LO . ^-^7^ Jt
sin LGO (
cos
x _ ; cos s
s -- 1 - T> .
COS-^A COS^A
or for computation without logarithms
^ f ^ (cos ^ cosJ-A ) / x
^ "cosfA"
Example. Let Z> f = 7 20', A = 42, and for the
spiral let n 9, ^ = 23, giving ^ = 7 30', and for
(0 + i), .A - 7 T 5' 4"-
Eq. (10) a; log 0.978743
^A 21 a. c. log cos 0.029848
10.200 1.008591
K 7 20' log 2.893118
s 7 30' log cos 9.996269
-JA2I a. c. log cos 0.029848
830.300 2.919235
sum 840.500
./?' 7 20' 781.840
s 58.660
ELEMENTARY PROBLEMS. 21
20. Given : The angle A at the vertex and the dis-
tance VH = to determine the radius R' of a
circular curve with spirals connecting the tangents
and passing through the point H. Fig. 5.
Solving eq. (n) for R' we have
K _ E, cos -j- A x
cos s cos 4- A \ )
But as this expression involves x and s of a spiral de-
pendent on the value of R' we must first find R' approxi-
mately, then select the spiral, and finally determine the
exact value of R' by eq. (12). The radius R of a simple
curve passing through the point H is a good approxima-
tion to R '. It is found by eq. (27) Field Engineering:
R
exsec -J-A '
or the degree of curve D may be found by dividing the
external distance of a i curve for the angle A by the
given value of E s . But evidently the value of D' will
be greater than D y and we may assume D' to be from
10' to i c greater according to the given value of A, the
difference being more as A is less. We now select from
Table III. a value of D K suited to D' so assumed, and
corresponding at the same time to any desired length of
spiral. Since D s so selected corresponds to (n 4- i) we
take the values of n and x from the next line above
D s in the table, find the value of s from Table IV., and
by substituting them in eq. (12) derive the true value of
R 1 for the spiral selected.
Example. Let A 42 and E s = 70, to find the value
of R' with suitable spirals.
From table of externals for i curve, when A = 42
E 407--64, which divided by 70 gives 5.823 ; or D =
22 THE RAILROAD SPIRAL.
5 50'. Assume D' say 20' greater, giving D' 6 10'
approx. If we desire a spiral about 300 feet long we
find, Table III., n = 10, c = 30, and for (n -f i) D s
6 06' 49". For 72 = 10, s = 9 10'.
Eq. (12) cos^A, 21 -9335 8
^ 7
65-35 o6
x 16.768
48.5826 log 1.686481
cos s 9 10' .98723
COsiA 21 .93358 .05365 log 8.729570
.*. R' = rad. (say) 6 20' curve. 905.55 2.956911
Proof. Take the exact radius of a 6 20' curve and
the above spiral and calculate E a by eq. (10) or (n).
We shall obtain E s = 69.97. Again : if we desire a
spiral of 200 feet, we find, Table III., n = 8, c = 25, and
for (n + i) D, 6, and by eq. (12) R' rad. of (say)
6 02' curve ; and by way of proof we find E, = 69.96.
Again : if we desire a spiral of about 400 feet, we find,
Table III., n = 12, c = 33, s 13, and for (n -f i)
D, 6 34' 07". Hence by eq. (12) R' rad. of (say)
6 50' curve. By way of proof we find eq. (TO) E, =
Remark. It is thus evident that a variety of curves
with suitable spirals will satisfy the problem, but D' is
increased as the spiral is lengthened for in the ex-
ample, with a 200 ft. spiral, D' 6 02' ; with a 300 ft.
spiral, D' = 6 20'; and with a 396 ft. spiral, I)'
6 50'. Therefore the length of spiral, as well as the
value of A, must be considered in first assuming the
value of I}' as compared with D of a simple curve.
ELEMENTARY PROBLEMS. 23
21. In case the value of R ', as calculated by eq. (12),
should give a value to D' inconsistent with the spiral
assumed, we may easily ascertain by consulting the
table what spiral will be suitable. Choosing a spiral of
the same number of chords, but of a different chord-
length c, we may calculate R' (a new value) as before ;
or the work may be somewhat abbreviated by the fol-
lowing method :
Given : a change in the value of x^ eq. (12) to find the
corresponding change in the value of R'\ n being con-
stant.
If the values of E s , A, and s remain unchanged, we
find, by giving to x any two values, and subtracting one
resulting value of R' from the other,
- -
COS S COS - A
that is, R' increases as x decreases, and the differences
bear the ratio of - = .
cos s cos YA
Example. Let A = 42, E s = 70, and for the spiral
let n 10, c ~ 30, s = 9 10', as in the last example,
giving R' = 905.55 ; to find the change in R' due to
changing c from 30 to 29.
Eq. (13) for c 30, x = 16.768
for c 29, x = 16.209
diff. x .559 log 9.7474
cos s COS^A (as before) .05365 log 8.7296
.'. diff. R' 10.42 1.0178
old value 95-55
. *. new R' 9 T 5-97 -&' ( sa y) 6 J 6',
24 THE RAILROAD SPIRAL.
which agrees well with D s 6 19' 29" for (n 4- i) in
the new spiral.
If we prove this result by calculating the value of
E s for these new values by eq. (10) we shall find E, =
69.93.
The slight discrepancy between these calculated
values of E s and the original is due solely to assuming
the value of D' at an exact minute instead of at a
fraction.
SPECIAL PROBLEMS.
CHAPTER IV.
SPECIAL PROBLEMS.
22. Given : two tangents joined by a simple curve, to
find a circular arc with spirals joining the same tan-
gents, that will replace the simple curve on the
same ground as nearly as may be, and preserve the
same length of line. Fig. 6.
To fulfill these conditions it is evident that the new
curve must be outside of the old one at the middle
point H, since the
spirals are inside
of the simple
curve at its tan-
gent points ; also,
the radius of the
new "curve must
be less than that
of the old one,
otherwise the cir-
cle passing out- ox
side of H would
cut the given tan-
gents.
Let SV, Fig. 6
be one tangent,
and V the vertex. FlG - 6 -
Let AH be one half the simple curve, and O its centre.
Let SL be one spiral, LH' one half the new circular
26 THE RAILROAD SPIRAL.
arc, and O' its centre. Draw the bisecting line VO, the
radii AO = R and LO' = R ', and the perpendicular
LM = x. Then MS =y. Produce the arc H'L to A'
to meet the radius O'A' drawn parallel to OA, and let
| A = the angle AOH = A'O'H'. Let s = the angle
A 'OX = the angle of the spiral SL. Let h the radial
offset HH' at the middle point of the curve. Draw
O'N and LF perpendicular to OA, LF intersecting O'A
at I.
a. To find the radius R 1 of the new arc LH' in terms
of a selected spiral SL.
We have from the figure AO = ML -f FN + NO.
But AO = R, ML = x, FN = LO' cos s = R' cos s
and NO = O'O cos A = (OH' - O'H') cos A =
(h + R R'} cos \ A ; and substituting we have
R = x + R' cos s + (h + R R') cos A . (14.)
whence
, _ R vers ^ A h + cos \ A 4- x . ,
cos s cos A cos ^ cos \ A
It is found in practice that h bears a nearly constant
ratio to x for all cases under the conditions assumed in
this problem. Let k = the ratio and the last equa-
tion may be written
, _ R vers \ A __ (/EcosjA + i)x , , x
cos s cos 4 A cos s cos -J A
which gives the radius of the new arc LH' in terms of
s, ~v and k.
SPECIAL PROBLEMS. 27
b. To find the offset h = HH' :
From eq. (14) we derive
h cos \ A R ( i cos 4- A ) R ( i vers s) 4-
R' cos Y A x
= jB(i - cos| A) ^'O- cosiA)-h
-#' vers s x
(R R') vers J A + -#' vers .$ #.
Hence
*g=X - *') exsec 4- A + -- (17.)
cos i A cos i A
which gives the value of h in terms of s, x and -#'.
C. 70 /// ///^ ^//^ of d AS :
We have from the figure SM = SA + NO' + IL.
But SM=j;, SA = 4 NO'= OO ; sin | A and IL =
LO' sin s, and by substitution,
y d + (h + R R') sin -J- A +7?' sin j.
Hence
^=7 ~ [( ;/ + R R') sin| A + ^'sin^] (18.)
\
which gives the distance on the tangent from the point
of curve A to the point of spiral S.
d. To compare the lengths of the new and old lines :
SAH = SA + AH = d+ 100^-, . .(19.)
in which D is the degree of curve of AH ;
SLH' = SL + LH' = n . c + 100 *
in which D' is the degree of curve of LH'.
28 THE RAILROAD SPIRAL.
If the spiral and arc have been properly selected, the
two lines will be of equal length or practically so.
The last two equations assume the circular curves to
be measured by 100 foot chords in the usual manner,
but when the curves are sharp it is often desirable that
they should agree in the length of actual arcs, especially
where the rail is already laid on the simple curve. For
this purpose we use the formulae
SAH(arc) = </+*. A. ~ (21.)
SLH' (arc) = n .c + R - j ~ (22.)
in which the angle is expressed in degrees and decimals.
If the odd minutes in the angle cannot be expressed by
an exact decimal of a degree, the angle should be re-
duced to minutes, and the divisor of ft changed from
180 to 10800.
The value of - is .0174533 log 8.241877
IoO
it
- is .00029089 : 6.463726.
10800
The length of spiral is given by chord measure in the
last equations, since the chords are so short and subtend
such small angles that the difference between chord and
arc is not material to the problem.
e. To select a spiral in a given case, we require to
know approximately the value of D', and to select the
spiral (n . c) such that the value of D s for (n + i) shall
not differ greatly from the value of D '. To aid in find-
SPECIAL PROBLEMS. 29
ing approximate values of D' and k, Table V. has been
prepared for curves ranging from 2 to 16 and central
angles (A) ranging from 10 to 80.
Assume s at pleasure (less than i A ), which fixes the
value of n. Then inspect Table V. opposite n for
values of D and A next above and below the values of
D and A in the given problem, and by inference or in-
terpolation decide on the probable values of k and D'.
Then in Table III. select that value of c which gives
D s for (n -h i) most nearly agreeing with D' . Now
calculate R' by eq. (16), and as this will usually give
the degree of curve D' fractional, take the value of
D' to the nearest minute only, and assume the corre-
sponding value of R' as the real value of R'. A table
of radii makes this operation very simple.
But should it happen that D' differs too widely from
from D s(n + ^ to make an easy curve, increase or di-
minish the chord-length c by i, thus giving a new value
to x in eq. (16), and also a new value of J} s ( n + I )
with which to compare the resulting D'. In changing
x only the last term of eq. (16) is affected, and the first
term does not require recalculation.
f. When the value of R 1 is decided, substitute it in
eq. (17) and calculate h. But if it happens that the
value of R' selected differs not materially from the result
of eq. (16), we have at once h = kx ; or in case the value
of R' is changed considerably from the result of eq. (16),
the corresponding change in h will be
. ....... . cos s cos -J-A ,. D , , 1X
diff. h - diff. R , . (22t)
cosf A
which may therefore be applied as a correction to h kx,
and we thus avoid the use of eq. (17). Eq. (22^) is de-
30 THE RAILROAD SPIRAL.
rived from eq. (15) by supposing h to have any two
values, and subtracting the resulting values of R' from
each other. Note that h diminishes as R! increases, and
vice versa.
When R' and h are found, proceed to find d by eq.
(18), and the length of lines by eq. (19), (20), or by
(21), (22), as may be preferred. But to produce equal-
ity of actual arcs, k must be a little greater than when
equality by chord-measure is desired.
Should the lines not agree in length so nearly as de-
sired, a change of one minute in the value of D 1
may produce the desired result, but any such change
necessitates, of course, a recalculation of h and d.
The values of k in Table V. appear to vary irregu-
larly. This is due to the selection of D' to the nearest
minute, and also to the choice of spiral chord-lengths,
c, not in an exact series. The reader is recommended
to supplement this table by a record of the problems he
solves, so that the values of R' and k may be approxi-
mated with greater certainty.
Example. Given a 6 curve, with a central angle of
A 50 12', to replace it by a circular arc with spirals,
preserving the same length of line. Assume s = 7 30'
giving n = 9.
Since 6 is an average of 4 and 8, while 50 12' is
nearly an average of 40 and 60, we examine Table V.
under 4 curve and 8 curve, and opposite A 40
and 60 on the same line as s = 7 30', and take an
average of the four values of Z> s (n + r), thus found;
also of the four values of k ; we thus find approx. k =
.0885, and D' = 6 18' . Now looking in Table III^
opposite n 9, we find that when c = 26, D s ( n + i)
6 24' 48", we therefore assume c = 26, and proceed to
calculate R' by eq. (16).
SPECIAL PROBLEMS. 31
Eq. (16) cos j 7 30' -99 X 44
COS^A 25 06' -9055^
.08587 a. c. log 1.066159
R 6 log 2.980170
vers^-A 25 o6 r log 8.975116
1050.6 log 3.021445
cos s cos -JA a. c. log 1.066159
I '-f y^COS 4- A = 1. 080 0.033424
x 1.031989
135-4 2.131572
.'. R' (say 6 16') 9 Z 5- 2
Eq.(i 7 ) JK6 955-366
R' 6 1 6' 914-75
(R R') 40.616 log 1.608697
exsec ^A 25 06' log 9.018194
4.235 log 0.626891
R' 6 16' log 2.961303
vers s , 7 30' log 7.932227
cos -}A 25 06' a. c. log 0.043079
8.642 log 0.936609
12.877
log 1.031989
i 25 .06' a. c. log 0.043079
11.887 1-075068
.*. h 0.990
Eq. (18) (R - R') 40.616
41.606 log 1.619156
sin ^-A 25 06' log 9.627570
17.649 log 1.246726
3 2
THE RAILROAD SPIRAL.
