Encyclopaedia Britannica, 11th Edition, "Baconthorpe" to "Bankruptcy"

Chapter XX: Front Matter (20)

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Example.--Find the alteration of elevation required at a range of 3000 yds. in the exchange of fire between a ship and a fort 1200 ft. high, a 12-in. gun being employed on each side, firing a shot weighing 850 lb with velocity 2150 f/s. The complete ballistic table, and the method of high angle fire (see below) must be employed.

[v.03 p.0273]

Range. s. s/C. S(v). v. T(v). t/C. t. T(v_0). v_0
0 0 0 20700.53 2150 28.6891 0.0000 0.000 28.6891 215
500 1500 518 20182.53 1999 28.4399 0.2492 0.720 28.5645 207
1000 3000 1036 19664.53 1862 28.1711 0.5180 1.497 28.4301 199
1500 4500 1554 19146.53 1732 27.8815 0.8076 2.330 28.2853 191
2000 6000 2072 18628.53 1610 27.5728 1.1163 3.225 28.1310 184

Range. v_0. D(v_0). [phi]/C. [phi].[beta]/C. [beta].
0 2150 50.9219 0.0000 0.000 0.0000 0.000
500 2071 50.8132 0.1087 0.315 0.1135 0.328
1000 1994 50.6913 0.2306 0.666 0.2486 0.718
1500 1918 50.5542 0.3677 1.062 0.4085 1.181
2000 1843 50.4029 0.5190 1.500 0.5989 1.734

A. Remaining Velocity.
B. To strike an object 10 ft. high range must be known to
C. Slope of Descent.
D. 5' elevation or depression alters
point of impact (Range ... Laterally or Vertically).
E. Elevation.
F. Range.
G. Fuse scale for T. and P. middle No. 54 Marks I., II., or III.
H. 50% of rounds should fall in: (Length x Breadth x Height).
I. Time of Flight.
J. Penetration into Wrought Iron.

-----+-----+----+----------+------+-----+-----+------------+------+-----
A. | B. | C. | D. | E. | F. | G. | H. | I. | J.
-----+-----+----+----------+------+-----+-----+------------+------+-----
f/s. | yds.|1 in|yds. yds. |deg ' | yds.| |yds yds yds |secs. | in.
2154 | .. | .. | .. 0.00 | 0 0 | 0 | .. | .. .. .. | 0.00 | 13.6
2122 |1145 |687 | 125 0.14 | 0 4 | 100 | 1/4| .. 0.4 .. | 0.16 | 13.4
2091 | 635 |381 | 125 0.29 | 0 9 | 200 | 3/4| .. 0.4 .. | 0.31 | 13.2
2061 | 408 |245 | 125 0.43 | 0 13 | 300 |1 | .. 0.4 .. | 0.47 | 13.0
2032 | 316 |190 | 125 0.58 | 0 17 | 400 |1-1/4| .. 0.4 .. | 0.62 | 12.8
2003 | 260 |156 | 125 0.72 | 0 21 | 500 |1-3/4| .. 0.5 0.2 | 0.78 | 12.6
1974 | 211 |127 | 125 0.87 | 0 26 | 600 |2 | .. 0.5 0.2 | 0.95 | 12.4
1946 | 183 |110 | 125 1.01 | 0 30 | 700 |2-1/4| .. 0.5 0.2 | 1.11 | 12.2
1909 | 163 | 98 | 125 1.16 | 0 34 | 800 |2-3/4| .. 0.5 0.2 | 1.28 | 12.0
1883 | 143 | 85 | 125 1.31 | 0 39 | 900 |3 | .. 0.6 0.3 | 1.44 | 11.8
1857 | 130 | 78 | 125 1.45 | 0 43 |1000 |3-1/4| .. 0.6 0.3 | 1.61 | 11.6
1830 | 118 | 71 | 125 1.60 | 0 47 |1100 |3-3/4| .. 0.6 0.3 | 1.78 | 11.4
1803 | 110 | 66 | 125 1.74 | 0 51 |1200 |4 | .. 0.6 0.3 | 1.95 | 11.2
1776 | 101 | 61 | 125 1.89 | 0 55 |1300 |4-1/2| .. 0.7 0.4 | 2.12 | 11.0
1749 | 93 | 56 | 125 2.03 | 0 59 |1400 |4-3/4| .. 0.7 0.4 | 2.30 | 10.8
1722 | 86 | 52 | 125 2.18 | 1 3 |1500 |5 | .. 0.7 0.4 | 2.47 | 10.6
1695 | 80 | 48 | 125 2.32 | 1 7 |1600 |5-1/2| 25 0.8 0.5 | 2.65 | 10.5
1669 | 71 | 43 | 125 2.47 | 1 11 |1700 |5-3/4| 25 0.9 0.5 | 2.84 | 10.3
1642 | 67 | 40 | 100 2.61 | 1 16 |1800 |6-1/4| 25 1.0 0.5 | 3.03 | 10.1
1616 | 61 | 37 | 100 2.76 | 1 22 |1900 |6-1/2| 25 1.1 0.6 | 3.23 | 9.9
1591 | 57 | 34 | 100 2.91 | 1 27 |2000 |7 | 25 1.2 0.6 | 3.41 | 9.7
-----+-----+----+----------+------+-----+-----+------------+------+-----

The last column in the Range Table giving the inches of penetration into wrought iron is calculated from the remaining velocity by an empirical formula, as explained in the article ARMOUR PLATES.

_High Angle and Curved Fire._--"High angle fire," as defined officially, "is fire at elevations greater than 15deg," and "curved fire is fire from howitzers at all angles of elevation not exceeding 15deg." In these cases the curvature of the trajectory becomes considerable, and the formulae employed in direct fire must be modified; the method generally employed is due to Colonel Siacci of the Italian artillery.

