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department of Ballistics of the U. S. Artillery School.
EXTERIOR BALLISTICS
IN THE
PLANE OF FIRE
BY
JAIVIKS ISA, INOAIvIvS,
Cai'tain P'ikst Aktilleky, U. S. Army,
Instructor,
NEW YORK:
D. VAN NOSTRAND, PUBLISHER,
23 MURRAY AND 27 WARREN STREETS,
1886,
■I^*'"
^V'
HEADQUARTERS UNITED STATES ARTILLERY SCHOOL.
Fort Monroe, Va., February, 1885.
Approved and Authorized as a Text- Book.
Pat. 26, Regulations U. S. Artillery School, appioved 1882, viz.:
" To the end that the school shall keep pace with professional progress, it
is made the duty of Instructors and Assistant-Instructors to prepare and
arrange, in accordance with the Programme of Instruction, the subject-matter
of the courses of study committed to their charge The same shall be sub-
mitted to the Staff, and, after approval by that body, the matter shall become
the authorized text-books of the school, be printed at the school, issued, and
adhered to as such." _ -, ^y
By order of Lieutenant-Colonel Tidball.
Tasker H. Bliss,
First Lieutenant ist Artillery, Adjutant.
Copyright, 1886,
By D. van NOSTRAND.
PREFACE.
This work is intended, primarily, as a text-book for
the use of the officers under instruction at the U. S.
Artiller}^ School, and the arrangement of the matter has
been made with reference to the wants of the class-room.
The aim has been to present in one volume the various
methods for calculating range-tables and solving impor-
tant problems relating to trajectories, which are in vogue
at the present day, developed from the same point of
view and with a uniform notation. The convenience of
this is manifest.
It is hoped, also, that the practical artillerist will find
here all that he may require either for computing range-
tables for the guns already in use, or for determining
in advance the ballistic efficiency of those which may
be proposed in the future.
ERRATA
Page 54, line 27 :
For - read -.
u V
Page 64, line 4 :
For (i) and {(f) read {i\ and (^X
Page 72, line 18:
4 i
For sec ^ read sec 5 f. /
Page 73, line 22 :
' -^4- ^^^^ V*
Page 93, line 11 :
For g read j.
Page 116, equation {78):
For r— read
cos'' (p 2 cos ip
CONTKNTS
INTRODUCTION.
Object and Definitions,
•AGE
5
CHAPTER I.
RESISTANCE OF THE AIR.
Normal Resistance to the Motion of a Plane,
Oblique Motion, .......
Pressure on a Surface of Revolution, ....
Applications, .......
Resistance of the Air to the Motion of Ogival-headed Projectiles,
CHAPTER II.
EXPERIMENTAL RESISTANCE.
Notable Experiments, ....
Methods of Determining Resistances,
Russian Experiments with Spherical Projectiles,
Mayevski's Deductions from the Krupp Experiments,
Ilojel's Deductions from the Krupp Experiments,
Bashforth's Coefficients,
Law of Resistance deduced from Bashforth's K, .
Comparison of Resistances,
Example, .....
7
9
9
10-13
13-16
17
19
23
28
29
31
35
37
39
CHAPTER III.
DIFFERENTIAL EQUATIONS OF TRANSLATION — GENERAL PROPERTIES OF
TRAJECTORIES.
Preliminary Considerations,
Notation, ......
DifFerential Equations of Translation,
Minimum Velocity, .....
Limiting Velocity, .....
Limit of the Inclination in the Descending Branch,
Asymptote to the Descending Branch,
Radius of Curvature, ....
4.1
41
42
46
47
48
49
50
CONTENTS.
CHAPTER IV.
RECTILINEAR MOTION.
Relation between Time, Space, and Velocity,
Projectiles differing from the Standard,
Formulas for Calculating the T- and ^'-Functions,
Ballistic Tables, ....
Extended Ranges, ....
Comparison of Calculated with Observed Velocities,
PAGE
52
53
54
57
59
60
CHAPTER V.
RELATION BETWEEN VELOCITY AND INCLINATION.
General Expressions for the Inclination in Terms of the Velocity
Bashforth's Method, .
High-Angle and Curved Fire,
Siacci's Method,
Niven's Method,
Modification of Niven's Method,
CHAPTER VI.
HIGH-ANGLE FIRE.
Trajectory in Vacuo,
Constant Resistance,
Resistance Proportional to the First
Euler's Method,
Bashforth's Method, .
Modification of Bashforth's Method,
Power of the Velocity,
64
65
66
68
73
75
77
80
81
91
95
97
CHAPTER VII.
DIRECT FIRE.
Niven's Method,
.
102
Sladen's Method,
. 106
Siacci's Method,
. 108
Practical Applications,
.118
Correction for Altitude,
. 127
EXTERIOR BALLISTICS
IN THE PLANE OF FIRE.
INTRODUCTION.
Definition and Object. — Ballistics, from the Greek
l^aUw, I throw, is, in its most general signification, the
science which treats of the motion of heavy bodies pro-
jected into space in any direction ; but its meaning is usu-
ally restricted to the motion of projectiles of regular form
fired from cannon or small arms.
The motion of a projectile may be studied under three
different aspects, giving rise to as many different branches
of the subject, called respectively Interior Ballistics, Ex-
terior Ballistics, and Ballistics of Penetration,
1. Interior Ballistics.— Interior Ballistics treats of
the motion of a projectile within the bore of the gun while
it is acted upon by the highly elastic gases into which the
powder is converted by combustion. Its object is to deter-
mine by calculation the velocity of translation and rotation
which the combustion of a given charge of powder of
known constituents and quality is capable of imparting to
a projectile, and the effect upon the gun.
2. Exterior Ballistics. — Exterior Ballistics considers
the circumstances of motion of a projectile from the time
it emerges from the gun until it strikes the object aimed
at. Its data are the shape, caliber, and weight of the pro-
jectile, its initial velocity both of translation and of rotation.
6 EXTERIOR BALLISTICS.
the resistance it meets from the air, and the action of grav-
ity.
3. Ballistics of Penetration. — This branch of the
subject has reference to the effect of the projectile upon
an object; the data being the energy and incHnation with
which the projectile strikes the object, the nature of the re-
sistance it encounters, etc.
The above is not the order in which the three divisions
of the subject are usually presented to the practical artil-
lerist, but the reverse. He desires to penetrate or destroy
a given object — say the side of an armored ship. Ballistics
of penetration enables him to determine the minimum en-
ergy which his projectiles must have on impact, and the
proper striking angle, to accomplish the desired result.
Exterior Ballistics would then carry the data from the ob-
ject to be struck to the gun, and determine the necessary
initial velocity and angle of elevation. Lastly, Interior
Ballistics would ascertain the proper charge and kind of
powder to be used to give the projectile the initial velocity
demanded.
The following pages treat only of Exterior Ballistics;
and this subject will be limited, at present, to motion in the
vertical plane passing through the axis of the piece.
CHAPTER I.
RESISTANCE OF THE AIR.
Preliminary Considerations. — The molecular the-
ory of gases is not yet sufficiently developed to be made
the basis for calculating the resistance which a projectile
experiences in passing through the air. We know, how-
ever, that if a body moves in a resisting medium, fluid or
gaseous, the particles of the fluid must be displaced to allow
the body to pass through ; and hence momentum will be
communicated to them, which must be abstracted from the
moving body. From the assumed equality of momenta
lost and gained Newton deduced the law of the square of
the velocity to express the resistance of the air to the mo-
tion of a body moving in it.
The following, which is the ordinary demonstration,
supposes the particles of air against which the body im-
pinges to be at rest, and takes no account of the reaction of
the molecules upon each other, nor of their friction against
the surface of the body. The result will therefore be but an
approximation, which must be estimated at its true value by
means of well-devised and accurately-executed experiments.
Normal Resistance to the Motion of a Body
presenting a Plane Surface to the Medium.— Let
a moving body present to the particles of a fluid against
which it impinges, and which are supposed to be at rest, a
plane surface whose area is 5, and which is normal to the
direction of motion. Let w be the weight of the moving
body, V its velocity at any time t, d the weight of an unit-
volume of the fluid, and ^ the acceleration of gravity. The
plane 5 will describe in an element of time dt a path v d t,
and displace a volume of fluid Svdt ; therefore the mass
of fluid put in motion during the element of time is- Svdt.
8 EXTERIOR BALLISTICS.
And as this moves with the velocity v, its momentum is
— Sv'dt; and this has been abstracted from the moving
body, whose velocity has thereby been decreased by dv.
Therefore
dvzm-Sv'dt
g g
or IV dv b ^ ^
~ - -y- = - 5 z/"
g dt g
The first member of this last equation is the momentum-
decrement of the body, due to the pressure of the fluid
upon the plane face 5, and is therefore a measure of this
pressure. Calling this latter P, we have
_, w dv <5 - ,
g dt g
or, per unit of mass,
w dt w
As before stated, several circumstances have been omit-
ted in this investigation ^vhich, if taken into account, would
probably increase the pressure somewhat, at least for high
velocities. We will therefore introduce into the second
member of the above equation an undetermined multiplier
k {k y i), and we have
P = k-Sv' ..
g (0
The pressure is, therefore, proportional to the area of
the plane surface, to the density of the medium, and to the
square of the velocity.
If in equation (i) we make 5=1, the second member
will then express the normal pressure upon an unit-surface
moving with the velocit}^ v; calling this /o,' we have
and
P^AS
EXTERIOR BALLISTICS.
Oblique Motion. — If the surface 5 is oblique to the
direction of motion, let f be the angle which the normal to
the plane makes with that direction ; and resolve the velo-
city 7.' into its components v cos f, perpendicular, and v sin f,
parallel, to vS. This last, neglecting friction, having no re-
tarding effect, we have for the normal pressure upon 5 the
expression
P=/^-'z^''5cos' 6=/o 5cos'f /
Poncelet {Mecanique Industrielle, 403) cites the following
empirical formula for calculating the normal pressure, viz. :
i"fsec' e ^ '
derived by Colonel Duchemin from the experiments of
Vince, Hutton, and Thibault. As this expression satisfied
the whole series of experiments upon which it was based
better than any other that was proposed, we will adopt it in
what follows.
Pressvire on a Surface of
Revolution. — Let A D B, Fig.
I, be the generating curve of a
surface of revolution, which we
will suppose moves in a resisting
medium in the direction of its
axis,^_(9 A. \{ m in' in" = <^5 be
an element of the surface, inclined ^
to the direction of motion by the
angle Ninv=^e, it will suffer a
pressure in the direction of the
normal N in, equal, by (2), to
2p,dS
I + sec' e
Resolving this pressure into two components.
2 p^d S cos> 8
2p^d Ssin s
i°+sec'. ' P^""^'^"'' ^"^ i+^c^' Pe'-Pe"dic"la'-.
lO EXTERIOR BALLISTICS.
to OA, it is plain that this last will be destroyed by an
equal and contrary pressure upon the elementary surface
n n' n" situated in the same meridional section as ;;/ in' m" , and
making the same angle with the direction of motion. It is
only necessary, therefore, to consider the first component,
2p^d S cos £
I + sec' e
It is evident that expressions identical with this last are
applicable to every element of the zone ;// m' n n' described
by the revolution of m in' ; and we may, therefore, extend
this so as to include the entire zone by substituting its area
for dS. If we take O A for the axis of A'', this area will be
expressed by 2 it yds, in which ds is an element of the gene-
rating curve ; therefore, the pressure upon any elementary
zone will be
, y ds cos f
dx'
Substituting — dy for ds cos f , and 2 -|- -r-, for i -\- sec* e, and
integrating between the limits 4- = /, and x^o, we have
/' y<iy
As. all service projectiles are solids of revolution, this
last equation may be used to calculate the relative pressures
sustained by projectiles having differently shaped heads, sup-
posing their axes to coincide with the direction of motion at
each instant. In appl3nng the formula, y will be eliminated
by means of the equation of the generating curve. The
superior limit of integration (/) will be the length of the
head. R will denote the radius of the projectile.
Application to Conical Heads. — Let /^ i? be the length
of the conical head, the angle at the point being
2 tan
■(0
EXTERIOR BALLISTICS.
The equation of the generating line is
II
y=--^R
whence
y dy
i+i
and, therefore,
d^ n\2-^fe)
~df
{n R — x) dx
47tp,
= 7tR'p,
R
x)dx
When n=:o, the head becomes flat, and the above equa-
tion reduces to
P^nR'p,
as it should.
Application to a Prolate Hemi-Spheroidal Head,
with Axes in the Ratio of one to two. — The equation
of the generating ellipse is
4/ + ;l;' = 47?^
whence
y dy
x^ dx
4(8i^^
and, therefore, since / = 2 7?,
p— 111 I ^'^-^
:=7tR^P,{2\0g2-l)
Application to Ogival Heads. J^i
—Let A B D (Fig. 2) be a section of ^
an ogival head made by a plane pass-
ing through the axis of the projectile.
Let A O = Rbe the radius of the pro-
jectile, and A £ = 7t R be the radius
.\UJ
12 EXTERIOR BALLISTICS.
of the generating circle, whose equation is, if we make O the
origin and O B the axis of X,
y^ire R^-x'Y-in- \)R
Making j^^o, we find
O B = l=:R V2n ^i
Let the angle A E B=iy ; therefore
V2n— I
tan y =
n— I
which serves to determine the length of the arc of the
ogive, A B.
The differential of the equation of the generating circle
is
, X dx
^^~~ {it'R'-xY
whence
, J , (n — i) Rx dx
and
, , dx"" n^ R'-X-x'
dy" 2 x"
therefore
^RV^;r:r, I 2{n— i)Rx' 2 x' )
^=-2^A/^ I {n'R'+x'){n'R'-xy ~ n'R'+x j
„, ( , n(n—i) . ^ + V2+ I
( 1/2 «— V2+^
-«Mog ^'+^:-' }
= 7rR^AF{n),{say) (3)
If « is the angle at the point of the projectile, the expres-
sion for dj/ gives
2 n — i\
n — I J
a
EXTERIOR BALLISTICS.
13
When n= i, A D B becomes a semi-circle and the head a
hemisphere.
The following- table gives the values oi F (n)^ the lengths
of head in calibers, and the angles at the point, for integral
values of n from i to 6 :
n
Fin)
LENGTH OF HEAD
(0
ANGLE AT POINT
I
0.6137
0.5000
180° 00' 00''
2
0.4187
0.8660
120° 00' 00''
3
0.3176
I.II8O
96° 22' 46''
4
0.2560
1.3229
82° 49' 09''
5
0.2146
1.5000
73° 44' 23''
6
0.1848
1.6583
6f & ^2"
Resistance of the Air to the Motion of Ogival-
heacled Projectiles. — The expression
P=7zR'p,F(n)
which, by substituting for/^ its value, becomes
g
serves to determine the pressure, as deduced by the above
theory, upon an ogival head ; and requires that this pressure
should be proportional to the density of the air, to the area
of the cross-section of the body of the projectile, and to the
square of the velocity. The truth of the first two of these
deductions may be considered as fully established by expe-
riment, and is admitted by all investigators. The relation
between the front pressure and the velocity has not been
satisfactorily determined by experiment, and we are there-
fore unable to verify directly the law of the square deduced
above. It seems probable, however, from experiments made
to determine the resistance of the air to the motion of pro-
14 EXTERIOR BALLISTICS.
jectiles, as well as from theory, that this law is approxi-
mately true for all velocities.
If we represent the pressure of the air upon the rear
part of the projectile by P' , and the resistance by />, we shall
evidently have ~
p^P-P'
It is evident that P' will be zero whenever the velocity
of the projectile is greater than that of air flowing into a
vacuum. In this case, and also when P' is so small rela-
tively to P that it may be neglected, we have approxi-
mately
p^P
Application to Ogival Heads struck witli Radii
of one and a half Calibers. — Experiments have proven
that for practicable velocities exceeding about 1300 f. s. the
resistance of the air is sensibly proportional to the square of
the velocity ; and a discussion of the published results of
Professor Bashforth's experiments has shown that, within
the above limits, the resistance to elongated projectiles
having ogival heads struck with radii of one and a half cali-
bers may be approximately expressed by the equation,
pz=z- d'v'
S
in which d is the diameter of the projectile in inches, ^ the
acceleration of gravity (32.19 ft.), and log A = 6,1525284 —
10. Whence
p — o.o'44i37^'z;'
Making b = 534.22 grains, which is the weight of a cubic
foot of air adopted by Professor Bashforth, and F{n)=:F{^)
= 0.3176, we find for the corresponding expression for P
P = o.o%io6g k d' v'
A comparison of the second members of these two equa-
tions seems to warrant the conclusion that for velocities
greater than about 1300 f. s., the rear pressure is either zero
or so small relatively to the front pressure that it may be
EXTERIOR BALLISTICS. 1 5
neglected without sensible error. Equating the two mem-
bers, we find for velocities greater than 1300 f. s.
k— 1.0747
In the following table the first and second columns give
the velocities and corresponding resistances, in pounds, to
an elongated projectile one inch in diameter and having an
ogival head of one and a half calibers. They were deduced
from Bashforth's experiments by Professor A. G. Greenhill,
and are taken from his paper published in the Proceedings
of the Royal Artillery Institution, No. 2, Vol. XIII. The
third column contains the corresponding pressures upon the
head of the projectile computed by the formula
576^
in which the constants have the values already given. The
fourth and fifth columns are sufficiently indicated by their
titles.
These results are reproduced graphically in Plate I.
A is the curve of resistance (^), drawn by taking the velo-
cities for abscissas and the corresponding resistances, in
pounds, for ordinates. This curve is similar to that given
by Professor Greenhill in his paper above cited. B is the
curve of front pressures (P), and is a parabola whose equa-
tion is given above. It will be seen that while the velocity
decreases from 2800 f. s. to 1300 f s., the two curves closely
approximate to each other; the differences (P— />) for the
same abscissas being relatively small and alternately plus
and minus. As the velocity still further decreases, the curve
of resistance falls rapidly below the parabola B, showing
that the resistance now decreases in a higher ratio than the
square of the velocity. This continues down to about 800
f. s., when the parabolic form of the curve is again resumed,
but still below B. The differences P— p from z/= 1300 f. s.
to 2/= 100 f s. are shown graphically by the curve (7, which
may represent, approximately, the rear pressures iox decreas-
ing velocities, and possibly account, in a measure, for the
i6
EXTERIOR BALLISTICS.
sudden diminution of resistance in the neighborhood of the
velocity of sound.
V
p
P
P-P
P-P
V
p
P
P-P
P-P
P
P
2800
2750
2700
35.453
33.586
31.846
34.603
33.378
32.176
-0.850
-0.208
+ 0.330
1080
1070
1060
3-999
3.756
3.478
5.148
5.053
4.959
+ 1.149
1.297
1. 481
0.223
0.256
0.298
2650
2600
2550
30.241
28.613
27.243
30.995
29.836
28.700
+ 0.754
+ 1.223
+ 1.457
1050
1040
1030
3.139
2.823
2.604
4.866
4.774
4.684
1.727
1. 951
2.080
0.355
0.409
0.444
2500
2450
2400
26.406
25.898
25.588
27.585
26.493
25.422
+ 1.379
+-0.595
-0.166
1020
lOIO
1000
2.482
2.404
2.330
4.592
4.502
4.414
2. 114
2.098
2.084
0.459
0.466
0.472
2350
2300
2250
25.242
24.760
23.566
24.374
23.347
22.344
-0.868
-1. 413
— 1.222
990
980
970
2.261
2.193
2.127
4.326
4.239
4.153
2.065
2.046
2.026
0.477
0.483
0.488
2200
2150
2100
22.158
20.811
19.504
21.362
20.402
19.464
-0.796
-0.409
—0.040
960
950
940
2.061
1.998
1.935
4.068
3.983
3.900
2.007
1.985
1.965
0.493
0.498
0.504
2050
2900
1950
18.229
17.096
16.127
18.548
17.654
16.783
+ 0.319
+ 0.558
+ 0.656
930
920
910
1.874
1. 814
1.756
3.817
3.736
3.655
1.943
1.922
1.899
0.509
0.515
0.520
1900
1850
1800
15.364
14.696
14.002
15.934
15.106
14.300
+ 0.570
+ 0.410
+0.298
900
850
800
1.699
1. 431
1. 212
3.575
3.189
2.825
1.876
1.758
1. 613
0.525
0.551
0.580
1750
1700
1650
13.318
12.666
12.030
13.517
12.766
12.016
+ 0.199
+ 0.100
—0.014
750
700
650
1.043
0.905
0.784
2.483
2. 163
1.865
1.440
1.258
1. 081
0.580
0.581
0.580
1600
1550
1500
II. 416
10.829
10.263
11.298
10.604
9.930
—0.018
-0.225
-0.333
-0.342
-0.273
— 0.141
600
550
500
450
400
350
0.674
0.572
0.473
0.381
0.294
0.221
1.589
1.335
1. 103
0.894
0.706
541
0.915
0.763
0.630
0.513
0.412
0.320
0.576
0.572
0.571
0.574
0.583
0.592
1450
1400
1350
9.622
8.924
8.185
9.280
8.651
8.044
1300
1250
1200
7.413
6.637
5.884
7.459
6.896
6.356
+ 0.046
0.259
0.472
0.006
0.038
0.070
300
250
200
0.162
0.112
0.072
0.397
0.276
0.177
0.235
0,164
0.105
0.592
0.595
0.591
II50
IIOO
1090
5.179
4.420
4.221
5.837
5.340
5.244
0.658
0.920
+ 1.023
0.113
0.172
0.195
150
100
0.040
0.018
0.099
0.044
0.059
+ 0.026
0.594
0.591
CHAPTER 11.
EXPERIMENTAL RESISTANCE.
Notable Experiments. — Benjamin Robins was the
first to execute a systematic and intelligent series of experi-
ments to determine the velocity of projectiles and the effect
of the resistance of the air, not only in retarding but in de-
flecting them from the plane of fire. He was the inventor
of the ballistic pendulum, an instrument for measuring the
momenta of projectiles and thence their velocities. He also
invented the Whirling Machine for determining the resistance
of air to bodies of different forms moving with low velo-
cities. His *' New Principles of Gunnery," containing the
results of his labors, was published in 1742, and immediately
attracted the attention of the great Euler, who translated it
into French.
The next series of experiments of any value were made
toward the close of the last century by Dr. Hutton, of the
Royal Military Academy, Woolwich. He improved the
apparatus invented by Robins, and used heavier projectiles
with higher velocities. His experiments showed that the
resistance is approximately proportional to the square of
the diameter of the projectile, and that it increases more
rapidly than the square of the velocity up to about 1440 f. s.,
and nearly as the square of the velocity from 1440 f. s. to
1968 f. s.
In 1839 ^"<^ 1840 experiments were conducted at Metz,
on a hitherto unprecedented scale, by a commission ap-
pointed by the French Minister of War, consisting of MM.
Piobert, Morin, and Didion. They fired spherical projec-
tiles weighing from 11 to 50 pounds, with diameters varying
from 4 to 8.7 inches, into a ballistic pendulum, at distances
of 15,40,65,90, and 115 metres; by this means velocities
I8 EXTERIOR BALLISTICS.
were determined at points 25, 50, 75, and 100 metres apart,
the velocities varying from 200 to 600 metres per second.
From these experiments General Didion deduced a law
of resistance expressed by a binomial, one term of which is
proportional to the square, and the other to the cube, of the
velocity. This gave good results for short ranges ; but with
heavy charges and high angles of projection the calculated
ranges were much greater than the observed.
Another series of experiments was made at Metz, in the
years 1856, 1857, and 1858, by means of the electro-ballistic
pendulum invented by Captain Navez, of the Belgian Artil-
lery. This, unlike the ballistic pendulum, affords the means
of measuring the velocity of the same projectile at two
points of its trajectory. The results of these elaborate ex-
periments may be briefly stated as follows: The resistance
for a velocity of 320 m. s. does not differ sensibly from that
deduced from the previous experiments at Metz; but the
resistances decrease with the velocity below 320 m. s., and
increase with the velocity above 320 m. s., more rapidly than
resulted from the former experiments. The commission
having charge of these experiments, whose president was
Colonel Virlet, expressed the resistance of the air by a
single term proportional to the cube of the velocity for all
velocities.
In 1865 the Rev. Francis Bashforth, M.A., who had then
been recently appointed Professor of Applied Mathematics
to the advanced class of artillery officers at Woolwich,
began a series of experiments for determining the resistance
of the air to the motion of both spherical and oblong projec-
tiles, which he continued from time to time until 1880. As
the instruments then in use for measuring velocities were
incapable of giving the times occupied by a shot in passing
over a series of successive equal spaces, he began his labors
by inventing and constructing a chronograph to accomplish
this object, which was tried late in 1865 in Woolwich
Marshes, with ten screens, and with perfect success. It was
afterwards removed to Shoeburyness, where most of his
EXTERIOR BALLISTICS. I9
subsequent experiments were made. He employed rifled
guns of 3, 5, 7, and 9-inch calibers, and elongated shot hav-
ing ogival heads struck with radii of i^ calibers; also
smooth-bore guns of similar calibers for firing spherical
shot. From the data derived from these experiments he
constructed and published, from time to time, extensive
tables connecting space and velocity, and time and velocity,
which for accuracy and general usefulness have never been
excelled. The first of these tables was published in 1870,
and his Final Report, containing coefficients of resistance
for ogival-headed shot, for velocities extending from 2800
f. s. to JOG f. s., was published in 1880. These experiments
will be noticed more in detail further on.
General Mayevski conducted some experiments at St.
Petersburg, in 1868, with spherical projectiles, and in the
following year with ogival-headed projectiles, supplement-
ing these latter with the experiments made by Bashforth in
1867 with 9-inch shot. An account of these experiments,
with the results deduced therefrom, is given in his " Traite
Balistique Exterieure," Paris, 1872.
General Mayevski has recently (1882) published the re-
sults of a discussion of the extensive experiments made at
Meppen in 1881 with the Krupp guns and projectiles.
These latter, though varying greatly in caliber, were all
sensibl}^ of the same type, being mostly 3 calibers in length,
with an ogive of 2 calibers radius. General Mayevski's
results, together with Colonel HojeFs still more recent dis-
cussion of the same data, will be noticed again.
-7? Methods of Determining Resistances. — If a prO'-*
jectile be fired horizontally, the path described in the first
one or two tenths of a second may, without sensible error,
be considered a horizontal right line ; and, therefore, what-
ever loss of velocity it may sustain in this short time will be
due to the resistance of the air, since the only other force
acting upon the projectile, gravity^ may be disregarded, as
it acts at right angles to the projectile's motion. For ex-
ample, an 8-inch oblong shell, having an initial velocity of
20 EXTERIOR BALLISTICS.
1400 f. s., will describe a horizontal path, in the first two-
tenths of a second after leaving the gun, of 278 ft., while its
vertical descent due to gravity will be less than 8 inches.
Moreover, if its velocity should be measured at the distance
of 278 ft. from the muzzle of the gun, it would be found to
be but 1380 f. s., showing a loss of velocity of 20 f. s., due to
the resistance of the air.
The relation between the horizontal space passed over
by a projectile and its loss of velocity may be determined
as follows :
Let w be the weight of the projectile in pounds, V and
V its velocities, respectively, at the distances^ and a' from
the muzzle of the gun, in feet per second, and g the accele-
ration of gravity. The vis viva of the projectile at the dis-
. wV" zv V"
tance a from the gun is , and at the distance a\ :
^ g
consequently the loss of vis viva in describing the path
vu
a' —a^ is -( F^— V ^) ; and this, by the principle of vis viva, is
equal to twice the work due to the resistance of the air. If
the distance a'— a is not too great, say from 100 to 300 ft.,
according to the velocity of the projectile, it may be as-
sumed that for this distance the resistance will not vary
perceptibly ; and if p is the mean resistance for this short
portion of the trajectory, we shall have
'^{V'-V'^) = 2{a'-a)p
whence
P- 2g{a'-a)
As, the resistance of the air is proportional to its density,
which is continually varying, it is necessary, in order to
compare a series of observations made at different times, to
reduce them all to some mean density taken as a standard.
If b is the density of the air at the time the observations are
made, and b^ the adopted standard density to which the ob-
/' =
EXTERIOR BALLISTICS. 21
servations are to be reduced, the second member of the
preceding equation shoidd be multipHed by ~^ which gives
' 2g{a' — a) d
We may take for the value of (\ the weight of a cubic
foot of air at a certain temperature and pressure; o will then
be the weight of an equal volume of air at the time of mak-
ing the experiments, as determined by observations of the
thermometer, barometer, and hygrometer.
As ft is the mean resistance for the distance a^ — a, it may
VA-V
be considered proportional to the mean velocity, v^ — ;
and substituting this in the above expression, it becomes
wv{V- V') d, ,
By varying the charge so as to obtain different values
for Fand V, the resistance corresponding to different ve-
locities may be determined, and thence the /aw of resistance
deduced.
In order to compare the results obtained with projec-
tiles of different calibers, the resistance per unit of surface
(square foot) is taken ; and, to make the results less sensible
to variations of velocity, Didion proposed to divide the
values of o by -J^ and compare the quotients (p') instead of
/>. Therefore, making ^t — — ^2-^, equation (4) becomes
^ gTzF^via' -a) 3 ^^^
It will be observed that since p is divided by ^'', the
values of f/ will be constant when the resistance varies as
the square of the velocity ; when this is not the case // will
evidentl}^ be a function of the velocity; or f/ = A' f{v)
(suppose), where the constant A', and the form of the fune-
tion,/(2/), are both to be determined.
3
/
22 EXTERIOR BALLISTICS.
Two assumptions have been made in deducing the ex-
pression for (), neither of which is exactly correct: ist, that
the resistance can be considered constant while the pro-
jectile is describing the short path a' — a ; and, 2d, that this
assumed constant resistance is that due to the mean velo-
city, V. The nature of the error thus committed may be
exhibited as follows:
The exact expression for p is
w dv wv dv
^'~~~g~dt~~gds
Comparing this with (4), it will be seen that we have made
\V- V _ _dv
a' — a ds
which is true only when the path described by the projec-
tile is infinitesimal.
To determine the amount of error committed, we can re-
calculate the values of// by means of the law of resistance
deduced from the experiments; and it will be found that in
the most unfavorable cases the two sets of values of />' will
not differ from each other by any appreciable amount. For
example, suppose the law of resistance deduced by this
method is that of the square of the velocity ; what is the
exact expression for // in terms of F~ V and a — a? We
have
, p _ w dv
^' ~^:^"J~ '^g^^' vds
and therefore
, , w dv
p' dsz=L — —
whence, integrating between the limits Fand V , to which
correspond a and a' , we have, since p' is constant in this
case,
^' "^ gTzR^oT^) ^^^ Y'
To test the two expressions for //, take the follow
EXTERIOR BALLISTICS. 23
ing data from Bashforth's ''Final Report," page 19, round
486:
F=2826 f. s. ; F' = 2777 f. s. ; 7e' — Solbs. ; 7? = 4 in. = ^ft.;
F — F' = 49 ; ^^= 32.191 ; a' — a=^ i$o ft., and z^ =
V+ V
2
= 2801.5.
We find ■ — ^-i^^-y—, v=: 0.047463; and this is a factor in
^-rrR'ia -a)
both expressions for />'. Therefore, by the approximate
method,
f/ = 0.047463 28^-T = 0.00083
and by the exact method,
^ 1 2826
f)' = 0.047463 log = 0.00084.
For a second example, suppose the law of resistance to
be that of the cube of the velocity. In this case f/ varies as
the first power of the velocity, or f/ =^ A^ v. Therefore
A, 1 2v dv
^^ TT R V
whence
II
^,^ ee. F^~"F
gTzK' a' - a
and
.'-A'^^-- "^ v{V-V'y
' — "~ gT.k'ia' -a) W
Comparing this with (5), it will be seen that (omitting the
factor ^0 the two equations are identical, if we assume
z;^ = VV ; and this is very nearly correct when, as in the
present case, V — V is very small compared with either
For v.
As an example of this method of reducing observations,
the experiments made at St. Petersburg in 1868 by General
24
EXTERIOR BALLISTICS.
Mayevski, with spherical projectiles, have been selected.
In these experiments the velocities were determined by
two Boiilenge chronographs, and the times measured were
in every case within the limits of o.''io and o.'' 15.
X
*
<
•f
*
it
♦
\
dq\.
'
\
\>
§ w
The experiments were made with 6 and 24-pdr. guns
and 120-pdr. mortars, and the velocities ranged from 745
f. s. to 1729 f. s. At least eight shots were fired with the
EXTERIOR BALLISTICS.
25
same charge; the value of// was calculated for each shot,
and the mean of all the values of />' so calculated was taken
as corresponding to the mean velocity of all the shots fired
with the same charge. The values o^ a' — a varied from
164 ft. to 492 ft., the least values being taken for the
heaviest charges, and the greatest values for the smallest
charges. The greatest loss of velocity {V — V) was 131
ft., and the least 33 ft.
The values of {/ deduced from these experiments are
given in the following table. For convenience English
units of weight and length are employed ; that is, the
weights of the projectiles are given in pounds, the veloci-
ties in feet per second, and the radii of the projectiles and
the values of <^' — ^ in feet.
Values of p for Si'Herical Projectiles, deduced from the Experi-
ments MADE AT St. Petersburg in 1868.
Mean
Mean
Kind of Gun.
Velocity
Values of
P'
Kind of Gun.
Velocity
Values of
6-pdr. gun
745 f. s.
0.000561
24-pdr. gun
1247 f. s.
0.001054
24-pdr. gun
768 "
508
0-pdr. gun
1260 "
"45
120-pdr. mort.
860 "
687
120-pdr. mort.
1339 "
1117
6-pdr. gun
912 "
807
6-pdr. gun
1362 "
1189
24-pdr. gun
942 "
782
24-pdr. gun
1499 "
1138
120-pdr. mort.
1083 "
934
120-pdr. mort.
I5I9 "
1 163
24-pdr. gun
TII9 "
987
6-pdr. gun
1558 "
1189
6-pdr. gun
II22 "
0.001107
24-pdr. gun
1729 "
0.001178
These results are reproduced graphically in Fig. 3, the
velocities being taken for abscissas, and the corresponding
values of// for ordinates. It will be seen that the trend ot
the last seven points is nearly parallel to the axis of ab-
scissas, and may, therefore, be represented approximately
by the right line A, whose equation is
/>'z= 0.00116
in which the second member is the arithmetical mean of the
last seven tabulated values of />'.
