INTERIOR BALLISTICS
BY
JAMES M. INGALLS
Colonel United States Army, retired
Formerly Instructor of Ballistics at the U. S. Artillery School ; Author of
Treatises on Exterior and Interior Ballistics, Ballistic
Machines, Ballistic Tables, Etc.
THIRD EDITION
NEW YORK
JOHN WILEY & SONS
LONDON: CHAPMAN & HALL, LIMITED
1912
COPYRIGHT 1912
BY JAMEvS M. INGALLS
PREFACE TO THE EDITION OF 1894
(SECOND EDITION)
WHEN, in the summer of 1889, it was decided by the Staff
of the Artillery School to add to the curriculum a course of
interior ballistics, the instructor of ballistics, knowing of no
text-book on the subject in the English language entirely suited
to the needs of the school, employed the time at his disposal
before the arrival of the next class of student officers in studying
up and arranging a course of instruction upon this subject, so
important to the artillery officer. The text-book then planned
was partially completed and printed on the Artillery School
press, and has been tested by two classes of student officers.
In the summer of 1893 the author again had leisure to work
on the unfinished text-book, but in the meantime he had found
so much of it which admitted of improvement that, with the
encouragement of Lieutenant-Colonel Frank, Second Artillery,
the Commandant of the School, it was decided to rewrite nearly
the entire work as well as to complete it according to the original
plan by the addition of the last two chapters.
With the exception of portions of Chapters IV and V, the
author claims no originality. He has simply culled from various
sources what seemed to him desirable in an elementary text-
book, arranged it all systematically from the same point of view
and with a uniform notation.
ARTILLERY SCHOOL,
February 15, 1894.
258715
PREFACE TO THE THIRD EDITION
THE second edition of this work was used as a text-book at
the Artillery School until the School suspended operations at
the outbreak of war with Spain, in April, 1898. This edition,
having become exhausted, the author has been induced by the
request of officers for whose wishes he has great respect, to
prepare a new edition embodying the results-of some investiga-
tions which were published in Volumes 24, 25 and 26 of the
Journal of the United States Artillery, and which have been
favorably received by Artillery Officers, both at home and abroad.
The Journal articles have been rewritten, and many improve-
ments attempted, suggested by friendly criticisms for which the
author wishes to express his thanks.
As most of the formulas of interior ballistics in the present
state of our knowledge of the subject are more or less empirical
in their nature, many applications of the formulas deduced in
Chapter IV are given in the following chapter to show their
agreement with the results of actual firing with guns of widely
different calibers. In this connection it is gratifying to be able
to quote from an article published in the Journal of the Royal
Artillery, vol. 36, No. 9, by Captain J. H. Hardcastle, R.A.,
who states, with reference to the formulas of Chapter IV as
applied to firing-practice with English guns loaded with cordite,
that " After many dozens of calculations I can find no serious
disagreement between the results of calculation and experiment."
VI PREFACE
In this paper Captain Hardcastle has very ingeniously adapted
the formulas of Chapter IV to slide- rule operations, thereby
lessening the labor of calculation somewhat, though at the
expense of accuracy in some cases. For the benefit of those who
are accustomed to use the slide rule in making logarithmic com-
putations a supplemental table of the X functions has been
added to Table I, omitting the function X & , which is not used in
Captain Hardcastle's method.
This work was prepared primarily for the officers of our
Coast Artillery Corps; but it is hoped that gun-designers and
powder-manufacturers may find in it something useful to them.
The author desires to express his indebtedness to Lieut.-
Colonel Ormand M. Lissak and Major Edward P. O'Hern of
the Ordnance Department for valuable suggestions and for
data employed in the " applications." Also to Captains Ennis
and Bryant, for assistance in computing Table I.
PROVIDENCE, R. I.,
September 20, 1911.
TABLE OF CONTENTS
CHAPTER I
PAGE
Definition and object. Early history of gunpowder. Robins' experi-
ments and deductions. Mutton's experiments. D'Arcy's method.
Ram ford's experiments with fired gunpowder. Rodman's inventions
and experiments. Modern explosives. Density of powder. Inflam-
mation and combustion of a grain of powder. Inflammation and
combustion of a charge of powder I to 14
CHAPTER II
Properties of perfect gases. Marriotte's law. Specific volume. Specific
weight. Law of Gay-Lussac. Characteristic equation of the gaseous
state. Thermal units. Mechanical equivalent of heat. Specific heat.
Specific heat of a gas under constant pressure. Specific heat under
constant volume. Numerical value of R for atmospheric air. Law of
Dulong and Petit. Determination of specific heats. Ratio of specific
heats. Relations between heat and work in the expansion of a per-
fect gas. Isothermal expansion. Adiabatic expansion. Law of tem-
peratures. Law of pressures and volumes. Examples. Theoretical
work of an adiabatic expansion in the bore of a gun. Noble and
Abel's researches on fired gunpowder in close vessels. Description
of apparatus employed. Summary of results. Pressure in close ves-
sels deduced from theoretical considerations. Value of the ratio of
the non-gaseous products to the volume of the charge. Determina-
tion of the force of the powder, and its interpretation. Theoretical
determination of the temperature of explosion of gunpowder. Mean
specific heat of the products of combustion. Pressure in the bore
of a gun derived from theoretical considerations. Table of pressures.
Theoretical work effected by gunpowder. Factor of effect. Actual
work realized as expressed by muzzle energy 1 5 to 54
CHAPTER III
Combustion of a grain of powder under constant atmospheric pressure.
Notation. Definition of the vanishing surface. General expression
for the burning surface of a grain of powder. Expression for the
VIII TABLE OF CONTENTS
PAGK
volume consumed in terms of the thickness burned. Definition of the
form characteristics. Fraction of grain burned. Applications.
Spheres. Cubes. Strips. Solid cylinders. Pierced cylinders.
Multiperf orated grains. General expression for surface of combus-
tion of multiperforated grains. Maximum surface of combustion.
Slivers. Expression for volume of slivers. Proposed ratio of dimen-
sions of multiperforated grains to web thickness. Expression for
weight of charge burned at any instant. Expressions for initial
volume and surface of combustion of a charge of powder. Expression
for specific gravity of grain. Initial surface of unit weight of powder.
Volume of charge. Gravimetric density. Density of loading.
Reduced length of initial air space. Working formulas for English and
metric units. Examples 55 to 78
CHAPTER IV
Combustion and work of a charge of powder in a gun. Introductory
remarks. Sarrau's law of burning under a variable pressure and
reason for adopting it. Expression connecting the velocity of
burning of grain with velocity of projectile in bore. Expression for
fraction of charge burned in terms of volumes of expansion of the
gases generated. Expression for velocity of projectile while powder
is burning. Velocity of projectile after powder is all burned. Pres-
sure on base of projectile while powder is burning. Pressure after
powder is burned. Expression for the initial pressure upon the sup-
position that the powder was all burned before the projectile had
moved from its seat, and the relation of this pressure to the force
of the powder. Method of computing the X functions. Special
formulas. Expressions for maximum pressure. Formula for velocity
of combustion under atmospheric pressure. Working formulas. Eng-
lish units. Metric units. Characteristics of a powder. Expressions
for constants in terms of the characteristics for English and metric
units. Expressions for force of powder when weights of charge
and projectile vary 79 to 97
CHAPTER V
Applications. Formulas which apply only while powder is burning. For-
mulas which apply only after powder is all burned. Formulas which
apply at instant of complete combustion. Discontinuity of pressure
curve for certain forms of grain. Monomial formulas for velocity
and pressure. Typical pressure and velocity curves. Example of
TABLE OF CONTENTS ix
PAGE
monomial formulas, as applied to the 8-inch B. L. R. Comparison
of computed velocities and maximum pressures with observed values.
Determination of travel of projectile at point of maximum pressure,
and also when powder is all burned. Expression for fraction of
charge burned for any travel of projectile. Examples. Greatest
efficiency when charge is all consumed at muzzle. Application to
hypothetical 7-inch gun. Binomial formulas for velocity and pressure.
Forms of grain for which binomial formulas must be employed.
Methods for determining the constants from experimental firing.
Applications to Sir Andrew Noble's experiments with a 6-inch gun.
Description of the experiments. Discussion of the data for cordite,
0.4", 0.35", and 0.3" diameter. Remarks on the so-called "force of
the powder" as deduced from the calculations. Examples. Applica-
tion to the Hotchkiss 57-mm. rapid-firing gun. Data obtained by
D'Arcy's method. New method for determining the form char-
acteristics of the grains. Application to the magazine rifle, model of
1903. Powder characteristics. Formulas for designing guns for
cordite, with application to a hypothetical 7-inch gun. Trinomial
formulas. Grains for which these formulas are necessary. Spherical
and cubical grains. Formulas for computing the constants. Ap-
plication to Noble's experiments with ballistite in a 6-inch gun.
Table of computed velocities and pressures. Remarks on the
velocity and pressure curves. Examples. Multiperforated grains.
Special formulas required for these grains. Discussion of the data
obtained by the Ordnance Board with the 6-inch Brown wire gun.
Remarks on the discontinuity of the pressure curves. Examples.
Superiority of uniperforated to multiperforated grains. Application
to the 14-inch rifle. Effect of increasing the volume of the chamber
upon the maximum pressure. Better results can be obtained by
lengthening the powder grains. Table of pressures. . . . 9810169
CHAPTER VI
On the rifling of cannon. Advantages of rifling. The developed groove.
Uniform twist. Increasing twist. General expression for pressure
on the lands. Angular acceleration. Pressure for uniform twist.
Increasing twist. Semi-cubical parabola. Common parabola. Rel-
ative width of grooves and lands. Application to the lo-inch
B. L. R., model of 1888. Application to the 14-inch gun. Retarding
effect of a uniform twist of one turn in twenty-five calibers. . 170 to 186
Tables 18910215
INTERIOR BALLISTICS
CHAPTER I
INTRODUCTION
Definition and Object. Interior ballistics treats of ihe\
formation, temperature and volume of the gases into which the
powder charge, in the chamber of a gun, is converted by com-
bustion, and the work performed by the expansion of these
gases upon the gun, carriage and projectile. Its object is the
deduction and discussion of rules and formulas for calculating
the velocity, both of translation and of rotation, which the
gases of a given weight of powder of known composition and
quality are able to impart to a projectile and their reaction
upon the gun and carriage. The discussion of the formulas
deduced will bring out many important questions, such as the
proper relation of weight of charge to weight of projectile and
length of bore, the best size and shape of the powder grains for
different guns and their effect upon the maximum and muzzle
pressures, the velocity of recoil, etc. The most approved
formulas for calculating the pressures upon the surf ace of the
bore will be given ; but the methods which have been devised
for building up the gun, so as best to resist these pressures, will
not be entered upon here as their consideration belongs to
another branch of the subject.
Early History of Interior Ballistics. For more than five
hundred years gunpowder an intimate mixture of nitre,
sulphur and charcoal, was used almost exclusively as the pro-
2 INTERIOR BALLISTICS
pelling agent in firearms; and though it has been entirely
superseded within the last quarter of a century by gun-cotton,
mtro-glycerine, and their various compounds, yet it possessed
many admirable qualities which the modern powders do not
as yet so fully enjoy. It ignited easily without deflagration;
its effects were regular and sure; its manufacture was economical,
rapid and comparatively safe; it produced but little erosion
in the bore. Finally, it kept well in transportation, and in-
definitely in properly ventilated magazines. It is on record
that experiments made with gunpowder, manufactured more
than two centuries before, showed that it had lost none of its
ballistic qualities. The principal objection to gunpowder, as
compared to nitrocellulose powders, are the dense volumes of
smoke accompanying its explosion, the fouling of the bore,
and the comparatively large charges required to give the
desired muzzle velocity, necessitating an abnormal enlarge-
ment of the powder chamber or an impracticable lengthening
of the gun.
Robins' Experiments and Deductions. The celebrated Ben-
jamin Robins seems to have been the first investigator who had a
tolerably correct idea of the circumstances relating to the action
and force of fired gunpowder. In a paper which was read before
the Royal Society in 1743 entitled, "New principles of gunnery,"
Robins described among other things some experiments he had
made for determining the velocities of musket balls when fired
with given charges of powder. These velocities were measured
by means of the ballistic pendulum invented by Robins, "the
idea of which is simply that the ball is discharged into a very
large but movable block of wood, whose small velocity, in conse-
quence of that blow, can be easily observed and accurately
measured. Then, from this small velocity thus obtained, the
large one of the ball is immediately derived from this simple
proportion, viz., as the weight of the ball is to the sum of the
weights of the ball and the block, so is the observed velocity of
INTRODUCTION 3
the last to a fourth proportional, which is the velocity of the ball
sought." *
The deductions which Robins makes from these experiments,
so far as they relate to interior ballistics, may be summarized
as follows:
(1) Gunpowder fired either in a vacuum or in air produces,
by its combustion, a permanent elastic fluid or air.
(2) The pressure exerted by this fluid is, cateris paribus,
directly as its density.
(3) The elasticity of the fluid is increased by the heat it
has at the time of explosion.
(4) The temperature of the fluid at the moment of combus-
tion is at least equal to that of red-hot iron.
(5) The maximum pressure exerted by the fluid is equal to
about 1,000 atmospheres.
(6) The weight of the permanent elastic fluid disengaged by
the combustion is about three-tenths that of the powder, and its
volume at ordinary atmospheric temperature and pressure is
about 240 times that occupied by the charge.
These deductions, considering the extremely erroneous ancj
often absurd opinions that were entertained by those who
thought upon the subject at all in Robins' time and even down
to the close of the century show that Robins is well entitled
to be called the "father of modern gunnery."
Button's Experiments. Dr. Charles Hutton, professor of
mathematics in the Royal Military Academy, Woolwich, con-
tinued Robins' experiments at intervals from 1773 to 1791. He
improved and greatly enlarged the ballistic pendulum so that
it could receive the impact of i -pound balls, whereas that used
by Robins was adapted for musket balls only. Button's
experiments are given in detail in his thirty-fourth, thirty-fifth,
thirty-sixth, and thirty-seventh tracts. They verify most of
* Hutton's " Mathematical Tracts," vol. 3, p. 210 (Tract 37), London, 1812.
4 INTERIOR BALLISTICS
Robins' deductions, but with regard to Robins' estimate of the
temperature of combustion and the maximum pressure Hutton
says: "This was merely guessing at the degree of heat in the
inflamed fluid, and, consequently, of its first strength, both
which in fact are found to be much greater." * His own estimate
of the temperature is double that of Robins, and he places the
maximum pressure of fired gunpowder at 2,000 atmospheres.
Hutton gives a formula for the velocity of a spherical projectile
at any point of the bore, upon the assumption that the combus-
tion of the charge is instantaneous and that the expansion of
the gas follows Mariotte's law no account being taken of the
loss of heat due to work performed a principle which at that
time was unknown.
D'Arcy's Method. In 1760 the chevalier D'Arcy sought to
determine the law of pressure of the gas in the bore of a musket
by measuring the velocity of the projectile at different points
of the bore. This he accomplished by successively shortening
the length of the barrel and measuring for each length the velocity
of the bullet by means of a ballistic pendulum. Having obtained
from these experiments the velocities of the bullets for several
different lengths of travel, the corresponding accelerations could
be calculated, and then the pressures, by multiplying the
accelerations by the mass. This was the first attempt to
determine the law of pressures dynamically.
Rumford' s Experiments with Fired Gunpowder. The first
attempt to measure directly the pressure of fired gunpowder
was made, in 1792, by our countryman, the celebrated Count
Rumford. A most interesting account of his experiments is
given in his memoir entitled "Experiments to determine the
force of fired gunpowder," f which must be regarded as the most
important contribution to interior ballistics w r hich had been
* Tracts, vol. 3, p. 211.
t Philosophical Transactions, London, 1797, p*. 222; also "The Complete
Works of Count Rumford," Boston, 1870, vol. I, p. 98.
INTRODUCTION 5
made up to that time. The apparatus used by Rumford con-
sisted of a small and very strong wrought-iron mortar (or
eprouvette), which rested with its axis vertical upon a solid
stone foundation. This mortar (or barrel, as Rumford calls it),
was 2.78 inches long and 2.82 inches in diameter at its lower
extremity and tapered slightly toward the muzzle. The bore
(or chamber) was cylindrical, one-fourth of an inch in diameter
and 2.13 inches deep. At the centre of the bottom of the barrel
there was a projection 0.45 inch in diameter and 1.3 inches long,
having an axial bore 0.07 inch in diameter connecting with the
chamber above, but closed below, forming a sort of vent, but
having no opening outside.
By this arrangement the charge could be fired without any
loss of gas through the vent by the application of a red-hot
ball provided with a hole, into which the projecting vent-tube
could be inserted, which latter would thus become in a short
time sufficiently heated to ignite the powder. The upper part
of the bore or muzzle was closed by a stopper made of compact,
well-greased sole leather, which was forced into the bore, until
its upper surface was flush with the face of the mortar, and upon
this was placed the plane surface of a solid hemisphere of hard-
ened steel, whose diameter was 1.16 inches. "Upon this
hemisphere the weight made use of for confining the elastic
fluid generated from the powder in its combustion reposed.
This weight in all the experiments, except those which were
made with very small charges of powder, was a piece of ordnance
of greater or less dimensions or greater or less weight, according
to the force of the charge, placed vertically upon its cascabel
upon the steel hemisphere which closed the end of the barrel;
and the same piece of ordnance, by 'having its bore filled by a
greater or smaller number of bullets, as the occasion required,
was made to serve for several experiments."
* Rumford's Works, vol. i, p. 121.
6 INTERIOR BALLISTICS
As one of the objects of Rumford's experiments was to
determine the relation between the pressure of the powder gases
and their density, he varied the charge, beginning with i grain,
and for each charge placed a weight, which he judged was about
equivalent to the resulting pressure, upon the hemisphere. If,
on firing, the weight was lifted sufficiently to allow the gases to
escape, it was increased for another equal charge; and this was
repeated until a weight was found just sufficient to retain the
gaseous products that is, so that the leathern stopper would
not be thrown out of the bore, but only slightly lifted. The
density of the powder gases could easily be determined by
comparing the weight of the charge with the weight of powder
required to completely fill the chamber and vent, which latter
was about 25^2 grains troy. Rumford increased the charges a
grain at a time from i grain to 18 grains, and from a mean of
all the observed pressures he deduced the empirical formula,
i +.0004*
in which p is the pressure in atmospheres and x the density of
loading to a scale of 1000 that is, for a full chamber x = 1000;
for one-half full x = 500, and so on. This formula gives
29,178 atmospheres for the maximum pressure that is, when
the powder entirely fills the space in which it is fired. In this
case the value of x is 1000, and Rumford's pressure formula
becomes
p = 1.841 X 1000 I<4 = 29178
Nearly a century later Noble and Abel (see Chapter II)
found by their experiments, which are entirely similar in charac-
ter to those of Rumford, that the maximum pressure of fired
gunpowder is but 6,554 atmospheres, or 43 tons per square inch;
and this result has been accepted by all writers on interior
INTRODUCTION 7
ballistics as being very near the truth. Their formula for the
pressure in terms of Rumford's x is
2.818*
^ '" 1 0.00057*
in which p and x are denned as before. If in this formula we
make x = 1000, we have, as already stated,
2.818 X 1000
P = "" = 6554
i 0.57
For small densities of loading, Noble and Abel's formula
gives greater pressures than Rumford's principally because the
powder used by the later investigators was the stronger; but
as the densities increase this is reversed. With a charge of 18
grains, for which x = 702, Noble and Abel's formula gives a
pressure of 3,298 atmospheres, while Rumford's gives 8,140
atmospheres. To enable us to understand the cause of this
great difference in the results obtained by these eminent savants
(which is very instructive), we will go a little into detail. Two
experiments were made by Rumford with a charge of 18 grains
of powder. In the first of these a 24-pounder gun, weighing
8,08 1 pounds, was placed vertically on its cascabel upon the
steel hemisphere closing the muzzle of the barrel. When ihe
charge was fired "the weight was raised with a very sharp
report, louder than that of a well-loaded musket." The barrel
was again loaded with 18 grains as before, and enough shot were
placed in the bore of the 24-pounder gun to increase its weight
to 8,700 pounds. Upon firing the powder by applying the
red-hot ball "the vent-tube of the barrel was burst, the explosion
being attended with a very loud report." These experiments
were the eighty-fourth and eighty-fifth, and closed the series.
In the eighty-fourth experiment a weight of 8,081 pounds
was actually raised by the explosion of 18 grains of powder
(about one-fourth the service charge of the Springfield rifle),
8 INTERIOR BALLISTICS
acting upon a circular area one-quarter of an inch in diameter.
To raise this weight under the circumstances would require a
pressure of more than 11,200 atmospheres, while, as we have
seen, the actual pressure due to this density of loading, according
to Noble and Abel's formula, is but 3,298 atmospheres. Evident-
ly then the weight in this experiment was not raised by mere
pressure; but we must attribute a great part of the observed
effect (in consequence of the position of the charge at the bottom
of the bore) to the energy with which the products of combustion
impinged against the leathern stopper, which had only to be
raised 0.13 inch (the thickness of the leather) to allow the gases
to escape. In Noble and Abel's experiments there was no such
blow from the products of combustion because the apparatus
for determining the pressure (crusher gauge) was placed within
the charge. Had the leathern stopper in Rumford's experiments
been a little longer, it is probable that his conclusions would
have been entirely different.
Rodman's Inventions and Experiments. We have space
only to mention the names of Gay-Lussac, Chevreul, Graham,
Piobert, Cavalli, Mayevski, Otto, Neumann, and others, who
did original work, of more or less value, for the science of interior
ballistics prior to the year 1860. We will, however, dwell a
few moments on the important work done by Captain (after-
wards General) T. J. Rodman, of our own Ordnance Department,
between the years 1857 and 1861.* The objects of Rodman's
experiments were: First, to ascertain the pressure exerted
upon different points of the bore of a 4 2 -pounder gun in firing
under various circumstances. Second, to determine the press-
ures in the y-inch, Q-inch, and n-inch guns when the charges
of powder and the weight of projectiles were so proportioned
that there should be the same weight of powder behind, and
* " Experiments on Metal and Cannon and Qualities of Cannon Powder,"
by Captain T. J. Rodman, Boston, 1861.
INTRODUCTION 9
the same weight of metal in front of each square inch of area
of cross-section of the bore. Third, to determine the differences
in pressure and muzzle velocity due to the variations in the
size of the powder grains; and, fourth, to determine the pressures
exerted by gunpowder burned in a close vessel for different
densities of loading.
For the purpose of carrying out these experiments Rodman,
instead of using the system of varying weights employed by
Rumford, invented what he called the "indenting apparatus/'
which has since been extensively used, not only in this country
but in all foreign countries, under the name of Rodman's pressure
(or cutter) gauge; and which is too well known to require a
description.
The maximum pressure of gunpowder when exploded in its
own space, as determined by Rodman by the bursting of shells
filled with powder, ranged from 4,900 to 12,600 atmospheres;
the mean of all the experiments giving 8,070 atmospheres, or
53 tons per square inch. These results, though much nearer the
truth than those deduced by Rumford, are still about 25 per
cent, greater than Noble and Abel's deductions; and this is
undoubtedly due to the position of the pressure gauge, which
was placed near the exterior surface of the shell, so that when
the products of combustion had reached the gauge they had
acquired a considerable energy which probably exaggerated the
real pressure. The same causes, it will be remembered, vitiated
Rumford's experiments. In both cases it was as if a charge of
small shot had been fired with great velocity against the leathern
stopper in the one case, or the end of the piston of the indenting
tool in the other.
General Rodman was the first person to suggest the proper
shape for powder grains, in order to diminish the initial velocity
of emission of gas and to more nearly equalize the pressure in
the bore of the gun/ For this purpose he employed what he
termed a "perforated cake cartridge" composed of disks of
10 INTERIOR BALLISTICS
compressed powder from i to 2 inches thick and of a diameter
to fit the bore. Rodman demonstrated that such a form of
cartridge would relieve the initial strain by exposing a minimum
surface at the beginning of combustion, while a greater volume
of gas would be evolved from the increasing surfaces of the
cylindrical perforations as the space behind the projectile be-
came greater; and this would tend to distribute the pressure
more uniformly along the bore. Rodman's experiments with
this powder in the 15 -inch cast-iron gun which he had recently
constructed for the government and which is without doubt the
most effective and the best smooth-bore gun ever made fully
confirmed his theory; but for many reasons he found it more
convenient and equally satisfactory to build up the charge by
layers of pierced hexagonal prisms about an inch in diameter
fitting closely to one another, instead of having them of the
diameter of the bore.
The war of the rebellion which was inaugurated while
General Rodman was in the midst of his discoveries and in-
ventions, put an end forever to his investigations, but his ideas
were speedily adopted in Europe, and his " prismatic powders,"
but slightly modified, are extensively used.
Modern Explosives. Gun-cotton, made by immersing
cleaned and dried cotton waste in a mixture of strong nitric and
sulphuric acids, was discovered by Schonbein of Basel, in 1846,
who immediately proposed to employ it as a substitute for
gunpowder. General von Lenk made many experiments with
gun-cotton by compressing it into cubes or cylinders, with the
idea of employing it for artillery use. But all his efforts failed
from the fact that, no matter how much it was compressed, it
was still mechanically porous; and when ignited in a gun the
flame and hot gases speedily penetrated the mass causing it to
detonate, or, at least, to approach dangerously near to detona-
tion. It was not until the discovery in the early eighties that
gun-cotton could be dissolved or made into a paste, or colloid,
INTRODUCTION 1 1
by acetone and other so-called solvents, that it was possible to
employ it as a propellant. In this condition, when moulded
into grains and thoroughly dried, it loses its mechanical porosity
and burns from the surface in parallel layers, the grain retaining
its original form until completely consumed.
Gun-cotton is mixed in certain proportions with nitro-
glycerine to form nearly all the powders employed for war
purposes. For example, the powder used in the British army
and navy (called cordite), consists of 65 per cent, of gun-cot-
ton, 30 per cent, of nitro-glycerine and 5 per cent, of mineral
jelly or vaseline, this latter being used as a preservative. This
is also very nearly the composition of the powder used in the
United States army and navy. For a full account of the
properties, manufacture and uses of gun-cotton and nitro-
glycerine, the reader is referred to General Weaver's "Notes
on Explosives."
Density of Powder. By density of a powder is meant its
specific gravity, or the ratio of the weight of a given volume of
the powder to the weight of an equal volume of water at the
standard temperature. It is sometimes referred to as mercurial
density, since it may be determined by art apparatus which
utilizes the property of mercury of filling the interstices between
the grains without penetrating into the pores or uniting chemi-
cally with the powder. The density varies somewhat according
to the pressure to which the grains were subjected during the
manufacture and ranges from about 1.56 to 1.65.
Inflammation and Combustion of a Grain of Powder.
Inflammation is the spreading of the flame over the free surface
of the grain from the point of ignition. Combustion is the
propagation of the burning into the interior of the grain. Igni-
tion is produced by the sudden elevation of the temperature
of a small portion of the grain to about 180 C. (in the case of
cordite) either by contact with an ignited body, by mechanical
shock or friction, or by detonation of a fulminate. The velocity
12 INTERIOR BALLISTICS
of inflammation depends upon the nature of the source of heat
which ignites it, upon the state of the surface of the grain and
upon its density and dryness. The combustion of a grain takes
place in successive concentric layers, and in free air equal
thicknesses are burned in equal times. As the mass of gas
disengaged in any given time is proportional to the quantity of
powder burned during the same time, and, therefore, propor-
tional to the surface of inflammation, it follows that the emission
of gas is largely influenced by the form of the grain. For
example, if the grain is spherical the surface of inflammation
decreases rapidly up to the end of its burning where it is zero.
On the other hand, the surface of inflammation (or of combus-
tion) of a multi-perforated grain increases until it is nearly
consumed.
Inflammation and Combustion of a Charge of Powder.
The inflammation of a charge of powder involves the trans-
mission of the flame from one grain to another. Its velocity
depends not only upon the inflammability of the grain but also
upon the facility with which the gases first generated are able to
penetrate the charge. This is assisted by a proper arrangement
of the grains composing the charge and also by placing an igniter
of fine rifle powder at each end of the cartridge. The com-
bustion of a charge composed of grains of the same form and
dimensions should, from what has been said, practically termi-
nate at the same time with each or any grain; and, therefore,
the time of combustion of a charge increases with the size of the
grains, and is in all cases with service powders much longer than
the time of inflammation.
If a charge of powder be confined in a close vessel and ignited,
its combustion takes place silently, and permanent gases and a
certain amount of solid matter are produced which can be
collected for analysis by opening the vessel, as in the experiments
of Noble and Abel described in Chapter II. In this case no
work is performed by the gases, and the accompanying phe-
INTRODUCTION 13
nomena are comparatively simple. But if the combustion takes
place in a chamber of which one of the walls is capable of moving
under the tension of the gases, which condition is realized in
cannon, the resulting phenomena are much more complicated,
as a little consideration will show.
When the charge of powder in the chamber of a loaded gun
is ignited at both ends of the cartridge, all the grains will be
inflamed practically simultaneously. The first gaseous products
formed will expand into the air-spaces of the chamber and
almost immediately acquire a tension sufficient to start and
overcome the forcing of the projectile. This latter once in
motion will encounter no resistances in the bore comparable with
those which opposed its start, and its velocity will rapidly in-
crease under the continued action of the pressure of the gases.
This pressure will also increase at first; for, though the displace-
ment of the projectile gives a greater space for the expanding
gases, this is more than compensated for by a more abundant
disengagement of gas. But the pressure soon reaches its
maximum; for if, on the one hand, the disengagement of gas
is accelerated by the increase of pressure, on the other hand the
increasing velocity of the projectile offers more and more space
for the gases to expand in. The velocity itself would soon
reach a maximum if the bore were sufficiently prolonged; for
in addition to the friction and the resistance of the air, both of
which retard the motion of the projectile, the propulsive force
decreases by the expansion and cooling of the gases. Therefore
the retarding forces will in time predominate and the projectile
be brought to rest. Its velocity starting from zero passes to its
maximum and if the gun terminated at this point the projectile
would leave the bore with the greatest velocity the charge was
capable of communicating to it.
So far only charges in general have been considered. Take,
now, a charge composed of small grains of slight density. The
initial surface of inflammation will be very great and the emission
14 INTERIOR BALLISTICS
of gas correspondingly abundant. The pressure will increase
rapidly, and, in consequence, the velocity of combustion. It
results from this that the grains will be consumed nearly as soon
as inflamed, and this before the projectile has had time to be
displaced by a sensible amount. Hence all the gases of the
charge, disengaged almost instantaneously, will be confined an
instant within the chamber; their tension will be very great,
and they will exert upon the walls of the gun a sudden and violent
force which may rupture the metal, and which in all cases will
produce upon the gun and carriage shocks which are destructive
to the system and prejudicial to accuracy of fire. On the other
hand the projectile will be thrown quickly forward, as by a
blow from a hammer.
If, on the contrary, the charge is made up of large grains of
great density, the total weight of gas emitted will be the same as
before; but the mode of emission will be different. The initial
surface of inflammation will be less, and the initial tension of the
gas not so great. The combustion will take place more slowly,
and will be only partially completed when the projectile shall
have begun to move. The pressure of the gases will attain a
maximum less than in the preceding case, but the pressure will
decrease more slowly. Under the continued action of this
pressure, the velocity of the projectile will be rapidly accelerated
and at the muzzle will differ but little from that obtained by the
fine powder, without producing upon the gun and carriage the
destructive effects mentioned above.
CHAPTER II
PROPERTIES OF PERFECT GASES
Mariotte's Law. When a mass of gas is subjected to pressure
the volume diminishes until the increased tension just balances
the pressure; and it was found by experiment that if the tem-
perature of the gas remains constant, the tension, or pressure,
is inversely proportional to the volume. Thus, if Vi and v 2
represent different volumes of the same mass of gas and pi and
p 2 the corresponding tensions, or pressures, then if the tem-
perature is the same for both volumes we have the proportion:
Hence
Vi pi = v- 2 p- 2 = constant.
That is, for every mass of gas at invariable temperature the
product of the volume and tension is constant. This law is
generally called Mariotte's law, though it was first discovered
by the English chemist Robert Boyle, in 1662, and verified by
Mario tte in 1679.
Specific Volume. The specific volume of a gas is the volume
of unit weight at zero temperature and under the normal atmos-
pheric pressure. Designate the specific volume by v and the
normal atmospheric pressure by p . Then we have by Mariotte's
law
Specific Weight. The specific weight of a gas is the weight
of unit volume at zero temperature and under the pressure p .
It is therefore the reciprocal of the specific volume v .
15
1 6 INTERIOR BALLISTICS
Law of Gay-Lussac. The coefficient of expansion of a gas
is the same for all gases, and is sensibly constant for all tem-
peratures and pressures. Let, as before, v be the specific
volume, v t the volume at temperature / and a the coefficient
of expansion. Then the variation of volume by Gay-Lussac's
law will be expressed by the equation
Vt - v o = atv ;
whence
v t = V (i + at)
The value of the coefficient a is approximately - - for
2 73
each degree centigrade. The last equation may, therefore, be
written
' i H ^
Characteristic Equation of the Gaseous State. The last
equation, which expresses Gay-Lussac's law, may be combined
with Mariotte's law, introducing the pressure p. The problem
may be enunciated as follows : Having given the specific volume
of a gas v to determine its volume v t at a temperature t under
the corresponding pressure p t .
Let x be the volume v t would become at o C., under the
pressure p t . Then by Gay-Lussac's law
v t = x ( i + a t)
and by Mariotte's law
Pt* = P O V O ',
whence eliminating x,
Pt /= PoVo (i + = (273 +
c . oo.
Since - - is constant, put
273.
7? **.
K = ~
273
PROPERTIES OF PERFECT GASES 17
whence
p t v t = (273 + /);
or, dropping the subscripts as no longer necessary,
pv = R(2J3 + t)
The temperature (273 + /) is called the absolute tempera-
ture, and is reckoned from a zero placed 273 degrees below the
zero of the centigrade scale. Calling the absolute temperature
T there results finally
pv = RT ...... ( i)
which is called the characteristic equation of the gaseous state.
It is simply another expression of Mariotte's law in which the
temperature of the gas is introduced.
Equation (i) expresses the relation existing between the
pressure, volume and absolute temperature of a unit weight of
gas. For any number y units of weight occupying the same
volume v the relation evidently becomes
pv=yRT ...... (2)
A gas supposed to obey exactly the law expressed in equation
(i) is called a perfect gas, or is said to be theoretically in the
perfectly gaseous state. This condition represents an ideal
toward which gases approach more nearly as their state of
rarefaction increases. Of all gases, hydrogen approximates
most closely to such an hypothetical substance, though at
ordinary temperatures the simple gases, nitrogen, oxygen and
atmospheric air, may for most practical purposes be considered
perfect gases.
Thermal Unit. The heat required to raise the temperature
of unit weight of water at the freezing point one degree of the
thermometer is called a thermal unit. There are two thermal
units in general use, namely: the British thermal unit (B. T. U.),
which is the heat required to raise the temperature of one pound
1 8 INTERIOR BALLISTICS
of water from 32 F. to 33 F.; and the French thermal unit
(called a calorie), which is the heat required to raise the tem-
perature of one kilogram of water from o C. to i C. There
is still another thermal unit of frequent use, namely: the heat
required to raise the temperature of one pound of water from
o C. to i C., and which may be designated as the pound-
centigrade (P. C.) unit.
Mechanical Equivalent of Heat. The mechanical equivalent
of heat is the work equivalent of a thermal unit, and will be
designated by E. According to Rowland the value of E is
778 foot-pounds for a B. T. U. Since a degree of the centi-
grade scale is of a degree of the Fahrenheit scale, we have
for a P. C. thermal unit, E = X 778 = 1400.4 foot-pounds.
3
Also since there are 3.280869 feet in a metre, the value of E for
a calorie is
1400 4
3. 280869 = 42<5 ' 84 kil g ram - metres -
Specific Heat. The quantity of heat, expressed in thermal
units, which must be imparted to a unit weight of any sub-
stance to increase its temperature one degree of the thermometer,
or the quantity of heat given up by the substance while its
temperature is lowered one degree, is called its specific heat.
