1=C(T(U) -T(u)], x=C cos n [S(U)—S(u)], y=C sin n [S(U) −S(u)]. -8=C cos n [I(U) −I(u)}, tan-tan-C sec n (1(U) -1 (u)}, while, expressed in degrees, (73) °-°C cos n [D(U) - D(u)]. 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 cannot be exactly the same mean angle in all these equations; but if ʼn is the same in (69) and (70), (74) y/x = tan 7, so that 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 to be known with accuracy, as 1% variation in causes more than 1% variation in tan 77. The difficulty is avoided by the use of Siacci's altitude-function A or A(u), by which y/x can be calculated without introducing sin or tan, but in which ŋ occurs only in the form eos or sec n which varies very slowly for moderate values of n. so that need not be calculated with any great regard for accuracy, the arithmetic mean (+0) of and being near enough for over any arc of moderate extent. Now taking equation (72), and replacing tan 0, as a variable final tangent of an angle, by tan i or dy/dx, FIG. 2, Also the velocity ", at the end of the arc is given by (87) 4 sec 8 cos n. Treating this final velocity , and angle as the initial velocity and angle 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 - of an arc should be so chosen that dye, the height ascended, should be limited to about 1000 ft., equivalent factor r. to a fall of 1 inch in the barometer or 3% diminution in the tenuity 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 two-thirds 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 two-thirds 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 9-2-in. 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. Wolley Dod, R.A., in the Proceedings R.A. Institution, 1888, employing Siacci's method and about twenty arcs; and Captain Ingalls, by assuming a mean tenuity-factor 70-68, 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 altitude-function is useful in direct fire, for giving immediately the angle of elevation 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 8. In direct fire the pseudo-velocities U and u, and the real velocities V and v, are undistinguishable, and sec may be replaced by unity so that, putting y=o in (79), (88) Also (89) so that (90) tan - C tan -tan 8=C[I (V) −L(v)] ̧ · tan (77) S1(u)dx = S_1(u)dxdu u du gf() =C cos (A(U) – A(u)] in Siacci's notation; so that the altitude-function A must be calculated by summation from the finite difference AA, 'where (78) u Au AA=I() =I(u)AS, or else by an integration when it is legitimate to assume that f(e)k in an interval of velocity in which m may be supposed constant. Dividing again by x, as given in (76), (79) tan -2=C sec n -A (u) "[1(U)-5(U)-5(4) 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 with velocity to, the curvature of the arc -0 is first settled upon, and now (80) 7-1(6+0) is a good first approximation for or, as (88) and (90) may be written for small angles, (91) (92) sin 26=2C [I(V)-AA]. sin 28=2C · [-1]· To simplify the work, so as to look out the value of sin 24 without the intermediate calculation of the remaining velocity, a doubleentry table has been devised by Captain Braccialini Scipione (Problemi del Tiro, Roma, 1883), and adapted to yd., ft., in. and b | ball, and the deviation is in the opposite direction of the drift units by A. G. Hadcock, late R.A., and published in the Proc. R.A. observed in artillery practice, so artillerists are still awaiting Institution, 1898, and in Gunnery Tables, 1898. theory and crucial experiment. sin 24 Ca, 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 and B, 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 ô, 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 left-handed 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 yacuum; 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 right-handed twist. The effect is as if there was a mean sidelong thrust w tan 8 on the shot from left to right in order to deflect the plane of the trajectory at angle & 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 8. An effect similar to drift is observable at tennis, golf, base-ball and cricket; but this effect is explainable by the inequality of pressure due to a vortex of air carried along by the rotating 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 climinated, such as variations in the air and in the muzzle-velocity, 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 accordin the range table (see PROBABILITY.) ance with the laws of probability of the 50% zones shown Anest Irguel in feet Pressure Curves, from Chronoscope experiments in 6 inch gun of 100 calibres, with various Explosives. FIG. 4. 1000 steam-engine, representing graphically by a curve CPD the relation between the volume and pressure of the powder-gas; and in addition the curves AQE of energy . AvV of velocity v, and A/T of time t can be plotted or derived, the velocity and energy at the muzzle B being denoted by V and E. After a certain discount for friction and the recoil of the gun, the net work realized by the powder-gas as the shot advances AM is represented by the area ACPM, and this is equated to the kinetic energy e of the shot, in foot-tons, +4tan's) in which the factor 4(d)tan's represents the fraction due to the rotation of the shot, of diameter d and axial radius of gyration k, and & 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 rd, the cross-section 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 1, the length in inches of the travel AB of the shot up the bore, gives the work realized in inch-tons; which work is thus equal to the M.E.P. multiplied by td-B-C, 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 powder-chamber. Working this out for the 6-in. gun of the range table, taking L=216 in., we find B-C-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 6-in. 