# 1911 Encyclopædia Britannica/Ballistics

BALLISTICS (from the Gr. βάλλειν, to throw), the science of throwing warlike missiles or projectiles. It is now divided into two parts:—Exterior Ballistics, in which the motion of the projectile is considered after it has received its initial impulse, when the projectile is moving freely under the influence of gravity and the resistance of the air, and it is required to determine the circumstances so as to hit a certain object, with a view to its destruction or perforation; and Interior Ballislics, in which the pressure of the powder-gas is analysed in the bore of the gun, and the investigation is carried out of the requisite charge of powder to secure the initial velocity of the projectile, without straining the gun unduly. The calculation of the stress in the various parts of the gun due to the powder pressure is dealt with in the article Ordnance.

I. Exterior Ballistics.

In the ancient theory due to Galileo, the resistance of the air is ignored, and, as shown in the article on Mechanics (§ 13), the trajectory is now a parabola. But this theory is very far from being of practical value for most purposes of gunnery; so that a first requirement is an accurate experimental knowledge of the resistance of the air to the projectiles employed, at all velocities useful in artillery. The theoretical assumptions of Newton and Euler (hypotheses magis mathematicae quam naturales) of a resistance varying as some simple power of the velocity, for instance, as the square or cube of the velocity (the quadratic or cubic law), lead to results of great analytical complexity, and are useful only for provisional extrapolation at high or low velocity, pending further experiment.

The foundation of our knowledge of the resistance of the air, as employed in the construction of ballistic tables, is the series of experiments carried out between 1864 and 1880 by the Rev. F. Bashforth, B.D. (Report on the Experiments made with the Bashforth Chronograph, &c., 1865-1870; Final Report, &c., 1878-1880; The Bashforth Chronograph, Cambridge, 1890). According to these experiments, the resistance of the air can be represented by no simple algebraical law over a large range of velocity. Abandoning therefore all a priori theoretical assumption, Bashforth set to work to measure experimentally the velocity of shot and the resistance of the air by means of equidistant electric screens furnished with vertical threads or wire, and by a chronograph which measured the instants of time at which the screens were cut by a shot flying nearly horizontally. Formulae of the calculus of finite differences enable us from the chronograph records to infer the velocity and retardation of the shot, and thence the resistance of the air.

As a first result of experiment it was found that the resistance of similar shot was proportional, at the same velocity, to the surface or cross section, or square of the diameter. The resistance R can thus be divided into two factors, one of which is d², where d denotes the diameter of the shot in inches, and the other factor is denoted by p, where p is the resistance in pounds at the same velocity to a similar 1-in, projectile; thus R =d²p, and the value of p, for velocity ranging from 1600 to 2150 ft. per second (f/s) is given in the second column of the extract from the abridged ballistic table below.

These values of p refer to a standard density of the air, of 534·22 grains per cubic foot, which is the density of dry air at sea-level in the latitude of Greenwich, at a temperature of 62° F. and a barometric height of 30 in.

But in consequence of the humidity of the climate of England it is better to suppose the air to be (on the average) two-thirds saturated with aqueous vapour, and then the standard temperature will be reduced to 60°F., so as to secure the same standard density; the density of the air being reduced perceptibly by the presence of the aqueous vapour.

It is further assumed, as the result of experiment, that the resistance is proportional to the density of the air; so that if the standard density changes from unity to any other relative density denoted by τ, then R = τd²p, and τ is called the coefficient of tenuity.

The factor τ becomes of importance in long range high angle fire, where the shot reaches the higher attenuated strata of the atmosphere; on the other hand, we must take τ about 800 in a calculation of shooting under water.

The resistance of the air is reduced considerably in modern projectiles by giving them a greater length and a sharper point, and by the omission of projecting studs, a factor κ, called the coefficient of shape, being introduced to allow for this change.

For a projectile in which the ogival head is struck with a radius of 2 diameters, Bashforth puts κ= 0·975; on the other hand, for a flat-headed projectile, as required at proof-butts, κ= 1·8, say 2 on the average.

For spherical shot κ is not constant, and a separate ballistic table must be constructed; but κ may be taken as 1·7 on the average.

Lastly, to allow for the superior centering of the shot obtainable with the breech-loading system, Bashforth introduces a factor σ, called the coefficient of steadiness.

This steadiness may vary during the flight of the projectile, as the shot may be unsteady for some distance after leaving the muzzle, afterwards steadying down, like a spinning-top. Again, σ may increase as the gun wears out, after firing a number of rounds.

Collecting all the coefficients, τ, κ, σ, into one, we put

(1) ${\displaystyle R=nd^{2}p=nd^{2}f(v),\,}$ where
(2) ${\displaystyle n=\kappa \sigma \tau ,\,}$

and n is called the coefficient of reduction.

By means of a well-chosen value of n, determined by a few experiments, it is possible, pending further experiment, with the most recent design, to Utilize Bashforth's experimental results carried out with old-fashioned projectiles fired from muzzle-loading guns. For instance, n= 0·8 or even less is considered a good average for the modern rifle bullet.

Starting with the experimental values of p, for a standard projectile, fired under standard conditions in air of standard density, we proceed to the construction of the ballistic table. We first determine the time t in seconds required for the velocity of a shot, d inches in diameter and weighing w lb to fall from any initial velocity V(f/s) to any final velocity v(f/s). The shot is supposed to move horizontally, and the curving effect of gravity is ignored.