The specific heat of different substances varies greatly. Thus,
if a pound of mercury and a pound of water receive the same
quantity of heat the temperature of the former will be much
greater that the latter. Indeed, it requires about 32 times as
much heat to raise the temperature of water i as it does to
raise the temperature of mercury by the same amount.
The heat imparted to a substance is expended in three
different ways: i. Increasing the temperature, which may
be called vibration work; 2. In doing internal or disgregation
work; 3. In doing external work by expansion. If it were
PROPERTIES OF PERFECT GASES IQ
possible to eliminate the two latter, we should get the true
specific heat, or the heat necessary to increase the temperature
simply. For a perfect gas, however, the disgregation work is
zero, and for all substances the disgregation work is small in
comparison with the vibration work. The specific heat of a
gas may be determined in two different ways, giving results
which are of fundamental importance in thermodynamics,
namely: Specific heat under constant pressure, and specific
heat under constant volume.
Specific Heat of a Gas Under Constant Pressure. To
fix the ideas suppose a unit weight of gas to be confined in a
spherical envelope capable of expanding without the expenditure
of work and which allows no heat the gas may have to escape,
and to be in" equilibrium with the constant pressure of the at-
mosphere. Under these conditions let a certain quantity of
heat be applied to the gas just sufficient to raise its temperature
one degree of the thermometer after it has expanded until
equilibrium is again restored. This quantity of heat, in thermal
units (designated by C p ), is called specific heat under constant
pressure.
Specific Heat Under Constant Volume. Next repeat the
experiment just described, but replacing the elastic envelope,
which by hypothesis permitted the gas to expand freely, by a
rigid envelope, thus keeping the volume of the gas constant
while heat is applied. It will now be found that there will less
heat be required to raise the temperature of the gas one degree.
The quantity of heat required in this case is called the specific
heat under constant volume, and in terms of the thermal unit
employed, is designated by C v .
The number of molecules of gas being the same in both
experiments and the temperatures being equal, it follows that
the quantity of heat absorbed by the gas, or the vibration work,
is the same in both experiments. But in the experiment made
under constant volume the heat absorbed is necessarily equal
20 INTERIOR BALLISTICS
to the total heat supplied, namely, C v thermal units, since the
envelope is considered impermeable to heat. Therefore in the
first experiment there is a loss of heat equal to C p C v thermal
units. This last heat must then have been expended in over-
coming the atmospheric pressure in expanding; and the work
done will be found by multiplying C p C v by the mechanical
equivalent of heat. That is, for an increase of one degree of
temperature,
Work of expansion = (C p - C v ) E.
The work of overcoming a constant resistance is measured
by the product of the resistance into the path described. In
the case of the expanding gas just considered the constant re-
sistance is the atmospheric resistance p \ and the path described
is measured by the increase of volume of the gas. To determine
this latter Gay-Lussac's law gives for the centigrade scale
tV
Vt VQ = ' -
2 73
and therefore for an increase of temperature of one degree there
is an increase of volume equal to ^0/273. The work of expansion
for one degree is, therefore,
*
2 73
The quantity R is, then, the external work of expansion
performed under atmospheric pressure by unit weight of gas
when its temperature is raised one degree centigrade. But
this work of expansion has already been found equal to (C p C v )
E. There results, therefore, the important equation
(C V -C V }E = ^ = R . . . . (3)
for the centigrade scale of temperature. For the Fahrenheit
scale the equation becomes
(C p -C V )E = ^- fa 1 )
491.4
PROPERTIES OF PERFECT GASES 21
Numerical Value of R. The numerical value of R for any
particular gas depends upon the units of length and weight
adopted, the atmospheric pressure, the specific weight of the
gas and the scale of temperature. Throughout this chapter
the foot and pound will be employed for the units of length and
weight, respectively; and generally the centigrade scale of
temperature will be used. The adopted value of the atmos-
pheric pressure is
Po = 10333 kgs. per m. 2 log = 4-01423-
p = 2116.3 Ibs. per ft. 2 lo g = 3-3 2 55 8 -
p = 14.6967 Ibs. per in. 3 log = 1.16722.
As an example, find the numerical value of R for atmospheric
air. The specific weight of this gas, according to the best
authorities, is 0.080704 Ibs. The specific volume is the recip-
rocal of this; or V = 12.3909 c. ft. Therefore,
2116.3 X 12.3000
R = JV = 96.056 foot-pounds.
273
Therefore, for one pound of this gas,
p v = 96.056 T]
and for y pounds
p v = 96.056 y T.
Law of Dulong and Petit. The product of the specific heat
of a perfect gas under constant volume, by its density, is a constant
number.
By the density of a gas is meant its specific weight expressed
in terms of the specific weight of atmospheric air taken as unity.
If C va is the specific heat of air at constant volume and C v and d
the specific heat at constant volume, and density, respectively,
of any other gas, then in accordance with this law,
C,d = C m .
Determination of Specific Heats. The specific volume
and the specific heat at constant pressure of a gas can both be
22
INTERIOR BALLISTICS
determined with great accuracy by experiment; but the specific
heat under constant volume is almost impossible to measure
directly on account of the dissipation of heat through the sides
of the vessel containing the gas. It can, however, be computed
by equation (3) which gives
C v = C p - - (4)
By means of this equation and the direct determination of
specific heats under constant pressure, Regnault has deduced
the following law for perfect gases :
The specific heats under constant pressure and constant volume
are independent of the pressure and volume.
The following table gives the specific weights, volumes and
heats of those gases which approximate most nearly to the
theoretically perfect gas. The values of R were computed
by (i) and those of C v by (4). The temperature is supposed
to be o C., and the barometer to stand at 760 mm. = 29.922 in. :
Gas
Specific
Weight
Specific
Volume
R
C P
C v
Atmospheric air
Pounds
0.080704
Cubic Feet
12.3909
96 . 056
0.23751
o. 16892
Nitrogen
0.078394
12.7569
98.887
0.24380
O.I73I9
Oxygen
Hydrogen
0.089230
o OOSSQO
II .2070
178 8910
86.878
1386.8
0.21751
3 . 40900
0.15547
2.41873
Ratio of Specific Heats. In the study of interior ballistics
the values of C p and C v for the gases given off by the explosion
of the charge are of little importance. It suffices generally to
know their ratio which is constant for perfect gases and approxi-
mately so for all gases at the high temperature of explosion.
That this ratio is constant for perfect gases may be shown as
follows : Since
PROPERTIES OF PERFECT GASES 23
R = P V = P = P
273 273 w 273 d w a
in which w a is the specific weight of atmospheric air, we shall
have for two gases distinguished by accents, the relation
that is, the values of R for two perfect gases are inversely as
their densities. But by the law of Dulong and Petit we have
C, d" R'
T^r = ~r f ~n, (as shown above).
Therefore
R' R"
-~r = r^r = constant.
^ V ^ V
Therefore from equation (4),
c* /?
-:- = i + -T^-E = constant = n (say).
C v C^zi
If we compute n by means of atmospheric air, we shall have
96.056
n = i + -77) - = 1.406.
0.16892 X 1400.4
Relations Between Heat and Work in the Expansion of
Perfect Gases. The relations which exist between the varia-
tions of the volume and pressure of a given weight of gas and
the heat necessary to produce them, may now be determined
from equation (i) as follows: This equation is
pv = RT
and contains three arbitrary variables p, v and T. If we suppose
an element of heat, d q, to be applied to the gas, the temperature
will generally be augmented by an elementary amount d T,
and this may be accomplished in three different ways :
i. The volume may increase by the element dv without
altering the pressure. 2. The pressure may increase by d p
24 INTERIOR BALLISTICS
while the volume remains constant. 3. The volume and
pressure may both vary at the same time. We will consider
each of these cases separately.
i. Differentiating (i), supposing p constant, we have
and therefore the quantity of heat communicated to the gas
will be, in thermal units, from the definition of specific heat,
,/ r IT C pP dv
dq = CpdT
2. If, the volume v remaining constant, the pressure is
varied by d p, we shall have, proceeding as before,
A r ir C v vd P
a q = L v a 1 5
3. If the volume and pressure vary together, the corre-
sponding element of heat will be the sum of the partial variations
given above. That is
dq = -j(C p pdv + C v vdp) ... (5)
The differential of (i) is
RdT = pdv + vdp; . . . . (6)
whence, eliminating v d p between (5) and (6), there results
dq = C v dT + Cp ~ R Cv pdv ... (7)
Whence, since C p , C v and R are constants for the same gas,
/c c r
dT+-^ J pdv.
The first integral represents the change of temperature and
the second the external work of expansion. Denoting by 7\
and T the initial and final temperatures of the expanding gas
and by W the external work, we have
q = (7\ - T) C v + -^ W . . . (8)
PROPERTIES OF PERFECT GASES 25
Isothermal Expansion. If we suppose the initial temperature
TI to remain constant, that is, that just sufficient heat is im-
parted to the gas while it expands to maintain its initial tem-
perature, equation (8) becomes
We see in this case that the quantity of heat absorbed by
the gas is proportional to the external work done. The quantity
r>
-~r - ^r is, therefore, the ratio of the effective work of a unit
weight of gas to the quantity of heat absorbed, or the mechanical
equivalent of heat, E. Therefore
E = c^c,
a result already established by another method.
The work performed, therefore, by the isothermal expansion
of unit weight of gas is given by the equation
W = E q = 1400.4 q foot-pounds, ... (9)
where q is expressed in P. C. thermal units.
The work of an isothermal expansion may also be expressed
in terms of the initial and final volumes or pressures. Thus,
substituting in the general equation of the work of expansion,
W =-- fpdv,
the value of p from (i) and integrating between the limits Vi
and v, we have
W = R 7\ log, ^ = #1 i log, ~ v (10)
where v is the greater volume and Vi the less.
Since from (i)
^__!i
Vi ' p
26 INTERIOR BALLISTICS
we also have
W = p l v l log e ^ ..... (n)
in which pi is the greater tension and p the less.
The reciprocal of E may be called the heat equivalent of
work, that is, the quantity of heat equivalent to a unit of work.
Therefore from (9), (10) and (n), we have
W Pl Vl W v '
q ~- = ~E : ~E~ log ^|
p^ Pl ' ' ' (>
loge p j
Equations (10) and (n), by inverting the ratios of volumes
or pressures, evidently hold good when the initial volume Vi
and initial tension pi are changed by compression under constant
temperature into the less volume v and greater tension p.
Adiabatic Expansion. If a gas expands and performs work
in an envelope impermeable to heat, so that it neither receives
nor gives up heat during the expansion, the transformation is
said to be adiabatic. In such an expansion the temperature
and tension of the gas both diminish and the work performed
must be less than for an isothermal expansion, other things
being equal. For an adiabatic expansion, q is zero in (8) and,
therefore, since the temperature diminishes,
P F = 7 r(r I -r)i
l^p I-'?
= _*_ . ....
= C V E (iV- T)\
Therefore in an adiabatic expansion the work done is pro-
portional to the fall of temperature.
Next consider equation (7), where, if we make dq zero, it
becomes
PROPERTIES OF PERFECT GASES 27
/? T
which, by dividing by C v and replacing p by its value , re-
duces to
_ dT dv
~T ~ ~ ~v
Integrating between limits, we have
T
T, "
Again, making d q zero in (5), we have
o = C p p d v + C v v d p,
which may be written (dividing by C v p v)
d v dp
Integrating between limits, we have
Combining (14) and (15) gives the important relations
- <>
By means of (16) the work of an adiabatic expansion given
by (13) may be expressed either in terms of the initial and
terminal volumes, or of the initial and terminal pressures. Thus,
since
the last of equations (13) may be written,
28 INTERIOR BALLISTICS
EXAMPLES.
1 . Determine the volume of 5 pounds of oxygen at a pressure
of 50 pounds per square inch by the gauge, and at a temperature
of 60 C.
The real pressure is gauge pressure plus the atmospheric
pressure = 50 + 14.6967 = 64.6967 Ibs. per in. 2 Therefore,
p = 144 X 64.6967 Ibs. per ft. 2 T = 273 + 60 = 333. R =
86.878. Therefore from (2),
5 X 86.878 X 333 ,, 3
144 X 64.6967
2. One pound of atmospheric air occupying a volume of
one cubic foot has a tension of 50,000 Ibs. per ft. 2 What is its
temperature by the Fahrenheit scale ?
For the centigrade scale we have R = 96.056, v = i, p =
50,000 and y = i.
Therefore T = ~ = 5 2O. 54 C. - 9 68. 97 F.
.*. / = 968.97 - 491.4 = 477-57 F.
3. A gas-receiver having a volume of 3 cubic feet contains
half a pound of oxygen at 70 F. What is the pressure by the
gauge ?
Here y = ^, v = 3, R = 86.876, t = 21 - C., and T = 294!.
Therefore,
86.876 X 2941-
^ = % v 7 v TA,~ ~ J 4-697 = 14-876 Ibs. per in.
A 3 A 144
4. A spherical balloon 20 feet in diameter is to be inflated
with hydrogen at 60 F., when the barometer stands at 30.2 in.,
so that gas may not be lost on account of expansion when the
PROPERTIES OF PERFECT GASES 29
balloon has risen till the barometer stands at 19.6 in., and the
temperature falls to 40 F. How many pounds and how many
cubic feet of gas are to be run in?
Here v = -- * X io 3 = 4188.8 ft. 3
= *9- 6 X *" 6 -3 = I3 86.8 lbs. per ft.'
29.9215
T = 277! C.
R = 1386.8.
p V
' y = ~D~T = I 5-9 2 lbs.
To determine the number of cubic feet of gas run. in, we have
yR T
v = S-j = 2827.4 ft. 3 ,
where
30. 2 X 2116.3
29.9215
= 2136.0.
2P= -1 (60 _ 32) + 273 = 2 88f.
5. "The balloon in which Wellman intends to seek the North
Pole has a capacity of 2 24,244 cubic feet, and weighs, with its
car and machinery, 6,600 lbs. What will be its lifting capacity
when filled with hydrogen at 10 C. and 760 mm. of the ba-
rometer ?" (Lissak's " Ordnance and Gunnery," p. 61.)
The balloon, when inflated, will hold at 10 C., 17,458 lbs.
of air and 1,209 Mbs. f hydrogen. Its lifting capacity will.
therefore, be 17,458 (1,209 + 6,600) = 9,649 lbs.
6. Two pounds of air expand adiabatically from an initial
temperature of 60 F., and a pressure of 65.3 lbs. per in. 2 to a
pressure of 50 lbs. per in. 2 Determine the initial and terminal
volumes, the terminal temperature and the external work
done.
30 INTERIOR BALLISTICS
Here p, = 144 X 65.3 = 9403-2; p = 50 X 144 == 7200;
T! = 288! C.; R = 96.056; y = 2. Take n = 1.4
.'.0i= T - = 5-8954 ft. 3
ri
T = T\~) 7 = 267.37 C. = 481.266 F.
:.t = 21.87 F.
W = ^ - (Ti- T) = 10177 ft.-lbs.
7. Compute the work of expansion of 2 pounds of air at
temperature 100 C., which expands adiabatically until it
doubles its volume. Also determine the temperature after
expansion and the ratio of the initial and terminal pressures.
Answers: W = 43378 ft.-lbs.
/ = i9.68 C.
P = 0.3789 />L
8. A mass of air occupying a volume of 3 ft. 3 expands adiabati-
cally from an initial temperature of 70 F., and pressure of
85 Ibs. per in. 2 , until external work of 8,000 ft.-lbs. has been
done. Compute the terminal volume, pressure, temperature,
and weight of air.
Answers: v = 3.768 ft. 3
p = 61.78 Ibs. per in. 2
t = 2 3 .86 F.
y = i. 3 Ibs.
Theoretical Work of an Adiabatic Expansion in the Bore of
a Gun. If, in the first of equations (17), we replace C v E 7\, by
its equal - it becomes for y pounds of gas
ni
n i
( v
PROPERTIES OF PERFECT GASES 3!
This equation gives the work of y pounds of gas at the initial
temperature 7\, expanding from the initial volume v to volume
v. Suppose the mass of gas to occupy the chamber of a gun
with the projectile at its firing seat; and to expand by forcing
the projectile along the bore. In this case v\ will be the volume
of the chamber, which is an enlargement of the bore, and is
measured by what is called the reduced length of the chamber;
that is, by the length of a cylinder whose cross-section is the
same as the bore and whose volume is that of the chamber.
If u is the reduced length required, V c the volume of the chamber
and d the diameter corresponding to the area of cross-section
of bore, and which on account of the rifling is slightly greater
than the caliber, we evidently have
~d 2
The variable volume v is the volume of the chamber plus
the volume of the bore in rear of the projectile after it has
moved any distance u; and is, therefore, measured by u -f u.
Therefore the above expression for the work of expansion be-
comes
W = y ^\i -
n-i (
There is some uncertainty as to the proper value of n for
the gases of fired powder. As we have seen, the value of this
ratio for perfect gases is approximately 1.4; and it has been
generally assumed that at the high temperature of combustion
of powder the gases formed may be regarded as possessing all
the properties of perfect gases ; and therefore most of the earlier
writers on interior ballistics employed this value of n in their
deductions. But more recent experiments have shown that
this value is too great, but have not fixed its true value. The
experiments of Noble and Abel with the gases of fired gunpowder,
at or near the temperature of combustion, made n = i^ nearly;
32 INTERIOR BALLISTICS
and this is the value which, for want of a better, we will adopt
in what follows. Introducing this value of n into the above
expression for the work of expansion; and making R T l = f
and the ratio u/u = x, we have
s/*|'--TrnFl
The work of expansion in the bore of a gun is expended in
many ways, but chiefly in the energy of translation imparted
to the projectile. If we assume that the entire work is thus
expended, we shall have
W W ( (l +JC 3 )
It is evident from (2) that / is the pressure per unit of
surface of unit weight of gas at temperature 7\. The ratio x
is the number of volumes of expansion of y pounds of gas due
to the travel u.
The assumption that the work of expansion is measured by
the energy of translation of the projectile does not change the
form of the second member of (19); and it is evident that by
giving to / a suitable value determined by experiment, the
equality expressed in (19) may be strictly true. But in this
case / ceases to have the value R 7\ and becomes simply an
experimental coefficient.
In English units (pound and foot) , / would be theoretically
the pressure in pounds per square foot of one pound of gas at
temperature T\ confined in a volume of one cubic foot.
In metric units (kilogramme and decimetre), / would be
defined as above, making the proper change of units.
We may deduce a second approximation to the velocity
impressed upon the projectile by the expansion of the gas by
taking into account the work performed upon the gun and
carriage, as well as upon the projectile. We will suppose the
gun mounted upon a free-recoil carriage. Let M be the mass
PROPERTIES OF PERFECT GASES 33
of the gun and carnage, V their velocity at any period of mo-
tion and m the mass of the projectile. The expression for the
work of expansion will now be
2 W = m IT + M V\ . . . . . (20)
A second equation between the velocities v and V can be
deduced by equating the momenta of the system proiected upon
the axis of the gun. We thus obtain
m v = M V ...... (21)
Eliminating V from (20) and (21) there results
2W
This expression for v 2 is the same as that given by (19) with
the exception of the small fraction m/M which can be safely
neglected in comparison with unity. Similarly it may be shown
that the work expended upon the projectile in giving it rota-
tion about its axis is small in comparison with the work of
translation.
Noble and Abel's Researches on Fired Gunpowder. Noble
and Abel's experiments on the explosion of gunpowder in close
vessels were given to the world in two memoirs which were
read before the Royal Society in 1874 and 1879, respectively.
These experiments have an important bearing upon the subject
of interior ballistics, since they furnish the most reliable values
we possess of the temperature of combustion of fired gun-
powder, the mean specific heat of the products of combustion
(solid as well as gaseous), the ratio of solid to gaseous products,
and, lastly, what is known as the force of the powder, all of
which are important factors in computing the work done by the
gases of a charge of gunpowder exploded in the chamber of a gun.
The vessels in which the explosions were produced were of
two sizes, the smaller one for moderate charges and for experi-
34
INTERIOR BALLISTICS
ments connected with the measurement or analysis of the gases,
while in the larger one Captain Noble states that he has succeeded
in absolutely retaining the products of combustion of a charge of
23 pounds of gunpowder.* These vessels consisted of a steel
barrel open at both ends, the two open ends being closed by
carefully fitted screw plugs (firing plug and crusher plug),
furnished with gas checks to prevent any escape of gas past the
screw. In the firing plug was a conical hole closed from within
by a steel cone which was ground into its place with great
exactness, and which, when the cylinder was prepared for firing,
was covered with very fine tissue paper to give it electrical insula-
tion from the rest of the apparatus. The two wires from a
Leclanche battery were attached, the one to the insulated cone
and the other to the firing plug, and were connected within the
powder chamber by a fine platinum wire passing through a
glass tube filled with mealed powder. This platinum wire
became heated when the electric current passed through it, and
the charge was thus fired. At the opposite end of the cylinder
from the firing plug was another plug fitted with a crusher
gauge for determining the pressure of the gases. The vessel
was also provided with an arrangement for collecting the gases
after an explosion for analysis, measurement of quantity, or
for other purposes.
Results of the Experiments. It was found that about
I 57 per cent, by weight of the products of combustion were non-
gaseous, consisting principally of potassium carbonate, potassium
j sulphate, and potassium sulphide, the first named greatly
preponderating. The remaining 43 per cent, were permanent
gases, principally C0 2 , CO and N. These gases, when brought
to a temperature of o C., and under the normal atmospheric
pressure of 760 millimetres, occupied about 280 times the volume
of the unexploded powder.
* Lecture on Internal Ballistics, by Captain Noble, London, 1892, p. 12.
PROPERTIES OF PERFECT GASES
35
Pressure in Close Vessels, Deduced from Theoretical Con-
siderations. The expression for the pressure of the gases
developed by the combustion of gunpowder in a close vessel
is deduced upon the following suppositions :
i st. That a portion of the products of combustion is in a
liquid state.
2d. That the pressure due to the permanent gases can only
be calculated by deducting the volume of the liquid products
from the volume of the vessel.
Upon these hypotheses the expression for the pressure may
be deduced as follows :
Let A B C D be a section of a
close vessel of volume v in which a
given charge of powder is exploded.
Let A E F D represent the space
(vi) occupied by the charge, and A G
H D the space (v 2 ) occupied by the
non-gaseous products. Let A t be the
so-called density of the products of
combustion, that is AI = -- ; and
B
OL the ratio of the non-gaseous prod-
ucts to the volume of the charge, or
a =
V,
2-. The gases after ex-
v
D
plosion will occupy the space v v 2 = v AI z; = v (i a AI).
Let pi be the pressure that would be developed if the volume
of the vessel were A E F D (or z;,). In this case the density
of the products of combustion (A t ) (the charge remaining the
same) would be unity; and the space occupied by the gases
would be Vi v 2 = Vi (i a) = A t v (i ). Now if p is the
pressure when the volume of the vessel is t>, we have by Mariotte's
law (assuming that the temperature is the same for all densities
of the products of combustion),
36 INTERIOR BALLISTICS
A I T^(I a r )
or, making
we have
I - a
The factor /is called the force of the powder.
Value of the Ratio . Let p 2 and p s be the pressures in
the same vessel produced by two different charges, and A 2 and
A 3 the corresponding densities of the products of combustion.
Then from equation (23) (assuming/ to be the same for all values
of AO,
A-,
and
whence by division,
I-A 2 ^ 3 A 2 *
Therefore
3 A 2 -/> 2 A 3 )
=
A A
by means of which the mean value of a can be determined when
a sufficient number of pressures, corresponding to different
values of AI, have been found by experiment. The value of
finally adopted by Noble and Abel is 0.57.
Determination of the Force of the Powder. To determine/
we have from equation (23),
i .=57 A,
f-t- A ;
**1
from which / may be found by means of a single measured
pressure corresponding to a given density of the products of
PROPERTIES OF PERFECT GASES 37
combustion. When A t = i, that is, when the vessel is completely
filled by the charge, p was found to be 43 tons per square inch,
and therefore / = 43 (i .57) = 18.49 tons or 41417.6 pounds
per square inch. Therefore Noble and Abel's formula for the
pressure in a close vessel is, for different densities of the products
of combustion,
A!
p = 18.49 ~~ r tons per sq. in.
~t
= 41417.6 - - Ibs. per sq. in.
1 ~ -57 **i
To transform this equation so that it shall express the press-
ure in kilos per dm. 2 we may employ a simple rule which, as it is
of frequent use, is here inserted for convenience :
RULE: To reduce a pressure expressed in tons per square
inch to the same pressure expressed in kilos per dm. 2 , add to the
logarithm of the former the constant logarithm 4.1972544 and
the sum is the logarithm of the pressure required.
If the pressure to be reduced is in pounds per in. 2 then the
constant logarithm to be added is 0.8470064.
Applying this rule the expression for the pressure of the
products of combustion of a charge of gunpowder fired in a close
vessel is found to be
p = 291200
-.57 !
. ' . / 291200 kilos per dm. 2
It will be seen from the definition given to A t that it is the
density of loading as defined in Chapter III when the gravi-
metric density of the powder is unity, that is, when a kilo of
the powder fills a volume of a dm. 3 ; or, what is the same thing,
when a pound occupies a volume of 27.68 cubic inches; and in
this case, when A t is unity the charge just fills the receptacle.
Noble and Abel were careful to keep the gravimetric density
of the powder they experimented with as near unity as possible.
38 INTERIOR BALLISTICS
Interpretation of f . It will be seen from Equation (23) that
the quantity designated by / is the pressure of the gases when
(i- a
that is, when the space occupied by the gases is equal to the
volume of the charge, which requires that the vessel should have
i -j- OL units of volume. Thus if the kilogramme and litre are the
units of weight and volume, respectively, the volume of the
vessel must be 1.57 litres in order that the gases may occupy a
volume of one litre, and have a tension equal to /. From this
/ may be denned to be the pressure of the gases of unit weight of
powder occupying unit volume at the temperature of combustion
r,.
If e is the weight of gas furnished by the combustion of unit
weight of powder we have from Equation (2),
p! 1)i = R Tij
and if Vi is the unit of volume, there results
p l =f= RT 1 ..... (25)
If the pound is the unit of weight the unit of volume is 27.68
cubic inches. In this case the definition of / requires that the
volume of the vessel should be 1.57 X 27.68 = 43459 cubic
inches.
The value of e, according to Noble and Abel, is 0.43; and
therefore the pressure of unit weight of the gases of fired gun-
powder at temperature TI is
0-43*
From this it follows that the pressure of one pound of the gases
of fired gunpowder at temperature of combustion, confined in a
volume of 27.68 cubic inches, is
41417-6
- = 96320 Ibs. per square inch.
PROPERTIES OF PERFECT GASES 39
Also, the pressure of one pound of the gases of the paragraph
immediately preceding, confined in a volume of one cubic foot,
is, in pounds per square foot,
06320 X 27.68
= 222180 &g.
12
If the gravimetric density of the powder be unity, and y and
v be taken in pounds and cubic inches, respectively, then Equa-
tion (23) becomes
Solving with reference to y and to v gives
pv
2 7 .68 (a ;
and
v = 27 .6 & y(a P+ f) (2g)
These equations are useful in questions involving the bursting
of shells, etc.
Theoretical Determination of the Temperature of Explosion
of Gunpowder. Having determined the value of / from the
experiments, we can deduce the temperature of explosion by
means of the formula
T! 2 ?3 f
According to Noble and Abel's experiments, if the gravi-
metric density of the powder is such that a kilogramme occupies
one litre, the gases furnished by its combustion will fill a volume
of 280 litres at o C. under the normal atmospheric pressure of
103.33 kgs. per square decimetre. We therefore have
280
v ~ y *
and
Po = I0 3-33
40 INTERIOR BALLISTICS
whence
273 X 291200
C '
> 103.33 X 280
This is the absolute temperature of combustion of gunpowder
according to Noble and Abel's latest deductions from their ex-
periments. Subtracting 273 from this temperature we have
temperature of explosion = 2475 C. (4487 F.).
Mean Specific Heat of the Products of Combustion. From
equation (8), we have when W = o, that is, when no external
work is performed,
Q = C, (T, - 273)
in which Q is the heat of combustion; that is, the quantity of
heat that unit of weight of the explosive substance evolves, under
constant volume, when the final temperature of the products of
combustion is o C. From this equation we find
c
- 273
The heat of combustion was determined by Noble and Abel
in the following manner:
"A charge of powder was weighed and placed in one of the
smaller cylinders, which was kept for some hours in a room of
very uniform temperature. When the apparatus was through-
out of the same temperature, the thermometer was read, the
cylinder closed, and the charge exploded.
1 ' Immediately after explosion the cylinder was placed in a
calorimeter containing a given weight of water at a measured
temperature, the vessel being carefully protected from radiation,
and its calorific value in water having been previously deter-
mined.
' ' The uniform transmission of heat through the entire volume
of water was maintained by agitation of the liquid, and the
thermometer was read every five minutes until the maximum
PROPERTIES OF PERFECT GASES 41
was reached. The observations were then continued for an equal
time to determine the loss of heat in the calorimeter due to
radiation, etc.; the amount so determined was added to the
maximum temperature."
In this way the heat of combustion of R. L. G. and F. G.
powders was found to be 705 heat-units; that is, the combustion
of a unit weight of the powder liberated sufficient heat to raise
the temperature of 705 unit- weights of water i C. We there-
fore have
This result is accepted by Noble and Abel, and also by Sarrau,
as a very close approximation to the mean specific heat of the
entire products of combustion. If we assume that the mean
specific heat of gunpowder of different compositions is constant,
we can compute the temperatures of combustion when the heat
of combustion has been determined by the calorimeter, by the
formula
T Q
0.285
in which T is given by the centigrade scale.
Pressure in the Bores of Guns Derived from Theoretical
Considerations. "At an early stage in our researches, when we
found, contrary to our expectation, that the elastic pressure de-
duced from experiments in close vessels did not differ greatly
(where the powder might be considered entirely consumed, or
nearly so) from those deduced from experiments in the bores of
guns themselves, we came to the conclusion that this departure
from our expectation was probably due to the heat stored up in
the liquid residue. In fact, instead of the expansion of the per-
manent gases taking place without addition of heat, the residue,
in the finely divided state in which it must be on the ignition of
the charge, may be considered a source of heat of the most per-
42 INTERIOR BALLISTICS
feet character, and available for compensating the cooling effect
due to the expansion of the gases on production of work.
"The question, then, that we now have to consider is What
will be the conditions of expansion of the permanent gases when
dilating in the bore of a gun and drawing heat, during their ex-
pansion, from the non-gaseous portions in a very finely divided
state?"*
Let c t be the specific heat of the non-gaseous portion of the
charge, which we can assume, without material error, to be con-
stant. We shall then have c t d T for the elementary quantity of
heat yielded to the gases per unit of weight of liquid residue. If
there are w t units of weight of liquid residue it will yield to the
gases w v c t d T units of heat; and if there are w 2 units of weight
of gas we shall have in heat-units,
in which
that is, is the ratio between the weights of the non-gaseous
and gaseous portions of the charge. The negative sign is given
to the second member because T decreases while q increases.
Substituting the above value of d q in Equation (7), it be-
comes
-(C v + p Cl )dT = C -^pdv . . . . (29)
and this combined with Equation (6), gives, by a slight reduction,
- (ft c, + )? -(fa+CJ^ . . . (30)
Since C p , C VJ Ci and p are, by hypothesis, constant during the
expansion, the integration of Equation (30) between the limits v 2
* Noble and Abel, Researches, etc., page 98.
PROPERTIES OF PERFECT GASES
43
and v s the former being the initial volume occupied by the per-
manent gases and the latter their volume after the projectile has
been displaced by a distance u, gives
in which
r =
Equation (31), it will be seen, becomes identical with Equa-
tion (15), when /? = o; that is, when there is no liquid residue.
To introduce Vi and v, that is the volumes occupied by the
charge and the entire volume in the rear of the projectile, into
Equation (30) in place of v 2 and z> 3 , proceed as follows: Let
ABC D
a?!
E F G H
A C EG represent the chamber of the gun, which we will suppose
filled with powder without compression, and further that one
pound of the powder fills a space of 27.68 cubic inches. The
gravimetric density and density of loading are each unity; and
if Vi is the volume of the chamber, it follows that
Vi = 27.68 w.
co being the weight of charge.
Suppose the powder to be entirely consumed before the pro-
jectile moves any perceptible distance; and that the non-gaseous
products fill the space A B E F, whose volume is a z>,. The gases,
44 INTERIOR BALLISTICS
therefore, which by their expansion give motion to the projectile
will occupy the space B C F G before perceptible motion begins.
The volume of the space B C F G is evidently v 2 = v t a ^ =
Vi (i a). Let D H be the base of the projectile after it has
moved a distance ; and, designating the volume A D E H by v,
we evidently have v 3 = v a v lt Substituting these values of
v 2 and v s in Equation (31) gives
In this equation ^ is the pressure produced by the combustion
of a charge of powder in a close vessel when the density of load-
ing is unity. The values of the constants are given by Noble and
Abel as follows:*
pi = 43 tons per square inch
a = 0.57
^ = 1-2957
C p = 0.2324
C v = 0.1762
c t = 0.45
Vi = 27.68 <o
from which we find r = 1.074. Substituting these values in the
expression for p it becomes
.
= 43 -
which gives the pressure in tons per square inch.
If, as in a close vessel, we let
then
~ -57 A!
Researches, etc., page 167,
PROPERTIES OF PERFECT GASES 45
A j / r - 74
REMARKS: The value of ft = 1.2957, adopted by Noble and
Abel, gives for unit weight, Wi = 0.5644 and w 2 = 0.4356, while
the values of these quantities adopted in our equations are 0.57
and 0.43, respectively. These last-named values would make
P = 1.3256.
Noble and Abel's values of the specific heats of the permanent
gases of combustion, namely, C p = 0.2324 and C v = 0.1762, make
n = 1.32; while for perfect gases, as has been shown, n = 1.4
very nearly.
Table of Pressures. In the following table of pressures the
third column gives the pressures in the bore of a gun correspond-
ing to the values of AI in the first column. They were computed
by Equation (34) upon the assumption that the permanent gases
in expanding, and thereby doing work, borrow heat from the non-
gaseous residue; and also that the combustion is complete before
the projectile has moved perceptibly; and finally that there is no
conduction of heat to the walls of the gun. The tensions in the
fifth column were computed by Equation (24) and agree with
Noble and Abel's experiments.*
Temperatures of Products of Combustion in Bores of Guns.
The temperature in the bore of a gun during the expansion of
the products of combustion, may be determined from Equation
(29), which replacing R by its value from Equation (i), becomes
dT C-C<
p C/ + C v
whence integrating between the same limits as before, and ob-
serving that
7T~7T^ = r - i,
'V
* For table of pressures see page 46.
46
INTERIOR BALLISTICS
TABLE OF PRESSURES.
f
Mean density
of products
of combustion.
\
Corre-
sponding ex-
pansions.
I
"A^
Tensions calculated by
Equation (33).
Tensions in close cylinders,
or where gases expand
without doing work.
Tons per square
inch.
Differences.
Tons per square
inch.
Differences.
1 .00
.000
43-oo
5.01
43.00
4.69
95
053
37-99
4.40
38.31
4.14
.90
.in
33-59
3-88
34-17
3.68
85
.176
29.72
3-44
30-49
3-30
.80
.250
26.28
3-06
27.19
2.97
75
333
23.22
2.76
24.22
2.68
.70
.429
20.46
2.48
21.54
2-45
65
539
17.98
2.25
19.09
2.23
.60
.667
15-73
2.04
16.86
2.05
55
.818
13.69
.86
14.81
1.88
50
2.000
11.83
.70
12.93
i-74
45
2 .222
10.13
-56
II .19
1.61
.40
2.500
8-57
-43
9-58
50
35
2.857
7-H
31
8.08
-39
30
3-333
5-83
.21
6.69
30
25
4.000
4.62
I .11
5-39
.22
.20
5 . ooo
3-51
I .02
4-1?
H
15
6.667
2.49
93
3-03
.07
.10
10.000
1-56
.84
1.96
.01
05
20 . ooo ! . 72
-95
....
we have
Ti l-r
r i
Replacing v 2 by Vi (i a), and v s by v v i} for reasons
already given, we have for the absolute temperature of the gases
during expansion, the equation
T .
