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) Rδt = loss of momentum in second-pounds,

= w(v+½δv)/g - w(v-½δv)/g = wδv/g

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

(4) δt = wδv/nd2pg.

We put

(5) w/nd2 = C,

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

(6) δt = CδT, where

(7) δT = δv/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) T(v) = sum of all the preceding values of δT plus an arbitrary constant, expressed by the notation

(9) T(v) = ∑(δv)/gp + a constant, or ∫dv/gp + 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) T(V) - T(v) = ∑Vδv/gp or ∫Vdv/gp;

and for a shot whose ballistic coefficient is C

(11) t = C[T(V) - T(v)].

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 v+½δv to v-½δv, we have

(12) Rδs = loss of kinetic energy in foot-pounds

=w(v+½δv)2/g - w(v-½δv)2/g = wvδv/g, so that

(13) δs = wvδv/nd2pg = CδS, where

(14) δS = vδv/gp = vδT,

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) S(v) = ∑(δS) + a constant, by which S(v) is calculated from δS, and then between two assigned velocities V and v,

(16) S(V) - S(v) = V

v
δT = vδv or V

v
vdv,
gpgp

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

(17) s = C[S(V) - S(v)].

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) 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) v(di/dt) = g, or di/dt = g/v;

so that we can put

(20) δi/δt = 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) δ/180 = i/π, so that

(22) δδ = 180δi = 180gδt;
ππv

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

(23) δI = gδT=δv, δD =180gδT, and
vvpπv
(24) I(V) - I(v) = V

v
δv or V

v
dv, D(V) - D(v) =180[I(V) - I(v)].
vpvpπ

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

.0239

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

v1.55

× log-1 3.6090480

2600

1.7

2.9038022

v1.7

× log-1 3.0961978

1800

2

3.8807404

v2

× log-1 4.1192596

1370

3

7.0190977

v3

× log-1 8.9809023

1230

5

13.1981288

v5

× log-1 14.8018712

970

3

7.2265570

v3

× log-1 8.7734430

790

2

4.3301086

v2

× log-1 5.6698914

The numbers have been changed from kilogramme-metre to pound-foot 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) T(V) - T(v) = k ∫V v-m dv, S(V) - S(v) = k ∫V vm+1 dv, I(V) - I(v) = gk ∫V v-m-1 dv,

and the corresponding integration.