If Δt seconds is the time during which the resistance of the air, R lb, causes the velocity of the shot to fall Δv(f/s), so that the velocity drops from v+½Δv to v-½Δv in passing through the mean velocity v, then

 (3) ${\displaystyle R\Delta t\,}$ ${\displaystyle =\,}$ ${\displaystyle {\mbox{loss of momentum in second-pounds,}}\,}$ ${\displaystyle =\,}$ ${\displaystyle w(v+{\frac {1}{2}}\Delta v)/g-w(v-{\frac {1}{2}}\Delta v)/g=w\Delta v/gp}$

so that with the value of R in (1),

(4) ${\displaystyle \Delta t=w\Delta v/nd^{2}pg.\,}$

We put

(5) ${\displaystyle w/nd^{2}=C,\,}$

and call C the ballistic coefficient (driving power) of the shot, so that

(6) ${\displaystyle \Delta =C\Delta T,{\mbox{ where}}\,}$
(7) ${\displaystyle \Delta T=\Delta v/gp,\,}$

and ΔT is the time in seconds for the velocity to drop Δv of the standard shot for which C=1, and for which the ballistic table is calculated.

Since p is determined experimentally and tabulated as a function of v, the velocity is taken as the argument of the ballistic table; and taking δv=10, the average value of p in the interval is used to determine ΔT.

Denoting the value of T at any velocity v by T (v), then

(8) ${\displaystyle T(v)\,}$=sum of all the preceding values of ΔT plus an arbitrary constant, expressed by the notation
(9) ${\displaystyle T(v)=\Sigma (\Delta v)/gp+{\mbox{ a constant, or}}\int dv/gp+{\mbox{ a constant}},}$ in which p is supposed known as a function of v.

The constant may be any arbitrary number, as in using the table the difference only is required of two tabular values for an initial velocity V and final velocity v; and thus

(10) ${\displaystyle T(V)-T(v)=\Sigma _{v}^{V}\Delta v/gp{\mbox{ or }}\int dv/gp;}$

and for a shot whose ballistic coefficient is C

(11) ${\displaystyle t=C\left\lbrack T(V)T(v)\right\rbrack .}$

To save the trouble of proportional parts the value of T(v) for unit increment of v is interpolated in a full-length extended ballistic table for T.

Next, if the shot advances a distance Δs ft. in the time Δt, during which the velocity falls from vv to vv we have

 (12) ${\displaystyle R\Delta s\,}$ ${\displaystyle =\,}$ ${\displaystyle {\mbox{loss of kinetic energy in foot-pounds,}}\,}$ ${\displaystyle =\,}$ ${\displaystyle w(v+{\frac {1}{2}}\Delta v)^{2}/g-w(v-{\frac {1}{2}}\Delta v)^{2}/g=wv\Delta v/g,{\mbox{ so that}}}$
(13) ${\displaystyle \Delta s=wv\Delta v/nd^{2}pg=C\Delta S,{\mbox{ where}}\,}$
(14) ${\displaystyle \Delta S=v\Delta v/gp=v\Delta T,\,}$

and ΔS is the advance in feet of a shot for which C=1, while the velocity falls Δv in passing through the average velocity v.

Denoting by S(v) the sum of all the values of ΔS up to any assigned velocity v,

(15) ${\displaystyle S(v)=\Sigma (\Delta S)+{\mbox{ a constant}}\,}$, by which S(v) is calculated from ΔS, and then between two assigned velocities V and v,
(16) ${\displaystyle S(V)-S(v)={\begin{matrix}\sum _{v}^{V}\Delta T\end{matrix}}=\sum {\frac {v\Delta v}{gp}}{\mbox{or}}\int _{v}^{V}{\frac {vdv}{gp}},}$

and if s feet is the advance of a shot whose ballistic coefficient is C,

(17) ${\displaystyle s=C\left\lbrack S(V)-S(v)\right\rbrack .\,}$

In an extended table of S, the value is interpolated for unit increment of velocity.

A third table, due to Sir W.D. Niven, F.R.S., called the degree table, determines the change of direction of motion of the shot while the velocity changes from V to v, the shot flying nearly horizontally.

To explain the theory of this table, suppose the tangent at the point of the trajectory, where the velocity is v, to make an angle i radians with the horizon.

Resolving normally in the trajectory, and supposing the resistance of the air to act tangentially,

(18) ${\displaystyle v(di/dt)=g\cos i\,}$, where di denotes the infinitesimal decrement of i in the infinitesimal increment of time dt. In a problem of direct fire, where the trajectory is flat enough for cos i to be undistinguishable from unity, equation (16) becomes
(19) ${\displaystyle v(di/dt)=g,{\mbox{ or }}di/dt=g/v;\,}$

so that we can put

(20) ${\displaystyle \Delta i/\Delta t=g/v,\,}$

if v denotes the mean velocity during the small finite interval of time Δt, during which the direction of motion of the shot changes through Δi radians.

