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 powdergas 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., 18651870; Final Report, &c., 18781880; 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 1in, 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 sealevel 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) twothirds 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 flatheaded projectile, as required at proofbutts, 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 breechloading 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 spinningtop. 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) where
 (2)
and n is called the coefficient of reduction.
By means of a wellchosen 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 oldfashioned projectiles fired from muzzleloading 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 v½Δv in passing through the mean velocity v, then


(3)

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

 (4)
We put

 (5)
and call C the ballistic coefficient (driving power) of the shot, so that

 (6)

 (7)
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) =sum of all the preceding values of ΔT plus an arbitrary constant, expressed by the notation

 (9) 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)
and for a shot whose ballistic coefficient is C

 (11)
To save the trouble of proportional parts the value of T(v) for unit increment of v is interpolated in a fulllength extended ballistic table for T.
Next, if the shot advances a distance Δs ft. in the time Δt, during which the velocity falls from v+½v to v½v we have


(12)


 (13)

 (14)
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) , by which S(v) is calculated from ΔS, and then between two assigned velocities V and v,

 (16)
and if s feet is the advance of a shot whose ballistic coefficient is C,

 (17)
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) , 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)
so that we can put

 (20)
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)

 (22)
and if δ and i change to D and I for the standard projectile,

 (23)

 (24)
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 v^{m}, 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 v^{m}/k or its equivalent Cr, where r is the retardation.
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  283090  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)=v^{m}/k. 

3600  1.55  2.3909520  
2600  1.7  2.9038022  
1800  2  3.8807404  
1370  3  7.0190977  
1230  5  13.1981288  
970  3  7.2265570  
790  2  4.3301086 
The numbers have been changed from kilogrammemetre 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)
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 6in. shot weighing 100lb, taking n=0.96,
 (b) of a rifle bullet, 0.303in. 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 coordinates x and y being measured in feet,

 (26)

 (27)
The first equation leads, as before, to

 (28)

 (29) :(28)
The integration of (24) gives

 (30)
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)
and denoting Tt by t', and taking g=32f/s^{2},

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

 (34)
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 onetenth 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)
and then the time of flight T by

 (36)
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)
To share the δ between φ and β, the vertex A is taken as the point of halftime (and therefore beyond halfrange, because of the continual diminution of the velocity), and the velocity v_{0} at A is calculated from the formula

 (38)
and now the degree table for D(v) gives

 (39)

 (40)
This value of φ is the tangent elevation (T.E); the quadrant elevation (Q.E.) is φS, 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)
or, for a small angle, expressed in minutes, taking the radian as 3438',

 (42)
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 6in. 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 12in. 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(v_{0}).  v_{0}.  D(v_{0})  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 



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)

 (44)
and eliminating r,

 (45)
and this, in conjunction with

 (46)

 (47)
reduces to

 (48)
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)
equation (43) becomes

 (50)
and therefore by (48)

 (51)
It is convenient to express r as a function of v in the previous notation

 (52)
and now

 (53)
an equation connecting q and i.
Now, since v=q sec i

 (54)
and multiplying by dx/dt or q,

 (55)
and multiplying by dx/dx or tan i,

 (56)
also

 (57)

 (58)
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, 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 righthand side of equations (54)(56) by some mean value η we introduce Siacci's pseudovelocity u defined by

 (59)
so that u is a quasicomponent parallel to the mean direction of the tangent, say the direction of the chord of the arc. Integrating from any initial pseudovelocity U,

 (60)

 (61)

 (62)
and supposing the inclination i to change from φ to θ radians over the arc.

