See also:

• BALLISTICS (from the Gr. fMXkee)

BALLISTICS (from the Gr. fMXkee)  , to throw), the See also:

• SCIENCE (Lat. scientia, from scire; to learn, know)

science of throwing warlike missiles or projectiles . It is now divided into two parts:—Exterior See also:

• BALLISTICS (from the Gr. fMXkee)

Ballistics, in which the See also:

• MOTION (Lat. motio, from movere, to move)
• MOTION, LAWS OF

motion of the projectile is considered after it has received its initial impulse, when the projectile is moving freely under the See also:

• INFLUENCE (Late Lat. influentia, from influere, to flow in)

influence of gravity and the resistance of the See also:

• AIR (from an Indo-European root meaning " breathe," " blow ")
• AIR, or ASBEN

air, and it is required to determine the circumstances so as to See also:

• HIT

hit a certain See also:

• OBJECT

object, with a view to its destruction or perforation; and Interior Ballistics, in which the pressure of the See also:

• POWDER (through O. Fr. puldre, modern poudre, from Lat. pulvis, pulveris, dust)

powder-See also:

• GAS

gas is analysed in the See also:

• BORE

bore of the See also:

• GUN

gun, and the investigation is carried out of the requisite See also:

• CHARGE
• CHARGE (through the Fr. from the Late Lat. carricare, to load in a carrus or wagon; cf. " cargo ")

charge of powder to secure the initial velocity of the projectile, without straining the gun unduly . The calculation of the stress in the various parts of the gun due to the powder pressure is dealt with in the See also:

• ARTICLE (from Lat. articulus, a joint)

article See also:

• ORDNANCE (a syncopated form of " ordinance " or " ordonnance," so spelt in this sense since the 17th century)

ORDNANCE . I . EXTERIOR BALLISTICS . In the See also:

• ANCIENT
• ANCIENT (also spelt ANTIENT; derived, through the Fr. ancien, old, from the late Lat. antianum, from ante, before)

ancient theory due to Galileo, the resistance of the air is ignored, and, as shown in the article on See also:

• MECHANICS

MECHANICS (§ 13), the trajectory is now a See also:

• PARABOLA

parabola . But this theory is very far from being of See also:

• PRACTICAL

practical value for most purposes of gunnery; so that a first requirement is an accurate experimental knowledge of the resistance of the air to the projectiles employed, at all velocities useful in See also:

• ARTILLERY (the O. Fr. artiller, to equip with engines of war, probably comes from Late Lat. articulum, dim. of ars, art, cf. " engine " from ingenium, or of artus, joint)

artillery . The theoretical assumptions of See also:

• NEWTON
• NEWTON, ALFRED (1829–1907)
• NEWTON, JOHN (1725-1807)
• NEWTON, JOHN (1823-1895)
• NEWTON, SIR CHARLES THOMAS (1816-1894)
• NEWTON, SIR ISAAC (1642-1727)

Newton and See also:

• EULER, LEONHARD (1707-1783)

Euler (hypotheses magis mathematicae quam naturales) of a resistance varying as some See also:

• SIMPLE

simple See also:

• POWER [WILLIAM GRATTAN] TYRONE (1797-1841)

power of the velocity, for instance, as the square or See also:

• CUBE (Gr. K46os, a cube)

cube of the velocity (the quadratic or cubic See also:

• LAW
• LAW (0. Eng. lagu, M. Eng. lawe; from an old Teutonic root lag, " lie," what lies fixed or evenly; cf. Lat. lex, Fr. loi)
• LAW, JOHN (1671—1729)
• LAW, WILLIAM (1686-1761)

law), See also:

• LEAD
• LEAD (pronounced iced)

lead to results of See also:

• GREAT

great See also:

• ANALYTICAL

analytical complexity, and are useful only for provisional extrapolation at high or See also:

• LOW, SETH (1850- )
• LOW, WILL HICOK (1853- )

low velocity, pending further experiment . The See also:

• FOUNDATION (Lat. fundatio, from fundare, to found)

foundation of our knowledge of the resistance of the air, as employed in the construction of ballistic tables, is the See also:

• SERIES (a Latin word from serere, to join)

series of experiments carried out between 1864 and -188o by the Rev . F . Bashforth, B.D . (See also:

• REPORT (O.Fr. report or raport, modern rapport, from O.Fr. reporter, mod. rap porter, Lat. reportare, to bring back, in poetical use only, of bringing back an account, news, &c.)

Report on the Experiments made with the Bash-forth See also:

• CHRONOGRAPH

Chronograph, &c., 1865—187o; Final Report, &c., 1878—1880; The Bashforth Chronograph, See also:

• CAMBRIDGE
• CAMBRIDGE, EARLS AND DUKES OF
• CAMBRIDGE, RICHARD OWEN (1717-1802)

Cambridge, 1890) .

