302 Eliminating t, we get v/ = x tan e - t> cos e which is the equation to the trajectory, and is a parabola. To find the range, put y = ; the two resulting values of x give the position GUNNERY go? of the gun, and the end of the range, thus, OR= In Robins experiments in 1742 and subsequent years, it became apparent that up to a certain point the resist ance of the air increased with an increasing power of the velocity. Hutton in 1790 placed this point at 1600 f. s., and considered that the ratio decreased after this velocity was passed. He proposed as a formula for the resistance Roc av + bv" 2 . General Didion, in 1840, deduced from the experiments at Metz the formula Roc av 2 + bv*. The ex periments with rifled projectiles carried on at Gavre in 1861 led Professor H^lie to the conclusion that the resistance of the air at practical velocities was more nearly proportional to the cube of the velocity than to any other working ex pression. He constructed a formula for the trajectory by empirically modifying the formula given above for the path TABLE III. Loss of Velocity of a Projectile in Times and Distances. >rmula. O f a projectile in vacua thus y = x tan e - grar 2 COS 2 eV z from which he deduces an expression for the range which at low elevations for velocities between 800 f. s. and 1400 f. s. is very fairly accurate, and which furnishes the simplest method yet devised of roughly constructing a range table : Bash- brth s nethods. where . r = range in feet ; v = velocity in feet per second ; E = angle of elevation ; & =0-0000000458 ; 10 rf = diameter of projectile in inches ; w = weight of projectile in pounds; </ = acc. of gravity = 32 -19. The above value of k is given for ogival heads ; with hemispherical, pointed, or flat heads a different coefficient of resistance would be required. The ogival is the form which suffers least retardation from the air. The most complete and valuable series of experiments yet carried out is that of Professor Bashforth (18G5 to 1370) at Woolwich and Shoeburyness. He found that, velocities and forms being equal and similar, the resistance varies exactly as the square of the diameter of the projectile. He further arrived at the conclusion that between 900 and 1100 f. s. Roc v 2 , between 1100 and 1350 f. s. Roc v 3 , and above 1350 f. s. Roc v 2 . These results confirm in great measure those pre viously obtained, but unfortunately do not lend themselves readily to the construction of mathematical formulae. Bashforth adopted the cubic law as his basis, as it offers the least difficulty in manipulation, and, putting R = cy 3 , made the coefficient c a variable, the values of which he has tabulated. His method consists in building up the arc of the trajectory bit by bit, taking each portion so that a mean value of c for that portion may be used without important error. The path of the projectile may thus be constructed with almost any desired amount of accuracy, but the process is excessively laborious, and requires the use of a lengthy set of tables which cannot be reproduced here. For a com plete knowledge of the subject the reader is referred to Mr Bashforth s work. Mr Bashforth, however, tabulated the loss of velocity due to the resistance of the air with regard to the time and distance of the shot s flight, and subse quent experiments carried out by the War Office in 1878- 79 have now confirmed and extended the results he had previously obtained. As most of the problems of modern gunnery can be approximately solved by the use of these tables, we give in Table III. an abstract with explanation. F.S. Feet. Seconds. F.S. Feet. Seconds. F.S. Feet. Seconds. 400 5000 5-000 1100 20898 30-531 1800 24441 33-091 410 5416 6-028 1110 20974 30-600 1810 24481 ! 