Starting with the exact equations of motion in a resisting medium,

(43) d^2x/dt^2 = -r cos i = -r dx/ds,

(44) d^2y/dt^2 = -r sin i - g = -r dy/ds - g,

and eliminating r,

(45) dx/dt d^2y/dt^2 - dy/dt d^2x/dt^2 = -g{dx/dt};

and this, in conjunction with

(46) tan i = dy/dx = {dy/dt}/{dx/dt},

(47) sec^2 i{di/dt} =
({dx/dt}{d^2y/dt^2} - {dy/dt}{d^2x/dt^2}) / (dx/dt)^2,

reduces to

(48) di/dt = -{g/v} cos i, or {d tan i}/dt = -g/{v cos i},

the equation obtained, as in (18), by resolving normally in the trajectory, but di now denoting the _increment_ of i in the increment of time dt.

Denoting dx/dt, the horizontal component of the velocity, by q, so that

(49) v cos i = q,

equation (43) becomes

(50) dq/dt = -r cos i,

and therefore by (48)

(51) dq/di = {dq/dt} {dt/di} = {rv}/g.

It is convenient to express r as a function of v in the previous notation

(52) Cr = f(v),

and now

(53) dq/di = {v f(v)}/{Cg},

an equation connecting q and i.

Now, since v = g sec i

(54) dt/dq = -C sec i / f(q sec i),

and multiplying by dx/dt or q,

(55) dx/dq = -C q sec i / f(q sec i),

and multiplying by dy/dx or tan i,

(56) dy/dq = -C q sec i tan i / f(q sec i);

also

(57) di/dq = Cg / {q sec i . f(q sec i)},

(58) d tan i/dq = C g sec i / {q . f(q sec i)},

from which the values of t, x, y, i, and tan i are given by integration with respect to q, when sec i is given as a function of q by means of (51).

Now these integrations are quite intractable, even for a very simple mathematical assumption of the function f(v), say the quadratic or cubic law, f(v) = v^2/k or v^3/k.

But, as originally pointed out by Euler, the difficulty can be turned if we notice that in the ordinary trajectory of practice the quantities i, cos i, and sec i vary so slowly that they may be replaced by their _mean_ values, [eta], cos [eta], and sec [eta], especially if the trajectory, when considerable, is divided up in the calculation into arcs of small curvature, the curvature of an arc being defined as the angle between the tangents or normals at the ends of the arc.

Replacing then the angle i on the right-hand side of equations (54) - (56) by some mean value [eta], we introduce Siacci's pseudo-velocity u defined by

(59) u = q sec [eta],

so that u is a quasi-component parallel to the mean direction of the tangent, say the direction of the chord of the arc.

[v.03 p.0274] Integrating from any initial pseudo-velocity U,

(60) t = C[Integral,u:U] du/f(u),
(61) x = C cos [eta] [Integral] u du/f(u),
(62) y = C sin [eta] [Integral] u du/f(u);

and supposing the inclination i to change from [phi] to [theta] radians over the arc,

(63) [phi] - [theta] = Cg cos [eta] [Integral] du/{u f(u)},
(64) tan [phi] - tan [theta] = Cg sec [eta] [Integral] du/{u f(u)}.

But according to the definition of the functions T, S, I and D of the ballistic table, employed for direct fire, with u written for v,

(65) [Integral,u:U] du/f(u) = [Integral] du/gp = T(U) - T(u),
(66) [Integral] u du/f(u) = S(U) - S(u),
(67) [Integral] g du/u f(u) = I(U) - I(u);

and therefore

(68) t = C[T(U) - T(u)],
(69) x = C cos [eta] [S(U) - S(u)],
(70) y = C sin [eta] [S(U) - S(u)],
(71) [phi] - [theta] = C cos [eta] [I(U) - I(u)],
(72) tan [phi] - tan [theta] = C sec [eta] [I(U) - I(u)],

while, expressed in degrees,

(73) [phi]deg - [theta]deg = C cos [eta] [D(U) - D(u)],

The equations (66)-(71) are Siacci's, slightly modified by General Mayevski; and now in the numerical applications to high angle fire we can still employ the ballistic table for direct fire.

It will be noticed that [eta] cannot be exactly the same mean angle in all these equations; but if [eta] is the same in (69) and (70),

(74) y/x = tan [eta].

so that [eta] is the inclination of the chord of the arc of the trajectory, as in Niven's method of calculating trajectories (_Proc. R.S._, 1877): but this method requires [eta] to be known with accuracy, as 1% variation in [eta] causes more than 1% variation in tan [eta].

The difficulty is avoided by the use of Siacci's altitude-function A or A(u), by which y/x can be calculated without introducing sin [eta] or tan [eta], but in which [eta] occurs only in the form cos [eta] or sec [eta], which varies very slowly for moderate values of [eta], so that [eta] need not be calculated with any great regard for accuracy, the arithmetic mean 1/2([phi] + [theta]) of [phi] and [theta] being near enough for [eta] over any arc [phi] - [theta] of moderate extent.

Now taking equation (72), and replacing tan [theta], as a variable final tangent of an angle, by tan i or dy/dx,

(75) tan [phi] - dy/dx = C sec [eta] [I(U) - I(u)],

and integrating with respect to x over the arc considered,

(76) x tan [phi] - y = C sec [eta] [xI(U) - [Integral,0:x] I(u)dx],

But

(77) [Integral,0:x] I(u)dx = [Integral,U:u] I(u) dx/du du
= C cos [eta] [Integral,x:U] I(u) {u du}/{g f(u)}
= C cos [eta] [A(U) - A(u)]

in Siacci's notation; so that the altitude-function A must be calculated by summation from the finite difference [Delta]A, where

(78) [Delta]A = I(u) u[Delta]u / gp = I(u)[Delta]S,

or else by an integration when it is legitimate to assume that f(v)=v^m/k in an interval of velocity in which m may be supposed constant.

Dividing again by x, as given in (76),

(79) tan [phi] - y/x =
C sec [eta] [I(U) - {A(U) - A(u)}/{S(U) - S(u)}]

from which y/x can be calculated, and thence y.

In the application of Siacci's method to the calculation of a trajectory in high angle fire by successive arcs of small curvature, starting at the beginning of an arc at an angle [phi] with velocity v_[phi], the curvature of the arc [phi] - [theta] is first settled upon, and now

(80) [eta] = 1/2([phi] + [theta])

is a good first approximation for [eta].