If the inclination or change of inclination in degrees is denoted by δ or Δδ,

(21) ${\displaystyle \delta /180=i/\pi ,{\mbox{ so that}}\,}$
(22) ${\displaystyle \Delta \delta ={\frac {180}{\pi }}\Delta i={\frac {180g}{\pi }}{\frac {\Delta t}{v}};}$

and if δ and i change to D and I for the standard projectile,

(23) ${\displaystyle \Delta I=g{\frac {\Delta T}{v}}={\frac {\Delta v}{vp}},\Delta D={\frac {180}{\pi }}{\frac {\Delta T}{v}},{\mbox{ and}}}$
(24) ${\displaystyle I(V)-I(v)=\sum _{v}^{V}{\frac {\Delta v}{vp}}{\mbox{ or }}\int _{v}^{V}{\frac {dv}{vp}},\,\,D(V)-D(v)={\frac {180}{\pi }}\left\lbrack I(V)-I(v)\right\rbrack .}$

The differences ΔD and ΔI are thus calculated, while the values of D(v) and I(v) are obtained by summation with the arithmometer, and entered in their respective columns.

For some purposes it is preferable to retain the circular measure, i radians, as being undistinguishable from sin i and tan i when i is small as in direct fire.

The last function A, called the altitude function, will be explained when high angle fire is considered.

These functions, T, S, D, I, A, are shown numerically in the following extract from an abridged ballistic table, in which the velocity is taken as the argument and proceeds by an increment of 10 f/s; the column for p is the one determined by experiment, and the remaining columns follow by calculation in the manner explained above. The initial values of T, S, D, I, A must be accepted as belonging to the anterior portion of the table.

In any region of velocity where it is possible to represent p with sufficient accuracy by an empirical formula composed of a single power of v, say vm, the integration can be effected which replaces the summation in (10), (16), and (24); and from an analysis of the Krupp experiments Colonel Zabudski found the most appropriate index m in a region of velocity as given in the following table, and the corresponding value of gp, denoted by f(v) or vm/k or its equivalent Cr, where r is the retardation.