26 EXTERIOR BALLISTICS.
It was found that the remaining points could be best
represented by a curve B, of the second degree, of the
form (/ =: p-\-q 7>^, containing two constants p and q whose
values were determined by the method of least squares,
each tabular value of // and the corresponding value of
V furnishing one " observation equation." it was found
that the most probable values of/ and q were^/ = 0.012
and ^ = 0.00000034686 ; or, reducing to English units of
k k
weight and length by multiplying / by - ^, and q by «,
where k is the number of pounds in one kilogramme, and m
the number of feet in one metre, we have
^>' = 0.00022832 -[-0.00000000061309 v"
or, in a more convenient form,
^/ = 0.00022832 )i+(g-^ J [
To find the point of intersection of the right line A with
the curve B, equate the values of />' given by their respective
equations, and solve with reference to v. It will be found
that v^ 1233 f. s., at which velocity we assume that the law
of resistance changes.
In strictness there is probably but one laiv of resistance^
and this might be, perhaps, expressed by a very complicated
function of the velocity, having variable exponents and co-
efficients, depending, upon the ever- varying density of the
air, the cohesion of its particles, etc. ; but, however compli-
cated it may be, we can hardly conceive of its being other
than a continuous function. But, owing to the difficulties
with which the subject is surrounded, both experimental
and analytical, it is usual to express the resistance by in-
, tegral powers of the velocity and constant coefficients, so
chosen, as in the above example, as to represent the mean
resistance over a certain range of velocity determined by
experiment.
* Mayevski, " Traite de Balistique Exterieure," page 41.
EXTERIOR BALLISTICS.
Expression for />. — The expression for /> in terms of
which, since [/ is generally a function of 7', may be written
The resistance per unit of mass, or the retarding force, will
therefore be
or, taking the diameter of the projectile in inches,
The first member of this equation expresses the retarding
force when the air is at the adopted standard density and
the projectile under consideration is similar in every respect
to those used in making the experiments which determined
//. To generalize the equation for all densities of the at-
mosphere we must introduce into the second member the
factor IT ; and we will also assume, at present, that the equa-
tion will hold good for different types of projectiles if d'^ be
multiplied by a suitable factor {c), depending upon the kind
of projectile used. For the standard projectile and for
spherical projectiles, 6=1; for one offering a greater re-
sistance than the standard, 6'>i; and if the' resistance
offered is less, r < i. Making, then,
576
and
^~ d cd'
we have for all kinds of projectiles
p- dv A ^ , . ,^.
C is called the ballistic coefficient, and c the coefficient of
reduction.
i.
28 EXTERIOR BALLISTICS.
For the Russian experiments with spherical projectiles
the standard density of air to which the experiments were
reduced was that of air half saturated with vapor, at a tem-
perature of 1 5° C, and barometer at o"'. 75. In this condition
of air the weight of a cubic metre is 1^.206; and, therefore,
the weight of a cubic foot ( = o) is 0.075283 lbs. = 526.98 grs.
The value of ^ taken was 9"\8i = 32.1856 feet. Applying
the proper numbers, we have the following working expres-
sions for the retarding force for spherical projectiles.
Velocities greater than 1233 f. s. :
^ /? = — 7/%- log A = 6.3088473 - 10
Velocities less than 1233 f. s. :
f- r = c ''' V "^ ?' / ' ^^^ ^ ^ 5.6029333 - 10
r = 612.25 ^^•
Oblong Projectiles: Oeneral Mayevski's For-
mulas. — General Mayevski, by a method similar in its gen-
eral outline to that given above, the details and refinements
of which we omit for want of space, has deduced the fol-
lowing expressions for the resistance when the Krupp pro-
jectile is employed, viz. : '^'
700™ >V> 419™, /> = 0.0394 TT R^ -^ v^
419'" >v> 375"\ ^o = 0.0^94 r R' -yv'
375"" > -^ > 295"^, p — o.o'67 7: R" -^v"
295^^ > z; > 240^ /> = 0.0^583 ;r /?^ y 7^^
240™ > v> o™, /> = 0.014 7: T?'^ -^ v"
Changing these expressions to the form here adopted
* Revue d^Artilleriey April, 1883.
EXTERIOR BALLISTICS. 29
[equation (6)], and reducing to English units of weight and
length, they become
2300 ft. > z/> 1370 ft. :
ir-
= ^T/%- log yi =6.1192437 -
1370 ft. >^'> 1230 ft.:
- ID
i'-
--^ 7>\- log ^ = 2.9808825 -
1230 ft. > •z/>97o ft.:
ID
ir-
- J, v" ; log A = 6.8018436 -
970 ft. > z/> 790 ft.:
•20
i"-
^ 3 ,
790 ft. > 7' > ft. :
• ID
i"
= ^7^- log ^=5.6698755 -
ID
Colonel Hojel's Deductions from the Krupp Ex-
periments.— Colonel Hojel, of the Dutch Artillery, has
also made a study of the Krupp experiments discussed by
General Mayevski : and, as it is interesting and instructive
to compare the resistance formulas deduced by each of these
two experts, both using the same data, we give a brief syn-
opsis of Colonel Hojel's method and results.
He expresses the resistance by the following formula,
easily deduced from equation (6):
in which, from (4),
It is assumed that the loss of velocit}^ V — V\ is some func-
tion of the mean velocity v, which can be expressed approx-
imately, for a limited range of velocity, by a monomial of the
form
4
30 EXTERIOR BALLISTICS.
in which A and n are constants to be determined. The
method of procedure is analogous to that followed m deter-
mining fj', and need not be repeated. Colonel Hojel has
considered it necessary to employ fractional exponents,
thereby sacrificing simplicity without apparently gaining
in accuracy. The results he arrived at are as follows: "
700^ >v> soo'", /{v) = 2A 868 v'-''
500™ > -6^ > 400™, / (tj) = 0.29932 z/'"
400™ >v> 350'", / (v) = o.o'205 524 7/'-''
350°^ > ^ > 300"\ / (v) = o.o'2 1692 V*
300™ >v> I40"\ /{z') = 0.033814 v'-'
Substituting these values oi /{v) in the equation
w^ zv -^ ■' 4w -^ ^ ^
and reducing the results to English units, that is, taking w
in pounds, v in feet, and d in inches, we have as the equiva-
lents of Hojel's expressions, all reductions being made, the
following :
2300 ft. > 7^ > 1640 ft. :
a- A
±- p z=z ~ v'''\- log y4 =6.4211771 — ID
1640 ft. > 7^> 1310 ft. :
-|.«-^^^-"; iog^ = 5.3923859- 10
1 3 10 ft. > 7/ > II 50 ft. :
0- A
-|-^ = — 7/^«%- log ^ = 0.4035263 - 10
1150 ft. >7'>98o ft.:
a- A
^p = —v\- log ^ = 6.8232495 - 20
980 ft. > 7.' > 460 ft. :
<r A
^ f)= — v"-" ; log A — 4.3060287 — 10
Comparison of Resistances dedviced from tlie
above Formnlas. — Making ^= i and f^, = o, in the above
* Revue iV Artilleries June, 1884.
EXTERIOR BALLISTICS.
31
formulas, gives the resistance in pounds per circular inch at
the standard density of the air. Calling this ^o^, we have
A ^
The following table gives the values of p^ for different
velocities according to Mayevski's and Hojel's formulas re-
spectively ; and also the same derived from " Table de
Krupp," Essen, 1881:
Velocity
in feet
per sec.
According
to
Mayevski.
P/
According
to
Hojel.
p'
According
. to
Krupp.
Velocity
in feet
per sec.
P/ .
According
to
Mayevski.
According
to
Hojel.
p'
According
to
Krupp.
2300
2250
2200
21.629
20.699
19.789
21.598
20.710
19.840
21.637
20.643
19.738
1250
1200
II50
5.807
4.899
3.960
5.715
4.888
4.160
5-753
4.904
3-943
2150
2100
2050
18,900
18.031
17.183
18.987
18.153
17.337
18.900
17.962
17.091
1 100
1050
1000
3-171
2-513
1.969
3.331
2.640
2.068
3-105
2.480
2.044
2000
1950
1900
16.355
15.547
14.760
16.538
15-757
14-995
16.287
15-359
14. 611
950
900
850
I. 581
1.344
1. 132
1-749
1.527
1.324
1.720
1.486
1. 318
1850
1800
1750
13-993
13.247
12.521
14-250
13-523
12.815
13.929
I3-I81
12.500
800
750
700
0.944
0.817
0.712
1. 138
0.969
0.815
1. 162
0.983
0.804
1700
1650
1600
II. 816
II. 131
10.467
12.125
11.453
10.713
II. 818
11.059
10.400
650
600
550
0.614
0.523
0.439
0.677
0.554
0.446
0.648
0.514
0.413
1550
1500
1450
9.823
9.199
8.596
9.981
9.277
8.601
9-752
9.126
8.490
500
450
400
0.364
0.294
0.232
0.351
0.270
0.201
0.313
1400
1350
1300
8.014
7.315
6.535
7.954
7-334
6.6^1
7.920
7.238
6.445
i
Bashforth's Coefficients. — Professor Bashforth adopt-
ed an entirely different method from that just developed to
determine the coefficients of resistance, of which we will
give an outline, referring for further particulars to his
work,* which is well known in this country.
* " Motion of Projectiles," London, 1875 ^n^ 1881,
32 EXTERIOR BALLISTICS.
ds
We have v =: — , whence, differentiating and making s
the equicrescent variable,
dv ds d^i
'~dt~ 'df~
dv
and this value of -r substituted in (6) gives
g_ _ ds d't _ /dsV d'^t __ ^ d't
w^'~ df ~~\dt) ds''~^'' ds'
From this it follows that if the resistance varied as the cube
of the velocity, — would be constant; and we should have
ds"^
--,=2^, (say);
whence, integrating twice,
t -=1 bs" -\- a s -\- c
which is the relation between the time and space upon this
hypothesis. When the resistance is not proportional to the
cube of the velocity, - in the equation
^ ds' ^
— /> = — -- ir — 2b V
w * ds
will be variable, and its value must be so determined by ex-
periment as to satisfy this equation for each value of v,
Bashforth's method of deducing these values is briefly as
follows :
Ten screens are placed at equal distances (150 feet) apart
in the plane of fire, and the exact time of the passage
of a projectile through each screen is measured by the
Bashforth chronograph. The first, second, third, etc., dif-
ferences of these observed times are taken, which call
d,, d,,d,, etc.
Let s be the distance the projectile has moved from
some assumed point to any one of the screens, say the first ;
EXTERIOR BALLISTICS. 33
/the constant distance between the screens; and /,^ /,+/^ /,+2/,
etc., the observed times of the projectile's passing succes-
sive screens. Then from a well-known equation of finite
differences we have
, , n(n— 1) . , n(n— i)in — 2) , . ,
ts.ni=^t,-\- ltd, + ^——-^ d, H ^^ ~\\ ^- d, + etc.
1.2 I • - • 3
in which ;/ is an arbitrary variable. Arranging the second
member according to the powers of//, we have
ts.ni^t,-\-n \d, — \d^-\-\d^ — -d,\ etc. )
\ 2 3 4 /
-f etc., etc.,
terms multiplied by the cube and higher powers of ^/.
Since / is a function of s, we have t^—f{s) and t,^„i:=
f{s + 111). Expanding this last by Taylor's formula, we have
, dt, nl , dU, n'l' ^
whence, equating the coefficients of the first and second
powers of 71 in the two expansions of /^ + „/, we have
/^^ = ^-i^,+ i-^3_i^^ + etc.
ds ' 2 ' ' 3 4
and
,„ ^V, , , , II - 10 7 1 X
:r-^^=:^,-^3 + — <- — < + etC.
The first of these equations gives
ds I
dt, ^-d,-\d,-^^\d,-\d,
and the second
7;
ds-" '
where i\ is the velocity and - /> the resistance per unit of
mass at the distance s from the gun.
.J,,_-q^^_^^ + iL^._I|^. + etc.)
34
EXTERIOR BALLISTICS.
As an example take the following experiment made with
a 6.92-inch spherical shot, weighing- 44.094 lbs., fired from a
7-inch gun."^ The times of passing the successive screens
were as follows :
Screens.
Passed at,
Seconds.
d.
d^
^3
I
2 . 90068
8431
306
10
2
2.98499
8737
316
TO
•3
3.07236
9053
326
10
4
3. 16289
9379
336
10
5
3.25668
9715
346
10
6
3.35383
10061
356
I I
7
3-45444
10417
367
II
8
3.55861
10784
378
9
3.66645
11162
10
3.77807
To find, for example, the velocity at the first screen, we
have
150
1 1.4 t. s.,
= 1465.3 f"- s.
' 0.08431—^0.00306-1-^0.00010
and at the seventh screen
150
' 0.10417 — ^0.003674-^0.00011
The retarding forces at the same screens are as follows:
or V^
^ f)^=z - — ^- (o . 00306 — o . oooio) = o . ooooooi 3 1 56 Z^,' = 2<^, V,^
and
- Pt= 7 — '-^(0.00367 — 0.00011) = 0. ooooooi 5822 z'/ = 2^, z;/.
As these small numbers are inconvenient in practice,
* Bashforth, page 43.
EXTERIOR BALLISTICS. 35
Bashforth substituted for them a coefficient K, defined by
the equation
A-=24J(.ooo)'.
In the experiment selected above the weight of a cubic
foot of air was 553.9 grains = (?, while the standard weight
adopted was 530.6 grains = d^. Therefore we have
(150) (6.92) 553.9
and
j^ 0.00356 ^^ ^„
A; = ^\K,— 139.6*
0.00296 ' ^^
That is to say, when the velocity of a spherical projectile
is 1811.4 f. s., A"=ii6.i; and when its velocity is 1465.3
f. s., A'= 139.6. By interpolation the values of K, after
having been determined for a sufficient number of velo-
cities, are arranged in tabular form with the velocity as
argument.
Bashforth determined the values of K by this original
and beautiful method for both spherical and ogival-headed
projectiles ; and for the latter for velocities extending from
2900 f. s. down to 100 f. s. The experiments upon which
they were based were made under his own direction at
various times between 1865 and 1879, ^'ith his chronograph,
probably the most complete and accurate instrument for
measuring small intervals of time yet invented.
Law of Resistance deduced from Bashforth's
K. — It will be seen, by examining Bashforth's table of A" for
ogival-headed projectiles, that as the velocity decreases
from 2800 f. s. down to about 1300 f. s., the values of K
gradually increase, then become nearly constant down to
about ii3of. s., then rapidly decrease down to about 1030
f. s., become nearly constant again down to about 800 f. s.,
and then gradually increase as the velocity decreases, to the
♦ Bashforth's " Mathematical Treatise," page 97.
36 EXTERIOR BALLISTICS.
limit of the table. These variations show that the law of
resistance is not the same for all velocities, but that it
changes several times between practical limits. We may
use Bashforth's K for determining these different laws of
resistance as follows :
We have for the standard density of the air,
^ p = 2bv' 3= _ _— — (7)
w ^ zv (looo)
and
from which we get
,_ S76Kv
^^ ;r^(ioooy
The values of />' have been computed by means of this
formula, for ogival-headed projectiles, from if — 2900 f. s. to
V = 100 f. s., and their discussion has yielded the following
results :
Velocities greater than 1330 f. s. :
o A
^p — —,7>\- log ^ =: 6.1525284— 10
1330 f. s. > 't^> II20 f. s. :
^/> = -^7^'; log ^=: 3.0364351 - 10
1 1 20 f . s. > 2/ > 990 f . s. :
^/> = -^-^"/ log yi = 3.8865079 -20
990 f. s. > 7^ > 790 f. s. :
fv^^'c'^'' log ^=2.8754872 -10
790 f. s. > 7/ > 100 f. s. :
p- A
^pz=.-v\- log 7^ = 5.7703827- 10
These expressions, derived as they are from Bashforth's
EXTERIOR 15ATXISTICS. 37
coefficients, give substantially the same resistances for like
velocities as those computed directly by means of equation
(7). The agreement between the two for high velocities is
shown graphically by Plate I., in which A is Bashforth's
curve of resistance, while that part of the parabola, B, com-
prised between the limits ^^2800 f. s. and 7'= 1330 f. s., is
the curve of resistance deduced from the first of the above
expressions. If, hovvever, we compare these expressions
with those deduced by Mayevski or Hojel from the Krupp
experiments, it will be found that these latter give a less
resistance than the former for all velocities.
This is undoubtedly due to the superior centring of the
projectiles in the Krupp guns over the English, and to the
different shapes of the projectiles used in the two series of
experiments, particularly to the difference in the shapes of
the heads. The English projectiles, as we have seen, had
ogival heads struck with radii of i| calibers, while those
fired at Meppen had similar heads of 2 calibres, and,
therefore, suffered less resistance than the former indepen-
dently of their greater steadiness.
Comparison of Resistances. — Let f) and {>^ be the re-
sistances of the air to the motion of two different projectiles
of similar forms ; w and zv^ their weights ; 5 and S^ the areas
of their greatest transverse sections; d and d^ their dia-
meters ; and D and D^ their densities. Then, if we suppose,
in the case of oblong projectiles, that their axes coincide
with the direction of motion, we shall have from (6) for the
same velocity, since 5 and S^ are proportional to the squares
of their diameters.
i"
s
w
A V ^
; and ^ = —
i''~
that is, for the same velocity the resistances are proportional
to the areas of the greatest transverse sections, while the
retardations are directly proportional to the areas and in-
5
38 EXTERIOR BALLISTICS.
versely proportional
tiles we have
to
the )
A^eights.
For
spher
ical
projec-
.5=i;^^^ S,.
= i^
^A
, 7i>-
= i;r^^A
an
d w^ =
:t-;r
^;^.;
therefore
_d^D,
~ dD
that is, for spherical projectiles the retardations are in-
versely proportional to the products of the diameters and
densities. This shows that for equal velocities the loss of
velocity in a unit of time will be less, and, therefore, the
range greater, cceteris paribus, the greater the diameter and
density of the projectile.
As the weight of an oblong projectile is considerably
greater than that of a spherical projectile of the same caliber
and material, it follows that the retardation of the former
for equal velocities is much less than the latter, indepen-
dently of the ogival form of the head of an oblong projectile
which diminishes the resistance still more. Indeed, the re
tarding effect of the air to the motion of a standard oblong
projectile, for velocities exceeding 1330 f. s., is less than for a
spherical projectile of the same diameter and weight, and
moving with the same velocity, in the ratio of 14208 to
20358. As an example, if d and w are the diameter and
weight of a solid spherical cast-iron shot which shall suffer
the same retardation as an 8-inch oblong projectile weighing
180 lbs. and moving with the same velocity, we shall have,
since we know that a solid shot 14.87 inches in diameter
weighs 450 lbs.,
,_ (14.87)' X 180X20358
and
^ „ — 29.65 inches
450X64X 14208 ^ •"
450 X (20.65)' . ,,
w— ^^ . W, ^^ —3567 lbs.
(14.87 ' ^^ ^
The retarding effect of the air to the motion of projectiles
EXTERIOR BALLISTICS.
39
of different calibers but having the same initial velocity and
angle of projection, is shown graphically in Fig. 4, which was
carefully drawn to scale. A is the curve which a projectile
would describe in vacuo, B that actually described b}^ a
spherical projectile 14.87 in diameter weighing 450 lbs., and
C that described by a spherical shot 5.9 inches in diameter
weighing 26.92 lbs. The initial velocity of each is 1712.6
f. s., and angle of projection 30°.
Example. — Calculate the resistance of the air and the re-
tardation for a 15-inch spherical solid shot moving with a
velocity of 1400 f. s. Here ^= 14.87 in., 7(^ = 450 lbs., and
A ~ 20358X iQ-l
Substituting these values in equation (6), we have
and
dv
dt
(14.87)' 20358 ,
^-^-^ X -^ X (1400)^
32.16 10 ^ -1- /
(i4.87r
450
X '-'^^ X (.400)'
2743 lbs..
196.07 f. s. ;
that is, at the instant the projectile was moving with a
velocity of 1400 f. s. it suffered a resistance of 2743 lbs. ;
and if this resistance were to remain constant for one second
the velocity of the projectile would be diminished by 196.07
ft. As, however, the resistance is not constant, but varies as
the square of the velocity, it will require an integration to
determine the actual loss of velocity in one second.
We have from (6)
dt IV
40 EXTERIOR BALLISTICS.
or
dv iV . ,
-^,=- Adt
whence, integraling between the limits F, 7>, we have
Now, making V= 1400 and t=zi, we find v — 1228 f. s. ;
and the loss of velocit}^ in one second is 1400 — 1228 = 172 ft,
CHAPTER III.
DIFFERENTIAL EQUATIONS OF TRANSLATION — GENERAL
PROPERTIES OF TRAJECTORIES.
Preliiiiiiiary Considerations.— A projectile fired from
a gun with a certain initial velocity is acted upon during its
flight only by gravity and the resistance of the air; the
former in a vertical direction, and the latter along the tan-
gent to the curve described by the projectile's centre of
gravity. It will be assumed, as a first approximation, that
the projectile, if spherical, has no motion of rotation ; and,
in the case of oblong projectiles, that the axis of the pro-
jectile lies constantly in the tangent to the trajectory ; also
that the air through which it moves is quiescent and of uni-
form densit}'. xA.s none of these conditions are ever fulfilled
in practice, the equations deduced will only give what may
be called the normal trajectory^ or the trajectory in the plane
of fire, and from which the actual trajectory will deviate
more or less It is evident, however, that this deviation
from the plane of fire is relatively small ; that is, small in
comparison with the whole extent of the trajectory, owing
to the very great density of the projectile as compared with
that of the air.
Notation.— In Figure 5, let (9, the point of projection,
be taken for the origin of rectangular co-ordinates, of which
let the axis of X be horizontal and that of F vertical. Let
O A be the line of projection, and O B E the trajectory de-
scribed. The following notation will be adopted:
o denotes the acceleration of gravity, which will be taken
at 32.16 f. s. ;
IV the weight of the projectile in pounds;
d its diameter in inches;
(p the angle of projection, A O E ;
42 EXTERIOR BALLISTICS.
V the velocity of projection, or muzzle velocity ;
[/ the horizontal velocity of projection = Fcos (f ;
V the velocity of the projectile at any point M of the
trajectory ;
3 the angle included between the tangent to the curve
at any point Jf and the axis of X, = T M H ;
CO the angle of fall, CEO;
Y
D
u the horizontal velocity = ^' cos ^;
/ the time of describing any portion of the trajectory
from the origin ;
s the length of any portion of the arc, as O in ;
X the horizontal range, O E ;
T the time of flight ;
ft the resistance of the air, or the resistance a projectile
encounters in the direction of its motion, in pounds.
Dift'erential Equations of Translation. — The ac-
celeration'^" in the direction of motion due to the resistance
of the air is — //; and the correspondins^ acceleration due to
gravity is ^ sin /> ; therefore the /<?/<:?/ acceleration in the
direction of motion is expressed by the equation,
f =-^-,-,-sin,> (8)
The velocities parallel to X and Y are, respectively,
* The term "acceleration" is here used for retardation. To avoid multiplying terms re-
tardation will be regarded as negative acceleration.
EXTERIOR BALLISTICS. 43
V COS <>and v sin /> ; and the accelerations parallel to the same
o g
axes are -- /> cos /> and j^ + -- p sin />.
Therefore
^ (^ cos '>) ^ g , ,
^ ,, = - — /' COS /!/ (9)
at zu ' ^^^
and
d {v sin /5^) ^.
dt "- - '
-^ ^ f> sin ^y
7£/
Performing- the differentiations indicated in the above
equations, multiplying the first by sin & and the second by
cos />, and taking their difference, gives
— ^ = -^^cos/V (10)
Introducing the horizontal velocity u = 7' cos /> in (9) and
(10), and substituting for •_'^ ft its value from (6), they become,
making/ (7/) = 7^",
du A u"" • ,
rf7 "" "~ 6'^ cos""-^^^ ^^^^
and
-^^ = — a- cos' fi (12)
whence, eliminating <3^/,
<^ /> g C du
(13)
cos"^'<> ^ ?/**^'
Symbolizing the integral of the first member of (13) by
(/>)„, that is, making
^^^" J "cos^'^"^
71 k^ C
and writing for the sake of symmetry, for -^, we shall
have
rdu k^
id-\ = n k' I = h C
44
EXTERIOR BALLISTICS.
If (z) is the value of (f"^) when // is infinite, we have
C=(0; and therefore
whence
and
={t\-{»).
k
k sec (5*
From (ii) we have
C ^ , Kidn
(it ■-— cos"-' f>—-
A ?/"
and this substituted in the equations
c/x = ?/ d/, dy = // tan ^ dt, ds = // sec li dt,
gives
^,= __^cos«-,>_
./j/= --^sln/^cos'-^/^f-,
^^ =
From (12) w^e have
C
cos"
,>i^
dt= -
It d />
^^ cos'' ^>
d tan ^y
whence, as before,
dx=^ d tan &
g
dy
ds
tan b- d tan /V
— sec ^ ^ tan ^^
(14)
(•5)
(>-)
(18)
(19)
(20)
(21)
(22)
(23)
(24)
EXTERIOR BALLISTICS.
45
Eliminating- u from these last four equations by means
of (15), they take the following eleg-ant forms :
it= ~ ~
k_ d tan d^ (25)
, ^ ^t an d^ (26)
, k- tan ^y^/ tan /V (27)
_ _ /P sec d d tan /> (28)
7?^' w<7r/'j-.— Subject to the conditions specified in the pre-
liminary considerations, equations (16) to (20) or (25) to (28)
contain the whole theory of the motion of translation of a
projectile in a medium whose resistance can be expressed by
an integral power of the velocity. Equation (16) gives the
velocity in terms of the inclination; (18) and (19) or (26) and
(27), could they be integrated generally, would give the co-
ordinates of any point of the trajectory, while the time would
depend upon the integration of (17) or (25). But, unfortu-
nately, the ''laws of resistance" which obtain in our atmo-
sphere do not admit of the integration of these equations ;
we are, therefore, obliged to resort to indirect solutions
giving approximations more or less exact. Of these many
have been proposed by different investigators; but, with
few exceptions, they are either too operose for practical use
or not sufficiently approximate.
General Didion, in the fifth section of his '* Traite de
Balistique," gives a full and interesting ri^sumi^ o{ \\\q labors
of mathematicians upon this difficult problem up to his
time (1847), ^"<^ ^" the same work gives an original solution
of his own of great value. Within the last quarter of a cen-
tury much has been accomplished to improve and simplify
6
46 EXTERIOR BALLISTICS.
the methods for calculating tables of fire and for the solution
of the various problems relating to trajectories ; and we will
endeavor in the following pages to present such of these
methods as are of recognized value, developed after a uni-
form plan and based upon the preceding differential equa-
tions.
Oeiieral Properties of Trajectories. — Though it
is impossible with our present knowledge to deduce the
equation of the trajectory described by a projectile, there
are certain general properties of such trajectories which
may be determined without knowing the law of resistance,
if we admit that the resistance increases as some power of
the velocity greater than the first, from zero to infinity;
whence, making — = /(^)> we shall have f {v) > o, and
/(x)= X.
Variation of tlie Velocity — Miniiniim Velocity.
— The acceleration in the dircctio7i of motion is [equation (8)]
f =-."T/(^') + sin*]
in which — ^sin d is the component of gravity in the direc-
tion of motion; and, therefore, whether the velocity is in-
creasing or decreasing with the time at any point of the tra-
jectory, depends upon the algebraic sign of the second mem-
ber; and this, since f {v) \ = — ) is considered positive,
depends upon the sign of sin d^. In the ascending branch
sin & is positive, and, therefore, from the point of projection
to the summit the velocity is decreasing. At the summit
sin ?^ = o, and at this point gravity, which has hitherto con-
spired with the resistance to diminish the velocity, ceases
to act for an instant in the direction of motion, and then, as
sin d- changes sign in the descending branch, begins to act
in opposition to the resistance ; that is, its action tends to
increase the velocity. The component of gravity acting
perpendicular to the projectile's motion (^ cos d), and which
EXTERIOR BALLISTICS. 47
is a maximum at the summit, tends to increase the in-
clination in the descending branch, and thus to increase
(numerically) — sin &, until at a certain point of the de-
scending branch where the inclination is (say) — &' the
acceleration of gravity in the direction of motion has in-
creased until it just equals the retardation due to the re-
sistance of the air, which latter has continually decreased
with the velocity. Beyond this point, as the component of
gravity in the direction of motion still increases with the
inclination while the resistance remains constant for an in-
stant, the velocity also increases; and, therefore, at the
point where
w
the velocity is a minimum, and — = o.
Passing the point of minimum velocity, the acceleration
of gravity and the retardation due to the resistance of the
air both increase; but that there is no maximum velocity,
properly speaking, may be shown as follows :
Differentiating the above expression for the acceleration,
we have
d'^v ' dv ^ d&
and putting in place of ^ its value from (10), we shall have
d'v ^,. ^dv , .^-^cos^/>
df '*'' ' V/ ' V
dv
and this is necessarily positive whenever — == o. The velo-
city, therefore, can only be a minimum ; but it tends towards
1- • • 1 • ^ P 10 '^
a hmitmp: value, viz., when — - =: i, and //=^ .
Liinitiiig" Velocity. — As the limiting velocities of all
service spherical projectiles are less than 1233 f. s., we can
48
EXTERIOR liALLlSTlCS.
determine these velocities by means of the expression for
the resistance given in Chapter II., from which we get
A (P
o zv
(.+o>)
:= I
where ^4 = 0.000040048 and r = 610.25. Solving with re-
ference to 7' we ore t
Z'+^^f--'
which gives tlie limiting velocity.
The following table contains the limiting velocities of
spherical projectiles in our service calculated by the above
formula :
Solid Shot.
Inches.
Lbs.
Final
Velocity. '
Feet. i
Shells
Unfilled.
Inches.
7('
Lbs.
Final
Velocity.
iFeet.
20-inch
19.87
1080
859 :
15-inch
14.87
330
1
726 1
15-inch
14.87
450
783
13-inch
12,87
216
682
13-inch
12.87
283
743
lo-inch
9.87
101.75
635
lo-inch
9.87
128
684
8-inch
7.88
45
561
i2-pdr.
4.52
12.3
526
i2-pdr.
4.52
8.34
458
Limit of the Iiieliiiatioii of the Trajectory in the
Desceiidinj>- Braiicli. — We have assumed above that tlie
descending branch of tlie trajectory ultimately becomes
vertical. To prove this, take equation (10), viz. :
and integrating from a point of the trajectory where ^y = if
and ^ = o, we have
As the velocity v, between the limits / = o and / = x , is
XY )
EXTERIOR BALLISTICS. 49
finite and continuous, and cannot become zero, we have,
since v is a function of />,
where A^ is some value of v greater than its least, and less
than its greatest value between the limits of integration.
As (^ is negative in the descending branch, the above
equation shows that, when / is infinite, /V is equal to — '-.
2
From (24) we have
tan('' + ^'
z^^/i \4 2
A log
ot/s
= —v\ .
COS />
and,
therefore
I, when
t is
infinite.
c —
2
C0
cos/y-
: K'
<b-^ —
tan
\4 4.
tan o
where K' is some value of -ir greater than its least, and
less than its "freatest value between the limits of inte-
?5
»(i+-9.,.
tan
gration ; and, as log ^ — ^ ^ is infinite, so is the arc
tan o
which conesponds to / = x .
Asymptote to the Descending- Branch.— As the
tangent to the descending branch at infinity is vertical, if it
can be shown that it cuts the axis of X at a finite distance,
it is an asymptote. To determine this, take equation (22)
which ofives
.'•-=/_'. -'^'' = ^"'('f
where K" is a finite quantity, since v^ is finite between the
limits of integration. Therefore the descending branch has
a vertical asymptote.
50 EXTERIOR BALLISTICS.
Radius of Curvature. — Designate the radius of curva-
ture by y. We have by the differential calculus v = -^
(since the trajectory is concave toward the axis ofX); we
also have ds^=zvdt ; consequently y z=z —- — —, and therefore
from (12)
y zzz — sec (J \
g
The radius of curvature is therefore independent of the
resistance of the air, and at any point of the trajectory de-
pends only upon the velocity and the inclination, and, there-
fore, has the same value for the corresponding points of a
parabola described by a projectile in vacuo. The above ex-
pression shows that the radius of curvature decreases from
the point of projection to the summit of the trajectory,
since v and sec d- both decrease between those limits. Be-
yond the summit v still decreases, but as sec d- increases we
cannot determine by simple inspection where y ceases to
decrease and becomes a minimum. Differentiatinof the ex-
pression for y, we have
dy 2v sec d^ dv , 7/ ,, .,
-TTT = — — 777 + — tan tJ sec o-
d(> g dty g
From (13) and (6) we have
d{v cos d) ft
■fe
V
dd- w
whence, differentiating and reducing,
+ sin &
dv '' f (^
d& cos d- \ zv
Substituting this in the expression for -^ gives
dy zr
—^ = — sec
^'^ g
^.>g + 3sin,y)
This equation shows that beyond the summit --~r is posi-
EXTERIOR BALLISTICS. 5 I
tive up to the point where — + 3 sin d =r o, and then
changes its sign. At this point, therefore, the radius of
curvature becomes a minimum and afterwards increases to
infinit}^
At the point of maximum curvature we have, in conse-
20
quence of the condition ^^ + 3 sin & zzz o,
-777 = V tan If
dt^ 2
and therefore, since 0- is negative in the descending branch,
-777- is positive at that point, and v is decreasing with d-]
au
in other words, the velocity has not yet become a mini-
mum. Therefore the point of maximum curvature is near-
er the summit of the trajectory than the point of minimum
velocity.
CHAPTER IV.
RECTILINEAR MOTION.
Relation between Time, Space, and Velocity. —
For many practical purposes, and especially with the heavy,
elongated projectiles fired from modern guns, useful results
may be obtained by considering the path of the projectile
a horizontal right line, and therefore unaffected by gravity.
Upon this supposition f'i- becomes zero, and equations (17),
(18), and (20) become
C dv
and
n dx ^=- (is ^=^ — — —
A v^-^
whence integrating, and making / and ^ zero when 7^= F,
we have
t-c\ ^ L__l
((;/- \)Av^-' {n- \)A F^'M
and
,^r S I L_ ^
\{n — 2)Av^-^ {71— 2) A F«-^f
Writing, for convenience,
T {%>) for ; r ^ ^ -, and 5 {v) for ; , . „_„
these equations become
t^C\T{v)-T{.^\ 7\ (29)
and rji ' '
s=C\S{v)-S{zt)\ ' (30)
When n = 2, the above expression for s becomes inde-
terminate. In this case we have
EXTERIOR BALLISTICS. 53
, C dv
whence
s = —\\og V-\ogv I
and therefore, when n = 2,
Equations (29) and (30) (or their equivalents) were first
given by Bashforth in his *' Mathematical Treatise," Lon-
don, 1 873. He also gave in the same work tables of 5 (v) and
T{7>) for both spherical and elongated shot; the former ex-
tending from V =: 1900 f. s. to v = 500 f. s., and the latter
from V = 1700 f. s. to v =: 540 f. s. In a " Supplement " to his
work above cited, published in 1881, he extended the tables
for elongated projectiles to include velocities from 2900 f. s.
to 100 f. s.