Now calculate the pseudo-velocity u_[phi] from

(81) u_[phi] = v_[phi] cos [phi] sec [eta],

and then, from the given values of [phi] and [theta], calculate u_[theta] from either of the formulae of (72) or (73):--

(82) I(u_[theta]) =
I(u_[phi]) - {tan [phi] - tan [theta]}/{C sec [eta]},
(83) D(u_[theta]) =
D(u_[phi]) - {[phi]deg - [theta]deg}/{C cos [eta]}.

Then with the suffix notation to denote the beginning and end of the arc [phi] - [theta],

(84) _[phi]t_[theta] = C[T(u_[phi]) - T(u_[theta])],
(85) _[phi]x_[theta] = C cos [eta] [S(u_[phi]) - S(u_[theta])],
(86) _[phi](y/x)_[theta] =
tan [phi] - C sec [eta] [I(u_[phi]) - [Delta]A/[Delta]S];

[Delta] now denoting any finite tabular difference of the function between the initial and final (pseudo-) velocity.

Also the velocity v_{[theta]} at the end of the arc is given by

(87) v_[theta] = u_[theta] sec [theta] cos [eta].

Treating this final velocity v_[theta] and angle [theta] as the initial velocity v_[phi] and angle [phi] of the next arc, the calculation proceeds as before (fig. 2).

In the long range high angle fire the shot ascends to such a height that the correction for the tenuity of the air becomes important, and the curvature [phi] - [theta] of an arc should be so chosen that _[phi]y_[theta] the height ascended, should be limited to about 1000 ft., equivalent to a fall of 1 inch in the barometer or 3% diminution in the tenuity factor [tau].

A convenient rule has been given by Captain James M. Ingalls, U.S.A., for approximating to a high angle trajectory in a single arc, which assumes that the mean density of the air may be taken as the density at two-thirds of the estimated height of the vertex; the rule is founded on the fact that in an unresisted parabolic trajectory the average height of the shot is two-thirds the height of the vertex, as illustrated in a jet of water, or in a stream of bullets from a Maxim gun.

The longest recorded range is that given in 1888 by the 9.2-in. gun to a shot weighing 380 lb fired with velocity 2375 f/s at elevation 40deg; the range was about 12 m., with a time for flight of about 64 sec., shown in fig. 2.

A calculation of this trajectory is given by Lieutenant A. H. Wolley-Dod, R.A., in the _Proceedings R.A. Institution_, 1888, employing Siacci's method and about twenty arcs; and Captain Ingalls, by assuming a mean tenuity-factor [tau]=0.68, corresponding to a height of about 2 m., on the estimate that the shot would reach a height of 3 m., was able to obtain a very accurate result, working in two arcs over the whole trajectory, up to the vertex and down again (Ingalls, _Handbook of Ballistic Problems_).

Siacci's altitude-function is useful in direct fire, for giving immediately the angle of elevation [phi] required for a given range of R yds. or X ft., between limits V and v of the velocity, and also the angle of descent [beta].

In direct fire the pseudo-velocities U and u, and the real velocities V and v, are undistinguishable, and sec [eta] may be replaced by unity so that, putting y = 0 in (79),

(88) tan [phi] = C [I(V) - [Delta]A/[Delta]S].

Also

(89) tan [phi] - tan [beta] = C [I(V) - L(v)]

so that

(90) tan [beta] = C [[Delta]A/[Delta]S - I(v)],

or, as (88) and (90) may be written for small angles,

(91) sin 2[phi] = 2C [I(V) - [Delta]A/[Delta]S],
(92) sin 2[beta] = 2C [[Delta]A/[Delta]S - I(v)].

To simplify the work, so as to look out the value of sin 2[phi] without the intermediate calculation of the remaining velocity v, a double-entry table has been devised by Captain Braccialini Scipione [v.03 p.0275] (_Problemi del Tiro_, Roma, 1883), and adapted to yd., ft., in. and lb units by A. G. Hadcock, late R.A., and published in the _Proc. R.A. Institution_, 1898, and in _Gunnery Tables_, 1898.

In this table

(93) sin 2[phi] = Ca,

where a is a function tabulated for the two arguments, V the initial velocity, and R/C the reduced range in yards.

The table is too long for insertion here. The results for [phi] and [beta], as calculated for the range tables above, are also given there for comparison.

_Drift_.--An elongated shot fired from a rifled gun does not move in a vertical plane, but as if the mean plane of the trajectory was inclined to the true vertical at a small angle, 2deg or 3deg; so that the shot will hit the mark aimed at if the back sight is tilted to the vertical at this angle [delta], called the permanent angle of deflection (see SIGHTS).

This effect is called _drift_ and the reason of it is not yet understood very clearly.

It is evidently a gyroscopic effect, being reversed in direction by a change from a right to a left-handed twist of rifling, and being increased by an increase of rotation of the shot.

The axis of an elongated shot would move parallel to itself only if fired in a vacuum; but in air the couple due to a sidelong motion tends to place the axis at right angles to the tangent of the trajectory, and acting on a rotating body causes the axis to precess about the tangent. At the same time the frictional drag damps the nutation and causes the axis of the shot to follow the tangent of the trajectory very closely, the point of the shot being seen to be slightly above and to the right of the tangent, with a right-handed twist. The effect is as if there was a mean sidelong thrust w tan [delta] on the shot from left to right in order to deflect the plane of the trajectory at angle [delta] to the vertical. But no formula has yet been invented, derived on theoretical principles from the physical data, which will assign by calculation a definite magnitude to [delta].

An effect similar to drift is observable at tennis, golf, base-ball and cricket; but this effect is explainable by the inequality of pressure due to a vortex of air carried along by the rotating ball, and the deviation is in the opposite direction of the drift observed in artillery practice, so artillerists are still awaiting theory and crucial experiment.