ABRIDGED BALLISTIC TABLE.
v. p. ΔT. T. ΔS. S. ΔD. D. ΔI. I. ΔA. A.
f/s
1600 11.416 .0271 27.5457 43.47 18587.00 .0311 49.7729 .000543 .868675 37.77 8470.36
1610 11.540 .0268 27.5728 43.27 18630.47 .0306 49.8040 .000534 .869218 37.63 8508.13
1620 11.662 .0265 27.5996 43.08 18673.74 .0301 49.8346 .000525 .869752 37.48 8545.76
1630 11.784 .0262 27.6261 42.90 18716.82 .0296 49.8647 .000517 .870277 37.35 8583.24
1640 11.909 .0260 27.6523 42.72 18759.72 .0291 49.8943 .000508 .870794 37.21 8620.59
1650 12.030 .0257 27.6783 42.55 18802.44 .0287 49.9234 .000500 .871302 37.09 8657.80
1660 12.150 .0255 27.7040 42.39 18844.99 .0282 49.9521 .000492 .871802 36.96 8694.89
1670 12.268 .0252 27.7295 42.18 18887.38 .0277 49.9803 .000484 .872294 36.80 8731.85
1680 12.404 .0249 27.7547 41.98 18929.56 .0273 50.0080 .000476 .872778 36.65 8768.65
1690 12.536 .0247 27.7796 41.78 18971.54 .0268 50.0353 .000468 .873254 36.50 8805.30
1700 12.666 .0244 27.8043 41.60 19013.32 .0264 50.0621 .000461 .873722 36.35 8841.80
1710 12.801 .0242 27.8287 41.41 19054.92 .0260 50.0885 .000453 .874183 36.21 8878.15
1720 12.900 0.239 27.8529 41.23 19096.33 .0256 50.1145 .000446 .874636 36.07 8914.36
1730 13.059 .0237 27.8768 41.06 19137.56 .0252 50.1401 .000439 .875082 35.94 8950.43
1740 13.191 .0234 27.9005 40.90 19178.62 .0248 50.1653 .000432 .875521 35.81 8986.37
1750 13.318 .0232 27.9239 40.69 19219.52 .0244 50.1901 .000425 .875953 35.65 9022.18
1760 13.466 .0230 27.9471 40.53 19260.21 .0240 50.2145 .000419 .876378 35.53 9057.83
1770 13.591 .0227 27.9701 40.33 19300.74 .0236 50.2385 .000412 .876797 35.37 9093.36
1780 13.733 .0225 27.9928 40.19 19341.07 .0233 50.2621 .000406 .877209 35.26 9128.73
1790 13.862 .0223 28.0153 40.00 19381.26 .0229 50.2854 .000400 .877615 35.11 9163.99
1800 14.002 .0221 28.0376 39.81 19421.26 .0225 50.3083 .000393 .878015 34.96 9199.10
1810 14.149 .0219 28.0597 39.68 19461.07 .0222 50.3308 .000388 .878408 34.86 9234.06
1820 14.269 .0217 28.0816 39.51 19500.75 .0219 50.3530 .000382 .878796 34.73 9268.92
1830 14.414 .0214 28.1033 39.34 19540.26 .0216 50.3749 .000376 .879178 34.59 9303.65
1840 14.552 .0212 28.1247 39.17 19579.60 .0212 50.3965 .000370 .879554 34.46 9338.24
1850 14.696 .0210 28.1459 39.01 19618.77 .0209 50.4177 .000365 .879924 34.33 9372.70
1860 14.832 .0209 28.1669 38.90 19657.78 .0206 50.4386 .000360 .880289 34.25 9407.03
1870 14.949 .0207 28.1878 38.75 19696.68 .0203 50.4592 .000355 .880649 34.14 9441.28
1880 15.090 .0205 28.2085 38.61 19735.43 .0200 50.4795 .000350 .881004 34.02 9475.42
1890 15.224 .0203 28.2290 38.46 19774.04 .0198 50.4995 .000345 .881354 33.91 9509.44
1900 15.364 .0201 28.2493 38.32 19812.50 .0195 50.5193 .000340 .881699 33.80 9543.35
1910 15.496 .0199 28.2694 38.19 19850.82 .0192 50.5388 .000335 .882039 33.69 9577.15
1920 15.656 .0197 28.2893 38.01 19889.01 .0189 50.5580 .000330 .882374 33.55 9610.84
1930 15.809 .0196 28-3090 37.83 19927.02 .0186 50.5769 .000325 .882704 33.40 9644.39
1940 15.968 .0194 28.3286 37.66 19964.85 .0184 50.5955 .000320 .883029 33.26 9677.79
1950 16.127 .0192 28.3480 37.48 20002.51 .0181 50.6139 .000316 .883349 33.12 9711.05
1960 16.302 .0190 28.3672 37.26 20039.99 .0178 50.6320 .000311 .883665 32.94 9744.17
1970 16.484 .0187 28.3862 36.99 20077.25 .0175 50.6498 .000305 .883976 32.71 9777.11
1980 16.689 .0185 28.4049 36.73 20114.24 .0172 50.6673 .000300 .884281 32.48 9809.82
1990 16.888 .0183 28.4234 36.47 20150.97 .0169 50.6845 .000295 .884581 32.26 9842.30
2000 17.096 .0181 28.4417 36.21 20187.44 .0166 50.7014 .000290 .884876 32.05 9874.56
2010 17.305 .0178 28.4598 35.95 20223.65 .0163 50.7180 .000285 .885166 31.83 9906.61
2020 17.515 .0176 28.4776 35.65 20259.60 .0160 50.7343 .000280 .885451 31.57 9938.44
2030 17.752 .0174 28.4952 35.35 20295.25 .0158 50.7503 .000275 .885731 31.32 9970.01
2040 17.990 .0171 28.5126 35.06 20330.60 .0155 50.7661 .000270 .886006 31.07 10001.33
2050 18.229 .0169 28.5297 34.77 20365.66 .0152 50.7816 .000265 .886276 30.82 10032.40
2060 18.463 .0167 28.5466 34.49 20400.43 .0149 50.7968 .000260 .886541 30.58 10063.33
2070 18.706 .0165 28.5633 34.21 20434.92 .0147 50.8117 .000256 .886801 30.34 10093.80
2080 18.978 .0163 28.5798 33.93 20469.13 .0144 50.8264 .000251 .887057 30.10 10124.14
2090 19.227 .0160 28.5961 33.60 20503.06 .0141 50.8408 .000247 .887308 29.82 10154.24
2100 19.504 .0158 28.6121 33.34 20536.66 .0139 50.8549 .000242 .887555 29.59 10184.06
2110 19.755 .0156 28.6279 33.02 20570.00 .0136 50.8688 .000238 .887797 29.32 10213.65
2120 20.010 .0154 28.6435 32.76 20603.02 .0134 50.8824 .000234 .888035 29.10 10242.97
2130 20.294 .0152 28.6589 32.50 20635.78 .0132 50.8958 .000230 .888269 28.88 10272.07
2140 20.551 .0150 28.6741 32.25 20688.28 .0129 50.9090 .000226 .888499 28.66 10300.95
2150 20.811 .0149 28.6891 32.00 20700.53 .0127 50.9219 .000222 .888725 28.44 10329.61
v. m. log k. Cr=gp=f(v)=vm/k.
3600 1.55 2.3909520 ${\displaystyle v^{1.55}\times \log ^{-1}{\bar {3}}.6090480}$
2600 1.7 2.9038022 ${\displaystyle v^{1.7}\times \log ^{-1}{\bar {3}}.0961978}$
1800 2 3.8807404 ${\displaystyle v^{2}\times \log ^{-1}{\bar {4}}.1192596}$
1370 3 7.0190977 ${\displaystyle v^{3}\times \log ^{-1}{\bar {4}}.9809023}$
1230 5 13.1981288 ${\displaystyle v^{5}\times \log ^{-1}{\bar {14}}.8018712}$
970 3 7.2265570 ${\displaystyle v^{3}\times \log ^{-1}{\bar {8}}.7734430}$
790 2 4.3301086 ${\displaystyle v^{2}\times \log ^{-1}{\bar {5}}.6698914}$

The numbers have been changed from kilogramme-metre to poundfoot units by Colonel Ingalls, and employed by him in the calculation of an extended ballistic table, which can be compared with the result of the abridged table. The calculation can be carried out in each region of velocity from the formulae:—

(25) ${\displaystyle T(V)-T(v)=k\int _{v}^{V}v^{-m}dv,\ S(V)-S(v)=k\int _{v}^{V}v^{m+1}dv}$
${\displaystyle I(V)-I(v)=gk\int _{v}^{V}v^{-m-1}dv,}$

and the corresponding integration.