R r 6 16'
sin s 7 30'
119.399
log
log
log
2.961303
9.115698
2.077001
137.048
y
.'. d
n ( r r\ I - . *
233-579
A T R 1 TJ
.'. SAH
514.864
Eq. (20) (^A s) 1056' X 100
D' 376'
n . c 9 x 26
.--. SLH'
Difference
h
actual k = = 0.092
x
280.851
234-
SM-^i
-.013
log 5.023664
lo g 2.575188
log 2.448476
Comparison of actual arcs.
Eq. (21) 25.1 log 1.399674
i log 8.241877
R 6 log 2.980170
418.525 log 2.621721
96-531
Eq. (22) 17.6 log 1.245513
i log 8.241877
R' 6 i6 7 log 2.961303
280.991 log 2.448693
n.c 234.
5T4-99 1
Difference = 0.065
SPECIAL PROBLEMS.
33
23 Given : a simple curve joining two tangents, to
move the curve inward along the bisecting line VO
so that it may join a given spiral without change
of radius. Fig. 7.
Let SL be the given
spiral, AH one-half of the
given curve, and HL a
portion of the same curve
in its new position, and
compounded with the
spiral at L.
To find the distance
h = HH' = OO 7 :
Since the new radius is
equal to the old one, or
^?'= R, we have from eq.
(17) by changing the sign
of h, since it is taken in the opposite direction,
x R vers s
COS
To find the distance d = AS :
Changing the sign of h in eq. (18) and making R!
R we have
d y (R sin s h sin -J A) ( 2 4-)
This problem is best adapted to curves of large
radius and small central angle.
Example. Given, a curve D = i 40' and A
26 40', and a spiral s = i, n = 3, and c = 40, to find
// and d and the length LH'.
Eq. (23) R i 40' log 3-5363
vers s i log 6. 1 82 7
cos i A 13 20' a. c. log 0.0119
34 - THE RAILROAD SPIRAL.
538 log 9-7309
x log 9.9109
cos-J-A* a. c. log 0.0119
.837 9.9228
.'.// .299
Eq. (24) R i 40' log 3.536289
sin s i u " 8.241855
59.999 x -77 8l 44
- 2 99 log 9-4757
sin 4 A 13 20 9.3629
.069 8.8386
59-93
y 119.996
. * . d 60.066
H'O'L ^ (| A - s) = 12 20' .-. H'L = 740 feet.
24. Given, a simple curve joining two tangents, to
compound the curve near each end with an arc
and spiral joining the tangent without disturbing the
middle portion of the curve. Fig. 8.
Let H be the middle point of the given curve, Q the
point of compounding with the new arc, and L the
point where the new arc joins the spiral SL.
Let s = the spiral angle, and let = AOQ. Now in
this figure AOQS will be analogous to AOH'S of Fig. 6,
if in the latter we suppose H' to coincide with H or
// = o. If, therefore, in eq. (15) we write for -J- A and
make // = o, we have for the new radius O'Q,
, _ R vers x . .
~ cos s cos 0' '
SPECIAL PROBLEMS.
35
in terms of and the
spiral assumed. But
as the value of D'
resulting is likely to
be fractional and
must be adhered to,
it is preferable to as-
sume jR' a little less
than R, select a suit-
able spiral and cal-
culate the angle 0.
Resolving eq. (17)
after making h = o
and replacing \ A
by 0, we have
FIG. 8.
vers :=
r-JT'
vers s
(26.)
The angle so found must be less than -3- A , and in-
deed for good 'practice should not exceed 3- A. If too
large, may be reduced by assuming a smaller value of
^', and repeating the calculation with a suitable spiral.
Otherwise it will be preferable to use one of the forego-
ing problems in place of this. This problem is specially
useful when the central angle is very large.
To find the distance d AS, we have only to write
for 4 A and make h = o in eq. (18), whence
dy - \(R - J?') sin + R' sin^] . . . (27.)
Example. Given a curve D ~ 2 30', A =35, to
compound it with a curve D' = 2 40' and a spiral s =
2 30', n = 5, < = 37.
THE RAILROAD SPIRAL.
Eq. (26) R 2 30' 2292.01
R' 2 40' 2148.79
R- R 1
X
143.22
R- R'
vers s 2 30'
o r
2 40
R'
.-. versO 6 28' 30"
Eq. (2 1 }R-R'
sin <* 6 28' 30"
R'
sin s
2 U 40'
2 30'
.-. d
AH
log 2.156004
log 0.471203
0020663 log 8.315199
a. c. log 7.843996
log 6.978536
* lo g 3-33 2I 93
log 8. 154725
.014280
.006383
16.151
93.729
109.880
184.962
log 2.156004
9.052192
1.208196
3-33 2I 93
8.639680
1.971873
775.082
SL, n.c 185.00
LQ, - s = 3 58' 30" 149.06
QH, i A = n oi' 30" 441.00 775- 6
Difference
.022
SPECIAL PROBLEMS.
37
25. Given: a compound curve joining two tan-
gents, to replace it by another with spirals, pre-
serving the same length of line. Fig. 9.
Let A 2 = AO 2 P,
the angle of the arc
AP, and A, =
PdB, the angle of
the arc PB. Let
^ 2 A O 2 , and
R, = BO,.
Adopting the
method of 22,
the offset h must
be made at the
point of compound
curve P instead of
at the middle point.
Cons idering first
the arc of the
larger radius AO 2 ,
the formulae of 22
will be made to
FIG. 9.
apply to this case by writing A 2 in place of \ A , and
Ri in place of R, whence eq. (16)
vers A g
cos s cos A 2
(k cos A 2 + i) x
cos s cos A 2
,
and eq. (17)
7 / r> r> f\ A , * vers s X t \
h (R^ R^) exsec A 2 + - (20.)
COS A 2 COS A 2
and eq. (18)
d=y. - [(k + R, - RJ) sin A 2 + R* sin s] . . (30.)
38 THE RAILROAD SPIRAL.
But in considering the second arc PB, we must retain
the value of // already found in eq. (29) in order that
the arcs may meet in P'. We therefore use eq. (15)
which, after the necessary changes in notation, becomes
n , R l vers A l h cos A , + x , x
. . .
cos s cos A j cos s cos A !
which value of lt\ must be adhered to.
The spiral selected for use in the last equation is in-
dependent of the spiral just used in connection with J?./.
It should be so selected that while suitable for ^?/ its
value of x may be equal to as nearly as may be, the
K
value of k being inferred from Table V. for D' and
2 A!.
Assuming the value of 7?/ found by eq. (31)? even
though DI be fractional, we may verify the value of h by
h = (JP, - ft) exsec A , + L - - ( 32 .)
cos A j cos A j
and then proceed to find d' = BS' by
d j -y- [(h + ^ -J?,') sin A, f.tf/sin.f] (33.)
Example. Given the compound curve D == 8., A ,
29 and Z> 2 = 6, A 2 = 25o6' : to replace it by an-
other compound curve connected with the tangents by
spirals.
Considering first the 6 branch of the curve, we may
assume the spiral s = 73o', n = 9, c = 26. This part
of the problem is then identical with the example given
in 22, by which we find h .990 and d 96.531.
To select a spiral for the 8 branch, having reference
at the same time to this value of h ; we find in Table V.
SPECIAL PROBLEMS. 39
under D = 8 and opposite A 2 A l = 58 or say 60,
that the given value of h falls between the tabular
values of 7* for nc = 9 x 20, and //^ = io"x 22. We there-
fore infer that the spiral ^ 9x21 is most suitable to
this case. Adopting this, we have
Eq. (31) COS.T 73o'. 99144
COS A! 29. 87462
.11682 log 9. 0675 1 7 a>c - l
R, 8 " 2.855385
vers A^ " 9.098229
769.302 " 2.886097
h cos 29 .866
x 8.694
9.560 :< 0.980458
cos s cos A! a.c. " 0.932483
8i.8 3 5
" 1.912941
\
" 1.481471
"9-685571
.-. US 82o'3o r 687.467
[. (33) (h + .#,) 717.769
30.302
sin A ! 29
14.691
J?/ 687.467
sin s 73o f
89-732
' 1.167042
" 2.837251
9. 115698
1.952949
104.423
188.660
y
'. d 84.237
40 THE RAILROAD SPIRAL.
For the methods of computing the lengths of lines,
see 22.
26. Given : a compound curve joining two tangents,
to move the curve inward along the line PO 2 so that
spirals may be introduced without changing the ra-
dii. Fig. 10.
The distance h = PP' is found for the arc of larger
Fig. 10.
radius AO 2 by the following formula derived by analogy
from eq. (23):
, ___ x R* vers s . , ^
cos A 2
and for the distance d = AS we have analogous to eq.
d y ( z sin, /fcsin A 2 ) . (35.)
SPECIAL PROBLEMS. 41
. Now the same value of /*, found by eq. (34) must be
used for the arc PB, and a spiral must be selected which
will produce this value. To find the proper spiral, we
have from eq. (34) after changing the subscripts,
x R l vers s + h cos A ^ . . (36.)
The last term is constant. The values of x and s must
be consistent with each other, and approximately so with
the value of R^. Assume s at any probable value, and
calculate x by eq. (36). Then in Table III. look for
this value of x opposite n corresponding to j, and note
the corresponding value of the chord-length c. Com-
pare D s of the table with Z>i and if the disagreement is
too g^eat select another value of s and proceed as be-
fore.
The term JR^ vers s may be readily found, and with
sufficient accuracy for this purpose, by dividing the value
of R i versj Table IV. by Z> lm If the calculated value
of x is not in the Table III., it may be found by inter-
polating values of c to the one tenth of a foot, since for
a given value of s or/n the values of x and y are pro-
portional to the values of c.
When the proper spiral has been found and the value
of c determined, it only remains to find the value of d =
BS'by
d y (Ri sin s h sin A i), . (37.)
in which the value of y will be taken according to the
values of c and s just established.
Example. Given: Z> 2 i4o', A 2 = 13 20', Z> 1 = 3,
and A = 224o', to apply spirals without change of
radii. Fig. 10.
Assume for the i 40' arc the spiral s i, n = 3,
c 40. This part of the problem is then identical with
the example given in 23, from which we find h = 0.299.
THE RAILROAD SPIRAL.
For the second part, if we assume s i 40', n = 4,
and find by Table IV. ^ vers s = ^ = 0.808, we
o
have by eq. (36)
x = 0.808 -f 0.277 1.085,
the nearest value to which in Table III. is under c =
25, giving Ds =2 40', or for (n + i), D = 3 20', which
is consistent with D l 3. By interpolation we find
that our value of x corresponds exactly to c = 24.85,
n = 4, and therefore the spiral should be laid out on the
ground by using this precise chord.
In order to find d = BS' we first find the value of y
by interpolation for c =24.85, when by eq. (37) we^ have
*T= 99-39 1 - (55-554 ~ 0.115) = 43-95 2 -
27. Given : a compound curve joining two tan-
gents, to introduce spirals without disturbing
the point of
^^
B
compou nd
curvature P.
Fig. ii.
a. The radius
of each arc may
be shortened, giv-
ing two new arcs
compounded at
the same point
P. Having se-
lected a suitable
spiral, we have
for the arc AP
s by analogy from
eq- (15), since
Fig.
SPECIAL PROBLEMS.
_ Rv vers A 2 x
cos ^ cos A 2 '
43
(38.)
and, similarly, after selecting another spiral for the arc
PB,
_ ^, vers A , x
cos s cos Aj " " V39-J
From eq. (18) we have for the distance AS,
d~y \_(R* RI'] sin A 2 + RJ sin s], . (40.)
and for the distance BS',
/
d ~ y [(XL Ri') sin A! + RI sin s] . (41.)
The values of
DI and Dj re-
sulting from eq.
(39) and (40)
must be adhered
to, even though
involving a frac-
tion of a minute.
b. Either arc
may be again com-
pounded at some
point Q, leaving
the portion PQ
undisturbed, as
explained in 24.
Fig. 12.
Let e = the an-
Fig. 12.
gle AO 2 Q, and we have from eq. (26), after selecting a
suitable spiral and assuming .#./,
vers
vers s
44 THE RAILROAD SPIRAL.
For the distance AS, we have from eq. (27)
d = y - [(. - A') sin + RJ sin j] . (43.)
Similar formulae will determine the angle = BOjQ'
and the distance BS' for the other arc PB in terms of a
suitable spiral : thus,
x R\ vers s
d y - [(^ - ^/) sin + ^/ sin s] . (45.)
The method a may be adopted with one arc and the
method b with the other if desired, since the point P is
not disturbed in either case. The former is better
adapted to short arcs, the latter to long ones.
These methods apply also to compound curves of
more than two arcs, only the extreme arcs being altered
in such cases.
FIELD WORK. 45
CHAPTER V.
FIELD WORK.
28. HAVING prepared the necessary data by any of
the preceding formulae, the engineer* locates the point S
on the ground by measuring along the tangent from V
or from A. He then places the transit at S, makes the
verniers read zero, and fixes the cross-hair upon the tan-
gent. He then instructs the chainmen as to the proper
chord c to use in locating the spiral, and as they meas-
ure this length in successive chords, he makes in succes-
sion the deflections given in Table II. under the
heading "Inst. at S," lining in a pin or stake at the end
of each chord in the same manner as for a circle.