After all care has been taken in laying and pointing, in accordance with the rules of theory and practice, absolute certainty of hitting the same spot every time is unattainable, as causes of error exist which cannot be eliminated, such as variations in the air and in the muzzle-velocity, and also in the steadiness of the shot in flight.

To obtain an estimate of the accuracy of a gun, as much actual practice as is available must be utilized for the calculation in accordance with the laws of probability of the 50% zones shown in the range table (see PROBABILITY.)

II. INTERIOR BALLISTICS

The investigation of the relations connecting the pressure, volume and temperature of the powder-gas inside the bore of the gun, of the work realized by the expansion of the powder, of the dynamics of the movement of the shot up the bore, and of the stress set up in the material of the gun, constitutes the branch of interior ballistics.

A gun may be considered a simple thermo-dynamic machine or heat-engine which does its work in a single stroke, and does not act in a series of periodic cycles as an ordinary steam or gas-engine.

An indicator diagram can be drawn for a gun (fig. 3) as for a [v.03 p.0276] steam-engine, representing graphically by a curve CPD the relation between the volume and pressure of the powder-gas; and in addition the curves AQE of energy e, AvV of velocity v, and AtT of time t can be plotted or derived, the velocity and energy at the muzzle B being denoted by V and E.

After a certain discount for friction and the recoil of the gun, the net work realized by the powder-gas as the shot advances AM is represented by the area ACPM, and this is equated to the kinetic energy e of the shot, in foot-tons,

(1) e = {w /2240} (1 + {4k^2 / d^2} tan^2 [delta]) {v^2 / 2g},

in which the factor 4(k^2/d^2)tan^2[delta] represents the fraction due to the rotation of the shot, of diameter d and axial radius of gyration k, and [delta] represents the angle of the rifling; this factor may be ignored in the subsequent calculations as small, less than 1%.

The mean effective pressure (M.E.P.) in tons per sq. in. is represented in fig. 3 by the height AH, such that the rectangle AHKB is equal to the area APDB; and the M.E.P. multiplied by 1/4[pi]d^2, the cross-section of the bore in square inches, gives in tons the mean effective thrust of the powder on the base of the shot; and multiplied again by l, the length in inches of the travel AB of the shot up the bore, gives the work realized in inch-tons; which work is thus equal to the M.E.P. multiplied by 1/4[pi]d^2l = B - C, the volume in cubic inches of the rifled part AB of the bore, the difference between B the total volume of the bore and C the volume of the powder-chamber.

Equating the muzzle-energy and the work in foot-tons

(2) E = w/2240 V^2/2g = {B - C} / 12 x M.E.P.

(3) M.E.P. = w/2240 V^2/2g 12/{B - C}

Working this out for the 6-in. gun of the range table, taking L = 216 in., we find B - C = 6100 cub. in., and the M.E.P. is about 6.4 tons per sq. in.

But the maximum pressure may exceed the mean in the ratio of 2 or 3 to 1, as shown in fig. 4, representing graphically the result of Sir Andrew Noble's experiments with a 6-in. gun, capable of being lengthened to 100 calibres or 50 ft. (_Proc. R.S._, June 1894).

On the assumption of uniform pressure up the bore, practically realizable in a Zalinski pneumatic dynamite gun, the pressure-curve would be the straight line HK of fig. 3 parallel to AM; the energy-curve AQE would be another straight line through A; the velocity-curve AvV, of which the ordinate v is as the square root of the energy, would be a parabola; and the acceleration of the shot being constant, the time-curve AtT will also be a similar parabola.

If the pressure falls off uniformly, so that the pressure-curve is a straight line PDF sloping downwards and cutting AM in F, then the energy-curve will be a parabola curving downwards, and the velocity-curve can be represented by an ellipse, or circle with centre F and radius FA; while the time-curve will be a sinusoid.

But if the pressure-curve is a straight line F'CP sloping upwards, cutting AM behind A in F', the energy-curve will be a parabola curving upwards, and the velocity-curve a hyperbola with center at F'.

These theorems may prove useful in preliminary calculations where the pressure-curve is nearly straight; but, in the absence of any observable law, the area of the pressure-curve must be read off by a planimeter, or calculated by Simpson's rule, as an indicator diagram.

To measure the pressure experimentally in the bore of a gun, the crusher-gauge is used as shown in fig. 6, nearly full size; it records the maximum pressure by the compression of a copper cylinder in its interior; it may be placed in the powder-chamber, or fastened in the base of the shot.

In Sir Andrew Noble's researches a number of plugs were inserted in the side of the experimental gun, reaching to the bore and carrying crusher-gauges, and also chronographic appliances which registered the passage of the shot in the same manner as the electric screens in Bashforth's experiments; thence the velocity and energy of the shot was inferred, to serve as an independent control of the crusher-gauge records (figs. 4 and 5).

As a preliminary step to the determination of the pressure in the bore of a gun, it is desirable to measure the pressure obtained by exploding a charge of powder in a closed vessel, varying the weight of the charge and thereby the density of the powder-gas.

The earliest experiments of this nature are due to Benjamin Robins in 1743 and Count Rumford in 1792; and their method has been revived by Dr Kellner, War Department chemist, who employed the steel spheres of bicycle ball-bearings as safety-valves, loaded to register the pressure at which the powder-gas will blow off, and thereby check the indications of the crusher-gauge (_Proc. R.S._, March 1895).

Chevalier d'Arcy, 1760. also experimented on the pressure of powder and the velocity of the bullet in a musket barrel; this he accomplished by shortening the barrel successively, and measuring the velocity obtained by the ballistic pendulum; thus reversing Noble's procedure of gradually lengthening the gun.

But the most modern results employed with gunpowder are based on the experiments of Noble and Abel (_Phil. Trans._, 1875-1880-1892-1894 and following years).

A charge of powder, or other explosive, of varying weight P lb, is fired in an explosion-chamber (fig. 7, scale about 1/5) of which the volume C, cub. in., is known accurately, and the pressure p, tons per sq. in., was recorded by a crusher-gauge (fig. 6).