- a\r - i
a
Introducing the density of the products of combustion (Ai),
and the numerical values of and r into this last equation, it
becomes
PROPERTIES OF PERFECT GASES 47
= 0.93946 7\ | i _ l A ~ | ... (35)
The value of T for any given density of the products of com-
bustion (represented by A x ) will depend upon their initial tem-
perature (or absolute temperature of combustion), T v . Its
theoretical value, based upon Noble and Abel's latest deductions
from their experiments, as published in their second memoir,
has already been found to be 2748 C. But there are very great
difficulties in the way of verifying by experiment the theoretical
value of TI , and Captain Noble in his Greenock lecture (February
1 2th, 1892) takes the absolute temperature of combustion at
2505 C., as deduced in their first memoir. Therefore making
r, = 2505 c.,
the expression for T becomes
r = 2 353-3J I _ Al 57Ai [' 74 (36)
The temperatures in degrees Centigrade and Fahrenheit, cal-
culated from Equation (36), are given in the following table.
"It is hardly necessary to point out that the values given in this
table are only strictly accurate when the charge is ignited before
the projectile is sensibly moved; but in practice the correction
due to this cause will not be great."
Theoretical Work Effected by Gunpowder. The theoretical
work which a charge of gunpowder is capable of effecting during
the expansion of its volume from ^ to any volume v is expressed
by the definite integral
W = pdv,
For table of temperatures see next page.
4 8
INTERIOR BALLISTICS
TABLE OF TEMPERATURES.
Mean density of prod-
ucts of combustion.
A!
Number of volumes of
expansion.
I
Al
TEMPERATURES.
Centigrade.
Fahrenheit.
I.OO
.0000
2231
4048
.95
.0526
2210
4010
.90
.1111
2189
3972
.85
1765
2168
3934
.80
.2500
2147
3897
.75
3333
2126
3859
.70
.4286
2106
3823
.65
.5385
2085
3785
.60
.6667
2063
3745
55
.8182
2041
3706
50
2 . OOOO
2018
3664 '
45
2.2222
1994
3621
.40
2 . 5000
1968
3574
35
2.8571
1940
3524
30
3-3333
1909
3468
25
4 . oooo
I8 74
345
.20
5 . oooo
1834
3333
15
6.6667
1785
3245
.10
10.0000
1719
3126
05
20 . OOOO
1615
2939
.00
00
or, substituting for p its value from (32),
v r dv
= pi vAl a) I -, - ^-',
' A>! (v aVj)"
whence, integrating, we have
W = ^- i
-. r i
Multiplying and dividing the second member by hi (i )]' ',
we have
(r -
PROPERTIES OF PERFECT GASES 49
If, in this last equation, pi be expressed in kilogrammes per
square decimetre, and Vi be made unity (one litre), the work will
be expressed in decimetre-kilogrammes per kilogramme of
powder burned. To express the work in foot-tons per pound of
powder burned, we must make Vi = 27.68 cubic inches; and
then, since pi is given in tons per square inch, divide the result
by 12, the number of inches in a foot. Making these substitu-
tions and replacing <* and r by their values already given, we
have, in foot-tons,
( / 43^ N 1 - 1
W = 576.369 i - L '
D /
or, in terms of A 1?
i
= 576.369! i - 0.93946 (-^^) j" (37)
Substituting in Equation (37) from Equation (35) we have
or, since, according to Noble and Abel,
ro
i = 2 55
we have
W = 0.23008 (Tt-T) ..... (38)
which gives the work in terms of the loss of temperature of the
products of combustion.
Table III gives the work of expansion of the gases of one
pound of gunpowder of the normal type and free from moisture,
computed by Equation (37) . By means of the work given in this
table, and by the use of a proper factor of effect determined by
experiment, Noble and Abel consider that the actual work of a
given charge of powder upon a projectile may be computed with
considerable accuracy. Their method of using this table will be
clearly seen by the following extract:
"If we wish to know the maximum work of a given charge,
4
50 INTERIOR BALLISTICS
fired in a gun with such capacity of bore that the charge suffered
five expansions (A! = 0.2) during the motion of the projectile in
the gun, the density of loading being unity, the table shows us
that for every pound in the charge, an energy of 91.4 foot- tons
will as a maximum be generated.
"If the factor of effect for the powder and gun be known, the
above values, multiplied by that factor, will give the energy per
pound that may be expected to be realized in the projectile.
"But it rarely happens, especially with the very large charges
used in the most recent guns that densities of loading so high as
unity are employed; and in such cases, from the total energy
realizable must be deducted the energy which the powder would
have generated, had it expanded from a density of unity to that
actually occupied by the charge. Thus in the example above
given, if we suppose the charge instead of a density of loading
of unity to have a density of 0.8, we see from Table 3, that from
the 91.4 foot-tons above given, there must be subtracted 19.23
foot- tons; leaving 72.17 foot- tons as the maximum energy realiz-
able under the given conditions, per pound of the charge."
To apply these principles practically for muzzle "velocities, let,
as before,
Vi be the volume occupied by the charge, in cubic inches.
v the total volume of bore and chamber, in cubic inches.
V b the volume of the bore.
V c the volume of the chamber, in cubic inches.
Then
v=V b +V c ;
and, if the gravimetric density of the powder be unity,
Vi = 27.68 co,
where co is the weight of the charge in pounds. Therefore the
* Noble and Abel, Researches, page 176.
PROPERTIES OF PERFECT GASES 51
number of volumes of expansion of the products of combustion
will be, at the muzzle,
JL s .i _A_ ,i.
l\ ~ A! 27.68 co "V
which may be written, if the gravimetric density of the powder
be unity,
= 0.0361263-^+
A! u A
in which A is the density of loading as denned in Chapter III.
If the gravimetric density of the powder be not unity, let v 2
be the volume in cubic inches of one pound of powder not pressed
together except by its own weight; and let
27.68
- = m\
^
then we have in all cases,
^ = m \ 0.0361263^+^
in which is the number of volumes of expansion of the prod-
ucts of combustion.
Let W 2 be the work taken from Noble and Abel's table (Table
III) of the gases of one pound of powder for a given value of
, and Wi the work due to the expansion -. Also, let F
be the factor of effect. Then if we assume that the work of
expansion is all expressed in the energy of translation of the
projectile, we shall have approximately,
- FW * ...... (39)
in which w is the weight of the projectile and
W = W 2 - W l
From (39) the muzzle velocity v may be computed when the
52 INTERIOR BALLISTICS
factor of effect is known; or, we may determine the factor of
effect when the muzzle velocity has been measured by a chrono-
graph. These two equations reduced to practical forms are the
following :
= 379-57 \IFW- .... (40)
\ w
and
F = 0.000006041 -fjT . . . . (41)
w &
As an illustrative application of these formulas to interior
ballistics take the following data from Noble and Abel's second
memoir, relative to the English 8-inch gun: It was found by
firing a charge of 70 pounds of a certain brand of pebble powder,
with a projectile weighing 180 pounds, that a muzzle velocity of
1694 foot-seconds was obtained. What was the factor of effect
(F) pertaining to this gun and brand of powder ? For this
particular gun and charge we have w = 70 pounds, w = 180
pounds, A! = 0.1634, A = 0.605 andw = i. In Noble and Abel's
table of work (Table III) the first column gives values of ,
increasing by a common difference, while the second column con-
tains the corresponding values of A t . By a simple interpolation
we find for the values of A t and A given above, W 2 = 99.4 and
Wi = 37.6; whence W = 61.8 foot-tons. Substituting these
values in Equation (41) we have
1 80 X (i694) 2
F = 0.000006041 2 5~~ = 0.8287.
70 X 61.08
That is, the actual work realized, as expressed and measured
by the projectile's energy of translation, as it emerges from the
bore, is nearly 83 per cent, of the theoretical maximum work
which the powder gases are capable of performing, leaving but
17 per cent, for the other work done by the gases, namely, the
work expended upon the charge, the gun and carriage, and in
PROPERTIES OF PERFECT GASES 53
giving rotation to the projectile; the work expended in overcom-
ing passive resistances, such as forcing the rotating band into
the groove, the subsequent friction as the projectile moves along
the bore, and the resistance of the air in front of the projectile;
and lastly, the heat communicated to the walls of the gun. It is
very difficult to evaluate these non-useful energies, but it is prob-
able that they do not consume more than 17 per cent, of the
maximum work of the gases. Longridge finds by an elaborate
calculation that this lost work in a lo-inch B. L. Woolwich gun
amounts to 30 per cent, of the maximum work; * but it is believed
that he has greatly overestimated the work required to give
motion to the products of combustion. Colonel Pashkievitsch
makes the lost work rather less than 17 per cent, of that expressed
in the energy of translation of the projectile.!
To test the correctness of Equation (40) for determining
muzzle velocities we will apply it to the same gun by means of
which the factor of effect was determined, increasing the charge
from 70 to 90 pounds, and again to 100 pounds, and compare the
computed velocities with those measured with a chronograph.
For a charge of 90 pounds of powder we have A! = 0.210 and
A = 0.780; whence W 2 = 89.3, Wi= 20.86, and W = 68.44
1.8287X90X68.44 = 2Q2i f
'\ 180
The measured velocity with this charge was 2027 foot-seconds.
In a similar way we find by the formula that for a charge of 100
pounds v = 2174 foot-seconds, while the measured velocity was
2182 foot-seconds. The differences between the computed and
observed velocities in these examples are about one-third of one
* " Internal Ballistics." By Atkinson Longridge. London, 1889.
Chapter V.
f " Interior Ballistics. " By Colonel Pashkievitsch. Translated from
the Russian by Captain Tasker H. Bliss, U. S. Army. Washington, 1892.
54 INTERIOR BALLISTICS
per cent., and are well within the limits of probable error in
measuring them.
The factor of effect increases with the caliber of the gun, as is
shown by experiment. Thus with the English lo-inch gun fired
with charges of 130 and 140 pounds of the pebble powder we
have been considering, the factor of effect is 0.855; while with
the n-inch gun, and charge of 235 pounds, the factor of effect
is 0.89.
CHAPTER III
COMBUSTION UNDER CONSTANT PRESSURE
Combustion of a Grain of Powder Under Constant Atmos-
pheric Pressure. In what follows it is assumed that the powder
grain is of some regular geometrical form to which the elementary
rules of mensuration can be applied. It will also be assumed as
the result of observation, that the combustion of the grain takes
place simultaneously on all sides and that, under the constant
pressure of the atmosphere, parallel layers of equal thickness are
burned away in equal successive intervals of time that is, that
the velocity of combustion under constant pressure is uniform.
The form and dimension of each grain of powder constituting
the charge are of the utmost importance, as upon them depends
the proper distribution of the mean effective pressure within
the bore. If the initial surface of combustion of the charge be
large and the web thickness of the grains small, then the maxi-
mum pressure will be excessive and the muzzle velocity inade-
quate. On the other hand, if the web thickness be too great
the chase pressure may prove destructive to the gun. More
than one of our heavy guns it is believed have been wrecked
during the past ten years simply from excessive web thickness.
Many forms of grain have been adopted by different manu-
facturers in this and foreign countries, but they may all be
divided into two general groups, viz.: those burning with a
continuously decreasing surface, and those in which the surface
of combustion may increase (or decrease) to a certain stage, the
grain then breaking up into other forms entirely dissimilar to
the original and which are then consumed with a rapidly de-
creasing surface. To the first group belong spherical, cubical,
ribbon-shaped, and indeed all solid grains of whatever form,
55
56 INTERIOR BALLISTICS
and cylindrical grains with an axial perforation. To the latter
group belong pierced prismatic and the so-called multiperforated
grains employed by both our army and navy.
Notation. Let
/ = thickness of layer burned in time /.
1 = one-half the least dimension of the grain. Since com-
bustion takes place on all sides of a grain at once, it may be
assumed that when / = 1 all grains of the first group are totally
consumed. This, of course, is not the case with m. p. grains.
S = the total initial surface of combustion of the grain.
S = surface of combustion at time t, corresponding to /.
S' = the total burning surface when I = I ' y that is, when the
grain, as a grain, is about to disappear. This surface may be
called the vanishing surface of combustion.
V = the initial volume of a grain.
V = volume of grain burned at time t. That is, the volume
comprised between the surfaces S and 5.
V
k = fraction of grain burned in time /. That is, k = TT-
* o
The general expression for the burning surface of a grain of
powder moulded into any one of the simple geometrical forms
adopted by powder manufacturers may take the form,
S = S + a I + bF (i)
where / is the thickness of layer burned from instant of
ignition. At that instant / is zero and 5 the initial surface of
combustion S . In the course of burning when / is about to
become 1 , S is about to become S f . Therefore
S-' = S + al + bi: (2)
In these two equations a and b are constants for the same
form of grain, whose values will be deduced later.
The general expression for the volume consumed while a
thickness / is burned away, is
COMBUSTION UNDER CONSTANT PRESSURE 57
whence substituting for 5 its general value from (i) and inte-
grating,
The initial volume V is evidently what V becomes when the
grain is completely consumed, that is, when / = 1 . Therefore
This, of course, gives the entire original volume only for
those grains which are completely consumed when I = 1 , or, in
other words, when the web thickness is burned. It need hardly
be said that it does not apply to m. p. grains. In this latter
case, it gives the original volume minus the " slivers," so called.
If, in (4), we substitute for S its value from (2), namely,
S = S' - a 1 - b 1 2
it becomes
F.-S').-4V-^.' .... (5)
o
From (4) and (5) we readily find
... (6)
and
b =j-,(S. + S') --T^ .... (7)
^o l o
These equations give a and b when S , S' and V can be
computed by the rules of mensuration. It will be observed that
a is a linear quantity while b is of zero order of magnitude. These
properties afford tests, as far as they go, as to whether the work
of deducing a and b in any particular case has been correctly
performed.
58 INTERIOR BALLISTICS
Fraction of Grain Burned for any Value of 1. We have by
definition
k = _L = 2 _ _3
This may be transformed into
V
ju, M a_t _L , *v L
V 'I - "
o v o
Put for convenience,
Then
7 ( / 72 )
.... (9)
For all grains of the first group k becomes unity when / = 1 ,
that is, when the grain is all burned ; in this case (9) reduces to
i = a (i + \+ /*) (10)
This relation always subsists between these numerical con-
stants and serves to test the correctness of their derivation in
any case.
The following relations which are easily established will be
useful :
aV
S' = (i + 2\ + 3 tiS -$.-f-jT (aX+3J); (n)
or, more generally,
We also have
a (X + 2 /.) = ~- - i.
' n
COMBUSTION UNDER CONSTANT PRESSURE 59
Therefore for all grains whose vanishing surface (S') is zero,
we have
I + 2\+ 3 fJ. = O.
and
a (\+ 2f*) =- i (12)
Applications. We will now apply these formulas to a dis-
cussion of various forms of grain now in use or which may
come into use.
i. Sphere. For a spherical grain 1 is evidently the radius.
Then by mensuration
S = 4^lo
Substituting these in (6) and (7) we readily find
a = 8 TC 1 and b = 4 n.
Therefore from (i)
and, therefore, 5 is a decreasing function of /.
From (8) we find
a = 3, X = - i and fJ. = -;
o
and these substituted in (9) give
/\ 3
/ i r- j / i\
T + ~ 71 = I - ( I - T)
In S "ft I x I'n'
which is the fraction of grain burned in terms of the thickness
of the layer /.
If we divide the thickness of web (radius of grain) into five
equal parts the following table may be computed, which will be
useful for comparing this form of grain with others to be given:
6o
INTERIOR BALLISTICS
/
I.
k.
First Differences.
0.0
o.ooo
O.2
0.488
0.488
0.4
0.784
0.296
0.6
0.936
0.152
0.8
0.992
0.056
I.O
I.OOO
O.OO8
The second column gives the entire fraction of grain burned
and the third column the fraction of grain burned for each layer.
It will be observed that nearly one-half the initial volume of the
grain is in the first layer.
2. Parallelopipedon. Let 2 1 be the least dimension of the
parallelopipedon and m and n the other two dimensions. Then,
by the rules of mensuration,
S = 4l m + 4l n + 2mn
S' = 2 (m - 2 1 ) (n - 2 Q = 2 m n - 4 1 m - 4 1 n + 8 / 2
V = 2l mn
Substituting these values of S , S f and V in (6) and (7), gives
a = 8 (2 1 + m -\- n) and b = 24.
Making the following substitutions, viz. :
2 L
2 I
- u i u
- = x and = y
m n
in which x and y are generally less than unity, we have, finally,
x + y + xy ocy
y
It may be noted that these values of a, X, // satisfy equation (10).
COMBUSTION UNDER CONSTANT PRESSURE 6 1
There are three special parallelopipedons worthy of separate
notice :
(a) Cube. The cubical form has been used for ballistite
and for some other powders. For this form we evidently have
x = y = i.
Therefore
a = 3; \ = i; p = -.
o
These are the same as were found for spherical grains, as
might have been inferred. They also apply approximately
to sphero-hexagonal, mammoth and rifle powders (old style).
(b) Square Flat Grains. For these grains (still used with
certain rapid-firing guns), m and n are equal and greater than
2 1 . Therefore x and y are equal and less than unity. There-
fore,
If these grains are very thin, x becomes a very small fraction
and may be omitted in comparison with unity. In this case X
and IJL are approximately zero and a unity. This gives
or, a constant emission of gas during the burning; but the grain
would be consumed in a very short interval of time.
(c] Grains Made Into Long Slender Strips (or "Ribbons"),
with Rectangular Cross-Section. These grains are approximately
those of the new English powder called "axite." Also of the
French "B N" powders, and others. If we suppose the width
of the strip to be five times, and the length one-hundred and
fifty times, its thickness (which corresponds nearly with the
" B N " powders) , we shall have x = and y = -- . Therefore
5
a = 1.207; ^ = 0.172; /z = o.ooi;
62 INTERIOR BALLISTICS
and the expression for k becomes
I
k = 1. 2O7T--J I 0.172
In (
The following table illustrates the progressiveness of this
particular grain :
I
lo
k.
First Differences.
0.0
o.ooo
0.2
0.233
0-233
0.4
0.450
0.217
0.6
0.650
0.200
0.8
0.833
0.183
I.O
I. 000
o. 167
I. 000
These strips, made up into compact bundles or fagots to
form the charge, seem well adapted for rapid-firing guns of
moderate caliber. In the application of the expression for
k for computing velocities and pressures in the gun, fj, may be
regarded as zero, and thus greatly shorten the calculations with-
out impairing their accuracy.
If the cross-section of the strip is square, we shall have
2 I,
m = 2 1 , x = i and y = ,
n being the length of the strip. Therefore, in this case,
1 + 2 y y
a = 2 + y; X = ; ft = - .
2 + y ' 2 + y
If the strip be very long in comparison with the linear
COMBUSTION UNDER CONSTANT PRESSURE 63
dimension of cross-section, y may be considered zero, and we
have
a = 2 ; X = --;/ = o.
Therefore
k = 2-\I ~--} = I -(l -^
3. Solid Cylinder. For this form of grain there are two
cases to be considered: (a) When the diameter of cross-section
of the cylinder is the least dimension, (b) When the length of
the cylinder is the least dimension. That is, a cylinder proper
and a circular disk.
(a) Cylinder Proper. In accordance with the notation
adopted, 1 will be the radius and m the length of the cylinder.
We have by mensuration,
S = 2 TT (1 m + / 2 ); S f = o; V = * 1 2 m\
whence
a = 2 TT (4 1 -f m) and b = 6 n.
2 1
Putting, as before, - = x, there results
These are the same expressions for , X, /* as was found for
a strip with square cross-section, as might readily be inferred.
If x be small in comparison with unity, that is, if the grains
are long slender cylinders (thread like), like cordite, we have
very approximately,
a = 2 ; X= - j;/* = o;
and, as before shown,
IV
k
=-(-)
64 INTERIOR BALLISTICS
The following table was computed by this formula:
J_
k.
First Differences.
0.0
O.2
0.00
0.36
0.36
0.4
0.64
0.28
0.6
0.84
O.2O
0.8
0.96
0.12
0.04
I.O
I. 00
I .00
Comparing this table with that given for "strips," it will
be seen that the burning of cordite is not so progressive as that
of axite.
If the length of the solid cylindrical grain be the same as its
diameter, then x = i ; and we have
as for spherical and cubical grains.
(b) Circular Disk. With this form of grain the thickness
becomes the least dimension instead of the diameter. Let 2 1
be the thickness of the disk and R its radius.
Then
S = 2 TT R (2 1 + #>; S' = 2 K (R - / ) 2 ; V = 2 n 1 R\
Whence
a = 4 TT (2 R + 1 ) and b = 6 n.
2 I
Therefore making -^ = ~ = x, we have, as has already
COMBUSTION UNDER CONSTANT PRESSURE 65
been found for square flat grains,
x (2 + x) x 2
a = i + 2 #: X =
I + 2JC ' I + 20C
4. Cylinder with Axial Perforation. Let R = radius of
grain, r = radius of perforation, and m = its length. We then
have i
2l = R-r,
and
R + r = 2(R~1 ). . ' . R 2 - r 2 = 4 1 (R - Q.
By the rules of mensuration, we find, after reduction,
S = 2 TT m (R + r ] + 2 TT (R- r) =4x( m + 2 1 ) (R - 1 \
s f - 4 *(- Wit* -4)
V =4*l (R-lo)
Therefore
a = 16 T: (7? / ) and b = o.
2 /
Making, as before, x = - - we have
Therefore
As an example of this form of grain, suppose the length to
be three hundred times the thickness of web. Then
x = - ; a = - ; X = - ; fi = o.
300' 300' 301"
The expression for k is
_ _
300 / / 301 / 5 ^ ( 300 300 /
66 INTERIOR BALLISTICS
The following table was computed by this formula :
I
c
k.
First Differences.
o o
O.QQOO
O.2
0.2005
0.2005
0.4
0.4O08
0.2003
0.6
0.6008
O.2OOO
0.8
0.80O5
0.1997
O IQQS
I .0
I .OOOO
I. OOOO
This form of grain is very progressive, much more so than
any other form that has been proposed, and seems well adapted
for guns of all calibers. The first differences show that for all
practical purposes the emission of gas may be considered constant
during the entire burning of the grain.
From (n) we have, when /* = o,
Therefore in this example, when x = -- .we have
3
and the burning surface during the entire combustion lies be-
tween its initial value S and its final value -- S .
\J
5. Multiperf orated Grains. These grains, which are used
exclusively with the heavy artillery of the army and navy of the
United States, are cylindrical in form and have seven equal
longitudinal perforations, one of which coincides with the axis
COMBUSTION UNDER CONSTANT PRESSURE 6 7
of the grain, while the others are disposed symmetrically about
the axis, their centres joined forming a regular hexagon. The
web thickness (2 Q is the distance between any two adjacent
circumferences ; and therefore, if R is the radius of the grain and
r the radius of each of the perforations, we have the relation
2l - R ~ 3r
2
From the geometry of the grain as denned above we have
the following relations:
S = 2 r: [F - 7 r> + m(R + 7 r)} . . . (13)
S' = S + 4*lo(3- 2(R + 7 r) - 9 1 ) . (14)
V = r. m (R 2 - 7 r) = {S - 2 r, m (R + 7 r) } (15)
V' = / S + 27r/ 2 ( 3 m- 2(R + 7 r) -61 ) . (16)
In these expressions S and V are the initial surface of
combustion and volume, respectively, while S f and V' are the
vanishing surface and volume burned, when / is about to become
1 and the grain to break up into slivers. If we substitute the
values of S , S' and V' from the above equations in (6) and ( 7 ),
they reduce to
a = 4 TT (3 m - 2 (R -f 7 r))
and
b = - 36 TT
These values of a and b, in equations (8), give
__ _ ( \
' R 2 - 7 r 3 + m (R + 7 r)
These values of a, X, and ^ satisfy the equation of condition
a (i + X + ,) = i,
68 INTERIOR BALLISTICS
since when I = 1 the volume V' has been consumed. When
this occurs, the original form of the grain disappears and there
remain twelve slender, three-cornered pieces with curved sides
technically called "slivers." These of course must be treated
differently. In the applications of these formulas given in
Chapter V, the form characteristics of the slivers are assumed
to be OL = 2, X = and /* = o, with good results.
The form characteristics deduced in (17), (i 8), and (19),
if substituted in (9), will give the fraction of volume V ' burned.
But what is required in practice is the fraction of the entire
grain (or charge). This is found by employing V instead of
V . By this means we find
and this value of OL will be used in all the applications. The
expressions for A and /*, being independent of the volume (see
equations (8)), are those deduced above.
Substituting the form characteristics in (9) and making
l = lo
we shall have the fraction of the entire grain burned when the
web thickness is burned. Calling this fraction k r it will be
found that
,_!.( (n-^(R + ,r + 3 U
m ( R 2 - 7 r 2 )
This expression for k' would also be obtained by dividing
(16) by (15).
It will be seen that for the same web thickness a and k'
decrease as m increases, but within moderate limits, their limit-
ing values, when m is infinite, being
R 2 - 7 r 3
COMBUSTION UNDER CONSTANT PRESSURE 69
and
, _ 2l (R + jr+3l )
R 2 - 7 r*
For the grains employed in the United States service, the
D
ratio varies but little from n. If we adopt this ratio, the
expressions for the form characteristics a, X, /* and k' become
12 2/
19 ' m
2(m-6 1
X =
19/0 + 6 w
6m
k > =
19 19 m
We also have R = 5.5 1 and r = 0.5 1 .
If, in addition, we make m = n 1 we have
12 2
<* = 1
19 n
a ( - 6)
6w + 19
4
M = -
+ 19
6
19 19 n
It may be noted that the limiting values of these form
characteristics, as the length of the grain is indefinitely increased,
are,
a = ; X = ; /z = o and k f = .
J 9 3 19
70 INTERIOR BALLISTICS
Also that X is zero when n = 6 and becomes negative when the
grain is still further shortened.
It will be seen that the percentage of slivers can never be
greater than about 16.
For the grains in use n is approximately 26, which gives
a = = 0.70850
247 / *
8
X = = 0.22857
35
/ = - -~^ = - 0.022857
*' = ^7 = - 85425
There seems to be no valid reason why these, or other simple
ratios, for R/r and m/l should not be adopted by powder
manufacturers for all sizes of m.p. grains, making the diameter
of the grain and perforations, and also its length, depend upon
the web thickness adopted for a particular gun. For example,
the web thickness adopted for the i4-inch gun is 0.1454 inch.
Therefore the dimensions of the grains would be
Diameter = 5.5 X 0.1454 = 0.7997 in.
Diameter of perforation = 0.1454/2 = 0.0727 in.
Length = 13 X 0.1454 = 1.89 in.
These dimensions are practically the same as those of the
actual grains. From eqjation (26') of this chapter it will be
seen that the initial surface of one pound of these grains would
vary inversely as the web thickness.
For these grains, equations (13) to (16) reduce to
S = $2$*l*
S'= 729 *l?
V = 741 * lo
COMBUSTION UNDER CONSTANT PRESSURE 71
The vanishing surface is therefore about 39 per cent, greater
than the initial surface.
Captain Hamilton has shown conclusively that the m.p.
grains now in use are much too short to secure a proper alignment
in the powder chamber, and that this lack of alignment conduces
to excessive pressure.*
If we make n = 200, that is, make the length of the grains
100 times the web thickness, we should have
a = 0.64158
X = 0.31829
/z = 0.00328
k' = 0.84368
This value of n would make the length of the grains for the
i4-inch gun 14.52 inches; which would not only secure a good
alignment of the grains in the containing bag, but would also
give a much less initial surface of combustion to the charge and
would thus reduce the maximum pressure.
The general expression for the surface of combustion of an
m.p. grain with 7 perforations, in terms of the thickness of web
burned, is by (i),
5 = S + 4 * (3 ~ 2 (R + 7 ')) I ~ 36 n P
Differentiating twice, we have
= 4 TT (3 m - 2 (R + 7 r)) - 72 nl
There is, therefore, a maximum value of S which occurs when
3 m - 2 (R + 7 r)
I =
18
* Journal U. S. Artillery, July-August, 1908, page 9.
INTERIOR BALLISTICS
and the maximum surface of combustion is
7r( 3 w- 2(#
From these formulas are easily deduced the following:
1. When 3 m 2 (R + 7 r) = o, S is a decreasing function
of / during the entire burning of the web thickness.
2. When 3 m 2 (R + 7 r) is equal to, or greater than,
iBl the grain burns with an increasing surface.
3. When 3 m 2 (R + 7 r) lies between o and 18 1 the sur-
face of combustion is at first increasing and then decreasing.
Expression for Weight of Charge Burned. If we assume
that the entire charge is ignited at the same instant, which is
practically the case with an igniter at both ends of the cartridge,
the combustion of the charge will be expressed by the same
function that applies to a single grain. Therefore if y is the
weight of the charge burned at any period of the combustion
and o> the weight of the entire charge, we may assume the
equality (since the weights are proportional to the volumes)
In this equation a is always positive from its definition,
S I
viz.: a = ~TF^- It varies in value from 3 (spheres and cubes)
* o
to less than unity (service multiperf orated grains). The
smaller OL is, cczteris paribus, the less will be the maximum
pressure for a given charge. Of the other characteristics, X
and fij either may be positive, negative, or zero, but not both at
the same time.
Expressions for Initial Volume and Surface of Combustion
of a Charge of Powder. Let N be the number of grains in
unit weight of powder, V the volume of unit weight of water,
and the specific gravity of the powder. Then, from the
definition of specific gravity,
COMBUSTION UNDER CONSTANT PRESSURE 73
V
NV
' (23)
since we may assume that the weights are proportional to the
volumes. The number of grains in unit weight of powder can
be counted, and, with the carefully moulded grains now in use,
V can be calculated with great accuracy. Thus (23) can be
employed to determine the specific gravity of a powder when it
is not given by the manufacturer, as is usually the case. For
the large grains designed for seacoast guns the number of grains
in 100 units should be counted, estimating the fraction of a
grain in excess. For small-arms powder, if the specific gravity
of the mass of which the grains are made is known, the number
of grains in unit weight may be computed by the formula
V
N = Jy-^^ (24)
The units to be used in these and other formulas that will
be deduced will be considered later.
Initial Surface of Unit Weight of Powder and of the Entire
Charge. Let S be the initial surface of the grains of unit weight
of powder. Then if S is the surface of one grain, we have, by
(24)
Si = NS.= j^. ..... (25)
But by (8)
^o /
l o
Therefore
d 1 1
for one unit weight of powder; and for w units weight,
74 INTERIOR BALLISTICS
This simple formula was first published in the Journal U.
S. Artillery for November-December, 1905. It shows that for
two charges of equal weight and made up of grains of the same
density and thickness of web, but of dissimilar forms, the entire
surfaces of all the grains in the two charges are proportional to
the corresponding values of . It also shows that if the initial
surfaces of two charges of equal weight but made up of grains of
dissimilar forms, are to be the same, the web thicknesses must be
inversely as the values of <*. For example, if the two charges
are made up, the one of cubes and the other of long slender
cylinders (axite and cordite), the web thickness of the former
must be one-half greater than the latter to obtain the same
initial surface for each charge. These principles are important
since the maximum pressure in a gun varies very nearly with the
initial surface of the charge.
Volume of Entire Charge. Let Vs. be the volume of a
charge of o> units weight supposed to be reduced to a single
homogeneous grain. For a single grain of unit weight (23) gives
V -
d
and for d> units
*VF ^
Gravimetric Density. Gravimetric density is the density
of a charge of powder when the spaces between the grains are
considered. It is measured by the ratio of the weight of any given
volume of the powder grains to the weight of the same volume
of water. Since one pound of water fills 27.68 cubic inches we may
say that the gravimetric density of a powder is the weight in
pounds of 27.68 cubic inches of the powder not pressed together
except by its own weight. Or, if we take a cubic foot as the
unit and designate the gravimetric density by ?% the weight of
a cubic foot of the powder grains by ', and by w the weight of
COMBUSTION UNDER CONSTANT PRESSURE 75
a cubic foot of water, we shall have by definition,
w 1728/27.68 62.427*
It is evident that f will vary not only with the density of
the individual grains but also with the volume of the interstices
between them; and this latter varies with the general form of
the grains, or, in other words, with their ability to pack closely
or the reverse. It is evident that the maximum value of f is
the weight of a cubic foot of solid powder, in which case the above
ratio would be the specific gravity of the powder, designated by
8. The gravimetric density is therefore always less than the
specific gravity. For modern powders gravimetric density is
of very little importance.
Density of Loading. Density of loading is defined to be
the "ratio of the weight of charge to the weight of a volume of
water just sufficient to fill the powder chamber." Let A be the
density of loading and V c the volume of the powder chamber.
Since V is the volume of unit weight of water it is evident that
V c / V is the weight of a volume of water equal to the volume of
the chamber. Hence by definition,
From (27) we have,
V
and this substituted in (28) , gives
^> /
A = ~y~ ( 2 9)
c
From this last equation the density of loading may be defined
as the ratio of the volume of the powder grains supposed to be
reduced to a single grain, to the volume of the chamber, multi-
plied by the density of the powder. If V^ = V CJ that is, if the
76 INTERIOR BALLISTICS
chamber is filled by a single grain, then A = d ; and this is the
superior limit of density of loading. The inferior limit is, of
course, zero, namely, when V& = o. If the density of loading
is unity it follows from (28), that
V c
= F 7 '
that is, the weight of charge equals the weight of water that
would fill the chamber.
Reduced Length of Initial Air Space. By initial air space
is meant that portion of the volume of the chamber not occupied
by the powder grains constituting the charge. The reduced
length of the. initial air space is the length of a cylinder whose
cross-section is the same as that of the bore, and whose volume
is equal to the initial air space. Denote this length by z and
the area of cross-section of the bore by &>. Then as V c V^
is the volume of the air space we have
Substituting for V c and V& their values from (28) and (27),
we have
Zo= ~VvA~
Put
i i d -A
A Ad
Then
(30)
Working Formulas for English and French Units. The
English units used with formulas (23) to (30), inclusive, are the
pound and inch. Therefore
V = 26.78 cubic inches, nearly.
COMBUSTION UNDER CONSTANT PRESSURE 77
The French units employed with the same formulas are the
kilogramme and decimetre. For these units we have
V f = i cubic decimetre.
The two sets of formulas in working form are therefore:
d
N
V,
A
7.
English
27.68
Units
(230
(240
a * (->fr'\
French
* J
Units
(23")
(24")
(26")
(27")
(28")
IW
" NV
27.68
~NV
N '
" dV
27.68
^ w.
a co
27.68*
(20 )
>
" <W^
F - -
o-
27.68
\ 2 7 )
& (<?%'}
V d
CO
A = V
9.
v c
27.68
(2Q )
- * (***\
EXAMPLES
i. Compute the number of grains in a pound of the
powder used with the service magazine rifle. Also the initial
surface.
The grains of this powder are pierced cylinders of the follow-
ing dimensions:
R = o.".o45
r = 0.015 .'. 2 1 = o/'.o3
i
m = - - in.
21
* = 1.65
OL = 1.63
CO = I
78 INTERIOR BALLISTICS
From (24') and (26'), we have
27.68
N = -- ; - 75 -- rr-T = 62300
4 TT 1 m (R - 1 ) d
27.68 X 1.63
Si = -~- -- - = 1823 in. 2
1.65 X 0.015
It is officially stated that the number of grains per pound
varies from 83,000 to 93,000. This discrepancy is partly due to
shrinkage and partly to the breaking and chipping of the grains.
Possibly also to the method of counting.
2. What is the entire initial surface of a charge of 70 Ibs. of
the m.p. powder designed for the 8-inch rifle ? For this powder
we have
R = o".256; r = o.".o255; m = i".O29; d = 1.58
- - = o".044875;
27.68 X 0.72667 X 70
1 = - - = o".044875; = 0.72667
, = 19813 in."
1.58 X 0.044875
3. Suppose the powder of example 2 to be made into cubes
having the same thickness of web. What would be the initial
surface of the charge?
For a cube OL = 3. Therefore
To make the initial surface of the latter charge the same as
the former the web thickness would have to be
3 X 0.08975
2 1 0.72667 >37
4. The volume of the chamber of the 1 2-inch rifle is 17487
cubic inches. If the charge is 400 Ibs. what is the density of
loading? Ans.: A = 0.633.