When the point Las reached by (n) chords, the tran-
sit is brought forward and placed at L ; the verniers are
made to read the first deflection given in Table II.
under the heading " Inst. at n " (whatever number n may
be), and a backsight is taken on the point S. If the
verniers are made to read the succeeding deflections, the
cross-hair, should fall successively on the pins already
set, this being merely a check on the work done, until
when the verniers read zero, the cross-hair will define the
tangent to the curve at L. From this tangent the cir-
cular arc which succeeds may be located in the usual
manner.
In case it became necessary to bring forward the tran-
sit before the point L is reached, select for a transit-
point the extremity of any chord, as point 4, for
46 THE RAILROAD SPIRAL.
example, and setting up the transit at this point, make
the verniers read the first deflection under " Inst. at 4,"
Table II., and take a backsight on the point S. Then,
when the reading is zero, the cross-hair will define the
tangent to the curve at the point 4, and by making the
deflections which follow in the table opposite 5, 6, &c.,
those points will be located on the ground until the
desired point L is reached by n chords from the begin-
ning S.
The transit is then placed at L, and the verniers set
at the deflection found under the heading " Inst. at n "
(whatever number // may be), and opposite (4) the point
just quitted. A backsight is then taken on point 4,
and the tangent to the curve at L found by bringing the
zeros together, when the circular arc may be proceeded
with as usual.
29. To locate a spiral from the point L running toward
the tangent at S : we have first to consider the number of
chords (n) of which the spiral SL is composed. Then,,
placing the transit at L, reading zero upon the tangent
to the curve at L, look in Table II. under the heading
" Inst. at #," and make the deflection given just above
o oo' to define the first point on the spiral from L
toward S ; the next deflection, reading up the page, will
give the next point, and so on till the point S is
reached.
The transit is then placed at S ; the reading is taken
from under the heading "Inst. at S," and on the line n
for a backsight on L. Then the reading zero will give
the tangent to the spiral at the point S, which should
coincide with the given tangent.
If S is not visible from L, the transit may be set up
at any intermediate chord-point, as point 5, for example.
The reading for backsight on L is now found under the
FIELD WORK. 47
heading " Inst. at 5," and on the line n corresponding to
L ; while the readings for points between 5 and S are
found above the line 5 of the same table. The transit
being placed at S, the reading for backsight on 5, the
point just quitted, is found under " Inst. at S " and
opposite 5, when by bringing the zeros together a tan-
gent to the spiral at S will be defined.
30. Since the spiral is located exclusively by its
chord-points, if it be desired to establish the regular 100-
foot stations as they occur upon the spiral, these must be
treated asflusses to the chord-points, and a deflection
angle will be interpolated where a station occurs. To
find the deflection angle for a station succeeding any chord-
point : the differences given in Table II. are the deflec-
tions over one chord-length, or from one point to the
next. For any intermediate station the deflection will
be assumed proportional to the sub-chord, or distance
of the station* from the point. We therefore multiply
the tabular difference by the sub-chord, and divide by the
given chord-length, far the deflection from that point to
the station. This applied to the deflection for the point
will give the total deflection for the station.
This method of interpolation really fixes the station
on a circle passing through the two adjacent chord-
points and the place of the transit, but the consequent
error is too small to be noticeable in setting an ordinary
stake. Transit centres will be set only at chord-points,
as already explained.
31. It is important that the spiral should join the
main tangent perfectly, in order that the full theoretic
advantage of the spiral may be realized. In view of
this fact, and on account of the slight inaccuracies
inseparable from field work as ordinarily performed, it
is usually preferable to establish carefully the two points
48 THE RAILROAD SPIRAL.
of spiral S and S' on the main tangents, and beginning
at each of these in succession, locate the spirals to the
points L and L'. The latter points are then connected
by means of the proper circular arc or arcs. Any slight
inaccuracy will thus be distributed in the body of the
curve, and the spirals will be in perfect condition.
32. A spiral may be located without deflection angles,
by simply laying off in succession the abscissas y and
ordinates x of Table III. corresponding to the given
chord-length c. The tangent EL at any point L, Fig. 4,
is then found by laying off on the main tangent the dis-
tance YE = x cot s, and joining EL. In using this
method the chord- length should be measured along the
spiral as a check.
33. In making the final location of a railway line
through a smooth country the spirals may be introduced
at once by the methods explained in Chapter III. But
if the ground is difficult and the curves require close ad-
justment to the contour of the surface, it will be more
convenient to make the study of the location in circular
curves, and when these are likely to require no further
alterations, the spirals may be introduced at leisure by
the methods explained in Chapter IV. The spirals
should be located before the work is staked out for con-
struction, so that the road-bed and masonry structures
may conform to the centre line of the track.
34, When the line has been first located by circular
curves and tangents, a description of these will ordi-
narily suffice for right of way purposes ; but if greater
precision is required the description may include the
spirals, as in the following example :
" Thence by a tangent N. 10 i5'E., 725 feet to station
1132 + 12; thence curving left by a spiral of 8 chords,
288 feet to station 1 135; thence by a 4 12' curve (radius
FIELD WORK. 49
1364.5 feet), 666.7 feet to the station 1141 +66.7; thence
by a spiral of 8 chords 288 feet to station 1144 + 54.7
P.T. Total angle 40 left. Thence by a tangent N. 29
45' W.," &c.
35. When the track is laid, the outer rail should re-
ceive a relative elevation at the point L suitable to the
circular curve at the assumed maximum velocity. Usu-
ally the track should be level transversly at the point S,
but in case of very short spirals, which sometimes can-
not be avoided, it is well to begin the elevation of the
rail just one chord-length back of S on the tangent.
36. Inasmuch as the perfection of the line depends
on adjusting the inclination of the track proportionally
to the curvature, and in keeping it so, it is extremely im-
portant that the points S and L of each spiral should be
secured by permanent monuments in the centre of the
track, and by witness-posts at the side of the road. The
posts should be painted and lettered so that they may
serve as guides to the trackmen in their subsequent
efforts to grade and "line up " the track. The post op-
posite the point S may receive that initial, and the post
at L may be so marked and also should receive the
figures indicating the degree of curve.
37. The field notes may be kept in the usual manner
for curves, introducing the proper initials at the several
points as they occur. The chord-points of the spiral
may be designated as plusses from the last regular sta-
tion if preferred, as well as by the numbers i, 2, 3, &c.,
from the point S. Observe that the chord numbers
always begin at S, even though the spiral be run in the
opposite direction.
TABLE
ELEMENTS OF THE SPIRAL
Inclina-
Point
Degree
of curve
Spiral
angle
tion of
chord
Latitude of each
chord.
Sum of the lati-
tudes,
to axis
i
n.
Ds.
s.
of Y.
TOO x cos Incl.
?-
o oo'
o oo'
o oo'
I
10'
10'
05'
99.99989423
99.99989423
2
20'
30'
20'
99.99830769
199.99820192
3
30'
1
45'
99.99143275
299.98963467
4
40'
1 40'
1 20'
99.97292412
399.96255879
5
50'
2 30'
2 05'
99-93390 07
499 89645886
6
1
3 30'
3
99.8629535
599.7594123
7
1 10'
4 4'
4 05'
99.7461539
699.5055662
8
1 20'
6
5 20'
99.5670790
799.0726452
9
I' 30'
7 30'
6 45'
99.3068457
898.3794909
10
I 4 0'
g 10
8 2o|
98.944164
997-3236549
ii
I' SO'
11
10 05'
98.455415
1095.779070
12
2
13
12
97.814760
1193.593830
13
2 10'
15 10'
14 05'
96.994284
1290.588114
14
2 20'
I7 I 3
1 6 20'
95.964184
1386.552298
15
2 30'
20
i84 5 '
94.693014
1481.245312
16
2 40'
22 40'
21 20'
93-147975
1574-393287
17
2 50'
25 30
24 05'
91.295292
1665.688579
18
3
28 30''
27
89.100650
1754.789229
19
3 10'
31 40
3o 05'
86.529730
1841.318959
20
3 20'
35
33 20'
83.548730
1924.867739
Point.
Log^ =
Deflection angle,
.
log tan /.
it
I
7.1626964
o 05' oo. 'oo
2
7.5606380
o 12' 30. 'oo
3
7.831709!
23' 20. '00
4
8.0377730
o 37' 29. '99
5
8.2041217
o 54' 59- '97
6
8.3436473
i 15' 49. '90
7
8.4638309
i 39' 59- '75
8
8.5694047
2 07' 29. '45
9
8.6635555
2 38' IS. '90
10
8.7485340
3 12' 27. '95
OF CHORD-LENGTH, 100.
Departure of
Sum of the depart-
Logarithm,
Logarithm,
Point
each chord.
ures,
100 x sin Incl.
X.
logjj/.
log jr.
n.
.1454441
.1454441
1.9999995
9.1626960
I
.5817731
.7272172
2.3010261
9.8616641
2
I.308Q593
2.0361765
2.4771063
0.3088154
3
2.3268960
4.3630725
2.6020194
0.6397924
4
3.6353009
7.9983734
2.6988800
0.9030017
5
5-23359 6
13.231969
2.7779771
1.1216244
6
7.120730
20.352699
2.8447911
1.3086220
7
9. 29499 [
29.647690
2.9025862
.4719909
8
11-75374
41.40143
2.9534598
.6170153
9
14.49319
55.89462
2.9988361
.7473701
10
17.50803
73.40265
3.0397231
.8657117
n
20.79117
94.19382
3.0768567
.9740224
12
24.33329
118.52711
3.1107877
2.0738177
13
28.12251
146.64962
3.1419362
2.I6628II
14
32.14395
. 178.79357
3.1706269
2.2523519
15
36.37932
215.17289
3.1971131
2.3327875
16
40. 80649
255.97938
3.2215938
2.4082049
17
45.39905
301.37843
3.2442250
2.4791121
18
50.12591
35L50434
3.2651291
2.5459307
19
54.95090
406.45524
3.2844009
2.6090128
20
T *
Deflection an-
Point
L S y =
gle,
n.
log tan i.
t.
II
8.8259886
349'56."39
12
8.8971657
43o'43."95
13
8.9630300
5 14' 50."28
14
9.0243449
6 02' I4."93
15
9.0817250
652'57."3*
16
9.1356744
746'56."7i
17
9.1866111
8 44' I2."26
IS
9.2348871
9 44' 42. "92
19
9.2808016
10 48' 27. "44
20
9.3246119
n55'24."34
TABLE IT.
DEFLECTION ANGLES, FOR LOCATING SPIRAL CURVES IN THE
FIELD.
Rule for finding a Deflection.
Read under the heading corresponding to the point at which the
instrument stands, and on the line of the number of the point
observed.
INSTRUMENT AT S.
s = o.
No. of Point,
Deflection
from
Tangent,
Difference
of Deflec-
n.
i.
tion.
oo'
I
05
05
2
3
4
5
6
7
8
9
10
ir
12
13
14
15
16
17
18
19
20
i
i
2
2
3
3
4
5
6
6
7
8
9
10
ii
12
23
37
55
15
40
07
38
12
49
30
14
02
52
4 6
44
44
48
55
30"
20
30
00
50
00
29
19
28
56
.44
SG
15
57
57
12
43
27
24
07
10
14
17
20
24
27
30
34
37
40
44
47
50
54
57
60
63
66
30'
50
10
30
50
10
29
50
09
28 -
48
06
25
42
oo
15
31
44
57
5 2
TABLE II. DEFLECTION ANGLES.
INST. AT i. s = o 10'.
INST. AT 2. s = o 30'.
No. of Deflection 'from
Diff . of
No. of
Deflection from
Diff. of
Point.
aux. tan.
Deflection.
Point.
aux. tan.
Deflection.
O
I
05'
00
05'
I
17' 30"
10
7' 30"
IO
10
2
IO
2
00
12 30'
15
3
4
5
6
7
8
9
10
ii
12
13
14
15
16
17
18
- 19
20
22 30"
38 20
57 3o'
I 9 20 00
i 45 50
2 15 00
2 47 29
3 23 18
4 02 27
4 44 55
5 30 42
6 19 47
7 12 ii
8 07 51
9 06 49
10 09 01
ii 14 28
12 23 08
15 50
19 10
22 30
25 50
29 10
32 29
35 49
39 09
- 42 28
45 47
49 05
52 24
55 40
58 58
62 12
65 27
68 40
3
4
5
6
8
9
10
ii
12
13
14
15
16
17
18
19
20
15
32 30
53 20
i c 17 30
i 45 oo
2 15 50
2 49 59
3 27 29
4 08 18
4 52 26
5 39 54
6 30 40
7 24 44
8 22 06
9 22 45
10 26 39
ii 33 49
12 44 12
17 30
20 50
24 10
27 30
30 50
34 09
37 30
40 49
44 08
47 28
50 46
54 04
57 22
60 39
63 54
67 10
70 23
INST. AT 3. s= i oo'.
INST. AT 4. j 1 40'.
No. of
Deflection from
Diff. of
No. of
Deflection from
Diff. of
Point.
aux. tan.
Deflection.
Point.
aux. tan.
Deflection.