[v.03 p.0277] The result is plotted in figs. 8 and 9, in a curve showing the relation between p and D the _gravimetric density_, which is the specific gravity of the P lb of powder when filling the volume C, cub. in., in a state of gas; or between p and v, the reciprocal of D, which may be called the _gravimetric volume_ (G.V.), being the ratio of the volume of the gas to the volume of an equal weight of water.

The results are also embodied in the following Table;--

TABLE 1.
Pressure in Tons per sq. in.
G.D. G.V. Pebble Powder. Cordite.
0.05 20.00 0.855 3.00
6 16.66 1.00 3.80
8 12.50 1.36 5.40
0.10 10.00 1.76 7.10
12 8.33 2.06 8.70
14 7.14 2.53 10.50
15 6.66 2.73 11.36
16 6.25 2.96 12.30
18 5.55 3.33 14.20
20 5.00 3.77 16.00
22 4.54 4.26 17.90
24 4.17 4.66 19.80
25 4.00 4.88 20.63
26 3.84 5.10 21.75
30 3.33 6.07 26.00
35 2.85 7.35 31.00
40 2.50 8.73 36.53
45 2.22 10.23 42.20
50 2.00 11.25 48.66
55 1.81 13.62 55.86
60 1.66 15.55 63.33

The term _gravimetric density_ (G.D.) is peculiar to artillerists; it is required to distinguish between the specific gravity (S.G.) of the powder filling a given volume in a state of gas, and the specific gravity of the separate solid grain or cord of powder.

Thus, for instance, a lump of solid lead of given S.G., when formed into a charge of lead shot composed of equal spherules closely packed, will have a G.D. such that

(4) G.D. of charge of lead shot
--------------------------- = 1/6 [pi] [sqrt]2 = 0.7403;
S.G. of lump of solid lead

while in the case of a bundle of cylindrical sticks of cordite,

(5) G.D. of charge of cordite
-------------------------- = 1/6 [pi] [sqrt]3 = 0.9067.
S.G. of stick of cordite

At the standard temperature of 62deg F. the volume of the gallon of 10 lb of water is 277.3 cub. in.; or otherwise, 1 cub. ft. or 1728 cub. in. of water at this temperature weighs 62.35 lb, and therefore 1 lb of water bulks 1728 / 62.35 = 27.73 cub. in.

Thus if a charge of P lb of powder is placed in a chamber of volume C cub. in., the

(6) G.D.= 27.73P/C, G.V. = C/27.73 P.

Sometimes the factor 27.68 is employed, corresponding to a density of water of about 62.4 lb per cub. ft., and a temperature 12deg C., or 54deg F.

With metric units, measuring P in kg., and C in litres, the G.D. = P/C, G.V. = C/P, no factor being required.

From the Table I., or by quadrature of the curve in fig. 9, the work E in foot-tons realized by the expansion of 1 lb of the powder from one gravimetric volume to another is inferred; for if the average pressure is p tons per sq. in., while the gravimetric volume changes from v - 1/2[Delta]v to v + 1/2[Delta]v, a change of volume of 27.73[Delta]v cub. in., the work done is 27.73p[Delta]v inch-tons, or

(7) [Delta]E = 2.31 p[Delta]v foot-tons;

and the differences [Delta]E being calculated from the observed values of p, a summation, as in the ballistic tables, would give E in a tabular form, and conversely from a table of E in terms of v, we can infer the value of p.

On drawing off a little of the gas from the explosion vessel it was found that a gramme of cordite-gas at 0deg C. and standard atmospheric pressure occupied 700 ccs., while the same gas compressed into 5 ccs. at the temperature of explosion had a pressure of 16 tons per sq. in., or 16 x 2240 / 14.7 = 2440 atmospheres, of 14.7 lb per sq. in.; one ton per sq. in. being in round numbers 150 atmospheres.

The absolute centigrade temperature T is thence inferred from the gas equation

(8) R = pv / T = p_0v_0/273,

which, with p = 2440, v = 5, p_0 = 1, v_0 = 700, makes T = 4758, a temperature of 4485deg C. or 8105deg F.

In the heading of the 6-in. range table we find the description of the charge.

Charge: weight 13 lb 4 oz.; gravimetric density 55.01/0.504; nature, cordite, size 30.

So that P = 13.25, the G.D. = 0.504, the upper figure 55.01 denoting the specific volume of the charge measured in cubic inches per lb, filling the chamber in a state of gas, the product of the two numbers 55.01 and 0.504 being 27.73; and the chamber capacity C = 13.25 x 55.01 = 730 cub. in., equivalent to 25.8 in. or 2.15 ft. length of bore, now called the equivalent length of the chamber (E.L.C.).

If the shot was not free to move, the closed chamber pressure due to the explosion of the charge at this G.D. (= 0.5) would be nearly 49 tons per sq. in., much too great to be safe.

But the shot advances during the combustion of the cordite, and the chief problem in interior ballistics is to adjust the G.D. of the charge to the weight of the shot so that the advance of the shot during the combustion of the charge should prevent the maximum pressure from exceeding a safe limit, as shown by the maximum ordinate of the pressure curve CPD in fig. 3.

Suppose this limit is fixed at 16 tons per sq. in., corresponding in Table 1. to a G.D., 0.2; the powder-gas will now occupy a volume b = 3/2 x C = 1825 cub. in., corresponding to an advance of the shot 3/2 x 2.15 = 3.225 ft.

Assuming an average pressure of 8 tons per sq. in., the shot will have acquired energy 8 x 1/4[pi]d^2 x 3.225 = 730 foot-tons, and a velocity about v = 1020 f/s, so that the time over the 3.225 ft. at an average velocity 510 f/s is about 0.0063 sec.

Comparing this time with the experimental value of the time occupied by the cordite in burning, a start is made for a fresh estimate and a closer approximation.

Assuming, however, that the agreement is close enough for practical requirement, the combustion of the cordite may be considered complete at this stage P, and in the subsequent expansion it is assumed that the gas obeys an adiabatic law in which the pressure varies inversely as some m^{th} power of the volume.