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 pressure-curve would be the straight line HK of fig. 3 parallel to AM; the energycurve AQE would be another straight line through A; the velocitycurve ADV, 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 time-curve A/T will also be a similar parabola. If the pressure falls off uniformly, so that the pressure-curve is a straight line PDF sloping downwards and cutting AM in F, then the energy-curve will be a parabola curving downwards, and the velocity-curve can be represented by an ellipse, or circle with centre F and radius FA; while the time-curve will be a sinusoid. Plugs 4 10 10 12 12 14 16 14 15 10 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 Velocity Curves, from Chronoscope experiments in 6 inch gun of 100 calibres, with Cordite. FIG. 5. But if the pressure-curve is a straight line F'CP sloping upwards, | cutting AM behind A in F', the energy-curve will be a parabola curving upwards, and the velocity-curve a hyperbola with center at F These theorems may prove useful in preliminary calculations where the pressure-curve is nearly straight; but, in the absence of any observable law, the area of the pressure-curve 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 crusher-gauge 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 powder-chamber, 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 crusher-gauges, and also chronographic appliances which regis tered 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 crusher-gauge 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 powder-gas. 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 ball-bearings as safety- But the most modern results employed with gunpowder are based on the experiments of Noble and Abel (Phil. Trans., 1875-1880-18921894 and following years). is FIG. 5. A charge of powder, or other explosive, of varying weight P lb. fired in an explosion-chamber (fig. 7, scale about) of which the volume C, cub. in., is known accurately, and the pressure p, tons per sq. in., was recorded by a crusher-gauge (fig. 6). Sometimes the factor 27.68 is employed, corresponding to a density of water of about 62-4 lb per cub. It., 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/P, no factor being required. From the Table I., or by quadrature of the curve in fig. 9, the work E in foot-tons realized by the expansion of 1 th 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 -JAD to v+Jav, a change of volume of 27.73 Av cub. in., the work done is 27-73 p Av inch-tons, or (7) AE=2.31 pad foot-tons; and the differences AE 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 cordite-gas at o° 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 16X2240414-7= 2440 atmospheres, of 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 R = po/T=p/273. *(8) which, with p=2440, -5, Po=1, % -700, makes T-4758, a temperature of 4485° C. or 8105° F. PRESSURE IN A CLOSED VESSEL OBSERVED AND CALCULATED 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 PRESSURE IN TONS PER SO INGA. 45 60 25 30 در 40 45 50 55 43 10 75 GRAVIMETRIC DENSITY OF PRODUCTS OF EXPLOSION FIG. 8. in of water at this temperature weighs 62-35 lb, and therefore 1 lb of water bulks 1728 +62·35 × 27·73 cub. in. GRAVIMETRIC VOLUME FIG 9. In the heading of the 6-in. 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 b, 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-13-25X55-01-730 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 powder-gas will now occupy a volume b=C=1825 cub, in, corresponding to an advance of the shot 1X2-15-3-225 ft. Assuming an average pressure of 8 tons per sq. in., the shot will have acquired energy 8X1 X3-225-730 foot-tons, and a velocity about 1020 Ls, so that the time over the 3.225 ft. at an average velocity 510 f/s is about 0-0063 scc. Comparing this tirae 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 cunsidered 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" 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/(m-1) inch-tons, or to any volume | dropped according to choice into the right receptacle; but 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 6-in. 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) B 216+25-8 -3.75, we find the work realized by expansion is 2826 foot-tons, sufficient 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 o 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., 1905-1906 and following years. AUTHORITIES.-Tartaglia, Nova Scientia (1537); Galileo (1638): Robins, New Principles of Gunnery (1743): Euler (trans. by Hugh Brown), The True Principles of Gunnery (1777); Didion, Hélie, Hugoniot, Vallier, Baills, &c., Balistique (French); Siacci, Balistica (Italian); Mayevski, Zabudski, Balistique (Russian): La Llave, Ollero, Mata, &c., Balistica (Spanish); Bashforth, The Motion of Projectiles (1872); The Bashforth Chronograph (1890): Ingalls, Exterior and Interior Ballistics, Handbook of Problems in Direct and Indirect Fire: Bruff, Ordnance and Gunnery; Cranz, Compendium der Ballistik (1898); The Official Text-Book of Gunnery (1902); Charbonnier, Balistique (1905): Lissak, Ordnance and Gunnery (A. G. G.) (1907). BALLOON, a globular bag of varnished silk or other material impermeable to air, which, when inflated with gas lighter than common air, can be used in aeronautics, or, according to its size, &c., for any purpose for which its ability to rise and float in the atmosphere adapts such a mechanism. "Balloon" in this sense was first used in :783 in connexion with the invention of the brothers Montgolfier, but the word was in earlier use (derived from Ital. builone, a large ball) as meaning an actual ball or ball-game, a primitive explosive bomb or firework, a form of chemical retort or receiver, and an ornamental globe in architecture; and from the appearance and shape of an air balloon the word is also given by analogy to other things, such as a "balloon skirt" in dress, "balloon training" in horticulture. (See AERONAUTICS, and FLIGHT AND FLYING). BALLOT (from Ital. ballotta, dim. of balla, a ball), the modern method of secret-voting employed in political, legislative and judicial assemblies, and also in the proceedings of private clubs and corporations. The name comes from the use of a little ball nowadays it is used for any system of secret-voting, even though Great Britain.-In Great Britain the ballot was suggested for For a description of Grote's card-frame, in which the card was punctured through a hole, and was thus never in the voter's hands, see Spectator, 25th February 1837 |