CHAPTER IV
COMBUSTION AND WORK OF A CHARGE OF POWDER IN
A GUN
IT has been established by experiment that a grain of modern
powder burns in concentric, parallel layers, and that the velocity
of combustion under constant pressure is uniform. Let 1 be
one-half the web thickness of a grain and r the time of burning
this thickness under the constant pressure of the atmosphere.
We then have, since the web burns on both sides,
7 = velocity of combustion = constant = v c (say) . (i)
In the bore of a gun, however, the pressure surrounding the
grain is very far from being constant and greatly exceeds the
atmospheric pressure. All writers on interior ballistics agree
that the velocity of combustion may be regarded at each instant
as proportional to some power of the pressure; but they differ
widely among themselves as to what this power is. Sainte-
Robert, Vieille, Gossot, and Liouville give reasons (based, how-
ever, upon experiments made with a small quantity of powder
exploded in an eprouvette of a few cubic inches capacity) for
2 Q
adopting the exponent -. Centervall makes the exponent
for "Nobel N K" powder. Sebert and Hugoniot, from ob-
servations of the recoil of a lo-cm. gun mounted on a free-recoil
carriage, deduced a law of burning directly proportional to the
pressure. This law is the most simple of all and allows an easy
and complete integration of the equations entering into the
problem.* But simple as is this law of Sebert and Hugoniot,
* See Journal U. S. Artillery, vol. 7, pp. 62-82.
79
80 INTERIOR BALLISTICS
we prefer to make use of Sarrau's law of the square root of the
pressure, because the resulting formulas are easily worked
and give results which "agree very well with facts" as stated by
Sarrau, and as has been repeatedly shown by the writer and
others.
Sarrau's law of burning under a variable pressure p leads
directly to the equation,
dl 1 P\" ...
in which p is the atmospheric pressure and / the thickness of
layer burned in time t.
It will be assumed that the variable pressure p in the bore
is measured by the energy of translation imparted to the
projectile (which is many times the sum of all the other energies
entering into the problem) ; and it will be taken for granted that
all the other work done by the expansion of the powder gas may
be accounted for by giving suitable values to the constants so
as to satisfy the firing data by means of which they are deter-
mined. This procedure will be fully illustrated further on.
If p is the variable pressure per unit of surface upon the base
of the projectile at any instant, & the area of the base, and u
the corresponding distance travelled by the projectile from its
firing seat, we have from the principle of energy and work,
in which w is the weight of the projectile.
But from mechanics and calculus,
d 2 u dv d v du dv id (v 2 )
dt 2 ~ dt ~ du dt V du "~ 2 du '
in which v from now on represents velocity.
Therefore
w d(*)
COMBUSTION OF A CHARGE OF POWDER IN A GUN 8l
Combining (2) and (3), we have
*! = !*.( w --\* (<LW1\*
dt ~~ T \2gupJ \ du ) U)
Since z is the reduced length of the initial air-space and u
the distance travelled by the projectile from its firing seat, we
may say very approximately, by the principle of the covolume,
that u/z is the number of volumes of expansion of the gas during
the travel u, and this whether the charge is all converted into
gas or not. If we make u/z = x, and therefore du/d x = z ,
(3) and (4) become
w d
and
co p = -- r ..... (c)
2 g Z dx
dt ' r\2 uz \ dx
It will be seen that (6) connects the velocity of burning of
the grain with the velocity of travel of the projectile in the
bore. It will be better to make x the independent variable in
the first member as well as the second.
We have from calculus,
dl dl d x du v dl
dt dx du dt ~ z dx
Therefore, substituting in (6),
wz
dx r \2 gu> po' \ d x v
Integrating between the limits o and x, we have
In order to perform the integration indicated, we must know
the relation existing between v and x, that is between the velocity
of the projectile in the bore at any instant and the corresponding
6
82 INTERIOR BALLISTICS
number of volumes of expansion of the gas. We get this relation
from (19) Chapter II, which is
w
From this equation we deduce by simple differentiation
dx v
Substituting this in (7) and making
r* d x
x > = J voT^FTcrr
we have
It will be observed that X is a function of a ratio and is in-
dependent of any unit, and may therefore be tabulated with x
as the argument.
If we put
K I- --- ) ..... (n)
r V6# a /
we have
j = KX. ...... (12)
1
Substituting the value of Ill from (12) in (22), Chapter III,
we have
k =-1 = aKX (i + \KX + v(KX o y-) . (13)
CO
an equation which gives the fraction of the charge burned at any
instant in terms of the volumes of expansion of the gases gener-
ated. When the powder is all burned in the gun (if it be all burned
before the projectile leaves the bore), we have y = co and I = 1 .
COMBUSTION OF A CHARGE OF POWDER IN A GUN 83
If, therefore, we distinguish X by a dash when / = 1 0) (12)
becomes
KX = i, ...... (14)
and (13) reduces to
i = a(i + X + /0
a fundamental relation established in Chapter III.
Substituting the value of K from (14) in (13), we have, while
the powder is burning, the relation
X X
Expression for Velocity of Projectile while the Powder is
Burning. Substituting the value of y from (15) in (8) and
making
*'=*('- (FT^ji) ^
we have
co X j ( X fXo\ 2 \
tf =6gaf- '=-Ji+X =- + M = ) \. -
w Xo ( X \A / )
This equation holds only while the powder is burning and
ceases to be true when X > X .
Velocity of Projectile when y = co. When X = X and,
therefore, X l = Xi, equation (17) reduces to
V = 6g/-=r<*(i + X + //);
w X 2
or, since
" (i + X + /) = i,
it becomes
This equation is, of course, the same as (8) from which it is
84 INTERIOR BALLISTICS
derived as is evident from (16). Putting
A"t _-
^ ~ X2>
or, generally,
the expression for v 2 becomes
It should be remembered that all symbols employed in this
work affected with a dash refer to the position of the projectile,
either in the bore on in the bore prolonged, when the powder
has all been burned, and therefore where y = o>.
From (19), we have
, , v 2 w
6 gf = = r ; (20)
and this substituted in (17), gives
.^vMi+xfs+>Y|fyt ()
For convenience, put
A-l
Then, finally, while the powder is burning,
A-l J*- ** ^
tf=MXi{i+NX +N'X *} . . . (22)
Velocity of Projectile after Powder is all Burned. The
velocity of the projectile after the powder is all burned is given
by (8), substituting & for y. Reducing by means of (16), (18),
and (20), and denoting velocity after the powder is all burned
by capital F, equation (8) becomes
COMBUSTION OF A CHARGE OF POWDER IN A GUN 85
The velocity then after the powder is all burned varies
directly as the square root of X 2 . From (16) and (18), we have
and therefore the superior limit as x (or ) increases indefi-
nitely is unity. On this supposition (23) becomes
r- = = = V? (say) .... (25)
A 2
We may regard Vi then as the theoretical limiting velocity
after an infinite travel. In terms of V\ (23) becomes
F 2 =7 1 2 JT 3 ...... (26)
Since from (25) v 2 = X 2 V\ ! , therefore
and, therefore,
,, F t 2
^ : -= (27)
Pressure on Base of Projectile while Powder is Burning.
Differentiating (17) with respect to the independent variable x
and putting for simplicity
we have
Therefore, from (5)
W
Combining the constants outside the brackets into one
multiplier by making
- - M',
86 INTERIOR BALLISTICS
we have the following expression for the pressure per unit of
surface, on the base of the projectile :
Pressure after the Powder is all Burned. Differentiating
(26) with reference to x and substituting the differential co-
efficient in (5) we have, employing capital P to express pressure
in this case,
wV* dX 2
2 g co z dx
But from (24)
dx 3
(i +x)*'
wV, 2
p'
do)
6 g oj 2
P
P'
\^ u /
(*T\
whence, putting
we have finally
If we make x = o in (31), we have
p = f. (32)
Therefore P' is the pressure per unit of surface at the origin
supposing the powder to be all burned before the projectile
moves from its seat.
Relation Between f and P'. From (19) and (25) we get
Combining this with (30) there results
P' =J~ (34)
Z co
Since / is (at least theoretically) the pressure per unit of
surface of the gases of one pound of powder at temperature of
combustion, occupying unit volume, it follows from (34) that
COMBUSTION OF A CHARGE OF POWDER IN A GUN 87
P' is the pressure per unit of surface of the gases of co pounds
of powder (the entire charge), occupying a volume equal to the
initial air space z % as has already been shown by equation (32).
Equation (31) is, therefore, the equation of the pressure curve
upon the supposition that the charge is all converted into gas
before the projectile has moved from its seat. From equation
(30'), Chapter III, we have,
27.68
z u = r- a co cubic feet.
Therefore, from (34)
27.68 a
Values of the X Functions. These values may be most
easily and simply expressed by means of auxiliary circular
functions. Thus let
(i -f x)* = sec0 (36)
Then, by trigonometry,
sin 2 = i TT = X 2 . (from (22))
and
tan = V(i -f- #)i i
Also
d x = 6 sec 6 tan d
Substituting these values in the expression for X we have
X = 6J o sec 3 0</0 (37)
Integrating, we have
X = 3 sec tan0 + 3 log e (sec + tan0) . (38)
By substituting the values of sec and tan given above in
(38), we get an expression for X in terms of x. But it is of no
practical interest. The definite integral in (37) is a well-known
function of and has been extensively used in exterior ballistics.
88 INTERIOR BALLISTICS
A table of this function computed for every minute of arc up to
87 was first published by Euler, and. has recently been re-
printed (1904) at the Government Printing Office and issued
as "Supplement No. 2, to Artillery Circular M." By means of
this table it is easy to compute X for any value of x. First
compute by (36), and then take the definite integral corre-
sponding to 0, which has been symbolized by (0), from the table
just mentioned. We then have
X = 6 (0) ...... (39)
Since
X 2 = sin 2 ....... (40)
we have from (18)
X^X^m't ..... (41)
By definition
x dXl
x >-
But from (41)
dX
sin* cos*
From (9) and (36) we deduce
dX .
-7 sm = sin cos 0.
d x
Also, from what precedes,
d sin cos cos 8
2 sin cos -j = ^ = .
djc 3sec 8 0tan0 3
Therefore
i
A 3 = sin cos + X cos 0.
o
Let
X = -
i + i X cos 4 cosec
COMBUSTION OF A CHARGE OF POWDER IN A GUN 89
Then we have 'v/cuiQ ,
sin cos 4
X a = Y~ (^
From the foregoing equations we find
X 3 dx
by means of which are easily deduced from the definitions of
X 4 and X*, the following simple equations:
X* = X (i + X) (43)
and
V V 2 ( I ~ V\ f . .\
X^= X 2 (i + 2 A) .... (44)
By means of equations (39), (40), (41), (42), (43) and (44),
the table of the logarithms of the X-functions given at the end
of the volume was computed.
Some Special Formulas. Dividing (21) by (15) and reducing
by (25), we have, since yl&= k,
7 .2 _ z, yi _ z, y 2 v- / \
V - K V - K V i vV 2 . . . . V45/
That is, the velocity of the projectile at any travel before the
charge is all burned is equal to what the velocity would have
been at the same travel had all the charge been converted into
gas before the projectile moved, multiplied into the square root
of the fraction of charge burned.
For spherical, cubical, and certain other forms of grain, we
have a = 3, \ = i and n = . Substituting these in (15),
o
we have by obvious reductions,
k = i ( i -=2 j (46)
and therefore
x = x { i -(i -*)M .... (47)
QO INTERIOR BALLISTICS
For cordite and similar grains we have a= 2, X = - - and ^
= o. Substituting these in (15), gives
*-'-('-=;)' .... (48)
and
X =X { i-(i -*)} .... (49)
Equations (46) and (48) give the fraction of the charge consumed
for any given travel of the projectile, and, conversely, (47) and
(49) enable us to determine the travel of projectile for any given
fraction of charge burned. For any other forms of grain the
solution of a complete cubic equation is necessary to determine
X when k is given. See equation (15).
Expressions for Maximum Pressure. It is well known
that the maximum pressure in a gun occurs when the projectile
has moved but a comparatively short distance from its seat, or
when u and x are relatively small. The position of maximum
pressure is not fixed but varies with the resistance encountered.
As a rule it will be found that the less the resistance to be over-
come by the expanding gases the sooner will they exert their
maximum pressure, and the less will the maximum pressure be.
The differentiation of (29) gives an analytical expression for
the maximum value of p; but it is too complicated to be of any
practical use. A reference to the table of the X functions shows
that Xz is approximately a maximum when x = 0.64, while X
and X 5 increase indefinitely. When X is negative it is evident
that p is a maximum when x is less than 0.64; and when X is
positive, when x is greater than 0.64. Therefore there will be
two cases depending upon whether the grains burn with an in-
creasing or a decreasing surface. These will be considered
separately.
(a) When the grains burn with a decreasing surface; or what
is the same thing, when X is negative. A function at, or near,
its maximum changes its value slowly. Therefore a moderate
COMBUSTION OF A CHARGE OF POWDER IX A GUN 91
variation of the position of maximum pressure will have no
practical effect upon its computed value. It has been found
by trial in numerous cases that x = 0.45 gives the position of
maximum pressure when X is negative with great precision.
For this value of x the table gives,
log X 3 = 9.85640 10
log X 4 = 0.48444.
logX b = 0.93587.
Substituting these in (29) and designating the maximum
pressure by p ml we have approximately, when X is negative,
p m = [9.85640 -io]M'{i- [0.48444] ^V + [0.93587] N' } (50)
or, ,
p m = 0.71846 M'{i - 3.0510^ + 8.6273^'] . (50')
(b) When the grains burn with an increasing surface. When
the grains burn with an increasing surface X is generally positive,
and it will not be far wrong to assume that the maximum pressure
occurs when x = 0.8. For this value of x the table gives:
log X 3 = 9.86027-10.
log X = 0.60479
Substituting these in (29), we have,
p m = [9.86027 -io]M'{ i + [0.60479] #-[1.17352] #') (51)
or,
p m = 0.72489^(1 +4-0252 N- 14.911 #'} . (51')
Expressions for Computing r and the Velocity of Combustion.
From (n) and (14) we have
( *o \*= , .
:=l<^t/* ..... (52)
If v c is the velocity of combustion under atmospheric press-
ure we shall have
/,
*<=->
92 INTERIOR BALLISTICS
and therefore
(6g<*Po\*lo , .
V < = -(-^ Z -)Y O - - - (53)
Let v' c be the velocity of combustion at any instant under
the varying pressure p. Then from (2) we have
Working Formulas. English Units. It is customary in
our service, following the English practice, to express the volumes
of the powder chamber and bore in cubic inches; the various
pressures in pounds per square inch; the caliber, reduced
length of initial air space, and travel of the projectile in the
bore, in inches; while the velocity of the projectile is expressed
in foot-seconds and its weight in pounds and ounces. These
units are apt to cause confusion and error in the applications of
ballistic formulas; and to avoid this as much as possible it will
be well to reproduce the most important of the formulas deduced
in the preceding pages with all the reductions made and the
mathematical and physical constants introduced and combined
into one numerical coefficient. The physical constants adopted
for English units (foot-pound), are the following:
g = 32.16 f.s. (mean for the United States)
p = 14.6967 Ibs. per in. 2
V = 27.68 cubic inches.
/ is taken in pounds per square inch. The formulas are re-
numbered for convenience.
A = 27.68^- = [1.44217]^- ...... (54)
' c v c
I i 8 - A
a = T~i = TF ...... (S5)
4X27.68 a Si r oi' 7 "/- u \ / *\
Z = - ~ -^ = I* -54708] js- (inches) . (56)
COMBUSTION OF A CHARGE OF POWDER IN A GUN Q3
w-OLCl
(57)
Ff = 144 X 6 g f -^ = [4.44383]^ (foot-seconds) . . (58)
_r_ V* = MX = \M = sMP' _v>
kX 2 X 2 a aN ' M f X 2
av- a V, 2
M = -rr=-=- ............ ( )
AI A
6 iv M r , w M
M'= - -= 7.82867 - 10 - . . . . . . (61)
27.68^ as, as,
aP' 1728
T = ^68 ............. (65)
X 27.68 X Vaws, r X Va w a ,, ,
' ^ =[8-56006-10] --- -
7 J /72
v c = - = [i.43994 ] -^ /- = (inches per second) . . (67)
r X Va w a,
v e ' = v c 2 = [941639 - 10] v c V7 .... (68)
v fi -43994] r^ 3 [1.43994] lo d 2
XQ= - -- -j==-- ....... (69)
V aw w v c \/ aw &
It must be remembered that v and /> refer to the period when
the powder is burning and V and P to the period after the powder
is all burned.
94 INTERIOR BALLISTICS
FRENCH UNITS
In metric units we shall take V c in cubic decimetres, p in
kilogrammes per square centimetre, d in centimetres and z , u
and v in metres. Also g = 9.80896 m.s. With these units- our
formulas become,
A = - (71)
40 a co r , a s,
Z = -jT = 11.10491] -# (72)
f ~
Vf = [2.76977] -^ (f in kilos, per cm. 2 ) . . . (73)
w M
M'= L7.70735 - 10] T (tilos. per cm. 2 ) . . (74)
d CO
wV*
P' '= [7.23023 10] - (kilos, per cm. 2 ) . . (75)
a co
wV 2
f = [7.23023 10] - - (kilos, per cm. 2 ) . . (76)
CO
v c = [0.63128] (cm. per. sec.) . . (77)
X V a w co
/ d 2 T d 2
X = [0.63128] = [0.631-28]=. . (78)
V aw &
T = [9.36872 - 10] -^ (79)
Characteristics of a Powder. The quantities /, T, a, X
and M were called by Sarrau the characteristics of the powder be-
cause they determine its physical qualities. Of these quan-
tities / depends principally upon the composition of the powder,
and, with the same gun, for service charges, is practically constant
for all powders having the same temperature of combustion.
The value of r depends generally upon the density and least
COMBUSTION OF A CHARGE OF POWDER IN A GUN 95
dimension of the grain. The factors a, X and (JL called "form
characteristics" depend upon the form of the grain, and for
the carefully moulded powders now employed their values may
be determined with great precision. They are constant so long
as the grain in burning retains its original form.
Expressions for M, M', N and N' in Terms of the Charac-
teristics of the Powder. When / and v c are known from ex-
perimental firings or otherwise, for any gun and powder, the
quantities V? and X can be determined either from (58) and
(69), or (73) and (78). Substituting these in the proper expres-
sions for M, M', N and N' they become
For English Units
.... (80)
If- [0.83356] -- -... (81)
N = [8.56006- IO]-KJ v 7 aw n, . . . (82)
AT'=A" (83)
For Metric Units
lrfx * (84)
-, <-*. / u c I w w \ *
tf'=[o.igoo3]-jM--) . . . (85)
N = [9.71291 - io]Vawu . . . (86)
A T '= %N* (87)
If we substitute the value of M' from (81) in (50) or (51), and
reject the terms within the brackets, we have in effect Sarrau's
96 INTERIOR BALLISTICS
monomial formula for maximum pressure. But it is evident
there can be no monomial formula for velocity or pressure unless
X and fj, are approximately zero. Equations (80) to (87) are
useful for determining the values of M and N (upon which all
the other constants depend), when the charge varies or when
there are variations in the weight of the projectile. In these
formulas a-, X, n and 1 are independent of & and d and are strictly
grain constants. v c is a powder constant, varying only with
the composition and density of the powder. / is approximately
constant for full service charges of the same kind of powder, in
guns of all calibers. For example the magazine rifle, caliber
0.3 inches, and the 1 6-inch B. L. R. give approximately the same
value to/ when computed by equation (64) or (65). This factor,
however, varies with the charge in the same gun, for it is evident
that its effective value as measured by projectile energy must
decrease with the charge. Indeed if the charge be sufficiently
reduced it is obvious that / becomes zero since we have omitted
from our formulas all consideration of the force necessary to
start the projectile. The law of variation is not known; but
we will assume provisionally that/ varies with the charge accord-
ing to the law expressed by the equation
/-/. ..... (88)
where co is the service charge by means of which M and N were
determined and f the corresponding value of / computed by
(64) or (76). If the weight of the projectile also varies we will
assume that / may be determined by the equation
The exponents n and n f must be determined from experi-
mental data. If we make
K -~-r> (90)
co W
COMBUSTION OF A CHARGE OF POWDER IN A GUN 97
(89) becomes
/ = K " w n> (90')
Substituting this expression for / in (80) and (84) gives, for
English units:
^b^^-ff^)*. . . (91)
and for metric units:
M = [2.48268] K -^yV^^r^J' - (91')
In the applications of these equations f must be computed
by (64) or (76) and v c by (67) or (77).
7
CHAPTER V
APPLICATIONS
THE principal formulas deduced in Chapter IV are here re-
produced for convenience of reference. They are the following:
(a) Formulas which Apply Only While Powder is Burning.
v 2 = MX, [i + NX +N'X 2 } . . . (i)
p = M'X 3 {i+NXt+N'Xs} ... (2)
It will be observed that these formulas for velocity and
pressure are identical in form, and that the constants within the
brackets are common to both. Also that M' is a simple multi-
ple of M. Moreover, from the manner of deriving p from v 2 , the
velocity and pressure deduced from these formulas correspond
at every point so that one can be easily and exactly computed
from the other without the necessity of laying down velocity
curves in order to obtain the pressures.
(b) Formulas which Apply Only After the Powder has All
Been Burned.
... (3)
a
p =(iw = p '(<-^ ... - (4)
(c) Formulas Which Apply at the Instant of Complete
Combustion.
?= MX,{i + NX + N'X 2 }, (from (i)): . . (5)
and _ Mlti , ,.
v 2 = (from (3)) .... (5 )
Equations (5) and (5') give the same value to zr, since the
APPLICATIONS 99
former equation reduces to the latter_at the point w. But (i)
and (3) are not tangent at the point w unless the vanishing sur-
face (5 r ) of the grain is zero, as with cubes, spheres, solid cylin-
ders, etc.
From (2) we have at the travel
P = M f X*{ i + N Xt + N'X 5 } . . . (6)
and from (4)
''
Equations (6) and (6') give the same value to p for all grains
whose vanishing surface is zero, as may be thus shown :
Substituting for M' in (6) its value from (59), Chapter IV,
and giving to N and N' their values in terms of X , we have
But = i + X and =^ = i + 2 X.
For all grains whose vanishing surface is zero we have the
relations (Equs. (10) and (12), Chapter III.)
a (i + X + M ) = i
and
a (X + 2 M) = -i
which readily reduces to (6'). Therefore for all forms of grain
whose vanishing surface is zero the pressure curves (2) and (4)
are tangent at u. This is not true for grains for which 5">o.
100 INTERIOR BALLISTICS
For these the pressure at travel u given by (2) is greater than
that given by (4), and this difference increases with 5".
Monomial Formulas for Velocity and Pressure While the
Powder is Burning. The expressions for velocity and pressure
while the powder is burning (equations (i) and (2)) are generally
trinomials because equations (9) and (22), Chapter III, are tri-
nomials. And these are so because of the geometrical character-
istics a, X and IJL. In order to have a monomial expression for
velocity or pressure X and n must both be zero. But upon this
supposition (22), Chapter III, would become
or, the fraction of grain burned would be directly proportional
to the thickness of layer burned; which is impossible, since the
grain burns on all sides. This same supposition would also
make
I.S.-V,
which is not true, at least for finite volumes.
It has been shown in Chapter III, that all grains which, under
the parallel law of burning, retain their original form until wholly
consumed and for which 5">o, have one or the other of the
following expressions for a, namely, i + x or 2 + x, x being the
ratio of the thickness of web to the length (or breadth) of the
grain. Only grains for which a = i + x can give approximate
monomial expressions for velocity and pressure, and this by
making x so small that it may be omitted in comparison with
unity, in which case a becomes practically unity and X and /*
zero. To this class belong thin, flat grains and long cylindrical
grains with axial perforation.
When a = i and X and /z are zero, equations (i) and (2) be-
come
v 2 = M X, . (7)
APPLICATIONS 10 1
and
p = M'X Z = [7.82867 - 10] M X 3 (8)
a &
Also, by equation (3), since a = i, we have, after the powder
is all burned,
It will be seen that the pressures by the monomial formula
are directly proportional to X 3j which therefore gives, to the
proper scale, the typical pressure curve. Its maximum value,
as seen from the table of the X functions, occurs when x = 0.64,
and its logarithm is 9.86390 10.* Applying this in equation
(8) gives for the maximum pressure,
IV
p m = [7.69257 - io]M .... (10)
If the maximum pressure, assumed' to be the crusher-gauge
pressure, is known by experiment, we may compute M from the
last equation. Thus we have
,, r i a * P
M = [2.30743] ^- .... (n)
Substituting this in (7) and (8) we have while the powder is
burning,
and
p = [0.13610] p m X, ..... (13)
Since by (12) the velocity is proportional to \/X lt this func-
tion represents the typical velocity curve while the powder is
* The maximum value of X 3 occurs when x = 0.6336+- But the value
of x given above is near enough for all practical purposes. It may be
noted here that the curve of X 3 has a point of inflection when x 1.3891.
102 INTERIOR BALLISTICS
burning. After the powder is all burned the monomial formulas
and
_P^_ M'X , }
~ '
are to be employed.
Example. As an example of monomial formulas for velocity
and pressure take the following data from "Notes on the Con-
struction of Ordnance," No. 89, pages 43-47:
Gun: 8-inch B. L. R., Model 1888. ^ = 3617 in. 3 ; u m
= 205.25 in.
Powder: Nitrocellulose composition, single-perforated grains
of the following dimensions: length (m) 47.69 in.; outside dia-
meter (2 R) 0.4455 in.; diameter of perforation (2 r) 0.1527 in.
From these dimensions we find by the formulas of Chapter
in.,
2 jf. - - (0.4455 - 0-1527) = 0-1464 in.
* = ^f = 0.0030525.
a = I + x = 1.0030525.
x
X = ^ ^ = 0.0030433.
fj, = O.
We may, therefore, in this case, assume a = i and X = o
without material error, and employ the monomial formulas (7)
and (8), computing M either by (9) or (n), according as we take
the observed muzzle velocity or crusher-gauge pressure for this
purpose. If the crusher-gauge pressure (assumed to be p m ) is
employed equations (12) and (13) may be used. If it is known
that the powder is all burned at, or near, the muzzle (9) becomes
M = ^ (9')
APPLICATIONS 103
in which both symbols in the second member refer to the muzzle.
If the charge is not all consumed at the muzzle and we know
the value of v c for the powder used, X can be found by (69),
Chapter IV., and then ^V can be computed by (9). Finally if
v c is not known equations (12) and (13) must be employed.
As an example one shot was fired with a charge of 78 Ibs.,
and a projectile weighing 318 Ibs. The observed muzzle velo-
city was 2040 f. s., and crusher-gauge pressure 30450 Ibs. per in. 2 ,
and it was known that the combustion of the charge was practi-
cally complete at the muzzle. From the given data we find
(taking 5 = 1.567), A = 0.5969, log a = 0.01584, and z =
44.548 in. Therefore,
u m 205.25
and from Table i, for this value of x m ,
\QgX = 0.77147
logXi = 0.41207.
\QgX 2 = 9.64060 10.
.'. log If = 2 log 2O40 0.41207 = 6.20719.
Also by (61), Chapter IV, log M' = 4.63036.
The equations for the velocity and pressure curves for this
shot are, therefore,
v = [3-10359] V^i ..... (16)
and
/> = [4.63036] X 8 . . . . (17).
The first of these equations gives, of course, the observed
muzzle velocity; and the second gives (by taking x = 0.64) a
maximum pressure of 31208 Ibs. per in. 2 , exceeding the crusher-
gauge pressure by 758 Ibs.
If we determine the value of M by means of the crusher-gauge
pressure we shall have by (n), log M = 6.19652; and the
equations for velocity and pressure now become
v = [3.09826] VX l
104
INTERIOR BALLISTICS
and
p = [4.61969] X 3
This last equation gives the observed crusher-gauge pressure
while the first makes the muzzle velocity 25 f. s. less than the
observed. As muzzle velocities can be more accurately measured
than maximum pressures, the first set of formulas are probably
the more accurate and will be used in what follows in preference
to the other set.
The expression for fraction of charge burned at any travel
of projectile is found from (70), Chapter IV., and is for this
example,
v 2
k = [3.02134 -- 10] Y" ..... ( l8 )
The travel of projectile is given by the equation
u = z x = 44.548 x inches .... (19)
The following table computed by means of equations (16),
(17), (18), and (19), is represented by the curves v and p in
Fig.
i.
X
Travel
U
inches
Velocity
V
ft. sees.
Pressure
P
Ibs. per in. 2
Fraction
of charge
burned.
k
Pressure
P
Ibs. per in. 2
Velocity
ft.- sees.
0.0
O.O
0.0
00
0.0
84084
0.0
O.2
S.QIO
379-7
26II5
0.257
65939
749-3
0.4
17.819
600.2
30260
0-357
53687
1005 . i
0.6
26.729
770-3
3H94
0.430
44931
1175.0
0.8
35.638
910.1
30949
0.489
38402
1301.6
I.O
44-548
1029.2
30214
0-539
33369
1401.5
1.2
53-457
II33-I
29281
0-583
29387
1483-4
i-4
62.367
1225.2
28285
0.623
26167
1552.3
1.6
71.277
1308.0
27288
0.659
23519
1611.5
1.8
80.186
1383-2
26322
0.692
21306
1663.1
2.0
89.096
1452.0
25401
0.722
19434
1708.6
2-5
111.370
1602.2
23322
0.790
15823
1802.8
3-0
I33-644
1729.3
21548
0.849
13242
1877.0
3-5
I55-9I8
1839.1
20034
0.901
II3I8
1937-5
4.0
178.192
1936.0
18733
0.948
9834
1988.2
4-5
200 . 466
2022 . 5
17606
0.991
8661
2031.5
4.6073
205.250
2040 . o
17384
1. 000
8441
2040 . o
APPLICATIONS
I0 5
The last two columns in the table represented by the curves
V and P, Fig. i, show the velocity and pressure upon the supposi-
tion that the powder was all converted into gas at the tempera-
FlG I.
ture of combustion before the projectile had moved. They
were computed by the formulas
V* = V*X 2 = [6.97866]*, .... (20)
and
[4.92471]
_
'
, .
The force of the powder (/) and the velocity of combustion
in free air (v c ) for this particular charge and brand of powder
can now be computed by equations (64) and (67), Chapter IV.
We find/ = 1396.9 Ibs. per in. 2 and v c = 0.13614 in. per sec.
If we wish to compute velocities and pressures in this 8-inch
gun when the charge varies K must be computed by (90) and M
by (91), Chapter IV. Since the weight of projectile is constant
2
n' is zero; and for an 8-inch gun we will assume that n = ,
O
106 INTERIOR BALLISTICS
this assumption to be tested by experiment. With these
values of n and n' we have
and therefore,
/ = 76.519 & ..... (22)
Substituting this value of K, and the gun and powder con-
stants in (91), we have
M = [2.09974] a" w v ..... (23)
Also, from (61), Chapter IV,
M'= [2.43084] ^ .... (24)
Therefore from (7), (8), and (10),
v = [1.04987] a 1 u^VX'j. .... (25)
" X
p = [2.43084] r ...... (26)
and
w = [2.29474]! ..... (27)
which are the formulas for velocity and pressure for this gun and
brand of powder in terms of the weight of charge.
As an example, what would be the maximum pressure with
a charge of 79^ I DS ? We first find A = 0.6084 an d then log a =
0.00238. We then have by (27)
log p m = 2.29474 + log 79.5 - - log a = 4-5 Io6 5
* Pm = 3 2 4o8 Ibs. per in. 2
This agrees very closely with observation.
We have the means of testing the accuracy of these equations
to a limited extent, since there were four shots fired with charges
of 70, 78, 85, and 88 Ibs. The following table gives -the results
of all the necessary preliminary calculations for the four shots
APPLICATIONS
107
fired and also for two others " estimated from prolonged empirical
curves." The data from the shot fired with a charge of 78 Ibs.
have been taken as the basis of the calculations. The gun
constants will be found on page 102.
AT
MUZZLE
CO
A
log a
log Z
Ibs.
X
logXo
logX t
logX 2
60
0.4592
0.18742
.70647
4-0347
0.74978
0.36944
9.61965-10
70
0-5357
0.08940
.67540
4-3339
0.76150
0.39260
9.63110
78
0.5969
0.01584
.64883
4.6073
0.77147
0.41207
9.64060
85
0.6505
9.95382
.62414
4.8769
0.78066
0.42987
9.64919
88
0.6734
9-92777
6I3I5
5.0018
0.78475
0.43770
9.65295
95
0.7370
9.86767
.58629
5-3210
0.79467
0.45663
9.66196
The computed muzzle velocities and maximum pressures in
the following table were obtained (witn the exception of the
first two muzzle velocities) by equations (25) and (27). The
values of / were computed by (22) and X by (69), Chapter
IV.
CO
Ibs.
log X
f
Ibs. per
inch 2
MUZZLE VELOCITY,
F. S.
MAXIMUM PRESSURES,
LBS. PER IN. 2
Observed
Com-
puted
O.-C.
Observed
Com-
puted
O.-C.
60
70
78
85
88
95
0.74265
0.75819
0.77147
0.78382
0.78931
0.80274
H73
1306
1397
1479
I5H
1593
1600
1839
2040
2200
2275
2454
1600
1844
2040
2205
2276
2441
~~ 5
- 5
i
+ 13
18000
24889
30450
35600
39301
47280
18859
25272
31206
37051
39756
46583
-859
-383
-756
-1451
-455
+697
For the first two shots the powder was all burned before the
projectile had reached the muzzle, as is shown by the values of
log X . For these the muzzle velocities were computed by
equation (14).
It will be observed that the equations by means of which
the muzzle velocities and maximum pressures given in this
table were computed depend for their constants upon one
io8
INTERIOR BALLISTICS
measured velocity only, due to a charge of 78 Ibs. The measured
crusher-gauge pressure for this charge has not been made use
of at all. The constant M upon which all the other constants
depend might have been determined by equation (n) in which
the muzzle velocity does not enter. But muzzle velocities can
be more accurately measured than maximum pressures and
are, therefore, better adapted to the determination of ballistic
constants. The computed maximum pressures in the table are
probably nearer the actual pressures on the base of the projectile
than those given by the crusher gauge. The agreement between
the computed and measured muzzle velocities is all that could
be expected from any ballistic formulas.
To determine the travel of projectile when all the charge was
burned we take x by interpolation from the table of the X
functions corresponding to the values of log X . We then have :
u = .x z .
For the travel of projectile when the pressure is a maximum.
we have, calling this travel u f ,
u' = 0.64 z .
The following table gives the values of u' and u for all the
shots:
ft
u'
u
u m u
k
Remarks
Ibs.
inches
inches
inches
60
32.56
196.56
8.69
70
30-3I
201.13
4.12
78
28.51
205.25
0.00
1. 0000
85
26.94
209.30
- 4.05
0.9926
88
26.26
2II.I6
- 5.91
0.9895
95
24.69
215.88
-10.63
0.9813
It will be observed that as the charge increases the sooner
it exerts its maximum pressure. The last column gives the
fraction of the charge burned at the muzzle and shows that
APPLICATIONS
approximately the entire charge for the series was consumed
at the muzzle, k was computed by the formula
v 2
k = [6.I7483-IO] ,1 y
s A 2
In order to determine the velocity and pressure curves for
any given charge we should compute M and M ' by equations
(23) and (24), and then employ (7) and (8) as has already been
done for a charge of 78 Ibs. For example, determine the velocity
and pressure curves for a charge of 95 Ibs. We have, from (23)
and (24),
log M = 2.09974 + log a + log co = 6.31864
log M' = 2.43084 + ^ log - - log a = 4.80435
Therefore
v = b^spa 2 ! VXi
and
p = [4-80435] X*
are the equations required.
Example. Suppose the thickness of web of the grain we have
been considering to be increased 10 per cent., all other conditions
remaining the same. Deduce the velocity and pressure curves
for a charge of 78 Ibs. In this case it is evident that all the
charge would not be burned in the gun and that in consequence
both the maximum pressure and muzzle velocity would be
diminished.
It will be seen from (69), Chapter IV, that, other things being
equal, the value of X varies directly with the web thickness.