Projectiles differing from tlie Standard. — It will
be seen that the value of the functions T{v)a.nd S (v) depend
upon those of v and A, the former of which is independent
of the nature of the shot, while the latter depends partly
upon the form of the standard projectile, which in this
country and England has an ogival head struck with a
radius of ij calibers, and a body 2^ calibers long. The fac-
tor 6^ ( or -^ ~~) depends upon the weight and diameter of
\ o ca / •
the projectile, the density of the air, and the coefficient c ;
which latter varies with the type of projectile used. The
factor^ varies, therefore, with c ; but by the manner in which
A and c enter the expressions for / and s, it will be seen
that the results will be the same if we make A constant,
and give to ^ a suitable value determined by experiment for
each kind of projectile. By this means the tables of the
functions T{v) and 5(z/), computed upon the supposition
that ^ = I, can be used for all types* of projectiles. We
will now show how these tables may be computed for ob-
long projectiles, making use of the expressions for the re-
7
54 EXTERIOR BALLISTICS.
sistance derived from Bashforth's experiments given in
Chapter I.
Oblong Projectiles, Velocities greater than 1330
f. s. — For velocities greater than 1330 f. s. we have ^^ = 2
and log ^ =6.1525284 — 10; therefore
r(.) = -iandr(F)=-i,
or, since the value of t depends upon the difference of T{y)
and T{V), we may, if convenient, introduce an arbitrary
constant into the expression for T{v). Therefore we may
take
and, similarly,
•S (^) = ^ (- log r + log 0',) = ^ log ^
To avoid large numbers and to give uniformity to the
tables we will determine the constants Q, and Q\ so that
the functions shall both reduce to zero for the same value
of v; and it will be convenient to begin the table with the
highest value of v likely to occur in practice, which we will
assume (following Bashforth) to be 2800 f. s.
We therefore have
A \2800 ' ---V
2800
I 1... Q\
loR -S^ =0 G'l = 2800
^ 2800 '
Substituting the above values of A, Q^, and Q\ in the
expressions for T{v) and S (v), and reducing, we have for
velocities between 2800 f. s. and 1330 f. s.
T{v) = [3.8474716] -^- - 2.5137
and
S{v) =155866.12 — [4.2096873] log V.
The numbers in brackets are the logarithms of the nu-
merical coefficients of the quantities to which they^^are
EXTERIOR BALLISTICS. 55
prefixed ; and the factor lo^ v is the common logarithm of
z', the modulus being included in the coefficient.
Velocities between 1330 f. s. and 1130 f. s.— For
velocities between 1330 f. s. and 11 20 f. s. we have n — i
and log A = 3.0364351 — 10; therefore, as before,
Arbitrary Constants. — To deduce suitable values for
the arbitrary constants Q^ and ^'2, we must recollect that
the function representing the resistance of the air changes
its form abruptly when the velocity is 1330 f. s. ; and to
prevent a correspondingly abrupt change in our table at
the same point — that is, to make the numbers in the table a
continuous series — we must give to g^ and Q^ such values
as shall make the second set of functions equal in value to
the first when z;=i330. They will, therefore, be deter-
mined by the following relations:
_L/ L_ - ^) = lf_J L_^
2 A V(i33o)' ^ ^7 A V1330 2800/
and
I / I I ^/ \ I 1 2800
in which the A in the first member must not be confounded
with that in the second. Making the necessary reductions,
we have
and
Tizi) = [6.6625349] A-+ 0.1791
V
5(z/)= [6.9635649]-^ - 1674.:
Velocities between 1130 f. s. and 990 f. s.— For
velocities between 1120 f. s. and 990 f. s. we have ;/ = 6
and log A — 3.8865079 — 20 ; therefore
^(-^ = 5i-(^+^")
56 EXTERIOR BALLISTICS.
and . V
The constants must be determined as before, by equating
the above expressions to the corresponding ones in the case
immediately preceding, making ^= 1120. The results are,
all reductions being made,
r(T^) = [15.4145221] ■;:^ + 2.3705
and
5(2;)=[i5.5ii432i] I- + 4472.7
Velocities between 990 f. s. and 790 f. s.— For
velocities between 990 f. s. and 790 f. s. we have // = 3 and
log A = 2.8754872 — 10; whence
and
Proceeding as before, we have
r(2;) = [6.8234828] i,- 1.6937
and
*^(^) = [7-i245i28] i- 5602.3
Velocities less than 790 f. s. — For velocities less
than 790 f. s. we have « 1= 2 and log ^4 = 5.7703827 — 10 ;
therefore
^(-)=z(^ + a)
and
whence, as before,
T{v) = [4.2296173] -^ - 12.4999
S{v) — 124466.4— [4-59' 8330] log 2/.
EXTERIOR BALLISTICS. 57
Ballistic Tables. — Table I. gives the values of the time
and space functions for oblong projectiles, computed by the
above formulas, and extends from v zizz 2800 f. s. to v = 400
f s. The first differences are given in adjacent columns;
and as the second differences rarely exceed eight units of
the last order, it will hardh^ ever be necessary to consider
them in using this table.
Table II. gives the values of these functions for spherical
projectiles, and is based upon the Russian experiments dis-
cussed in Chapter II.
EXAMPLES OF THE USE OF TABLES I. AJ^D II.
Example i. — The velocity of an 8-inch service projectile
weighing 180 lbs. was found by the Boulenge chronograph
to be 1398 f. s. at 300 ft. from the' gun. What was the
muzzle velocity ?
Here C == ^, v = 1398, and s = 300, to find V. From
64
(30) we have
and from Table 1. yXW"*
5(1398) =4903.8 - ^^^^^'^ =4888.7
also
s 64 ^
C = 300 X-^= 106.7
whence
5(F) = 4782.0
. •. F= 141 5 + ^1^ = 14^9-4 f- s.
Example 2. — Determine the remaining velocity and the
time of flight of the 12-inch service projectile, weighing 800
lbs., at 1000 yds. from the gun, the muzzle velocity being
1886 f. s.
58 EXTERIOR BALLISTICS.
1. Fand s are given, to find v; where </= 12, 2£/ = 800
800
F= 1886, J =: 3000, and C ^^
144
We have
5(.) = 5(,886) + 32^^4
From Table I.,
5 (1886) = 2803.7 - 0.6 X 37.4 = 2781.3
3000-X 144^
800 ^^
, 10 X 27.0 ^ ^
. •. z/= 1740 H — = 1746.7 f. s.
^ 40.3
2. Fand z/ are given, to find // from Table I.,
T{v)= 1.5 16
r(F)=i.2i7
r(z;)- r(F) = 0.299
.-./ = 0.299 X ^=1^66
144
" Example 3. — Suppose we wish to determine the value
of the coefficient of reduction, c, for a particular projectile
whose form differs from the standard projectile. From (30)
we have
^ w s
u Td}'' S(v)-S(V)
whence ^ ^ ^ ^
jv S{v)-S {V)
d-" s
It is, therefore, only necessary to measure the velocity
of the projectile at two points of its trajectory as nearly in
the same horizontal line as practicable, and at a known dis-
tance apart, and substitute the values thus obtained in the
above formula. For example, the 40-centimetre (71 -ton)
Krupp gun fires a projectile weighing 17 15 lbs. with a
muzzle velocity of 1703 f. s. By experiment it is found
that the velocity at 1800 ft. from the gun is 1646 f s. What
is the value of c for this projectile ?
EXTERIOR BALLISTICS.
59
Here
F= 1703, V == I
646, ^ == 1 80
15.748.
From Table I.,
^(^) = 3742.2
5(F) 3= 3499-7
log 242.5
= 2.3846580
- w
= 0.8397959
clogs
== 6.7447275
2t/= 1715, and d
log ^ = 9.9691814 ^ = 0.9315
.-. log (7= 0.87061451
Extended Ranges. — For the heaviest elongated pro-
jectiles, fired with high initial velocities, the remaining
velocities and times of flight may be determined by this
method with sufficient accuracy for quite extended ranges;
that is to say, for ranges due to an angle of projection of
10° or 12°, or, in other words, when the least value of cos ^
for the entire trajectory does not depart very much from
unity, its assumed value.
Example 4. — Compute the remaining velocities, with the
data of the last example, at 1800 ft., 3600 ft., 5400 ft, ... up
to 18000 ft. from the gun.
The work may be arranged as follows:
S{v) = 3499-7» log 6^= 0.8706145.
J
c
Siv)
V
Computed by
Krupp's Formula.
1800 ft.
3600 -
5400 ''
7200 "
242.47
484.9
727-4
969-9
3742.2
3984.6
4227.1
4469.6
1645 f. s.
1589 ''
1536 -
1484 "
1646 f.
1590 '
1536 *
1484 '
S.
9000 "
10800 ''
12600 ''
I2I2.3
1454.8
1697.3
4712.0
4954-5
5197.0
1434 "
1385 ''
1338 "
1434 '
1385 '
1338 '
14400 ''
16200 "
1939.8
2182.2
5439-5
5681.9
1293 "
1250 "
1293 *
I25I '
18000 "
2424.7
5924.4
1211 *'
I2II "
60 EXTERIOR BALLISTICS.
The numbers in the second column are simple multiples
of the first number in that column; those in the third column
are found by adding S {V) = 3499.7 to the numbers on the
same lines in the second column, and the velocities in the
fourth column are taken from Table I. with the argument
S{v).
The velocities in the last column were computed by
Krupp's formula. They are copied, as also the data of the
problem, from " Professional Papers No. 25, Corps of En-
gineers, U. S. A.," page 41.
In this example the angles of projection and fall for a
range of 18000 feet are, respectively, 7° 18' and 9° 20'; while
an 8-inch shell weighing 180 lbs. would require for the same
range, with the same initial velocity, an angle of projection
of 11° 5^ and the angle of fall would be 19° 40'.
In this latter case the velocity computed by the above
method would not be a very close approximation.
Comparison of Calculated with Observed Velo-
cities, — The following table, taken, with the exception of
the last two columns, from "Annexe a la Table de Krupp,"
etc., Essen, 1881, shows the agreement between the observed
and calculated velocities for projectiles having ogives of 2
calibers. The sixth column gives the distances, in metres,
between the points at which the velocities were measured
{X^ and X^). The seventh and eighth columns give the
observed velocities at the distances from the gun X, and X^
respectively. The ninth column gives the velocities at the
distances X^ from the gun computed by Krupp's table and
formula. The tenth column gives the velocities at the dis-
tances X^ computed by equation (30), using Table I. of this
work. The coefficient of reduction (c) was taken at 0.907,
which is its mean value for velocities between 2300 f. s. and
1200 f. s., as determined by a comparison of Bashforth's and
Krupp's tables of resistances given in Chapters I. and II.
The only discrepancies of any account between the calcu-
lated velocities in this column and the observed velocities
occur when the curvature of the trajectory is considerable,
EXTERIOR BALLISTICS.
6i
JU
-H.S
c«
rt
.
i
•§^
tn
>,
>,
&
V
u
>» /
13
C
s
6
s
s
B
1
^ c<
53
Is
r
■0 c
2
}
1 '^
a
S
I
240
2.8
125
1-245
1450
467
380
379.9
380.7
380.6
2
240
2.8
161
1.245 1450 1
454.5
390
388.3
•387.7
387.5
3
172.6
2.8
61.5
1.226
1389
477
388
388.7
389.3
388.7
4
172.6
2 8
61.5
1.226
1429
514.7
416.6
417.^
417.6
415.7
5
149. 1
2.8
39-3
1.260 1429 1
518
401.6
402.1
403.0
401.2
6
149. 1
2.5
33.5
1.240
1429
507.7
380
380.7
379.9
379.1
7
149. 1
2.8
31-3
1.265
924
475.8
387.8
388.2
387.7
387.3
8
355
2.8
525
1.200
1884
495-9
432.7
433-1
433.8
432.6
9
355
2.8
525
1.200
2384
490
415
411. 8
414.4
412.3
lO
355
2.8
525
1.200
2389
488.5
409.6
410.4
412.3
410.9
II
149. 1
2.8
31.3
1.265
1950
609-
394
393-9
395.4
392.7
12
149. 1
4
51
1.206
1929
505-2
394.6
393.3
393.4
392.3
13
152.4
4
51.5
1.205
1450
472.4
391.3
389-3
389.1
388.6
14
152.4
2.8
32.5
1.205
1450
577
422
422.0
424.2
421.5
15
149-1
2.8
31.3
1.230
1450
632.4
460.9
460.3
462.8
459.8
i6
240
3.8
215
1.208
1904
480.4
412.8
412.0
412.4
411. 1
17
400
2.8
777
1. 180
2384
499.4
433.7
432.1
433 -o
431.7
i8
400
2.8
643
1. 190
2384
533.4
443-8
447.0
448.2
446.6
19
400
2.8
643
I.I90
2384
531.5
444-5
445-4
446.6
445.0
20
84
2.8
6.55
1. 197
2447
446.9
266
267.2
259-7
267.4
21
120
2.8
16.4
1. 211
2447
463.3
284.1
289.2
281.6
289.3
22
149. 1
2.8
31-3
1.285
3448
536.6
294.8
290.6
283.7
290.5
23
105
3.5
16
1.300
3436
481.5
282
278-4
271.2
279.6
24
96
3.5
12
1.340
3439
425.8
256.2
.250.5
244.1
254.4
25
107
2.7
12.5
I. 218
777.5
205.1
188.2
189.8
187.7
189.8
20
152.4
2.8
31.5
1.206
966.5
203
188
187.4
185.9
188.0
27
105
3.5
16
1.222
950
514.2
426.9
421. 1
422.2
420.4
28
149. 1
2.8
39
I. 218
1429
470
369-5
370.4
369.1
369.3
29
283
2.5
234-7
1.206
4450
464.7
321.2
31S.9
311-3
317.6
30
283
2.5
234-7
1.205
|i879
465-3
403.9
403.3
404.6
403.7
31
283
2.5
234.7
1.200
11919
465.9
385.4
384.7
384-0
383.8
32
283
2.5
234.7
1.200
12425.5
466.5
370.6
368.0
366.6
367.0
33
283
2.5
234-7
1.220
2921.5
464.8
347-8
350.9
347-7
349.7
34
283
2.5
234-7
1.227
3426.0
463-7
336.0
337.6
331-4
336.6
35
283
2.5
234.7
1.220
I4446.5
460.0
316.6
316.6
308.6
315.0
36
283
2.5
234.7
1. 192
,5945.0
1
455.8
295.0
293.9
285.6
293-0
37
283
2.5
234.7
1.206
5945.0
453.1
294.7
291.5
283.2
291.4
62 EXTERIOR BALLISTICS.
as in'the last four rounds, and one or two others. Equation
(30) is based upon the supposition that the path of the pro-
jectile is a horizontal right line, and, of course, gives only
approximate results when this path has any appreciable
curvature. It will be shown subsequently that, to obtain
the real velocity, the " v " computed by (30) should be mul-
tiplied by the ratio of the cosines of the angles of projec-
tion and fall. In No. 37, for example, it will be found that
to attain a range of 5945 metres (3! miles) the angle of pro-
jection would have to be 12° 37', and the angle of fall would
be 17° 40'. Making the necessary correction, we should
find the velocity to be 290.7 m.
The last column gives the remaining velocities computed
by Mayevski's formulas. They follow very closely those
computed by Krupp.
In the absence of tables we ma}^ determine remaining
velocities which exceed 1300 f. s. as follows: We have
found, when n = 2,
C , V
•^ = -X log —
V ^ , As ^ UAs^ , ^
V
. As . i/AsV .
As
As -yr is usually a small quantity, all its powers higher
than the first may be neglected, and we may put
V 6
V As
V
1+4
For oblong projectiles having ogival heads of i J- calibers
A ^0.000142. If the ogive is of 2 calibers, A =0.0001316.
This method gives correct results for distances of a mile, or
even more, especially for the heavy projectiles used with
modern seacoast guns. If the data are given in French units
— that is zv, d, and d^ in kilogrammes, din centimetres, and s
and V in metres — the value of A will be 0.000030357.
EXTERIOR BALLISTICS. 63
Example. Let dz=.io.^ cm., 2e/ = 455 kg., <5 = 1.274 kg.,
<5^ = 1.206 kg., F= 520.8 m., and .^=1900 m. [Krupp's
Bulletin, No. 31.]
We have
^ 455 X 1.206 ,
C = , \^ =■ 0.46301
(30.5r X 1.274
and
520.8 520.8 .
0.000030357 X 1900 1. 12457
^ ~^ 0.46301
The measured velocity in this example was 465.5 m.,
while the velocity computed by Krupp was 460.1 m.
CHAPTER V.
RELATION BETWEEN VELOCITY AND INCLINATION.
Expressions for the Velocity. — Equation (15), which,
since {i) = --— -|- (^), may be written
(f)»-(#)» = ^-{jr--^4 (31)
gives the relation between the horizontal velocities ^and 11
and the corresponding inclinations ^ and d^\ and of these
four quantities any three being given, the fourth can be ac-
curately computed, provided, of course, that the value of k
has been accurately determined by experiment. The func-
tions (^)„ and (?^)„ are the integrals of ;^^-^-, and the fol-
COS t/"
lowing are the forms they take for the values of n here
adopted :
('>), = i { tan » sec » + log tan (^ + y) }
{»), = tan » + i tan' »
+ A|og.„g+|)
It is evident that all these expressions become o when
/> — o, negative when f'^ is negative, and infinite when
?^ =: ' ; or, in symbols, (o) = o,(— ^) = — (/5^), and r' j = x
If there were buto^ie " law of resistance" — in other words,
\^ n had but one value for all velocities — it would be easy to
calculate the velocity for any given value of /> by means of
EXTERIOR BALLISTICS.
65
(31). It would only be necessary to tabulate the values of (/>)„
for all practical values of d- as the argument, and to pro-
vide a similar table of (-j with ?/ as the argument. But, as
we have seen, ;i may change its value two or three times in
the same trajectory ; and though it would be possible to
ascertain by trial the exact point of the trajectory where
this change occurred, yet the labor involved would be very
great.
Basliforth's Method. — Professor Bash forth overcomes
this difficulty by giving to 71 the constant value 3, and
making /r' to vary in such a manner as to satisfy (31) for all
velocities. His method of procedure is as follows: making
« = 3 and /> = o, (31) becomes
C/'
i^tan ^ + i tan>^
in which 6^ and (f are the horizontal velocity and inclination,
respectively, at the beginning of an}^ arc of the trajectory
we may be considering; and v^ the velocity at the summit.
In Bash forth 's notation
3^ ^'
^(loooy w'
substituting this in the above equation and multiplying by
(1000)^ to avoid the inconvenience of very small numbers,
we have
/iooo\^ /iooo\' K d' i ^ , . 3 )
by means of which either z^^, [/, or (p can be determined
when the other two are known. When the resistance can
be taken proportional to the cube of the velocity, K is con-
stant; but for all other velocities it is a variable, and we
must take a certain mean of its values for the arc under con-
sideration. Prof. Bashforth takes the arithmetical mean,
which will generally give very accurate results for arcs of
66 EXTERIOR BALLISTICS.
lo or 15 degrees in extent. In his work he gives the ne-
cessar}^ tables for suitably determining — for all velocities
from 100 f. s. to 2900 f. s., and also tables giving values of
3 tan ip -\- tan^ ip for all practical values of ^ .
Other approximate methods involving less labor will be
given further on.
High Aiig:le and Curved Fire. — When the initial
velocity does not exceed 800 f s., which includes nearly all
mortar and howitzer practice, the law of resistance for
oblong projectiles is that of the square of the velocity;
whence, making n ^ 2, and dropping the subscript, (31) be-
comes
or, writing / (u) for
(^)-(^)=^{/W-/(C/)) (32)
The value of /(?/) for any given value of ti can be taken
directly from Tables T. and II., the method of construction
of which will be given further on. Table III. gives (^) and
extends from ^ = o to (^ = 60°.
To use (32) for computing low velocities (and also for
high velocities, exceeding 1330 f. s.), we have
/«=f I (?)-('?)} + /(f^) (33)
2
in which u and B are the only variables; -^, (^), and I{U)j
having been determined, do not change their values for the
same trajectory.
To illustrate the ease with which velocities may be cal-
culated by (33), take the following data from Bashforth's
"Treatise," page 115:
EXTERIOR BALLISTICS.
67
V^ 751 f. s. ; ^ == 30°; w = 70 lbs., and (i' = 6.27 inches.
Here C/== 751 cos 30° ^ 650.385 f. s. ; and from Table
I., /(r7) = a93354; -g. = -^ = 1. 12323.
We will, following Bashforth, compute the velocities for
^ = 28°, 24°, 20^
40°. The work may be conve-
niently arranged as follows:
{f) z= 0.60799 I{U) = 0.93354.
e
(»)
(<<,) - (0)
~({6) - (0))
/(«)
(Table I.)
u
« sec 6 = V
Bash-
forth's
Differ-
ence.
30°
0.60799
. 00000
0.00000
0.93354
650.38
751-0
75I.O
0.0
28"
.55580
.05219
.05862
0.99216
636.09
720.4
720.4
0.0
24°
.45953
.14846
.16675
I . 10029
612.03
669.5
670.2
- -7
20°
.37185
.23614
.26524
I. 19878
592.33
630.3
630.5
.2
16°
.29063
.31736
.35647
I . 29001
575.69
598.9
598.9
0.0
12°
.21415
.39384
•44237
1.37591
561.23
573.8
573.5
+ .3
8°
. 14100
.46699
.52454
1.45808
548.38
553.8
553.1
•7
4°
. 06998
.53801
.60431
i^537S5
536.71
538.0
537.0
i.o
o"^
.00000
.60799
.68291
I. 61645
525-91
525.9
524.6
1.3
-4"
— .06998
.67797
.76151
1.69505
515.74
517.0
515.5
1.5
8°
.14100
• 74899
.84129
1.77483
505.99
511.0
509.3
1.7
12°
.21415
.82214
•92345
1.85699
496.52
507.6
505.7
1.9
16°
. 29063
.89862
1.00935
1.94289
487.15
506.8
504.7
2.1
20°
.37185
.97984
. I. 10056
2.03410
477.77
508.4
506.2
2.2
24"
•45953
1.06752
I . 19906
2. 13260
468.22
*5i2.5
510.2
2.3
28°
•55580
I. 16379
1.30720
2 . 24074
458.38
5i9^i
516.8
2.3
32°
.66343
I. 27142
1.42809
2.36163
448.06
528.3
525.9
2.4
36°
.78617
I. 39416
1.56596
2.49950
437.11
540.3
537.9
2.4
40°
.92914
1.53713
1.72654
2.66008
425.32
555.2
552.8
2.4
The numbers in the second column are taken directly
from Table III. for the values of f^ given in column i. Sub-
tracting the numbers in column 2 from {(p) (=0.60799) gives
2
those in column 3; and these multiplied by -^ {= 1. 12323)
are written in column 4. Adding I (U) (=0.93354) to
these last gives the values of / (ti) in column 5.
The values of u are then taken from Table I., and these
multiplied by sec '^ give the velocities sought. For com-
parison the velocities computed by Bashforth, by his method
already explained, are also given ; and it will be seen that
68 EXTERIOR BALLISTICS.
the differences between his velocities and those computed
by (33) are practically nil.
This method of determining velocities may be used
without material error when the initial velocity is as great
as 1000 f. s.
Example. — The 8-inch howitzer is fired with a quadrant
elevation of 23°; muzzle velocity, 920 f. s. ; weight of shell,
180 lbs.; diameter, 8 inches. What will be the velocity in
the descending branch when /> = — 27° 54' ? (See Mac-
kinlay's '' Text-Book," page 109.)
Here
F=920, Z7= 920 cos 23°= 846.86
/(/7) =0.40884; log ^ = 9.85 194
The computation is as follows:
(23°) = 0.43690
(-27° 540=— 0.55327
log 0.99017 = 9.99571
C
log - = 9.85194
log 0.70412 = 9.84765
I{U)= 0.40884
I {11) := I.I 1296 . • . ^27. 5^. = 609.4 f. S.
Mackinlay gets by Niven's method, dividing the tra-
jectory into two parts, 6^270 54' = 610.6 f. s. It will be seen
that by the method developed above for calculating veloci-
ties, the length of the arc taken makes no difference in the
accuracy of the results.
Siacci's Method. — Equation (13) may be written
/^ dd- _gC r ^ sec' & d u
Since ^ is a function of u, there must be some constant
mean value of sec d- which will satisfy the above definite
EXTERIOR BALLISTICS. 69
integral. Representing this mean value of sec d- by a, and
writing U' and u' for af/and au respectively, we have
n d& _agC_ f^ _duf__
Making
^(«') = ^7^ + e
(34) becomes
tan ^ - tan ,!' = ^{ /(«')- 7(^7)} (35)
The values of I {ii') are given in Table 1. for oblong pro-
jectiles, and in Table II. for spherical projectiles. The
method of computing the /-function is entirely similar to
that already described for the 5 and /-functions, and need
not be repeated. For oblong projectiles the formulae areas
follows, in which, for uniformity, / (z^) is employed as the
general functional symbol:
2800 f. s. > 7^ > 1330 f. s. :
/ W = [5.3547876] ^ — 0.028872
1 330 f. s. > 7/ > II 20 f. s. :
/(t/) =[8.2947896] ^ + 0.015293
1120 f. s. > T^ > 990 f. s.:
7(7') = [17.1436868] -^ +0.085087
990 f. s. > z/ > 790 f. s. :
7(2;) = [8.4557375] -L — 0.061373
790 f. s. > z^ > o :
I{v)^ [5.7369333] -^ — 0.356474
70
EXTERIOR BALLISTICS.
If we compare (34) with (31) it will be seen that
„_ i (f).-w« I ^
( tan (p — tan § )
and this value of a renders (34) and (35) exact equations; in
fact, reduces them to (31). It would seem at first as if
nothing had been gained by introducing a into (35), since
its value depends upon that of ^2, and must, therefore, change
when n changes. The following table gives the values of a
for the arcs contained in the first column, when ;/ = 2, w = 3,
and n=z6, computed by the above formula :
Arc
Mo*
30° to 20°
30^
10^
ic-
30° '' —20^
30^
30^
I . 1066
1-0741
I. 0531
I. 0419
I .0409
I. 0531
I . 1069
1.0749
I. 0541
I .0429
I .0418
I. 0541
1079
0772
0573
0460
0443
0573
It is evident from this table that when the angle of pro-
jection is as great as 30°, the velocity at any point of the tra-
jectory may be computed with sufficient accuracy by using
either set of values «; since the greatest difference between
those in the second and fourth columns on the same line is
but 0.0042, and this would make but a slight difference in
the values of U' or u'. MoreoVer,"since U' — a Fcos ^, and
u' =^ av cos ^?, it is apparent that U' and u' differ less from V
and V respectively than do U and u; and this is important
when, as is usually the case, the law of resistance is different
for the initial and terminal velocities.
If in the above expression for a we make n = 2, we have
Didion's expression for «, viz. :
^^ (y)-W
tan (p — tan ^
EXTERIOR BALLISTICS. 7 1
in which
(i?) — i I tan ^ sec & + log tan ^- + — ) I
Example. — 'A 12-inch service projectile, weighing 800 lbs.,
is fired at an angle of projection of 30° and a muzzle velocity
of 1886 f. s. Required its velocity when (a) the inclination
of the trajectory is 15°, and (b) when the inclination is — 15°.
Here^=i: 12, w — 800, V = 1886, and ip = 30°. From (35)
we get
/ {u') =: /(^') -f -^ I tan ^ - tan ?? i
(a) ^ = 15°. From our data we have
„^ (30°)-(.5°) ^^gd3g821^,.o888
tan 30° — tan 15° 0.30940
U' = a Fcos30° = 1778.34 .'./{[/') = 0.04270
^ _ w 800
d^ 144
and
tan 30° — tan 15° =0.30940
Tt>\ X 288 X 0.30940
.-./(.)= 0.04270 + 3^ ^ ^3^3^ - 0.14500
.-. «'= 1149.77.
1149.77 .
. • . z/,.o == — ^^-^ = 1093.3 I- s.
" a cos 15° ^^ ^
(b) x^= — 15°. We have
^^ (30°) + (15°) ^ 0.87911 ^ J ^.^
tan 30° + tan 15° 0.84530 "^
C/' — « Fcos 30°= 1698.65 .-. 7(^0 = 0.04958
tan 30° + tan 1 5" == 0.84530
.-. I {u') =0.04958 + 0.29260 = 0.34218
.-. ?/' = 891.14
.-. z;,,,. =887.1 f. s.
The values of v^^^ and z/_,^, computed by (31) are 1097.6
and 892.9 respectively.
72 EXTERIOR BALLISTICS.
Siacci's Modification of (35) for Direct Fire.—
Since in direct fire the angle of projection does not exceed
15°, and is generally much less, the values o^ a for this kind
of fire will not differ much from unity. For example, with
10° elevation, and an angle of fall of — 12°, we shall have
for a
,,_ (10°) + (12°) _ Q.39i39 _^^^.,
■" tan 10° + tan 12° ~ 0.38889 "~ "^
It is manifest, therefore, that for sucli small angles no
material error would result in making «= i ; the following,
however, is a closer approximation. If we consider that
part of the trajectory lying above the horizontal plane
passing through the muzzle of the gun, it will be seen that *
a should be greater than unity and less than sec co. Siacci
makes
W-2
a = (sec^)«-i
therefore, when « = 2, a =1 ; when « = 3, a = V sec ^, and
when n =^ 6^ a =z sec f ; and the average value of a for the
whole trajectory generally fulfils the above condition.
This value of a substituted in (34) gives, by an easy
reduction,
•tan c^ - tan ^ =: /^ , \ . L-— - -L \
- — ^ n A cos (p { {u sec cpf F" )
or, writing u' for u sec ^, and proceeding as already ex-
plained,
' Example. — Take the follow^ing data:
800
^= 12 ; 7£/ = 800; 6'= ; F= 1886 ; ^ =1 10°. Compute
144
the remaining velocity in the descending branch when
t?=^ 13°. We have
/ {u') = -^ cos" (f (tan ^ - tan ^) + / (F)
EXTERIOR BALLISTICS.
and the computation will be as follows:
log (tan 10° + tan 13°) = 9.60980
log- -^ = 9.55630
2 log- cos 10° = 9.98670
log 0.142 1 7 = 9.15280
7(1886)1=0.03477
/ (//') z= 0. 1 7694 // = 1 07 1 . 76
IO7T.76 COS lo"^
n
COS 13°
= 1083.2 f. s.
The velocity at the same point computed by (31), divid-
ing- the trajectory into three arcs, with the points of division
corresponding- to velocities of 1330 f. s. and 1120 f. s. respec-
tively, is 7' =: 1081.55 f. s. This agreement is very close;
but if we make if = 30° and ^ == 15°, as in the preceding ex-
ample, we should find by tiiis method ?''i5« = 1113.1; and if
d- =z — 15°, we should find 7^_i^« = 859.3, which differ consid-
erably from their true values.
Mven's Method.— W. D. Niven, Esq., M.A., F.R.S.,
has given the following method for determining velocities
in terms of the inclination :
Equation (13) may be written
J , ^ Aj , («sec.?r" "-A J „. «'"*■
in which, as before, a is some mean value of sec d^ between
the limits sec <p and sec ??, and 1/ =z av cos d^ and U' ^=.a Fcos (p.
Integrating, we have
' an A \ //« 'U'-S ~ a InA u'^ nA U'^S ^^^^
iplying botl"
degreed, and making
Multiplying both members by - — to reduce ^ — ?^ to
i^(,-^) = Z)
74 EXTERIOR BALLISTICS,
and
n TT A n"" ^ ^
the above equation becomes
n = ^\D{,/)-j?{u')\* (38)
which is the equivalent of Niven's expression for the velo-
city and inclination. Mr. Niven has published a table of the
/>>• function for velocities extending- from 400 f. s. to 2500 f. s.
(See Table VI. ii Mackinlay's "Text-Book.") It will be
seen by comparing the expressions for D {v) and I [v) that
we have the relation
and, therefore, in terms of the /-function, (38) becomes
.^ = t|'{^M-/(^0} (39)
log ^==1.4570926
Comparing (37) with (31), it is apparent that to make (37)
or (38) exact equations we must have
-\^^r
For direct fire Didion's value of a may be used ; but for
high-angle firing- the following gives more accurate results,
obtained from the above equation by making a/ = 2 :
a =
i'^f[
Example. — Take the following data:
df=l2; w=r8oo; F=: 1886; ^ — 30° and <> = — 30° ;
/? = 30° + 30° = 60° ; to find v,^..
%
* If we use Niven's tables, in which the functions decrease with the velocity. (38) should be
written
i>«£j/?(i/0-^(«')[
EXTERIOR BALLISTICS.
75
We have from (38)
D{u') = D{U') + ^D
The computation may be conveniently arranged as fol-
lows :
log (ip) = 978390
constant = 1.758 12
c log 30 = 8.52288
log
3) 0.06490
log a = 0.02163
log D^ 177815
r log (7= 9.25527
11.3516= 1.05505
log F= 3.27554
log a = 0.02163
log cos ip = 9.93753
log U' = 3.23470
6^'= 1716.74
(Niven's Table) D {U') = 84.6090
C
D{u')
Dzzz II. 3516
73-2574
.*.«' = 827.12 =:« Z/ cos ??
. • . v.j^^. = 908.7 f. S.
Siacci's method, using Table I. of this work, gives
^_3oo = 907.5 f. s. ; while equation (31) gives v_^^ = 913.2 f. s.
Modificatioii of (38) for Direct Fire. — If we make
a = (sec (fY^
we shall have, by a process similar to that already employed
in Siacci's method, the following modified form of (38),
which can be used in all problems of direct fire, viz.:
C
j9 =
cos (f
in which u' ^=^u sec ip.
Example, — Let <a^ = 1 2 ;
\^D iu') - D {y)\^
w
800; V:
(40)
\
886; ^ = 10°;
= — 13®. The computation is as follows :
y6 EXTERIOR BALLISTICS.
log Z^= 1. 36173
log cos (f =z 9.99335
c\oo; C= 9-25527
log 4.0771 = 0.6T035
i; (1886) = 84.9966 ^^^^
D («0 = 80.9195 .-.//= 1068. 14 = z'-^;^^
.' . V = 1079.6 f. s.
which is within 2 feet of the value of z/ computed by the exact
formula (31). This modified form of Niven's method, for sim-
plicity and accuracy, seems to leave nothing to be desired.
