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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+&FRAC12;δv to v-&FRAC12;δv in passing through the mean velocity v, then

(3) Rδt = loss of momentum in second-pounds,

= w(v+&FRAC12;δv)/g - w(v-&FRAC12;δv)/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+&FRAC12;δv to v-&FRAC12;δv, we have

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

=w(v+&FRAC12;δv)2/g - w(v-&FRAC12;δv)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 | , |

gp | gp |

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 = | 180 | δi = | 180g | δt | ; |

π | π | v |

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

(23) δI = g | δT | = | δv | , δD = | 180g | δT | , and |

v | vp | π | v |

(24) I(V) - I(v) = | ∑ | V v | δv | or | ∫ | V v | dv | , D(V) - D(v) = | 180 | [I(V) - I(v)]. |

vp | vp | π |

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.

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