36' 40"
Q' 10"
O
1 O2' 30"
10' 50"
I
27 30
12 30
I
51 40
14 10
2
3
15
00
15
2O
2
3
37 30
20
17 30
2O
4
20
4
00
22 30
2 5
5
6
7
8
9
10
ii
12
13
14
15
16
42 30
l p 08 20
i 37 30
2 10 OO
2 45 50
3 24 59
4 07 28
4 53 17
5 42 25
6 34 52
7 30 37
8 29 40
25 50
29 10
32 30
35 50
39 09
42 29
45 49
49 08
52 27
55 45
59 3
62 21
5
6
8
9
10
ii
12
13
M
15
16
25
52 30
I 23 20
i 57 30
2 35 oo
3 ID 50
3 59 59
4 47 28
5 38 16
6 32 24
7 29 50
8 30 34
27 30
30 50
34 10
37 30
40 50
44 09
47 29
50 48
54 08
57 26
60 44
64 02
17
18
19
20
9 32 01
10 37 37
ii 46 29
12 58 35
6 5 36
68 52
72 06
17
18
19
20
9 34 36
10 41 55
ii 52 29
13 06 i 8
67 19
70 34
73 49
53
TABLE II. DEFLECTION ANGLES.
INST. AT 5. s = 2 30'.
INST. AT 6. j = 3 30'.
No. of
Deflection from
Diff. of
No. of Deflection from Diff. of
Point.
aux. tan.
Deflection.
Point.
aux. tan. Deflection.
I
i 35' oo"
I 22 30
. .12' 30"
I
2 14' 10"
2 OO OO
14' 10"
2
3
4
5
I 06 40
47 30
25
00
15 50
19 10
22 30
25
2
3
4
5
I 42 30
I 21 40
57 30
3
17 30
20 50
24 10
27 30
6
3
30
6
oo
30
7
8
9
10
ii
12
13
14
15
16
17
18
20
1 02 30
I 38 20
2 17 30
3 oo oo
3 45 50
4 34 59
5 27 28
6 23 15
7 22 23
8 24 48
9 30 31
10 39 32
ii 51 48
13 07 20
32 3O
35 50
39 10
42 30
45 50
49 09
52 29
55 47
59 08
62 25
65 43
69 01
72 16
75 32
7
8
9
10
ii
12
13
14
15
16
17
18
19
20
35
I 12 30
i 53 20
2 37 30
3 25 oo
4 15 49
5 09 58
6 07 27
7 08 15
8 12 21
9 19 46
10 30 28
ii 44 27
13 oi 41
35
37 30 ,
40 50
44 10
47 30
50 49
54 09
57 29
60 48
64 06
67 25
70 42
73 59
77 14
INST. AT 7. ,$ = 4 40'.
INST. AT 8. s = 6 co'.
No. of
Deflection from
Diff. of
No. of
Deflection from
Diff. of
Point.
aux. tan.
Deflection.
Point.
aux. tan.
Deflection.
I
2
3
4
6
7
3 00' 00"
2 44 10
2 25 00
2 O2 30
I 36 40
I 07 30
35
00
* 5' 50"
19 10
22 30
25 50
29 10
32 30
35
f
2
3
4
5
6
7
352 / 3i //
3 35 oo
3 14 10
2 50 OO
2 22 30
I 51 40
I 17 30
40
17' 31"
20 50
24 10
27 30
30 50
34 10
37 30
8
4
40
8
OO
40
9
10
it
12
13
14
15
16
17
18
I 22 30
2 08 20
2 57 30
3 50 oo
4 45 49
5 44 58
6 47 26
7 53 14
9 02 19
10 14 43
42 30
45 50
49 10
52 30
55 49
59 9
62 28
65 48
69 05
72 24
75 4i
9
10
ii
12
13
14
15
16
17
18
45
i 32 30
2 23 20
3 17 30
4 15 oo
5 15 49
6 19 58
7 27 26
8 38 13
9 52 18
45
47 30
50 50
54 10
57 30
60 49
64 09
67 28
70 47
74 05
77 22
19
20
ii 30 24
12 49 21
78 57
19
20
ii 09 40
12 30 20
80 40
J
54
TABLE II. DEFLECTION ANGLES.
INST. AT g. ^ = 7 30'.
INST. AT 10. j = 9 10'.
No. of
Deflection from
Diff. of
No. of Deflection from
Diff. of
Point.
aux. tan. ,
Deflection.
Point. aux. tan.
Deflection.
I
2
3
4
5
6
7
8
9
4 5i' 41"
4 32 31
4 10 oi
3 44 10
3 15 oo
2 42 30
2 06 40
I 27 30
45
00
19' 10"
22 30
25 51
29 10
32 30
35 50
39 10
42 30
45
I
2
3
4
6
7
8
9
557 / 32 //
5 36 42
5 12 31
4 45 oi
4 14 10
3 40 oo
3 02 30
2 21 40
i 37 30
50
20' 50"
24 II
27 30
30 51
34 10
37 30
40 50
44 10
47 30
10
50
50
10
CO
5
ii
12
13
14
15
16
17
18
19
20
i 42 30
2 38 20
3 37 30
4 40 oo
5 45 49
6 54 57
8 07 25
9 23 ii
10 42 16
12 04 38
52 30
55 50
59 i
62 30
65 49
69 08
72 28
75 46
79 5
82 22
ii
12
13
15
16
17
18
19
20
55
i 52 30
2 53 20
3 57 30
5 05 oo
6 15 49
7 29 57
8 47 24
10 08 10
ii 32 14
55
57 30
60 50
64 10
67 30
70 49
74 08
77 27
80 46
84 04
INST. AT ii. s = 11 oo'. INST. AT 12. j = 13 oo'.
No. of
Deflection from
Diff. of
No. of
Deflection from
Diff. of
Point.
aux. tan.
Deflection.
Point
aux. tan.
Deflection.
I
2
3
4
7 jo' 04"
6 47 33
6 21 42
5 52 32
5 20 oi
22' 31"
25 51
29 10
32 31
1? c T
O
I
2
3
4
8 29' 16"
8 05 05
7 37 34
7 06 43
6 32 32
24' Ii"
27 31
30 51
34 ii
5
6
4 44 10
4 05 oo
39 I0
6
5 55 oi
5 T 4 IT
40 50
8
9
10
ii
3 22 30
2 36 40
i 47 30
55
00
42 3
45 50
49 10
52 30
55
7
8
9
10
ii
4 30 oo
3 42 30
2 51 40
i 57 30
I OO OO
44 ii
47 30
50 50
54 10
57 30
Co
60
12
I OO OO
12
^ oo
62 30
65
13
14
15
16
17
18
20
2 02 30
3 08 20
4 17 30
5 29 59
6 45 48
8 04 57
9 27 24
10 53 09
65 50
69 10
72 30
75 49
79 9
82 27
85 45
13
14
16
17
18
19
20
i 05 oo
2 12 30
3 23 20
4 37 30
5 54 59
7 15 48
8 39 56
10 07 23
67 30
70 50
74 10
77 29
80 49
84 08
87 27
55
TABLE II. DEFLECTION ANGLES.
INST. AT 13. j= 15 10'.
INST. AT 14. j = 17 30'.
No. of Deflection from; Diff. of
No. of Deflection from Diff. of
Point. aux. tan.
Deflection.
Point.
aux. tan. Deflection.
O
I
2
9 55' 10"
9 29 18
9 oo 06
25' 52"
29 12
O
I
2
n 27' 45"
ii oo 13
10 29 20
27' 32"
30 53
3
4
5
6
7
8
9
10
8 27 35
7 5i 44
7 12 32
6 30 02
5 44 ii
4 55 oo
4 02 30
3 06 40
32 31
35 5i
39 12
42 30
45 5i
49 ii
52 30
55 50
59 1
3
4
5
6
7
8
9
10
9 55 08
9 17 36
8 36 45
7 52 33
7 05 02
6 14 ii
5 20 oo
.4 22 30
34 12
37 32
4 5i
44 12
47 31
50 51
' 54 ii
57 30
60 50
ii
2 07 30
62 ^o
ii
3 21 40 .
64 10
12
13
'or r *:
12
13
2 17 30
1 IO OO
67 30
70
70
14
I 10 00 i _
14
00
15
16
17
18
19
20
2 22 30
3 38 20
4 57 30
6 19 59
7 45 48
9 14 56
72 30
75 50
79 i
82 29
85 49
89 08
15
16
17
IB
'9
20
i 15 oo
2 32 30
3 53 20
5 17 30
6 44 59
8 15 48
75
77 30
80 50
84 10
87 29
90 49
INST. AT 15. j 20 co'.
INST. AT 16. s = 22 40'.
No. of
Point.
Deflection from
aux. tan.
Diff. of
Deflection.
No. of
Point.
Deflection from
aux. tan.
Diff. of
Deflection.
O
I
2
3
4
6
8
1 3 07' 03"
12 37 49
12 05 16
ii 29 23
10 50 TO
10 07 37
9 21 45
8 32 34
7 40 02
29' 14"
32 33
35 53
39 J 3
42 33
45 52
49 ii
52 32
O
I
2
3
4
5
6
7
8
14 53' 03"
14 22 09
13 47 54
13 10 20
12 29 26
ii 45 12
10 57 39
10 06 46
9 12 34
30' 54"
34 15
37 34
40 54
44 14
47 33
50 53
54 12
9
10
ii
12
13
14
15
6 44 ii
5 45 oi
4 42 30
3 36 40
2 37 30
i 15 oo
oo
55 5 1
59 10
62 31
65 50
69 10
72 30
75
So
9
10
ii
12
13
14
15
8 15 03
6 14 ii
6 10 qi
5 02 30
3 5i 40
2 37 30
I 2O OO
57 3 T
60 52
64 10
67 3i
70 50
74 10
77 30
80
16
I 20 OO
16
00
17
18
19
20
2 42 30
4 08 20
5 37 30
7 09 59
85 50
89 10
92 29
17
18
19
20
i 25 oo
2 52 30
4 23 20
5 57 30
87 30
90 50
94 10
56
TABLE II. DEFLECTION ANGLES.
INST. AT 17. j = 25 30'.
INST. AT 18, -y 28 30'.
No. of Deflection from! DiiL of
No. of
Deflection from
Diff. of
Point.
aux. tan.
Deflection.
Point.
aux. tan.
Deflection.
I
2
3
4
5
6
7
8
9
10
ii
12
16 45' 48'
16 13 ii
15 37 15
14 57 59
14 15 24
13 29 29
12 40 14
ii 47 41
10 51 47
9 52 35
8 50 03
' 7 44 12
6 35 oi
32' 37"
36 56
39 i 6
42 35
45 55
49 15
52 33
55 54
59 12
62 32
65 51
69 IT
O
I
2
3
4
5
6
7
8
9
10
ii
12
IS 10 59
17 33 21
16 52 23
16 08 05
15 20 28
14 29 32
13 35 17
12 37 42
ii 36 49
10 32 36
9 25 03
8 14 12
34' 1 8"
37 38
40 58
44 18
47 37
50 56
54 15
57 35
60 53
64 13
67 33
70 51
13
5 22 30
72 31
13
7 oo oi
74 it
14
15
16
17
4 06 40
2 47 30
i 25 oo
oo
75 5
79 10
82 30
85
14
15
16
17
5 4 2 30
4 21 40
2 57 30
i 30 oo
77 3 r
80 50
84 10
87 30.
18
i 30 oo
9
18
00
9
19
20
3 02 30
4 38 20
92 30
95 50
19
20
i 35 oo
3 12 30
95
97 30
INST. AT 19. j = 31 40'.
INST. AT 20. j- = 35 oo' .
No. of Deflection from
Diff. of
! No. of
Deflection from
Diff. of
Point.
aux. tan.
Deflection .
i Point.
aux. tan.
Deflection.
I
2
3
4
5
6
7
8
9
10
n
12
13
20 5 1' 33"
20 15 32
19 36 II
18 53 3i
18 07 31
17 IS 12
16 25 33
15 29 36
14 30 20
13 27 44
12 21 50
II 12 36
10 oo 04
8 44 12
36' 01 "
39 21
42 40
46 oo
49 J 9
52 39
55 57
59 16
62 36
65 54
69 14
75 32
75 52
70 II
I
2
3
4
5
6
8
9
10
ii
12
13
23 04' 36"
22 26 52
21 45 48
21 01 25
20 13 42
19 22 40
18 28 19
17 30 39
16 29 40
15 25 23
14 17 46
13 06 51
ii 52 37
10 35 04
37' 44"
41 04
44 23
47 43
51 02
54 21
57 40
60 59
64 17
67 37
7'^> 55
74 -14
77 33
80 52
M
15
16
17
18
19
7 25 oi
6 02 30
4 36 40
3 07 30
i 35
oo
82 31
85.50
89 10
92 30
95
IOO
14
15
16
17
18.
19
9 r 4 12
7 50 oi
6 22 30
4 51 40
3 17 30
i 40
84 ii
87 31
90 50
94 10
97 30
1 60
20
i 40
20
00
57
TABLE III.
DEGREE OF CURVE AND VALUES OF THE COORDINATES x AND
y, FOR EACH CHORD-POINT OF THE SPIRAL FOR VARIOUS
LENGTHS OF THE CHORD.
f. CHORD-LENGTH = IO.
n.
nc.
/>.
y>
Xt
Log x .
j
10
i 40' oo"
IO.OOO
0.0145
8.162696
2
20
3 20 02
20. ooo
.0727
8.861664
3
30
5 oo 06
29.99-)
. 2036
9.308815
4
40
6 40 13
39.996
.4363
9.639792
5
50
8 20 26
49.990
.7998
9.903002
6
60
10 oo 45
59-976
1.323
0.121624
7
70
II 41 12
69.951
2.035
0.308622
8
80
13 21 48
79.907
2.965
0.471991
9
9
ID 02 34
89.838
4.140
0.617015
10
100
16 43 3i
99.732
5-589
0.747370
ii
no
18 24 42
109 578
7-340
0.805712
12
120
20 06 07
119-359
9.419
0.974022
13
130
21 47 48
129.059
11.853
1.073818
14
140
23 29 46
138.655
14.665
1.166281
15
150
25 12 02
148.125
17.879
.252352
16
160
26 54 39
157.439
21.517
.332788
17
170
28 37 38
166.569
25-598
.408205
18
1 80
3O 21 01
175-479
30 138
.479112
19
i go
32 04 48
184.132
35.150
545931
20
200
33 49 02
192.487
40.645
.609013
35 33 46
TABLE III.
r. CHORD-LENGTH = n.
f c.