The following exercises will show the application of the ballistic table. A slide rule should be used for the arithmetical operations, as it works to the accuracy obtainable in practice.

Example 1.- Determine the time t sec. and distance s ft. in which the velocity falls from 2150 to 1600 f/s

(a) of a 6-in. shot weighing 100lb, taking n=0.96,
(b) of a rifle bullet, 0.303-in. calibre, weighing half an ounce, taking n=0.8.
V. v. T(V). T(v). t/C. S(V). S(v). s/C.
2150 1600 28.6891 27.5457 1.1434 20700.53 18587.00 2113.53
d. w. C. t/C. t. s/C. s.
(a) 6 100 2.894 1.1434 3.307 2113.53 6114 (2038yds.)
(b) 0.303 1/32 0.426 1.1434 0.486 2113.53 900 (300yds.)

Example 2.- Determine the remaining velocity v and time of flight t over a range of 1000 yds. of the same two shot, fired with the same muzzle velocity V=2150f/s.

S. s/C. S(V). S(v). v. T(V). T(v). t/C. t.
(a) 3000 1037 20700.53 19663.53 1861 28.6891 28.1690 0.5201 1.505
(b) 3000 7050 20700.53 13650.53 920* 28.6891 23.0803 5.6088 2.387
* These numbers are taken from a part omitted here of the abridged ballistic table.

In the calculation of range tables for direct fire, defined officially as "fire from guns with full charge at elevation not exceeding 15°," the vertical component of the resistance of the air may be ignored as insensible, and the actual velocity and its horizontal component, or component parallel to the line of sight, are undistinguishable.

 Fig. 1.

The equations of motion are now, the co-ordinates x and y being measured in feet,

(26) ${\displaystyle {\frac {d^{2}x}{dt^{2}}}=-{\mbox{r}}r=-{\frac {gp}{C}},}$
(27) ${\displaystyle {\frac {d^{2}x}{dt^{2}}}=-g.}$

The first equation leads, as before, to

(28) ${\displaystyle t=C\left\{T(V)-T(v)\right\},}$
(29) :(28) ${\displaystyle x=C\left\{S(V)-S(v)\right\},}$

The integration of (24) gives

(30) ${\displaystyle {\frac {dy}{dt}}={\mbox{constant}}-gt=g\left({\frac {1}{2}}T-t\right),}$

if T denotes the whole time of flight from O to the point B (fig. 1), where the trajectory cuts the line of sight; so that ½T is the time to the vertex A, where the shot is flying parallel to OB.

Integrating (27) again,

(31) ${\displaystyle y=g\left({\frac {1}{2}}T-{\frac {1}{2}}t^{2}\right)=3/4gt(Tt)={\frac {1}{2}}gt(T-t);}$

and denoting T-t by t', and taking g=32f/s2,

(32) ${\displaystyle y=16tt^{\prime },\,}$

which is Colonel Sladen's formula, employed in plotting ordinates of a trajectory.

At the vertex A, where y=H, we have t=t'=½T, so that

(33) ${\displaystyle H={\frac {1}{8}}gT^{2},}$

which for practical purposes, taking g=32, is replaced by

(34) ${\displaystyle H=4T^{2},{\mbox{ or }}(2T)^{2}.\,}$

Thus, if the time of flight of a shell is 5 sec., the height of the vertex of the trajectory is about 100 ft.; and if the fuse is set to burst the shell one-tenth of a second short of its impact at B, the height of the burst is 7.84, say 8 ft.

The line of sight Ox, considered horizontal in range table results, may be inclined slightly to the horizon, as in shooting up or down a moderate slope, without appreciable modification of (28) and (29), and y or PM is still drawn vertically to meet OB in M.

Given the ballistic coefficient C, the initial velocity V, and a range of R yds. or X=3R ft., the final velocity v is first calculated from (29) by

(35) ${\displaystyle S(v)=S(V)-X/C,\,}$

and then the time of flight T by

(36) ${\displaystyle T=C\left\{T(V)-T(v)\right\}.}$

Denoting the angle of departure and descent, measured in degrees and from the line of sight OB by φ and β, the total deviation in the range OB is (fig. 1)

(37) ${\displaystyle \delta =\phi +\beta =C\left\{D(V)-D(v)\right\}.}$

To share the δ between φ and β, the vertex A is taken as the point of half-time (and therefore beyond half-range, because of the continual diminution of the velocity), and the velocity v0 at A is calculated from the formula

(38) ${\displaystyle T(v_{0})=T(V)-{\frac {{\frac {1}{2}}T}{C}}={\frac {1}{2}}\left\{T(V)+T(v)\right\};}$

and now the degree table for D(v) gives

(39) ${\displaystyle \phi =C\left\{D(V)-D(v_{0})\right\},}$
(40) ${\displaystyle \beta =C\left\{D(v_{0})-D(v)\right\},}$

This value of φ is the tangent elevation (T.E); the quadrant elevation (Q.E.) is φ-S, where S is the angular depression of the line of sight OB; and if O is h ft. vertical above B, the angle S at a range of R yds. is given by

(41) ${\displaystyle \sin S=h/3R,\,}$

or, for a small angle, expressed in minutes, taking the radian as 3438',

(42) ${\displaystyle S=1146h/R.\,}$

So also the angle β must be increased by S to obtain the angle at which the shot strikes a horizontal plane — the water, for instance.