For small angles of projection, say not exceeding 5°, we
may put ?/ = v, and cos'^ = i ; and (40) becomes
% Example, — In the preceding example suppose ^ = 3°.
What will be the value of d- when the velocity is reduced
to 1500 f. s.?
(a) By Niven's Table :
Z> (1886) = 84.9966
Z> (1500)== 83^9359,
log 1.0607 =: 0.02560
log 6^ = 0.74473
log D = 0.77033
/>=5°.89 = 3°-'>
.-. ??= - 2°.89
(b) By Table 1. :
7(1500)= 0.07173
/(i886) = 0.03477
log 0.03696 = 8.56773
log?= 145709
log (7 = 0.74473
log 7^ = 0.76955
D = 5°.88
.-. ^= -2°.88
CHAPTER VI.
TRAJECTORIES— HIGH-ANGLE FIRE.
As we have seen, the differential equations for x,y^ t, and
s do not generally admit of integration in finite terms for
any law of resistance pertaining to our atmosphere ; that
is, for any recognized value of ?i. It is true that Professor
Greenhill has recently* succeeded, by a profound analysis,
in deducing exact finite expressions for x and y by means of
elliptic functions, when ?^ = 3 ; but these results, though of
great interest to the mathematician, are far too complicated
for the practical use of the artillerist. When /^ = 2 the ex-
pression for ds can be integrated and useful results deduced
therefrom, as will be seen further on.
For low velocities, such as are generally employed in
high-angle and curved fire, the effect of the resistance of
the air upon heavy projectiles is comparatively slight; and
for a first (though rough) approximation we may, in such
cases, omit the resistance altogether, or, better still, we may
suppose the projectile subject to a mean constarit resistance.
A still closer approximation may be obtained by taking a
resistance proportional to the first power of the velocity.
As the differential equations for the co-ordinates and time
are susceptible of exact integration upon each one of these
hypotheses, we will consider them in turn.
TRAJECTORY IN VACUO.
Making p = o, (9) becomes
duzuzo
and therefore, in vacuo, the horizontal velocity is constant, or
/^= U
Integrating (21), (22), (23), and (24) between the limits
(p and d- gives, \i ti ^ U,
* " Proceedings of the Roj^al Artillery Institution," Vol. XI.
10
78 EXTERIOR BALLISTICS.
and
/ = —(tan f — tan d) (4O
;ir = — (tan ip — tan d) (42)
772
7 = — '(tan' ip ~ tan' />) (43)
((^) - ('>)) (44)
Equation of Trajectory in Vacuo. — Eliminating
tan d- from (42) and (43) gives
y ^=L X tan (p
2W
which is the equation of a parabola whose axis is vertical
A parabola, therefore, is the curve a projectile would de-
scribe in vacuo.
Since a parabola is symmetrical with respect to its axis,
the ascending branch is similar in every respect to the de-
scending branch, the angle of fall being equal to the angle
of projection ; and generally, for the same value of j, tan d^
has numerically the same value, but with contrary signs, in
both branches; being positive in the ascending branch,
negative in the descending branch, and zero at the vertex.
If we make ?^ = — ^ in (42) it becomes
^- 2 C/' V sin 2ip
X = tan <p = i
g g
and this, for a given velocity, is evidently a maximum when
f = 45°.
Subtracting (42) from the above equation, and reducing,
gives
X — X— — (tan <p + tan ^)
2 tan ^ ^ ^ ' ^
also, dividing (43) by (42) gives
f = ^(tan^ + tane?)
whence
:^ = ^(-^-^)tan^ (45)
EXTERIOR BALLISTICS. 79
Making ?? = — ^ in (41), we have
^ 2U ^ 2V .
Y rzz tan w = — sin <p
g g
Subtracting (41) from this last equation gives
T — t=— (tan (p + tan d)
also, (43) divided by (41) gives
-7 = 7 (tan ^ + tan??)
^{T-t) f46)
whence
2
Dividing (44) by (42) gives
s^ (y)-(^) ^^
X tan (f — tan d^
Didion's «, then, is the ratio of a parabolic arc whose
extremities have the same inclination as the arc of the tra-
jectory under consideration, to its horizontal projection.
Expression for the Velocity. — From (43) we have,
since V cos (p zz^v cos d- = U,
v" sin'' d-z^zV sm^ (p — 2gy.
Adding v" cos'' d^ to the first member, and its equal,
V cos"* (f, to the second member, and reducing, we have
v'' = V - 2gy
If h is the vertical height through which the projectile
must fall to acquire the velocity of projection (F), we shall
have Tza 7
V^ zzz2 gk
and this substituted in the above formula gives
v'' = 2g{h-y)
that is, the velocity of the projectile at any point of the
trajectory is that which it would acquire by falling through
a vertical distance equal to ^ — j^.
All the properties of the trajectory in vacuo may be
easily and elegantly determined by means of the funda-
mental equations (41) to (44) inclusive.
80 EXTERIOR BALLISTICS.
CONSTANT RESISTANCE.
P _
Suppose the resistance constant, and put ~ ^ m ; then
zv
the elimination of dt from (9) and (12) gives
du d&
in
u cos ?>
whence ^
log u = m log tan 1 - -] I -f- ^.
Let v^ be the velocity when ?> = o, that is, at the summit
of the trajectory ; then C — log v^, and we have
(f+4) <«)
r= 2/„ tan
4
Substituting this value oi u in equations (21) to (24), and
integrating so that /, x, y, and s shall all be zero at the
origin, that is, when -& =npy we have, making the necessary
reductions,
TT sm w — m sin i^ — m
j^2 COS (p (sin ^ — 2m) ^ cos ?^ (sin d — 2m)
~ g{i — 4^') "" ^ <^ (i - 4^^')
j^2 I + sin ^ (sin (p — 2m) ^9 1 + sin d- (sin ?^ — 2/;?)
^ ~ 4^(1 -Iff) "" 4^(1 -f^)
,j, cos^ (f -{-2m (sin ip — in) ^ ^ cos'' d- ~\-2m (sin z? — ??/)
4w^(i — ^^z'') 4^«^(i — ^^^)
When 2?;^ = i, the differential expression for x becomes
logarithmic, as do those for /, y, and s when in — \. The
integrations are easily obtained for these values of m^ but
are omitted on account of their length, and as being of no
great practical importance. In the application of these for-
mulae it will be necessary, since the resistance of the air is
not constant, but varies with the velocity, to determine a
proper mean value for m between the limits of integration ;
and this we may do as follows : After having computed the
horizontal velocities u^ and u^ by means of (33), corre-
sponding to the inclinations a and /9, the value of in may be
determined by the following equation deduced from the
above expression for 71 :
EXTERIOR BALLISTICS. 8 1
^jj^ ^ log u^ - log U^
logta„(^ + f)-logta„(^+4)
Example. — Compute the values of t, .r, y, and s, from
(p = 30° to ?? = o, with the data given on pag-e 6y. We have
,,, _ log 75 1 + log cos 30" - log 525.9 1 _^^.^_
'"^ - log tan 60° - ^-^^^^^
Substituting in the above formulae, we find
^ = 3-I073 + 7.4295 = 10^537
X = 16908 — 10557 = 6351 ft.
y = 4446 — 2526 = 1920 ft.
^=: III55 — 4578 = 6577 ft.
Bashforth gets, by dividing the arc into 8 parts,
t = io''.4i3, X = 6074 ft., and 7 = 1882 ft.
It is easy to see how by suitable tables, the construction
of which offers no difficulty, the time and co-ordinates ma}^
by this method be readily, and for arcs of limited extent
accurately, computed. For example, we have
x = A V~A' v'
A being a function of m and ^, and A^ the same function of
m and d^.
RESISTANCE PROPORTIONAL TO THE FIRST POWER OF THE
VELOCITY.
Differential Equations. — When ;/ = i, the differential
equations (13), (17), (18), and (19) become respectively, since
tiA
CO^ ^^
dt=-
k du
g u
dx =
k
du
dy = -
k
tan -& du
82 EXTERIOR BALLISTICS.
Time and Co-ordinates. — The integration of the first
three of these equations between the limits {cp, d) and {[/, 7i)
gives (supposing k constant)
tan <p — tzin^ = k(~ — ^') {48)
or, using common logarithms,
f=M-\og- (49)
in which M = 2.30259; and
:^=:^{U-u) (50)
Substituting for tan ?^ in the expression for dj/ its value
from (48), it becomes
dy= (7-, -^ tsiu <p) du -\
or
dj/= IjY -{- tan ^ j<^4r — /&^^
whence, supposing y to vanish with x and if,
7 = (-^ + tan <pj x — kt (51)
Determination of h. — In the above integrations we
have assumed k to be constant, whereas it varies with the
velocity ; but our results will be correct if we give to ^ a
proper mean of all its values between the limits of integra-
tion ; and as k varies slowly and with considerable regularity
for all velocities for which this method will be used, we will
take for k the value corresponding to the arithmetical mean
of the two velocities at the extremities of the arc under
consideration. It is evident that the smaller the arc of the
trajectory over which we integrate, the less will be the
error committed in taking this value for k. But it will be
EXTERIOR BALLISTICS.
83
shown by examples that no material error will result for
velocities less than about 1000 f. s., when the whole tra-
jectory is divided into two arcs with the point of division at
the summit.
When ;/ = I, we^have
w
whence from (6) and (7)
C
(loooy
C m (say)
The following table gives the values of tn for velocities
extending from 900 f. s. to 500 f. s., with first differences :
TABLE OF 7n.
V
m
d,
V
tn
d,
500
32.814
66^
710
23.700
346
510
32.146
618
720
23.354
357
520
31.528
572
730
22.997
340
530
30.956
554
740
22.657
323
540
30.402
539
750
22.334
335
550
29.863
527
760
21.999
376
560
29.336
490
770
21.623
388
570
28 . 846
427
780
21.235
372
580
28.419
392
790
20.863
358
590
28.027
387
800
20.505
344
600
27.640
384
810
20.161
384
610
27.256
381
820
19.777
448
620
26.875
382
830
19.329
433
630
26.493
382
840
18.896
442
640
26. I II
356
850
18.454
426
650
25-755
388
860
18.028
412
660
25.367
365
870
17.616
398
670
25.002
343
880
17.218
385
680
24.659
321
890
16.833
372
690
24.338
300
900
16.461
359
700
24.038
338
84
EXTERIOR BALLISTICS.
The value of k in the ascending- branch will be assumed
to be that due to the velocity | {y-\-v^\ and in the descend-
ing branch, to \ (^o + ^e)* ^e being the velocity at the point of
fall. The first step, then, is to compute v^ and Vq ; and this
can readily be done by means of (33), as already explained.
Expressions for the Ascending and Descending-
Branches. — It will be seen that x, y, and t are functions of
6^ and u; and these latter depend upon (p and ?^, as shown in
equation (48).
From this equation we have
■^ + tan ^
- +tan?^ = -
in which u^ is the value of 21 at the summit; whence
k
U
-f- tan <p
and, since d- is negative in the descending branch,
k
^fl =
+ tan d-
(52)
(53)
The following expressions for t, x, and y for the ascend-
ing and descending branches are easily deduced from (49),
(50), and (51), in connection with (52) and (53):
ASCENDING BRANCH.
'0 = ^— log —
=}{--)
DESCENDING BRANCH.
k
JJ^o = ■— ^0
kL
ye
(^o— ^e)
kto
In using these formulae, u^ and Uq are to be computed by
means of (52) and (53).
The zero subscript is to be interpreted " from the origin
to the summit"; and the theta subscript "from the summit
EXTERIOR BALLISTICS. 85
to a point in the descending branch where the inclination
is^^/'
The method of computing a trajectory by these simple
formulas will be best exhibited by examples, which we will
select from those that have been worked out by other
methods of recognized accuracy, or which have been tested
by firing.
Example i. — -Calculate the trajectory with the data on
page 6^^ viz. :
F=75if. s. ; ^ = 30° (whence C/'= Fcos^ = 650.385); d:=.
2 2(1''
6.27 inches; w=.jo lbs. (whence -^ = = 1.12323).
Assuming — 37° to be the angle of fall, we will divide
the trajectory into two arcs, the first extending from 30° to
0°, and the second from 0° to —37°. The velocities v,, and
z/_370 are computed as follows :
From Table III. we take out (30°) = 0.60799, and (37°) =
0.81977; and from Table I., /(^) = / (650.385) = 0.93354.
Then
2
— (30°) = 1. 1 2323 X 0.60799 = 0.68291
/(^):zz 0.93354
I{v^= 1.61645
(Table I.) v^ = 525.91
2
-^(37°) = 1-12323 X 0.81977 = 0.92079
I (y^ = 1.61645
/(^^_3,o) = 2.53724
z/_3,« = 434.25 sec 37° == 543.74 f. s.
The mean velocity from which to determine k in the
ascending branch is i (751 + 525.91) = 638 f. s. ; whence
m = 26.187. The remaining calculations may be conve-
niently arranged as follows:
II
86 EXTERIOR BALLISTICS.
log m = 1.4180857
log (:: = 0.2505630
log ^= 1.507 7210 U= 32.19)
log k = 3-1763697
log C/= 2.8131705
log 2.3078 = 0.3631992 = log jj
[Equation (52)] tan <p = 0.5774
log 2.8852 = 0.4601759 (sub. from log X^
log ?^o = 2.7161938
u^ = 520.228
^=r 650.385
log 130.157 = 2.1144675
log - = 1.6686487
s •
log;ro = 3.783 1162
x^ = 6069 ft.
Bashforth gets by 8 steps, 6074
Difference, 5 ft.
log 17= 2.8 1 3 1705
log «;== 2.7161938
log 0.0969767 = 8.9866674
log J/=: 0.3622157 (add log-— j
. . log /o =1.0175318
Bashforth'gets 10^413
Difference, o''.ooi
log — ^ = 1.0669224 (add log k)
4.2432921 = log 175 10
log kt\ — 4.1939015 — log 15628
y,^ 1882
Bashforth gets 1882
Difference, o
EXTERIOR BALLISTICS. 87
These results, being practically identical with those de-
duced with vastly greater labor by Prof. Bashforth, pnn^e
that when the law of resistance is that of the square of the
velocity, as in this example, we may get quite as close an
approximation to the true trajectory by assuming that the
resistance is proportional to the first power of the velocity
as we can upon the hypothesis of the law of the cube, and
with a great gain in simplicity and labor.
We have next to compute the descending branch from
f^ =0° to 3 = — 37°. The mean velocity from which to
determine k in this branch is
i (525.91 + 543.74) = 534.8 f. s.
whence m = 30.690.
log m = 1.4869969
log 6' = 0.2505630
log £ = 1. 5077210
log k = 3.2452809
log z^o = 2.7209 114
k
[Equation (53)] log 3.34480 = 0.5243695 = log —
tan 37° = 0.75355
log 4.09835 = 0.6126090
log 2^.3,0 = 2.6326719
f^_3,o = 429.21
^0=525.91
log 96.70 = 1.9854265
log 7= 1-7375599
log ^-„« = 3.7229864
^_3,0= 5284 ft.
88 EXTERIOR BALLISTICS.
log 2^0 = 2. 7209 114
log ?/_3,o = 2.6326719
log 0.0882395 = 8.945663 1
log ^]/ = 0.3622157
log/-3,o= 1.0454387
/_3,o3:zIlM03
k
log— ^.3,0 = 4.2473 5 59 = log 17675
log k t.,,. = 4.2907196 = log 1953 1
J/_3,,= - 1856 ft. >
The projectile is still 1882 — 1856 — 26 ft. above the level
of the gun = Ay. If Ax and At are the corresponding addi-
tions to the range and time of flight, we shall have approxi-
mately
Ax
Ax = 26 cot 37° =r 35 ft. ; and At = = o''.o8o.
We therefore have
^=6069 + 5284 -f- 35 = 11388 ft.
r=i 10.412 -[- II. 103 + 0.080 = 2i'^595
These values agree almost exactly with those deduced
by interpolation from the table on page 117 of Bashforth's
work.
Example 2. — The 8-inch howitzer is fired with a quad-
rant elevation of 23°. Muzzle velocity, 920 f. s. ; weight of
shell, 180 lbs. ; diameter, 8 inches. Find the range and
time of flight. (Mackinlay's "Text-Book of Gunnery,"
page 107.)
Assuming the angle of fall to be — 27° 54', we find by the
above method
X=z 7886 + 7108— 13 = 14981 ft.
T = 10.183 + 10.801 — 0.022 = 20^^.962
Mackinlay gets, using Niven's method,
X— 14787 ft, and r=2o".8i3
He states that "the published range-table gives 15000ft.
as the range, and 2i'^5 for the time of flight."
EXTERIOR BALLISTICS. 89
Example 3. — Let V =. 2g% m. = 977.71 ft., d ^= \^ cm.,
w = io k.g-., f = 35° 21', o — 1.270 k.g-., and 0,= 1.206 kg.
Find Xand T. (Krupp's Bulletin, No. 55, December, 1884.)
For the Krupp projectiles and low velocities we will
take for c the ratio of the coefficients of resistance deduced
from the Krupp and Bashforth experiments respectively,
and which are given in Chapter II. Let these coefficients
be represented by A and A' . Then for velocities less than
790 f s. we have
10^^ = 5.6698755- 10
log ^'=5.7703827 — 10
log c = 9.8994928
.•.^=0.7934
To find cT, expressed in English units, when w and d are
given in kilogrammes and centimetres respectively, we have
^ _ loooo k w
~ \^m^ c d'
in which k is the number of pounds in one kilogramme, and
in the number of feet in one metre. Reducing, we have
C-= [1.2534887] J
As the initial velocity in this example is considerable,
we will take into account the density of the air at the time
the shots were fired, and also the diminution of density due
to the altitude attained by the projectile; and for this pur-
pose we will assume the mean value of y for the whole tra-
jectory to be 2000 ft.
The complete expression for (7 is (Chapter VII.),
from which we determine log 6" as follows:
log w =: 1.4771213
^ log ^^ = 7.6478175
constant log = 1.2534887
log 0, = 0.0813473
^ log ^ = 9.8961963
z
log eh =1:0.0312468
log (7=0.3872179
90 EXTEklOR BALLISTICS.
Assuming the angle of fall to be — 44° 40', and proceed-
ing as in the first example, we find
X= 10408 + 8736 + 104 = 19248 ft.
7^r=: 15.088 + 16.324 + 0.221 == 3i''.633
Krupp gives the ranges of three shots fired with the
initial velocity and angle of departure of this example, and
the ranges reduced to the level of the mortar, as follows:
NO. OF SHOT.
RANGE IN FEET.
18
19039
19
19265
20
19364
Mean of the three shots = 19223 ft.
Computed — mean = 25 ft.
Example 4. — Given F= 206.6 m. — 677.834 ft., d = 21
cm., w = gi k.g., and (f = 60°, to find Jfand T. (Krupp's
Bulletin, No. 31, Dec. 30, 1881.)
It will be found that (assuming the angle of fall to be
— 63° 30', and taking no account of atmospheric conditions)
^ = 5390 + 4945 + 67 = 10402 ft.
T= 17.016+17.543+0.250 = 34''.8o9
Krupp gives the observed ranges of five shots, with the
above data, as follows :
NO. OF SHOT. OBSERVED RANGE.
22 10332 ft.
23 10305 "
24 10384 "
25 10463 "
26 10440 **
Mean of the five shots = 10385 ft.
Computed — mean = 17 ft.
Example 5. — Given F=: 204.1 m. = 669.63 ft., </ = 21 cm.,
z£/ = 91 k.g., and (p — 45°, to find X and T. (Krupp's Bul-
letin, No. 31, January 19, 1882.)
Assuming the angle of fall to be — 49°, we find as fol-
lows:
^=6152 + 5678 + 56 = 1 1886 ft.
r= 13.817 + 14.238 + 0.147 = 28^202
EXTERIOR BALLISTICS. 9 1
The following ranges were measured at Meppen :
NO. OF SHOT. OBSERVED RANGE.
71 11923 ft.
72 I 1920 "
73 11841 "
74 1 1 808 ''
75 11749 *'
Mean of the five shots = 11 848 ft.
Computed— mean = 38 ft.
Example 6. — Compute JTand 2" with the data of the pre-
ceding example, except that ^ — 30°.
Assuming the angle of fall to be — 33°, we find as follows :
X= 5478 + 5 143 + 26 = 10647 ft.
r= 9.908 -)- 10.183 -f 0.054 = 20". 145
Krupp gives as the mean of five measured ranges,
Jf = 10779 ft.
Computed — mean = — 132 ft.
euler's method.
Expression for s. — If we make nz=z2, that is, suppose
the resistance of the air proportional to the square of the
velocity, we shall have from (20)
C du
^ — ~ 'aH.
whence, integrating and supposing j = o when u =z U, we
have
therefore (page 52)
s = CiS{u)-S{U)^ (54)
which gives the length of any arc of a trajectory when the
resistance is proportional to the sqiiare of the velocity, by
means of the table of space functions.
We may also obtain another expression for s, better
suited to our purpose, as follows:
92 EXTERIOR BALLISTICS.
Since
' J c
COS'
we have, when n=^2,
d(d) = -^^ = sec ?^ ^ tan &
^ ^ cos ?7
and this substituted in (28) gives
in which
(?^) = i I tan?? sec ^? + log tan ^- + ^^ I
whence, integrating between the limits ip and ??, we have
or, if we use common logarithms,
in which J/= 2.30259.
Expressions for a? and t/.— Equation (55) gives the
value of s from the origin. If / is the length of an arc of
the trajectory from the origin to where the inclination is d-\
and s" the length to some other point further on where the
inclination is d-" (??'> W), we shall have from (55)
/=i:^— log ^ ^ -
and
whence
s" — s' — As—'M — los-
If??'' differs but little from ??' (say one degree), the cor-
responding values of Ax and Ay can be calculated with sufli-
«■
-(f)
«-
-(*")
w-
-(f)
(0-
- (^")
EXTERIOR BALLISTICS. 93
cient accuracy by multiplying Js by cos ^ {d-' -\- d-") for the
former, and sin ^ {&' -\~ &") for the latter; or,
Ax^M— log ^f^.^ ~ ^/2 cos I (ir + ^^'0 = M—J^ (sav)
Jj/ = M J log ll^^j sin i {^r + &") = M^A: (say)
For the entire range we evidently have
X= y Jx =r M~ I A^=M-^
K g
the summation extending from ^ = ^ to ^ = w, w being the
angle of fall.
To determine the value of co we have, since the sum of
the positive increments of ^'^ in the ascending branch is equal
(numerically) to the sum of the negative increments in the
descending: branch,
Expression for the Time.— For the time of flight we
have, when dx is small,
u
in which u is the mean horizontal velocity corresponding
to Ax ; but, from (15), when n = 2,
__ k
whence
\{i)-{^)\
I Ax ^ ( '^^ — ^^^'^ ^ ^
At =
k
or, substituting for Ax its value given above,
M = ^AZS,.. ,.A\
If we put
je=Jf{(0-w[*
12
94 EXTERIOR BALLISTICS.
we may have
log J0=log J^ + i log [(0 - i^)]
The two values of log [{i) — (^)] corresponding to the
extremities of the arc Js, are
log [ (0 -(<?')]. and log [(0 -(#")]
the first of which is too small and the second too great;
whence, taking their arithmetical mean,
log Je=\og J?+ilog[{i)-{d')] + i\og[{i)-{9")l
by means of which may be computed, and we then have
Tables. — General Otto, of the Prussian Artillery, has
published extensive tables* of the values of {&), q, C, and 6 ■—
the last three double entry tables with i and (p for the argu-
ments — by means of which it is easy to solve many of the
problems of high-angle fire.
Determination of k^, — General Otto, in the work
above cited, gives the following method for determining k"" :
We have
and
whence
an equation independent o{ P.
independent of X and T, being functions of the angle i and
the angle of projection cp ; and their ratio -^ may be tabu-
lated with these angles for arguments. General Otto has
inserted such a table in his work calculated for angles of
* " Tafeln fiir den Bombenwurf." Translated into French by Rieffel with the title " Tables
Balistiqucs Generales pourie tir eleve." Paris, 1844.
X^
g
/
T'z=.
MX
e
gT^"
6'
dent oiU
\ Moreover
e
and 6"
are
both
EXTERIOR BALLISTICS. 95
projection beginning at 30° and proceeding by intervals of
5° up to 75°.
Now, suppose a certain projectile is fired with a known
angle of projection (p, and its horizontal range X, and time
of flight 7", are carefully measured. With this data we
compute-^ by means of the above equation; and entering
Otto's Table III. with the argument ^5 find in the proper
column the computed value of -^, and take out the corre-
sponding value oil. Next, with (p and ? as arguments, take
from Table IF. the value of ?^, from which k^ can be computed
by the following formula, derived from the expression for X
given above :
^ ~ M e
bashforth's method.
For all values of n greater than unity the differential
equations of motion take their simplest form when ?/ = 3.
For this reason Professor Bash forth assumes the cubic law
of resistance throughout the whole extent of the trajectory,
and employs variable coefficients to make the results con-
form to the actual resistance.
Making ;/ == 3, equation (25) becomes
- k d tan d-
at =^
g
{ « - W j *
in which
{&) = tan &-\-^ tan^ ^
From (14) we have, when ;2 = 3 and ?? = o,
... k'
W = -3
and this substituted in the above expression for dt gives, by
a slight reduction,
96 EXTERIOR BALLISTICS.
^^ _ _ 3 d tan d^
^ {1-^^(3 tan .^ + tan^^)p
Introducint^ Bashfortli's coefficient K, making
g %v Viooo/
to correspond with his notation, and integrating between
the limits (^, d) and (o, /), we have
<^/ I i_;k(3 tan/^ + tan^^)i * ^ "
Operating in the same way upon (26) and {2^)^ we obtain
1~7 ^^
z;,' /^-^ d tan />
S j I I _ -^ (3 tan /> + tan^ if) \
v'' r^ tan d- d tan d-
y-irf T^ — — T;T* = T^n
and ^.. ^ tan d- d tan ^ _ t^^
-y{2> tan ,> + tan^ (>) U' ^
Professor Bashforth has published extensive tables of the
definite integrals ''*7^,*X:J,and '^F^^ for values of ^ extending
from +60° to — 60°, and of y from o to 100, calculated by
quadratures; by means of which the principal elements of a
trajectory may be accurately determined as follows:
As the coefficient of resistance K generally varies with
the velocity, the trajectory must be divided into arcs of such
limited extent that the value of K for each arc may be con-
sidered constant ; and it should be so taken as to give, as
nearly as possible, its mean value for the arc under con-
sideration.
In the equation given on page 65, viz.:
/loooX-' /iooo\' . K d" { ^ , ♦ . I
suppose U and ^ to be the initial horizontal velocity and
angle of projection respectively, and both known ; and let
&, also known, be the inclination of the forward extremity
\a^.
— *• » TJ J. X
EXTERIOR BALLISTICS. 97
of the first arc into which the trajectory is divided. Now,
assuming a mean velocity for this arc, take out the corre-
sponding value of K from the proper table and compute
(I ooo\ ^
- — 1 ; then, in the same equation, changing (p to />, 17 be-
comes tlie horizontal velocit}^ at the forward extremity ot
the arc, which can also be computed.
Next compute y by means of the equation given above,
with which and the known values of ^ and ^ enter the
tables and take out '^T^ , ^X^ , and ^Y^-^ lastly, multiplying
2
the first by — ^, and each of the others by -— , we have the
^ g ^ g
time of describing the first arc of the trajectory and the co-
ordinates of its for vvard extremity. By repeating the process
with the second and following arcs into which the trajectory
may be divided, the whole trajectory becomes known.
Professor Bashforth gives various other tables in his
work, besides those we have mentioned, for facilitating the
calculation of trajectories by his method, with examples of
their application and full directions for their use.
Modiflcatioii of Bashfortli's Method for low Velo-
cities. — When the initial velocity. does not exceed 790 f. s.
the law of resistance is that of the square of the velocity for
the entire trajectory; and even when the initial velocity is
as great as 1000 f. s. examples show that no material error
results if we still retain the law of the square in our calcu-
lations ; and this furnishes a very easy method for calcu-
lating trajectories for high angles of projection and for the
initial velocities usually employed in high-angle fire, and
which, it is believed, gives as accurate results as by any
other method, and with less labor.
Making ;/ =2, equation (25) becomes
k d tan &
dt
in which
6 {(,-)_(^)|4
((?) = !{ tan d sec & + log tan g + |^) |
98 EXTERIOR BALLISTICS.
We also have from (15), when « = 2, and ^ = 0,
k^ I
( = — = — (say)
and this substituted in the above expression for ^/ gives
Vc. d tan />
dt— —
? ji_^(#)|i
whence
tan ^ V,
In the same wa}^ we obtain from (26) and (27) the follow-
ing expressions for x and y :
d tan d- v^
< Pi
r(^>)
"^x,'
^"^ - . ^</> tan ^ ^ tan &
It will be seen that this method depends upon tables of
definite integrals which must be calculated by quadratures
as in Bashforth's method, and with the same number of
arguments; but the great advantage of these formulas over
Bashforth's is in the fact that y is constant for a given tra-
jectory, and, therefore, the labor of calculation is the same
for all angles of projection.
To determine the value of k^ for oblong projectiles of
the standard type we have
2A
Taking the value of A derived from the Bashforth experi-
ments for velocities less than 790 f. s., and making ^^=1 32.16,
-« fi"d k' ^ [5.4359033] c
For the Krupp projectiles we should have, taking May-
evski's value of A,
'^' = [5-5367564] c
The numbers between brackets are the logarithms of the
factors by which C is to be multiplied.
EXTERIOR BALLISTICS. 99
For computing- v^ we have from (32), when ^ = 0,
/(t/„)=-J(^)+/(f/) (56)
in which ^ may be the inclination at any point in either
branch, and U the corresponding horizontal velocity. The
values of (^) are given in Table III.
To show the practical working of this method, we will
take the example from Bashforth already given (see
page 6j). The data are: V:^j^\ f. s. ; ^ = 30°; ^ = 6.27
inches, and w:=z'/o lbs.; whence ^=650.385 f. s., and
70
C^^-rp — - == 1.78059. Determine the range, time of flight,
angle of fall, and terminal velocity.
First compute v^. We have from Table III.
(30°) = 0.60799
whence, from (56),
^^^S~ "^ ""(^50.385) = 0.68291 + 0.93354 = I.61645
therefore, from Table I., '
^0 = 525.91 f. s.
Computation of y :
log (7=0.2505630
constant log = 5 -435903 3
log /^'' = 5.6864663
log z/q' = 5.4418228
log r = 97553565
r = 0.56932
As general tables of the definite integrals '^7'^, '^X^ , and
** V^ have not yet been prepared, the following table has
been calculated for this particular example, merely to illus-
trate the method :
lOO
EXTERIOR BALLISTICS.
r =
= 0.56932
T
X
V
30°
0.63676
. 70486
0.21775
24
.47838
.51493
.12039
18
.34169
.35965
.06045
12
.21944
.22662
. 02460
+ 6
+ .10673
+ .10838
+ .00575
.00000
. 00000
. 00000
- 6
- .10358
— . 10208
+ .00531
12
.20647
.20061
.02091
18
.31104
.29793
.04701
24
.41977
. 39620
.08479
30
•53551
.49759
.13656
36
.66179
.60449
.20615
37
.68417
.62303
.21987
The value of ^°°V° by the above table is 0.21775, and as
this must be equal to °F" we see at a glance that co lies
between — 36° and — 37° ; and by interpolation we get
w=r— 36°5i'; and therefore °X'J = 0.62025 and ^T'J 0.68081.
Adding to these the numbers corresponding to the argument
30°, we get "PX- = 1.32511, and *^7:y^ = I.3I757- Lastly,
multiplying the first of these by -^,and the second by — , we
obtain
X= 11396 ft.
and
r= 21^546
which agree with Bashforth's calculations.
The terminal velocity is found from (32), viz.:
and
V... = «,.. sec CO
We find
and
^o. = 434.7 f. s.
^«o = 543-2 f. s.
It will be seen that the inverse problem, namely, Given
EXTERIOR BALLISTICS. lOI
the terminal velocity and angle of fall, to determine the
initial velocity, angle of projection, range, and time, can be
solved by this method with the same ease and accuracy as
the direct problem. We should first compute the summit
velocity by the equation
/W = /(0-|=H (57)
and then all the other elements would be determined, as
already explained.
In calculating trajectories by this method with the help
of tables of the definite integrals '^T^ , etc., it will generally
be necessary, as in Bashforth's method, to interpolate with
reference to y as well as d-, and for this purpose the integrals
must be tabulated for different values of y proceeding by
constant diff'erences, and including the highest and lowest
values of y likely to be needed in practice, which are, ap-
proximately, I and O.2.
13
CHAPTER VII.
TRAJECTORIES CONTINUED — DIRECT FIRE.
Niven's Method. — If a is some mean value of sec d
between the limits of integration ; that is, if we make
a = sec ^? (say)
then equations (17) to (20) may be written as follows:
_ 6; d {a li)
^ A (a uf
C -T-d (a m)
^^=-^cos^-^^^^^ (58)
C . -. d{au)
dy— — -7- smd . .„_\
A {a lif '
C d (a u)
A (auf-'
Making a u=iu\ and integrating so that t, x, y, and s
shall each be zero when u' = U\ we have
t = —£_\_l L_l
C
cos i)^
y =
{n-2)A "-''^^ ( u'^-' — U'""--
c
11 — 2) A \ u"-' U'"^ )
Comparing these equations with those deduced in Chap-
ter IV. for rectilinear motion, it will be evident that we
have as follows:
t = ciT{u^)- r{U')-] (59)
x=Cco?>^[S {u') - S ( U')~j (60)
y=C sin J [S (?/) - S{U'\^x tan 5 (61)
s=ClS{u')-S{U')'] (62)
EXTERIOR BALLISTICS. IO3
The first three of these equations (or their equivalents)
were first published by Mr. Niven in 1877, and in connection
with equation (38), viz.:
D=Ccos.J[D {u') - D ( U')] (63)
constitute what is known as "Niven's Method."
If we use the /-function instead of the Z>-function, equa-
tion (63) becomes
/} = 25_^ cos &U W) - I ( U')-\ (64)
or, better still, for direct fire (see Chapter V.),
D = 25_^ sec (f [/ (u sec f) - I { F)] (65)
in which
log ^=:l. 45 70926*
The values of ^ adopted by Mr. Niven are as follows:
For the /^-integral
-— tan <p 4- tan ^
tan *. = —^
For the X-, V-, and T-integrals
- - U
for the ascending branch, and
U + ic 3
for the descending branch of the trajectory. For the
method of deducing these expressions for ^, see a paper by
Professor J. M. Rice, U. S. Navy, in the eighth volume of
" Proceedings Naval Institute," page 191.