D s .
y-
x.
Log x.
I
II
i 30' 55"
1 1 . CO3
0.0160
8.204089
2 j 22
3 oi 50
22.OOO
.0800
8.903057
3
33
4 32 48
32.999
.2240
9.350208
4
44
6 03 48
43.996
4799
9 681185
5
55
7 34 52
54.989
.8798
9.944394
6
66
9 06 01
65.974
1.456
0.163017
7
77
10 37 16
76.946
2.239
0.350015
8
88
12 08 37
87.898
3.261
0-513384
9
99
13 40 06
98.822
4-554
0.658408
JO
no
15 ii 44
109.706
6.148
0.788763
IT
121
16 43 3i
120.536
8.074
0.907104
12
132
18 15,29
I3L295
10.361
1.015415
13
143 ! 19 47 39
141.965
13-038
1.115210
14
154 I 21 20 01
152.521
16.131
1.207674
15
165 22 52 38
162.937
19.667
.293745
16 | 176
24 25 29
173.183
23.669
.374180
17
I8 7
25 58 36
183.226
28.158
.449598
18
198
27 32 01
193.027
33.152
.520505
19
20g
29 05 45
202.545
38.665
587323
20
220
30 39 48
2H.735
44.710
.650405
32 14 ii
. c. CHORD-LENGTH = 12.
;/.
11C.
D s . -
y-
jr.
Log jr.
I
12
i 23' 20"
12.000
0.0175
8.241877
2
24
2 46 41
24.OOO
.0873
8.940845
3
36
4 10 03
35-999
2443
9o87997
4
4 8
5 33 28
47.996
.5236
9.718974
5
60
6 56 55
59-988
.9598
9.982183
6
72
8 20 26
71.971
1.588
o. 200806
7
8 4
9 44 oi
83.941
2.442
0.387803
8
9 6
ii 07 42
95.889
3.558
0.551172
9
108
T2 31 28
107.806
4.968
0.696196
10
120 13 55 21
119.679
6.707
0.826551
ii
132
15 19 22
I3L493
8.8c8
0.944893
12
144
16 43 31
143.231
11-303
.053204
13
156
18 07 48
154.871
14.223
.152999
14
168 19 32 15
166.386
17-598
.245462
15
180 20 56 53
177-749
21-455
.331533
16 192 22 21 43
188.927
25.821
.411969
17 204
23 46 44
199.883
30.718
.487386
18 216 25 ii 59
210.575
36.165
558293
19 228 26 37 28
220.958
42.181
.625113
20
240
28 03 12
230.984
48.774
.688194
29 29 12
59
TABLE III.
c. CHORD-LENGTH = 13.
n.
nc.
Ds.
y-
X.
Log x.
l
13
i 16' 55"
13.000
0.0189
8.276639
2
26
2 33 52
26.000
0945
8.975607
3
39
3 50 49
38.999
.2647
9 422759
4
52
5 07 48
51-995
5672
9.753736
5
65
6 24 49
64.987
1.040
0.016945
6
78
7 4i 53
77.969
1.720
0.235568
7
91
8 59 oo
90.936
2.646
0.422565
8
104
10 16 12
103.879
3.854
0.585934
9
117
ii 33 28
116.789
5-382
0.730959
JO
130
12 50 49
129.652 -
7.266
0.861313
ii
143 ! 14 08 16
142.451
9-542
0-979655
12
156
15 25 50
155.167
12.245
.087966
13
169
16 43 30
167.776
15.409
.187761
14
182
i3 01 18
180.252
19.064
.280224
15
195
IQ 19 14
192.562
23.243
.366295
16
208
20 37 20
204.671
27.972
.446731
17
221
21 55 34
216.540
33.277
.522148
18
234
23 14 oo
228.123
39-179
.593055
19
247
24 32 35
239.371
45.696
.659874
20
260
25 5i 23
250.233
52.839
1.722956.
27 10 23
c. CHORD-LENGTH = 14.
71.
11C.
A
y-
X.
Log x.
I
14
i n' 26"
14.000
0.0204
8.308824
2
28
2 22 52
28. coo
.1018
9.007792
3
42
3 34 19
41.999
.2851
9-454943
4
56
4 45 48
55.995
.6108
9.785920
5
70
5 57 18
69.986
I.I2O
0.049130
6
8 4
7 "8 51
83.966
1.852
0.267752
7
9 8
8 20 26
97.931
2.849
0.454750
8
112
9 32 04
111.870
4-I5I
0.618119
9
126
10 43 47
125 773
5.796
0.763M3
10
140
ii 55 33
139.625
7.825
0.893498
TI
154
13 07 24
153-409
10.276
1.011840
12
1.68
14 19 20
167.103
13.187
1.120150
13
182
15 31 22
180.682
16.594
1.219946
14
196
16 43 29
194.117
20.531
1.312409
15
2IO
*7 55 44
207.374
25.031
1,398480
16
224
19 c6 05
220.415
30.124
1.478915
17
238 i 20 20 34
233.196
35-837
1-554333
1-8
252 21 33 ii
-245-670
42.193
1.625240
J 9
266 22 45 56
257./35
49.211
1.692059
20
280
23 53 51
269.481
56.903
I.755I4I
25 ii 55
60
TABLE III.
c. CHORD-LENGTH = 15.
?/. j nc.
D s .
y.
x.
Log x.
i IS
i 06' 40"
15.000
0.0218
8.338787
2
30
2 13 2O
30. ooo
.1091
9-037755
3-
45
3 20 02
44.998
.3054
9.484907
4
60
4 26 44
59.994
.6545
9.815884
5
75
5 33 28
74.984
1.200
0.079093
6
90
6 40 13
89.964
1.985
0.297716
7
105
7 47 oi
104.926
3-053
0484713
8
1 20
8 53 5i
119.861
4-447
0.648082
9
135
10 oo 45
134.757
6.2IG
0.793107
10
150
ii 07 41
149-599
8.384
o 923461
ii
165
12 14 41
164.367
II.OIO
1.041803
12
1 80
13 21 47
179.039
14.129
1.150114
13
195
14 28 56
193.588
17-779
.249909
14
210
15 36 09
207.983
21.997
.342372
15
225
16 43 28
222.187
26.819
428443
16
2 4
17 50 54
236.159
32.276
.508879
17
255
18 58 25
249.853
38.397
.584296
18
270
20 06 02
263 218
45.207
.655203
19
285
21 13 47
276.198
52.726
1.722022
20
3 00
22 21 39
288.730
60.968
1.785104
23 29 48
c. CHORD-LENGTH = 16.
;/.
nc
D s . -
y-
jr.
Log jr.
I
16
i 02' 30"
1 6 ooo
0.0233
8.366816
2
32
2 05 00
32.000
.1164
9.065784
3
48
3 07 31
47.998
.3258
9-5I2935
4
64
4 10 03
63.994
.6981
9.843912
5
80
5 12 36
79-983
1.260
0.107122
6
96
6 15 ii
95.961
2.117
0.325744
7
112
7 17 47
111.921
3-256
0.512742
8
128
8 20 26
127.852
4-744
0.676111
9
144
9 23 07
143-74I
6. 624
o 821135
10
1 60
10 25 51
159-572
8-943
0.951490
ii
I 7 6
ii 28 37
175.325
11.744
.069832
12
IQ2
12 31 28
190.975
15.071
.178142
13
208
13 34 21
206.494
18.964
.277938
14
224
14 37 20
221.848
23.464
.370401
15
240
15 4 21
236.999
28.607
.456472
16
256
16 43 28
251.903
34.428
53 f >97
17
272
17 46 40
266.510
40.957
.612325
18
288
18 49 57
280.766
48.221
.683232
19
304
19 53 20
294.611
56.241
.750051
20
320
20 56 49
307.979
65.032
813133
22 00 23
61
TABLE III.
t. CHORD-LENGTH = 17.
//.
11C.
>*.
y+
-V.
Log x.
i
17
o 58' 49"
17.000
0.0247
8.393M5
2
34
i 57 33
34.000
.1236
9.092113
3
51
2 56 27
50.998
.3461
9 539 2 64
4
68
3 55 19
67.994
7417
9.870241
5
85
4 54 12
84.982
1.560
O.I3345I
6
IO2
5 53 06
101.959
2.249
0.352073
7
II 9
6 52 oo
118.916
3-460
0.539071
8
I 3 6
7 50 57
1 3 5- -842
5.040
o. 702440
9
153
8 49 55
152.725
7.038
0.847464
10
I 7
9 48 56
169.545
9.502
0.977819
ii
I8 7
10 48 oo
186.282
12.478
1.096161
12
204
ii 47 07
202.911
16.013
1.204471
13
221
12 46 15
219.400
20.150
1.304267
14
238
13 45 27
235.7H
24-930
396730
15
255
14 44 44
251.812
30.395
.482801
16
272
15 44 03
267.647
36.579
.563236
17
289
16 43 27
283.167
43.5I6
.638654
18
306
17 42 56
298.314
5L234
.709561
19
323
18 42 29
313.024
59.756
.776380
20
340
19 42 07
327.228
69.097
.839462
20 41 49
f. CHORD-LENGTH = 18.
;/.
ftf.
$
y.
a~.
Log^r.
I
18
o 55' 33"
18.000
0.0262
8.417968
2
36
i 51 07
36.000
.1309
9.116937
3
54
2 46 40
53.998
.3665
9.564088
4
72
3 42 16
71.993
.7853
9.895065
5
90
4 37 5i
89.981
1.440
0.158274
6
1 08
5 33 28
107-957
2.382
0.376897
7
126
6 29 05
125.911
3-663
0.563894
8
144
7 24 45
143.833
5-337
0.727263
9
162
8 20 26
161.708
7-452
0.872288
10
180
9 16 08
179.518
10. 06 1
1.002643
ii
198
10 ii 54
197.240
13.212
1.120984
12
216
ii 07 41
214.847
16.955
1.229295
13
234
12 03 31
232. 3c6
21-335
1.329090
14
252
12 59 24
249.579
26.397
I.42I554
15
270
13 55 20
266.624
32.183
1.507624
16
288
14 51 18
283.391
38.731
1.588060
17
306
15 47 20
299.824
46.076
1.663477
18
324
16 43 27
315.862
54.248 1.734385
*9
342
17 39 37
33L437
63.271 1.801203
20
360
18 35 5i
346.476
73.161
1.864285
19 32 08
62
TABLE III.
c. CHORD-LENGTH = 19.
n.
lie.
D s .
y>
x.
Log^r.
i
19
o 52' 38"
19.000
0.0276
8.441450
2
3^
i 45 16
38.000
.1382
9.140418
3
57
2 37 54
56.998
.3869
9-587569
4
76
3 30 34
75-993
.8290
9-9i 8 546
5
95
4 23 13
94.980
1.520
0.181755
6
H4
5 15 54
113-954
2.514
0.400378
7
133
6 08 36
132.906
3.867
0.587376
8
152
7 01 19
151.824
5.633
0.750744
9
171
7 54 03
170.692
7.866
0.895769
10
190
8 46 49
189.491
10.620
1.026124
ii
209
9 39. 36
208.198
13.947
1.144465
12
228
10 32 26
226.783
17.897
1.252776
13
247
ii 25 18
245.212
22.520
.352571
14
266
12 18 12
263.445
27.863
445035
15
285
13 II 09
281.437
33-971
53H05
16
304
14 04 09
299.135
40.883
.611541
17
323
14 57 ii
316.481
48.636
.686958
18
342
15 50 16
333-410
57.262
.757866
19
361
16 43 25
349- 8 5i
66.786
.824684
20
380
17 36 33
365-725
77.226
.887766
18 29 54
c. CHORD-LENGTH = 20.
11.
nc.
D s . *
}'-
X.
Log x.
i
20
o 50' oo"
20.000
0.0291
8.463726
2
40
i 40 oo
40.000
.1454
9.162694
3
60
2 30 01
59.998
.4072
9.609845
4
80
3 20 02
79-993
.8726
9.940822
5
IOO
4 10 03
99-979
i. 600
0.204032
6
120
5 oo 05
119-952
2.646
0.422654
7
140
5 5 8
139.901
4.071
0.609652
8
1 60
6 40 13
159 8i5
5.930
0.773021
9
180
7 30 18
179.676
8.280
0.918045
10
200
8 20 26
199.465
11.179
1.048400
ii
22O
9 i 34
219.156
14.681
1.166742
12
240
10 oo 44
238.719
18.839
1.275052
13
260
10 50 56
258.118
23-705
1.374848
14 1 280
ii 41 10
277-310
29.330
1.467311
15
3OO
12 31 26
296.249
35-759
1.553382
16
320
13 21 45
314.879
43-035
1.633817
17
340
14 12 c6
333.138
51.196
1.709235
18
360
15 02 29
350.958
60.276
1.780142
19
380
15 52 55
368.264
70.301
1.846961
20
400
16 43 25
384.974
81.290
1.910043
17 33 &
TABLE III.
c. CHORD-LENGTH = 21.
i
n.
nc.
D s .
}'
X.
Log. x.