A systematic exercise is given here of the compilation of a range table by calculation with the ballistic table; and it is to be compared with the published official range table which follows.

A discrepancy between a calculated and tabulated result will serve to show the influence of a slight change in the coefficient of reduction n, and the muzzle velocity V.

Example 3.- Determine by calculation with the abridged ballistic table the remaining velocity v, the time of flight t, angle of elevation φ, and descent β of this 6-in. gun at ranges 500, 1000, 1500, 2000 yds., taking the muzzle velocity V=2150 f/s, and a coefficient of reduction n=0.96. [For Table see p.274.]

An important problem is to determine the alteration of elevation for firing up and down a slope. It is found that the alteration of the tangent elevation is almost insensible, but the quadrant elevation requires the addition or subtraction of the angle of sight.

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.

Range. s. s/C. S(v). v. T(v). t/C. t. T(v0). v0. D(v0) φ/C. φ. β/C. β.
0 0 0 20700.53 2150 28.6891 0.0000 0.000 28.6891 2150 50.9219 0.0000 0.000 0.0000 0.000
500 1500 518 20182.53 1999 28.4399 0.2492 0.720 28.5645 2071 50.8132 0.1087 0.315 0.1135 0.328
1000 3000 1036 19664.53 1862 28.1711 0.5180 1.497 28.4301 1994 50.6913 0.2306 0.666 0.2486 0.718
1500 4500 1554 19146.53 1732 27.8815 0.8076 2.330 28.2853 1918 50.5542 0.3677 1.062 0.4085 1.181
2000 6000 2072 10628.53 1610 27.5728 1.1163 3.225 28.1310 1843 50.4029 0.5190 1.500 0.5989 1.734
RANGE TABLE FOR 6-INCH GUN.
 Charge ${\displaystyle {\Bigg \{}}$ weight, 13 lb 4 oz. ${\displaystyle {\frac {55.01}{0.504}}.}$ gravimetric density, nature, cordite, size 30.
${\displaystyle {\Bigg |}}$
 Projectile ${\displaystyle {\bigg \{}}$ Palliser shot, Shrapnel shell. Weight, 100 lb.
${\displaystyle {\Bigg |}}$
 Muzzle velocity, 2154 f/s. Nature of mounting, pedestal. Jump, nil.
Remaining Velocity To strike an object 10 ft. high range must be known to Slope of Descent. 5' elevation or depression alters point of impact. Elevation. Range. Fuse scale for T. and P. middle No. 54 Marks I., II., or III. 50% of rounds should fall in. Time of Flight. Penetration into Wrought Iron.
Range. Laterally or Vertically. Length. Breadth. Height.
f/s yds. 1 in. yds. yds. ° ' 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 15°," and "curved fire is fire from howitzers at all angles of elevation not exceeding 15°". 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) ${\displaystyle {\frac {d^{2}x}{dt^{2}}}=-r\cos i=-r{\frac {dx}{ds}},}$
(44) ${\displaystyle {\frac {d^{2}y}{dt^{2}}}=-r\sin i-g=-r{\frac {dy}{ds}}-g,}$

and eliminating r,

(45) ${\displaystyle {\frac {dx}{dt}}{\frac {d^{2}y}{dt^{2}}}-{\frac {dy}{dt}}{\frac {d^{2}x}{dt^{2}}}=-g{\frac {dx}{dt}},}$

and this, in conjunction with

(46) ${\displaystyle \tan i={\frac {dy}{dx}}={\frac {dy}{dt}}{\Bigg /}{\frac {dx}{dt}},}$
(47) ${\displaystyle \sec ^{2}i{\frac {di}{dt}}=\left({\frac {dx}{dt}}{\frac {d^{2}y}{dt^{2}}}-{\frac {dy}{dt}}{\frac {d^{2}x}{dt^{2}}}\right){\Bigg /}\left({\frac {dx}{dt}}\right)^{2},}$

reduces to

(48) ${\displaystyle {\frac {di}{dt}}=-{\frac {g}{v}}\cos i,{\mbox{ or }}{\frac {d\tan i}{dt}}=-g{\frac {g}{v\cos i}},}$

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) ${\displaystyle v\cos i=q,\,}$

equation (43) becomes

(50) ${\displaystyle dq/dt=-r\cos i,\,}$

and therefore by (48)

(51) ${\displaystyle {\frac {dq}{di}}={\frac {dq}{dt}}{\frac {dt}{di}}={\frac {rv}{g}}}$

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

(52) ${\displaystyle Cr=f(v)\,}$

and now

(53) ${\displaystyle {\frac {dq}{di}}={\frac {vf(v)}{Cg}},}$

an equation connecting q and i.