We will now apply these formulae to the solution of a
problem of direct fire; and, as we wish to compare the re-
sults obtained with those to be deduced from other methods
we will use Table I. of this work instead of Niven's tables,
and we will also perform the calculations with more accu-
racy than is generally necessary in practice.
104 EXTERIOR BALLISTICS.
Example of Niven s Method. — A 1 2-inch service projectile
is fired at an angle of departure of io°, and an initial velocity
of 1886 f. s. Find v, x, y, and / (a) when & = o, and (b)
when & = — 1;^°.
Here d= 12 in., z£'= 800 lbs., C — ,<c^ 10°, V^ 1886
144
f. s., U— 1886 cos 10° = 1857.33.
(a) ?? = o .-, n= 10°. We have first
tan ?^, = i tan 10° = 0.0831635
.-. 5, = 5° 2' i8'^ and U' = Usgc'J,=^ 1864.56
Next compute u' by means of the equation
nu') = ^^sec&, + I{U')
or
/ (u^) = 0.06308 + 0.03624 = 0.09932
.-. u' = 1328.96 = ?^ sec d-^
.-. ti^= 132372
Next compute the value of ^ to be used with the X-,
F-, and ^-integrals. We have
^ = 5° 2' 18'' + ^^57-33 - 132372 X 1? .. 5° 35' 51''
^1857.33+132372 3
The new values of U^ and 1/ are, therefore,
U' = 1866.25, and u' = 1330.06
From Table I. we find
5(^/0-2855.3 5 (?0 = 5239-2
' T{U')= 1.258 T{u')= 2.778
.•.4 = ^{2.778- 1.258} ::=8^444
■^o = Y^cos5| 5239-2 - 2855.3 | — 13180.7 ft.
yo = ^ tan d^ = 1291.8 ft.
(b) ^= — 13°. It will be necessary in this case to take
a new origin at the summit of the trajectory, as thei'e is no
EXTERIOR BALLISTICS. I05
provision made in this method for calculating an arc of a
trajectory lying- partly in the ascending and partly in the
descending branches. Indeed, since the differential ex-
pression for J/ contains sin ^ as a factor, which becomes zero
at the summit and changes its sign in the descending branch,
equation (61) does not hold true, unless the limits of integra-
tion {if and d^) are both positive or both negative.
We have, then, for this arc of the trajectory the follow-
ing data :
F= U^ 1323.72, ip =0°, ^= - 13°, and D= 13°
tan ^j = — i tan 13°= — o.ii5434i_ .' .^,=z — 6° 35' s"
[/'= 1332. SI 7(^0 = 0.09860
7 {u') = 0.08222 + 0.09860 = 0.18082
.' . u' — 1064.39 = Vq cos d^ sec d^^
. • . vq^ 1085.18, and Uq = 1057.37
^ :^ - 6° 35' ^" - ^32372-1052:37 jz_I3 _ _60 6'o''
1323.72 + 1057.37 3
The new values of U^ and u^ are, therefore,
f/^= 1331.26, and u' z= 1063.39.
From Table I. we get
5 {[/') = 5232.9 5 («0 = 701 1.7
tIu') = 2.773 ^M = 4-282
.•.^ = ^{4.282-2.7731 = 8^383
X = — cos ?? j 7011.7 — 5232.9 [ = 9826.3 ft.
144
y =z X tan ?? 1= — 1050. 1
The co-ordinates of the point of the trajectory whose in-
clination is— 13°, taking the origin at the point of projec-
tion, are therefore
X= 1 3 180.7 + 9826.3 = 23007.0 ft.
F== 1291.8 — 1050.1 = 241.7 ft.
And the time,
7^=8.444 + 8.383 = 16^827
For comparison we have computed the same elements
I06 EXTERIOR BALLISTICS.
directly from equations (i6), (25), (26), and (27), dividing the
whole arc into three parts, with the points of division corre-
sponding to velocities of 1330 f. s. and 1120 f. s. respectively.
The integrals for each arc were computed by quadratures,
and the following are the final results:
^^=1081.55 f. s.; Xi= 23025.7 ft.; F=: 243.14 ft., and
r= 16^843.
The agreement between these two sets of values is re-
markably close, and shows that for the purpose of com-
puting co-ordinates of different points of a trajectory,
Niven's method is all that could be desired so far as ac-
curacy is concerned. For high angles of projection the
trajectory should be divided into arcs not exceeding 10° or
15° each, and always with one point of division at the sum-
mit.
Example 2. — Given d ^ 12 in., 'w=^ 800 lbs., F= 1886 f. s.,
and ip = 30°. Compute the time and co-ordinates when
d- = 24°.
Answer :
BY niven's method. BY QUADRATURES.
^, = 27° 4' 29''
"5 = 2f 19' 4"
Xq = 8482.0 ft. 8481.4 ft.
_;/, = 4381.2 ft. 4381.9 ft.
/^ = 5''.889 5^888
Vff = 1400.58 f. s. 1400.4 f. s.
In the same manner, by successive steps, can the whole
trajectory be computed. In practice it is never necessary
to divide a trajectory into arcs of less than 10°.
Sladen's Method for Low-Aiigle Firing/'^ — When
the angle of projection is small, say not exceeding 3°, the
time corresponding to a given range can be computed with
great accuracy by means of (29) and (30). We should first
find V by means of the equation
* "Principles of Gunnery," by Major J, Sladen, R.A., London, 1879, Chapter VI.
EXTERIOR BALLISTICS.
107
and then with th.is value of v compute T by means of (29).
In the same manner we could find the value of / for a given
value of Xy less than X ; and these values of T and t substi-
tuted in (46), viz.,
would give the value of jj^ corresponding to x ; since, under
the conditions supposed, the vertical component of the velo-
city would be so small as to produce no appreciable resist-
ance to the projectile in that direction.
Example i — Required the following co ordinates of the
trajectory described by a 500-grain bullet fired from a
Springfield rifle, for a range of 600 ft., viz. : when ;ir= 150 ft.,
300 ft., and 450 ft. respectively ; r^ = 524.29, ^^ = 534.22.
Here ^== 0.45 in., zv = 500 grains = J^ lb., F= 1280 f. s.,
and X= 600 ft. We first find 5 (K) = 5509.70; r(F) = 2.985;
and
i\ X 534.22
C
0.35942
(0.45) X 524-29
The principal steps of the remaining calculations are
given in the following table :
(ft.)
X
C
S{v)
(f. s.)
/
(inches.)
(inches.)
(inches.)
150
417-34
5727.04
1209.72
0". 12055
9-365
9.406
7-950
300
834.69
6344-39
1146.76
0". 24814
13-167
12.987
10.600
450
1252.03
6761.73
I09I .31
0". 38235
10.386
9-956
7-950
Coo
1669.38
7179-08
1046.55
0". 52313
{T)
0.000
0.000
0.000
The sixth column gives the computed values oiy, and the
seventh the mean of five trajectories measured with great
care at Creedmoor by Mr. H. G. Sinclair, in charge of the
" Forest and Stream Trajectory Test." The last column
gives the corresponding values of j in vacuo, computed by
(45)-
t08 EXTERIOR BALLISTICS.
SIACCl'S METHOD FOR DIRECT FIRE.
dy
Expression for y. — We have from (35), since tan ^ = -7^
dy ^^ a C
dx
or
2 j dy
tan^-^{/(«')-/(^')[
^{-^-ta„,}-/(i/')=-/(«')
We also have from (58)
a . du'
ax =:
C A u'^-'
whence multiplying the last two equations together, mem-
ber by member,
2
C
Integrating and making x and j both zero at the origin,
where //' = U\ we have
{</^-u„,^.(-|./(^V- = ^|3^
2 j ^ \ a ^,^„^ I fl{u')du'
Making for convenience
-.K)=i/^^'
(in which the ^'s must not be confounded) the above equa-
tion becomes
^,{>/-^tan^} -^I{U')x=- \a{u')-A{U')\
From (60) we have
^x = S{u')-S{U')
whence, by division,
^ \y nn<.l nu'\- A{u')-A{U')
_|__tan^}-/(f/)_- ^-(„,)_5(f;,)
aC iA(u')-A(C/') ,,rml iA^\
or
z
X
EXTERIOR BALLISTICS. IO9
Calculation of the ^-Function.— We have (Chap-
ter V.)
and therefore
_ g Q ,
^ l2\n- I) A' u"^'^ + {n-2)Au'^-' + ^'
which becomes, when ;/ = 2,
The constants Q, corresponding to the five different ex-
pressions for the resistance, are given in Chapter V., and
the values of Q' are to be determined as explained in Chapter
IV. Making the necessary substitutions, and using A {v) as
the general functional symbol, we have for standard oblong
projectiles the following expressions for calculating the A-
functions :
2800 f. s. > -t^ > 1330 f. s. :
A {v) = [8 9012292] -^, + [2.6701589] log v - 1714-55
1330 f. s. > z' > II 20 f. s. :
A (7;) = [14.6562945] ^ + [5.1480576] i - 53.13
1 1 20 f. s. > z' > 990 f . s. :
A (v) = [32.2571789] ^„ + [14.4412953] ^ + 126.68
990 f. s. > z^ > 790 f . s. :
A (v) = [14.9781903] — , — [5.9124902] ^ + 449.89
790 f. s. > -z/ > 100 f. s. :
A {v) = [9.6655206] ^ + [4.1438598] log V - 45916.40
The values of A {v) calculated by the above formulae are
given in Table I.
14
no EXTERIOR BALLISTICS.
Equation {66), together with (35), (59), and (60), are the
fundamental equations of *' Siacci's method." This method,
by Major F. Siacci, of the Italian Artillery, was published
in the Revue d' Artillerie for October, 1880. A translation
of this paper by Lieutenant O. B. Mitcham, Ordnance De-
partment, U. S. A., was printed in the report of the Chief
of Ordnance for 1881. Lieutenant Mitcham added to his
translation a ballistic table adapted to English units, and
based upon the coefficients of resistance deduced by Gene-
ral Mayevski from the Russian and English experiments
noticed in Chapter IL In this table he gives for the first
time the values of T{v).
We will, for convenience, collect thesd equations to-
gether and renumber them :
They are :
tan ^ - tan ?? = ^ I / (;/) — I{U') | {^7)
x^^-\s{u')-S{U')\ (68)
: — : I >i 1 1 in — ^
S{u') — i,{U')
t =ClT{u')-T{U')^ (70)
u' — av cos d^ (71)
As the origin of co-ordinates is at the point of departure,
y is zero at the origin and also at the point in the descend-
ing branch where the trajectory pierces the horizontal plane
passing through the muzzle of the gun. Calling the velo-
city at this point v^, we shall have, making — d^ z:z w,
u'o> = « ^ ^ cos CO (72)
From (69) we have
aC \A (u'^)-A (U') .. ^„^ ) ._s
^ 2 [ S {u'^) — S {[/') ^ )
and from {6y)
a C
~ ^ 2 \s{u') — :^{U') ^'^^M ^ ^^
tan
^ =^ "^ I / {ti'^) - 7(^0 I - tan CO (74)
EXTERIOR BALLISTICS. Ill
Eliminating^ tan ^ from these last two equations gives
tan ^^ = — I ^ (^^ ^) - ■5l/?.)-5(^0 ^
From (68) and (70) we have
X=z— \s{2/^)-S{U')\ (76)
and a I ^ ^ ' )
T= C[T(u'^)^T{U')] {77)
By means of equations {67) to {77) all problems of ex-
terior ballistics in the plane of fire may be solved. If we
wish to compute the co-ordinates of the extremities of any
arc of a trajectory having the inclinations (f and d^, we should
make use of equations {67) to (71). If the object is to deter-
mine the elements of a complete trajectory lying above the
horizontal plane passing through the muzzle of the gun, at
one operation, we should employ equations (72) to {77). We
will give an example of each, using Didion's value of a.
Example i. — Given F= 1886 f. s. ; <^= 12 in.; 2£; = 800
lbs., ^ = 10°, and ?? = — 13°; to find z/^, ;ir0, je, and /e. (See
example i, Niven's method.)
We have first
(10°) + (13°)
a = — ^ i \ I o = 1.00723 1
Next ^^" 10° + tan 13° ^ ^
U' = 1886 « cos 10° = 1870.78
From Table I.,
5(^0=-2838.3;zJ (^0=44.06; /(^0=O-O358i; r<^U')^i.2t,o
From {67^ we have
/(^/.) =^ 1 ^"'^^ '""^ + ^""^ '^° 1 +^(^')
= 0.14554 + 0.03581 =0.18135
.-. 2^'=: 1063.42; 5(/0==7Oii<4; -^K)=440.44; 7 V) =4-282.
These values substituted in (68), (69), and (70) give
xq^=. 23017 ft.
yQ = 248.06 ft.
tQ z= i6''.844
112 EXTERIOR BALLISTICS.
From (71) we have
Ve = K = 1083.6 f. s.
a cos a
These results are quite as accurate as those deduced by
Niven's method by two steps.
Example 2. — Required the horizontal range, time ot
flight, and striking velocity, with the data ot Example i.
In computing « we will assume an angle of fall of — 14° 30',
which gives
«== 1.008645
.-. ^'=1873.40
5(^0=2828.5; A{U')=Al-7^\ 7(^/0=0.03563; n^0=i-243-
From (73) we have
^^^^g|i=^^tan, + /(t/') = o.09856
from which to calculate ic'^. As the relation between the
S-function and y^-function does not admit of a direct solu-
tion of this equation, it will be necessary to determine the
value of z/o, by successive approximations; and for this pur-
pose the rule of ** Double Position" is well adapted. This
rule is deduced as follows : Let u^ and u^ be two near values
of?/ (or the quantity to be determined), one greater and the
other less ; and e^ and e^ the errors respectively, when n^ and
u^ are substituted for u in the equation to be solved. Tiien,
upon the hypothesis that the errors in the results are pro-
portional to the errors in the assumed data, we have
e^\ e^W u — //j : u — u^
whence, by division,
e^— e^\ e^W u^ — u^'. u — u^
or
e^ — e^\ e^\ : u^ — u^\ ii — ti^
from which is derived the following rule: As the difference
of the errors is to the difference of the assumed numbers, so
is the lesser of the two errors (numerically) to the correc-
tion to be applied to the corresponding assumed number.
EXTERIOR BALLISTICS. II 3
If 11^ and u^ are selected with judgment, the resulting
value of II will generally be sufficiently correct by a single
application of the rule, or, at most, by two trials.
In our example assume it^ =z 1050, for a first trial ; whence
5 (1050) = 7143.7, and A (1050) =464.94 ; and these in the
above equation give
464.94 — 43.71 ^
^ ^ ^ -to / _ 0.09762
7143.7-2828.5
If we had taken for 71^ the correct value of u^^, the second
member would have been 0.09856, and hence ^, = — 0.00094.
Whenever ^, is negative the assumed value of u' ^ is too
great; we will, therefore, next suppose 2/2= 1040, and pro-
ceeding in the same way we find ^^ = +0-00128. The cor-
rect value of u^^ is, then, between 1050 ft. and 1040 ft. Ap-
plying the rule, we have the following proportion :
222 : 10 :: 94 : 4.23
consequently u'^ = 1050 — 4.23= 1045.77 f- s. : and this satis-
fies the above equation.
We next find
5(?/^) = 7i87.i; ^ (//^)=473.2o; 7(2^'^) =0.1 9 154; T{u'^)=4.44S
We now have from (75)
tan io = j o. 191 54 — 0.09856 i = 0.2605 1
.-..(>= 14" 36^ (By Table III.)
From {26) and (yy)
I 7187.1 — 2828.5 I =24007 ft.
r= c:[ 4.448 - 1.243] = i7".8o6
From (72)
^« = ^=i07i.4f. s.
a cos
Various other problems may be solved by a suitable com-
bination of equations {6y) to (71). Indeed, if a velocity,
a
114 EXTERIOR BALLISTICS.
either initial or terminal, and one other element be given,
all the other elements may be computed, though in certain
cases this can only be accomplished by successive approxi-
mations. Most of these problems, for direct fire, will be
solved further on.
Api>licatioii of Siacci's Equations to Mortar-
Firing. — For low velocities, such as are used in mortar-
firing, we may take for a in all cases the following value :
tan ^
This simplifies the calculations, and gives results sufficiently
accurate for most practical purposes, as the fbllowing ex-
amples will show :
Example i.— Given F=75i f. s. ; ^ = 30°; and log C zizz
0.25056. Required X, T, w, and v^. (See Example i, Chap-
ter VI.)
We have, Table III., {(f) = 0.60799.
log {(f) = 9.78390
log tan <p = 9.76144
log a == 0.02246
log V=: 2.87564
log cos ^ = 9.93753'
log U' = 2.83563 U' = 684.90
5(£/0=i368i.i; ^(^0-= 344443;/ (^0=0-80679; T{U')=^
12.274.
log 2 = 0.30103 [Equation (73)]
c. log a — 9.97754
' c. log 6'= 9.74944 (^dd log t^" f)
log 0.61 581 =9.78945
7 (£/') = 0.80679
1.42260
• ^ (»'.)- 3444-43 ^,fo
5(«'„)— 13681.1 *
EXTERIOR BALLISTICS. II5
By double position we find from this equation
7/^ = 45978
.-. 5 (?/<,) = 20443.1 ; /(?/'^) = 2.22481 : T {2/^) =24,4.04
X=— I 20443.1 - 13681.1 I =11434 ft.
Tzzz (7 [24.404 — 12.274] = 2 I ''.60
tan (o = j 2.22481 — 1.42260 [ [Eq. (75)]
.•... = 36° 57'
?/
^co = ~ — = 546.3 f. s. [Eq. (72)]
a cos CO -^^ ^ L n \/ /J
Example 2. — Given f^== 977.71 f. s., ip — 35° 21', and
log C'^: 0.38722. Required X, 7", w, and t/^^. (See Example
3, Chapter VL)
Answer:
^=19328 ft.
^-31^63
?/'„ =517.63
CO = 44° 44'
v^ =1675.65 f. s. ^ ^
Example 3. — Given F:^ 609.63 f. s. ; ^ = 45°, and log 6^ =
0.56809; required X, T, oj, and ?^^. (See Example 5, Chap-
ter VI.)
Answer:
X= 11984 ft.
r= 28^30
?/a. = 436.52
CO = 49° 10'
7;^ = 581.64
Siacci's Equations for Direct Fire.— As already
stated, a is some mean value of the secants of the inclina-
tions of the extremities of the arc of the trajectory over
which we integrate ; and consequently if we take the whole
r
Il6 EXTERIOR BALLISTICS.
trajectory lying above the level of the gun, a will be greater
than I and less than sec co. To illustrate, suppose we have for
our data a given projectile fired with a certain known initial
velocity and angle of projection, and we wish to calculate
the angle of fall, terminal velocity, range, and time of flight.
If we calculate these elements by means of (75), {ji), (76),
and {j']^^ making a = i, they will be too great ; while if a is
made equal to sec co, or even sec ^, they will be too small ;
and the correct value of each element would be found by
giving to a some value intermediate to the two. Moreover,
the value of a which would give the exact range would not
give the exact time of flight or terminal velocity. These
principles are further illustrated by the follow^ing numerical
results, calculated from the data, F= 1404 f. s. ; ^=10°;
w z=z 183 lbs., and <^= 8 in. :
a = 1 a = sec <p
X= 13752 ft. X— 13622 ft.
v^ = 892.2 f. s. v^ =881.4 f. s.
co=-ifif fti = -i3°23'
T=if.04 T=12\SS
As the true values of these elements lie between those
we have computed, it will be seen that either set of values
is correct enough for most purposes. It is, therefore, ap-
parent that in direct fire we may give to a that value which
shall reduce the above equations to their simplest forms, pro-
vided it lies between the limits a= i and a = sec (p.
As we have already seen (Chapter V.\ Major Siacci
gives to a the value
n-2
a = (sec (f) «^i
by means of which equation (37) was obtained, viz.:
tan^ = tany-— J- j /(«')- /(F)} (78)
in which
, cos ^
cos (p
EXTERIOR BALLISTICS. II7
Making the same substitution in (68), (69), and (70), they
become respectively
x- = C[S{u')-S{V)-] (79)
;i' ^2 cos' (f { S (?/) — 5 ( F) ^ M ' ^
When ^ and d- are so small that the ratio of their cosines
does not differ much from unity, we may put
and the above equations become
tan & = tan <p ^ ^ ^ (^) - /(H [ (82)
^ 2 cos <p { ^ ^ M
;r=6^[5(7;)-.V(F)] (83)
-^'-tanc 6- • i ^e.)- ^(F) [ ,3.
--tan <p - ^ ^^^^, ^ j ^^^^-^_- - /(^^) j l«4j
/-=— ^] r(^)-r(F)i (85)
cos ^ ( ^ ^ ^ M
We shall retain this form of the ballistic equations in
what follows, though when very accurate results are de-
sired we must use ?/ instead of z^
When J/ = 0, we have from (84)
Substituting for tan (p in (84) its value from (82), and re-
ducing, we have, when j' = o,
2 cos= <p tan « = C I / (V) - ^ ^^.^ _ ^ ^y^ \
For small angles of projection we may put
2 cos''' (f
and, therefore.
cos <p
2 cos (p tan o) = 2 sui co cos co ~ = sui 2ft>
^ cos ^6»
1 A{v)-A(V)l , .
sin 2<. = C- I / (.•) - s{v)-S{V) \ ^^7)
For the larger angles of projection employed in direct
^5
Il8 EXTERIOR BALLISTICS.
fire, if accurate results are desired, we must determine (o by
the equation
tan CO = tan w ^ \ I {v) - I (V) \
^ 2 COS if \ ^ ■' ^ ^ \
using ?/ instead oi v, as already explained.
Practical Applications. — We will now apply Siacci's
equations to the solution of some of the most important
problems of direct fire.
Problem i. — Given the initial velocity and angle of pro-
jection, to determine the range, time of flight, angle of fall, and
terminal velocity.
We have [equation (86)]
A{:v)-A {y) _ sin 2^
S{v)--S{l^) C "T" ^^^
from which to calculate v by '' Double Position," as already
explained. Having found v^ the remaining elements are
computed by the equations
x=c[se.o-5(F)]
r=— ^ I T{v)-T{V)\
COS ^ \ ' ' )
For curved fire we may proceed as follows: We have,
from the origin to the summit,
Now, if we assume tiiat the time from the point of pro-
jection to the summit is one-half the time of flight, we shall
have, from the above expressions for 7' and 4,
r(7;) = 2 T{v^-T{y)
which gives z^ by means of the 7^-functions, v^ being computed
bv the equation
derived from (82).
Example i. — The 8-inch rifle (converted) fires an ogival-
EXTERIOR BALLISTICS. II9
headed shot weig-hing 183 lbs. If the angle of projection
is 10°, and the initial velocity 1404 f. s., find the range, time
of flight, angle of fall, and terminal velocity.
We have F^ 1404 f. s. ; ^=zio°; ze; = 183 lbs.; d=Z
inches, whence log C — 0.45627 : to find X, T, od, and v.
From Table I. we find
5(F) = 4878.6 -0.8 X 25.1 =4858.5
A {V) = 163.96 — 0.8 X 2.16 = 162.23
/ (F) = 0.08661 — 0.8 X 0.00082 = 0.08599
T{V) = 2.514—0.8 X 0.018=2.500.
Next compute v:
log sin 2^ = 9.53405
log (7=0.45627
log o. II 96 1 = 9.07778
/(F) = 0.08599
0.20560
The value of v satisfying this equation is found to be
V = 873.8 ft., whence
5 {v) = 9641.8 A (z/^) = 1 145.65
/ (v) = 0.36668 T {v'^) = 7.030
X, T, (0, and z' are now computed as follows :
log C = 0.45627
log[S(2.)- 5(F)] = 3^7973
log X= 4.13600'
X= 13677 ft. ==4559 yc^s.
\og[T{vy- r(F)] = 0.65610
log sec <p = 0.00665
log r= I.I 1902
^''- I ^^''^ -s{z;)-SjV)\ = 9-^0704
log sin 2co =z 9.66331
2C0 = 27° 25' 30''
CO
= 13° 42' 45'
I20 EXTERIOR BALLISTICS.
The value of o; computed by the more exact foi inula
tan CO — — - — ^— \ I {v) -^-^ —~j^ \
ig 2 cos' (f I ^ ^ ^ (v) — ^ {V) )
..==13° 21^ 30''
differing by 21' from the less approximate value.
We have found above
z;= 873.8 f. s.
but this is only an approximation. To determine its true
value, that is, i^s true value so far as the formulce are eonee?ned,
we should have
cos 10° „^ r
V — 873.8 5 — -, — 7, = 884.45 f . s.
'^ cos 13° 21' 30'
differing from the approximate value by about 10 feet.
Example 2. — "A 6-inch projectile leaves the gun at an
angle of departure of 4°, with an initial velocity of 2100 f. s. ;
7e^= 64 lbs., </= 6 inches. Find the range in horizontal plane
through the muzzle of the gun, and time of flight." ('' Ex-
terior Ballistics," by Lieutenants Meigs and Ingersoll,
U.S.N.)
We have (Table I.)
5(F) = 2024.8;^(F) = 20.57; /(F) = o.02246; r(F) = o.838
Takin^: <:=i i, we have
Next we have 3^
AM^.20-57 ^ 36 ,i„ go + / (F) ^ 0..0074
5 {v) — 2024.8 64 \ \ J
from which equation we readily find
V = 993.77 f- s.
.' . S {v) = 7801.8, and T {v)=z 5.051
X— C [7801.8 — 2024.8] = 10270 ft.
Problem 2. — Given the angle of fall a7id terminal velocity, to
determine the initial velocity, angle of projection, range, and time
of flight.
'i
EXTERIOR BALLISTICS. 12
We have [equation (87)]
A(v)~A{V) _ . / X sin 2co
S {v) - S{V) - ^""^ C~~
from which to calculate Fby double position.
We may also determine V by the equation (see Prob-
lem i)
r(F) = 2 r(zO- T{z')
v^ being found by the equation
/ (vo) = / {v) ^—
derived from (82).
Having found F by either method, <p, X, and Tare com-
puted by the equations
^\A(v) — A{V) ,,,^J
X^C\S{v)-S{V)-]
r=-^i T{v) — T{V)\
COS (p { )
Example i.^Given </=4.5 inches; 7e'= 35 lbs. ; w=: 15°,
and z/= 772.74 f. s. ; to determine ^, X, and T.
It will be found that we have the following equation from
which to find V :
2058.17 — ^(F)
> -r -F777V = 0.26807
11633.6 — ^ (F) ^
For the first trial assume V ^z 1500, and, substituting in
the first member of the above equation, it reduces it to
0.26691, which is too small by 0.00116 = ^j. Next make
F= 1480, and we shall find that the first member now be-
comes too great by 0.00140 =: e^; then
256 : 20 : : 116 : 9.1
The correct value of Fis therefore 1500 — 9,1 =: 1490.9 f. s.,
from which are easily found
^ = 9° 51^ X= 12440 ft. ; T=i2".72.
Example 2. — " In attacking a place with curved fire it
was required to drop shell into the place with an angle of
122 EXTERIOR BALLISTICS.
descent of 12°, and terminal velocity of 600 f. s., using the
8-inch howitzer and a projectile of 180 lbs.; find the requi-
site position of the battery, and the requisite elevation and
charg-e of powder."'^
Here <3f=8 inches; zv =: iSo lbs.; 7^=600 f. s., and
co^ 12°; to find X, V, and (p. We have
log sin 2C0 = 9.60931
log (7 = 0.44909
log 0.14462 = 9.16022
/ (7;) = 1. 15929
I{v^ = 1. 01467 v^ = 630.85 f. s.
whence we find
T{V) = 2X 14-396— 15779 = 13-012
F:= 665.1 f. S.
5 (v) = 15926.6
5(F) 3^ 141 78.9
log 1747.7 = 3-24247
log A"=: 3.69156
X=49i5 ft. = 1638 yds.
I{z'o)= 1. 01 467
/(F) = 0.87708
log 0.13759 = 9.13859
log sin 2(p = 9.58768
2<p = 22° 46' ^ = 11° 23'
Problem 3. — Given the range and initial velocity, to deter-
mine the other elements of the trajectory.
This is by far the most important of the ballistic prob-
lems, and it happens, fortunately, to be one of those most
easily solved by Siacci's formulae.
For the terminal velocity we have
* Prof. A. G. Greenhill in " Proceedings Royal Artillery Institution," No. 2, vol. xiii. page 79.
EXTERIOR BALLISTICS.
123
and then, with Fand v known, all the other elements can be
computed by formulcE already considered.
Example i.— Find the elevation required for a range of
2000 yards with the i6-pdr. M. L. R. i^im, the muzzle velo-
city being 1355 f. s. ; find also the time of flight and angle
of descent.
Here <^r=: 3.6; 7(y = 16; log (7 = 0.09152 ; F= 1355, and
X = 6000.
Answer :
4° 41
T = 5^91
0^ =6° 13^
Example 2. — Compnte a range table for the Z-inch rifle {con-
verted), up to 15000 ft.
We have for chilled shot, 7£^ = 183 lbs.; <^=: 8 in. (whence
log (7 = 0.45627), and V— 1404 f. s. First take from Table
I. the following numbers, which are to be used in all the
calculations :
5 (r) = 4858.5, y^(F)r= 162.23, /(F) = 0.08595, r(F)=: 2.500
The remainder of the work niay be concisely tabulated
as follows :
X
ft.
X
c
S{v)
v
A{v)
/(v)
T(v)
1500
524-59
5383-1
1303.0
212.04
0. 10442
2.884
3000
1049
2
5907.7
I2I2.8
272.28
.12579
3
305
4500
1573
8
6432.3
II34-3
344 -^o
.15038
3
753
6000
2098
4
6956.9
I 69.2
430.79
.17826
4
230
7500
2622
9
7481.4
1019.2
532.14
. 20929
4
732
9000
3147
5
8006.0
978.8
650.68
.24314
5
257
10500
3672
I
8530.6
942.5
787.72
-27973
5
804
12000
4196
7
9055.2
908.8
944.68
•31914
6
371
13500
4721
3
9579.8
877-4
1123.07
.36148
6
959
15000
5245-9
10104.4
848.1
1324.47
. 40684
7.567
The numbers in the first column are the ranges for which
the elements of the trajectory are to be computed. The
numbers in the second column are simple multiples of the
first number in the column. Adding S {V) to the numbers
124
EXTERIOR BALLISTICS.
in the second column ^ives those in the third column, and
with these we take from Table 1. the values of v, and at the
same time those of A {^v), I {v), and T {v).
The time of fliglit, angle of departure, and angle of fall
are then computed by the following formulas:
'T= — ^ \ T{v)- T{V)\
cos (p
and
sm 2lp:
tan Col =
ciiM^Am-ii^V)
S{v)-S{V)
C ( , , , A (v]
-,{n^)
A{V))
2 cos' (p { ^ ' S (v) — S {V) )
Lastly, the values of v, tabulated above, a?-e to be multi-
plied by cos (f sec &-> to obtain the correct striking velocities.
In our example the results are as follows:
yds
<!>
a,
T
500
o°44'
o°47'
1303
I^IO
1000
i°33'
i°43'
I2I3
2^30
1500
2° 2/
2° 50^
II35
3''-59
2000
3° 2/
4° 08'
1070
4^96
2500
4° 32'
5° 38'
I02I
6^40
3000
5° 43'
7° 14'
982
7^92
3500
6° 59'
9° 01^
947
9^52
4000
8° 21^
10° 58^
916
11^19
4500
9° 49'
13° 06'
888
12^94
5000
11° 24'
15° 25'
862
14^78
By interpolation, using first and second differences, the
interval between successive values of the argument {X) may
be reduced from 500 yards to 100 yards.
Example 3. — Given d — 20 93 cm. ; if> = 140 kg. ; V = 521
m. s. ; d^ = 1.206; d= 1.233 ; X=4097 m.; angle o( jump = 8';
required the angle of elevation == ^ — 8', the angle of fall,
the striking velocity, and the time of flight.^
Making the ballistic coefficient {c) =0.907, we have for
* '■ Ballistische Formeln-von Mayevski nach Siacci. Fur Elevationen unter 15 Grad," Essen,
Fried. Krupp'sche Buchdruckerei, 1883, page 22. Also quoted by Siacci in " Rivista di Artiglieria
e Genio," vol. ii. page 414, who solves the example, using Mayevski's table.
EXTERIOR BALLISTrcS.
125
computing C in English units, when <^ is expressed in centi-
metres and w in kil(3grarames, the following expression :
C-[i..953743]f ^
The following are the results obtained by experiment,
by Mayevski's calculations, by Siacci's calculations, and by
Table I. of this work :
T
Angle of
Elevation.
Angle of
Fall.
Striking Velocity,
f. s.
By experiment
Mayevski...
Siacci
Table I
9"-7
9".6
9".675
9".66
5° 30'
5° 32'
5° 31'
5° 29' 30"
7° 16'
I 176
I 169
Example 4. — Given ^=24 cm.; 7e/ = 2i5 kg.; F= 529
m. s. = 1735.6 f. s. ; required the angle of departure for each
of the horizontal ranges contained in the first column of the
followintr table :
Horizontal
Range.
in
5/
J
Computed by
Table I.
Observed
value of
Values of <f> computed by
Mayevski's
Table.
Hojel's
Table.
2026
0.9569
2°.;'
2° .9'
2° 18'
2° 14'
3000
0.9407
3° 36'
3° 41'
3° 37^
3° 35'
4000
0.9756
5° 5'
5° 10'
5° 6'
5° 5'
5964
0.9560
8" 41'
8° 35'
8° 44^
8° 44'
7600
0.9461
12° 31'
12° 5'
12° 31'
12° 32'
The data in the first, second, and fourth columns are
taken from Krupp's Bulletin, No. 56 (February, 1885), page
4. The values of <p in the third column were computed by
Siacci's method, using Table I. of this work. In the last
two columns are given the values of ^ computed by Siacci's
method with Mayevski's and Hojel's tables respectively.
Problem 4. — With a given initial velocity^ required the angle
16
126 EXTERIOR BALLISTICS.
of projection necessary to cause a projectile to pass through a
given point.
Let X and y be the co-ordinates of the given point. Then
from (83) and (84) we have
and ^
Example. — An 8-inch service projectile is fired with an
initial velocity of 1404 f. s. from a point 33 feet above the
water; find the necessary angle of projection to attain a
range on the water of 3000 yards.
