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) , so that
- (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 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.
| 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 |