Now, since v=q sec i

(54) ${\displaystyle {\frac {dt}{dq}}=-C{\frac {\sec i}{f(q\sec i)}},}$

and multiplying by dx/dt or q,

(55) ${\displaystyle {\frac {dx}{dq}}=-{\frac {Cq\sec i}{f(q\sec i)}},}$

and multiplying by dx/dx or tan i,

(56) ${\displaystyle {\frac {dy}{dq}}=-{\frac {Cq\sec i\tan i}{f(q\sec i)}};}$

also

(57) ${\displaystyle {\frac {di}{dq}}={\frac {Cg}{q\sec i.f(q\sec i)}},}$
(58) ${\displaystyle {\frac {d\tan i}{dq}}={\frac {Cg\sec i}{q.f(q\sec i)}},}$

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)=v2/k or v3/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, η, cos η, and sec η, 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 η we introduce Siacci's pseudo-velocity u defined by

(59) ${\displaystyle u=q\sec \eta ,\,}$

so that u is a quasi-component parallel to the mean direction of the tangent, say the direction of the chord of the arc. Integrating from any initial pseudo-velocity U,

(60) ${\displaystyle t=C\int _{u}^{U}{\frac {du}{f(u)}},}$
(61) ${\displaystyle x=C\cos \eta \int {\frac {udu}{f(u)}},}$
(62) ${\displaystyle y=C\sin \eta \int {\frac {udu}{f(u)}},}$

and supposing the inclination i to change from φ to θ radians over the arc.

(63) ${\displaystyle \phi -\theta =Cg\cos \eta \int {\frac {du}{uf(u)}},}$
(64) ${\displaystyle \tan \phi -\tan \theta =Cg\sec \eta \int {\frac {du}{uf(u)}}.}$

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) ${\displaystyle \int _{u}^{U}{\frac {du}{f(u)}}=\int {\frac {du}{gp}}}$ ${\displaystyle =T(U)-T(u),\,}$ (66) ${\displaystyle \int _{u}^{U}{\frac {udu}{f(u)}}}$ ${\displaystyle =S(U)-S(u),\,}$ (67) ${\displaystyle \int _{u}^{U}{\frac {gdu}{uf(u)}}}$ ${\displaystyle =I(U)-I(u),\,}$

and therefore

(68) ${\displaystyle t=C\left\lbrack T(U)-T(u)\right\rbrack ,}$
(69) ${\displaystyle x=C\cos \eta \left\lbrack S(U)-S(u)\right\rbrack ,}$
(70) ${\displaystyle y=C\sin \eta \left\lbrack S(U)-S(u)\right\rbrack ,}$
(71) ${\displaystyle \phi -\theta =C\cos \eta \left\lbrack I(U)-I(u)\right\rbrack ,}$
(72) ${\displaystyle \tan \phi -\tan \theta =C\sec \eta \left\lbrack I(U)-I(u)\right\rbrack ,}$

while, expressed in degrees,

(73) ${\displaystyle \phi ^{\circ }-\theta ^{\circ }=C\cos \eta \left\lbrack D(U)-D(u)\right\rbrack ,}$

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 η cannot be exactly the same mean angle in all these equations; but if η is the same in (69) and (70),

(74) ${\displaystyle y/x=\tan \eta ,\,}$

so that η 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 η to be known with accuracy, as 1% variation in η causes more than 1% variation in tan η.

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 η or tan η but in which η occurs only in the form cos η or sec η, which varies very slowly for moderate values of η, so that η need not be calculated with any great regard for accuracy, the arithmetic mean ½(φ+θ) of φ and θ being near enough for η over any arc φ-θ of moderate extent.

Now taking equation (72), and replacing tan θ, as a variable final tangent of an angle, by tan ior dy/dx,

(75) ${\displaystyle \tan \phi -{\frac {dy}{dx}}=C\sec \eta \left\lbrack I(U)-I(u)\right\rbrack ,}$

and integrating with respect to x over the arc considered,

(76) ${\displaystyle x\tan \phi -y=C\sec \eta \left\lbrack xI(U)-\int _{0}^{\phi }I(u)dx\right\rbrack .}$

But

 (77) ${\displaystyle \int _{0}^{x}I(u)dx}$ ${\displaystyle =\int _{u}^{U}I(u){\frac {dx}{du}}du}$ ${\displaystyle =C\cos \eta \int _{u}^{U}I(u){\frac {udu}{gf(u)}}}$ ${\displaystyle =C\cos \eta \left\lbrack A(U)-A(u)\right\rbrack }$

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

(78) ${\displaystyle \Delta A=I(u){\frac {u\Delta u}{gp}}=I(u)\Delta S,}$

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

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

(79) ${\displaystyle \tan \phi -{\frac {y}{x}}=C\sec \eta \left\lbrack I(U)-{\frac {A(U)-A(u)}{S(U)-S(u)}}\right\rbrack }$

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 φ, with velocity , the curvature of the arc φθ is first settled upon, and now

(80) ${\displaystyle \eta ={\frac {1}{2}}(\phi +\theta )}$

is a good first approximation for η.