Here <^=: 8, ze/ = 180, F= 1404, x = 9000 ft., and j/= — 33 ft-
We have ^
^ (^^) = I^ ^ 9000 + 4858.5 = 8058.5
••• ^ = 975-07
In calculating tan ^ we will, at first, omit the factor cos" (p
in the second member.
33 , 180(663.56—162.23 „ I
= — 0.00367 -|- 0.09945 = 0.09578
Therefore the approximate value of ^ is 5° 28'. Complet-
ing the calculation by introducing cos"" ip we have
? = 5°3i'
which needs no further correction.
Problem 5. — Given the initial and terminal velocities, to
calculate the trajectory.
For the solution of this problem we have the following
equations: ^ A {v) - A {V) ,,„,!
^ \ ^'> s{v)-s(y)S
sm 2(«
X=C[5(^)-5(F)]
7-=-^ I T{v)- T{V)\
cos <f \ ' S
EXTERIOR BALLISTICS. I27
Example. — In experimenting- with the 15-inch S. B. gun,
it is desired to place a target at such a distance from the
gun that the projectile (solid shot weighing 450 lbs.) shall
have a velocity of 1000 f. s. when it reaches the target, and
this without diminishing the muzzle velocity, which is 1534
f. s. What is the required distance and the angle of pro-
jection ?
We readily find, using Table II.,
and ^ = 2° 33'
X=4678 ft.
CORRECTION FOR VARIATION IN THE DENSITY OF THE AIR.
The ballistic coefficient (Q is determined by the equation
r- ^ h.
cd' d
in which d^ is the adopted standard density of the air, and d
the density at the time of firing.
In computing Tables I. and II. the value of d^ was taken
as the weight, in grains, of a cubic foot of air at a tempera-
ture of 62° F. and a pressure of 30 inches of mercury. Ac-
cording to Bashforth we have
'^z = 534-22 grs.
For any other temperature (/), and barometric pressure
{b)j we may determine the value of d near enough for most
practical purposes by the following simple equation:
^_ 20.212 b
~ I -f .002178 t
Correction for Altitude. — When a projectile is fired
at such an angle of projection as to reach a great altitude in
its flight, the value of o, determined as above, will be too
great. We may calculate approximately, in this case, as
follows :
If o' is the density of the air at the height y above the
surface of the earth, we shall have
d'^de-'x
128
EXTERIOR BALLISTICS.
where ?. is the height of a homogeneous atmosphere of the
density <5, which would exert a pressure equal to that of the
actual atmosphere.'^'
d o ^
The factor -— becomes, therefore, ~ e^; and (7 must be
o o
multiplied b}^ this if we wish to take into account the dimi-
nution of density due to the height of the projectile, taking
for J a mean value for the arc of the trajectory which we are
computing.
y
The following table gives the values of ^-a. for every lOO
feet from j = o to /= 10,000 feet. In the computation ?.
was assumed to be 27800 feet, which is its approximate
value for a temperature of 15° C. and barometer at o"'.75.
The table is substantially the same as that given by Bash-
forth {" Motion of Projectiles," page 103), but in a moie con-
venient form.
y
100
200
300
400
500
6qo
700
800
900
I. 0000
0036
0072
0108
0145
0181
0218
0255
0292
0329
1000
1,0366
0403
0441
0479
0516
0554
0592
0631
0669
'0707
2000
1.0746
0785
0824
0863
0902
0941
0981
1020
1060
1 100
3000
I. I 140
1180
1220
1260
1 301
1 34 1
1382
1423
1464
1506
4000
I -1547
1589
1630
1672
1714
1756
1799
1841
1884
1927
5000
I. 1970
2013
2057
2100
2144
2187
2231
2276
2320
2364
6000
1.2409
2-154
2499
2544
2589
2634
2679
2725
2771
2817
7000
1.2863
2909
2956
3003
3049
3096
3H4
3191
3239
3286
8000
I 3334
3382
3431
3479
3528
3576
3625
3675
3724
3773
9000
1.3823
3873
3923
3973
4023
4074
4125
4176
4227
4278
* Chauvenet's " Practical Astronomy," vol. i. page 138.
BALLISTIC TABLES.
The term ''Ballistic Table" was applied by Siacci to
tlie tabulated values of the funclions S{v), A {v), I{v), and
7\v). Table L g-ives the values of these functions for ob-
long projectiles having ogival heads struck with radii of i|
calibers. It is based upon the experiments of Bashforth,
and was calculated by the formulas developed in the preced-
ing pages.
The table extends from z/=28oo to ^ = 400, which limits
are extensive enough for the solution of nearly all practical
problems of exterior ballistics. It may occasionally happen
in mortar practice that the horizontal velocity {v cos <f) may
be less than 400 (as in problem 4, Chapter V.) In such
cases we may employ the formulas by which this part of the
table was computed, viz.:
5 (v) = 124466.4 - [4.59i833(>] log ^
A (v) = [9.6655206] -^ -f [4.1438598] log V - 45916.40
/(^) = [5.7369333] ^ - 0.356474
T{v) = [4.2296173] ^ - 12.4999
Example i.— Let (^=8 in., w = 180 lbs., F= 700 f. s., and
^ = 60°. Find V when ^ = — 60°.
We have from (33)
and U ^=. joo cos 60° = 350, which is below the limit of
2 BALLISTIC TABLES.
the table. The operation may be concisely arranged as
follows :
const. log=:: 57369333
2 log f/= 5.0881360
0.6487973 = log 445448
(60) = 2.39053
log 4 (60°) = 0.9805542
log C= 0.4490925
0.5314617 = log 3.39987
0.895 1103 = log 7-85435
2)4.8418230
2 42091 15 =: log 263.6
. • . 7/ =: 263.6 X 2 = 527.2 f. S.
Example 2. — Given 5 {v) = 25496.8, to find v.
We proceed as follows:
1 24466.4
25496.8
log 98969.6 = 4.9954886
const, log = 4.5918330
log (log z/) = 0.4036556
.-. log 7;=2.533i2
£^=341.3
Table II. is the ballistic table for spherical projectiles,
and extends from z^= 2000 to ^^ = 450. It is based upon the
Russian experiments discussed in Chapter II., and is be-
lieved to be the only ballistic table for spherical projectiles
yet published.
Table III. is abridged from Didion's " Traite de Bal-
istique."
Forniulse for Interpolation. — To find the value of
f{z^ when V lies between v^ and v^, two consecutive values
of V, in Tables I. and II. Let v^ — v^r=^ h. Then, if d^ and d^
BALLISTIC TABLES.
are the first and second diflferences of the function, we shall
have, since y(?7) increases while v decreases,
2
by means of which f{v) can be computed. Conversely, if
f{7>) is given, and our object is to find v, we have
7\ — v\ d^
2
In using this last formula, first compute —^ — by omit-
Ti
ting the second term of the second member (which is usually
very small), and then supply this term, using the approxi-
mate value of-^-^^ — already found.
Ii ^
If the second differences are too small to be taken into
account, the above formulae become
/(z,)=/(t;,) + ^S-^rf,
and
which expresses the ordinary rules of proportional parts.
Example i. — Find from Table I. S{v) when z/= 1432.6.
We have v, = 1435, f{v^ = 4704.8, h — 5, and d, = 24.6.
.•.S{v) = 4704.8 + 1 435 - 1432.6 ^ ^^^^ ^ ^^j^^^
Example 2. — Given A (7/) = 229.89, to find v. Here 7/^ =
1274, /(7/,) = 229.29, </,= 1.25, and /^= 2.
2
. • . 7; = 1 274 (229.89 — 229.29) = 1 273.04
1.25
Example 3. — Find from Table II. A {v) when 77 = 517.8.
4 BALLISTIC TABLES.
We have e^, = 520, ^(^0 = 3755-9. >^^ = 5» ^, = 158.2, and
^,= 7.8.
2.2 2.2 / 2.2X7.8
.-. ^ (^) = 3755.9+ -X 158.2 ---(i--)^
= 3755-9 + 69-60 — 0.96 = 3824.5
Example 4.— Find from Table HI. the value of (^) when
^ = 54° 32'. Here ?^, == 54° 2o\ (^,) = 17619 1» h=z2o',d,z^
.02971, d^ = .00074.
.-. (^)z= 1.76191 +0.6X 0.02971 —0.6 X 0.4 X 0.00037
= 1.76191 +0.01783 —0.00009= 1.77965
TABLE I.
Ballistic Tabic for Ogival-Hcaded Projectiles.
V
6- (7')
Diflf.
A iv)
Diff.
1
Diff.
T{v)
Diff.
2800
2750
2700
j 000.0
126.8
[ 256.0
1268
1292
1315
0.00
0.07
0.28
7
21
36
0.00000
0.00106
0.00218
106
112
118
0.000
0.046
0.093
46
47
49
2650
2600
2550
387.5
521.6
658.3
1341
1367
1393
0.64
1. 18
1.89
54
71
93
0.00336
0.00461
0.00594
125
140
0.142
0.193
0.246
51
53
56
2500
2450
2400
797.6
939.8
1085.0
1422
1452
1481
2.82
3.97
5.37
115
140
166
0.00734
0.00883
0.01043
149
160
169
0.302
0.359
0.419
57
60
62
2350
2300
2250
I233.I
IJ84.5
1539.2
'514
1547
1582
7.03
9.00
11.31
197
231
266
O.OI2I2
0.01392
0.01584
180
192
205
0.481
0.546
0.614
65
. 68
72
2200
2190
2180
1697.4
1729.5
I76I.7
321
322
323
13.97
14.55
15.15
58
60
62
0.01789
0.01832
0.01876
43
44
44
0.686
0.700
0.715
14
^5
15
2170
2160
2150
1794.0
1826.5
1859.2
325
327
328
15.77
16.40
17.05
65
67
0.01920
0.01964
0.02010
44
46
46
0.730
0.745
0.760
15
15
15
2140
2130
2120
1892.0
1924.9
1958.0
329
331
17.72
18.40
19.10
70
73
0.02056
0.02102
0.02149
46
47
48
0.775
0.791
0.806
16
15
16
2IIO
2100
2090
I99I.3
2024.8
2058.4
335
336
337
19.83
20.57
21.33
74
76
79
0.02197
0.02246
0.02295
49
49
50
0.822
0.838
0.854
16
16
16
2080
2070
2060
2092.1
2126.0
2I60.I
339
341
343 >
22.12
22.92
23.74
80
82
85
0.02345
0.02396
0.02447
51
51
52
0.870
0.886
0.903
16
17
17
2050
2040
2030
2194.4
2228.8
2263.4
344
346
348
24.59
25.46
26.35
[
87
89
91
0.02499
0.02552
0.02606
53
54
54
0.920
0.937
0.954
17
17
17
2020
2010
2000
2298.2
2333.1
2368.2
349
351
353
27.26
28.20
29.16
94
96
98
0.02660
0.02715
0.02772
55
57
57 1
0.971
0.988
1.005
17
17
18
TABLE L— Continued.
V
S{v)
Diff.
A {V)
Diff.
7(7')
Diff.
T{v)
Diff.
1990
1980
1970
2403-5
2439.0
2474.6
355
^ 356
358
30.14
31-15
32.19
lOI
104
107
0.02829
0.02886
0.02945
57
59
60
1.023
1. 041
1-059
18
18
18
i960
1950
1940
2510.4
2546.4
2582.6
360
362
363
33-26
34-35
35-48
109
113
115
0.03005
0.03066
0.03127
61
61
62
1.077
1.096
1. 114
19
18
19
1930
1920
I9I0
2618.9
2655.5
2692.2
306
367
370
36.63
37-81
39.02
118
121
124
0.03189
0.03253
0.03318
64
65
65
I-I33
1. 152
1. 171
19
19
20
1900
1890
1880
2729.2
2766.3
2803.7
371
374
375
40.26
41-53
42.83
127
130
0.03383
0.03450
0.03517
67
69
1. 191
1. 210
1.230
19
20
20
1870
i860
1850
2841.2
2878.9
2916.9
377
380
382
44.16
1 45-53
46.93
137
140
143
0.03586
0.03656
; 0.03727
70
71
72
1.250
1.270
1. 291
20
21
20
1840
1830
1820
2955-1
2993-4
3032.0
386
388
48.36
49-83
51-34
147
151
155
0.03799
0.03872
0.03946
73 1
74
76
1. 311
1-332
1-353
21
21
22
I8I0
1800
1790
3070.8
3109.8
3149.0
390
392
394
52.89
54-47
56.09
158
162
167
' 0.04022
0.04099
10.04177
77
78
80
1-375
1.396
1.418
21
22
22
1780
1770
1760
3188.4
3228.0
3267.9
396
399
401
1 57-76
1 59-47
61.21
171
174
179
i
0.04257
0.04338
0.044.20
81 1
821
84!
1.440
1.463
1-485
23
22
23
1750
1740
1730
3308.0
3348.3
3388.9
403
406
409
63.00
64-83
66.71
183
188
193
0.04504
0.04589
0.04676
85 1
87
88!
1.508
1-531
1-555
23
24
23
1720
I7I0
1700
3429.8
3470-8
3512. 1
410
413
415
! 68.64
: 70.61
72.63
1
197
202
207
0.04764
0.04854
0.04945
90
9r\
1-578
1.602
1.626
24
24
25
1690
1680
1670
3553-6
3595-4
36374
418
420
423
1
74-70
76.83
79.01
213
218
223
0.05038
0.05133
0.05229
95
96 1
98 1
1. 651
1.676
1. 701
25
25
25
1660
1650
1640
3679-7
3722.2
3765-0
425
428
430
81.24
83-52
85.86
228
234
241
6
0.05327
0.05427
,0.05529
100
102
103 1
1.726
1-752
1.778
26
26
26
TABLE I.— Continued.
3808.0
3851-3
3894.9
3938.7
3960.7
3982.8
4005.0
4027.3
4049.6
4072.0
4094.4
4116.9
4139-5
4162.2
4185.0
4207.8
4230.7
4253-6
4276.7
4299.8
4323-0
4346.2
4369.6
4393-0
4416.5
4440.1
4463-8
4487-5
4511-3
I 4535-2
4559-2
4583.2
4607.4
4631.6
4655-9
4680.3
Diff. I
I
433 I
436 I
4381
220 1
221 j
222 I
223
223
224
224
225
226
227
228
228
229
229
231
231
232
232
234
234
235
236
237
237
238
239
240
240
242
242
243
244
245
A {7')
88.27
90-73
93-25
95-84
97.16
98.49
99.84
IOI.2I
102.60
104.00
105.42
106.86
108.32
109.79
111.29
112.80
114-33
115.88
117-45
119.04
120.65
123-93
125.60
127.29
129.01
130.75
132.50
134.28
136.09
137.92
139-77
141.65
T43-54
T45-47
147.42
Diff.
246
252
259
132
133
135 I
137
139
140
142
144
146
147
150
151
153
155
157
159
i6i
163
165
167
169
172
174
175
178
181
183
185
188
193
195
197
7
/{v)
0.05632
0.05738
0.05845
0.05955
0.06010
0.06066
0.06123
0.06180
0.06238
0.06296
0.06355
0.06414
0.06474
0.06534
0.06595
0.06657
0.06719
0,06782
0.06846
0.06910
0.06975
0.07040
0.07106
0.07173
0.07241
0.07309
0.07378
0.07447
0.07517
0.07588
0.07660
0.07732
0.07805
0.07879
0.07954
0.08029
Diff.
106
107
55
56
57
57
58
58
59
59
60
60
61
62
62
63
64
64
65
65
66
67
68
68
69
69
70
71
72
72
73
74
75
75
76
T{v)'
Diff.
1.804
1-831
1.858
27
27
27
1.885
1.899
14
14
1.913
14
1.927
14
1. 941
14
1-955
14
1.969
1.983
1.998
14
15
14
2.012
15
2.027
15
2.042
15
2.057
15
2.072
2.086
14
15
2.101
16
2. 117
15
2.132
15
2.147
2.162
15
16
2.J78
16
2.194
16
2.210
16
2.226
16
2.242
16
2.258
16
2.274
16
2.290
17
2.307
16
2.323
17
2-340
17
2-357
17
2.374
17
TABLE I. -Continued.
V
S{v)
Diff.
A {j^
Diff.
7(7.)
Diff.
r{v)
Diff.
1435
1430
1425
4704.8
1 4729-4
i 4754-1
246
247
247
149-39
151-39
153-42
200
203
205
0.08105
0.08182
0.08260
77
78
78
2.391
2.408
2.425
18
1420
I4I5
I4I0
1 4778.8
1 4803.6
, 4828.5
248
249
250
155-47
T57-55
159.66
208
211
214
0.08338
0.08418
0.08498
80
81
81
2.443
2.460
2.478
1
17
18
18
1405
1400
1395.
!
J 4853-5
i 4878.6
49P3-8
251
252
253
1 161.80
1 163.96
j 166.15
216
219
222
0.08579
0.08661
0.08744
82
83
84
2.496
2.514
2^-532
18
18
18
1390
1385
1380
4929-1
4954-5
j 4979-9
254
254
256
168.37
170.62
172.90
225
228
231
0.08828
0.08913
0.08999
85
86
87
2.550
2.568
2-587
18
19
18
1375
1370
'365
5005.5
■ 5031-1
5056.8
256
257
258
175.21
177-55
179.92
234
237
241
0.09086
0.09173
0.09262
87
89
89
2.605
2.624
2.643
19
19
19
1360
1355
1350
5082.6
! 5108.6
5134.6
260
260
261
182.33
184.76
187.23
243
247
250
0.09351
c. 09442
0-09533
91
91
93
2.662
2.681
2.700
19
19
.19
1345
1340
1335
5160.7
5186.9
5213-2
262 i
263'
263 ,
1
189.73
192.27
194.84
254
257
260
0.09626
0.09719
0.09813
94
94
95
2.719
2-739
2-758
20
•9
20
1330
1325
1320
5239-5
5265.8
5292.0
263!
262 j
106 ,
197.44
200.06
202.69
262
263
107
0.09908
0.10004
o.idioi
96 1
97
39
2-778
2.798
2.818
20
20
8
I3I8
I3I6
I3I4
5302.6
53^3-2
5323-8
106 :
106
107
203.76
204.84
205.92
108
108
109
0.10140
0.10179
0.10219
39 1
40 1
40'
2.826
2.834
2.842
8
8
8
I3I2
I3I0
.1308
5334-5
5345-2
5355-9
107
107
108
207.01
208.11
209.22
I 10
I r I
III!
1
0.10259
0.10299
0.10339
40
40
41
2.850
2.858
2.866
8
8
9
1306
1304
1302
5366.7
5377-5
108
108
109
210.33
211.45
•212.58
1
112
113
114
0.10380
0.10421
0.10462
41
41
41
2.875
2.883
2.892
8
9
8
1300
1298
1296
5399-2
5410.1
5421.0
109
109
no
213.72
214.87
216.02 1
115
115
117
0.10503
0.10544
0.10586
41
42
42
2.900
2.908
2.917
8
9
8
TABLE I.— Continued.
V
Six,)
1
Diff.
1
A{v)
Diff.
I{v)
Diff.
I
: T{v)
Diff.
1294
I 292
1290
5432.0
5443-0
5454.0
no
no
III
1
1 217.19
1 218.36
! 219.54
1
117
118
119
0.10628
0.10670
0.10713
42
43
43
1 2.925
2.934
1 2.942
9
8
8
1288
1286
T284
5465.1
5476.2
5487.3
III
III
112
220.73
221.93
223.13
120
120
122
0.10756
0.10799
0.10842
43
43
44
! 2.950
1 2.959
2.968
9
9
9
1282
1280
1278
549«-5
5509-7
5521.0
112
113
113;
224.35
225.57
226.80
122
123
124
0.10886
0.10930
0.10974
44
44
45
2.977
2.985
2.994
8
9
9
1276
1274
1272
5532.3
5543-6
5554-9
113 i
113!
114
228.04
229.29
230.54
125;
125
127 !
0.I10I9
0.11064
0.11109
45
45
45
3.003
3.012
3.021
9
9
9
1270
1268
1266
i 5566.3
5589-1
114
114
115
231.81
234.37
127
129
J29
0.11154
0.11200
0.11246
46
46
46
3-030
3-039
3.048
9
9
9
1264
J262
1260
5600.6
5612.1
5623.7
115
116
116!
1
235-66
236.97
238.28
13.1 1
131 1
132 1
0.11292
0.11338
O.I 1385
46
47
47
3-057
3.066
3-075
9
9
9
1258
1256
•1254
5635-3
5647.0
5658.6
117
116
117
239.60
240.94
242.28
134:
134 i
136!
0.11432
O.II479
0.11527
47
48
48
3.084
3-094
3-103
10
9
10
1252
1250
1248
5670.3
5682.1
5693-9
118
118
118
1
243-64
245.00
246.37
136 1
1371
139'
O.II575
O.I1623
0.11671
48
48
49
3-1^3
3.122
9
9
10
1246
1244
1242
5705-7
5717-6
5729-5
119
119
119
247.76
249-15
250-55
139;
140 i
142
O.II72O
O.II769
0.11819
49
50
50
3-141
3-150
3.160
9
10
9
1240
1238
1236
5741.4
5753-4
5765.4
120
120
121
251.97
253-39
254.83
142
144 j
144
0.11869
O.II919
0.11969
50
50
5^
3.169
3-179
3-189
10
10
9
1234
1232 ;
1230
5777-5
5789.6
5801.7
121
121
122
256.27
257.73
259.20
146!
147 1
148 ■
0.12020
O.T2071
0.12123
51
52
52
3-198
3.208
3.218
10
10
10
TABLE L— Continued.
V
S{v)
Diff. j
1228
1226
1224
5813.9
5826.1
5838.4
i
122
123
123!
1222
1220
I218
5850.7
5863.0
5875-4
123
124
124
I216
I 2 14
I2I2
5887.8
5900.3
1 5912.8
125 i
125 1
125
I2IO
1208
1206
5925.3
5937-9
1 5950.5
126
126
127
I 204
1202
1200
5963.2
5975-9
5988.6
127I
127
128
1
II98
II96
II94
6001.4
6014.2
6027.1
128
129
129
1
II92
I 1 90
I188
6040.0
6053.0
6066.0
130
130
131
I 186
I 184
I182
6079.1
6092.2
6105.3
131
131
132 !
1
I180
II78
II76
6118.5
6131-7
6145.0
132
II74
II72
II70
6158.3
6171.7
6185. I
134
134!
135 !
I168
I166
1 164
6198.6
6212. 1
6225.6
135
135
136'
A{v)
260.68
262.17
263.67
265.18
266.71
268.24
269.79
271-35
272.92
274.51
276.11
277.72
279.34
280.97
282.62
284.28
285.95
287.63
289.33
291.04
292.76
294.50
296.25
298.02
299.80
301.59
303-40
305.22
307.06
308.91
310.77
312.65
314-55
Diff.
i
149
150
151
I{v)
Diff.
r(z/)
O.I2I75
0.12227
0.12280
52
53
53
!
3.228
3-238
3-M8
'53
153
155
0.12333
0.12386
0.12439
53
53
54
3-258
3.268
3-278
156
\ 157
159
0.12493
0.12547
O.T2602
54
55
55
3.288
1 3-299
3-309
1 160
i 161
1 162
0.12657
O.I 27 I 2
0.12768
55
56
56
1 3-319
i 3-329
3-340
163
165
166
0.12824
0.12881
0.12938
57
57
57
3-350
3-361
3-371
167
168
170
0.12995
O.T3053
O.I3III
58
58
58
3-382
3-393
3.404
171
172
1.74
O.I3169
0.13228
0.13287
59
59
60
3.415
3.426
3.437
175
177
178
0.13347
0.13407
0.13467
60
60
61
3.448
3.459
3.470
179
181
182
0.13528
0.13589
0.13651
61
62
62
3.481
3.492
3.504
i
184
185
186
O.I3713
0.13776
0.13839
63
63
63
3.515
3-527
3-538 !
188
190
191
0.13902
0.13966
O.T403O
64
64
65
3-550
3-561
3-573'
Diff.
TABLE 1.— Continued.
S{v)
Diff.
162
160
159
158
157
156
1531
152
151
150
149
148
147
146
145
144
143
142
141
140
139
138
137
136
135
134
133
132
131
130
129
6239.2
6252.8
6259.7
6266.6
6273.4
6280.3
6287.2
6294.1
6301.0
6307.9
6314.8
6321.8
6328.8
6335-7
6342.7
63497
6356.7
63637
6370.7
6377.8
6384.8
6391.9
6399.0
6406. T
6413.2
6420.3
6427.4
6434.6
6441.7
6448.9
6456.1
6463.3
6470.4
136
69
69
68
69
69
69
69
69
69
70
70
69
70
70
70
70
70
71
70
71
71
7T
71
72
71
72
72
72
71
72
A {7')
316.46
318.39
31936
320.34
321.32
322.30
323.28
324.27
325.26
326.26
327.26
328.27
329.28
330.29
331-31
33^-33
333-3^
334-39
335-43
336.47
337-51
338.56
339-61
340.67
341.73
342.79
343-^6
344-94
346.02
347.10
348.19
349.28
350.38
)iff.
1(7')
1
Diff.
193
97
98
i
I
0.14095
O.I4160
O.I4I92
65
33
98
98
98
0.14225
0.14258
O.I429I
33
33
33
99
99
1 0.14324
1 0.14358
34
33 !
T(v)
Diff.
100
lOI
lOI
lOI
102
102
103
103
104
104
104
105
105
106
106
106
107
108
108
108
109
109
1 10
109
II
-.--too-!
10.14391
0.14425
0.14458
\ 0.14492
i 0.14526
1 0.14560
1 0.14594
i O.T4628
; 0.14662
0.14697
0.14731
0.14766
0.1 480 1
0.14836
0.14871
i 0.14906
0.14942
0.14977
0.15013
0.15049
0.15085
0.15121
^0.15157
io.15193
' 0.15229
34
33
34
34
34
34
34
34
35
34
35
35
35
35
35
36
35
36
36
36
36
36
36
36
3-584
3-596
3.602
3.608
3.614
3.62c
3.626
3-632
3-^3^
3-644
3-650
3656
3.662
3.668
3-674
3-680
3.686
3-693
3-699
3-705
3. 711
3-717
3-723
3-730
3-736
3-742
3-748
3-755
3-761
3-767
3-774
3-780
3.786
TABLE I.— Continued.
V
S{v)
Diff.
Ah)
Diff.
I{z)
Diff.
T (7')
Diff.
II28
1.27
II26
6477-6
6484.8
6492.1
72
73
72
351 47
352-57
353-68
no
III
111
0.15265
0.15302
0.15338
37
36
37
3-793
3-799
3.806
6
7
6
II25
II24
6499.3
6506.6
73
73
354-79
355-90
II I j
113
0.15375
O.I54I2
^7
37
3.812
3-818
6
7
II23
65139
73
357-03
113 1
0.15449
38 1
3-825
6
II22 i
II2I 1
II20
6521.2
6528.6
6536.0
74
74
74
358.16
35930
36045
114
115
115
0.15487
0.15524
15562
37 1
38;
38 1
3.831
3-838
3-844
7
6
7
III9
II18
I I 17
6543-4
6550-8
6558-3
74
75
75
361.60
362.76
36392
116
116
117
0.15600
0.15638
0.15676
38
38
39
3-851
3.858
3.864
7
6
7
II16
11 14
6565.8
^573-3
6580.8
75
75
76
365-09
366.28
367-47
119
119
120
0.15715
0.15754
0.15793
39
39
39
1 3.871
3.878
3-885
7
7
7
III3
II12
6588.4
6596.0
76
77
368.67
369.88
121
121
0.15832
0.15872
40
40
3.892
3.898
6
7
IIII
6603.7
77
37109
123
0.15912
40
3905
•
7
TIIO
6611.4
77
372.32
123
0.15952
41
3912
7
1 109
1108
6619. 1
6626.9
78
78
373-55
374-79
124
125
O.T5993
0.16033
40
41
3.919
3926
1
7
7
ITO7
I 106
1 105
6634.7
6642.5
6650.3
78
78
79
376.04
377-30
i 37857
126
127
128
0.16074
0.16115
0.16157
41
42
41
3-933
3940
3-947
7
7
8
1 104
I 103
II02
6658.2
6666.2
6674.1
80
79
80
379-85
381.14
382.44
129
130
131
0.16198
0.16240
0.16282
42
42
43
3-955
3-962
3-969
7
7
7
IIOI
I 100
1099
6682.1
6690.2
6698.3
81
81
81
j 383-75
38506
T31
132
0.16325
0.16367
0.16410
42
43
43
3-976
3-983
3-991
7
8
7
1098
1097
1096
6706.4
6714-5
6722.7
81
82
83
387-71
389.06
' 39041
135
135
137
0.16453
0.16497
0.16541
44
44
44
3-998
4.006
4013
8
7
8
TABLE I.— Continued.
V
S{v)
Diff.
A^zi)
Difif.
nv)
Diff.
T{v)
Diff.
1094 1
1093 i
6731.0
6739.2
6747-5
82
83
84
391-78
393-15
394-53
137
138
140
0.16585
0.16629
0.16674
44
45
45
4.021
4.029
4-036
8
7
8
1092
I09I
1090 1
6755-9
6764.3
6772.7
84
84
85
395-93
397.34
398.75
141
141
142
0.16719
0.16764
0.16810
45
46
46
4-044
4.051
4.059
7
8
8
1089
1088
1087
6781.2
6789.7
6798.2
85
85
86
400.17
401.60
403-05
143
145
145
0.16856
0.16902
0.16948
46.
46
47
4.067
4.075
4.083
8
8
8
1086
1085 j
1084
6806.8
6815.4
6824.1
86 1
87
87
1 404-50
405.97
407-45
147
148
149
0.16995
0.17042
0.17089
47
47
48
' 4.091
4.098
4.106
7
8
8
1083
T082
I08I
6832.8
6841.5
6850.3
87
88
88
408.94
410.44
411.95
150
151
152
0.17137
0.17185
0.17233
48
48
49
4.114
4.122
4.130
8
8
8
1080
1079
1078
6859.1
6867.9
6876.8
88
89
90
413-47
415.00
416.54
153
154
156
0.17282
0.17331
0.17380
49
49
49
4.138
4.146
4.155
8
9
8
1077
1076
1075
6885.8
6894.7
6903.7
89
90
91
418.10
419.66
421.24
156
158
159
0.17429
0.17479
0.17529
50
50
51
4.163
4.172
4.180
9
8
9
1074
ro73
1072
6912.8
6921.9
6931. 1
91
92
92
422.83
424.44
426.06
161
162
163
0.17580
0.17631
0.17682
51
51
51
4.189
4.197
4.206
8
9
8
1071
1070
1069
6940.3
6949.5
6958.8
92
93
93
427.69
429.33
430.98
164
165
166
0.17733
0.17785
0.17837
52
52
53
4.214
4.223
4.232
9
9
9
1068
1067
1066
6968.1
6977-5
6986.9
94
94
94
432.64
434.32
436.01
168
169
171
0.17890
0.17943
0.17996
53
53
53
4.241
4.250
i 4.259
9
9
9
1065
1064
1063
6996.3
7005.8
7015-4
95
96
96
437.72
439-44
441.17
172
173
175
0.18049
0.18103
0.18158
54
55
55
4.268
4.277
. 4.286
9
9
9
^3
TABLE I.— Continued.
V
S{v)
Diff.
A (v)
Diff.
I{v)
Diff.
1 T{v)
1
Diff.
1062
1061
1060
7025.0
7034.6
7044-3
96
97
97
442.92
444-68
446.45
176
177
178
0.18213
0.18268
0.18323
55
55
56
1
1 4-295
i 4-304
4-313
1
1
9
9
i 9
1059
1058
1057
7054.0
7063.8
7073.6
98
98
99
448.23
450-03
451.84
180
181
182
0.18379
0-18435
0.1 849 1
56
56
57
'] 4-322
4-332
4.341
10
9
9
1056
1055
1054
1 7083.5
7093-4
7103.4
99
100
100
453.66
455.50
457-36
184
186
187
0.18548
0.18605
0.18663
57
58
58
4-350
4-360
4-369
10
9
9
1053
1052
105 1
7113-4
7123.4
7133-5
100
lOI
I02
459.23
461.12
463.02
189
190
192
O.18721
0.18779
0.18838
58
59
59
4.378
4.387
4.397
9
10
9
1050
I049
1048
7143.7
7153-9
7164.1
102
I02
103
464-94
466.87
468.81
193
194
196
0.18897
0.18956
O.I 90 1 6
59
60
61
4.406
4.416
4.426
10
10
10
1047
1046
1045
7174-4
7184.7
7195-I
I03
104
105
470-77
472-74
474-73
197
199
201
0.19077
0.19138
0.19199
61
61
61
4.436
4.446
4-455
10
9
10
1044
1043
1042
7205.6
7216.1
7226.6
105
105
106
476.74
478.77
480.81
203
204
206
0.19260
0.19322
0.19385
62
63
4-465
4-475
4-485
10
10
10
1041
1040
1039
7237.2
7247.9
7258.6
107
107
107
482.87
484.95
487.04
208
209
211
0.19448
0.19511
0-19575
^3
64
64
4-495
4-505
4-516
10
II
10
1038
1037
1036
7269.3
7280.1
7291.0
108
109
109
489-15
491.28
493.42
213
214
216
0-19639
0.19703
0.19768
64
65
66
4-526
4.537
4.547
II
10
II
1035
1034
7301.9
7312.9
73239
no
no
III
495.58
497.76
499-95
218
219
222
0.19834
0.19900
0.19966
66
66
67
4.558
4-569
4-579
II
10
II
1032
1031
1030
7335.0
7346.1
7357.3
III
112
112
502.17
504.40
506.65 '
223
225
226 1
0.20033
0.20100
0.20168
67
68
68
4-590
4.600
4.611
10
II
II
14
TABLE I.— Continued.
V
S{v)
Diff.
A{v)
Diff.
I{v)
Diff.
T{.v)
Diff.
1029
1028
1027
7368.5
7379-8
739I-I
113
113
114
i
508.91
511.20
513-50
229
230
232
1
0.20236
0.20305
0.20374
69
69
69
4.622
4-633
4.645
II
12
II
1026
1025
1024
7402.5
7414.0
7425-5
115
115
116
515-82
518.17
520.54
235
237
238
0.20443
0.20513
0.20584
70
71
71
4.656
4.667
4.678
II
II
II
1023
1022
I02I
7437-1
7448.7
7460.4
116
117
117
522.92
525-32
527-75
240,
243
245
0.20655
0.20726
0.20798
71
72
73
4.689
4.701
4.712
12
II
II
1020
IOI9
IO18
7472.1
7483.9
7495.7
118
118
ii9|
530.20
532.66
535-14
!