The work done in expanding to infinity from p tons per sq. in. [v.03 p.0278] at volume b cub. in. is then pb/(m - 1) inch-tons, or to any volume B cub. in. is

(9) pb/{m - 1}[1 - (b/B)^{m-1}]

It is found experimentally that m = 1.2 is a good average value to take for cordite; so now supposing the combustion of the charge of the 6-in. is complete in 0.0063 sec., when p = 16 tons per sq. in., b = 1825 cub. in., and that the gas expands adiabatically up to the muzzle, where

(10) B/b = (216 + 25.8)/(2.5 x 25.8) = 3.75

we find the work realized by expansion is 2826 foot-tons, sufficient to increase the velocity from 1020 to 2250 f/s at the muzzle.

This muzzle velocity is about 5% greater than the 2150 f/s of the range table, so on these considerations we may suppose about 10% of work is lost by friction in the bore: this is expressed by saying that the _factor of effect_ is f = 0.9.

The experimental determination of the time of burning under the influence of the varying pressure and density, and the size of the grain, is thus of great practical importance, as thereby it is possible to estimate close limits to the maximum pressure that will be reached in the bore of a gun, and to design the chamber so that the G.D. of the charge may be suitable for the weight and acceleration of the shot. Empirical formulas based on practical experience are employed for an approximation to the result.

A great change has come over interior ballistics in recent years, as the old black gunpowder has been abandoned in artillery after holding the field for six hundred years. It is replaced by modern explosives such as those indicated on fig. 4, capable of giving off a very much larger volume of gas at a greater temperature and pressure, more than threefold as seen on fig. 8, so that the charge may be reduced in proportion, and possessing the military advantage of being nearly smokeless. (See EXPLOSIVES.)

The explosive cordite is adopted in the British service; it derives the name from its appearance as cord in short lengths, the composition being squeezed in a viscous state through the hole in a die, and the cordite is designated in size by the number of hundredths of an inch in the diameter of the hole. Thus the cordite, size 30, of the range table has been squeezed through a hole 0.30 in. diameter.

The thermochemical properties of the constituents of an explosive will assign an upper limit to the volume, temperature and pressure of the gas produced by the combustion; but much experiment is required in addition. Sir Andrew Noble has published some of his results in the _Phil. Trans._, 1905-1906 and following years.

AUTHORITIES.--Tartaglia, _Nova Scientia_ (1537); Galileo (1638); Robins, _New Principles of Gunnery_ (1743); Euler (trans. by Hugh Brown), _The True Principles of Gunnery_ (1777); Didion, Helie, Hugoniot, Vallier, Baills, &c., _Balistique_ (French); Siacci, _Balistica_ (Italian); Mayevski, Zabudski, _Balistique_ (Russian); La Llave, Ollero, Mata, &c., _Balistica_ (Spanish); Bashforth, _The Motion of Projectiles_ (1872); _The Bashforth Chronograph_ (1890); Ingalls, _Exterior and Interior Ballistics, Handbook of Problems in Direct and Indirect Fire_; Bruff, _Ordnance and Gunnery_; Cranz, _Compendium der Ballistik_ (1898); _The Official Text-Book of Gunnery_ (1902); Charbonnier, _Balistique_ (1905); Lissak, _Ordnance and Gunnery_ (1907).

(A. G. G.)

BALLOON, a globular bag of varnished silk or other material impermeable to air, which, when inflated with gas lighter than common air, can be used in aeronautics, or, according to its size, &c., for any purpose for which its ability to rise and float in the atmosphere adapts such a mechanism. "Balloon" in this sense was first used in 1783 in connexion with the invention of the brothers Montgolfier, but the word was in earlier use (derived from Ital. _ballone_, a large ball) as meaning an actual ball or ball-game, a primitive explosive bomb or firework, a form of chemical retort or receiver, and an ornamental globe in architecture; and from the appearance and shape of an air balloon the word is also given by analogy to other things, such as a "balloon skirt" in dress, "balloon training" in horticulture. (See AERONAUTICS, and FLIGHT AND FLYING).

BALLOT (from Ital. _ballotta_, dim. of _balla_, a ball), the modern method of secret-voting employed in political, legislative and judicial assemblies, and also in the proceedings of private clubs and corporations. The name comes from the use of a little ball dropped according to choice into the right receptacle; but nowadays it is used for any system of secret-voting, even though no such ball is employed. In ancient Athens, the dicasts, in giving their verdict, generally used balls of stone (_psephi_) or of metal (_sponduli_). Those pierced in the centre, or black in colour, signified condemnation; those unpierced, or white, signified acquittal. The boxes were variously arranged; but generally a brass box received both classes of votes, and a wooden box received the unused balls. In the assembly, cases of _privilegia_, such as ostracism, the naturalization of foreigners or the release of state-debtors, were decided by secret-voting. The petalism, or voting by words on olive-leaves, practised at Syracuse, may also be mentioned. At Rome the ballot was introduced to the comitia by the _Leges Tabellariae_, of which the _Lex Gabiana_ (139 B.C.) relates to the election of magistrates, the _Lex Cassia_ (137 B.C.) to _judicia populi_, and the _Lex Papiria_ (131 B.C.) to the enactment and repeal of laws. The wooden _tabellae_, placed in the _cista_ or wicker box, were marked U. R. (_uti rogas_) and A. (_antiquo_) in the case of a proposed law; L. (_libero_) and D. (_damno_) in the case of a public trial; in the case of an election, _puncta_ were made opposite the names or initials of the candidates. _Tabellae_ were also used by the Roman judices, who expressed their verdict or judgment by the letters A. (_absolvo_), C. (_condemno_), and N. L. (_non liquet_). In modern times voting by ballot is usually by some form of writing, but the use of the ball still persists (especially in clubs), and a "black ball" is the regular term for a hostile vote.