I
21
o 47' 37"
2 1 . OOO
0.0305
8.484915
2
42
i 35 14
42.000
.1527
9.183883
3
63
2 22 52
62.998
.4276
9-63I035
4
84
3 10 30
83.992
.9162
9.962012
5
105
3 58 08
104.978
1.680
0.225221
6
126
4 45 47-
125.949
2.779
0.443844
7
147
5 33 27
146. 896
4.274
0.630841
8
168
6 21 08
167.805
6.226
0.794210
9
189
7 08 50
188.660
8.694
0.939235
TO
2IO
7 56 33
209.438
11.738
.069589
II
231
8 44 18
230.114
15.415
.187931
12
252
9 32 03
250.655
19.781
.296242
13
273
10 19 51
271.023
24.891
396037
M
294
n 07 40
291.176
30.796
.488500
15
315
ii 55 3i
3II.062
37.547
.574571
16
336
12 43 24
330.623
45.186
.655007
17
357
13 31 20
349-795
53.756
.730424
18
378
14 19 17
368. 506
63.289
.801331
IQ
399
15 07 17
386.677
73.816
.868150
15 55 19
r. CHORD-LENGTH = 22.
;/.
nc.
D s .
y-
X.
Log. x.
i
22
45' 27"
22.000
0.0320 .
8.505119
2
44
i 30 53
44. ooo
.1600
9 204087
3
66
2 l6 22
65.998
.4480
9.651238
4
88
3 oi 50
87.992
9599
9.982215
5
no
3 47 18
109.977
1.760
0.245424
6
132
4 32 48
I3L947
2.911
0.464047
/
154
5 18 18
153.891
4.478
0.651045
8
176
6 03 48
I75-796
6.522
0.814414
9
198
6 49 19
197.643
9.108
0.959438
10
220
7 34 5i
219.411
12.297
.089793
ii
242
8 20 25
241.071
16.149
.208134
12
264
9 06 oo
262.591
20.7^3
.316445
13
286
9 5i 36
283.929
26.076
.416240
14
308
10 37 13
305042
32.263
.508704
15
330
ii 22 53
325.874
39-335
594775
1.6
352
12 08 34
346.367
47.338
.675210
17
374
12 54 16
366.451
56.3*5
1.750628
18
396
13 40 oi
386.054
66.303
1.821535
14 25 49
0-!-
TABLE III.
r. CHORD-LENGTH = 23.
n.
nc.
D*.
y-
X.
Log. x.
I
23
o'"' 43' 29"
23.000
0.0335
8.524424
2
46
I 26 58
46.000
.1673
9.223392
3
69
2 10 26
68.998
.4683
9-670543
4
92
2 53 56
91.991
1.004
0.001520
5
H5
3 37 26
114.976
.1.840
0.264729
6
138
4 20 56
137-945
3.043
0.483352
7
161
5 04 26
160.886
4.681
0.670350
8
184
5 47 58
. 183.787
6.819
0.833719
9
207
6 31 30
206.627
9.522
0.978743
10
230
7 15 04
229.384
12.856
1.109098
ii
253
7 58 38
252.029
16.883
.227439
12
276
8 42 13
274 527
21.665
.335750
*3
299
9 25 49
296.835
27.261
435545
14
322
10 09 27
318.907
33.729
.528009
15
345
10 53 06
340.686
41.123
.614080
16
368
1 1 36 47
362.110
49.490
694515
17
391
12 20 29
383.108
58.875
.769933
13 04 13
r. CHQRD-LENGTH = 24.
72.
nc.
D s .
;'
X.
Log. r.
I
24
41' 40"
24. ooo
0.0349
8.542907
2
48
1 23 20
48.000
.1745
9.241875
3
72
2 05 OO
71.998
.4887
9.689027
4
96
2 46 41
95.991
1.047
0.020004
5
1 20
3 28 22
119-975
1.920
0.283213
6
144
4 10 03
143.942
3.176
0.501836
7
1 68
4 5r 45
167.881
4.885
0.688833
8
192
5 33 28
191-777
7.115
0.852202
9
216
6 15 10
215.611
9.936
0.997226
10
240
6 5^ 54
239-358
13.415
1.127581
n
264
7 38 39
262.987
17.617
1.245923
12
288
8 20 25
286.463
22.607
i 354234
13
14
312
336
9 02 12
9 44 oo
309.741
332.773
28.446
35.196
.1.454029
1.546492
15
360
10 25 48
355.499
42.910
1.632563
16
384
Ji 07 30
377.854
51.641
1.712999
17
408
ii 49 31
399-765
61.435
1.788416
12 31 25
65
TABLE III.
C. L.tlU
IvU-I^l^INLrl
n = 25.
n.
nc.
D s .
y*
X.
Log. x.
I
25
o 40' oo"
25.000
0.0364
8.560636
2
50
I 2O CO
50.000
.1818
9.259604
3
75
2 00 00
74-997
.5090
9-706755
4
100
2 40 OI
99.991
1.091
0.037732
5
I2 5
3 20 02
124.974
2.OOO
0.300942
6
150
4 oo 03
149.940
3-308
0.519564
7
175
4 40 04
174.876-
5.088
0.706562
8
2GO
5 20 06
199.768
7.412
0.869931
9
225
6 oo 09
224-595
IO.35O
014955
10
250
6 40 13
249-33I
13-974
.145310
ii
275
7 20 17
273-945
18.351
.263652
12
300
8 00 22
298.398
23.548
.371962
13
325
8 40 28
322.647
29.632
.471758
14
350
9 20 35
346.638
36.662
.564221
'15
375
10 oo 43
37-3i i
44-698
.650292
16
400
10 40 52
393- 59 8
53-793
.730727
II 21 03
f. CHORD-LENGTH = 26.
n.
nc.
D s .
>'
X.
Log. x.
I
26
o 38' 28"
26.000
0.0378
8.577669
2
52
i 16 56
52.000
.1891
9.276637
3
78
i 55 24
77.997
.5294
9.723789
4
104
2 33 52
103.990
1. 134
0.054766
5
130
3 12 20
129.973
2.080
0.317975
6
156
3 50 48
155.937
3-440
0.536598
7
182
4 29 18
181.871
5.292
0.723595
8
208
5 07 48
207.759
7.708
0.886964
9
234
5 46 18
233.579
10.764
1.031989
10
260
6 24 48
259-304
14.533
1.162343
ii
286
703 20
284.903
19.085
1.280685
12
312
7 4 1 52
310.334
24.490
1.388996
13
338
8 20 25
335-553
30.817
1.488791
14
364
8 58 59
360.504
38.129
1.581254
15
39
9 37 33
385.124
46.486
1.667325
10 1 6 OQ
66
TABLE III.
c. CHORD-LENGTH = 27.
n.
11C.
D t .
}'
X.
Log. x.
I
27
o 37' 02"
27.000
0.0393
8.594060
2
54
i 14 04
54.000
.1963
9.293028
3
81
i 5i 07
80.997
.5498
9.740179
4
108
2 28 10
107.990
1.178
0.071156
5
135
3 05 12
I34-972
2.160
0.334365
6
162
3 42 15
161.935
3-573
0.552988
7
189
4 19 19
188.866
5-495
0.739986
8
216
4 56 23
215-750
8.005
0.903355
9
243
5 33 28
242.562
11.178
1.048379
10
270
6 10 32
269.277
15.092
LI78734
n
297
6 47 38
295.860
19.819
1.297075
12
324
7 24 44
322.270
25.432
1.405386
13
35i
8 or 51
348.459
32.002
1.505181
14 378
8 38 59
374-369
39-595
1.597645
15
405
9 16 07
399.936
48.274
1.683716
9 53 16
c. CHORD-LENGTH = 28.
n.
nc.
/&
y-
jr.
Log. jr.
I
28
o 35' 42"
28.000
0.0407
8.609854
2
56
i ii 26
55-999
.2036
9.308822
3
84
i 47 08
83.997
.5701
9-755973
4
112
2 22 52
in. 990
1.222
0.086950
5
140
2 58 36
139-97I
2.24O
0.350160
6
168
3 34 19
1 67.933
3-705 '
0.568782
7
196
4 10 03
195.862
5.699
o.755/So
8
224
4 45 48
223.740
8.301
0.919149
9
252
5 21 32
251.546
H.592
1.064173
10
280
5 57 17
279.251
15.650
1.194528
ii
308
6 33 03
306.818
20.553
1.312870
12
336
7 08 50
334.206
26.374
1.421180
13
364
7 44 36
361.365
33.188
1.520976
14
392
8 20 24
388.235
4I.O62
I.6I3439
8 56 13
67
TABLE III.
CHORD-LENGTH - 29.
n.
nc.
D s .
y-
X.
Log. oc.
I
29
o 34' 29"
29.000
0.0422
8.625094
2
53
i 08 58
57-999
.2109
9.324062
3
8?
I 43 27
86.997
595
9.771213
4
116
2 17 56
115-989
1.265
0.102190
5
145
2 52 26
144.970
2.320
0.365400
6
174
3 26 55
173-930
3-837
0.584022
7
203
4 01 26
202.857
5.902
0.771020
8
232
4 35 56
231.731
8.598
0.934389
9
261
5 10 26.
260.530
12.006
.079413
10
290
5 44 57
289.224
16.209
.209768
ii
319
6 19 29
317.776
21.287
.328110
12
348
6 54 01
346.142
27-316
.436420
13
377
7 28 34
374.271
34-373
.536216
14
406
8 03 07
402.100
42.528
.628679
8 37 40
CHORD-LENGTH = 30.
n.
nc.
Ds.
y-
jr.
Log. x.
T
3
o 33' 20"
30.000
0.0436
8.639817
2
60
i 06 40
59-999
.2182
9-338785
3
90
i 40 oo
89.997
.6108
9-785937
4
120
2 13 20
119.989
1.309
0.116914
5
150
2 46 41
149.969
2.400
0.380123
6
1 80
3 20 02
179.928
3-970
0.598746
7
210
3 53 22
209.852
6.106
0.785743
8
240
4 26 44.
239.722 .
8.894
0.949112
9
270
5 oo 05
269.514
12.420
.094137
10
3CO
5 33 27
299.197
16.768
.224491
ii
330
6 06 49
328.734
22.021
.^42833
12
360
6 40 12
358.078
28.258
.451144
13
39
7 13 36
387.176
35.558
.550939
7 47 oo
68
TABLE ,./
'
/
c. CHORD-LENGTH = 31.
n.
nc.
E ,
y
*.
Log x.
i
31
o 32' 15"
31.000
0.0451
8.654058
2
62
i 04 31
61.999
.2254
9.353026
3
93
i 36 47
92.997
.6312
9.800177
4
124
2 09 C2
123.988
1-353
0.131154
5
155
2 41 18
154.968
2-479
0.394363
6
1 86
3 13 34
185 925
4.102
0.612986
7
217
3 45 50
216.847
6.309
0.799984
8
248
4 1 8 07
247-713
9.191
0-963353
9
279
4 50 24
278.498
12.834
1.108377
10
310
5 22 41
309. 1 70
17.327
1.238732
ii
341
5 54 59
339.692
22.755
1.357073
12
372
6 27 17
370.014
29.200
1.465384
13
6 59 35
400.082
36.743
1.565179
7 3i 53
\
CHORD-LENGTH = 32.
n.
nc.
D s .
y-
X.
Log*.
I
3 2
o 31' 15"
32.000
0.0465
8.667846
2
64
I 02 30
63 999
.2327
9.366814
3
96
i 33 45
95-997
.6516
9.813965
4
128
2 05 00
127.988
1.396
0.144942
5
1 60
2 36 16
159.967
2-559
0.408152
6
192
3 07 31
101.923
4-234
0.626774
7
224
3 38 47
223.842
6.513
0.813772
8
256
4 10 03
255.703
9.487
0.977141
9
288
4 4i 19
287.481
13.248
1.122165
10
320
5 12 36
319.144
17.886
1.252520
ii
352
5 43 53
350.649
23.489
1.370802
12
384
6 15 10
381.950
30. 142
I.479 r 72
13
416
6 46 28
412.988
37.929
1.578968
7 17 46
69
TABLE III.
c. CHORD-LENGTH = 33.
n.
12C.
D s .
y-
X.
Log. x.
i
33
o 30' 19"
33.000
0.0480
8.681210
2
66
I OO 36
65.999
.2400
9.380178
3
99
i 30 55
98.997
.6719
9.827329
4
132
2 OI 13
131.988
1.440
0.158306
5
165
2 3 I 3 2
164.966
2.639
0.421516
6
198
3 oi 50
197.921
4-367
0.640138
7
231
3 32 09
230.837
6.716
0.827136
8
264
4 02 28
263.694
9.784
0.990505
9
297
4 32 48
296.465
13.662
1-135529
JO
33
5 03 07
329.117
18.445
1.265884
n
363
5 33 27
361.607
24.223
1.384226
12
396
6 03 47
393.886
31-084
1.492536
6 3.4 07
c. CHORD-LENGTH = 34.
n.
nc.
&*
y.
X.
Log. x.
I
34
o 29' 25"
34.000
0.0495
8.694175
2
68
o 58 49
67.999
.2473
9093I43
3
102
i 28 14
101.996
.6923
9.840294
4
136
i 57 39
135.987
1.483
O.I7J27I
5
170
2 27 04
169.965
2.719
0.434481
6
204
2 56 2()
203.918
4.499
0.653103
7
233
3 25 55
237.832
6 920
0,840101
8
272
3 55 20
271.685
10.080
1.003470
9
306
4 24 46
305.449
14.076
1.148494
10
340
4 54 12
339.090
19.004
1.278849
ii
374
5 23 38
372.565
24-957
1.397191
12
408
5 53 05
405.822
32.026
1.505501
6 22 II
70
TABLE III.
c. CHORD-LENGTH = 35.