246
248
251
0.20871
0.20944
O.21017
73
73
74
4.723
4.735
4-747
12
12
12
ICI7
IO16
IOI5
7507.6
7519-6
7531-6
120
120
121
537-65
540.17
542.72
252
255
258
O.21091
0.2 1 165
0.21240
1
74
75
76
4.759
4.771
4-782
12
II
12
IOI4
IOI3
IOI2
7543-7
7555-8
7568.0
121
122
123
545-30
547-89
550.51
259
262
265
O.21316
0.21392
0.21468
76
76
77
4.794
4.806
4.818
12
12
12
lOII
lOIO
1009
7580.3
7592.6
7605.0
123
124
124
553-i6
555-82
558-51
266
269
272
0.21545
0.21623
O.21701
78
78
79
4.830
4.842
4.855
12
13
12
1008
1007
1006
7617.4
7629.9
7642.5
125
126
1.6
561.23
563.96
566.71
273
275
278
0.21780
0.21859
0.21939
79
80
80
4.867
4.880
4.892
13
12
13
1005
1004
1003
7655-1
7667.8
7680.6
127
128
128
569-49
572.29
575.11
280
282
285
0.22019
0.22100
0.22182
81
82
82
4.905
4.918
4-930
13
12
13
1002
lOOI
1000
7693-4
7706.3
7719-3
129
130
131
577-96
580.83
583-72
287
289
292
0.22264
0.22347
0.22430
83
83
84
4.943
4.955
4.968
12
13
13
999
998
997
7732.4
7745-6
7758.8
132
132
133
586.64
589-59
592.56
295
297
300
0.22514
0.22599
0.22684
85
85
86
4.981
4.995
5.008
14
13
14
15
TABLE I.— Continued.
V
S\v)
Diff.
A{t^
Diff.
/(zO
Diff.
T{v)
Diff.
996
995
994
7772.1
7785.4
7798.7
134
595.56
598.59
601.65
303
306
Z^9
0.22770
0.22857
0.22944
87
87
87
5.022
5.035
5.048
T3
13
14
993
992
991
7812. 1
7825.5
7839.0
134
135
135
604.74
607.85
610.99
311
314
317
0.23031
0.23118
0.23206
87
88
89
5.062
5-075
5.089
13
14
13
990
989
988
7852.5
7866.1
7879.7
136
136
137
614.16
617.33
620.52
317
319
321
0.23295
0.23384
0.23474
89
90
90
5.102
5. 116
5-130
14
14
14
987
986
985
7893.4
7907.1
7920.8
137
^37
137
623.73
626.96
630.21
323
325
327
0.23564
0.23655
0.23746
91
91
91
5-144
5.158
5-171
14
13
14
984
983
982
7934-5
7948.3
7962.1
138
138
138
633-48
636.77
640.08
329
0.23837
0.23929
0.24021
92
92
92
5.185
5.199
5.213
14
14
14
981
980
979
7975-9
7989.8
8003.7
139
139
139
643-41
646.76
650.12
335
339
0.24T13
0.24206
0.24299
93
93
93
5.227
5-241
5-255
14
14
15
978
977
976
8017.6
8031.5
8045.5
139
140
140
653-51
656.92
660.35
341
343
345
0.24392
0.24486
0.24580
94
94
95
5.270
5.284
5-299
14
15
14
975
974
973
8059.5
8073-5
8087.6
140
141
141
663.80
667.26
670.75
346
349
351
0.24675
0.24770
0.24865
95
95
96
5-3^3
5-327
5.342
14
15
14
972
971
970
8101.7
8115.8
8129.9
141
141
142
674.26
677.80
681.35
354
355
357
0.24961
0.25057
0.25154
96
97
97
5.356
5.371
5-385
15
14
15
969
968
967
8144. 1
8158.3
8172.5
142
142
143
684.92
688.51
692.12
359
361
363 1
0.25251
0.25348
0.25446
97
98
98
5.400
5.415
5-429
15
14
15
966
965
964
8i86.8
8201. 1
8215.4
143
143
144
695.75
699.41
703-09
366
368
370
0.25544
0.25643
0.25742
99
99
99
5-444
5-459
5-474
15
15
, 15
16
TABLE 1.— Continued.
V
i
Diff.
A(v)
Diff.
/{v)
Diff.
T(v)
Diff.
963
962
961
: 8229.8
8244.2
8258.6
144
144
144
706.79
710.51
714.26
372
375
377
0.25841
0.25941
0.26041
100
100
lOI
5-489
5-503
j 5-518
1
14
15
15
960
959
958
1
8273.0
\ 8287.4
i 8301.9
144
145
145
718.03
721.81
1 725.62
378
381
384
0.26142
0.26243
0.26344
lOI
101
102
5-533
' 5-548
1 5-564
15
16
15
957
956
955
8316.4
! 8331.0
! 8345.6
146
146
146
729.46
1 733.32
737.20
386
3SS
390
0.26446
0.26549
0.26652
103
103
103
5-579
5-594
5.609
15
15
16
954
953
952
i 8360.2
8374.8
i 8389.5
146
147
J47
741.10
745-03
748.98
393
395
398
0.26755
0.26858
0.26962
103
T04
105
5-625
5-640
5-655
15
15
16
951
950
949
8404.2
i 8419.0
8433.8
148
T48
148
752.96
756.96
760.98
400
402
404
0,27067
0.27172
0.27277
105
105
106
5-671
5.686
5-702
T5
16
948
947
946
8448.6
1 8463.4
, 8478.3
148
149
149
765.02
769.09
773.18
407
409
412
0.27383
0.27489
0.27596
106
107
107
5-718
■5-733
5-749
15
16
16
945
944
943
8493.2
8508.1
! 8523.1
149
150
150
777-30
781.45
785.62
415
417
420
0.27703
0.278II
0.27919
108
108
108
5-765
5-781
5-797
16
16
15
942
941
940
1 8538.1
\ 8553.1
\ 8568.2
1
150
151
151
789.82
794.04
798.29
422
425
427
0.28027
0.28136
0.28246
109
no
no
5-812
5-828
5-844
16
16
16
939
938
937
8583.3
8598.4
8613.6
151
152
152
802.56
806.85
811. 17
429
432 1
435
0.28356
0.28467
0.28578
III
III
III
5.860
5-877
5-893
17
16
16
936
935
934
8628.8
8644.0
8659.2
152
152
153
815-52
819.89
824.30
437
441
443
0.28689
0.28801
0.28913
112
112
113
5-909
5-926
5-942
17
16
16
933
932
931
8674.5
8689.8
8705.2
153
^54
154
. 828.73
837-67
445
449
451
0.29026
0.29140
0.29254
114
114
114
5-958
5-974
•5-991
16
17
16
T7
TABLE I.— Continued.
V
S{v)
Diff.
A{v)
Diff.
/(Z')
Diff.
Tiv)
Diff.
930
929
928
8720.6
8736.0
8751.5
154
155
155
842.18
846.71
851.27
453
456
459
0.29368
0.29483
0.29598
^15
115
116
6.007
6.024
6.041
17
17
16
927
926
925
8767.0
8782.5
8798.0
155
155
156
855.86
860.48
865.13
462 1
465
468
0.29714
0.29830
0.29947
116
117
117
6.057
6.074
6.091
17
17
17
924
923
922
8813.6
8829.2
8844.9
156
157
157
869.81
874-51
879-25
470
474
477
0.30064
0.30T82
30300
118
118
119
6.108
6.125
6. 141
17
16
17
921
920
919
8860.6
8876.3
8892.0
157
157
158
884.02
888.81
479 1
4821
485 j
0.30419
0.30538
0.30658
119
120
120
6.158
6.175
6.192
17
17
18
918
917
916
8907.8
8923.7
8939-5
159
158
159
898.48
903.36
908.27
488
491
494
0.30778
0.30899
0.31020
121
121
122
6.210
6.227
6.245
17
18
17
915
914
913
8955-4
8971-3
8987-3
159
160
160
913.21
918.18
923.19
497
501
503
0.3II42
0.31264
0.31387
122
123
124
6.262
6.279
6.297
17
18
17
912
911
910
9003-3
9019.3
9035-4
160
161
161
928.22
933-28
938.37
506
509
513
0.3I5II
0.31635
0.31760
124
125
125
6.314
6.332
6.349
18
17
18
909
908
907
9051-5
9067.6
9083.8
161
162
162
943-50
948.65
953-84
515
519
522
0.31885
0.320II
0.32137
126
126
127
6.367
6.385
6.403
18
18
18
906
905
904
9100.0
9116.2
9132.5
162
163
163
959.06
964.31
969.60
525
529
532
0.32264
0.32392
0.32520
128
128
129
6.421
6.439
6.457
18
18
18
903
902
901
9148.8
9165.2
9181.6
.64
164
164
974-92
980.27
985-65
535
538
541
0.32649
0.32778
0.32908
129
130
130
6.475
6.493
6. 511
18
18
18
900
899
898
9198.0
9214-5
9231.0
165
165
165
991.06
996.51
1001.99
545
548
552
0.33038
0.33169
0.33300
131
131
132
6.529
6.548
6.566
19
18
19
iS
TABLE I.— Continued.
S(v)
9247-5
9264.1
9280.7
9297.3
9314-0
9330-7
9347.5
9364.3
9381. 1
9398.0
9414.9
9431-9
9448.9
9465.9
9483.0
9500.1
9517.2
9534.4
9551.6
9568.9
9586.2
9603.5
9620.9
9638.3
9655.8
9673.3
9690.8
9708.4
9726.0
9743.7
9761.4
9779.T
9796.9
Diff.
166
166
166
167
167
168
168
168
169
169
170
170
170
171
171
171
172
172
173
173
173
174
174
175
175
175
176
176
177
177
177
178
178
A{v)
007.51
013.06
018.65
024.27
029.92
035-61
041.34
047.10
052.90
058.73
064.60
070.52
076.47
082.45
088.47
094-53
100.62
106.75
112.92
119.13
125-38
131.67
138.00
144-37
150.78
157-23
163.72
170.25
176.82
183.44
190.09
196.79
203.54
Diff.
555
559
562
565
569
573
576
580
583
587
592
595
598
602
606
609
613
617
621
625
629
^33
637
641
645
649
653
657
662
665
670
675
678
I{v)
0.33432
0.33565
0.33698
0.33832
0.33966
0.34101
0.34237
0.34373
0.34510
0.34647
0.34785
0.34924
0.35063
0.35203
0.35344
0.35485
0.35627
0-35770
0.35913
0.36057
0.36202
0.36347
0.36493
0.36639
0.36786
0.36934
0.37083
0.37232
0.37382
0.37532
0.37683
0.37835
0.37988
Diff.
133
^33
134
134
135
136
136
137
137
138
139
139
140
141
141
142
143
143
144
145
145
146
146
147
148
149
149
150
150
151
152
^53
153
T{v)
6.585
6.603
6.622
6.640
6.659
6.677
6.696
6.714
6.733
6.753
6.772
6.791
6.811
6.830
6.849
6.868
6.888
6.907
6.927
6.947
6.966
6.986
7.006
7.026
7.046
7-065
7.085
7.105
7.126
7.146
7.167
7.187
7.208
Diff.
19
TABLE I.— Continued.
864
863
862
861
860
859
858
857
856
855
854
853
852
851
850
849
848
847
846
845
844
843
842
841
840
839
838
837
836
835
834
832
Siv)
9814.7
9832.6
9850-5
•4
9886.4
9904.4
9922.5
9940.6
9958.7
9976.9
9995-2
10013.5
10031.8
10050.2
10068.6
10087. 1
10105.6
10124.1
10142.7
10161.3
10180.0
10198.8
10217.5
T0236.3
10255.2
10274.1
10293.0
10312.0
10331.0
10350.1
10369.2
10388.4
10407.6
Diff.
179
179
1791
I
180
180
181
181
181
182
183
183
183
184
184
185
185
185
186
186
187
188
187
188
189
189
189
190
190
191
191
192
192
193
A{v)
210.32
217.15
224.02
230.93
237.89
244.89
251-94
259.04
266.18
273.36
280.59
287.87
295-19
302.56
309-98
317-44
324.96
332.52
340.13
347.79
355-50
363-26
371.07
378.93
386.84
394.80
402.82
410.89
419.01
427.18
435-41
443.69
452.02
Diff.
683
687
691
696
700
705
710
714
718
723
728
732
737
742
746
752
756
761
766
771
776
781
786
791
796
802
807
812
817
823
828
^33
839
I{v)
I0.38I4T
: 0.38295
; 0.38450
0.38606
0.38762
0.38919
0.39077
0.39235
0.39394
0.39554
0.39715
0.39877
0.40039
0.40202
0.40366
0.40530
0.40695
0.40861
0.41028
0.41196
0.41364
0.41533
0.41703
0.41874
0.42046
0.42218
0.42392
0.42566
0.42741
0.42917
0.43093
0.43271
0.43449
Diff.
154
155
156
156
157
158
158
159
160
161
162
162
163
164
164
165
166
167
168
168
169
170
171
172
172
174
174
175
176
176
178
178
180
T(v)
7.229
7.249
7.270
7.290
7-311
7-332
7-354
7-375
7-396
7.418
7-439
7.460
7.481
7.503
7-524
7-546
7-568
7-590
7.612
7-635
7-657
7-679
7.701
7-723
7-745
7-768
7-790
7-813
7.836
7-858
7.881
7-904
7.928
TABLE I.— Continued.
S{v)
Diff.
0426.9
0446.2
0465.6
[0485.0
[0504.4
0523-9
0543-4
0563.0
0582.7
[ 0602. 4
0622,1
[0641.9
0661.7
0681.6
0701.6
0721.6
0741.6
0761.7
:o78i.8
:o8o2.o
:o822.2
:o842.5
0862.8
: 0883. 2
10903.6
0924.1
0944.6
0965.2
0985.8
1006.5
1027.2
1048.0
1068.8
193
T94
194
194
195
195
196
197
197
197
198
198
199
200
200
200
201
201
202
202
203
203
204
204
205
205
206
206
207
207
208
208
209
Aiv)
460.41
468.85
477.35
485.90
494.51
503-18
511.90
520.69
529-52
538.42
547-38
556.39
565-47
574.61
583-80
593-05
602.37
611.75
621.20
630.70
640.27
649.90
659-60
669.36
679.19
689.08
699.04
709.07
719.16
729.32
739-55
749-84
760.21
Diff.
844
850
855
861
867
872
879
896
901
908
914
919
925
932
938
945
950
957
963
970
976
983
989
996
1003
1009
1016
1023
1029
1037
1043
/{v)
0.43629
0.43809
0.43990
0.44172
0.44354
0.44538
0.44722
0.44908
C.45094
0.45282
0.45470
0.45659
0.45849
0.46040
0.46231
0.46424
0.46618
0.46812
0.47008
0.47205
0.47402
0.47601
0.47800
0.48001
0.48202
0.48404
0.48608
0.48812
0.49018
0.49225
0.49432
0.49641
0.49850
Diff.
180
181
182
182
184
184
186
186
188
189
190
191
191
T93
194
194
196
197
197
199
199
201
201
202
204
204
206
207
207
209
209
211
T(v)
7-95T
7-974
7-997
8.021
8.044
8.068
8.091
8.115
8.139
8.163
8.187
8.211
8.235
8.259
8.284
8.308
^'333
8.357
8.382
8.407
8.432
8.457
8.482
8.507
8.533
8.558
8.584
8.610
8-635
8.661
8.687
8.713
8.739
Diff.
TABLE I.-rCONTINUED.
Siv)
798
797
796
795
794
793
792
791
790
789
788
787
786
785
784
783
782
781
780
779
778
777
776
775
774
773
772
771
770
769
768
767
766
1089.7
1 1 10. 7
1131.7
1152.7
1173.8
1195.0
1216.2
1237.5
1258.8
1280.3
1301.8
1323-4
1345-0
1366.6
1388.2
1409.8
1431-5
1453-3
I475-0
1496.8
1518.6
1540.4
1562.2-
1584.1
1606.0
1627.9
1649.9
1671.9
1693.9
1716.0
1738.0
1760. 1
1782.3
Diff.
210
210
210
211
212
212
213
213
215
215
216
216
216
216
216
217
218
217
218
218
218
218
219
219
219
220
220
220
221
220
221
222
222
A (7')
1770.64
1781.15
1791.72
1802.37
1813.10
1823.89
1834.76
1845.70
1856.71
1867.87
1879.08
1890.36
1901.70
1913.1i
1924.57
1936.10
1947.70
1959.36
1971.08
1982.87
1994.72
2006.64
2018.62
2030.68
2042.80
2054.98
2067.24
2079.56
2091.95
2104.41
2116.94
2129.54
2142.21
Diff.
051
057
065
073
079
087
094
lOI
116
121
128
134
141
146
153
160
166
172
179
185
192
198 j
206 i
212 I
1
2l8|
226 I
232 1
239
246
253
260
267
274
7(7.)
I 0.50061
10.50273
I 0.50486
! 0.50700
0-50915
0-51131
0.51348
0.51566
0-5
786
0.52008
0.52231
0.52454
0.52678
0.52904
0.53130
0.53357
0.53585
0.53813
0.54043
0.54273
0.54504
0.54736
0.54969
0.55203
0.55438
0.55674
0.55911
0.56148
0.56387
0.56626
0.56867
0.57108
0.57350
Diff.
212
213
214
215
216
217
218
220
222
223
223
224
226
226
227
228
228
230
230
231
232
233
234
235
236
237
237
239 I
239
241
241
242
244
T(v)
8.765
8.791
8.818
8.844
8.871
8.897
8.924
8.951
8.97.8
9.005
9.032
9.060
9.087
9.114
9.T42
9.170
9.197
9.225 !
9-2531
9.281 !
9-309 I
9-337 I
9365 i
9-394 i
9.422 j
9-450
9-479
9-507
9-536
9-565
9-593
9.622
9-651
TABLE I.— Continued.
S{v)
765
764
763
762
761
760
759
758
757
756
755
754
753
752
751
750
749
748
747
746
745
744
743
742
741
740
739
738
737
736
735
734
733
1804.5
1826.7
1848.9
1871.1
1893.4
1915-7
1938.0
1960.4
2005.3
2027.7
2050.2
2072.8
2095-3
2117.9
2140.5
2163. 1
2185.8
2208.5
2231. 2
2253-9
2276.7
2299.6
2322.4
2345-3
2368.2
2391-1
2414.1
2437-1
2460.1
Diff.
222
222
222
223
223
223
224
224
225
224
225
226
225
226
226
226
227
227
227
227
228
229
228
229
229
229
230
230
230
231
2^1
2483.2
2506.31 231
2529.4' 232
A{v)
Diff.
2154-95
2167.76
2180.64
2193-59
2206.62
2219.7 r
2232.88
2246.12
2259.44
2272.83
2286.30
2299.84
2313-45
2327.14
2340.91
2354-75
2368.67
2382.66
2396.74
2410.89
2425.12
2439-44
2453-83
2468.30
2482.86
2497-49
2512.21
2527.01
2541.89
2556.86
2571.91
2587.04
2602.25
281
288
295
303
309
3^7
324
332
339
347
354
361
369
377
384
392
399
408
415
423
432
439
447
456
463
472
/(v)
480 0.64271
488
497
505
513
521
530
0-57594
0.57838
0.58083
0.58330
0.58577
0-58825
0.59074
0.59324
0-59575
0.59827
0.60080
0.60334
0.60589
0.60845
0.61 103
0.61361
0.61620
0.61880
0.62142
0,62404
0.62667
0.62932
0.63198
0.63464
0.63732
0.64001
0.64542
0164814
0.65087
0.65361
0.65637
0.65913
Diff.
244
245
247
247
248
249
250
251
252
253
254
255
256
258
258
259
260
262
262
263
265
266
266
268
269
270
271
272
273
274
276
276
278
T{v)
9.680
9.709
9-738
9.767
9-797
9.826
9-855
9.885
5.914
9-944
9-973
10.003
10.033
10.063
10.093
10.123
10.153
10.184
10.214
10.244
10.275
10.306
10.336
10.367
10.398
10.429
10.460
10.491
10.522
10.554
10.585
10.616
10.648
Diff.
23
TABLE I.— Continued.
S(v)
732
731
730
729
728
727
726
725
724
723
722
721
720
719
718
717
716
714
713.
712
711
710
709
708
707
706
705
704
703
702
701
700
2552.6
2575-8
2599.0
2622.3
2645.6
2668.9
2692.3
2715.6
2739.0
2762.5
2786.0
2809.5
2833.1
2856.7
2880.3
2903.9
2927,6
2951-3
2975-1
2998.9
3022.7
3046.5
3070.4
3094-3
3118.3
3142.3
3166.3
3190.3
3214.4
3238.5
3262.7
3286.9
3311.I
Diff.
232
232
233
233
233
234
233
234
235
235
235
236
236
236
236
237
237
238
238
238
238
239
239
240
240
240
240
241
241
242
242
242
242
A{v)
2617.55
2632.94
2648.41
2663.97
2679.61
2695.34
2711.16
2727.07
2743-07
2759.16
2775-33
2791.60
2807.96
2824.41
2840.96
2857.60
2874.33
2891.15
2908.07
2925.08
2942.19
2959-39
2976.09
2994.09
3011.58
3029.17
3046.86
3064,66
3082.55
3100.54
3118.64
3136.84
3155-H
Diff,
539
547
556
564
573
582
591
600
609
617
627 ;
636 I
645
6551
664
673
682
692
701
711
720
730
740
749
759
769
780
789
799
810
820
830
841
/{v)
0.66I9I
0.66470
0.66750
0.67031
0.67313
0.67596
0.67881
0,68167
0.68454
0.68742
0,69031
0.69322
0.69614
0.69907
0.70201
0,70496
0.70793
0,71091
0.71390
0.7I69I
0.71993
0.72296
0.72600
0.72905
0.73212
0.73520
0.73830
0.74I4I
0.74453
0.74766
0.75081
0.75397
0.75715
Diff,
279
280
281
282
283
285
286
287
289
291
292
293
294
295
297
298
299
301
302
303
304
305
307
308
310
311
312
313
315
316
318
319
T{v)
0.679
0.711
0.743
0.775
:o.8o7
0.839
0.871
0.903
0.936
:o.968
1. 00 1
1-033
1,066
1.099
1-132
1. 165
1. 198
1.231
1.264
1.297
1-330
1.364
1.398
1-432
1.465
1.499
1-533
1.567
1.60T
1.636
1,670
1,704
1-739
24
TABLE I.— Continued.
S{v)
699
698
697
696
695
694
693
692
691
690
689
688
687
686
685
684
683
682
681
680
679
678
677
676
675
674
673
672
67.
670
669
668
667
3335-3
3359-6
3383-9
3408.3
3432.7
3457-1
3481.6
3506.1
3530.6
3555-2
3579-8
3604.4
3629.1
3653-8
3678.6
3703-4
3728.2
3753-1
3778.0
3802.9
3827.9
3852.9
3877-9
39030
3928.1
3953-3
3978.5
4003.7
4029.0
4054-3
4079.6
4105.0
4130.4
Diff.
243
243
244
244
244
245
245
245
246
246
246
247
247
248
248
248
249
249
249
250
250
250
25'
251
252
252
252
253
253
253
254
254
255
A (v)
3173-55
3192.06
3210.67
3229.39
3248.22
3267.15
3286.19
3305.33
3324.58
3343-95
3363-42
3383-00
3402.70
3422.50
3442.42
3462.45
3482.60
3502.86
3523-24
3543-73
3564.34
3585.07
3605.91
3626.88
3647.96
3669.17
3690.50
3711 94
373351
3755.21
3777.03
3798.98
3821.05
Diff.
1 85 I
1861
1872
1883
1893
1904
1914
1925
1937
1947
1958
1970
1980
1992
2003
2015
2026
2038
2049 !
2061 I
2073 I
1
2084
2097
2108
2121
2133
2144
2157
2170
2182
2195
2207
2219
I{v)
0.76034
0.76354
0.76675
0.76998
0.77322
0.77648
0.77975
0.78304
0.78634
0.78966
0.79299
0.79633
0.79969
0.80306
0.80645
0.80985
0.81327
0.81670
0.82015
0.82362
0.82710
0.83059
0.83410
0.83762
O.84116
0.84472
0.84829
0.85188
0.85549
O.85911
0.86274
0.86639
0.87006
Diff.
320
321
3 3
324
326
327
329
330
332
333
334
336
337
339
340
342
343
345
347
348
349
351
352
354
356
357
359
361
362
363
365
367
369
T{v)
11.774
11.809
11.844
11.879
II. 914
11.949
11.984
12.020
12.055
12.091
12.126
12.162
12.198
12.234
12.270
12.306
12.342
12.379
12.415
12.452
12.489
12.526
12.563
12.600
12.637
12.675
12.712
12.750
12.787
12.825
12.863
12.901
12.939
25
TABLE I.— Continued.
Siv)
666
665
664
663
662
661
660
659
658
657
656
655
654
653
652
65'
650
649
648
647
646
645
644
643
642
641
640
639
638
637
636
635
634
4155-9
4181.4
4206.9
4232.5
4258.1
4283.7
4309-4
4335.1
4360.9
4386.7
4412.6
4438.5
4464.4
4490.4
4516.4
4542.4
4568.5
4594-6
4620.8
4647.0
4673.2
4699.5
4725-9
4752.3
4778.7
4805.1
4831.6
4858.1
4884.7
4911.3
4938.0
4964.7
4991.4
Diff.
255
255
256
256
256
257
257
258
258
j
259
259
259
260
260
260
261
261
262
262
262
263
264.
264
264
264
265
265
266
266
267
267
267
268
A {7^)
3843-24
3865.57
3888.02
3910.60
3933-31
3956.16
3979-13
4002.24
4025.48
4048.86
4072.37
4096.01
4T19.79
4143-71
4167.77
Diff.
4340.12
4365-32
4390.67
4416.16
4441.81
4467.60
4493-55
4519.64
4545-89
4572.30
4598.86
4625.57
2233
2245
2258
2271
2285
2297
2311
2324
2338
2351
2364
2378
2392
2406
2419
4191.96 2434
4216.30 2448
4240.78 2462
4265.40 j 2476
4290.16 I 2491
4315.07 2505
2520
2535
2549
2565
2579
2595
2609
2625
2641
2656
2671
2687
/{v)
0-87375
0.87745
0.88II7
0.88490
0.88866
0.89243
0.89622
0.90002
0.90384
0.90768
0.9II53
0.9I54I
0.91930
0,92321
0.92715
0.931 10
0.93506
0.93904
0.94304
0.94706
0.95IIO
0.95516
0.95923
0.96333
0.96745
0.97158
0.97574
0.97991
0,98410
0.98831
0.99254
0.99680
1.00107
Diff.
370
372
373
376
377
379
380
382
384
385
388
389
391
394
395
396
398
400
402
404
406
407
410
412
413
416
417
419
421
423
426
427
429
T{v)
977
015
053
092
130
169
208
247
286
326
365
404
444
484
524
564
604
644
684
725
766
806
847
929
971
012
053
095
137
4 179
4.221
4-263
Diff.
26
TABLE I.— Continued.
V
S{zi)
Diff.
i
A (7-)
Diff.
I{v)
Diff.
T(v)
Diff.
633
632
63.
15018.2
i5C'45-o
15071.9
268
269
269
4652.44
4679.47
4706.65
2703
2718 '
2735
1.00536
1.00967
1. 01401
431
434
436
14.305
14.348
14.390
43
42
43
6.'?o
629
628
15098.8
15125.8
15152-8
270
2701
270 i
4734.00
4761.51
4789.18
2751
2767
2784
1. 01837
1.02274
I.027I3
437
439
442 i
14.433
14.476
14.519
43
43
43
627
626
625
15179-8
15206.9
15234.0
271
271
272
4817.02
4845.02
4873.18
2800
2816
2833
I 03155
J. 03598
1.04044
443
446
448
14.562
14.605
14.648
43
43
44
624
623
622
15261.2
15288.4
15315-7
272
273
273
4901.51
4930.00
4958.67
2849
2S67
2883
1.04492
1.04943
1.05395
451
452
455
14.692
14.735
14.779
43
44
44
621
620
619
15343.0
15370.3
15397.7
273
274
274
4987.50
1 5016.51
5045-69
2901
2918
2935
1 1.05850
1.06307
1.06766
457
459
461
14.823
14.867
14.91 1
44
44
45
618
617
616
15425. 1
15452.6
15480.1
275
275
276
5075-04
5104.57
5134-27
2953
2970
2988
1.07227
1.07690
1. 08156
463
466
468
,14.956
15.000
15.045
44
45
45
615
614
613
15507.7
15535.3
15563.0
276
277
277
5164.15
5194.21
5224.44
3006
3023
3042
1.08624
1.09095
1.09568
471
473
475
15.090
15-135
15.180
45
45
45
612
611
610
T5590.7
15618.4
15646.2
277
1 278
278
5254-86
i 5285.46
5316.24
3060
3078
3097
1 10043
1. 10520
I.IIOOO
477
48c
482
15-225
15.270
T5-316
45
46
45
609
608
607
1 15674.0
15701.9
15729.8
279
279
280
5347-21
5378.36
5409-71
31^5
3135
3153
1 I.II482
I.II966
1. 12452
484
486
489
15-361
15-407
15.453
46
46
46
606
605
604
15757-8
15785.8
15813-9
280
i 281
1 281
1
! 5441.24
5472.95
5504.86
3171
3191
3210
I.I294I
1. 13433
1 1.13927
492
494
497
■
15-499
15-546
15.592
47
46
46
603
602
601
15842.0
15870. 1
15898.3
281
282
' 283
1553696
i 556926
'5601.75
3230
3249
3268
1. 14424
1. 14923
1-15425
499
502
504
15.638
15.685
15.732
47
47
47
27
TABLE I.— Continued.
V
S{v)
Diff. I
1
1
A (v) Diff.
I{v)
Diff.
T{v)
Diff.
600
599
598
15926.6
15954.9
15983-2
283
283
284
5634.43 I 3288
5667.31 J3309
5700.40 3329
1. 15929
1. 16435
1.16944
506
509
5'2
15.779
15.826
15.873
47
47
48
597
596
595
160IT.6
1 6040. 1
16068.6
285
285
285 1
5733-69
5767.18
5800.87
3349
3369
3389
1. 17456
1. 17970
1. 18487
5H
517
519
15.921
15.968
16.016
47
48
48
594
593
592
16097. 1
16125.7
16154-4
286
287
287
5834-76
5868.85
5903.16
3409
3431
3451
1. 19006
1. 19528
1.20053
522
525
527
16.064
16. 113
16. 16 I
49
48
48
591
590
589
16183.I
16211.8
16240.6
287 1
288
288
5937-67
5972.39
6007.32
3472
3493
3515
T. 20580
I.2IIIO
1. 21643
530
533;
535 ,
16.209
16.258
16.307
49
49
49
588
587
586
16269.4
16298.3
16327.2
289
289
290
6042.47
6077.83
6113.41
3536
3558
3579
I. 22178
I. 22716
1.23257
538
541
544
16.356
16.405
16.454
49
49
50
585
584
583
16356.2
16385.2
16414.3
290
291
291
6149.20
6185.22
6221.46
3602
3624
3646
I. 23801
1.24348
1.24897
547
549
552
16.504
16.553
16.603
49
50
50
582
581
580
16443.4
16472.6
16501.8
292
292
293
6257.92
6294.61
6331-52
3669
3691
3714
1.25449
1.26004
1.26562
555
558
561
16.653
16.704
16.754
51
50
51
579
578
577
16531.1
16560.4
16589.8
293
294
294
6368.66
6406.01
6443-63
3735
3762
3783
1. 27123
1.27687
1.28253
564
566
570
16.805
16.855
16.906
50
51
52
576
575
574
16619.2
16648.7
16678.2
295
295
296
6481.46
65^9.52
6557.82
3806
3830
3854
1.28823
1.29396
I. 29971
573
575
579
16.958
17.009
17.060
51
51
52
573
572
571
16707.8
16737.4
16767. 1
296
297
298
6596.36
6635.14
6674.16
3878
3902
3926
1.30550
I.3II3'
I.31716
581
585
588
17.112
17.164
i 17.216
52
52
52
570
569
568
16796.9
16826.7
16856.6
298
299
299
671342
6752.93
6792.68
3951
3975
4000
1.32304
1-32895
1-33489
591
594
597
17.268
17.320
17-373
52
52
28
TABLE I.— Continued.
S{v)
567
566
565
564
563
562
561
560
559
558
557
556
555
554
553
552
551
550
549
548
547
546
545
544
543
542
541
540
539
538
537
536
535
6886.5
6916.4
6946.4
6976.5
7006.6
7036.8
7067.0
7097.3
7127.6
7158.0
7188.4
7218.9
7249.4
7280.0
7310.7
7341-4
7372.2
7403.0
7433-9
7464.8
7495-8
7526.8
7557.9
7589-1
7620.3
7651.6
7682.9
77H-3
7745-8
7777-3
7808.9
7840-5
7872.2
Diff.
299
300
301
301
302
302
303
303
304
304
305
305
306
307
307
308
308
309
309
310
310
311
312
312
313
313
314
315
315
316
316
317
317
Aiv)
6832.68
6872.93
6913-43
6954.18
6995.19
7036.46
7077.99
7119.78
7161.83
7204.15
7246.73
7289.58
7332.71
7376.11
7419.78
7463.74
7507-97
7552.48
7597.28
7642.36
7687.73
7733-39
7779-34
7825.58
7872.12
7918.96
7966.12
8013-55
8061.30
8109.36
8T57-73
8206.41
8255.41
Diff.
4025
4050
4075
4101
4127
4153
4179
4205
4232
4258
4285
43^3
4340
4367
4396
4423
4451
4480
4508
4537
4566
4595
4624
4654
4684
4716
4743
4775
4806
4837
4868
4900
4932
/{v)
34086
34686
35290
35897
36507
37120
37736
38356
38979
39606
40236
40869
41506
42146
42789
43436
44087
44741
45399
46060
46725
47394
48066
48742
49422
50106
50793
51484
52179
52878
53581
54287
54998
Diff.
600
604
607
610
613
616
620
623
627
630
637
640
643
647
651
654
658
661
665
669
672
676
680
684
687
691
695
699
703
706
711
715
Tiv)
7.425
7-478
7-531
7-584
7.638
7.691
7.745
7-799
7.853
7-908
7.962
8.017
8.072
8.127
8.183
8.238
8.294
8.350
8.406
8.462
8.519
8.576
8.633
8.690
8.747
8.805
8.921
8.979
9.038
9.096
9-155
9-215
Diff.