 (63)

 (64)
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) (66) (67)

and therefore

 (68)

 (69)

 (70)

 (71)

 (72)
while, expressed in degrees,

 (73)
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)
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 altitudefunction 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 φθ 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)
and integrating with respect to x over the arc considered,

 (76)
But


(77)

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

 (78)
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)
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 vφ, the curvature of the arc φ—θ is first settled upon, and now

 (80)
is a good first approximation for η.
Now calculate the pseudovelocity u_{φ} from

 (81)
and then, from the given values of φ, and θ, calculate u_{θ} from either of the formulae of (72) or (73):—

 (82)

 (83)
Then with the suffix notation to denote the beginning and end of the arc φθ,

 (84)

 (85)

 (86)
Δ now denoting any finite tabular difference of the function between the initial and final (pseudo) velocity.
Fig. 2. 
Also the velocity v_{θ} at the end of the arc is given by

 (87)
Treating this final velocity v_{θ} and angle θ as the initial velocity v_{φ} 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 φ—θ of an arc should be so chosen that φy_{θ} 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 twothirds 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 twothirds 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 9.2in. 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. WolleyDod, R.A., in the Proceedings R.A. Institution, 1888, employing Siacci's method and about twenty arcs; and Captain Ingalls, by assuming a mean tenuityfactor 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 altitudefunction 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 pseudovelocities 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)
Also,

 (89)
so that

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

 (91)

 (92)
To simplify the work, so as to look out the value of sin 2φ, without the intermediate calculation of the remaining velocity v, a doubleentry 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)
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 lefthanded 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 righthanded 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, baseball 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 muzzlevelocity, 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 powdergas 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 thermodynamic machine or heatengine which does its work in a single stroke, and does not act in a series of periodic cycles as an ordinary steam or gasengine.
An indicator diagram can be drawn for a gun (fig. 3) as for a steamengine, representing graphically by a curve CPD the relation between the volume and pressure of the powdergas; 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. 

 (1)
in which the factor 4(k^{2}/d^{2})tan^{2}δ 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 ¼πd^{2}, the crosssection 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 inchtons; which work is thus equal to the M.E.P. multiplied by ¼πd^{2}l=BC, 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 powderchamber.
Equating the muzzleenergy and the work in foottons

 (2)

 (3)
Working this out for the 6in., gun of the range table, taking L=216 in., we find BC=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 6in, 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 pressurecurve would be the straight line HK of fig. 3 parallel to AM; the energycurve AQE would be another straight line through A; the velocitycurve 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 timecurve AtT will also be a similar parabola.
If the pressure falls off uniformly, so that the pressurecurve is a straight line PDF sloping downwards and cutting AM in F, then the energycurve will be a parabola curving downwards, and the velocitycurve can be represented by an ellipse, or circle with centre F and radius FA; while the timecurve will be a sinusoid.
Fig. 5. 
But if the pressurecurve is a straight line F'CP sloping upwards, cutting AM behind A in F', the energycurve will be a parabola curving upwards, and the velocitycurve a hyperbola with center at F'.
These theorems may prove useful in preliminary calculations where the pressurecurve is nearly straight; but, in the absence of any observable law, the area of the pressurecurve 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 crushergauge 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 powderchamber, 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 crushergauges, 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 crushergauge 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 powdergas.
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 ballbearings as safetyvalves, loaded to register the pressure at which the powdergas will blow off, and thereby check the indications of the crushergauge (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., 1875188018921894 and following years).
A charge of powder, or other explosive, of varying weight P lb, is fired in an explosionchamber (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 crushergauge (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:—
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)
while in the case of a bundle of cylindrical sticks of cordite,

 (5)
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)
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 foottons 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 v½Δv to v+½Δv, a change of volume of 27.73 Δv cub. in., the work done is 27.73 p Δv inchtons, or

 (7)
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 corditegas 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)
which, with p=2440, v=5, p_{0}=1 v_{0}=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 6in, 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 powdergas 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 8×¼πd^{2}×3.225=730 foottons, 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. at volume b cub. in. is then pb/(m1) inchtons, or to any volume B cub. in. is

 (9)
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 6in, 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)
we find the work realized by expansion is 2826 foottons, 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., 19051906 and following years.