_Great Britain_.--In Great Britain the ballot was suggested for use in parliament by a political tract of the time of Charles II. It was actually used by the Scots parliament of 1662 in proceeding on the Billeting Act, a measure proposed by Middleton to secure the ostracism of Lauderdale and other political opponents who were by secret-vote declared incapable of public office. The plan followed was this: each member of parliament wrote, in a disguised hand, on a piece of paper, the names of twelve suspected persons; the billets were put in a bag held by the registrar; the bag was then sealed, and was afterwards opened and its contents ascertained in the exchequer chamber, where the billets were immediately burned and the names of the ostracised concealed on oath. The Billeting Act was repudiated by the king, and the ballot was not again heard of till 1705, when Fletcher of Saltoun, in his measure for a provisional government of Scotland by annual parliaments in the event of Queen Anne's death, proposed secret-voting to protect members from court influence. The gradual emancipation of the British parliament from the power of the crown, and the adoption of a strictly representative system of election, not only destroyed whatever reason may once have existed for the ballot in deliberative voting, but rendered it essential that such voting should be open. It was in the agitations for parliamentary reform at the beginning of the 19th century that the demand for the ballot in parliamentary elections was first seriously made. The Benthamites advocated the system in 1817. At the so-called Peterloo Massacre (1819) several banners were inscribed with the ballot. O'Connell introduced a bill on the subject in 1830; and the original draft of Lord John Russell's Reform Bill, probably on the suggestion of Lords Durham and Duncannon, provided for its introduction. Later on the historian Grote became its chief supporter in the House of Commons; and from 1833 to 1839, in spite of the ridicule cast by Sydney Smith on the "mouse-trap," and on Grote's "dagger-box, in which you stab the card of your favourite candidate with a dagger,"[1] the minority for the ballot increased from 106 to 217. In 1838 the ballot was the fourth point of the People's Charter. In the same year the abolition of the land qualification introduced rich commercial candidates to the constituencies. Lord Melbourne's cabinet declared the question open. The cause, upheld by Macaulay, Ward, Hume (in his resolutions, 1848) and Berkeley, was strengthened by the report of Lord Hartington's Select Committee [v.03 p.0279] (15th March 1870), to the effect that corruption, treating and intimidation by priests and landlords took place to a large extent at both parliamentary and municipal elections in England and Ireland; and that the ballot, if adopted, would probably not only promote tranquillity at elections, but protect voters from undue influence, and introduce greater freedom and purity in voting, provided secrecy was made inviolable except in cases where a voter was found guilty of bribery, or where an invalid vote had been given.

Meanwhile in Australia the ballot had been introduced by the Constitution Act of South Australia (1856), and in other colonies at the same date. In South Australia (Electoral Act of 1858) the returning-officer put his initials on the voting-card, which the voter was directed, under pain of nullity, to fold so that the officer might not see the vote which was indicated by a cross. In Victoria, under the Electoral Act of 1865, the officer added to his initials a number corresponding to the voter's number on the register. In Tasmania the chief peculiarity was that (as in South Australia) the card was not put directly by the voter into the box, but handed to the officer, who put it there (this being thought a security against double-voting or voting with a non-official card, and also against the voter carrying away his card). In 1869, at Manchester and Stafford in England, test-ballots were taken on the Australian system as practised in Victoria--the voting-card containing the names of all the candidates, printed in different colours (for the benefit of illiterate voters), and the voter being directed to score out the names of those he did not support, and then to place the card (covered by an official envelope) in the box. It was found at Manchester that the voting was considerably more rapid, and therefore less expensive, than under the old system; that only 80 cards out of 11,475 were rejected as informal; and that, the representatives of candidates being present to check false statements of identity, and the public outside being debarred from receiving information what voters had voted, the ballot rather decreased the risk of personation. At Manchester the cards were not numbered consecutively, as in Victoria, so that (assuming the officials to be free from corruption) no scrutiny could have detected by whom particular votes were given. At Stafford the returning-officer stamped each card before giving it to the voter, the die of the stamp having been finished only on the morning of the election. By this means the possibility was excluded of what was known as "the Tasmanian Dodge," by which a corrupt voter gave to the returning-officer, or placed in the box, a blank non-official ticket, and carried out from the booth his official card, which a corrupt agent then marked for his candidate, and gave so marked to corrupt voter No. 2 (before he entered the booth) on condition that he also would bring out his official card, and so on _ad libitum_; the agent thus obtaining a security for his bribe, unless the corrupt voter chose to disfranchise himself by making further marks on the card. At the close of 1870 the ballot was employed in the election of members for the London School Board under the Education Act of that year.

In 1872 W. E. Forster's Ballot Act introduced the ballot in all parliamentary and municipal elections, except parliamentary elections for universities; and the code of procedure prescribed by the act was adopted by the Scottish Education Board in the first School Board election (1873) under the Education (Scotland) Act 1872. The Ballot Act not only abolished public nominations of candidates, but dealt with the offence of personation and the expenses of elections.