29
TABLE I.— Continued.
S{v)
534
533
532
531
530
529
528
527
526
525
524
523
522
521.
520
519
518
517
516
5^5
514
513
512
511
510
509
508
507
506
505
504
503
502
7903.9
7935-7
7967.6
7999-5
8031.5
8063.5
8095.6
8127.8
8160.0
8192.3
8224.7
8257.1
8289.6
8322.1
8354-7
8387-4
8420.1
8452.9
8485.7
8518.6
8551-6
8584-7
8617.8
8651.0
8684.2
8717-5
8750-9
8784.3
8817.8
8851.4
8885.0
8918.7
8952.5
Diff
318
319
319
320
320
321
322
322
323
324
324
325
325
326
327
327
328
328
329
330
33^
33'^
332
332
333
334
334
335
33^
33^
337
33^
338
A'iv)
8304.73
8354.36
8404.32
8454.61
8505.22
8556.16
8607.44
8659.06
8711.01
8763.30
8815.94
8868.92
8922.25
8975.93
9029.97
9084.36
9139.11
9194.23
9249.71
9305.56
9361.79
9418.39
9475-38
9532.74
9590.49
9648.62
9707.15
9766.06
9825.38
9885.09
9945.21
10005.74
10066.67
Diff.
4963
4996
5029
5061
5094
5128
5162
5195
5229
5264
5298
5333
5368
5404
5439
5475
5512
5548
5585
5623
5660
5699
5736
5775
5813
5853
5891
5932
5971
6012
6053
6093
6134
I{v)
55713
56431
57154
57881
58612
59347
60086
60830
61578
62330
63086
63847
64612
65381
66155
66933
67716
68504
69296
70092
70894
71700
72510
73326
74146
74971
75801
76636
77476
78321
79171
80026
80886
Diff.
718
723
727
731
735
739
744
748
752
756
761
765
769
774
778
783
788
792
796
802
806
810
816
820
825
830
835
840
845
850
855
860
865
T{v)
19.274
J9.334
19-394
^9-454
19.514
19-574
19-635
19.696
19-757
19.819
19.881
19.943
20.005
20.067
20.130
20.193
20.256
20.319
20.383
20.447
20.511
20.575
20.640
20.705
20.770
20.835
20.901
20,967
21.033
21.099
21.166
21.233
21,300
30
TABLE I.— Continued.
501
500
499
498
497
496
495
494
493
492
491
490
489
488
487
486
485
484
483
482
481
480
479
478
477
476
475
474
473
472
471
470
469
■S(v)
3
9020.2
9054.2
9088.2
9122.3
9156.4
9190.6
9224.9
9259-3
9293.8
9328.3
9362.9
9397.6
9432.3
9467.1
9502.0
9536.9
9572.0
9607.1
9642.2
9677-5
9712.8
9748.2
9783.6
9819. 1
9854.7
9890.4
9926.2
9962.0
9997-9
20033.9
20070.0
20106,2
Difif.
339
340
340
341
341
342
343
344
345
345
346
347
347
348
349
349
351
351
351
353
353
354
354
355
356
357
358
358
359
360
361
362
362
A{v)
0128.01
0189.78
0251.9
0314.5
0377.6
0441.0
0504.9
0569-3
0634.1
0699.3
0765.0
0831. 1
0897.6
0964.7
1032.2
1 100. 1
1168.6
1237-5
1307.0
1376.9
1447-2
1518.1
1589.4
1661.3
1733-7
1806.6
1880.0
1953-9
2028.4
2103.4
2178.9
2254.9
2331-5
Diff.
6177
6219
626
631
634
639
644
648
652
657
661
665
671
675
679
685
689
695
699
703
709
•713
719
724
729
734
739
745
750
755
760
766
771
/{v)
.81751
.82622
.83498
•84379
-85265
-86157
•87054
-87957
.88865
.89778
.90697
.91622
•92552
-93488
.94430
•95378
.96332
.97292
.98258
.99230
2.00207
2.0II90
2.02180
2.03176
2.04179
2.05188
2,06203
2.07225
2.08253
2.09288
2.10329
2.II376
2.12430
Diff,
871
876
886
892
897
903
908
913
919
925
930
936
942
948
954
960
966
972
977
983
990
996
T003
1009
1015
1022
1028
1035
1041
1047
1054
1061
T{v)
21.367
21.435
21.503
21.572
21.641
21.710
21.779
21.848
2T.918
21.988
22.058
22.128
22.199
22.270
22.341
22.413
22.485
22.557
22,630
22,703
22.776
22.849
22.923
22.997
23.071
23.146
23.221
23.296
23.372
23.448
23-524
23.601
23.678
Diff.
31
TABLE I.— Continued,
V
S{v)
Diff.
A{v)
Diff.
/{v)
Diff.
T{v)
Diff.
468
467
466
20142.4
20J78.7
20215.0
363
365
12408.6
'12486.3
12564.6
777
783
788
2.13491
2.14559
2.15635
ic68 i
1076 i
1082
23.755
23.833
23.911
78
78
78
465
464
463
20251.5
20288.0
20324.7
365
367
367
12643.4
12722.8
12802.7
794
799
805
2.16717
2.17806
2.18902
1089 1
1096
1 104
23.989
24.068
24.147
79
79
79
462
461
460
20361.4
20398.1
20435.0
367
369
369
12883.2
12964.3
13045-9
811
816
822
2.20006
2.21116
2.22233
i
1110
1117
1124
24.226
24.306
24.386
80
80
80
459
458
457
20471.9
20508.9
20546.0
370
371
371
13128. 1
13211.O
13294.4
829
834
841
2.23357
2.24489
2.25629
1132
1140
1147
24.466
24.547
- 24.628
81
81
82
456
455
454
20583.1
20620.4
20657.7
373
373
374
13378.5
134633
13548.6
848
853
859
2.26776
2.27931
2.29094
1155
1163
1171
24.710
24.792
24.874
82
82
82
453
452
451
20695.1
20732.6
20770.2
375
376
377
13634.5
13721.1
13808.3
866
872
878
2.30265
2.31443
2.32628
1178
1185
1193
24.956
25-039
25.122
^3
83
84
450
449
448
20807.9
20845.6
20883.4
377
378
380
13896.1
13984.6
14073.7
885
891
898
2.33821
2.35022
2.36232
1201
1210
1218
25.206
1 25.290
25.374
84
84
85
447
446
445
20921.4
20959.4
20997.4
380
380
382
'4163.5
14254.0
14345. i
905
911
919
2.37450
2.38676
2.39911
1226
1235
1243
1 25.459
25.544
25.629
85
85
86
444
443
442
21035.6
21073.9
21112.2
385
14437.0
14529.5
14622.7
925
932
939
2.41154
2.42405
2.43665
1251
1260
1268
i 25.715
25.801
25.888
86
87
87
441
440
439
21150.7
21189.2
21227.8
385
386
387
14716.6
14811.2
14906.5
946
953
960
2.44933
2.46209
2.47494
1276
1285
1294
25.975
26.062
26.150
87
88
88
438
437
436
21266.5
213^5-3
2i34'4-2
388
389
389
15002.5
15099.3
15196.8
968
975
982'
2.48788
2.50091
2.51404
1303
1313
1322
26.238
26.327
' 26.416
89
89
89
32
TABLE I.— Continued.
S{v)
21383. 1
21422.2
21461.4
21500.6
21540.0
21579-4
21618.9
21658.5
21698.2
21738.0
21777.9
21817.8
21857.9
21898.1
21938.4
21978.7
22019.1
22059.6
Diff.
22100.2
22140,9
22
81.8
22222.7
22263.7
22304.8
22346.1
^2387.4
22428.8
22470.4
225 12.0
22553-7
22595.6
22637.5
22679.6
391
392
392
394
394
395
396
397
398
399
399
401
402
403
403
404
405
406
407
409
409
410
411
413
413
414
416
416
417
419
419
421
422
A{v)
5295-0
5394.0
5493-7
5594-2
5695-4
5797-3
5900.0
6003.5
6107.9
6213. 1
6319.1
6425.9
6533-5
6641.9
6751.2
6972.2
7084.1
7196.8
7310.5
7425.0
7540.5
7656.8
7774.1
7892.2
8011.3
813^-3
8252.4
8374.4
8497.4
8621.4
8746.4
8872.3
Diff.
990
997
1005
012
019
027
035
044
052
060
068
076
084
093
T09
119
127
137
145
155
163
173
181
191
200
211
220
230
240
250
259
270
I{v)
2.52726
2.54057
2.55397
2.56746
2.58104
2.59471
2.60848
2.62235
2.63632
2.65039
2.66456
2.67883
2.69320
2.70767
2.72225
2.73692
2.75169
2.76658
2.78158
2.79668
2,81190
2,82723
2.84267
2.85822
2.87388
2.88965
2.90554
2.92155
2.93768
2.95393
2.97030
2.98679
3.00341
Diff.
33^
340
349 i
358
367
377
387
397
407
417
427
437
447
458
467
477
489
500
510
522
533
544
555
566
577
589
601
613
625
637
649
662
674
T{v)
26.505
26.595
26.685
26.776
26.867
26.959
27.051
27.143
27.236
27.329
27.423
27.517
27.612
27.707
27.803
27-899
27-995
28.092
28.189
28.287
28.385
28.484
28.583
28.683
28.783
28.884
28.985
29.087
29.189
29.292
29-395-
29.499
29,603
33
TABLE I.— Continued.
V
Siv)
Difif.
A{v)
Diff.
I{v)
Diff.
Tiy)
Diff.
402
401
400
22721.8
22764.0
22806.4
422
424
424
18999.3
19127.3
19256.2
1280
1289
1300
3.02015
I 3-03701
' 3-05399
1686
1698
1710
29.708
29.813
29.919
105
106
106
34
TABLE II.
For Spherical Projectiles.
V
S{v)
Diff.
A(v)
Diff. 1
I{v)
Diff.
T{v)
Diff.
2000
25
0.00
I i
0.00000
40
0.000
12
1990
1980
25
49
24
25
O.OI
0.02
I !
2
00040
00080
40
41
0.012
0.025
13
12
1970
i960
1950
74
99
124
25
25
26
0.04
0.08
0.13
4
5
51
0.00121
00163
00205
42
42
43
0.037
0.050
0.063
13
13
13
1940
1930
1920
150
175
201
25
26
25
0.18
0.25
0.33
7J
8 1
9
0.00248
00292
00336
44
44
45
0.076
0.089
0.102
13
13
14
I9IO
1900
1890
226
252
278
26
26
26
0.42
0.53
0.65
13
0.00381
00427
00473
46
46
47
0.1 16
0.129
0.143
13
14
14
1880
1870
i860
304
357
26
27
26
0.78
0.92
1.07
14
15
17
0.00520
00568
00617
48
49
49
0.157
0.171
0.185
14
14
14
1850
1840
1830
383
409
436
26
27
27
1.24
1.43
1.63
19
20
21
0.00666
00716
00767
50
51
52
0.199
0.214
0.228
15
14
15
1820
181O
1800
463
490
517
27
27
28
1.84
2.07
2.31
23
24
26
0.00819
00872
00926
53
0.243
0.258
0.273
15
15
15
1790
1780
1770
545
572
600
27
28
28
2.57
2.84
3.14
27
30
31
0.00981
01036
01093
55
57
57
0.288
0.304
0.319
16
15
16
1760
1750
1740
628
656
684
28
28
28
3.45
3-78
4-13
35
37
0.01150
01209
01268
59
59
61
0.335
0.351
0.367
16
16
16
1730
1720
I7IO
712
741
769
29
28
29
4.50
4.89
5-30
39
41
43
0.01329
01390
01453
61
(>z
64
0.383
0.400
0.416
17
16
17
35
TABLE IL— Continued.
V
S{v)
Diff.
A{v)
Diff;
I{v)
Diff.
T{v)
Diff.
1700
1690
1680
798
827
856
29
29
30
5-73
6.18
6.65
45
47
50
O.OI5I7
01582
01648
65
66
67
0.433
0.450
0.468
17
18
17
1670
1660
1650
886
915
945
1
29
30
30
7.15
7.67
8.21
52
54
56
O.OI7I5
01783
01853
68
70
71
0.485
0-503
0.521
18
18
t8
1640
1630
1620
975
1005
1036
30
3T
30
8.77
9-35
9-97
58
62
64
0.01924
01996
02070
72
74
75
0.539
0.558
0.576
19
18
19
I6I0
1600
1590
1066
1096
T127
30
31
31
10.61
11.27
11.96
66
69
72
0.02145
02222
02300
79
0.595
0.614
0.633
19
19
20
1580
1570
1560
1158
1189
1220
31
31
32
12.68
1344
14.22
76
78
82
0.02379
02460
02542
81
82
84
0.653
0.673
0.693
20
20
20
1550
1540
1530
1252
1284
1316
32
32
32
15-04
15.90
16.78
86
88
92
0.02626
02712
02799
86
87
89
0.713
0.734
0.755
21
21
21
1520
I5I0
1500
1348
1380
1413
32
17.70
18.65
19.63
95
98
100
0.02888
02979
03072
91
93
94
0.776
0.797
0.819
21
22
22
1490
1480
1470
1446
1479
1512
Z2>
7>Z
34
20.63
21.68
22.77
105
109
114
03166
03262
03360
- 96
98
ICI
0.841
0.863
0.885
22
22
23
1460
1450
1440
1546
1580
1614
34
34
34
23.91
i 25.10
26.34
!
119
124
128
03461
03564
03669
103
105
107
0.908
0931
0.955
23
24
24
1430
T420
I4I0
1648
1682
1717
34
35
35
27.62
1 28.95
133
138
143
0.03776
03885
03997
109
112
114
0.979
T.003
1.028
24
25
25
1400
1390
1380
1752
1787
1823
35
35
31.76
33-25
34-79
149
154
160
0.041 1 1
04227
04346
116
119
122
1.053
1.079
1. 105
26
.6
26
36
TABLE II.— Continued.
V
S{v)
Diff.
A{v)
Diff.
I{v)
Diff.
T{v)
Diff.
1370
1360
1350
1858
1894
1931
36
37
36
36.39
38.03
39-73
164
170
175
0.04468
04592
04719
124
127
129
1. 131
1. 158
1. 185
27
27
27
1340
1330
1320
1967
2004
2041
37
37
37
41.48
43-29
45-14
181
185
191
0.04848
04981
05II7
136
139
1. 212
1.239
1.267
27
28
27
I3I0
1300
1290
j 2078
2Tl6
2T54
38
38
47-05
49.01
51-04
196
203
212
0.05256
05398
05542
142
144
148
1.294
1.322
I-351
28
29
30
1280
1270
1260
2192
2231
2269
39
38
39
53-16
55-37
57.67
221
230
240
0.05690
05842
05998
152
156
160
1.38T
1.411
1.442
30
31
31
1250
1240
1230
2308
2348
1 2388
40
40
40
60.07
62.56
65.14
249
258
267
0.06158
06323
06492
165
169
174
1.473
1-505
1.538
32
33
1220
I2IO
1200
2428
2470
2512
42
42
22
67.81
70.59
73-54
278
295
156
0.06666
06846
07033
180
187
97
1.571
1.605
1.640
34
35
18
I I 90
2534
2556
2578
22
22
22
75-IO
76.70
78.32
t6o
162
165
0.07130
07229
07329
99
100
102
1.658
1.676
1.694
18
18
18
I 180
II75
1170
1 2600
2623
2646
'23
23
23
79-97
81.66
83-39
169
173
177
0.07431
07535
07641
104
106
108
1.712
I-731
I-751
19
20
19
I165
I 160
i'55
j
i 2669
2692
2715
23
23
24
85.16
86.98
88.84
182
186
190
0.07749
07859
07972
no
113
115
1.770
1.790
1. 810
20
20
21
1150
1 145
1 140
2739
2763
2787
24
24
25
90.74
92.69
94.68
195
199
205
0.08087
08204
08324
117
120
122
1.831
1.852
1-873
21
21
22
1130
1125
2812
2837
2861
25
24
25
96.73
98.82
100.97
209
215
221
0.08446
08570
08697
124
127
130
1.895
1-917
1.940
22
23
23
37
TABLE II.— Continued.
V
Si^v)
Diff.
A{v)
Diflf.
I{v)
Diff.
T{v)
Diff.
II20
IITO
2886
2912
2938
26
26
26
103.18
J05-44
107.77
226
233
239
0.08827
08959
09094
132
'35
138
1.963
1.986
2.009
23
24
IIO5
IIOO
1095
2964
2991
3017
27
26
27
110.16
1 12,62
115-13
246
251
259
0.09232
09373
09516
141
'43
147
2.033
2.057
2.081
24
24
25
1090
1085
1080
3044
3071
3099
27
28
28
117.72
120.38
123.13
266
■275
283
0.09663
09812
09965
149
153
156
2.106
2.132
2.158
26
26
26
1075
1070
1065
3127
3155
3184
28
29
29
125.96
128.87
131.87
291
300
308
0.10121
10280
10443
159
163
166
2.184
2.210
2.237
26
27
28
io6o
1055
1050
3213
3243
3273
30
30
30
134.95
138.12
141.38
317
326
338
0.10609
10-79
10952
170
173
177
2.265
2.293
2.321
28
28
29
1045
T040
1035
3364
30
31
31
144.76
148.22
151-77
346
355
364
0.11129
11310
1 1495
181
185
189
2.350
2.379
2.409
29
30
31
1030
1025
1020
3395
3427
3459
32
32
32
155-41
159-15
162.99
374
384
394
0.11684
II877
12074
193
197
202
2.440
2.471
2.502
31
31
32
1015
lOIO
1005
3491
3524
3557
Z2>
34
166.93
170.99
175-17
406
418
430
0.12276
12482
12693
206
211
215
2.534
2.566
2.599
32
1000
995
990
3591
3625
3660
34
35
35
179-47
183.90
188.46
443
456
470
0.12908
13128
13354
220
226
231
2.632
2.665
2.699
zz
34
ZS
985
980
975
3695
3731
3767
36
193.16
198.00
202.98
484
498
513
0.13585
13821
14062
236
241
246
2.734
2.770
2.806
36
37
970
965
960
3803
3840
3877
37
37
38
208.1 1
213.40
218.86
529
546
563
0.14308
14560
14818
252
258
264
2.843
2.881
2.920
38
39
39
38
TABLE II.— Continued.
V
S{v)
Diff.
A{v)
Diff.
I{v)
Diff.
T{v)
Diff.
955
950
945
39^5
3953
3992
38
39
39
224.49
230.29
236.29
580
600
620
0.15082
15352
15628
270
276
283
2-959
2.999
3.040
40
41
42
940
935
930
4031
4070
41 10
39
40
41
242.49
248.86
255-43
637
657
676
O.I59II
I620I
16498
290
297
304
3.082
3-125
3.168
43
43
44
925
920
915
4151
4192
4234
41
42
43
262.19
269.17
276.37
698
720
743
0.16802
17113
17432
311
319
327
3-212
3-257
3-303
45
46
47
910
905
900
4277
4320
4363
43
43
44
283.80
291.47
299.40
767
793
819
0.17759
18094
18437
335
343
352
3-350
3-397
3.445
47
48
49
895
890
885
4407
4451
4496
44
45
46
307-59
316.04
324-77
845
873
901
0.18789
I9I49
I95I8
360
369
378
3-494
3.544
3.595
50
51
52
880
875
870
4542
4589
4636
47
47
48
333-78
343-06
352.67
928
961
997
0.19896
20283
20680
387
397
407
3-647
3.700
3-754
53
54
55
865
860
855
4684
4732
4781
48
49
49
362.64
372.96
1032
1064
1099
0.21087
21505
21933
418
428
439
3-809
3.865
3.922
56
57
58
850
845
840
4830
4880
4931
50
5»
52
394.59
405-96
417.71
"37
1175
1216
0.22372
22823
23285
451
462
476
3-980
4.039
4.100
59
61
61
835
830
825
4983
5036
5089
53
53
54
429.87
442.45
455-47
1258
1302
1347
0.23761
24248
24746
487
498
511
4. 161
4.224
4.288
64
820
815
810
5143
5198
5253
55
55
56
468.94
482.89
497-33
1395
1444
1495
0.25257
25783
26323
526
540
553
4-354
4-421
4.489
67
68
70
805
800
795
5309
5366
5424
57
58
59
512.28
527-77
543-81
1549
1604
1661
0.26876
27444
28031
568
587
601
4-559
4.630
4.702
71
72
74
39
TABLE II.— Continued.
V
Siv)
Diff.
A{v)
Diff.
nv)
Diff.
T{v)
Diff.
79Q
785
780
5483
5542
5602
59
60
61
560.42
577-64
595-48
1722
1784
1849
0.28632
29249
29883
617,
634
650;
4.776
4.852
4-929
76
77
79
775
770
765
5663
5725
5788
62
63
64
613-97
653.01
1916
1988
2062
0.30533
31203
31891
670
688
707
5.008
5.088
5-170
80
82
84
760
755
750
5852
5917
5983
65
66
67
673-63
695.01
717 19
2138
2218
2303
0.32-598
33325
34073
727
748
770
5254
5-340
5-427
86
87
90
745
740
735
6050
6118
6187
68
69
69
740.22
764.11
788.91
2389
248c
2574
0.34843
35634
36448
791
814
837
5-517
5-608
S-701
91
93
96
730
725
720
6256
6327
6399
71
72
73
814.65
841.38
869.14
2673
2776
2882
0.37285
38146
39033
861
887
912
5-797
5.894
5-994
97
100
102
715
710
705
6472
6546
6621
74
75
77
897.96
927.92
959.07
2996
3115
3238
0.39945
40885
41853
940
968
995
6.096
6.200
6.306
104
106
109
700
695
690
6698
6776
685s
78
79
80
991.45
1025.2
1060.2
3366
350
364
0.42848
43872
44926
1024
1054
1089
6.415
6.526
6.640
III
ri4
116
685
680
675
6935
7016
7098
81
. 82
84
1196.6
1134.4
1173.8
378
394
409
0.46015
47143
48302
1128
1159
1192
6.756
6.875
6.997
1
119
122
125
670
665
660
7182
7267
7354
85
87
88
1214.7
1257.4
1301.8
427
444
463
0.49494
50722
51989
1228
1267
1307
7.122
7-249
7-380
127
131
134
655
650
645
7442
7531
7622
89
91
92
1348.1
1396.3
1446.5
482
502
523
0.53296
54645
56037
1349
1392
1436
7-5M
7-651
1 7.79T
137
140
143
640
635
630
7714
7808
7903
94
95
97
1498.8
1553.4
1610.2
546
568
592
0.57473
58955
60484
1482
1529
1579
7 934
' 8.081
' 8.231
147
150
154
40
TABLE II.— Continued.
V
Siv)
Diff.
A {v)
Diff.
I{v)
Diff.
T{v)
Diff.
625
620
6.5
8000
8098
8198
98
100
lOI
1669.4
1731.2
1795-6
618
644
673
0.62063
63696
65386
1633
1690
1737
8.885
8-543
8.705
158
162
166
610
605
600
8299
8402
8507
103 1
105
107
1862.9
I933-I
2006.4
702
733
765
0.67123
68922
70781
1799
1859
1923
8.871
9.041
9.215
170
174
179
595
590
585
8614
8722
8833
108
III
1 12
2082.9
2162.9
2246.5
800
836
872
0.72704
74692
76747
1988
2055
2126
9.394
9.577
9-765
183
188
192
580
575
570
8945
9059
9175
114
116
118
2333-7
2424.8
2520.2
911
954
998
0.78873
81072
83348
2199
2276
2356
10.957
10.154
10.357
197
203
208
565
560
555
9293
9413
9535
120
122
124
2620.0
2724.3
2833.4
1043
1091
1142
0.85704
88144
90670
2440
2526
2617
10.565
10.778
10.997
213
219
225
550
545
540
9659
9785
9914
126
129
131
2947.6
3067.2
3192.4
1196
1252
1312
0.93287
95998
98808
2711
2810
2913
11.222
11-453
11.690
231
237
243
535
- 530
525
10045
10178
135
138
3323-6
3461.0
3605.0
1374
1440
1509
I.OI72I
1.04740
1.07873
3019
3247
11-933
12.183
12.440
250
257
264
520
515
510
10451
10591
10734
140
143
146
3755-9
3914.1
4080.1
1582
1660
1743
I.III20
1. 14486
I.I7981
ZZ(^(>
3495
3633
12.704
12.975
13-254
271
279
287
505
500
495
10880
11028
11179
.48
151
'53
4254-4
4437-3
4629.3
1829
1920
2017
I.21614
1.25393
T. 29312
3779
3919
4070
13-541
13.836
14.138
295
302
312
490
485
480
11332
11488
1 1648
156
160
162
4831.0
5042.8
5265-4
2118
2226
2340
1.33382
I.37614
1. 42013
4232
4399
4575
14-450
14.770
15.100
320
330
340
475
470
465
11810
11975
12143
168
172
5499-4
5745-5
6004.3
2461
2588
2724
1.46588
1. 51348
1. 56301
4760
4953
5157
15-440
15-790
16.150
350
360
370
41
TABLE II. —Continued.
V
S(v)
Diff.
A{v)
■
Diff.
I{v)
Diff.
T{v)
Diff.
460
455
450
12315
12490
12668
175
178
6276.7
6865.5
2868
3020
1. 61458
1.66826
! 1. 72419
1
5368
5593
16.520
16.902
17.296
382
394
42
TABLE III.
f)
(^)
Diff.
Tan^
Diff.
e
(^)
Diff.
TanO
Diff.
o° oo'
o 20
40
0.00000
00582
01164
i
582
582
582
0.00000
00582
01164
582
582
582
'
II 00
II 20
II 40
0.19560
20176
20794
616
618
621
0.19438
20042
20648
604
606
608
I 00
I 20
I 40
0.01746
02328
02910
582
582
583
0.01746
02328
02910
582
582
582
12 00
12 20
12 40
0.21415
22038
22663
623
625
627
0.21256
21864
22475
608
611
612
.2 00
2 20
2 40
0.03493
04076
04659
583
583
584
0.03492
04075
04658
583
583
583
13 00
13 20
13 40
0.23290
23920
24553
630
633
636
0.23087
23700
24316
613
616
617
3 00
3 20
3 40
0.05243
05827
06412
584
585
586
0.05241
05824
06408
583
584
585
14 00
14 20
14 40
0.25189
25827
26468
638
641
644
0.24933
25552
26172
619
620
623
4 00
4 20
4 40
0.06998
07585
08172
587
587
58S
0.06993
07578
08163
585
585
586
15 00
15 20
15 40
0.27112
27759
28409
647
650
654
0.26795
27419
28046
624
627
629
5 00
5 20
5 40
0.08760
09349
09939
589
590
591
0.08749
09335
09922
586
587
588
16 00
16 20
16 40
0.29063
29720
30380
657
660
663
0.28675
29305
29938
630
633
635
6 00
6 20
6 40
0.10530
II 122
11715
592
593
594
0.105 10
1 1099
11688
589
589
590
17 00
17. 20
17 40
0.31043
31710
32381
667
671
674
0.30573
31210
31850
637
640
642
7 00
7 20
7 40
0.12309
12905
13502
596
597
598
0.12278
12869
13461
591
592
593
18 00
18 20
18 40
0.33055
33733
34415
678
682
686
0.32492
33^36
33783
644
647
650
8 00
8 20
8 40
0.14100
14700
15301
600
601
603
0.14054
14648
1 15243
594
595
595
19 00
19 20
19 40
0.35101
35791
36486
690
695
699
0.34433
35085
35740
652
655
657
9 00
9 20
9 40
0.15904
16509
17116
605
607
608
0.15838
1 16435
1 17033
j
597
598
600
20 00
20 20
20 40
0.37185
37888
38596
703
708
713
0.36397
37057
37720
660
663
666
10 00
10 20
10 40
0.17724
18334
18946
610
612
614
0.17633
18233
18835
600
602
603
21 00
21 20
21 40
0.39309
40026
40748
717
722
728
0.38386
39055
39727
669
672
676
43
TABLE III.— Continued.
.
m
Diff.
Tan 6/
Diff.
6
(^)
Diff.
TanB
Diff.
22° Oo'
22 20
2 2 40
0.41476
42208
42946
732
738
744
0.40403
41081
41763
678
682
684
33
33
00'
20
40
0.69253
70245
71248
992
roo3
1015
0.64941
65771
66608
830
837
843
23 00
23 20
23 40
0.43690
44439
45193
749
754
760
0.42447
43136
43828
689
692
695
34
34
34
00
20
40
0.72263
73290
74330
1027
1040
1052
0.67451
68301
69157
850
856
864
24 00
24 20
24 40
0-45953
46719
47491
766
772
778
0.44523
45222
45924
699
702
707
35
35
35
00
20
40
0.75382
76447
77525
1065
1078
1092
0.70021
70891
71769
870
878
885
25 00
25 20
25 40
0.48269
49054
49845
785
791
798
0.46631
47341
48055
710
714
718
36
36
36
00
20
40
0.78617
79723
80843
1 106
1 120
1 1 34
0.72654
73547
74447
893
900
908
26 00
26 20
26 40
0.50643
51448
52260
805
812
818
0.48773
49495
50222
722
727
731
37
37
37
00
20
40
0.81977
83126
8429 1
1 149
1 165
1T82
0-75355
76272
77196
917
924
933
27 00
27 20
27 40
0-53078
53904
54738
826
834
842
0.50953
51688
52427
735
739
744
38
38
3^
00
20
40
0.85473
86670
87883
1197
1213
1231
0.78129
79070
80020
941
950
958
28 00
28 20
28 40
0.55580
56429
57286
849
857
865
0.53171
53920
54073
749
753
758
39
39
39
00
20
40
0.89114
90363
91629
1249
1266
1285
0.80978
81946
82923
968
977
^987
29 00
29 20
29 40
0.58151
59025
59907
874
882
892
0.55431
56194
56962
763
768
773
40
40
40
00
20
40
0.92914
94217
95541
1303
1324
1343
0.83910
84906
859'2
996
1006
1017
30 00
30 20
30 40
0.60799
61699
62608
900
909
919
0.57735
58513
59297
778
784
789
41
41
41
00
20
40
0.96884
98247
99632
•363
1385
1407
0.86929
87955
88992
1026
1037
1048
31 00
31 20
31 40
0.63527
64455
65394
928
939
949
0.60086
60881
61681
795
800
806
42
42
42
00
20
40
1.01039
02468
03920
1429
1452
U75
0.90040
91099
92170
1059
1071
T082
32 00
32 20
32 40
0.66343
67302
68272
959
970
981
0.62487
63299
64117
812
818
824'
43
43
43
00
20
40
1.05395
06894
08418
1499
1524
1550
0.93252
94345
95451
1093
1 106
1118
44
TABLE III.— Continued.
44 oo
44 20
44 40
45 00
45 20
45 40
m
1.09968
I-II544
1.13148
1. 14779
1. 16439
1.18129
46 00 1. 19849
46 20
46 40
47 00
47 20
47 40
48 00
48 20
48 40
49 00
49 20
49 40
50 00
50 20
50 40
51 00
51 20
51 40
r. 21600
1.23384
1. 25201
1-27053
1.28940
1.30863
1.32823
1-34823
1.36863
1.38944
1. 41068
1.43236
1.45450
1. 47710
1. 50019
1-52379
I-54791
Diff.
1576
1604
1631
1660
1690
1720
1751
1784
1817,
1852
1887
1923
i960
2000
2040
2081
2124
2168
2214
2260
2309
2360
2412
2466
Tan S
0.96569
97700
98843
1. 00000
1.01170
1-02355
1-03553
1.04766
1.05994
1.07237
1.08496
1.09770
1.II06
1. 12369
1. 13694
1. 15037
r. 16398
1. 17777
1.19175
1-20593
1. 22031
1.23490
1.24969
1. 26471
Diff.
1131
1 143
1157
1170
1185
1213
1228
1243
1259
1274
129
1308
1325
1343
1361
T379
1398
1418
1438
1459
1479
1502
1523
52 00
52 20
52 40
53 00
53 20
53 40
54 00
54 20
54 40
55 00
55 20
55 40
56 00
56 20
56 40
57 00
57 20
57 40
58 00
58 20
58 40
59 00
59 20
59 40
60 00
(^)
Diff.
1-57257
1.59779
1.62357
1.64995
1.67696
1.70460
1. 73291
1.76191
1. 79162
1.82207
1-85329
1.88530
1.91815
1.95186
1.98646
2.02199
2.05849
2.09600
2.13456
2.1742
2.21500
2.25697
2.3001
2.34468
2-39053
Tan^
Diff.
2522
2578
638
2701
2764
2831
2900
2971
3045
3122
3201
3285
3371
3460
3553
3650
3751
3856
3965
4079
4197
4321
4450
4585
4726
1.27994 1547
1. 29541 1569
1.31110 1594
1.32704 1619
1-34323 1645
1.35968 1670
1.37638
1.39336
1.41061
1.42815
1.44598
1.46411
1.48256
1.50133
1.52043
1.53986
1.55966
1.57981
1.60033
1. 62125
1.64256
1.66428
1.68643
1. 70901
1.73205
1698
1725
1754
1783
1813
1845
1877
1910
1943
1980
2015
2052
2092
2131
2172
2215
2258
2304
2351
45
THIS BOOK IS DUE ON THE LAST DATE
STAMPED BELOW
AN INITIAL FINE OF 25 CENTS
WILL BE ASSESSED FOR FAILURE TO RETURN
THIS BOOK ON THE DATE DUE. THE PENALTY
WILL INCREASE TO 50 CENTS ON THE FOURTH
DAY AND TO $1.00 ON THE SEVENTH DAY
OVERDUE.
S"^' '.1
V2^
^2M-
i\oM ST 184S
<'"?vB5lB
Apr'59BBl
V-V
MAR 26 1989
OCT 3.) 1943
FEB 1 1944
S^
^ea
JUL 16 li-40
Ak^m^-^
^■Ttj^ftl
^^
2e3w.»'^ *
"itV
t ^t. 4 db'^
LD 21-100m-7,'40 (6936s)
LIBRARY USE
RETURN TO DESK FROM WHICH BORROWED
LOAN DEPT.
THIS BOOK IS DUE BEFORE CLOSING TIME
ON LAST DATE STAMPED BELOW
JilKf^^' i
4Rfi2 IBo'-
JUN 61984
REC
C1RMAY2 9 1984
SENT ON It I
JUL 1 2 1 9 94
PNI U. C. BERKELEY
LD 62A-20m-9,'63
(E709slO)9412A
General Library
University of California
Berkeley