Therefore if this is increased by 10 per cent., or, what is the same
thing, is multiplied by i . i , X will also be multiplied by i . i ;
and from (60) and (61), Chapter IV, M and M f will be divided
by i.i. Therefore (16) and (17) will in this case become,
v = [3.08290]
HO INTERIOR BALLISTICS
and
P = [4.58897] *8.
These equations give v m = 1945 f. s., and p m = 28371 Ibs.
This is a loss of 95 f. s. in muzzle velocity and a diminution of
2079 Ibs. in maximum pressure. To determine the fraction of
the charge burned at the muzzle, we have from (45), Chapter IV,
k-
~
which gives, by employing the muzzle velocity just computed,
k = 0.909.
Therefore on account of the increased thickness of web, seven
pounds of the charge remained unburned when the projectile
left the gun.
We may next inquire what effect a decrease of 10 per cent, in
web thickness would have upon the muzzle velocity and maximum
pressure. In this case we must multiply the original value of X
by 0.9 and divide M and M ' by the same fraction. We thus get
log X = 0.72571
logM = 6.25295
log M' = 4.67612
Therefore, from (9), the muzzle velocity in this case is found
to be 2040 f. s.; and, by (10), the maximum pressure, 34675 Ibs.
That is, the muzzle velocity remains the same while the maximum
pressure is increased by 4225 Ibs. per in. 2 These examples show
that for the greatest efficiency (muzzle velocity and maximum
pressure both considered), the web thickness for this form of
grain should be such that the charge is all consumed at the
muzzle. From the value of X given above we find, by interpola-
tion, that x = 3.4890; and, therefore, u = 155.43 inches. For
this travel the above values of M and M' give v = 1936 f. s.,
and p = 22294 Ibs. The muzzle pressure comes out 8433 Ibs.
APPLICATIONS III
Suppose for a hypothetical 7 -inch gun we assume the follow-
ing data:
d = o". 7
V c = 4,000 c. i.
u m = 40 calibers = 280 inches.
A = 0.6.
5 = 1.5776
/ = 1396.9 Ibs.
v c = 0.13614 in. per sec.
w = 205 Ibs.
What muzzle velocity and maximum pressure would be
obtained, supposing the charge to be all consumed at the muzzle;
and what must be the thickness of web?
The weight of charge due to the given chamber capacity and
density of loading is found to be 86.7 Ibs. We next compute
the following numbers by formulas given in Chapter IV:
log a = 0.0122 1
Iogz = 1.80713
*m = 4-3 6 55
log X= log X om = 0.76269
log Xi= logX lm = 0.39492
log JV- 7-21529 (By (58), Chapter IV),
log M = 6.45260 (By (9))
Then by (7) and (10) we find
Muzzle velocity = 2653 f. s.
and
Maximum pressure = 32112 Ibs. per in. 2
The muzzle pressure, by (8), is 18413 Ibs.
The necessary thickness of web in order that the charge may
all be consumed at the muzzle, is 0.158 inches. The other
dimensions of the grains are immaterial.
If the volume of the chamber is taken at 3,000 c. i., all the
112 INTERIOR BALLISTICS
other data remaining the same, we should have the following
results :
w = 65.03 Ibs.
M . V. = 2413 f. s.
p m = 28868 Ibs. per in. 2
M . P. = 14105 Ibs. per in. 2
2l = 0.152 in.
If v c = 4500 c. i., we have the following:
co = 97.54 Ibs.
M. V. = 2753 f. s.
Pm= 33574 Ibs. per in. 2
M . P. = 24496 Ibs. per in. 2
2 1 = 0.160 in.
Binomial Formulas for Velocity and Pressure. Binomial
formulas pertain to grains for which /* is zero or so small that
it may be neglected, while X must be retained on account of its
magnitude. To this class belong all unperforated, long, slender
grains of whatever cross-section, such as strips, ribbons, cyl-
inders, etc. The binomial expressions for velocity and pressure
for these grains are
v 2 = MX, {i - NX } .... (28)
and
p = M'Xi{i- NX 4 ] .... (29)
The second term within the brackets has the negative sign
because X is always negative for these forms of grain.
Methods for Determining the Constants M and N. The
constants M and N can be determined when the given experi-
mental data are such that two independent equations can be
formed involving M and N. These data may be either two
measured velocities of the same shot at different positions in
the bore; or a measured muzzle velocity and crusher-gauge
pressure, the latter taken as the maximum pressure. In
APPLICATIONS 113
addition to these all the elements of loading, as well as the
powder and gun constants, are supposed to be known.
First Case. Let Vi and v 2 be two measured velocities in
the bore at the distances u^ and u 2 from the origin, which is the
base of the projectile in its firing position. From the gun and
firing constants compute z by (56), Chapter IV, and then Xi and
x 2 corresponding to Ui and u 2 by the equation
u
x =
Z
With these values of Xi and x 2 as arguments, interpolate
from the table of the X functions the corresponding values of
log X and log Xi, distinguishing them by accents. We then
have the two independent equations
r, 2 US V " CT AT V ff \
V 2 = M A! (i J\ X )
from which M and ^V may easily be determined. For simplicity
let
^ \ ~ I ' ~v7^ an d &' = ~v7T
VV A , A
We then have in a form well adapted to logarithmic computa-
tion
N T ~ b ( \
~ (i - bV) X" ( $ 0)
and
~ X\(i - N X' ) X'\(i - N X"
These equations are equally adapted to English or French units.
Second Case. When the powder is not all burned in the
gun let v m be the observed muzzle velocity and p m the crusher-
gauge pressure. We then have the two independent equations
U4 INTERIOR BALLISTICS
and ((50'), Chapter IV),
p m = [9.85640 - 10] M' (i - [0.48444] AO . (32)
Substituting for M' its value in terms of M ((61), Chapter IV),
and making, for English units,
WlPm
c == [7-68507 -- 10] ai - ) p m x i
we have
N " X - 3.051 c " (i -[o. 4 8 444 ] <A, . . (33)
\~ x '
and then M from (31). The X functions in these last two
formulas refer to v m . Any measured velocity within the bore
before the powder is all burned may be used instead of v m . For
French units the logarithmitic multiplier in the expression for
c is [7.56404 10].
Second Method. If the powder is all burned before the
projectile reaches the muzzle, we have from (3)
aF 2 w N
where V m is the muzzle velocity and X 2 corresponds to V m . N
must be determined either by a velocity Vi measured in the
bore before the powder is consumed, or by the crusher-gauge
pressure assumed to be p m . If by the former, we have from (28)
v? =MX l '(i - NX' }.
Substituting M from (34) in this equation and solving for
^V we have
4XX/z;U
In this equation X' and X' 2 correspond to the measured
velocity v lm The travel of projectile to the point where all the
powder is burned is found by the equation X = \/N and a
reference to the table of the X functions. In using these last
APPLICATIONS 115
two formulas N must first be computed, and then M. Equation
(35) is independent of the units employed.
If, as is usually the case, there is no interior measured velocity
available recourse must be had to the crusher-gauge pressure p m .
In this case we have by means of (32) and (34), and (61), Chap-
ter IV, for English units,
, T r T( / r ^4
N = -- [9.21453 - loj i -- (i - L2.79937]
aw y* m
and then M by (34). For metric units the logarithmetic
multiplier within the braces becomes [2.92040].
Application to Sir Andrew Noble's Experiments. These
very important experiments were made at the Elswick works,
Newcastle-on-Tyne, with a six-inch gun. They are thus de-
scribed by Sir Andrew *: "The energies which the new ex-
plosives are capable of developing, and the high pressures at
which the resulting gases are discharged from the muzzle of the
gun, render length of bore of increased importance. With the
object of ascertaining with more precision the advantages to be
gained by length, the firm to which I belong has experimented
with a six-inch gun of 100 calibers in length. In the particular
experiments to which I refer, the velocity and energy generated
has not only been measured at the muzzle, but the velocity and
pressure producing this velocity have been obtained for every
point of the bore, consequently the loss of velocity and energy
due to any particular shortening of the bore can at once be
deduced.
"These results have been attained by measuring the velocities
every round at sixteen points in the bore and at the muzzle.
" Report (1894) on methods of measuring pressures in the bore of guns";
and " Researches on Explosives, Preliminary Note." An abstract of these
papers is given in the "English Text-book of Gunnery," 1902; in Nature
for May 24, 1900; and in Encyclopaedia Britannica, nth edition, article
"Ballistics."
Il6 INTERIOR BALLISTICS
These data enable a velocity curve to be laid down, while from
this curve the corresponding pressure curve can be calculated.
The maximum chamber pressure obtained by these means is
corroborated by simultaneous observations taken with crusher
gauges, and the internal ballistics of various explosives have thus
been completely determined."
The velocities at the sixteen points in the bore were deter-
mined by registering the times at which the projectile passed
these points. The registering apparatus is thus described by
Sir Andrew in the " Report," page 1 1 : " The chronograph which
I have designed for this purpose consists of a series of thin disks
made to rotate at a very high and uniform velocity through a
train of geared wheels. The speed with which the circumference
of the disks travels is between 1200 and 1300 inches per second,
and, since by means of a vernier we are able to divide the inch
into thousandths, the instrument is capable of recording the
millionth part of a second.
"The precise rate of the disk's rotation is ascertained from
one of the intermediate shafts, which, by means of a relay,
registers the revolutions of a subsidiary chronoscope, on which,
also by a relay, a chronometer registers seconds. The subsidiary
chronoscope can be read to about the th part of a second.
5000
"The registration of the passage of the shot across any of the
fixed points in the bore is effected by the severance of the primary
of an induction coil causing a spark from the secondary, which
writes its record on prepared paper gummed to the periphery of
the disk. The time is thus registered every round at sixteen
points of the bore.
"I have ascertained by experiment that the mean instru-
mental error of this chronoscope, due chiefly to the deflection of
the spark, amounts only to about three one-millionths of a
second. Usually the pressures were deduced from the mean of
three consecutive rounds fired under the same circumstances."
APPLICATIONS
The following table gives the recorded experimental data
for the various kinds of smokeless powders employed at the
Elswick firings, and which will be used in the following dis-
cussions.
MEASURED VELOCITY WHEN PROJEC-
Weight
Density
Crusher-
TILE HAD TRAVELLED
Kind of Powder
of
Charge.
Ibs.
Loading
Pressure,
Ibs.
16.6 ft.
21.6 ft.
34-1 ft.
46.6 ft.
f. s.
f. S.
f. S.
f. s.
Cordite, o"-4 ....
27-5
0-55
47040
2794
2940
3166
3284
Cordite, o".35 . ..
22. O
0.44
30352
2444
2583
2798
2915
Cordite, o"-3 ....
2O. O
0.40
36960
2495
2632
2821
2914
Ballistite, o".3 . . .
20.0
0.40
33936
2416
2537
2713
2806
The cordite used in these experiments contained 37 per cent.
of gun-cotton, 58 per cent, of nitro-glycerine, and 5 per cent, of
a hydrocarbon known as vaseline. The ballistite was nearly
exactly composed of 50 per cent, of dinitrocellulose (collodion
cotton) and 50 per cent, of nitro-glycerine.
DISCUSSION OF THE DATA FOR CORDITE, o".4 DIAMETER.
The form characteristics of cordite are a. = 2, \ = and
ju = o. The equations for velocity and pressure are therefore,
V 2 = MXi(i - NX )
and
Since the second terms within the parentheses, which contain
X, have been made negative, X in subsequent calculations must
be regarded as positive.
For the preliminary calculations we have the following
data:
co = 27.5 Ibs.
w = 100 Ibs.
A = 0.55
5 = 1.56
n8
INTERIOR BALLISTICS
From these data we find by the proper formulas:
log a = 0.07084
Iogz = 0.42178 .'. z = 2.6411 ft.
To determine whether the powder was all burned in the gun,
the following table is formed which explains itself:
u
ft.
u
*"
V
(observed)
f. s.
log v "*
logX 2
log V,
log F a
v t
16.6
6.2853
2/94
6.89245
9.68496-10
7.20749
3-60374
4015
21.6
8.1784
2940
6.93669
9.71799
7.21870
3-60935
4068
34-1
12.9112
3166
7.00102
9.76657
7-23445
3.61722
4142
46.6
17.6446
3284
7.03281
9.79440
7.23841
3.61920
4161
The increase of Vi as shown in the last column, indicates
that the powder was all burned in the gun and between u =
34.1 and u = 46.6 ft.
We will compute M and N by means of the measured muzzle
velocity (V m ) and the mean crusher-gauge pressure (p m ), as these
data can always be obtained without sacrificing a gun.
For cordite equations (36) and (34) become
N = [ 9 . 21 453 - 10] i - (i - [*.79973l 1 j
and
4 V
(36')
(340
The numbers to be used in these formulas are
w = 27.5 Ibs.
w = 100 Ibs.
p m = 47040 Ibs. per in. 2
F m = 3 28 4 f. s.
logJ\T 2 = 9.79440 - 10 (at muzzle)
log a = 0.07084
APPLICATIONS IIQ
Performing the operations indicated in (36') and (34') we
have
log N = 8.73599 - 10
log M = 6.57646
Also, by (62), Chapter IV,
log M' = 4.89496.
The equations for the velocity and pressure curves while
the powder is burning are, therefore,
v-= [6.57646] Xj i - [8.73599 - io]X } . (37)
and
p = [4-89496] *i U -- [8.73599 - 10] ^4J . . (38)
After the powder is burned we have from (34'), dropping the
subscript from V mj and reducing,
V = [3.6i92o]\/A r 2 (39)
Also from (31) and (63), Chapter IV,
P [5.07979]
(i + *)* (4)
To determine the travel of projectile to the point where the
powder was all burned u we have, for cordite,
X ~ 2N
Therefore
logX = 0.96298;
and by interpolation from the table of the X functions,
x = 16.018.
Whence
u = x z = 42.30 ft.
The velocity v may be computed by either of equations (3 7)
and (39), as they both give the same value to v } namely,
v = 3261 f. s.
120
INTERIOR BALLISTICS
It will be seen that the increase of velocity from u = 42.26 ft.
to u = 46.6 ft., a travel of 4.34 ft., is only 23 f. s.
Since the vanishing surface of a grain of cordite is zero,
(38) and (40) give the same value to ~p. We find by either
equation,
p = 2745 Ibs. per in. 2
The muzzle pressure by (40) is 2431 Ibs. The distance
travelled by the projectile at point of maximum pressure is
0.45 X 2.64 = 1.19 ft. = 2.38 calibers.
Equations (37) to (40) give all the information that was
obtained by Noble's experiment with cordite, o".4. The only
question that can arise is as to their accuracy in giving the
velocity and pressure at every point of the bore. Equation (38)
gives the observed maximum pressure and (37) the correspond-
ing velocity. Equation (39) gives the observed muzzle velocity
and (40) the corresponding pressure. These equations may be
further tested by computing the velocities for 1 6. 6, 21.6, and
34.1 feet travel and comparing with the measured velocities.
The following table shows the results of this procedure. The
differences between the measured and computed velocities are
in all cases less than the probable error in measuring them, and
are entirely negligible.
Travel of
Projectile
Observed
Velocity
Computed
Velocity
O.-C.
Remarks
1 6.6 ft.
2794 f. s.
2781 f. s.
13
21.6
2940
2942
2
34-1
3166
3172
-6
46.6
3284
3284
The limiting velocity, F t , is 4161 f. s.
It only remains to compute the characteristics v c and / to
solve completely the problem pertaining to this round. These
APPLICATIONS 121
are found to be
v c = 0.38 inches per second,
and
/ = 2266 Ibs. per in. 2
This value of / would mean, if the problem under considera-
tion were completely solved, that one pound of the gases of this
powder, at temperature of combustion, confined in a volume of
one cubic foot, would exert a pressure of 2,266 pounds per
square inch. But the problem is very far from being solved
rigorously. In the deduction of equation (18), Chapter II,
which is the basis of all our formulas, there were neglected the
following energies:
1. The heat lost by conduction to the walls of the gun.
2. The work expended on the charge, on the gun and carriage,
and in giving rotation to the projectile.
3. The work expended in overcoming passive resistances,
such as forcing, friction along the grooves, the resistance of
the air, etc. In short the entire work of expansion was
supposed to be employed in giving motion of translation to
the projectile, and to be measured by the acceleration pro-
duced.
It may be seen, however, from a careful consideration of
equation (18) and the use made of it in deducing the X functions
that these functions are independent of the value of/; and that
when this factor has been determined so as to satisfy completely
such experiments as we have just been considering, these neglect-
ed energies are practically allowed for. Indeed, they are all
contained implicitly in the factors M and N. Similar remarks
apply to T whose deduced value from (2), Chapter IV, depends
upon the exponent of p /p, about which there is considerable
uncertainty. But these characteristics are unnecessary for
determining the equations of the velocity and pressure curves
from such data as we have been considering. But they are of
122 INTERIOR BALLISTICS
use in deducing the circumstances of motion when the charge
varies, as has been already shown.
We will now give a few illustrative examples which can be
solved by this one round.
Example i. What thickness of layer was burned from the
grains when the projectile had travelled 16.6 ft.?
Combining equations (12) and (14), Chapter IV, gives
X
That is, the thickness of layer burned from the surface of a
grain of powder of whatever shape, in the bore of a gun, varies
directly as the function X . In this example, applying the known
values of 1 , X and X (for u = 16.6), we find
/ = 0.1443 in. Ans.
Example 2. What was the velocity of combustion of the
grains at the point of maximum pressure?
We have (equation (53'), Chapter IV),
P
= '' 3795 B "47 m. per sec. Ans.
Example 3. What must be the diameter of the grains of
this powder in order that the charge of 27.5 Ibs. should all be
burned when the projectile has travelled 16.6 ft.?
When the only variation in the charge and conditions of
loading is in the thickness of web, equation (53), Chapter IV,
shows that the X function of the distance travelled by the pro-
jectile when the powder is all burned is directly proportional to
the web thickness. We therefore find, by employing known
numbers,
2 1 = 0.2885 m - Ans.
Cordite, o".35. Preliminary calculations show that the cor-
dite fired in this round was not quite all burned in the
APPLICATIONS 123
We will therefore compute N and M by (30) and (31) with the
following data:
log a = 0.21265
log z = 0.46666 .*. z = 2.929 ft.
zjj = 2583 f. s. = velocity for 21.6 ft. travel.
z; 2 = 2915 f. s. = muzzle velocity,
log X' = 0.84614
log X" = 0.96201
log X\ = 0.55164.
log^ = 0.74763
These numbers give
log N = 8.73582 10
logM = 6.48164
logM l = 4-755 I 4
The results of further calculations for this round are:
Pm= 34093
u = 46.96 ft.
/ = 2277 Ibs.
v c = 0.315 in. per sec.
The differences between the observed and computed veloc-
ities for u = 16.6 ft. and u = 34.1 ft., are, respectively, 12 and
6 f. s.
The limiting velocity for this round is 3731 f. s.
The mean crusher-gauge pressure was 30352 Ibs., which
is certainly erroneous. The force of the powder is practically
the same as for cordite, 0^.4; but this latter seems to be a quicker
powder than cordite o".35.
Cordite, o".3. The cordite fired with this charge was all
burned in the gun and we will, therefore, compute N and M
by equations (35) and (34), with the following data:
log a = 0.26928
log z = 0.48190 . ' . z = 3.033 ft.
I2 4 INTERIOR BALLISTICS
v,= 2495 f. s.
v m = 2914 f. s.
log X 2 = 9.78256 10
\ogX' 2 = 9.66596 10
\ogX f = 0.79917
By means of these numbers the various formulas give
log N = 8.80138 - 10
log M = 6.54986
log If = 4.80822
log P' ' = 4.92766
logFi 2 = 7.14642
log X = 0.89759
x_ = 10.319
u = 31.3 ft.
v = 2787.4 f. s.
? = 333 1. 2 Ibs. per in. 2
/ = 2521 Ibs.
v c = 0.309 in. per sec.
The computed maximum pressure is 37287 Ibs., which is but
327 Ibs. in excess of the mean crusher-gauge pressure. The
corresponding velocity is 877.2 f. s., and travel of projectile
1.33 ft. The differences between the observed and computed
velocities for u = 21.6 ft., and u = 34.1 ft. are i and 5 f. s.
respectively.
The limiting velocity is 3743 f. s.
This powder is apparently stronger than either the 0^.35 or
o".4 cordite. These latter are evidently of the same composi-
tion, known as "Mark i," while the former may have been the
so-called "Cordite M. D.," which is said to have a slightly
reduced rate of burning and to give higher velocities. Its
composition is gun-cotton 65 per cent., nitro-glycerine 30 per
cent., and mineral jelly 5 per cent.
Example. Suppose the cordite, o".3, to be moulded into
APPLICATIONS 125
cubes of the same web thickness. Determine the equations of
velocity and pressure. We have a = 3, X = i, and ju = ,
while FI and X remain the same as already found. We now
have M = 1 , N = =-, and N' = -=-;: The equations are
X X 3 X "
therefore,
ir = [6.72595] Xj ( i - [9.10241 - 10] X + [7.72770 - 10] X* }
and
p = [4.98431] X*{i - [9.10241 - io]X 4 + [7.72770 - 10] Y, }
The maximum pressure computed by this last formula is 45726
Ibs. The muzzle velocity is, of course, the same as before, as
is also the velocity v.
Application to the Hotchkiss 57 mm. Rapid-Firing Gun.
The data for the following discussion are taken from a paper by
Mr. Laurence V. Benet, printed in the Journal U. S. Artillery,
Vol. i, No. 3. The gun experimented with was a standard
pattern, all steel, 57 mm. Hotchkiss rapid-firing gun, and the
experiments consisted in "cutting off successive lengths from the
chase and observing the velocities of a series of rounds fired with
each resulting travel of projectile." The data necessary for
this discussion are the following:
GUN DATA.
Area of cross-section of bore, 0.2592 dm. 2
Equivalent diameter, 5.745 cm.
Net volume of powder chamber, 0.887 dm. 3
POWDER AND PROJECTILE.
"Two brands of the same type of smokeless powder were
employed, both of which were manufactured at the Poudrerie
Nationale de Sevran-Livry; they were designated as B NI
126
INTERIOR BALLISTICS
and B N 1M . These powders are in the form of thin strips, which
are scored longitudinally on one side with a series of parallel and
very narrow grooves. The chemical composition is unknown."
The grains were of the following dimensions and densities:
BN,.
76 mm.
1.4 mm.
0.5 mm.
B 1
85
1.6
0.6
1.78
mm.
mm.
mm.
Length of strips,
Distance between scores,
Thickness of strips,
Specific gravity,
The elements of loading were as follows:
Weight of charge, 0.460 kilos. 0.400 kilos.
Weight of projectile, 2.720 kilos. 2.720 kilos.
Density of loading, 0.519 0.451
The velocities were measured by means of two Boulenge"-
Breger chronographs on independent circuits; and the pressures
were determined by means of a crusher gauge seated in the breech
block of the gun. The mean pressures at the breech were for
B Ni powder, 2547 kilos, per cm. 2 ; and for B N 144 , 2543 kilos,
per cm. 2
From the firing records was obtained the following table
giving the velocity of the projectile corresponding to each length
of travel in the bore:
Velocity in
TRAVEL <
DF SHELL
Velocity in
Bore with
Bore with
Remarks
BN r
Metres
Calibers
N 144
Metres per Sec.
Metres per Sec.
543-1
0.880
15-44
503.7
574-4
.051
18.44
534-9
595-0
.222
21.44
553-1
612.6
393
24.44
565-0
622.3
564
27.44
573-5
636.5
.792
31-44
591.0
648.3
2.020
35-44
600.7
We will first consider the powder B Ni, and compute by the
APPLICATIONS
127
proper formulas already many times referred to, the values
of a, z and x for the given charge and travels of projectile.
Then take from the table of the X functions the logarithms of
A" , X } and X 2 . All these are given in the following table for
convenient reference:
log a = 0.11053. Iogz = 9-359 62 - io-
M
Metres
X
logXo
log X,
log X 2
Remarks
0.880
I.05I
1.222
1-393
1.564
1.792
2.O20
3-8447
4.5918
5.3389
6.0860
6.8362
7.8293
8.8254
0.74183
0.77092
0.79521
0.81604
0.83425
0.85540
0.87382
0.35356
0.4IIOO
045765
0.49670
0.53013
0.56819
0.60062
9.6II73
9.64008
9.66244
9.68067
9.69590
9.71279
9.72681
It is known that the powder was all burned in the gun, as
might be also inferred from the thinness of web; and the first
step is to determine the travel of projectile when this takes
place, in other words the value of u. On account of the "series
of parallel and very narrow grooves " with which the strips were
scored on one side, it is difficult to ascertain the form character-
istics from geometrical considerations. Their determination will
therefore be left until M and N are computed from the measured
velocities. /* will be considered zero.
The expression for F\, the limiting velocity, is
where V is any velocity after the powder is all burned and
X 2 a function of the corresponding travel of projectile. If then
we compute V l by this formula for all the measured velocities,
and find that it is approximately constant for a certain num-
ber of measured velocities nearest the muzzle, we shall have
an indication of the travel of projectile when the powder
128
INTERIOR BALLISTICS
is all burned. The following table gives the values of
computed:
so
u
Fi
0.880 m.
849 m. s.
.051
869
.222
878
393
885
564
883
.792
886
2.020
888
An examination of this table shows that u lies between 1.222
and 1.393, or that x lies between the numbers 5.3389 and 6.0860
and rather nearer the former than the latter.
We will assume x = 5-6 and Vi = 885.5 m - s -> which is a
mean of the last four values. Since
we find
v= 605.1 m. s.
N and M can now be computed by (30) and (31), which do
not contain the form characteristics.
The data are: ^= 543.1 m. s., v 2 = 605.1 m. s., logX' =
0.74183, log *",= 0.80284, log A"! =0.35356, and logX'\ =
0.47205.
The results of the calculations are:
log N = 8.68636 - 10 1
log M = 5.25171 I While powder is burning.
log M' ' = 3.62063
F = [2.94719] VX 2 }
_ [3.7^589 I After powder is all burned.
(* + *)* J
The following table shows the agreement between the ob-
served and computed velocities:
APPLICATIONS
129
i
Travel
VELOCITIES
of
o.-r.
Remarks
Projectile
Observed
Computed
O.88O m.
543.1 m. s.
543.1 m. s. o.o
1.051
5744
572-8
1.6
1.222
595-0
597-4
-2.4
1-393
612.6
613.1
-0.5
1.564
622.3
623.9
-1.6
1.792
636.5
636.3
0.2
2.O20
648.3
646.5
1.8
The greatest of these differences is less than one-half of one
per cent, of the observed velocity and the others are practically
nil. The maximum pressure computed by the formula
p m = [9.85640 - 10] M'{i - [0.48444] N}
is 2555 kilos, per cm. 2 , differing by less than one- third of one
per cent, of the mean crusher-gauge pressure. These results
show that the assumed value of x= 5.6 is practically correct.
Finally, we have,
/ = 7883 kg. per cm. 2
and
v c = 0.438 cm. per sec.
- 0.172 in. " "
The form characteristics a and X can be computed by the
formulas
MX.
V?
and X = N X,
From these we find a. = 1.4460 and X = 0.3045.
For the B $ 144 powder the equations for the velocity and
pressure curves are found, by a process entirely similar to the
above, to be, while the powder is burning,
v 2 = [5-24187] X, (i - [8.75166 - 10] X )
and
p = [3-56310] ^3 (i - [8.75166 - 10] X<)
9
130
INTERIOR BALLISTICS
After the powder is burned the equations become
V = [2.92007] VX 2
[3-68425]
and
The following table shows the agreement between the
observed and computed velocities:
Travel
Observed
Computed
of Projectile
Velocity
Velocities
o.-c.
Remarks
m.
m. s.
m. s.
0.880
5037
503.9
O.2
I.05I
534-9
532.2
2.7
1.222
553-1
553-6
-0-5
1-393
565-0
566.0
1.0
1.564
573-5
576.5
-3-o
1.792
591.0
588-4
2.6
2.020
600.7
598.5
2.2
The value of /for B N ni comes out 8001.5 kilos, per cm. 2 , and
D C is found to be 0.5268 cm. per sec. This powder is therefore
slightly "stronger" than B NI and about 22 per cent, quicker;
and this notwithstanding its greater density.
Application to the Magazine Rifle, Model of 1903. The
following data pertaining to this rifle were obtained partly
from a descriptive pamphlet issued by the Ordnance Depart-
ment, and partly through the courtesy of officers of the
Ordnance Department on duty at the Springfield Armory and
Frankford Arsenal, to whom the writer is under special obliga-
tions :
Caliber, 0.3 inches.
Volume of chamber, 0.252 cubic inches.
Total travel of bullet in bore, 22.073 inches.
Mean weight of powder charge, 44 grains.
Weight of bullet, 220 grains.
"The standard muzzle velocity of this ammunition is 2300
APPLICATIONS 131
f. s., with an allowed mean variation of 15 f. s. on either side of
the standard. The powder pressure in the chamber is about
49,000 pounds per square inch."
The powder used with this rifle is composed essentially of
70 per cent, nitrocellulose and 30 per cent, nitro-glycerine. "The
grains are tubular, being formed by running the powder colloid
through a die 0.09 inch in diameter, with a pin 0.03 inch in
diameter; and the string thus made is cut 21 to the inch."
There are considerable variations in the length and diameter of
the grains "due to the fact that the string is not cut exactly
perpendicular to its axis, and to irregularities in shrinking.
There are 83,000 to 91,000 grains per pound. The specific
gravity is about 1.65, and the gravimetric density is from 0.90
to 0.94."
On account of the tubular form of the grains the character-
istic M is zero, and therefore the equations for velocity and
pressure are binomials. We have reliable measured in-
terior velocities for this rifle, obtained at the Springfield
Armory in the fall of 1903, by firing with a rifle the barrel
of which was successively cut off one inch. Five shots (some-
times more) were fired for each length of barrel and the
velocities were measured at a distance of 53 ft. from the
muzzle, and reduced to muzzle velocity by well-known methods.
(See Table A.)
It is known that the charge in the magazine rifle is all burned
at, or very near, the muzzle. We may, therefore, take the two
extreme reduced velocities of the series for Vi and v 2 and thereby
minimize the effects of errors in measuring the velocities. The
firing data are then,
Vi= 1274 f.s.; v 2 = 2277.6 f. s.
u i = 3-73 m - ; ^2 = 20.073 m -
The weight of charge in these firings was 45.1 grains and
weight of bullet 220 grains. The preliminary calculations give
132
INTERIOR BALLISTICS
A = 0.7077
log a = 9.90686 10
log z = 0.30878 .'. z = 2.036 in.
Xl = 1.5093, loX' = 0.57969, logX\= 0.00146.
x 2 = 10.84125, \ogX" = 0.90504, \Q^X'\ = 0.65421.
These numbers and the velocities ^ and v z , substituted in
(30) and (31), give
log N = 8.73379 - 10
and
log If = 6.30896.
We also find
log If' = 4.91902.
The formulas for velocity and pressure are, therefore,
v* = [6.30896] X, {i - [8.73379 - 10] X ]
p = [4.91902] X 3 { i - [8.73379 - 10] X 4 }
We have
MX,
and this substituted in (15), Chapter IV, gives
y = > - -X [i - [8.73379 - 10] Xo\
V2
Since in this case v and X 2 refer to the muzzle, we have for
the powder burned, in grains,
y = [0.99736] X { i - [8.73379 - 10] X }
or, in another form more convenient for computation,
v 2
y = [4.68840 - 10] TT
A 2
Table A gives the measured and computed velocities for the
travels of projectile in the first column, and also the weight of
powder burned at each travel.
APPLICATIONS
TABLE A
Travel
of Projectile,
inches
Mean Velocity
53 Feet
from Muzzle,
f. s.
Muzzle Velocity
Deduced from
Measured
f. s.
Computed
Velocity,
f. s.
o.-c.
Powder
Burned,
grains
3-073
1253
1274
1274
29.99
4-073
1402
1426
1432
- 6
32.61
5-073
1531
1558
1555
3
34-63
6.073
1633
1662
1656
6
36.25
7-073
1742
1772
1740
32
37-59
8.073
1771
1802
1812
10
38.71
9-073
1860
1894
I8 74
20
39-66
10.073
1909
1943
1929
H
40.48
11.073
1957
1992
1976
16
41.19
12.073
1989
2023
2018
5
41.81
13-073
2016
2052
2057
- 5
42.36
I4-073
2050
2086
2091
- 5
42.83
15-073
2069
2105
2122
-17
43-25
16.073
2104
2140
2151
ii
43-63
17.073
2129
2165
2177
12
43-95
18.073
2183
2219
2200
19
44-25
19.073
2163
2200
2222
22
44-5
20.073
2201
2238
2242
- 4
44-73
21.073
2203
2240
226l
i
44-93
22.073
2240
' 2278
2278
o
45-io
Table B, on page 134, supplements Table A by giving com-
puted velocities and pressures from the origin of motion. The
velocity curve in the diagram, Fig. 2, on page 135, shows at a
glance the agreement between theory and observation.
It will be observed that the computed pressures depend
entirely upon two measured velocities. Also that the maximum
pressure occurs when x = 0.45, and agrees with the official
statement. The muzzle pressure is about 6,000 Ibs. per in. 2
Powder Characteristics. The form characteristics of these
grains according to the given dimensions are
a= 1.63 and X = 0.3865.
But these minute grains, of which there are 560 in the service
charge, shrink irregularly and many of them doubtless are more
or less abraded and perhaps broken, so that it is impossible to
determine the mean values of a and X geometrically with any
134
INTERIOR BALLISTICS
TABLE B
X
u,
inches
Computed
Velocity,
f. s.
Computed
Pressure,
Ibs. per inch 2
Powder
Burned,
grains
Pressure on
Base of
Projectile,
pounds
O.OOO
0.000
0.000
o.ooooo
0.000
OOO
0.001
0.002
8.590
4501
1.081
318
O.OI
0.020
47.854
13835
3.375
978
O.I
0.204
254-91
37008
10.139
2556
0.2
0.407
408.98
45117
13.842
3189
o-3
0.6II
531.70
48456
16.475
3425
0.4
0.814
635.14
49640
18.554
3509
0.45
0.916
681.48
49769
19.454
3518
0.5
I.OI8
724.89
49695
20.283
0.6
1.222
804.25
49118
21.766
0.8
1.629
939-77
47039
24.221
I.O
2.036
1052.5
44492
26.205
1.2
2-443
1148.8
41882
27.867
'.'.'.'.
1.4
2.850
1232.6
39369
29.291
1.6
3.258
1306.4
37023
30.533
2.O
4.072
32854
2.5
5.090
28^ SO
3
6.108
^^oo
25061
4
8.144
19812
5
I0.l8o
16086
6
I2.2I6
IT. T.IQ
7
14.252
OO :/
III88
8
16.288
9501
9
18.324
8134
10
20.360
7006
ii
22.396
604. S
T^\J
certainty. They may, however, be deduced from the values
of M and N. From these we find
Finally we find
and
a = 1.7710
x = -4353
/ = 1622.5 Ibs. per in. 2
v c = 0.28 in. per second.
Formulas for Designing Guns for Cordite. The caliber, of
course, is given, and the weight of the projectile of desired length
APPLICATIONS
and form of head can be computed by known methods.* The
grain characteristics and density of cordite and the values of /
ID 15
Travel, inches.
FIG. 2.
20 22.07
and v c are also known. The necessary formulas for this discussion,
given in the order in which they will be used, are the following:
& = [8.55783] A V c (a)
z = [1.54708] (c)
* See the author's "Handbook" (Artillery Circular N), chapter xi.
136 INTERIOR BALLISTICS
v:~= [4.44383]^ . . . .
. . . to
L v u ' )
in which
& _ [2.i28 9 o]/
This equation is deduced from (32), eliminating M' and N.