Now calculate the pseudo-velocity uφ from

(81) ${\displaystyle u_{\phi }=v_{\phi }\cos \phi \sec \eta ,\,}$

and then, from the given values of φ, and θ, calculate uθ from either of the formulae of (72) or (73):—

(82) ${\displaystyle I(u_{\theta })-I(u_{\phi })-{\frac {\tan \phi -\tan \theta }{C\sec \eta }},}$
(83) ${\displaystyle D(u_{\theta })=D(u_{\phi })-{\frac {\phi ^{\circ }\theta ^{circ}}{C\cos \eta }}.}$

Then with the suffix notation to denote the beginning and end of the arc φ-θ,

(84) ${\displaystyle {}_{\phi }t_{\theta }=C\left\lbrack T(u_{\phi })-T(u_{\theta })\right\rbrack ,}$
(85) ${\displaystyle {}_{\phi }x_{\theta }=C\cos \eta \left\lbrack S(u_{\phi })-S(u_{\theta })\right\rbrack .}$
(86) ${\displaystyle {}_{\phi }\left({\frac {y}{x}}\right){}_{\theta }=\tan \phi -C\sec \eta \left\lbrack I(u_{\phi })-{\frac {\Delta A}{\Delta S}}\right\rbrack ;}$

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

 Fig. 2.

Also the velocity vθ at the end of the arc is given by

(87) ${\displaystyle v_{\theta }=u_{\theta }\sec \theta \cos \eta .\,}$

Treating this final velocity vθ and angle θ as the initial velocity vφ and angle φ, 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 φθ of an arc should be so chosen that φyθ 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 τ.

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 40°; 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 τ=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 φ, 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 β.

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

(88) ${\displaystyle \tan \phi =C\left\lbrack I(V)-{\frac {\Delta A}{\Delta S}}\right\rbrack .}$

Also,

(89) ${\displaystyle \tan \phi -\tan \beta =C\left\lbrack I(V)-L(V)\right\rbrack \,}$

so that

(90) ${\displaystyle \tan \beta =C\left\lbrack {\frac {\Delta A}{\Delta S}}-I(v)\right\rbrack ,}$

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

(91) ${\displaystyle \sin 2\phi =2C\left\lbrack I(V)-{\frac {\Delta A}{\Delta S}}\right\rbrack ,}$
(92) ${\displaystyle \sin 2\beta =2C\left\lbrack {\frac {\Delta A}{\Delta S}}-I(v)\right\rbrack .}$

To simplify the work, so as to look out the value of sin 2φ, without the intermediate calculation of the remaining velocity v, a double-entry table has been devised by Captain Braccialini Scipione (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) ${\displaystyle \sin 2\phi =Ca,\,}$

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 φ and β, 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, 2° or 3°; so that the shot will hit the mark aimed at if the back sight is tilted to the vertical at this angle δ, 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 δ on the shot from left to right in order to deflect the plane of the trajectory at angle δ 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 δ.

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.

 Fig. 3.

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 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.

 Fig. 4.
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) ${\displaystyle e={\frac {w}{2240}}\left(1+{\frac {4k^{2}}{d^{2}}}\tan ^{2}\delta \right){\frac {v^{2}}{2g}},}$

in which the factor 4(k2/d2)tan2δ represents the fraction due to the rotation of the shot, of diameter d and axial radius of gyration k, and δ 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 ¼πd2, 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 ¼πd2l=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) ${\displaystyle E={\frac {w}{2240.}}{\frac {V^{2}}{2g}}={\frac {B-C}{12}}\times M.E.P.}$
(3) ${\displaystyle M.E.P.={\frac {w}{2240}}{\frac {V^{2}}{2g}}{\frac {12}{B-C}}.}$

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.

 Fig. 5.

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).

 Fig. 6.

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 1/6) 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). 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.

 Fig. 7.

The results are also embodied in the following Table:—

TABLE 1
G.D. G.V. Pressure in Tons per sq. in.
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) ${\displaystyle {\frac {\mbox{G.D. of charge of lead shot}}{\mbox{S.G. of lump of solid lead}}}={\frac {1}{6}}\pi {\sqrt {2}}=0.7403;}$

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

(5) ${\displaystyle {\frac {\mbox{G.D. of charge of cordite}}{\mbox{S.G. of stick of cordite}}}={\frac {1}{6}}\pi {\sqrt {3}}=0.9067;}$

 PRESSURES OBSERVED IN A CLOSED VESSEL WITH VARIOUS EXPLOSIVES Fig. 8.

At the standard temperature of 62° 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) ${\displaystyle G.D.=27.73P/C,\quad G.V.=C/27.73P.\,}$

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

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

From the Table 1., 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-½Δv to v+½Δv, a change of volume of 27.73 Δv cub. in., the work done is 27.73 p Δv inch-tons, or

(7) ${\displaystyle \Delta E=2.31p\Delta v{\mbox{ foot-tons}};}$

and the differences ΔE 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 0° 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 16×2240÷14.7=2440 atmospheres, or 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) ${\displaystyle R=pv/T=p_{0}v_{0}/273,\,}$

which, with p=2440, v=5, p0=1 v0=700, makes T=4758, a temperature of 4485° C. or 8105° F.

 PRESSURE IN A CLOSED VESSEL OBSERVED AND CALCULATED Fig. 9.

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=13.25×55.01=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=3/2C=1825 cub. in., corresponding to an advance of the shot 3/2×2.15=3.225 ft.

Assuming an average pressure of 8 tons per sq. in., the shot will have acquired energy 8×¼πd2×3.225=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 mth power of the volume.

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

(9) ${\displaystyle {\frac {pb}{m-1}}\left\lbrack 1-\left({\frac {b}{B}}\right)^{m-1}\right\rbrack .}$

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) ${\displaystyle {\frac {B}{b}}={\frac {216+25.8}{2.5\times 25.8}}=3.75,}$

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, Hélie, 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.)