As practised in the United Kingdom, a white paper is used on which the names of the candidates are printed in alphabetical order, the voter filling up with a X the blank on the right-hand opposite the name he votes for. The paper, before being given out, is marked by the presiding-officer on both sides with an official stamp, which is kept secret, and cannot be used for a second election within seven years. The paper is marked on the back with the same number as the counterfoil of the paper which remains with the officer. This counterfoil is also marked with the voter's number on the register, so that the vote may be identified on a scrutiny; and a mark on the register shows that the voter has received a ballot-paper. The voter folds up the paper so as to conceal his mark, but to show the stamp to the officer, and deposits it in the box, which is locked and sealed, and so constructed that papers cannot be withdrawn without unlocking it. Papers inadvertently spoiled by the voters may be exchanged, the officer preserving separately the spoiled papers. If a voter is incapacitated from blindness, or other physical cause, or makes before the officer a declaration of inability to read, or when the poll is on a Saturday declares himself a Jew, the officer causes the paper to be marked as the voter directs, and keeps a record of the transaction. A voter who claims to vote after another has voted in respect of the same qualification, obtains a (green) paper which is not placed in the box, but preserved apart as a "tendered" paper. He must, however, declare his identity and that he has not already voted. The presiding-officer at the close of the poll has to account to the returning-officer for the papers entrusted to him, the number being made up by--(1) papers in the box, (2) spoiled papers, (3) unused papers and (4) tendered papers. During the voting (for which schoolrooms and other public rooms are available, and for which a separate compartment must be provided for every 150 electors entitled to vote at a station) agents of candidates are allowed to be present in the polling-station, but they, as well as the officials, are sworn to secrecy as regards who have voted, and for whom; and they are prohibited from interfering with the voter, inducing him to show his vote, or attempting to ascertain the number on the back of the paper. These agents are also present with the returning-officer when he counts the papers and the votes, rejecting those papers--(1) which want the official mark _on the back_; (2) on which votes are given for more candidates than the voter is entitled to vote for; (3) on which anything except the number on the back is marked or written by which the voter can be identified; (4) which are unmarked, or so marked that it is uncertain for whom the vote is given. The counted and rejected papers, and also the "tendered" papers, counterfoils and marked register (which have not been opened), are, in parliamentary elections, transmitted by the returning officer to the clerk of the crown in chancery in England, or the sheriff-clerk in Scotland, who destroys them at the end of one year, unless otherwise directed by an order of the House of Commons, or of some court having jurisdiction in election petitions. Such petitions either simply dispute the accuracy of the return on the ground of miscounting, or wrongous rejection or wrongous admission of papers, in which case the court examines the counted and rejected papers; or make allegations of corruption, &c. on which it may be necessary to refer to the marked counterfoils and ascertain how bribed voters have voted. Since the elections of 1874 much discontent has been expressed, because judges have rejected papers with trifling (perhaps accidental) marks other than the X upon them, and because elections have been lost through the failure of the officer to stamp the papers. For this purpose the use has been suggested of a perforating instead of an embossing stamp, while a dark-ground paper with white voting-spaces would make _misplaced_ votes impossible.

The Ballot Act introduced several new offences, such as forging of papers or fraudulently defacing or destroying a paper or the official mark; supplying a paper without due authority; fraudulently putting into the box a non-official paper; fraudulently taking a paper out of the station without due authority; destroying, taking, opening or otherwise interfering with a box or packet of papers then in use for election purposes. These offences and attempts to commit them are punishable in the case of officers and clerks with imprisonment for two years, with or without hard labour. In other cases the term of imprisonment is six months.

The ballot was long criticized as leading to universal hypocrisy and deception; and Sydney Smith spoke of "voters, in dominos, going to the poll in sedan-chairs with closely-drawn curtains." The observed effect of a secret ballot has been, however, gradually to exterminate undue influence. The alarm of "the confessional" seems to be unfounded, as a Catholic penitent is not bound to [v.03 p.0280] confess his vote, and if he did so, it would be a crime in the confessor to divulge it.

_Continental Europe._--The ballot is largely employed in European countries. In France, where from 1840 to 1845 the ballot, or _scrutin_, had been used for deliberative voting in the chamber of deputies, its use in elections to the Corps Legislatif was carefully regulated at the beginning of the Second Empire by the Organic Decree of the 2nd of February 1852. Under this law the voting was superintended by a bureau consisting of the deputy returning-officer (called president of the section), four unpaid assessors selected from the constituency and a secretary. Each voter presents a polling-card, with his designation, date of birth and signature (to secure identity), which he had previously got at the Mairie. This the president mutilates, and the vote is then recorded by a "bulletin," which is not official, but is generally printed with a candidate's name, and given to the voter by an agent outside, the only conditions being that the bulletin shall be "sur papier blanc, sans signes exterieurs, et prepare en dehors de l'assemblee." The total number of votes given (there being only one member in each electoral district) is checked by reference to "la feuille _d'appel_ et inscription des votants," the law still supposing that each voter is publicly called on to vote. If the voter, when challenged, cannot sign his polling-card, he may call a witness to sign for him. The following classes of bulletins are rejected:--"illisibles, blancs, ne contenant pas une designation suffisante; sur lesquels les votants se sont fait connaitre; contenant le nom d'une personne n'ayant pas prete le serment prescrit" (_i.e._ of a person not nominated). Only the votes pronounced bad by the bureau in presence of representative scrutineers are preserved, in case these should be called for during the "Session pour verification des Pouvoirs." Practically the French ballot did not afford secrecy, for you might observe what bulletin the voter took from the agent, and follow him up the _queue_ into the polling-place; but the determined voter might conceal his vote even from the undue influence of government by scratching out the printed matter and writing his vote. This was always a good vote and scrutiny of good votes was impossible. The ballot is still used in the elections to the National Assembly, but in the Assembly itself only in special cases, as _e.g._ in the election of a "rapporteur." Under the law of 10th August 1871 the conseils generaux (departmental councils) are elected by ballot.

In Piedmont the ballot formed part of the free constitutional government introduced by Charles Albert in March 1848; it was extended to Italy in 1861. Voting for the Italian chamber of deputies takes place under the law of 20th November 1859, and in public halls (not booths), to which admission is gained by showing a certificate of inscription, issued by the mayor to each qualified voter. A stamped blue official paper, with a memorandum of the law printed on the back (_bolletino spiegato_), is then issued to the elector; on this he writes the name of a candidate (there being equal electoral colleges) or, in certain exceptional cases, gets a confidential friend to do so, and hands the paper folded-up to the president of the bureau, who puts it in the box (_urna_), and who afterwards presides at the public "squittinio dei suffragi." Greece is the only European country in which the ball-ballot is used. The voting takes place in the churches, each candidate has a box on which his name is inscribed, one half (white) being also marked "yes," the other half (black) "no." The voter, his citizenship or right to vote in the eparchy being verified, receives one ball or leaden bullet for each candidate from a wooden bowl, which a clerk carries from box to box. The voter stretches his arm down a funnel, and drops the ball into the "yes" or "no" division. The vote is secret, but there is apparently no check on "yes" votes being given for all the candidates, and the ball or bullet is imitable.

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Encyclopaedia Britannica, 11th Edition, "Baconthorpe" to "Bankruptcy"Chapter XX: Front Matter (20)

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