2 V 2 I
X ' ~ 2 X '
M'= [7.82867 - io]M^ (g)
_ _. __ p ,. _ "i / 7 \
# = M ' ^ 3 (i - N X,) (i)
p m = [9.85640 - 10] M' (i - [0.48444] N) . . (j)
1 = [8.56006 - 10]
Example. Take the hypothetical y-inch gun already con-
sidered on page in, for which d = j ff and w = 205 Ibs. For
cordite of o".3 diameter we found / = 2521 Ibs. per in. 2 , and
v c = 0.309 in. per sec. Also 5 = 1.56. The only assumptions
necessary are the volume of the chamber (V c ) and the density
of loading. And this last is not purely arbitrary, since considera-
tions of safety to the gun and its efficiency restrict its value to
narrow limits, say from 0.4 to 0.6. This latter value is often
exceeded, especially in our service; but it is believed that by
choosing the proper shape and size of grain this can always be
APPLICATIONS
137
avoided. As / and v c are unusually large for cordite, we will
take A = 0.4; and for a first assumption will give the chamber
a volume of 3,000 c. i.; which is less than the volume of the
chamber of the 6-inch wire- wound gun. Finally we will take
Pm= 37>oo Ibs. per in. 2 , leaving the muzzle velocity and travel
in the bore for later consideration.
From the given data we find, by means of the above formulas,
w = 43-35 2 Ibs-, lo g# = 0.26928, log z = 1.76317, log Fi 2 =
7.17066, log X = 0.90184, log M = 6.56985, log M f =4.80398
and log TV = 8.79713 10.
The equations for velocity and pressure are, therefore,
v*= [6.56985] X, (i - [8.79713 - 10] X )
and
p = [4-80398] X 9 (i - [8.79713 - 10] X 4 )
This last equation, which is the same as equation (/) when
x = 0.45, makes p m = 37000, thus verifying the calculations.
The muzzle velocity will, of course, depend upon where we
place the muzzle, in other words upon the value adopted for
u m . If we regard 40 calibers as a suitable travel in the bore, we
shall have
u m = 40 X 7" = 28o /r
whence
x m = ujz = 4-8305
For this value of x, Table I gives
log X = 0.77912
log Xi = 0.42689
and these in the above velocity equation give
v m = 2487 f. s.
We find, from Table I, taking log X as the argument,
x = 10.613;
138 INTERIOR BALLISTICS
and by (m)
u = 615.19 in.
Also by (k)
2/ =o".47
and by (/)
k m = 0-9394-
That is 94 per cent, of the charge was burned at the assumed
muzzle.
If the maximum pressure is increased to 38,000 Ibs., the
density of loading and volume of chamber remaining as before,
the velocity for a travel of 280 inches will be increased to 2503
f. s., and the thickness of web, or diameter of the grain, will be
reduced to 0^.45. This slight diminution in the diameter of
the grain increases the initial surface of combustion of the
charge about 3^ per cent., which fully accounts for the increased
maximum pressure.
If we take A =0.5, V c = 3,000 c. i. and p m = 37,000 Ibs.
per in. 2 , there results co = 54.19 Ibs., v 2S o = 2570 f. s., and
2 / =o".66.
Trinomial Formulas for Velocity and Pressure. Trinomial
formulas occur when the grains of which the charge is composed
are of such form and dimensions that the form characteristic /*
cannot be regarded as zero. Spherical, cubical, and multi-
perforated cylindrical grains are of this kind. For the first two
forms mentioned the second term is negative and the third
positive; while for m.p. grains (those used in our service), the
second term is positive and the third negative.
For spherical and cubical grains we may have, before the
powder is all burned, the two independent equations,
- N X' +-
v, 2 = MX," (i - N X" + -N* X'\]
APPLICATIONS 139
Put for convenience,
X
/f
o
3 (i a b) 2 c (i a)
C = 2(1 -ab*}X", = (i -abYx 7 ^ *
Then the quadratic equations give, using the sign applicable
to this problem,
--^) 5 ) . . ( 4 i)
The value of M may now be computed by either of the above
expressions for v 2 . Or, if V m is the muzzle velocity, that is,
if the powder is all burned in the gun, M may be computed by
the formula, derived from (3),
M = ~ (42)
^2
If the powder is not all burned in the gun and our data are
a muzzle velocity and the crusher- gauge pressure (assumed to
be the maximum pressure), N may be computed by the follow-
ing process: Compute the auxiliary quantities b, c, and d by
the formulas:
5 = [7.68507 - 10] ^r^x','- , bXi
2 X. li -^r
d = - 3(I -
Then
( / j\ *i
.... (43)
The functions X 4 and X 5 pertain to the tabular value of x
which gives the maximum pressure. If we take this to be 0.45
140
INTERIOR BALLISTICS
no material error will ensue. We therefore have
log Z 4 = 0.48444
log ^ 5 = 0.93587
The function X pertains to the muzzle.
It should be remembered that equations (41), (42), and (43)
are applicable to cubical and spherical grains only.
Application to Noble's Experiments with Ballistite. The
ballistite consisted of equal parts of dinitrocellulose and nitro-
glycerine and was in the form of cubes 0.3 of an inch on a side.
The gun, powder, and firing data are as follows : d = 6 inches,
A = 0.4, 6 = 1.56, w = 20 Ibs., and w = 100 Ibs. From these
we find log a = 0.26928 and Iogz = 0.48190. .'. z = 3.033 ft.
The following table, which explains itself, is formed for convenient
reference:
Observed
u,
X=U/Z
Velocity
log X
log*!
logX 2
Remarks
ft.
f. s.
16.6
5473
2416
0.79917
0.46513
9.66596-10
21.6
7.121
2537
0.84069
0.54I8I
9.70112
34-1
11.242
2713
0.91049
0.66339
9.75289
46.6
15-363
2806
0.95685
0.73940
9.78256
As the powder was not quite all burned in the gun the ex-
treme measured velocities are available for determining N and
M by means of (41). The data are Vi = 2416 f. s., v 2 = 2806
f. s., logJT = 0.79917, log X" = 0.95685, log X\ = 0.46513
and logXi" = 0.73940. Substituting these in (41), we find
log N = 9.03843 - 10
log N' = 7-59974 - 10
log M = 6.62918
log if ~ 4.88754
The equations for velocity and pressure are, therefore,
= [6.62918]*! {1-19.03843-10] *
and then
=[4.88754] * 3 { i -[9.03843-10] ^+[7.59974-10] x 6
APPLICATIONS 141
The equation for velocity will, of course, give the observed
velocities for u = 16.6 ft., and u = 46.6 ft. It should also give
the observed velocities, if our method is correct, for u = 21.6 ft.
and u = 34.1 ft., and indeed for every point in the bore from
the firing seat to the muzzle. The velocities computed for these
intermediate values of u are 2536.5 f. s., and 2710 f. s., respect-
ively, which differ so slightly from the measured velocities as
to be negligible. The maximum pressure, which occurs in this
case when x = 0.4, is 39,163 Ibs. per in. 2 ; and the corresponding
velocity 865.8 f. s., and travel of projectile 1.213 ft.
The distance travelled by a projectile to the point where the
powder is all burned is determined by means of X and the
table of the X functions. We have for cubical grains for which
X is unity, X = i/N. Therefore for this example,
log X = 0.96157
Corresponding to this value of log X , we find, by interpola-
tion from the table,
x = 15.8649, logXi 0.74694, and log X 2 = 9.78536 10.
To compute #, we have u = 15.8649 X 3.033 = 48.12 ft.
The charge was therefore not all burned in the gun, though the
fraction of the charge remaining unburned was exceedingly
small, practically zero. To get an expression for the fraction
of the charge burned for any travel of the projectile, we have,
from (45), Chapter IV,
v 2 = k VS X 2
But from (3), for cubic grains,
M
Fi 2 = -jj. .'. log Fi 2 = 7.11363 (for this example).
Therefore for cubic grains,
.
"
MX 2
I4 2 INTERIOR BALLISTICS
which for this example reduces to
k = [2.88637 - 10] -JJT
Applying different velocities and the corresponding values
of log X 2 in this formula it will be found that the charge was
practically consumed long before the projectile reached the
muzzle. Indeed nine teen- twentieths of the charge was burned
when the projectile had travelled 16.6 ft.
We may also determine k in terms of the travel of projectile
by means of the equation
k = i - ( i - ^) 3 (Eq. (46), Chapter IV).
Or, if we wish to know the distance travelled by the projectile
when a given fraction of the charge is burned, we have
As an example, suppose k = -. Then
Y
X =
Applying the value of N found for this round and completing
the calculations it will be found that one-half the charge was
burned when the projectile had travelled one foot.
The expressions for V and P for this round are
v = [3-55681] vx 2
and
[4.89487]
The last two formulas give the velocity and pressure upon
the supposition that all the powder was converted into gas at
temperature of combustion before the projectile had started.
APPLICATIONS
The initial value of P is found by making x zero. Whence
p f = 78,500 Ibs. per in. 2
Finally we find
; = 2338 Ibs.
and
v c = 0.266 in. per sec.
The following table was computed by the formulas de-
duced for this round for comparison with the deductions from
Sir Andrew Noble's velocity and pressure curves. Unfortunately
these curves, as published, are drawn to so small a scale and are
so mixed up with other curves that it is difficult to get the
velocities and pressures from them with much precision.
NOTE : The velocities and pressures in the second and third
columns were computed by formulas slightly different from those
deduced above. But the differences are so small as to be of no
account in the discussion.
X
u
ft.
Computed
Velocities,
f. s.
Computed
Pressures,
Ibs. per inch 2
Pounds of
Powder
Burned
V,
f. s.
P,
Ibs. per in. 2
0.000
o.ooo
0.0
0.0
o.o
78500
0.001
0.003
12.449
4196
0.716
65.786
78397
O.OI
0.030
68.889
12672
2.206
207.40
77466
0.05
0.152
221.52
25254
4-683
457.78
73557
O.I
0.303
359-32
32007
6-357
637.37
69132
0.2
0.607
569-36
37680
8.464
875.21
61560
0.4
1.213
869.33
39439
10.967
1174.0
50122
0.6
1.820
IO87.2
37567
12.549
1372.5
41948
0.8
2.427
1257-7
34846
13.686
1520.3
35852
1.0
3-033
1396.6
32050
14-557
1637.0
3H53
2.O
6.066
1842.4
21268
I7-043
1995-9
18143
3.0
9.100
2093.0
15013
18.225
2192.5
12363
4.0
12.133
2257.1
11164
18.892
2322.4
9181
5-o
15.166
2374.2
8645
19.299
2416.9
7200
5-473
16.6
2419.0
7742
19-435
2453.5
6507
7.121
21.6
2538.0
5500
17.742
2554-9
4809
11.242
34-1
27IO.O
2893
19.979
2711.8
2782
I5-363
46.6
2806.0
1890 19-999
2806.0
1890
The computed velocities in the third column of this table,
corresponding to the travels of projectile in the second column,
144 INTERIOR BALLISTICS
agree very well with those deduced from Sir Andrew Noble's
velocity curve, from the origin of motion to the muzzle, a distance
of 46.6 ft. As the velocities are thus shown to be correct, the
pressures in the fourth column are, from their manner of deriva-
tion as given in Chapter IV, necessarily correct also. That is,
they correspond to the energy of translation of a hundred-pound
projectile. In this respect they are more accurate than the
pressures given by Sir Andrew's pressure curve which was de-
rived from his velocity curve by graphic methods not sufficiently
precise for the great accelerations encountered in ballistic
problems.
The writer is indebted to Colonel Lissak, formerly Instructor
of Ordnance and Gunnery at West Point, for the accompanying
diagram (Fig. 3) of the velocity and pressure curves whose co-
ordinates are given in this table. Many interesting facts may be
gleaned from an examination of these curves, and the formulas
by which their coordinates were computed.
The two velocity curves v and V are both zero at the origin
but immediately separate, attaining their greatest distance apart
when the projectile has moved but a short distance. They
then approach each other very gradually and become tangent
at the point where the powder is all burned, practically at the
muzzle. Both curves are tangent to the axis of ordinates at the
origin and parallel to the axis of abscissas at infinity. The
pressure curve p begins at the origin, attains its maximum when
the projectile has traveled about 15 inches, changes direction
of curvature when u is about six feet and meets the axis of
abscissas at infinity. The pressure curve P is convex toward
the axis of abscissas throughout its whole extent. It lies above
the curve p from u = o to u = 30 inches (about), then passes
below p and the two curves become tangent at the point where
the powder is all consumed. Finally the areas under the curves
p and P are equal.
Example i. Suppose the charge in the example under
APPLICATIONS 145
consideration to be increased from 20 to 25 Ibs. Deduce the
equations for velocity and pressure.
In solving this example, we will compute the new constants
M, M f , N and N f by equations (80) to (83), Chapter IV; and as
the charge is increased by 25 per cent., a new value of /must be
found by (90) and (go'). For a six-inch gun we will take n = t
o
provisionally; and since the weight of the projectile remains the
same, n' must be zero. We therefore have
K = 86.12,
and the new value of / is 2518 Ibs.
The new values of a and z for co = 25 Ibs., are
log a = 0.13321
Iogz = 0.44277 .*. z = 2.772 ft.
Applying these numbers in the equations above mentioned
we find for a charge of 25 Ibs.,
logM= 6.73881
log If = 5.03633
log N = 9.01885
logN' = 7.56058
which give the equations required.
These constants give p m = 55676 Ibs., and a velocity of
2841 f. s., for a travel of 16.6 ft. That is, an increase of 5 Ibs.
in the charge increases the maximum pressure 16,800 Ibs. per
in. 2 , and the velocity at 16.6 ft. travel, 425 f. s. Taking the
reciprocal of N gives
log X = 0.98115
and from the table,
x = 18.1425
and
u = 18.1425 X 2.772 = 50.29^.
10
146 INTERIOR BALLISTICS
The limiting velocity and fraction of charge burned are given
by the equations
Therefore
log Fi 2 = 7.24284
and
y = k & = [4-I55 10 ~ IQ ] ~~
rt-a
From this last formula we find when u = 16.6 ft., y 24.18
Ibs.
The pressure at this point is found to be 10480 Ibs. per in. 2
It is interesting to compare these results with those found with
a charge of 20 Ibs.
In order to lessen the maximum pressure the grains must
be increased in size and thus diminish the initial burning surface.
Suppose we increase the size of the cubes from 0^.3 to 0^.5 on a
side. Determine the equations of velocity and pressure for a
charge of 25 Ibs. An examination of equations (80) to (82),
Chapter IV, will show that when the only change in the data is
in the thickness of web the new values of M, M' , and N will be
found by multiplying the previously determined values of these
constants by the ratio of the web thicknesses, in this case by
0.6. We therefore have for 25 Ibs. of 0^.5 cubes
log M = 6.51696
log M'= 4.81448
log N = 8.79700 10
log N'= 7.11688 - 10
From these we get
p m = 38437 Ibs. per in. 2
and
v = 2571 f. s., for u = 16.6 ft.
The measured velocity for this travel of projectile was 2416
APPLICATIONS 147
f. s., with a charge of 20 Ibs. of o" '.3 cubes. Therefore by
increasing the weight of charge 5 Ibs., and at the same time
enlarging the grain from o" '.3 to 0^.5 on a side the velocity is
increased 155 f. s., and this without increasing the maximum
pressure, though the mean pressure is, of course, considerably
increased.
The pressure for u = 16.6 ft., with a charge of 20 Ibs. of the
smaller grains, was 7741 Ibs.; and with a charge of 25 Ibs. of the
larger grains, the pressure for the same travel would be 11181
Ibs. The powder actually burned during this travel of projectile
is a little more in this latter case than in the former, and the
space in which it has been confined during its expansion is less,
both of which facts account for the greater work performed.
From equation (19'), Chapter III, it follows that for two equal
charges made up of grains of the same form and differing only
in their size, the entire initial surfaces of the two charges vary
inversely as the thickness of web. Therefore the initial surface
of the charge of o. r/ 5 grains is of the initial surface of the
same charge of o."3 grains. This accounts for the two charges
giving the same maximum pressure. It may be remarked that
the same results would have been obtained if the grains had
been spherical instead of cubical.
Application to Multiperforated Grains. A peculiar diffi-
culty arises in the application of any system of interior
ballistic formulas to multiperforated grains from the fact that
they do not retain their original form until completely con-
sumed as do all other forms of grain in use, but each grain
breaks up, when the web thickness proper is burned through,
into twelve slender rods, or " slivers," which burn according to
a different law; and thus two independent sets of formulas
become necessary to represent what actually takes place in the
gun. It was previously sought to overcome this difficulty by
supposing the web thickness to be slightly increased so as to
148 INTERIOR BALLISTICS
satisfy the equation of condition
a (i + X - M) = i
and thus ignoring the slivers.* This method represents quite
satisfactorily the actual circumstances of motion so long as the
grains retain their original form, but not afterward. It assumes
that the slivers are all burned with the fictitious web thickness ;
that is, when, in all our guns, the projectile has performed
approximately half its travel in the bore ; while it is certain that
in most cases with our service powders they are not completely
consumed when the projectile leaves the bore. It is necessary,
therefore, to divide the entire combustion of the grain into two
periods and to deduce formulas that shall represent the law of
burning, as well as the circumstances of motion, for each period.
From equation (22), Chapter III, we have, for m.p. grains
y I ( I
k = ~- = a-r-} I +\~j /i-
co 1 Q ^ i I
which gives the fraction of the charge consumed when any
thickness / of the web has been burned, and this without any
reference to the law of burning. When / = / Q , that is, when
the entire web thickness has been burned, this equation becomes
k' = a (i + X - /*)
in which k' is the fraction of the charge less the slivers. If we
substitute for a, X and n their values for any of our m.p. grains,
we shall find for this critical point,
k'= 0.85 (about),
and therefore the slivers constitute approximately 15 per cent,
of the charge. These slivers burn according to another law.
We may regard them as slender cylinders whose form character-
istics are very approximately
a = 2, X = i, n = c.
* See Journal U. S. Artillery, vol. 24, p. 196, and vol. 26, pp. 141 and 276.
APPLICATIONS 149
We will now deduce formulas for each period of burning.
Designate all symbols referring to the point where the grains
are converted into slivers by an accent, and those relating to
the muzzle including M, M' and N by a subscript m.
Equation (n), Chapter IV, becomes, by suitable reductions,
, v c \/ a w d>
A' = : [8.56006 loj -7j . . . (44)
(I IQ
in which v c is the velocity of combustion under atmospheric
pressure and 1 one-half the web thickness. From (12), Chap-
ter IV, we have
1/1 = KX
which, when the web thickness is burned through, gives for this
critical point,
KX' = i
Therefore from (44)
d*l
i) c v 7 a w <i
which gives X' when v c and 1 are known. Also
d 2 l
z> c = [1.43994] -== . . . (46)
A v dWu
and
/ = [8.56006 - I0 ]^^ (460
While the slivers are burning we have from (49) , Chapter IV,
K
in which X refers to any travel between u' and u m and
K = i - (i - k)t
If we know the values of X and k for any point, we can
150 INTERIOR BALLISTICS
determine N m by the equation
Therefore at the point of breaking up into slivers (48) be-
comes
~K f T^ f
N m = ~r or X' = -- .... (49)
.
The fraction k f which enters into K' can be computed for a
grain of given dimensions by (21), Chapter III; and log K' can
be taken from Table II with k' as the argument. Therefore
when N m is known X f can be found from (49), and then x' ', taken
from Table I with X' as the argument, locates the point where
the grains become slivers by the equation
u' =x' z ..... (50)
In order to determine N m it is necessary to assume a value
for k m , or the fraction of the entire charge burned at the muzzle,
and check this assumed value by the given maxim am (crusher-
gauge) pressure. By (45), Chapter IV, we have,
by means of which V l can be determined from muzzle data.
The constants M, N and N' to be used in the velocity and
pressure formulas from u = o to u' are given by the formulas
M = ^, N = and N' = ^T (S 2 )
A o A o X*
Finally the value of M m for the travel from u' to u m is given
by the formulas, deduced from (3),
M m =^N m VS = 4 N m VS . . . (53)
As an example of this method we will take the mean crusher-
gauge pressure and muzzle velocity of five shots fired March 14,
APPLICATIONS 151
1905, with the 6-inch Brown wire gun, by the Board of Ordnance
at Sandy Hook. The gun had been previously fired twenty-six
times with charges varying in weight from 32 Ibs. to 69 Ibs.,
and at this time was very little eroded. The gun data are as
follows :
V c = 3120 c. i.
d = 6 inches
u m = 252.5 inches (total travel in bore).
The firing charge for these five shots was 70 Ibs. of nitro-
cellulose powder, with 8 ounces of black rifle powder at each
end of the cartridge for a primer. As it is impossible to isolate
the action of each kind of powder, we will consider the charge
in its entirety and take w = 71 Ibs.* The projectiles varied
slightly in weight from 100 Ibs. (about one-quarter of one per
cent.) ; but no material error will result if we make w = 100 Ibs.
The mean muzzle velocity (v m ) was 3330.4 f. s., and the mean
crusher-gauge pressure (p m ) was 42497 Ibs. per in. 2 The
charges were made up of m.p. grains designed for an 8-inch
rifle, and of the following dimensions: R = 0^.256; r =
o".o255; m = i".c29. And, therefore, 1 = 0^.044875; a
0.72667; X = 0.19590; JJL = 0.02378; k f = 0.85174.
The granulation of this powder is 89 grains to the pound.
The volume of a single grain computed by (15), Chapter III, is
0.197144 c. i.; whence by (23'), Chapter III, d = 1.5776. From
these data are found by methods already fully illustrated,
A = 0.6299
log a = 9.97940 10
Iogz = 1.82144 .'. z = 66.289 in.
\ogX om = 0.74029
* Gossot recommends to increase the weight of charge by one-third that
of the igniter. But there is no practical difference in the results by the two
methods.
152 INTERIOR BALLISTICS
We will assume k m = 0.973. Therefore from Table II,
\ogK m = 9.92204 - 10, and log K f = 9.78885 - 10. From (48)
we have
K m E! ...
N = 7JT = Tx r (54)
* -^~om * ^ x o
whence
vf K X om , .
.'. \QgX' = 0.60710
We now find from the preceding formulas,
log VS= 7.44669
log M = 6.70093
log M ' = 4.69894
log N = 8.68493 ~~ I0
logA 7/ = 7.16201 10
Substituting these values of M', N and N' in equation (51),
Chapter IV, gives p m = 42521 Ibs. per in. 2 , differing insensibly
from the mean crusher-gauge pressure. The assumed value
of k m is therefore correct. We now find from (54) and (53),
\ogN m = 8.88072 - 10
logM w = 6.92947
The value of x' taken from Table i, by means of log X' , is
x'= 1.757.
Therefore u' = 1.757 X 66.289 = 116.47 inches.
The two sets of equations for velocity and pressure for a
charge of 70 Ibs., and primer of i pound, are:
From u = o to u' = 116.47 inches:
v*= [6.70093] X t {i + [8.68493 - 10] X - [7.16201 - io]X 2 t
p = [4.69894] X 3 {i + [8.68493 - 10] X*- [7.16201 -- 10] X b
From u' = 116.47 in. to muzzle:
&= [6.92947] X, {i - [8.88072 - 10] X }
p = [4-92748] X, {i - [8.88072 - 10] X 4 }
}
APPLICATIONS
P, 1000.J/
75000
70000-
v,V
60000-
-3000
55000-
-2750
50000-
10000-
30000-
1^000-
SWW
-250
FIG. 4.
154 INTERIOR BALLISTICS
The X functions for the travel u' are
log X' = 0.60710
log X\ = 0.06473
log X' 2 = 9-457 6 3 ~ 10
logJT 3 = 9-7933 2 - 10
log ^'4= 0.76498
logX' 5 = 1-48763
Both formulas for velocity give the same velocity for the
travel u', namely v f = 2614 f. s. The pressure at this point by
the first formula is 38431 Ibs.; and by the second 29324 Ibs.
per in. 2 The discontinuity shown by the two curves P and p
(see Fig. 4), at the travel u 1 ', where the grains break up
into slivers is due to the sudden diminution of the surface of
combustion of the grains at this point, whereby the rate of
evolution of gas and heat suddenly falls and with it also the
pressure. In this particular example the initial burning surface
of each grain is 3.2 in. 2 , and goes on increasing until at the point
of breaking up the vanishing surface is 4.2 in. 2 It then suddenly
falls to about 1.5 in. 2 , which is approximately the surface of the
twelve slivers. Of course there is no such absolutely abrupt
fall in the pressure as is indicated by the two pressure formulas.
Neither can it be supposed that all the grains maintain their
original form until the web thickness is completely burned.
Nevertheless the two pressure formulas give very approximately
the average pressure at or near this point. It might be possible
to connect the two pressure curves by another curve of very
steep descent; but this is hardly necessary.
The characteristics / and v c are / = 1418 and v c = 0.134.
These characteristics, computed with the firing data of an 8-inch
gun, were found on page 105 to be 1397 and 0.136, respectively.
The expression for y (powder burned) is by (45), Chapter IV,
y = [4-40457! -
APPLICATIONS 155
For the travel u', this formula gives y' = 60.472 Ibs. = 71 k'.
At the muzzle, by the above formula, y m = 69.085 Ibs. = 71 k m .
If it should be found in any case that the powder was all
burned in the gun, it would be necessary to compute X' by the
formula
X' =K'X, ...... (56)
In this case we should assume a value for X (or ~x), and
compute the maximum pressure for comparison with the crusher-
gauge pressure, following the same steps as before.
We will consider a few additional problems illustrative of
this method of treating m.p. grains.
Problem i. What must be the dimensions of the grains in
the example just considered in order that the combustion of the
entire charge may be completed at the muzzle? Also what
would be the muzzle velocity and maximum pressure?
In solving this problem we must first consider the second
period of combustion, namely, that of the slivers. It has
already been shown that for a charge of 71 Ibs., log VJ =
7.44669. We also have in this case, since k m = i,
2X
X being the muzzle value of X ; and by (53),
M m = 4 N m VS
We thus find for the second period of combustion,
v*= [7.00743] X, { i - [8.95868 - 10] X ) \ A
P = [5.00544] X, (l - [8.95868 - 10] X,} J
The muzzle velocity by the above equation is 3376 f. s., an
increase of 46 f. s., due to the combustion of the entire charge
in the gun. The muzzle pressure is 11433 ^s. per in. 2
In order to deduce equations for velocity and pressure for
the first period of combustion, it will be necessary to determine
156 INTERIOR BALLISTICS
the value of k' from which to compute X' and 2 1 . Suppose
we adopt grains for which R/r n and m/l = 30.
By the method given in Chapter III, we find for grains having
100 48 4 81
these ratios, a = -r- , X = , M = and k = . For
285 ' 199 199 95
this value of k' we find from Table II, log K' = 9.78966; and
since K m = i, we have from (55),
X' = K f X om
which gives
\ogX' = 0.52995.
By interpolation from Table i, we find x r = 1.15217, and
then log X\ = 9.88302 10, log X' 3 = 9.83966 io> log X\ =
0.67980 and \ogX' 6 = 1.32095. Next by equations (52), and
equation (61), Chapter IV, we deduce the following equations for
velocity and pressure, which apply from u = otow' = 1.1522 X
66.289 = 76-38 inches:
v*= [6.76075] ^{1 + 18.85245-10] * - [7.24333 -lo]* 3 .} -
p = [4-75876] X 3 { i + [8.85245-10] X,- [7.24333-10] X. }
Both sets of equations, A and B, give the same value to z>',
namely, 2319 f. s., while the pressures at u' by the two equations
are, respectively, 51723 and 39560 Ibs. per in. 2 , a drop of more
than 12000 Ibs. when the grains break up into slivers. The
maximum pressure (taking x = 0.8) is 52428 Ibs. per in. 2
The dimensions of the grains have yet to be determined.
We have found for this powder v c = 0.134 in. per sec. Sub-
stituting this and the value of log X' , given above, in (46') gives,
1 = 0.038 in.
and then
r l /2 = 0.019 m -
R = ii r = 0.209 in.
= 1-1 in.
APPLICATIONS 157
A grain of these dimensions fulfils all the conditions of the
problem. These calculations show in a striking manner the
great effect which minute variations (scarcely measurable) in
the dimensions of m.p. grains have upon the maximum pressure,
increasing it in this case by 10,000 Ibs. per in. 2
The cause of this great increase in the maximum pressure
is that the initial surface of combustion of the charge of the
smaller grains is about 15 per cent, greater than that of the
original grains, as is easily shown by equation (26'), Chapter
III.
Problem 2. What must be the dimensions of the grains of
a charge of 71 Ibs., in order that the burning of the web may
be .completed at the muzzle? Also determine the circumstances
of motion.
To solve this problem we obviously have u' = u m ; and
therefore x = x m = 3.8091. As all the X functions relate to
the muzzle only, we may drop the accents. We have from
Table i, log X = 0.74029, log X l = 0.35048, logX 2 = 9.61018,
logXz = 9.653 1 1, log X 4 = 0.91582, and log X b = 1.78077. Sub-
stituting the value of log X in (46), and making use of the
known value of v c , we find that for the new grains,
l o = ". 06098
Therefore, as in Problem i, r = = 0^.03049
R = 5-5*0= o"-33539
m = 3 lo^ i"-8294
k' = 0.85263 = k m
Since the limiting velocity Vi is independent of the dimen-
sions of the grains, we have as before, log V? = 7.44669; and
this, with the known values of a, X and /*, substituted in equations
(52), gives M, N and N'. We thus derive the following equa-
tions for velocity and pressure for a charge of 71 Ibs. of these
particular grains.
158 INTERIOR BALLISTICS
v*= [6.55041] X,(i + [8.64211 -- 10] X - [6.82262 - 10] X 2 }
p = [4.54842] X s { i + [8.64211 - 10] X*- [6.82262 - 10] X 5 }
From these formulas we get the following information:
Muzzle velocity, 3118 f. s.
Maximum pressure, 29897 Ibs. per in. 2
Muzzle pressure, 21014 Ibs. per in. 2
Powder burned in gun, 60.5 Ibs. = 71 k' .
The maximum pressure is quite moderate, owing to the
thickness of web which gives an initial surface of combustion
but 71 per cent, of that of the original grains. The pressure is
well sustained to the muzzle, where it would be considered
excessive for all except wire-wound guns.
If we suppose the length of the grains to be twelve times
the web thickness we should have a = 5, X = ^. ju = r-.
228' 163' 163'
k' = -7, and m = 0.916 in. Then, as before,
log M = 6.56065
logM' = 4.55866
log N = 8.60383 - 10
log N f = 6.90929 10
These constants give
Muzzle velocity, 3124 f. s.
Maximum pressure, 30163 Ibs. per in. 2
Muzzle pressure, 20875 Ibs. per in. 2
Powder burned in gun, 60.72 Ibs. = 71 k f
The initial surface of combustion of the shorter grain is about
2.4 per cent, greater than that of the longer grain, which fact is
shown in the maximum pressures.
Problem 3. Suppose the powder we have been considering
to be moulded into cylinders with an axial perforation.
If the length of the grain is 50 inches (approximately the
length of the cartridge), and the diameter of the axial perforation
APPLICATIONS 159
one-twentieth of an inch, what must be the diameter of the
grain and thickness of web in order that a charge of 71 Ibs. may
all be burned just as the shot leaves the muzzle? Also determine
the equations for velocity and pressure.
We have already found the thickness of web satisfying the
conditions of the problem to be o". 12 196. (See Problem i.)
Therefore, by means of the formulas pertaining to this form of
grain given in Chapter III, we find the diameter of the grains
to be o".294 and
a = 1.0024392
X = 0.0024333
fJL = O
Since log X = 0.74029 and log Fi 2 = 7.44669, we find
v 2 = [6.70746] X l {i - [6.64590 - 10] X }
p = [4.70547] X 3 { i - [6.64590 - 10] X 4 }
which are the equations required. The muzzle velocity and
maximum pressure by these formulas are
v m = 3376 f. s.
p m = 37040 Ibs. per in. 2
This latter, on account of the smallness of N, occurs when
x = 0.64. The muzzle pressure is 22750 Ibs. per in. 2
A comparison of these results with those deduced in Problem
i shows the great superiority of the uniperforated grain over
the multiperf orated grain so far as maximum pressure is con-
cerned. The muzzle velocity is the same in both cases since the
same weight of powder was burned in the gun. But the maxi-
mum pressure given by the m.p. grains is more than 15,000 Ibs.
greater, and the muzzle pressure 11,000 Ibs. less than with the
u.p. grains. For these latter grains the pressure is remarkably
well sustained from start to finish.
The monomial formulas for velocity and pressure for this
example are easily found to be
l6o INTERIOR BALLISTICS
^ = [3-353 20 ] V%
and
p = [4-70441] ^3
The first of these gives the same value for the muzzle velocity
as the complete formula; while the second gives maximum and
muzzle pressures differing about o.i per cent, of their former
values.
During the test- firing of the 6-inch Brown wire- wound gun
at Sandy Hook, shots were fired with charges varying from
32^ Ibs. to 75 Ibs., thus enabling us to determine whether our
formulas have any predictive value. Unfortunately the object
of the firing was simply to test the endurance of the gun and
no special effort was made to give to the results any scientific
value. Many of the recorded velocities and pressures are
inconsistent with each other as when, more than once, an increase
of charge gave a diminished velocity and pressure. Some of
the recorded muzzle velocities are so manifestly wrong that they
cannot be used in getting averages. They suggest that the
chronograph velocities were not always reduced to the muzzle.
We will compute the new values of / due to a change in the
weight of charge by (88), Chapter IV, taking = 71 Ibs., and
Jo = 1418 Ibs. per in. 2 , and for a six-inch gun, n = . We
o
therefore have
/ = [2.53451] of
To determine X' , we have
X'. = [1.43994!-
v c V aw a
which, by substituting the known values of d, 1 , v c and w, reduces to
[1.52243]
A = -=-
v a u
APPLICATIONS
161
N m is given by (54), which easily reduces to (since log K' =
9.78885 - 10)
^V w = [7.96539 - 10] \/ a a, .... (a)
Next we have from equation (58), Chapter IV, substituting
for/ its value given above and for w its value, 100 Ibs.,
F, 2 = [4.97834! tf
and lastly from (53),
Jf - 4 ^Fi ! = [3.54579] } a v ... (6)
The following table computed by these formulas shows the
agreement between the observed and computed velocities for
a range of charges between 75 Ibs. and 33^ Ibs. The differences
in the last column follow no apparent law and are unimportant.
dj
Ibs.
x m
logM m
**"m
Observed
Velocity
Computed
Velocity
o.-c.
75-o
3-9574
6.95290
8.87242-10
3455
3477
22
74-5
3.9383
6.95008
8.87347
3422
3459
-37
73-5
3-9005
6.94436
8-87557
3402
3423
21
72.5
3-8635
6.93849
8.87764
3380
3385
- 5
71.0
3-8091
6.92947
8.88072
3330
3330
69.0
3-7392
6.91693
8.88474
3254
3257
3
68.0
3-7052
6.91047
8.88672
3236
3220
16
59-0
3.4244
6.84548
8.90384
2879
2888
- 9
49.625
3.1742
6.76170
8.92037
2484
2536
-52
33-25
2.8146
6.55588
8.94643
1913
1896
17
The two sets of equations for velocity and pressure for the
charge of 75 Ibs. are:
From u = o to u' = 117.43 inches:
v 2 = [6.72436] X, {i + [8.67663 - 10] X -[ 7 .i45 4 i - 10] X 2 }
p = [4.73M * 8 {i + [8.67663 - 10] X, -[7.14541 - 10] X, }
From u 1 ' = 117.43 in. to muzzle:
ir= [6.95290] X l ]{i - [8.87242 - 10] X }
p = [4.96750] X 3 { i - [8.87242 - 10] X*}
ii
!62 INTERIOR BALLISTICS
By the first equation for pressure we find p m = 46509 Ibs. per
in. 2 And by the second, muzzle pressure = 15375 Ibs. per in. 2
Both expressions for velocity give v f = 2744 f. s.
For a charge of 62 Ibs., the two sets of equations are
From u = o to u' = H4-54 inches:
tf= [6.63999] X t {i + [8.70249 - 10} X - [
p = [4.60288] X 3 {i + [8.70249 - 10] X 4 - [
From u' to muzzle:
v 2 = [6.86853] Xi (i - [8.89828 - 10] X }
p = [4.83142] X s { i - [8.89828 - 10] X 4 }
These formulas give a muzzle velocity of 3,000 f. s., with a
maximum pressure of 34,263 Ibs., and a muzzle pressure of
11,784 Ibs. per in. 2 It would seem as if these last results are
all that could be desired for a 6-inch gun.