33-113 420 5820 7-003 1120 21048 30-667 1820 24520 33 135 430 6213 7-928 1130 21121 30-731 1830 24560 33-157 440 6595 8-808 1140 21191 30-794 1840 24599 33-178 450 6967 9-645 1150 21261 30-855 1850 24638 33-200 460 7330 10-443 11160 21330 30-914 1860 24677 33-221 470 7684 11-206 1170 21397 30-972 1870 24716 33-242 480 8031 11-936 1180 21463 31-028 1880 24755 33-262 490 8370 12-635 1190 21528 31-083 1890 24794 33-283 500 8701 13-306 1200 21592 31-137 1900 24832 33-303 510 9026 13-951 1210 21655 31-189 1910 24871 33-323 520 9345 14-570 1220 21717 31-240 1920 24909 33-343 530 9658 15-168 1230 21778 31 -290 1930 24947 33-363 540 9965 15-741 1240 21838 31-338 1940 24985 33-383 550 10267 16-295 1250 21897 31-386 1950 25022 33-402 560 10563 16-829 1260 21955 31-432 i 1960 25060 33-421 570 10854 17-345 1270 22012 31-477 1970 25097 33-440 580 11141 17-844 1280 22068 31-521 1980 25134 33-459 590 11423 18-327 1290 22124 31-565 1990 25171 33-477 600 11702 18796 1300 22179 31-607 2000 25207 33-496 610 11977 19-250 1310 22233 31-649 2010 25243 33-514 620 12247 19-690 1320 22286 31-689 2020 25279 33-532 630 12514 20-118 1330 22339 31-729 2030 25315 33-549 (Mi 12778 20-538 1340 22391 31768 J2040 25350 33-567 650 13037 20 -936 1350 22443 31-807 2050 25385 33-584 660 13293 21-364 1360 22494 31-844 2060 25420 33-601 670 13545 21-706 1370 22544 31-881 2070 25455 33-617 680 13793 22-074 1380 22594 31-918 2080 25489 33-634 690 14038 22-432 1390 22644 31-953 2090! 25523 33-650 700 14280 22-781 |1400 22693 31-988 2100 25556 33-666 710 14519 23-120 1410 22741 32-023 2110! 25590 33-682 720 14755 23-449 1420 22789 32-057 2120 25623 33-698 730 14987 23-769 1430 22837 32-090 2130 25656 33-713 740 15215 24-080 1440 22884 32-123 2140i25688 33-728 750 15440 24-383 1450 22931 32-156 2150125720 33-743 760 15662 24-677 1460 22978 32-188 2160 25752 33-758 770 15880 24-962 1470 23025 32-220 2170 25784 33-773 780 16095 25-239 1480 23071 32-251 2180 25815 33-787 790 16306 25-508 1490 23116 32-282 2190 25847 33-802 800 16513 25769 1500 23162 32-312 2200 258/6 33-816 810 16716 26 021 1510 23207 32-342 2210 25908 33-830 820 16916 26 267
- 1520
23252 32-372 2220 25939 33-844 830 17112 26-504 1530 23297 32-402 2230 25969 33-857 840 17303 26-734 1540 23342 32-431 2240 25999 33-871 8 50 17490 26-955 1550 23387 32-460 2250 26029 33-884 860 17672 27-169 1560 23431 32-488 2260 26058 33-897 870 17851 27-375 1570 23475 32-517 2270 26088 33-910 880 18025 27-575 1580 23519 32-544 2280 26117 33-923 890 18195 27-768 1590 23563 32-572 2290 26145 33-935 900 18362 27-954 iieoo 23607 32-599 12300 26174 33-948 910 18524 28-135 1610 23650 32-627 2310 26202 33-960 920 18684 28-309 1620 23693 32-653 2320 26230 33 972 930 18839 28-478 1630 23736 32-680 i2330 26258 33-984 940 18992 28-641 1640 23770 82-706 2340 26285 33-996 950 19141 28-799 1650 23822 82732 235C 26313 34-007 960 19287 28-953 1660 23864 32-758 2360 26340 34-019 970 19431 29-104 1670 23907 32-783 2370 26367 34-030 980 19571 29-295 1680 23949 32-808 2380 26393 34-041 990 19701 29-385 i <;<. < 23991 32-833 2390 26420 34-053 1000 19843 29-521 17(> 24033 32-858 2400 26446 34-064 1010 19975 29-652 1710 24074 32-882 2410 26472 34-074 1020 20104 29-780 17-21 24116 32-907 2420 26498 34-085 1030 20230 29-902 1730 24157 32-930 2430 26523 34-096 1040 20349 30-018 174C 24198 32-954 2440 26548 34-106 1050 20459 30-123 1750 24239 32-978 2450 26574 34-116 1060 20559 30-217 1760 24280 33-001 2460 26599 34-126 1070 20651 30-303 177C 24320 33-024 2470 26623 34-136 1080 20737 30-384 1780 24361 33-047 2480 26648 34-146 1090 20819 30-459 179C 24401 33-069 2-1 !i 26672 34-156 2500 26696 | 34-166 The rate of loss of velocity is here inversely recorded for an ogival projectile having a weight in pounds equal to the square, of its diameter in inches. Thus a 9-pounder shell of 3-inch calibre starting with a velocity of 2500 f. s. would retain a velocity of 400 f s at a distance of (26,696-5000 = ) 21,696 feet after a time of (34-166 - 5 -000 = ) 29 166 seconds. As the value of - decreases the resistance of the air retards the projectile less ; thus if the 3-inch shell weigh 18 ft it will overcome the resistance twice as well as 79
the 9-pounder, since its - =0 5 instead of TO.