;/.
11C.
D s .
y-
A\
Log a\
I
35
o 28' 34"
35.000
0.0509
8.706764
2
70
o 57 09
69.999
2545
9.405732
3
105
i 25 43
104.996
.7127
9.852883
4
140
I 54 17-
139.987
1.527
0.183860
5
175
2 22 52
174.964
2.799
0.447070
C
210
2 51 27
209.916
4.631
0.665692
7
245
3 20 01
244.827
7.123
0.852690
8
280
3 4S 36
279.675
10.377
.016059
9
3J5
4 17 12
314-433
14.490
.161083
10
35
4 45 47
349- 6 3
I9-563
.291438
ii
385
5 14 23
383-523
25.691
.409780
12
420
5 43 oo
417.758
32.968
.518090
6 09 36
\
c. CHORD-LENGTH = 36.
//. tic.
D s .
y-
X.
Log jc.
I
36
o 27' 47"
36.000
O.O524
8.718998
2
72
o 55 33
71.999
.26l8
9.417967
3
io3
I 23 20
107.996
7330
9.865118
4
144
I 51 07
143.987
I-57I
0.196095
5
1 80
2 18 54
179.963
2.879
0.459304
6
216
2 46 41
215.913
4- 764
0.677927
7
252
3 14 28
251.822
7.327
0.864924
8
2:8
3 42 15
287.666
10.673
1.028293
9
324
4 10 03
323-417
14.905
1.173318
10
360
4 37 5i
359037
20. 122
1.303673
ii
39 6
5 05 39
394.480
26.425
1.422014
5 33 27
TABLE III.
11.
11C,
z>*
/
x.
Log x.
I
37
o 27' 02"
37.000
0.0538
8.730898
2
74
o 54 03
73-999
.2691
9.429866
3
in
I 21 05
110.996
7534
9.877017
4
148
I 48 07
147.986
1.614
0.207994
5
185
2 1 5 09
184.962
2-959
0.471203
6
222
2 42 II
221.911
4.896
0.689826
7
259
3 09 ] 3
258.817
7-530
0.876824
8
296
3 36 15
295.657
10.970
1.040193
9
333
4 03 17
332.400
15-319
1.185217
10
370
4 30 20
369.010
20.681
L3I5572
ii
407
4 57 23
405-438
27.159
I.4339I3
5 24 26
e. CHORD-LENGTH = 37.
c. CHORD-LENGTH = 38.
11.
m\
D*
*
x
Log jr.
i
38 o'' 26' 19"
38.000
0-0553
8.742480
2
76
o 52 39
75-999
.2763
9.441448
3
114
i 18 57
113.996
7737
9.888599
4
152
i 45 16
151.986
1.658
0.219576
5
190
2 ii 35
189.961
3.039
0.482785
6
228
2 37 54
227.909
5.028
0.701408
7
266
3 04 14
265.812
7-734
0.888406
8
304
3 30 33
303.648
11.266
1.051774
9
342
3 56 53
34L384
15.733
1.196799
10
380
4 23 13
378.983
21.240
I.327T54
ii
418
4 49 33
416.396
27.893
1-445495
5 15 53
72
TABLE III.
c. CHORD-LENGTH = 39.
//.
nc.
D*.
>'
X.
Log x .
I
39
o 25' 38"
39.000
0.0567
8.753761
2
78
o 51 17
77-999
.2836
9.452729
3
117
I 16 55
116.996
.7941
9.899880
4
156
I 42 34
I55.9 8 5
1.702
0.230857
5
*95
2 08 13
194.960
5.119
0.494066
6
234
2 33 51
233.906
5.160
0.712689
7
273
2 59 30
272.807
7.938
0.899687
8
312
3 25 09
311.638
11.563
1.063055
9
35i
3 50 48
350.368
16.147
1.208080
10
390
4 16 28
388.956
21.799
L338435
4 42 07
c. CHORD-LENGTH ~ 40.
n.
1IC.
D s .
}'
X.
Log x.
I
40
o 25' oo''
40. ooo
0.0582
8.764756
2
80
o 50 oo
79-999
.2909
9.463724
3
120
I 15 oo
1 1 9. 996
.8145
9.910875
4
1 6O
I 40 oo
I59-985
1.745
0.241852
5
2OO
2 05 CO
199-959
3.199
0.505062
6
240
2 30 01
239.904
5.293
0.723684
7
280
2 55 oi
279.802
8.141
0.910682
8
32O
3 20 oi
319.629
11.859
1.074051
9
360
3 45 02
359-352
16.561
1.219075
10
4OO
4 10 03
398.929
22.358
1.349430
4 35 03
c. CHORD-LENGTH = 41.
n.
nc.
/>*
y-
X.
Log x .
i
41
o 24' 24"
41.000
0.0596
8.775480
2
82
o 48 47
81.999
.2982
9.474448
3
123
i 13 10
122.996
.8348
9.921599
4
164
i 37 34
163 985
1.789
0.252576
5
205
2 oi 57
204.958
3-*79
0.515786
6
246
2 26 21
245.901
5-425
o. 734408
7
287
2 50 45
286.797
8.345
0.921406
8
328
3 15 09
327.620
12.156
1.084775
9
369
3 39 33
368.336
16.975
1.229799
10
410
4 03 57
408.903
22.917
1.360154
4 28 21
73
TABLE III.
c. CHORD-LENGTH = 42.
n.
11 r.
D s .
/
X.
Log x .
i
42
o 23' 49"
42.000
0.0611
8.785945
2
84
o 47 37
83.999
.3054
9.484913
3
126
i n 26
125.996
.8552
9.932065
4
168
i 35 14
167.984
1.832
0.263042
5
2IO
i 59 02
209.957
3-359
0.526251
6
252
2 22 52
251.899
5-557
0.744874
7
294
2 46 41
293.792
8.548
0.931871
8
336
3- 10 30
335.6ii
12.452
1.095240
9
373
3 34 19
377.319
17-389
1.240265
10
420
3 58 08
418.876
23.476
1.370619
4 21 57
f. CHORD-LENGTH = 43.
.
nc.
/>*.
y>
X.
Log x.
i
43
o 23' 15"
43.000
0.0625
8.796164
2
86
o 46 31
85.999
3127
9-495I33
3
129
I 09 46
128.996
.8755
9.942284
4
172
i 33 02
171.984
1.876
0.273261
5
215
i 56 17
214-955
3-439
0.536470
6
2^8
2 19 33
257-897
5-690
0.755093
7
301
2 42 48
300.787
8-752
0.942090
8
344
3 06 04
343.601
12.749
1.105459
9
3S7
3 29 20
386.303
17.803
1.250484
10
430
3 52 35
428.849
24.035
1.380839
4 15 50
c. CHORD-LENGTH = 44.
n.
nc.
D s .
}'
X.
Logx.
i
44
o 22' 44"
44.000
0.0640
8.806149
2
88
o 45 27
87.999
.3200
9-505117
3
132
i 08 ii
131-995
.8959
9.952268
4
176
i 30 55
175.984
1.920
0.283245
5
220
i 53 38
219-954
3-519
0.546454
6
264
2 l6 22
263.894
5.822
0.765077
7
308
2 39 06
307.78,
8.955
0.952075
8
352
3 oi 50^.
351-592
13.045
1.115444
9
396
3 24 34
395.287
18.217
1.260468
3 47 18
74
TABLE III.
c. CHORD-LENGTH = 45.
ti.
7l('.
z
y-
jr.
Log jr.
45
o 22' 13"
45.000
0.0655
8.815908
2
9
o 44 27
89.999
.3272
9-5I4877
3
135
i 06 40
134-995
.9163
9.962028
4
i So
i 28 53
179.983
1.963
o. 293005
5
225
i 5i 07
224.953
3-599
0.556214
6
270
2 13 20
269.892
5.954
0.774837
7
315
2 35 34
314.778
9-159
0.961834
8
360
2 57 43
359-583
I3.34I
1.125203
9
405
3 20 01
404.271
18.631
1.270228
3 42 15
c. CHORD-LENGTH ~ 46.
;/.
nc.
D s .
y-
jr.
Log x.
I
46
2l' 44"
46. ooo
0.0669
8.825454
2
92
o 43 29
91.999
3345
9-524422
3
138
i 05-13
137.995
.9366
9.971573
4
184
i 26 58
183.983
2.007
0.302550
5
230
i 48 42
229.952
3.679
0.565759
6
276
2 10 26
275.889
6.087
0.784382
7
322
2 32 II
321.773
9.362
0.971380
8
368
2 53 56
367.573
13.638
I.I34749
9
414
3 15 40
413.255
I9- 45
1.279773
3 37 24
r. CHORD-LENGTH = 47.
.
nc.
Ds. '
y-
jr.
Log x.
i
47
o 21' 16"
47.000
0.0684
8.834794
2
94
o 42 33
93.999
.3418
9.533762
3
141
i 03 50
140.995
9570
9.980913
4
188
i 25 06
187.982
2.051
0.311890
5
235
i 46 23
234.951
3-759
0.575100
6
282
2 O7 40
281.887
6.219
0.793722
7
329
2 28 57
28.768
9.566
0.980720
8
376
2 50 14
.375^564
13-934
1.144089
9
423
3 ii 31
422.238
-9-459
1.289113
3 32 48
75
TABLE III.
f. CHORD-LENGTH - 48.
n.
nc.
zv
'?
jr.
Log^r.
i
48
o 20' 50"
48.000
o. 0698
8.843937
2
96
o 41 40
95-999
3491
9- 542905
3
144
I 02 30
143.995
9774
9.990057
4
192
I 23 20
191.982
2.094
0.321034
5
240
i 44 10
239-950
3.839
0.584243
6
288
2 05 OO
287.885
6.351
0.802866
7
336
2 25 51
335-7^3
9.769
0.989863
8
3S4
2 46 41
3S3.555
14.231
1.153232
3 06 31
f. CHORD-LENGTH = 49.
ft.
nc.
D s .
y-
jr.
Log x.
I
49
o 20' 25"
49.000
0.0713
8.852892
2
q8
o 40 49
97.999
.3563
9.551860
3
147
i 01 14
146.995
.9977
9.999011
4
196
I 21 38
195.982
2.138
0.329988
5
245
I 42 03
244.949
3.919
0.593198
6
294
2 02 27
293.882
6.484
0.811820
7
343
2 22 52
342.758
9-973
0.998818
8
39 2
2 43 17
39!-546
14.527
1.162187
3 03 31
c. CHORD-LENGTH = 50.
n.
nc.
/?*
y-
X.
Log x.
I
50
o 20' oo"
50.000
0.0727
8.861666
2
100
o 40 oo
99-999
.3636
9. 560634
3
150
I OO OO
149.995
1.018
0.007785
4
200'
I 20 00
199.981
2.182
0.338762
5
250
i 40 oo
249.948
3-999
0.601972
6
3OO
2 00 00
299.880
6.616
0.820594
7
35
2 2O OO
349-753
10.176
1.007592
8
400
2 40 OO
399.536
14.824
1.170961
3 oo oo
76
TABLE IV.
FUNCTIONS OF THE ANGLE s.
n.
S.
cos s.
log vers s.
R i x
vers s.
sin s.
log sin s.
s.
I
o 10'
99999
4.626422
.024
.00291
7.463726
o 10'
2
o 30
.99996
5.580662
.218
.00873
7.940842
o 30
/?
I 00
.99985
6.182714
873
-01745
8.241855
I 00
4
I 40
.99958
6.626392
2.424
. 02908
8.463665
I 40
5
2 30
.99905
6.978536
5-453
. 04362
8.639680
2 30
6
3 30
.99813
7.720726
10.687
.06105
8.785675
3 30
7
4 40
.99668
7.520498
18.994
.08136
8.910404
4 40
8
6 oo
.99452
7.738630
31 383
10453
9.019235
6 oo
9
7 30
.99144
7.932227
49.018
13053
9.115698
7 30
10
9 10
.98723
8.106221
73-173
.15931
9.202234
9 10
ii
II 00
.98163
8.264176
105.270
.19081
9.280599
II 00
12
13 oo
97437
8 408748
146.857
.22495
9.352088
13 oo
13
15 10
.9651718.541968
199.570
.26163
9.417684
15 10
14
17 30
.953728.665422
265.186
.30071
9.478142
17 30
15
20 oo
.93969
8.780370
345-540
. 34202
9-53405 2
20 00
16
22 40
.92276
8.887829
442-543
38537
Q-585877
22 40
17
25 30
90259
8.988625
558.153
43051
9.633984
25 30
18
28 30
.87882
9.083441
694-335
.47716
9.678663
28 30
19
31 40
.85112
9.172846
853-0501
.52498: 9.720140
31 40
20
35 oo
.81915
9- 2 573*4
1036.20 1
-57358
9-75859 1
35 oo
77
TABLE
SELECTED SPIRALS FOR A 2 CURVE, GIVING
A
s.
n x c.
-As(w+l).
>'.
flC
10
1 00'
3 x 32
2 05' OO"
2 03'
41.12
10
I 40
4 x 39
2 08 13
2 09
61.04
10
2 3O
5 x 43
2 19 33
2 IS
73.69
10
3 30
6 x 45
2 35 34
2 33
78.81
10
4 40
7 x 44
3 oi 50
2 40
70.47
20
I OO
3 x 33
2 01 13
2 OI
45.28
20
I 40
4 x 41
2 oi 57
2 02
73.85
20
2 30
5 x 48
2 05 00
2 05
99.99
20
3 30
6 x 50
2 20 00
2 06
109.52
30
I OO
3 x 34
i 57 39
2 OI
46.14
30
I 40
4 x 41
2 oi 57
2 01
75.16
30
2 30
5 x 49
2 02 27
2 02
109.78
30
3 30
6 x 50
2 2O OO
2 02
115.63
30
3 30
6 x 50
2 20 OO
2 03
110.90
40
I 00
3 x 35
I 54 17
2 01
46.90
40
I 40
4 x 42
i 59 02
2 01
76.96
40
2 30
.5 x 50
2 00 00
2 01
117.87
78
V.