APPLICATION TO THE FOURTEEN-!NCH RIFLE
The i4-inch rifle was designed by the Ordnance Department
to give a "muzzle velocity of 2,150 f. s..to a projectile weighing
i, 660 Ibs., with a charge of nitrocellulose powder of about 312
Ibs., and with a maximum pressure not to exceed 38,000 Ibs. per
square inch." The gun has a powder-chamber capacity of
13,526 cubic inches and a travel of projectile in the bore of
413.85 inches. The type gun has been fired to date 55 times
with charges varying from 102^ to 328 Ibs., producing muzzle
velocities ranging from 901 to 2,252 f. s., and crusher-gauge
pressures from 4,875 to 46,078 Ibs. per in. 2 , this latter with a
charge of 326 Ibs.
The powder employed was " International Smokeless powder,
lot i, 1906, for 1 2 -inch gun." The grains were cylindrical
multiperf orated (7 perforations), of the following dimensions:
APPLICATIONS 163
Outside diameter, 0.826 in.
Diameter of perforations, 0.0815 in.
Length, 1.883 m -
Thickness of web, 0.145375 in.
These dimensions give:
a = 0.71584
X = 0.20974
/A = 0.02151
k' = 0.85058
log^' = 9.78778-10.
The granulation of the powder is 20.6 grains to the pound,
which by (24'), Chapter III, makes the density (5) 1.4291.
We will base our calculations on round No. 55, fired January
23, 1911, with a charge of 328 Ibs. of nitrocellulose powder plus
an "igniter" of 9 Ibs. of rifle, or saluting, powder. This round
affords the following data:
co = 337 Ibs.
w = 1664 Ibs.
v m = 2252 f.s.
p m = 43640 Ibs. per in. 2
The preliminary calculations give
A = 0.68965
log a = 9.87523-10
Iogz = 1.65768
x m = 9.1025
log X om = 0.87855
log X im = 0.60885
By a few trials it will be found that the observed values of
v m and p m are satisfied when k m = 0.953 and therefore from
Table II, log K m = 9.89388-10. We also find log X' = 0.77245,
log V, 2 = 6.99575, x r = 4-6354 and u' = 210.75 inches.
164 INTERIOR BALLISTICS
The equations of the velocity and pressure curves are found
to be
Fromw = otow' = 210.75 in.:
v 2 = [6.07812] Xi {i + [8.54923-10] Xo ~ [6.78775-10] X 2 }
p = [4-72511] Xs{i + [8.54923-10] X, - [6.78775-10] X b }
From u f = 210.75 in. to muzzle:
v 2 = [6.31211] X l {i - [8.71430-10] X ]
p = [4.959*0] Xs{i - [8.71430-10] X*\
Both of the velocity formulas give v' = 1921 f. s. The first
formula for pressure gives p' = 27457 and the second 19,772
Ibs. per in. 2 The muzzle pressure comes out 9,485 Ibs. per in. 2
This round makes the powder characteristics, by (64) and
(67), Chapter IV,
/ = 1759-7
v c = 0.10214
For computing the velocity and pressure constants when the
charge varies, we will consider v c constant and assume / to vary
directly as the weight of charge. That is, we will compute / by
the formula
Equation (69), Chapter IV, becomes, by substituting the
values of J 2 , 1 and v c ,
[3.58446]
...... (b)
Also (58), Chapter IV, becomes, by employing the expression
for /given above,
\
APPLICATIONS
We then have
ad> \?
- f a& \
< VT^/
GO
and
~X
N' = -~ N* = [9.68929-10]
A""
By (49), we have
K'
Combining this with the expression for N we have
N m = -~ = (0.16507) N . . .
Finally we have from (53)
M m = 4 N m Ff = = (0.23399) M
(g)
(K)
The following table gives the computed muzzle velocities
and maximum pressures for certain charges, computed by these
formulas, together with the observed velocities and crusher-
gauge pressures for comparison:
M
Iha
w
It.-
Observed
Velocity,
Computed
Velocity,
O.-C.
f c
Observed
Pressure,
Computed
Pressure,
O.-C.
f.s.
f.s.
Ibs. per in.' J
Ibs. per in. 2
337
1664
2252
2252
43640
43628
12
335
1660
2238
2240
2
42811
42944
-133
334
1660
2232
2232
42877
42637
240
284
1662 y z
1857
1871
-14
25530
29142
3612
263
1660
1738
1724
14
21190
24431
-3241
239
1660
1567
1556
II
16795
19704
-2909
The greatest difference between the observed and computed
muzzle velocities is considerably less than one per cent, and
1 66 INTERIOR BALLISTICS
may be disregarded. The same is true of the differences of the
observed and computed maximum pressures of the first three
charges. Then, as the charges are greatly reduced, these differ-
ences are largely increased. This may be accounted for if the
same kind of copper cylinders were employed for all the charges.
For a charge of 314 Ibs. of service powder and an igniter of
9 Ibs. of rifle powder, making o> = 323 Ibs., and density of loading
0.66 1, the equations for velocity and pressure are as follows:
From u = o to u' = 208.75 inches.
ir = [6.05003] Xi!i + [8.55696-10] x -[6.80321 -- 10] x 2 .}
p = [4-67944] X 3 [i + [8.55696-10] X< - [6.80321 -- 10] X,}
From u' = 208.75 inches to muzzle.
v = [6.28402] X, {i - [8.72203-10] X }
p = [4.91343] ^3 {i ~ [8.72203-10] X 4 }
These formulas give
/ - 1686.6
Muzzle velocity = 2152 f. s.
Maximum pressure = 39351 Ibs. per in. 2
This muzzle velocity is that for which the gun was designed,
but the maximum pressure is about 3^ per cent, greater. The
muzzle pressure comes out 8714 Ibs. per in. 2
Example. Suppose the volume of the powder chamber to
be increased (as is proposed by the Ordnance Department) to
15,000 cubic inches, by lengthening the chamber 6.65 inches,
thereby reducing the travel of the projectile to 407.2 inches.
If the density of loading remain 0.66 1, what would be the
charge, the muzzle velocity, and maximum pressure?
Answers :
= 349-2 + 9 = 358.2 Ibs.
p m = 41683 Ibs. per in. 2
M . V. = 2233.5 f - s.
APPLICATIONS 167
With a charge of 337 Ibs. of service powder and an igniter
of 9 Ibs. of black powder, we should get, with the lengthened
chamber, a muzzle velocity of 2150 f. s., with a maximum
pressure of about 38,400 Ibs. per in. 2 These results are prac-
tically those sought for in designing the present 1 4-inch gun.
Example 2. Suppose, instead of enlarging the powder
chamber of the 1 4-inch gun, we lengthen the grains of powder,
and employ the ratios R/r = n and m/l = 200. These ratios
give, as is shown in Chapter III,
1210
a = - - = 0.64158
1900
= . 3l829
1219
fJL =
= 0.00328
1219
. . 84368
1900
logtf' = 9.78150 - 10. (By Table II.)
Employing these grains, what muzzle velocity and maximum
pressure may be expected with a charge of 314 Ibs. of service
powder and an igniter of 9 Ibs. of black powder, in the gun as
it is now, where V c = 13526 c. i., and u m = 413.85 in. ?
The preliminary calculations give:
A = 0.661
log a = 9.91017
Iogz = 1.67419
x m = 8.7630
log X om = 0.87273
log X lm = 0.59873
By equations (a) to (ti), inclusive, we find, the web thickness
remaining as before,
1 68 INTERIOR BALLISTICS
/ = 1686.6
log Fi 2 = 6.95993
logX f = 0.76472 .'. x' = 4-4201 and u' = 208.75 in -
The equations for velocity and pressure are
From u = o to u' = 208.75 m - :
v 2 = 6.00247 X, { i + [8.73812-10] X - [5.98664 - 10] X 2
p = 4.63188 X 3 { i + [8.73812-10] X, - [5.98664 - 10] X 6
From u' = 208.75 in. to muzzle:
v 2 = [6.27775] X l {i - [8.71576 - 10] X }
p = [4.90716] *i'fi - [8.71576 ~ 10] X,}
From these equations we find,
Maximum pressure = 37851 Ibs. per in. 2
Muzzle velocity = 2146 f. s.
Muzzle pressure = 8789 Ibs. per in. 2
v' = 1820.3 f. s.
. , 26299
The dimensions of these grains are found from the ratios
given above, and are as follows:
Diameter of perforations = 1 = 0.0727 in.
Diameter of grain = 1 1 1 = 0.8 in.
Length of grain = 200 1 = 14.54 in.
The following table gives the pressures (p') at different points
of the bore for a charge of 314 Ibs. of service powder plus an
igniter of 9 Ibs., making a> = 123 Ibs., and also the pressures
(p") of the same charge made up of the grains whose dimen-
sions are given above.
It will be seen from this table, and the previous calculations,
that increasing the length of the powder grains relieves the
maximum pressure more than is accomplished by lengthening
the powder chamber, for the same muzzle energy:
APPLICATIONS
169
X
u
Inches.
P'
Ibs. per in. 2
P"
bs. per in. 2
P'-P"
Remarks.
0. I
4.72
24325
22388
1937
0.2
9-45
31329
29138
2191
0-3
14.17
35084
32888
2196
0.4
18.89
37234
35I3I
2103
05
23.61
38453
36483
1970
0.6
28.34
39090
37275
1815
0.7
33-o6
39351
37691
1660
0.8
37-78
39351
37851
1500
Maximum pressure.
0.9
42.50
39I8I
37833
1348
I.O .
47-23
38892
37690
1202
i-5
70.84
36650
36065
585
2.0
94 45
34150
34019
131
2-5
118.07
31849
32060
211
3-0
141.68
29820
30293
- 473
3-5
165.29
28049
28726
- 677
4.0
188.91
26500
27341
841
4.4201
208.75
25344
26299
- 955
The web thickness is
5-0
236.13
16308
16295
+ 13
burned at this
point.
6.0
283.36
13566
13585
- 19
7.0
330-59
H45i
11496
45
8.0
377.82
9775
9838
- 63
8-763
4I3-85
8714
8789
- 75
Muzzle.
CHAPTER VI
ON THE RIFLING OF CANNON
Advantages of Rifling. The greater efficiency of oblong
over spherical projectiles is twofold. In the first place they
have greater ballistic efficiency, that is, for the same caliber,
muzzle velocity and range, an oblong projectile has a higher
average velocity during its flight than a spherical projectile.
This gives to the former a flatter trajectory which increases
the probability of hitting the target. Experimental firing has
demonstrated that the mean deviation of the shots from a rifled
gun at medium ranges, when all known and controllable causes
of deviation have been eliminated, is only one-third that from
a smooth bore. This advantage results both from the greater
sectional density of the oblong projectile whereby it is enabled
the better to overcome the resistance of the air, and also because
this resistance is diminished by the more pointed head.
In the second place the penetration of oblong projectiles,
other things being equal, is much greater than can be realized
with spherical shot, while the bursting charge of oblong shells is
as great or even greater than that of spherical shells on account
of their greater length. These are very substantial advantages;
but to secure them it is essential that the oblong projectile should
keep point foremost in its flight, otherwise it would have neither
range, accuracy nor penetration, but would waste its energy
beating the air.
The only way to secure steadiness of flight to an oblong
projectile is to keep its geometrical axis in the tangent to the
trajectory it describes by giving it a high rotary velocity about
this axis. This is accomplished by rifling, as it is called, that
170
ON THE RIFLING OF CANNON iyi
is, by cutting spiral grooves in the surface of the bore into which
a projecting copper band, securely encircling the projectile
near its base, is forced as soon as motion of translation begins,
thus giving to the projectile a rotary motion in addition to its
translation as it moves down the bore. The rifling may be
such that the grooves (or rifles) have a constant pitch, that is,
make a constant angle with the axis of the bore; or, this angle
may increase. In the first case the gun is said to be rifled with
a constant twist, and in the second case with an increasing
twist. In all cases the twist at any point of the bore is measured
by the linear distance the projectile would advance while making
one revolution supposing the twist at that point to remain
constant. This linear distance is always expressed in calibers,
and is therefore independent of the unit of length employed.
The Developed Groove. Uniform Twist. The element of a
groove of uniform twist developed upon a plane is evidently
a right line A C making, with the longitudinal element of the
surface of the bore A B, the constant angle B A C, whose
tangent is B C/A B* Suppose A B to be the longitudinal ele-
ment passing through the beginning of the groove at A, which is
near the base of the projectile when in its firing seat and directly
in front of the rotating band. Make A B = nd, n being the
number of calibers the projectile travels while making one
revolution. Then B C will be equal to the circumference of
the projectile; or, B C = TT d. If we designate the angle of
inclination of the groove, B A C by /?, we shall have
BC ird TT
tan )8 = -7-5 = i = - . . . . (i)
A B nd n
Increasing Twist. With a uniform twist the maximum
pressure produced on the lands (or sides of the grooves) occurs
(as will be shown presently) at the point of maximum pressure
* The simple diagrams required in this Chapter can easily be constructed
by the reader.
172
INTERIOR BALLISTICS
on the base of the projectile, which point, as we know, is near the
beginning of motion. From this point the pressure on the* lands
decreases to the muzzle where it is not generally more than one-
fourth of its maximum value. It is considered by gun-designers
a desideratum to have the pressure on the lands as uniform as
possible, and to this end recourse is had to an increasing twist-
that is, the angle which a groove makes with the axis of the bore,
instead of being constant as with a uniform twist, increases from
the beginning of rifling toward the muzzle. If this variable angle
be represented by 0, we shall still have, as before,
tan = - (i)
in which n is now a variable, decreasing in value as the distance
from the origin of rifling increases. If, as before, we take the
origin of rectangular coordinates at A, the beginning of rifling,
and suppose A B to be a longitudinal element of the bore and
B C the length of arc revolved through by a point on the surface
of the projectile while it travels from A to B, then the developed
groove, A C, is a curve convex toward A B, the axis of abscissas,
for the reason that by definition tan increases from A toward
B. The two forms of increasing twist that have been most
generally adopted are the parabolic and circular.
General Expression for Pressure on the Lands. Before
attempting to decide upon the best system of rifling it will be
necessary to deduce an expression for the pressure upon the
lands. Take a cross-section of the bore and suppose for sim-
plicity that there are only two grooves opposite to each other,
and let the prolongation of the bearing surface pass through the
axis of the bore, as is practically realized in the latest systems.
Let M represent the point of application of the bearing surface
of the upper groove. We will take three coordinate axes:
one axis (x) is coincident with the axis of the bore, while the others
(y and z) are in the plane of the cross-section of the bore and
ON THE RIFLING OF CANNON 173
perpendicular to each other. Let the axis y be directed along
O M, and z in a direction perpendicular to M.
The pressure at M , whatever may be its direction, can be
replaced by the three following mutually perpendicular com-
ponents: The first, perpendicular to the axis of the bore and
consequently acting along the radius through M; the second
lying in the plane tangent to the surface of the bore (normal
to the radius O M) and acting along the normal to the
groove; the third lying in this same plane and tangent to the
groove.
The first of these components will be destroyed by the
similar component of the opposite groove and does not enter
into the equation of motion of the projectile. The second
component, which is the normal pressure against the bearing
surface of the groove, we will designate by R. The third
component, being in the tangent to the groove, represents the
friction on the guiding side of the groove, and may be designated
by f R, in which / is the coefficient of friction. The forces R
and / R give the following components along the axes x and z:
Axis of x. Axis of z.
Force R . . - R sin 7? cos
Force fR . - f R cos -fRsm0
The positive direction of the axis of x is toward the muzzle;
that of z in the direction of the force R, and is the angle which
the groove makes with the axis of x. The full component for
the upper groove is :
On axis of x . . - R (sin + / R cos 0)
On axis of z . R (cos f R sin 0)
For the lower groove the component along the axis of x has
the same value and sign as the upper one; while the component
along the axis of z has the same value but the opposite sign.
Besides these forces the projectile is also subjected to the variable
174 INTERIOR BALLISTICS
pressure of the powder gases on its base acting along the axis of x
in the positive direction, which force call P. If we replace the
grooves by the forces enumerated above, we may consider the
projectile a free body and apply to it Euler's equations. These
equations are six in number; but, as is readily seen, they reduce
to two in the problem under consideration, namely: an equation
of translation along the axis of x, and of rotation about the same
axis, or, what is the same thing, the axis of the projectile. The
first equation is
M-^= P - 2 R (sin 6 + /cos0) ... (2)
and the second
-fsmd) ... (3)
in which r is the radius of the projectile, co its angular velocity
about its axis and k its radius of gyration.
The angular velocity co of a projectile about its geometrical
axis for an increasing twist, continually increases as it moves
along the bore from zero to its muzzle value, which is IT v/n r,
v being the muzzle velocity of translation and r the radius of
the projectile. Its magnitude at any instant is given by the
equation
where <p is the angle turned through from the beginning of
motion expressed in radians. The angular acceleration is
found by differentiating this equation with respect to the time,
which gives
A d u d?(p
Angular acceleration = -j- = -r- ... (4)
If we now take x and y as the rectangular coordinates of the
developed groove with the origin at the beginning of rifling and
the axis of abscissas parallel to the axis of the bore, then x will
ON THE RIFLING OF CANNON
175
represent at any instant the distance travelled by the shot, and
the corresponding value of y will be r<p, where r is the radius of
the projectile. Substituting y for <p in (4), gives
da} _ i d 2 y
~dt~"~rdP ....... (5 '
Substituting this value of d u/d t in (3), it becomes, putting
M fji-j = 2R (cos 6 - /sin d) . . . . (6)
Before these equations can be used for determining 2 R we
must eliminate d t; and this we can do by means of the equation
of the developed groove. Let
y=f(x)
be this equation. Then employing the usual notation we have
j| -/'(*) tan ..... (7)
and
&-*<*
Also, since
dy _ d y dx doc
~ii^Tx'~dl = f^~dt'
we have, by differentiating,
d 2 y d 2 x
Substituting this value of -7 , and also the value of -r^-
from (2) in (6), and solving for 2 R, we have
2 R = ( ,
' i -tan 0{/- M (/+ tan0)} '
In using this equation /" (x) and tan 6 are obtained from the
equation of the developed groove, M and M from the projectile,
INTERIOR BALLISTICS
V and P from the equations for velocity and pressure deduced
in Chapter IV, while / is determined by experiment. The
resulting value of 2 R will be sum of the rotation pressures on
all the lands.
Uniform Twist. For uniform twist f'(x) is zero and
becomes constant and equal to 0. Its value is given by (i).
Making these substitutions the expression for 2 R becomes
for uniform twist
IJL P tan sec , ^
2 R = i -/tan + M tan ft (f -ftan 0)
In the second member of (9), P, the pressure on the base of
the projectile, is the only variable; and therefore 2 R is directly
proportional to this pressure, and is a maximum when P is a
maximum. In equation (8) there are four variables in the
second member, namely, P, v, 6 and/" (x) ; and it is not obvious
on simple inspection where the point of maximum rotation is
located. It will be shown, however, by examples that for an
increasing twist this point is much nearer the muzzle than when
the twist is uniform.
Increasing Twist. Semi-Cubical Parabola. To continue
an increasing twist quite up to the muzzle must conduce to
inaccuracy of flight, and especially so when the projectile has
partially left the bore so that it has lost its centering. To
remedy this the acceleration of rotation near the muzzle is made
either zero or constant (preferably the former), in order to
relieve the rotating band as much as possible from pressure and
to reduce the torsional effect upon the gun, far removed from
its support at the trunnions. In all of our sea-coast guns the
final twist is made constant beginning at about 2^2 calibers from
the muzzle. The developed groove for the increasing twist is
a semi-cubical parabola whose general equation is
y + b = p (x + rf (10)
ON THE RIFLING OF CANNON 177
The axis of x is parallel to the axis of the bore and the origin
is at the beginning of the rifling, just in front of the rotating
band of the projectile when in its firing seat. The coordinates
of the vertex of the parabola are -a and -b, and these with the
parameter p are determined by the particular twist adopted
for the beginning and ending of the increasing twist. Suppose
the rifling to start with a twist of one turn in n v calibers, and
that at 2}/2 calibers from the muzzle where the rifling begins
to be constant it has a twist of one turn in n 2 calibers (HI > n^).
For these two points we have, by (i),
tan 61 = - - and tan 2 =
n, n 2
0! and 6 2 being the inclinations of the grooves with the axis of
x at the points considered. Differentiating (10), we have,
^ = tan0=|/>(* + <*) 2 . . . (n)
At the origin x = o, which gives
3 p \/a TT
At 2^2 calibers from the muzzle where the increasing twist
ends, x u 2 , and we have at this point
2 n%
From these two last equations we find
" . ()
and
1 78 INTERIOR BALLISTICS
Since at the origin x and y are both zero, we find from (10)
and (13),
2-n-a
Thus all the constants in the equation of the developed
groove are given in terms of HI, n 2 and u 2 . Lastly, differentiating
(n) gives
/'(*)= ^L= ..... (15)
4 V x + a
If the vertex of the semi-cubical parabola is at the origin, or
beginning of rifling, a and b are zero, and (10) becomes
y = P x* ....... (16)
In this case the twist is zero at the origin and increases to
one turn in n 2 calibers near the muzzle. The values of tan 0,
p and j" (x) for this particular form of rifling are deduced from
(n), (13), and (15), by making a zero. This form of groove is
that adopted by the navy for all their heavy guns of recent
construction.
Common parabola. The equation of the common parabola is
y + b = p (x + a) 2 ..... (17)
where a and b are the coordinates of the vertex. The
constants are determined as for the semi-cubical parabola, and
are as follows:
2 n
(19)
> Trr
i
7T
t 2n 2 (u 2 +a)
f" (*}=2 P (21)
ON THE RIFLING OF CANNON 179
Relative Width of Grooves and Lands. In our service
siege and sea-coast guns the number, N, of grooves (or lands)
is given by the equation
N = 6 d
in which d is the diameter of the bore in inches, and is a whole
number for each of these guns. If w g is the width of a groove
and Wi the width of a land, we have the relation
inches -
The best authorities lay down the rule that the width of a
groove should be at least double that of a land. In our guns
the lands are made 0.15 in. wide, and the grooves are therefore
0.5236 0.15 = 0.3736 in. wide.
Application to the lo-inch B. L. R. Model 1888. This gun
has 60 grooves which, beginning at 20.1 inches from the bottom
of the bore with a twist of one turn in 50 calibers, increase to
one in 25 at 20 inches from the muzzle, and from thence continue
uniform. We therefore have HI = 50 and n 2 = 25. The bore
is 22.925 ft. long, and therefore u 2 = 19.583 ft. The developed
groove is a semi-cubical parabola whose equation is (10). The
constants are
a = 19^3 = 6 . 52g ft .
6.S287T
b = -- = 0.27344 ft.
.75
P = " =5 = - l6 395-
.528
The equation of the developed groove (changing x to u to
indicate travel of projectile) is therefore,
y + 0.27344 = 0.16395 ( u + 6.528)* - (22)
in which y will be given in feet.
l8o INTERIOR BALLISTICS
From (n), we have
tan = 0.024592 \/ u + 6.528
Making u zero in this last equation gives
i=335'4*"
which is the inclination at which the groove starts. At 20 inches
from the muzzle where the twist becomes uniform (and which is
therefore a point of discontinuity on the developed groove) we
have u-i = 19.583; and at this point
*>=79'45"
This value of is retained to the muzzle.
From (15), we have
/" f x \ = o-oiffg^
Vu +6.528
This function decreases from the origin to the point of
discontinuity. From this point to the muzzle /" (x) is zero.
If we put
K = _ Msec
" i - tan 0{ f - n (f + tan 0) } '
equation (8) becomes
2 R = K [P tan + M ir f" (x) } . . (24)
Captain (now Sir Andrew) Noble, as the result of very careful
experiments made by him with i2-cm. quick-firing guns, found
/ = 0.2, and this value will be adopted in what follows. We
also have for cored shot /* = 0.5, nearly. Substituting these
values of / and /* in (23), it will be found that K increases very
slowly as increases. The values of K for u = oandw = 19.583
are, respectively, 0.5032 and 0.5064. We might therefore take
for K the arithmetical mean of these two values and write (24)
2 R = 0.5048 { P tan + M v~ f" (x) } (25)
ON THE RIFLING OF CANNON l8l
without any material error. This formula may be employed
for all our sea-coast guns.
If the lo-inch gun were rifled with the kind of groove given
by (16), we should find
2 7T
P = - X - -- = O.OI893I
tan = 0.028397 V u
In this form of rifling the initial inclination of the grooves
is zero and increases to 7 9' 45" at 20 inches from the muzzle,
where the twist becomes uniform. Between this point and the
muzzle, j" (x) is zero.
Uniform Twist. If we suppose the lo-inch gun to be rifled
throughout with a uniform twist of one turn in 25 calibers, we
have p = 7 9' 45". Employing the values of ju and / already
given, (9) reduces to
2R = 0.063624 P (26)
Working Expressions. If the equation of the developed
groove is (10), we have
( Mv 2 >
2 R = K tan B -( P + , r L . . (27)
L 2(u + a)}
and
TT / u + a \ \
tan 6 = -
25\w 2 + a /
If (16) is the equation of the developed groove, we have
and
tan 9 = -I-
l82 INTERIOR BALLISTICS
Pressure on the Lands of the lo-Inch B. L. R. The equations
for velocity and pressure for this gun are the following:
ir= [6.20536]*! {i - [8.59381 -- io]X } . . (29)
and
p = [4.720601*8 1 1 ~ [8-593 81 " io]* 4 } . . (30)
The gun and firing data were V c = 7064 c. i., u m = 22.925 ft.,
co = 250 Ibs. of brown cocoa powder, w = 575 Ibs., muzzle
velocity 1975 f. s., maximum pressure on base of projectile,
33300 Ibs. per in. 2 , A = 0.98, and z = 3.461 ft. It will be
convenient to change (30) so that it will give the entire pressure
(P) on the base of the projectile; and to avoid large numbers
we will adopt the ton as the unit of weight. We then have
-
~ 8960
and (30) becomes
P = [3.26544] M 1 - [8-5938i -- 10] X<\. . (30')
Finally, the mass of the projectile expressed in tons is
We have now all the formulas and data necessary for comput-
ing the pressures on the lands of the lo-inch B. L. R., by means
of (26), (27), and (28), for the three principal systems of rifling
adopted in our service. These calculations are given in the
table on page 183.
The last three columns of this table show that the maximum
pressure on the lands is greater for uniform twist than for either
form of increasing twist; but the difference between these max-
ima is not very great. Moreover, the maximum pressure for
uniform twist occurs at the trunnions where its torsional effect
upon the gun so far as deranging the aim is concerned is a
ON THE RIFLING OF CANNON
Pressures on lands required to produce rotation of shot in the
lo-inch B. L. R. for different systems of rifling. Charge 250
Ibs. Projectile 575 Ibs. Muzzle velocity 1975 f. s. Maximum
pressure on base of projectile 33300 Ibs. per square inch.
Travel
of
Velocity
of
Pressure
on Base of
PRESSL
RE ON LANDS.
TONS
X
Shot,
feet
Shot,
f. s.
Shot,
tons
Uniform
Twist
Increasing
Twist
Eq. (28)
Increasing
Twist,
Eq. (27)
0.0
0.
0.0
0.0
0.0
0.0
0.0
.1
0.3461
227.7
841.1
53-5
I2.O
27.0
.2
0.6922
366.7
1036.4
65.9
21.5
36.9
o-3
1.0382
478.1
1122.4
71.4
29.0
42.3
4
1.3843
572.4
1158.3
73-7
35-3
46.1
5
1.7304
654.7
1167.4
74-3
40.9
48.9
0.6
2.0765
727.8
1161.1
73-9
44.8
5LI
.7
2.4226
793-6
1145.6
72.9
48.5
52.9
.8
2.7686
853-4
1124.7
71.6
5L6
54-3
0.9
3-II47
908.2
1100.7
70.0
54-3
55-5
I.O
3.4608
958.7
1075.0
68.4
56.7
56.5
i.i
3.8069
1005.6
1048.5
66.7
58.7
57-3
.2
4.I530
1049.3
1021.9
65-0
60.5
58.1
3
44991
1090.2
995-5
63-3
62.1
58.7
4
4.8452
1128.6
969.7
61.7
63-5
59-2
5
5.I9I2
1164.7
944-5
60. i
64.7
59-7
.6
5-5372
1198.9
920.1
58.5
65.8
60. i
7
5.8833
1231.4
896.4
57-0
66.7
60.5
.8
6.2294
1262.1
873.6
55-6
67-5
60.8
9
6-5755
1291.4
851.7
54-2
68.2
61.1
2.0
6.9216
I3I94
830.5
52-8
69.0
61.3
3-0
10.3824
15434
6594
42.0
72.7
62.4
4.0
13.8432
1702.8
541-4
34-4
73.5
62.3
5-o
17.3040
1824.9
456.4
29.0
73-2
61.6
5.6586
19.5833
1891.6
412.4
26.2
72.6
61.0
6.0000
20.7648
1922.8
392.4
25.0
25.0
25.0
6.6242
22.9250
1975.0
359-9
22.9
22.9
22.9
1 84 INTERIOR BALLISTICS
minimum; while the position of the maximum pressure upon the
lands for either form of increasing twist is well down the chase.
It is difficult to see any superiority of an increasing twist over a
uniform twist, especially in view of the fact demonstrated by
Captain Noble's experiments, that the energy expended in giving
rotation to the projectile with rifling having an increasing twist
is nearly twice as great as with a uniform twist.
Application to the 1 4-inch Rifle. This gun has 126 grooves
and the same number of lands, in this respect differing from the
rule followed with the other seacoast guns. The values of n\ 9
HZ, 61 and 2 are the same as those found for the lo-inch rifle.
The rifling begins 7.05 inches from the base of the projectile
when in its firing seat and becomes uniform 22.8 inches from
the muzzle. Therefore
u 2 = 4i3- 8 5 ~ (7-5 + 22 -8) = 384 inches.
From (12), (13), and (14), we now find
a = 128
p = 0.0037024
b = 5.36165
Therefore the equation of the developed groove is
y + 5-36165 = 0.0037024 (u + 128)'
From (n) and (15), we have, finally,
tan = 0.0055536^ u + 128
VU + I2S
M = - = c
2240
and
in which P is the entire pressure on the base of the projectile in
ON THE RIFLING OF CANNON
tons while p is the pressure in pounds per square inch given by
the equation on page 166, for a charge of 314 Ibs.
Substituting these expressions for tan 0,/" (x), and M in (25)
and reducing, we have the working expression
I
128
2R = [7.44769-10] V I* -f 128 jP|4- [8.06145-10]
in which 2 R is the normal pressure on all the lands in tons and
u the travel of the projectile in inches. To determine the
normal pressure on each land 2 R must be divided by 126.
For a uniform twist 2 R is given by (26).
In the following table v and p were computed by the formulas
on page 166 for a charge of 314 Ibs., and P and 2 R by the
formulas given above:
M
inches.
V
L s.
p
tons.
2 R
Increasing T
2 R
Uniform T
O.I
4.72
198.7
1671.7
54-i
106.4
0.2
0-3
9-45
14.17
325.2
429.1
2153-0
2411 . I
71.1
81.1
137.0
153-4
0.4
0-5
0.6
18.89
23-61
28.34
518-9
598.5
670.3
2558.8
2642 . 6
2686.3
87-7
92.2
95-3
162.8
168.2
I7I.3
0.7
0.8
0.9
33-06
37-78
42.50
735-8
796.1
852.1
2704.3
2704.3
2692 . 6
97.6
99-2
100.4
I72.I
I72.I
I7L3
I.O
2.O
47-23
70.84
94-45
904-3
1123-3
1295.1
2672 . 8
2518.7
2346.9
IOI.2
102.5
101.8
I70.I
160.3
149 < 3
2-5
3-0
3-5
118.07
141.68
165.29
1436.7
1557-5
1662.9
2188.7
2049.3
1927.6
100.5
99.1
97-8
139-3
130-4
122.7
4.0
5-0
6.0
188.91
236.13
283.36
1756.5
1891.0
1981.2
1821.1
1120.7
932-3
96.5
66.0
59.3
II5-9
7i-3
59.3
7.0
8.0
8.763
330-59
377-82
413-85
Muzzle
2053-7
2H3-3
2152.0
787.0
671.8
598.8
53-6
48.8
50.1
42.7
1 86 INTERIOR BALLISTICS
Influence of the Rifling for a Uniform Twist. For a uniform
twist we have
where n is constant, and r is the radius of the projectile. Differ-
entiating with respect to / we have
do) TT dv TT d z jx
~dt " "nr dt~ nr dt z
Substituting this value of d u/d t in (3), it becomes
*-^*J = 2 IZ(cos/J-/sm0) . . . (31)
Eliminating 2 R between equations (2) and (31), gives
P M ,.
~-~dP\ M n i-/tan/3j'
If the bore were smooth the equation of translation of the
projectile would be
from which it appears that the effect of the grooves upon the
motion of the projectile for a constant twist is equivalent to
increasing the mass of the projectile by the quantity
TT_MM / + tan |8
n i tan
By making / = 0.2, n = 25, ^ = 0.5 and /3 = 7 9' 45", the
value of this supplemental term is found to be 0.021 M. That
is, the retarding effect of a constant twist of one turn in 25
calibers is equivalent to increasing the weight of the projectile
2 per cent.
* Artillery Circular, N, p. 201.
TABLES
PAGE
I. X Functions l8 9
II. K = 1- (l-fc)i 2I1
III. Work of Fired Gunpowder 2I2
TABLES
189
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INTERIOR BALLISTICS
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TABLES
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INTERIOR BALLISTICS
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TABLES
2OI
w
CQ
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ON cor-- cooo
hH HH O O ON
COOO CO ONTt-
ONOO OO t->- t^
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INTERIOR BALLISTICS
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TABLES
20 3
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204
INTERIOR BALLISTICS
w
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TABLES
205
n d ON t>- lO
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C4 W M CN M
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206
INTERIOR BALLISTICS
Cj
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w
PQ
W
w
TABLES
207
I
a
w
5
W
Q
CO COM
COHH N
hH hH
hH O '
O O ON
ONO
oo r^* r^*
*
.000
00 ON C
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t>.oo ON
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M M M
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CO CO CO
CO CO CO
cocOfO
***
208
INTERIOR BALLISTICS
q
t^vovo
10 10 Tj-
iO tf CO
rt- co CO
f^ co CN
04 N K.
CM HH NH
$
ON HH N
cOOO CO
ON ON ON
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131
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PL,
CXI
TABLES
20Q
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odd
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odd
q
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2IO
INTERIOR BALLISTICS
w
PQ
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w
PL,
PL,
q
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Ti-Ti-Tj-
rj-rj-rt-rf
*
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rt-n o ON
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odd
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Q
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Q
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00 00 00 00
X
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i^oo oo
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CO
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odd
-TJ- r^oo ON
Tt- 10 too
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C< CN M IN
TABLES
TABLE II
211
k
logK
D
k
logK
D
k
logK
D
0.60
9.56531
930
0.908
9.84304
2O6
0.979
9.93201
177
0.61
9.57461
922
0.910
9.84510
207
0.980
9.93378
181
0.62
9-58383
915
O.9I2
9.84717
209
0.981
9-93559
185
0.63
9.59298
9 08
0.914
9.84926
2IO
0.982
9.93744
189
0.64
9.60206
902
O.9I6
9.85136
212
0.983
9-93933
194
0.65
9.61108
896
0.918
9.85348
213
0.984
9.94127
199
0.66
9.62004
890
0.920
9-8556I
215
0.985
9-94326
205
0.67
9.62894
886
0.922
9.85776
217
0.986
9-94531
211
0.68
9.63780
882
0.924
9-85993
218
0.987
9.94742
219
0.69
9.64662
879
0.926
9.86211
221
0.988
9.94961
227
0.70
9-6554I
875
0.928
9-86432
222
0.989
9.95188
236
0.71
9.66416
872
0.930
9.86654
22 4
0.990
9-95424
122
0.72
9.67288
871
0.932
9.86878
226
0.9905
9^5546
125
0-73
9-68159
869
0-934
9.87104
229
0.9910
9.95671
128
0.74
9.69028
869
0.936
9.87333
231
0.9915
9-95799
132
0-75
9.69897
869
0.938
9.87564
234
0.9920
9-95931
135
0.76
9.70766
869
0.940
9.87798
236 0.9925
9.96066
139
0.77
971635
871
0.942
9.88034
239 0.9930
9-96205
144
0.78
9.72506
873
0.944 9-88273
242 | 0.9935
9.96349
150
0.79
9-73379
877
0.946 9-885I5
245
0.9940
9.96499
154
0.80
9.74256
880
0.948
9.88760
248
0-9945
9.96653
162
0.81
9-75I36
886
0.950
9.89008
252
0.9950
9-96815
I6 9
0.82
9.76022
893
0.952
9.89260
256
0-9955
9.96984
I 7 8
0.83
9.76915
900
0-954
9.89516
260
0.9960
9.97162
I8 9
0.84
977815
454
0.956
9.89776
264
0.9965
9-97351
203
0.845
9.78269
456
0.958
9.90040
269
0.9970
9-97554
218
0.850
9.78725
459
0.960
9.90309
274
3.9975
9.97772
241
0-855
9.79184
462
0.962
9.90583
28O
0.9980
9.98013
272
0.860
9.79646
465
0.964
9.90863
285
0.9985
9.98285
319
0.865
9.80111
469
0.966
9.91148
292
0.9990
9.98604
414
0.870
9.80580
473
0.968
9.91440
300
0.9995
9.99018
982
0.875
9.81053
478
0.970
9.91740
152
I.OOOO
o.ooooo
. . .
0.880
9.81530 482
0.971
9.91892
155
0.885
9.82012
488
0.972
9.92047
157
0.890
9.82500
492
0-973
9.92204
1 60
0.895
9.82992
500
0.974
9.92364
161
0.900
9.83492
2OI
0-975
9-92525
165
0.902
9-83693
2O2
0.976
9.92690
167
0.904
9.83895
204
0.977
9.92857
170
0.906
9.84099
205
0.978
9.93027
174
14
212
INTERIOR BALLISTICS
TABLE III. Giving the total work that dry gunpowder of
the W. A. standard is capable of performing in the bore of a
gun, in foot-tons per Ib. of powder burned. 1
Number of
volumes of
expansion.
Corresponding
density of
products of
combustion.
1*
Ms
!<2J
H S.g
Difference.
Number of
volumes of
expansion.
Corresponding
density of
products of
combustion.
Total work
per Ib. burned
in foot-tons.
Difference.
.OO
I .OOO
.56
.641
T.A !?OO
8lQ
.01
.990
.980
. 9 80
58
633
OT* O
35-30^
ul y
.801
.02
.980
1.936
-956
.60
.625
36.086
785
03
.971
2.870
-934
.62
.617
36.855
769
.04
.962
3.782
.912
.64
.610
37.608
753
05
952
4-674
.892
.66
.602
38.346
-738
.06
943
5-547
873
.68
595
39.069
723
.07
935
6-399
.852
.70
.588
39-778
.709
.08
.926
7-234
-835
.72
581
40.474
.696
.09
.917
8.051
.817
74
575
41.1.56
.682
.IO
909
8.852
.810
76
568
41.827
.67'!
.11
.901
9-637
-785
78
.562
42.486
659
.12
893
10.406
.769
.80
555
43-133
-647
13
.885
i i . 160
754
.82
549
43-769
.636
.14
877
11.899
739
.84
543
44-394
-625
15
.870
12.625
.726
.86
537
45.009
.615
.16
.862
13-338
713
.88
532
45-6I4
-605
17
855
14.038
.700
.90
526
46 . 209
595
.18
.847
I4-725
.687
.92
521
46.795
-586
19
.840
15.400
-675
94
515
47-372
577
.20
833
16.063
.663
.96
.510
47.940
.568
.21
.826
16.716
-653
.98
505
48.499
559
.22
.820
17-359
643
2.00
.500
49.050
55 1
23
.813
17.992
-633
2.05
.488
50-383
1 -333
.24
.806
18.614
.622
2.10
.476
5I-673
.290
s
.800
794
19.226
19.828
.612
.602
2.15
2.20
465
454
52-922
54-I32
.249
.,210
2 l
.787
20.420
-592
2.25
444
55-304
.172
.28
.781
21.001
-581
2.30
435
56.439
135
29
775
21.572
571
2-35
425
57-539
. 100
30
.769
22.133
-56i
2.40
4 T 7
58.605
.066
32
758
23-246
1.113
2-45
. .408
59-639
034
34
.746
24-324
i .078
2.50
.400
60 . 642
i .003
36
735
25-37I
1.047
2-55
392
61.616
974
38
725
26.389
1.018
2.60
384
62 . 563
947
.40
42
7H
.704
27-380
28.348
.991
.968
2.65
2.70
377
370
63-486
64-385
923
899
44
.694
29.291
943
2-75
363
65 . 262
.877
.46
.685
30.211
.920
2.80
357
66.119
-857
.48
50
.676
.667
31-109
31 986
.898
.877
2.85
2.90
351
345
66.955
67.771
.836
.816
52
54
.658
.649
32.843
33-681
857
-838
2-95
3-oo
339
333
68.568
69-347
797
779
1 From Noble and Abel's " Researches on Fired Gunpowder
TABLES
2I 3
Number of
volumes of
expansion.
Corresponding
density of
products of
combustion.
Total work
per Ib. burned
in foot-tons.
Difference.
Number of
volumes of
expansion.
Corresponding
density of
products of
combustion.
"% C
iis
321
3-
Difference.
3-05
.328
70.109
.762
7.10
.141
105.125
539
3.10
.322
70.854
745
7.20
-139
105-655
530
3-15
317
7L584
731
7-30
137
106.176
521
3.20
.312
72.301
.716
7.40
135
106.688
512
3-25
.308
73.002
.701
7-50
133
107. 192
504
3-30
.303
73.690
.688
7.60
131
107.688
.496
3-35
.298
74-365
675
7.70
.130
108.177
.489
3-40
.294
75.027
.662
7-80
.128
108.659
.482
3-45
.290
75.677
650
7.90
.126
109.133
474
3-50
.286
76.315
.638
8.00
125
.109.600
467
3-55
.282
76.940
.625 !
8.10
.123
110.060
.460
3.60
.278
77-553
.613
8.20
. 122
110.514
454
3-65
.274
78.156
.603
8.30
. 1 2O
110.962
.448
3-70
.270
78.749
593
8.40
.119
III. 404
.442
3-75
.266
79-332
583
8.50
.117
III. 840
436
3.80
.263
79-905
573
8.60
.116
112.270
430
3-85
.260
80 . 469
564
8.70
US
112.695
425
3-90
.256
81.024
555
8.80
.114
113.114
.419
3-95
-253
81.570
546
8.90
. 112
113.528
.414
4.00
.250
82.107
537
9.00
. Ill
113-937
.409
4.10
.244
83.157
1.050
9.10
. no
114 341
.404
4.20
.238
84.176
i .019
9.20
.109
IH 739
398
4-30
.232
85.166
990
9-30
.108
II5-I33
394
4.40
.227
86.128
.962
9.40
. I0o
115-521
.388
4-50
.222
87.064
.936
9-50
.105
115-905
384
4.60
.217
87.975
.911
9.60
.104
i 16.284
379
4.70
.213
88.861
.886 !
9.70
.103
116.659
375
4.80
.208
89.724
863
9.80
. 102
117.029
370
4.90
.204
90-565
.841
9.90
. 101
H7-395
366
5.00
.200
91-385
.820
10
. 100
H7-757
362
5.10
.196
92.186
.801
II
.091
121.165
3.408
5.20
.192
92.968
.782
12
.083
124.239
3-074
5-30
.188
93-732
-764
13
.077
127.036
2.797
5-40
.185
94-479
-747
H
.071
129.602
2.566
5-50
.182
95-2io
731
15
.066
131.970
2.368
5.60
.178
95-925
-715
16
.062
134-168
2.198
5-70
175
96.625
.700
17
059
136.218
2.050
5.80
.172
97.310
-685
i l8
.055
138.138
.920
5-90
.169
97.981
.671
! 19
.052
139-944
.806
6.00
.165
98-638
-657
20
.050
141.647
703
6.10
.154
99.282
.644
21
.047
143-258
.611
6.20
.161
99.9I5
633
22
045
144.788
530
6.30
159
100.536
.621
23
043
146.242
454
6.p
.I 5 5
! 101.145
.609
24
.042
147.629
-387
6.50
-154
101.744
599
25
.040
148.953
! 1-324
6.60
151
102.333
589
30
033
154.800
1 5-847
6.70
.149
102.912
579
35
.028
159.667
4.867
6.80
.147
103.480
.=68
40
.025
163.828
4. 161
6.90
145
104.038
558
45
.022
167.456
3.628
7.00
143
104.586
548
50
.020
170.671
3-215
WORKS CONSULTED
HUTTON: "Mathematical Tracts," London, 1812.
RUMFORD: "Experiments to Determine the Force of Fired Gunpowder,"
London, 1797.
RODMAN: "Experiments on Metal and Cannon and Qualities of Cannon
Powder," Boston, 1861.
NOBLE AND ABEL: "Researches on Explosives," London, 1874, 1879.
NOBLE: "On the Energy Absorbed by Friction in the Bores of Rifled Guns."
Reprinted as "Ordnance Construction Note," No. 60. "On Methods
that have been Adopted for Measuring Pressures in the Bores of Guns,"
London, 1894. "Researches on Explosives." Preliminary Note,
London, 1894.
OFFICIAL: "English Text-book of Gunnery." Editions of 1897 and 1902.
SARRAU: "Recherches sur les effets de la poudre dans les Armes," and "For-
mules pratiques des vitesses et des pressions dans les Armes." A
translation of these memoirs into English by Lieutenants Meigs and
Ingersoll is given in Vol. X of the Proceedings U. S. Naval Institute.
"Recherches theoriques sur le chargement des bouches a feu." Transla-
tion by Lieutenant Howard, O. D., in "Ordnance Construction Note,"
No. 42. "War Powders and Interior Ballistics." A translation by
Lieutenant Charles B. Wheeler, O. D., as "Notes on the Construction
of Ordnance," No. 67. Washington, 1895.
Gossox AND LIOUVILLE: "The Ballistic Effects of Smokeless Powders in
Guns." Translated by Major Charles B. Wheeler, O. D. "Notes on
the Construction of Ordnance," No. 88. Washington, 1906.
DUNN: "Interior Ballistics." Part I. "Notes on the Construction of
Ordnance," No. 89. Washington, 1906.
SOUICH: "Poudres de Guerre. Balistique Interieur," Paris, 1882.
BAILLS: "Traite de Balistique Rationnelle," Paris, 1883.
LONGRIDGE: "Internal Ballistics," London, 1889.
MEIGS AND INGERSOLL: "Interior Ballistics," Annapolis, 1887.
PASHKIEVITSCH: "Interior Ballistics." Translated from the Russian by
Captain Tasker H. Bliss, A. D. C. Washington, 1892.
BERGMAN: "Larobok i Artilleriteknik." Del I. Krutlara. Stockholm,
1908. A part of this work was translated for the author by Colonel
Lundeen, Coast Artillery Corps, U. S. Army.
GLENNON: "Velocities and Pressures in Guns," Annapolis, 1889.
CROZIER: "On the Rifling of Guns." Ordnance Construction. Note No. 49.
A. W.: "Des Armes de guerre Modernes et de leurs Munitions." Revue
Militaire Beige, Vol. II, 1888.
McCuLLOCH: "Mechanical Theory of Heat," New York, 1876.
PEABODY: "Thermodynamics of the Steam Engine," New York, 1889.
RONTGEN: "The Principles of Thermodynamics." Translated from the
German by Professor A. Jay Du Boise. New York, 1889.
LISSAK: "Ordnance and Gunnery," New York, 1907.
Encyclopaedia Britannica, eleventh edition, 1911.
215
INDEX
ABSOLUTE temperature, definition of, 17; of fired gunpowder, 40, 47.
Adiabatic expansion, definition of, 26.
Air space, initial, definition of, 76; expressions for reduced length of, 76, 77,
92, 94-
Angular acceleration, 174.
Applications of velocity and pressure formulas: To magazine rifle, 130 to 134;
to Hotchkiss 57 mm. rapid-firing gun, 125 to 130; to 6-inch English gun,
115 to 125 and 140 to 147; to 6-inch Brown wire gun, 150 to 162; to 8-inch
rifle, 102 to no; to lo-inch rifle, 179; to i^-inch rifle, 162 to 169 and 184;
to hypothetical 7 -inch gun, in, 136.
Artillery circulars M and N, references to, 88, 135, 186.
Atmospheric pressure, value of, 21.
Axite, form-characteristics of, 61.
BALLISTIC pendulum, 2, 3.
Ballistite, 61, 140.
Binomial formulas for velocity and pressure, 112.
Bliss, Captain Tasker H., 53.
Board of Ordnance, reference to, 151.
Boyle, Robert, 15.
B N powders, form-characteristics of, 61 ; computation of by velocity for-
mula, 129.
CAVALLI, reference to, 8.
Centervall, law of combustion, 79.
Chamber, reduced length of, 31 ; alignment of grains in, 71; effect cf varying
volume of, in, 112, 166.
Characteristic equation of gaseous state, 17.
Characteristics of a powder, 94.
Charge of powder, behavior of when ignited in a gun, 12, 13; in a close vessel,
12, 35; initial surface of, 73, 77.
217
21 8 INDEX
Chase, excessive pressure in, 55, 158.
Chevreul, reference to, 8.
Chronograph, Noble's, 116; Boulenge'-Breger, 126.
Coefficient of expansion of a perfect gas, 16.
Combustion of a grain of powder, 11; under constant pressure, 55, 79; under
variable pressure, 79, 80.
Composition: of gunpowder, i; of cordite, n, 117, 124; magazine rifle powder,
131; ballistite, 140.
Constants, physical, adopted, 92, 94.
Cordite, composition of, n, 117, 124; form-characteristics of, 63.
Cube, form-characteristics of, 61.
Cylindrical grains: solid, 63; with axial perforation, 65; with seven perfora-
tions (m.p. grains), 66 to 72.
D'ARCY'S method of experimenting, 4.
Density: of powder, 1 1 ; of a gas, 21 ; of loading, 37, 75, 77.
Dulong and Petit, law of, 21, 23.
ELSWICK works, mention of, 115.
Encyclopaedia Britannica, eleventh edition, reference to, 115.
Energies neglected in deducing equation for velocity, 121.
Energy of translation of projectile, 32, 51, 52, 53, 80, 144.
English Text-Book of Gunnery, reference to, 115.
Euler, mention of, 88; equations of, 174.
Examples: of expansion of gases, 28; of the formulas of Chapter III, 77;
relating to 8-inch rifle, 109; to 6-inch gun, 122, 124, 144, 155; to 14-inch
rifle, 1 66.
Expansion, work of: isothermal, 25; adiabatic, 26; in the bore of a gun, 30, 47.
FACTOR of effect, 49, 52, 54.
Force of the powder, 33, 36.
Formulas: Characteristic equation of gaseous state, 17. For specific heat under
constant volume, 22. For work: of an isothermal expansion, 25; adia-
batic expansion, 26, 27, 32; of gases of fired gunpowder, 49. For tempera-
ture: of an adiabatic expansion of a perfect gas, 26, 27; of gases of fired
gunpowder, 47. For pressure: isothermal, 15, 17; adiabatic, 27; gases
of fired gunpowder in close vessels, 6, 36, 39; in guns, 45. For pressure
in guns with smokeless powders: While powder is burning, 85, 86, 101, 106,
INDEX 219
112, 140, 152; after powder is all burned, 86, 102. Maximum pressure,
91, 101, 1 06. Initial pressure when powder is all burned before projectile
moves, 86, 87, 93, 94, 99. For velocity of projectile in guns with smokeless
powders: while powder is burning, 83, 84, 89, 100, 101, 103, 112; after
powder is all burned, 84, 85, 102. For limiting velocity, 85, 93, 94, 98,
101, 127, 141, 146, 150, 164. For computing f, 32, 36, 84, 93, 94, 96, 97,
106, 164. For v c , 91, 92, 93, 94. For k and k' ', 58, 59, 61, 62, 63, 65,
68, 69, 72, 89, 90, 109, 136, 141, 148. For M, M' , N and N', 84, 85, 93,
94, 95, 97, 101, 102, 106, 112, 113, 114, 115, 118, 136, 139, 150, 155, 161,
165. For y, 39, 83, 93, 132, 146. For a, A, fJ., 58, 59, 60, 61, 62, 63, 64,
65, 67, 68, 69, 84, 129, 134. For P', 86, 87, 93, 94, 98. For the X
functions, 87, 88, 89, For X r ^o' 84, 93, 94, 119, 136, 149, 150, 152,
164. Working formulas, 77, 93, 94, 95, 97. For inclination of groove,
171, 172, 177. For pressure on lands, 175, 176, 180, 181, 185. For
semi-cubical parabola, 176. Common parabola, 178.
Frankford arsenal, mentioned, 130.
GAS, perfect, 17.
Gay-Lussac, law of, 16; mentioned, 8.
Gossot, Colonel F., law of combustion, 79; igniter, 151.
Graham, mentioned, 8.
Grains of powder, combustion of under constant pressure, 55 ; vanishing-sur-
face, 56; volume burned, 57; form-characteristics, 58; their relation to
each other, 58, 59.
Granulation, 151, 163.
Groove, developed, 171; width of, 179.
Gun-cotton, 10, n.
Gunpowder, i, 2.
HAMILTON, Captain Alston, length of m.p. grains, 71.
Heat: mechanical equivalent of, 18; specific heats, 18, 19, 21, 22.
Hugoniot, law of combustion, 79.
Hutton, Dr. Charles, experiments with gunpowder, 3, 4.
INFLAMMATION of a grain and charge of powder, n, 12, 13.
Isothermal expansion, 25.
JOURNAL U. S. Artillery, references to, 71, 79, 125, 148.
22O INDEX
LANDS, width of, 179.
Lenk, General von, experiments with gun-cotton, 10.
Liouville, R., law of combustion, 79.
Lissak, Colonel O. M., ordnance and gunnery, 29; construction of velocity
and pressure curves, 144.
Longridge, Atkinson, loss of energy in gun, 53.
MAGAZINE rifle, description of, 130.
Marriotte, law of, 15.
Maximum pressure in a gun, 90, 91
Maximum value of X 3 , 90, 101.
Mayevski, mention of, 8.
Monomial formulas, 100.
Muzzle velocities and pressures, computed, 107, 120, 123, 124, 129, 130, 133,
134, 143, 161, 169.
NATURE, reference to, 115.
Neumann, mentioned, 8.
Nobel, N. K. powder, law of combustion for, 79.
Noble and Abel, experiments with fired gunpowder in close vessels, and
deductions therefrom, 33 to 54.
Noble, Sir Andrew, experiments with 6-inch gun, 115; coefficient of friction,
1 80.
Notation, 15, 17, 19, 23, 24, 31, 51, 56, 58, 60, 67, 72, 74, 75, 76, 79, 80, 81,
82, 83, 84, 85, 86, 91, 92, 108, 149, 171, 172, 174, 176.
Notes on the construction of ordnance, reference to, 102.
ORDNANCE Department, reference to, 130, 162, 166.
Otto, mentioned, 8.
PARALLELOPIPEDON, form-characteristics of, 60.
Pashkievitsch, Colonel, lost work in a gun, 53.
Piobert, mentioned, 8.
Point of inflection of X 3 , 101.
Powder grains. See Grains of powder.
Powder, smokeless. See Composition.
Pressure: of fired gunpowder in close vessels, 6, 7, 9, 35, 37; in guns, 41.
RADIUS of gyration of projectile, 174, 180, 186.
Retarding effect of uniform twist, 186.
INDEX 221
Rifling of cannon, advantages of, 170.
Robins, Benjamin, experiments with fired gunpowder, 2.
Rodman, Captain T. J., experiments with fired gunpowder, 8; perforated
grains, 9; cutter gauge, 9.
Rumford, Count, experiments with fired gunpowder, 4; comparison of re-
sults with those of Noble and Abel, 6.
SAINTE-ROBERT, Count de, law of combustion, 79.
Sandy Hook, mention of, 151, 160.
Sarrau, ET., law of combustion, 80; monomial formula for pressure in a gun, 95.
Schonbein of Basel, discoverer of gun-cotton, 10.
Sebert, law of combustion, 79.
Spherical grains, form-characteristics of, 59.
Springfield Armory, mentioned, 130, 131.
TABLES, in text: of specific heats of certain gases, 22; of pressures in guns of
fired gunpowder, 46; of velocities and pressures in guns, 104. 107, 129,
130, 133, 134, 143, 161, 165, 169, 183, 185; of pressure on lands, 183, 185.
Temperature of fired gunpowder, 45.
Trinomial formulas, 138.
Twist, uniform, 171; increasing, 171.
VIEILLE, law of combustion, 79.
WEAVER, General E. M.. Notes on explosives, referred to, n.
Work of fired gunpowder, 47.
Working formulas, 77, 92, 181, 185.
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CIVIL ENGINEERING.
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BRIDGES AND ROOFS.
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Influence Lines for Bridge and Roof Computations 8vo, 3 00
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Heller's Stresses in Structures and the Accompanying Deformations.. . .8vo, 3 00
Howe's Design of Simple Roof- trusses in Wood and Steel 8vo. 2 00
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Waddell's De Pontibus, Pocket-book for Bridge Engineers 16mo, mor. 2 00
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HYDRAULICS.
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Bazin's Experiments upon the Contraction of the Liquid Vein Issuing from
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Bovey's Treatise on Hydraulics 8vo, 5 00
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Church's Diagrams of Mean Velocity of Water in Open Channels.
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MATERIALS OF ENGINEERING.
Baker's Roads and Pavements > 8vo, 5 00
Treatise on Masonry Construction 8vo, 5 00
Black's United States Public Works Oblong 4to, 5 00
Blanchard and Drowne's Highway Engineering. (In Press.)
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Burr's Elasticity and Resistance of the Materials of Engineering 8vo, 7 50
Byrne's Highway Construction 8vo, 5 00
Inspection of the Materials and Workmanship Employed in Construction.
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Church's Mechanics of Engineering 8vo, 6 00
Mechanics of Solids (Being Parts I, II, III of Mechanics of Engineer-
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Du Bois's Mechanics of Engineering.
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Eckel's Building Stones and Clays. (In Press.)
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Fowler's Ordinary Foundations 8vo, 3 50
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Holley's Analysis of Paint and Varnish Products. (In Press.)
* Lead and Zinc Pigments Large 12mo, 3 00
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8
.Johnson's (C. M.) Rapid Methods for the Chemical Analysis of Special Steels,
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Johnson's (J. B.) Materials of Construction Large 8vo, 6 00
Keep's Cast Iron 8vo, 2 50
Lanza's Applied Mechanics 8vo, 7 50
Lowe's Paints for Steel Structures 12mo, 1 00
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Maurer's Technical Mechanics 8vo, 4 00
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Merriman's Mechanics of Materials 8vo, 5 00
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Metcalf 's Steel. A Manual for Steel-users 12mo, 2 00
Morrison's Highway Engineering 1 . 8vo, 2 50
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Patton's Practical Treatise on Foundations 8vo, 5 00
Rice's Concrete Block Manufacture 8vo, 2 00
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'Tillson's Street Pavements and Paving Materials 8vo, 4 00
'Turneaure and Maurer's Principles of Reinforced Concrete Construction.
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Waterbury's Cement Laboratory Manual 12mo, I 00
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"Wood's (De V.) Treatise on the Resistance of Materials, and an Appendix on
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Wood's (M. P.) Rustless Coatings: Corrosion and Electrolysis of Iron and
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RAILWAY ENGINEERING.
.Andrews's Handbook for Street Railway Engineers 3X5 inches, mor. 1 25
Berg's Buildings and Structures of American Railroads 4to, 5 00
Brooks's Handbook of Street Railroad Location 16mo, mor. 1 50
* Burt's Railway Station Service 12mo, 2 00
Butts's Civil Engineer's Field-book 16mo, mor. 2 50
Crandall's Railway and Other Earthwork Tables 8vo, 1 50
'Oandall and Barnes's Railroad Surveying 16mo, mor. 2 00
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Dredge's History of the Pennsylvania Railroad. (1879) Paper, 5 00
Fisher's Table of Cubic Yards Cardboard, 25
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Hudson's Tables for Calculating the Cubic Contents of Excavations and Em-
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-Ives and Hilts's Problems in Surveying, Railroad Surveying and Geodesy
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Molitor and Beard's Manual for Resident Engineers 16mo, 1 00
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Nagle's Field Manual for Railroad Engineers 16mo, mor. $3 00
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Philbrick's Field Manual for Engineers 16mo, mor. 3 00
Raymond's Railroad Field Geometry 16mo, mor. 2 00
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Taylor's Prismoidal Formulae and Earthwork 8vo, 1 50
Webb's Economics of Railroad Construction Large 12mo, 2 50
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Wilson's Elements of Railroad-Track and Construction 12mo, 2 00
DRAWING
Barr and Wood's Kinematics of Machinery 8vo, 2 50
* Bartlett's Mechanical Drawing 8vo, 3 00
* " Abridged Ed 8vo, 1 50
* Bartlett and Johnson's Engineering Descriptive Geometry 8vo, 1 50
Blessing and Darling's Descriptive Geometry. (In Press.)
Elements of Drawing. (In Press.)
Coolidge's Manual of Drawing 8vo, paper, 1 00
Coolidge and Freeman's Elements of General Drafting for Mechanical Engi-
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Durley's Kinematics of Machines 8vo, 4 00
Emch's Introduction to Projective Geometry and its Application 8vo, 2 50
Hill's Text-book on Shades and Shadows, and Perspective 8vo, 2 00
Jamison's Advanced Mechanical Drawing 8vo, 2 OO
Elements of Mechanical Drawing 8vo, 2 50
Jones's Machine Design:
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* Kimball and Barr's Machine Design 8vo, 3 00
MacCord's Elements of Descriptive Geometry 8vo, 3 00
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Moyer's Descriptive Geometry 8vo, 2 00
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Text-book of Mechanical Drawing and Elementary Machine Design.. 8vo, 3 00
Robinson's Principles of Mechanism ,.... 8vo, 3 00
Schwamb and Merrill's Elements of Mechanism 8vo, 3 00
Smith (A. W.) and Marx's Machine Design 8vo, 3 00
Smith's (R. S.) Manual of Topographical Drawing. (McMillan.) 8vo, 2 50
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Tracy and North's Descriptive Geometry. (In Press.)
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Elements of Machine Construction and Drawing 8vo, 7 50
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10
ELECTRICITY AND PHYSICS.
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Benjamin's History of Electricity 8vo, 3 00
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* Collins's Manual of Wireless Telegraphy and Telephony 12mo, 1 50
Crehore and Squier's Polarizing Photo-chronograph 8vo, 3 00
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Dawson's "Engineering" and Electric Traction Pocket-book. . . . 16mo, mor. 5 00
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Dingey's Machinery Pattern Making 12mo, 2 00
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MATERIALS OF ENGINEERING.
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Church's Mechanics of Engineering 8vo, 6 00
Mechanics of Solids (Being Parts I, II, III of Mechanics of Engineering).
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* Greene's Structural Mechanics 8vo, 2 50
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Metcalf 's Steel. A Manual for Steel-users 12mo, 2 00
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Thurston's Materials of Engineering 3 vols., 8voi 8 00
Part I. Non-metallic Materials of Engineering 8vol 2 00
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14
Waterbury's Laboratory Manual for Testing Materials of Construction.
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Wood's (De V.) Elements of Analytical Mechanics 8vo, $3 00
Treatise on "the Resistance of Materials and an Appendix on the
Preservation of Timber 8vo, 2 00
Wood's (M. P.) Rustless Coatings: Corrosion and Electrolysis of Iron and
Steel 8vo, 4 00
STEAM-ENGINES AND BOILERS.
Berry's Temperature-entropy Diagram. Third Edition Revised and En-
larged 12mo, 2 50
Carnot's Reflections on the Motive Power of Heat. (Thurston.). .'. ..12mo, 1 50
Chase's Art of Pattern Making If mo, 2 50
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Hirshfeld and Barnard's Heat Power Engineering. (In Press.)
Hutton's Heat and Heat-engines 8vo, 5 00
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Kent's Steam Boiler Economy 8vo, 4 00
Kneass's Practice and Theory of the Injector 8vo, 1 50
MacCord's Slide-valves 8vo, 2 00
Meyer's Modern Locomotive Construction 4to, 10 00
Miller, Berry, and Riley's Problems in Thermodynamics 8vo, paper, 75
Moyer's Steam Turbine 8vo, 4 00
Peabody's Manual of the Steam-engine Indicator 12mo, 1 50
Tables of the Properties of Steam and Other Vapors and Temperature-
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Thermodynamics of the Steam-engine and Other Heat-engines. . . . 8vo, 5 00
* Thermodynamics of the Steam Turbine 8vo, 3 00
Valve-gears for Steam-engines 8vo, 2 50
Peabody and Miller's Steam-boilers 8vo, 4 00
Pupin's Thermodynamics of Reversible Cycles in Gases and Saturated Vapors.
(Osterberg.) 12mo, 1 25
Reagan's Locomotives: Simple, Compound, and Electric. New Edition.
Large 12mo, 3 50
Sinclair's Locomotive Engine Running and Management 12mo, 2 00
Smart's Handbook of Engineering Laboratory Practice 12mo, 2 50
Snow's Steam-boiler Practice 8vo, 3 00
Spangler's Notes on Thermodynamics 12mo, 1 00
Valve-gears 8vo, 2 50
Spangler, Greene, and Marshall's Elements of Steam-engineering 8vo, 3 00
Thomas's Steam-turbines 8vo, 4 00
Thurston's Handbook of Engine and Boiler Trials, and the Use of the Indi-
cator and the Prony Brake 8vo, 5 00
Handy Tables 8vo, 1 50
Manual of Steam-boilers, their Designs, Construction, and Operation 8vo, 5 00
Manual of the Steam-engine 2 vols., 8vo, 10 00
Part I. History, Structure, and Theory 8vo, 6 00
Part II. Design, Construction, and Operation 8vo, 6 00
Wehrenfennig's Analysis and Softening of Boiler Feed- water. (Patterson.)
8vo, 4 00
Weisbach's Heat, Steam, and Steam-engines. (Du Bois.) 8vo, 5 00
Whitham's Steam-engine Design 8vo, 5 00
Wood's Thermodynamics, Heat Motors, and Refrigerating Machines. . .8vo, 4 00
MECHANICS PURE AND APPLIED.
Church's Mechanics of Engineering 8vo, 6 00
Mechanics of Fluids (Being Part IV of Mechanics of Engineering). .8vo, 3 00
* Mechanics of Internal Works 8vo, 1 50
Mechanics of Solids (Being Parts I, II, III of Mechanics of Engineering).
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15
Dana's Text-book of Elementary Mechanics for Colleges and Schools .12mo, $1 5O
Du Bois's Elementary Principles of Mechanics:
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Mechanics of Engineering. Vol. I Small 4t, 7 50
Vcl. II Small 4to, 10 00
* Greene's Structural Mechanics 8vo, 2 50
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James's Kinematics of a Point and the Rational Mechanics of a Particle.
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* King's Elements of the Mechanics of Materials and of Power of Trans-
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Lanza's Applied Mechanics 8vo, 7 50
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* Vol. II. Kinematics and Kinetics 12mo, 1 50
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Maurer's Technical Mechanics 8vo, 4 00
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Mechanics of Materials .8vo, 5 00
* Michie's Elements of Analytical Mechanics 8vo, 4 00
Robinson's Principles of Mechanism 8vo, 3 00'
Sanborn's Mechanics Problems Large 12mo, 1 50
Schwamb and Merrill's Elements of Mechanism 8vo, 3 00
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MEDICAL.
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Worcester and Atkinson's Small Hospitals Establishment and Maintenance,
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METALLURGY.
Betts's Lead Refining by Electrolysis 8vo, 4 00
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16
Iron Founder 12mo, $2 5O>
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Burgess and Le Chatelier's Measurement of High Temperatures. Third
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Douglas's Untechnical Addresses on Technical Subjects 12mo, 1 OO
Goesel's Minerals and Metals: A Reference Book 16mo, mor. 3 00
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Keep's Cast Iron 8vo, 2 50
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Smith's Materials of Machines 12mo, 1 00-
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Thurston's Materials of Engineering. In Three Parts 8vo, 8 00
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Ulke's Modern Electrolytic Copper Refining 8vo, 3 OO'
West's American Foundry Practice 12mo, 2 5O'
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MINERALOGY.
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Brush's Manual of Determinative Mineralogy. (Penfield.) 8vo, 4 OO'
Butler's Pocket Hand-book of Minerals 16mo, mor. 3 00'
Chester's Catalogue of Minerals 8vo, paper, 1 00
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Dana's First Appendix to Dana's New "System of Mineralogy". .Large 8vo, 1 00'
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Manual of Mineralogy and Petrography 12mo, 2 00
Minerals and How to Study Them 12mo, 1 50-
System of Mineralogy f Large 8vo, half leather, 12 50'
Text-book of Mineralogy 8vo, 4 00
Douglas's Untechnical Addresses on Technical Subiects 12mo, 1 OO
Eakle's Mineral Tables . 8vo, 1 25
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Groth's Introduction to Chemical Crystallography (Marshall) 12mo, 1 25
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Merrill's Non-metallic Minerals: Their Occurrence and Uses 8vo, 4 OO
Stones for Building and Decoration 8vo, 5 OO
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* Pirsson's Rocks and Rock Minerals 12mo, 2 50>
* Richards's Synopsis of Mineral Characters 12mo, mor. 1 25-
* Ries's Clays: Their Occurrence, Properties and Uses 8vo, 5 00>
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* Ries and Leighton's History of the Clay-working industry of the United
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* Rowe's Practical Mineralogy Simplified 12mo, 1 25
* Tillman's Text-book of Important Minerals and Rocks 8vo, 2 00
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MINING.
* Beard's Mine Gases and Explosions Large 12mo, 3 00
* Crane's Gold and Silver 8vo, 5 00
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Riemer's Shaft Sinking Under Difficult Conditions. (Corning and Peele.)8vo, 3 00
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Wilson's Hydraulic and Placer Mining. 2d edition, rewritten 12mo, 2 50
Treatise on Practical and Theoretical Mine Ventilation 12mo, 1 25
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Association of State and National Food and Dairy Departments, Hartford
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Jamestown Meeting, 1907 8vo, 3 00
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Gerhard's Guide to Sanitary Inspections 12mo, 1 50
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Leach's Inspection and Analysis of Food with Special Reference to State
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Mason's Examination of Water. (Chemical and Bacteriological) 12mo, 1 25
Water-supply. (Considered principally from a Sanitary Standpoint).
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* Mast's Light and the Behavior of Organisms Large 12mo, 2 50
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Ogden's Sewer Construction 8vo, 3 00
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Prescott and Winslow's Elements of Water Bacteriology, with Special Refer-
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Richards's Conservation by Sanitation 8vo, 2 50
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Richards and Woodman's Air, Water, and Food from a Sanitary Stand-
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Turneaure and Russell's Public Water-supplies 8vo, 5 00
Venable's Garbage Crematories in America 8vo, 2 00
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Ward and Whipple's Freshwater Biology. (In Press.)
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Value of Pure Water Large 12mo, 1 00
Winslow's Systematic Relationship of the Coccaceae Large 12mo, 2 50
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Emmons's Geological Guide-book of the Rocky Mountain Excursion of the
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* Fritz, Autobiography of John 8vo,
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Jacobs's Betterment Briefs. A Collection of Published Papers on Or-
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Metcalfe's Cost of Manufactures, and the Administration of Workshops. .8vo, 5 00
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Large 12mo, 3 00
* Rotch and Palmer's Charts of the Atmosphere for Aeronauts and Aviators.
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Rotherham's Emphasised New Testament Large 8vo, 2 00
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Thome's Structural and Physiological Botany. (Bennett). . 16mo, 2 25
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Gesenius's Hebrew and Chaldee Lexicon to the Old Testament Scriptures.
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UNIVERSITY OF CALIFORNIA LIBRARY
BERKELEY
Return to desk from which borrowed.
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