EQUAL LENGTHS BY CHORD MEASUREMENT.
old line.
^ new line.
Diff.
X.
h.
291.12
291.12
.00
.6516
.040
.061
311-04
311-04
.00
1.702
.187
.110
323.69
323.70
+ .01
3.439
354
.103
328.81
328.82
+ .01
5.954
59
.099
320.47
320.50
-h .03
8.955
.897
.100
545.28
545.28
.00
.6719
.122
.182
573-85
573.84
.01
1.789
.118
.066
599-99
600. oo
+ .01
3.839
.527
.137
609.52
609.52
.00
6.616
554
.084
796.14
796.22
+ .08
.6923
.566
.082
825.16
825.16
.00
1.789
.227
.127
859.78
859'75
- .03
3.919
377
.096
865.63
865.57
- .06
6.616
.249
.038
860.90
860.98
+ .08
6.616
1.013
.153
1046.90
1047.15
+ .25
.7127
1.222
1.715
1076.96
1077.09
+ .13
1.832
.848
.463
1117.87
1117.77
.10
3.999
.141
.035
79
TABLE
SELECTED SPIRALS FOR A 4 CURVE, GIVING
A
S.
72 X C.
->(n + l).
D\
d.
10
i oo'
3 x 16
4 10' 03"
4 07'
20.22
10
I 40
4 - 19
4 23 13
4 16
29.12
10
2 30
5 X 22
4 32 48
4 39
38.75
10
3 30
6 x 23
5 04 26
5 17
41.37
20
I 40
4 x 20
4 10 03
4 04
34.92
2O
2 30
5 x 24
4 10 03
4 09
50.72
20
3 30
6 x 27
4 !9 19
4 H
63-69
20
4 40
7 x 30
4 26 44
4 3i
78.07
20
6 oo
8 x 31
4 50 24
4 46
81.88
2O
7 30
9 x 32
5 12 36
5 16
85.40
30
i 40
4 x 20
4 I0 03
4 02
35-57
30
2 30
5 x 25
4 oo 03
4 04
57-39
30
3 30
6 x 28
4 10 03
4 07
72.37
30
4 40
7 x 32
4 10 03
4 14
93-09
30
6 oo
8 x 35
4 17 12
4 23
110.31
30
7 30
9 x 37
4 30 20
4 34
122. 2O
30
9 10
10 x 38
4 49 33
4 47
126.86
40
2 30
5 x 25
4 oo 03
4 02
58.91
40
3 30
6 x 28
4 10 03
4 04
73-75
40
4 40
7 x 32
4 10 03
4 08
94.65
40
6 GO
8 x 36
4 10 03
4 12
121.33
40
7 30
9 x 39
4 16 28
4 17
142.86
40
9 10
10 x 41
4 28 21
4 26
154.34
60
2 30
5 x 25
4 oo 03
4 01
59-68
60
3 30
6 x 29
4 01 26
4 02
81.04
60
4 40
7 x 32
4 10 03
4 03
99-59
60
6 oo
8 x 36
4 10 03
4 05
125.81
60
7 30
9 x 40
4 10 03
4 08
154-42
80
2 30
5 x 25
4 oo 03
4 or
58.29
80
3 30
6 x 29
4 01 26
4 01
82.82
80
4 40
7 x 33
4 02 28
4 02
106.99
80
6 oo
8 x 37
4 03 17
4 03
I35.6I
80
7 30
9 x 41
4 03 57
4 05
164.79
80
V.
f fuinvEESiTY-
Vv *,d o:H ' *t ^
EQUAL LENGTHS BY cfeQjto ^JTASUR^kENT.
old line.
| new line.
Diff.
X.
h.
k.
145-22
145.17
- -05
.3258
.045
135
154-12
I54-I3
+ .01
-.8290
.080
.100
163.75
163.76
+ .01
1.760
.177
.100
166.37
166.39
+ .02
3.043
305
.100
284.92
284.92
.00
.8726
.081
.100
300. 72
300.72
.00
1.920
.184
.096
313.69
313.75
+ .06
3-573
375
.iQ5>
328.07
328.08
+ .01
6.106
.598
.098
332.88
33L92
+ .04
9.191
.910
.092
335-40
335-47
+ .07
13-248
1.310
.099
410.57
410.57
.00
.8726
137
.157
432-39
432.38
.01
2.000
.147
.074
447-37
447-35
.02
3.705
.284
.077
468.09
468.09
.00
6.513
.687
.105
4S5.3I
485-32
s + .01
10-377
1.091
.105
497.20
497.23
+ .03
15.319
1.526
.100
501.86
5QJ.95
H- .09
2T.240
2.126
.100
558.91
558.88
.03
2.000
.109
.054
573-75
573-74
.01
3-705
.361
.097
594.65
594-66
+ .01
6.513
977
.150
621.38
621.33
.05
10.673
973
.091
642.86
642.83
- .03
16.147
1. 100
.086
654.34
654-36
+ .02
22.917
2.186
.095
809.68
809.67
.01
2.000
.180
.090
831.04
831.03
.01
3.837
.461
.120
849.59
849-52
.07
6.513
572
.088
875.81
875-76
- .05
10.673
1.074
.106
904.42
904.36
- .06
16 561
1.718
.104
1058.29
1058.61
-f- .32
2.OOO
979
.490
1082.82
1082.71
.11
3.837
.295
.074
1106.99
1107.03
+ .04
6.716
1. 000
.149
1135-61
H35.5I
.10
10.970
1.199
.109
1164.79
1164.92
+ .13
16.975
2.440
.144
81
TABLE
SELECTED SPIRALS FOR AN 8 CURVE, GIVING
A
S.
n X c.
#fi(7l + l).
D'.
d.
10
2 30'
5x11
9 06' 01"
9 06'
19-95
20
2 30
5 x 12
8 20 26
8 16
25.71
2O
3 30
6 x 14
8 20 26
8 34
34-86
20
4 40 7 x 15
8 53 5i
8 54
39-90
20
6 oo
8 x 16
9 23 07
9 24
45-52
30
2 30
5 x 12
8 20 26
8 07
26.50
30
3 30
6 x 14
8 20 26
8 14
36.16
30
4 40
7 x 16
8 20 26
8 26
47.01
30
6 oo
8 x 17
8 49 55
8 36
53-13
30
7 30
9 x 18
9 16 08
8 46
60.05
30
9 10
10 x 19
9 39 36
9 14
65.70
40
2 30
5 x 12
8 20 26
8 04
26.93
40
3 30
6 x 14 8 20 26
8 08
36.85
40
4 40
7 x 16
8 20 26
8 14
48.25
40
6 oo
8 x 18
8 20 26
8 22
61.35
40
7 30
9 x 19
8 46 49
8 30
68.07
40
9 10
10 X 20
9 10 34
8 40
75-01
40
II 00
II X 21
9 32 03
8 54
82.13
40
13 oo
12 X 22
9 5i 36
9 H
89.81
60
2 30
5 x 12
8 20 26
8 02
27.30
60
3 30
6 x 14
8 20 26
8 03
38.22
60
4 40
7 x 16
8 20 26
8 06
49-75
60
6 oo
8 x 18
8 20 26
8 10
62.87
60
7 30
9 x 20
8 20 26
8 16
77.16
- 60
9 10
10 X 22
8 20 25
8 24
93.05
60
II 00
II X 23
8 42 13
8 31
101.08
60
13
12 X 25
8 40 28
8 48
118.19
60
15 10
13 x 26
8 58 59
9 02
127.21
60
17 30
14 x 27
9 16 07
9 22
136.45
80
4 40
7 x 17
7 50 57
8 04
57.04
80
6 oo
8 x 19
7 54 03
8 06
71.78
80
7 30
9 x 20
8 20 26
8 oSJ-
79.18
80
9 10
IO X 22
8 20 25
8 13
95.23
80
II OO
ii x 24
8 20 25
8 19
112.67
80
13 oo
12 X 26
8 20 25 .
8 28
130.86
80
15 10
13 x 27
8 38 59
8 34
140.88
80
17 30
14 x 28
8 56 13
8 42
150.55
82
EQUAL LENGTHS BY CHORD MEASUREMENT.
% old line.
new line.
Diff.
X.
h.
k.
82.45
82.47
+ .02
.8798
.051
.058
150.71
150.72
+ .01
-9598
.051
.053
159.86
I5Q-88
+ .02
1.852
.117
.063
164.90
164.92
+ .02
3-C53
.185
.061
170.52
170.55
-f .03
4-744
.221
.047
214.00
214.00
.CO
.9598
.049
.051
223.66
223.68
+ .02
1.852
.142
.077
234.51
234.53
+ .02
3-256
.260
.080
240. 63
240.65
+ .02
5.040
.325
.065
247-55
247-55
.00
7-452
.287-
039
253.20
253.18
.02
10.620
590
.056
276.93
276.94
+ .01
.9598
.079
.082
286.85
286.87
+ .02
1.852
.181
.098
298.25
298.24
.01
3.256
293
.090
3H-35
3X1-33
.02
5.337
.330
.062
318.07
318.06
.01
7.866
.472-
.c6o
325-01
325.00
v .OI
11.179
.629
.056
332.13
332.12
.01
15.415
.840
054
339-81 '
339-81
.00
20.723
I.O24
.049
402. 30
402.32
+ .02
.9598
.136
.142
413.22
413.19
- .03
1.852
.083
045
424.75
424.76
+ .01
3-256
.317
.097
437.87
437.88
+ .01
5-337
539
.101
452.16
452.18
+ .02
8.280
.863 -
.104
468.05
468.02
- .03
12.297
I-I39
.093
476.08
476.09
+ .01
16.883
1.523
.090
493-19
493.18
.01
23.548
2.160
.092
502.21
502.21
.00
30.817
2.613
.085
5H-45
5H.45
.00
39-595
3-157
.080
557.04
557.02
.02
3.460
.366
.106
57L78
57L75
- .03
5*633
.408
.072
579^8
579- 18
.00
8.280
.860
.104
595.23
595.25
H- .02
12.297
1.346
.110
612.67
612.70
+ .03
17.617
1.719 -
.109
630.86
630.90
+ .04
,24.490
2.738
.112
640. 88
640.88
.00
32.002
3.H9
.098
650.55
650.62
+ .07
41.062
3.809
.093
TABLE
SELECTED SPIRALS FOR A 16 CURVE,
A
S.
n X c.
.>s(n + l).
D\
d.
30
4 4o'
7 x 10
13 2l' 48"
18 oo'
33-59
40
6 oo
8 x 10
15 02 34
17 14
36.14
60
7 3o
9 x 10
16 43 3i
16 32
38.47
60
9 10
10 X 11
16 43 3i
16 48
46.40
60
II 00
II X 12
16 43 3i
17 14
54-62
60
13 oo
12 X 12
18 07 48
17 22
54- M
60
15 10
13 x 13
18 oi 18
18 10
62.88
60
17 30
14 x 13
19 19 14
18 12
62.85
60
20 oo
15 x 14
19 06 05
20 00
72.14
80
7 30
9 x 10
16 43 31
16 16
39-74
80
9 10
IO X II
16 43 31
16 26
47-49
80
II OO
II X 12
16 43 31
16 38
56.19
80
13 oo
12 X 13
16 43 30
16 56
65.24
80
15 10
13 x 14
16 43 29
17 22
74.72
80
17 30
14 x 14
17 55 44
17 24
75.02
80
20 00
15 x 15
J7 50 54
18 06
85.15
So"
22 40
16 x 15
18 58 25
18 08
85-18
80
28 30
18 x 16
19 53 20
19 42
95.84
84
GIVING EQUAL LENGTHS OF ACTUAL ARCS.
old line.
new line.
Biff.
X.
h.
k.
127.64
127.64
.00
2.035
.388
.191
I6I.55
161.55
.00
2.965
.430
.145
226.58
226.56
.02
4.140
.436
.105
234.50
234.45
- .05
6.148
.576
.094
242.73
246.67
.06
8.808
.860
.099
242.25
242.26
+ .01
11-303
1.093
.097
250.99
250.99
\ .00
15.409
1.516
.098
250.96
250.97
+ .01
19.064
1-552
.081
260.25
260.25
.00
25-031
2.182
.087
290.55
290.47
- .08
4.140
,328
305
298.30
298.27
- -03
6.148
.680 1 .Hi
307.01
306.96
- .05
8.808
-943
.107
316.06
316.03
.03
12.245
1.384
113
325.53
325-54
-f .01
16.594
1-973
.119
325.83
325-81
.02
20.531
1-939
,094
335-97
335-96
.01
26.819
2.657
.099
336.00
335-99
.01
32.276
2.677
.083
346.65
346.66
+ .01
48.221
3.748
.078
1
THIS BOOK IS DUE ON THE LAST DATE
STAMPED BELOW
AN INITIAL PINlToF 25 CENTS
^!, L B BE ASSESSED F0 * FAILURE TO RETURN
THIS BOOK ON THE DATE DUE. THE PENALTY
DAV'-A^ RE T ASE T0 50 CENTS ON
OVERDUE^. " N THE 8EVEN
I8Apr52l?J
JUL 27 1943
Yft
UNIVERSITY OF CALIFORNIA LIBRARY