According to these experiments, the resistance of the air can be represented by no simple algebraical law over a large range of velocity . Abandoning therefore all a priori theoretical See also:

• ASSUMPTION, FEAST OF

assumption, Bashforth set to See also:

• WORK

work to measure experimentally the velocity of shot and the resistance of the air by means of equidistant electric screens furnished with See also:

• VERTICAL (from Lat. vertex, highest point)

vertical threads or See also:

• WIRE (A.S. wir, a wire; cf. Swed. vire, to twist, M.H.G. wiere, a gold ornament, Lat. viriae, armlets, ultimately from the root wi, to twist, bind)

wire, and by a chronograph which measured the instants of See also:

• TIME (0. Eng. Lima, cf. Icel. timi, Swed. timme, hour, Dan. time; from the root also seen in " tide," properly the time of between the flow and ebb of the sea, cf. O. Eng. getidan, to happen, " even-tide," &c.; it is not directly related to Lat. tempus)
• TIME, MEASUREMENT OF
• TIME, STANDARD

time at which the screens were cut by a shot flying nearly horizontally . Formulae of the calculus of finite See also:

• DIFFERENCES, CALCULUS OF (Theory of Finite Differences)

differences enable us from the chronograph records to infer the velocity and retardation of the shot, and thence the resistance of the air . As a first result of experiment it was found that the resistance of similar shot was proportional, at the same velocity, to the See also:

• SURFACE

surface or See also:

• CROSS

cross See also:

• SECTION
• SECTION (Lat. sectio, cutting, secare, to cut)

section, or square of the See also:

• DIAMETER (from the Cr. &a, through, µErpov, measure)

diameter . The resistance R can thus be divided into two factors, one of which is d2, where d denotes the diameter of the shot in inches, and the other See also:

• FACTOR (from Lat. facere, to make or do)

factor is denoted by p, where p is the _resistance in pounds at the same velocity to a similar 1-in. projectile; thus R=d2p, and the value of p, for velocity ranging from 1600 to 2150 ft. per second (f/s) is given in the second See also:

• COLUMN (Lat. columna)

column of the See also:

• EXTRACT (from Lat. extrahere, to draw out)

extract from the abridged ballistic table below . These values of p refer to a See also:

• STANDARD
• STANDARD, BATTLE OF THE

standard See also:

• DENSITY (Lat. densus, thick)

density of the air, of 534.22 grains per cubic See also:

• FOOT

foot, which is the density of dry air at See also:

• SEA (in O. Eng. sae, a common Teutonic word; cf. Ger. See, Dutch Zee, &c.; the ultimate source is uncertain)
• SEA, COMMAND OF THE

sea-level in the See also:

• LATITUDE (Lat. latitudo, latus, broad)

latitude of See also:

• GREENWICH

Greenwich, at a temperature of 62° F. and a barometric height of 30 in . But in consequence of the humidity of the See also:

• CLIMATE

climate of See also:

• ENGLAND
• ENGLAND, THE CHURCH OF

England it is better to suppose the air to be (on the See also:

• AVERAGE

average) two-thirds saturated with aqueous vapour, and then the standard temperature will be reduced to 6o° F., so as to secure the same standard density; the density of the air being reduced perceptibly by the presence of the aqueous vapour . It is further assumed, as the result of experiment, that the resistance is proportional• to the density of the air; so that if the standard density changes from unity to any other relative density denoted by r, then R=rd2p, and T is called the coefficient of tenuity . The factor r becomes of importance in See also:

• LONG, GEORGE (1800-1879)
• LONG, JOHN DAVIS (1838– )

long range high See also:

• ANGLE (from the Lat. angulus, a corner, a diminutive, of which the primitive form, angus, does not occur in Latin; cognate are the Lat. angere, to compress into a bend or to strangle, and the Gr. ayKOS, a bend; both connected with the Aryan root ank-, to

angle See also:

• FIRE (in O. Eng. f j'r; the word is common to West German languages, cf. Dutch vuur, Ger. Feuer; the pre-Teutonic form is seen in Sanskrit pu, pavaka, and Gr. irup; the ultimate origin is usually taken to be a root meaning to purify, cf. Lat. purus)

fire, where the shot reaches the higher attenuated strata of the See also:

• ATMOSPHERE (Gr. fir µ6r, vapour; orkaipa, a sphere)

atmosphere; on the other See also:

• HAND
• HAND (a word common to Teutonic languages; cf. Ger. Hand, Goth. handus)
• HAND, FERDINAND GOTTHELF (1786-185r)

hand, we must take r about 800 in a calculation of See also:

• SHOOTING

shooting under See also:

• WATER

water . The resistance of the air is reduced considerably in See also:

• MODERN

modern projectiles by giving them a greater length and a sharper point, and by the omission of projecting studs, a factor K, called the coefficient of shape, being introduced to allow for this See also:

• CHANGE (derived through the Fr. from the Late Lat. cambium, cambiare, to barter; the ultimate derivation is probably from the root which appears in the Gr. «a urmu', to bend)

change . For a projectile in which the ogival See also:

• HEAD (in 0. Eng. heafad; the word is common to Teutonic languages; cf. Dutch hoofd, Ger. Haupt, generally taken to be in origin connected with Lat. caput, Gr. KerbOvi7)
• HEAD, SIR EDMUND WALKER, BART
• HEAD, SIR FRANCIS BOND, BART

head is struck with a See also:

• RADIUS

radius of 2 diameters, Bashforth puts K=0•975; on the other hand, for a See also:

• FLAT (a modification of O. Eng. flet, an obsolete word of Teutonic origin, meaning the ground beneath the feet)

flat-headed projectile, as required at See also:

• PROOF (in M. Eng. preove, proeve, preve, &°c., from O. Fr . prueve, proeve, &c., mod. preuve, Late. Lat. proba, probate, to prove, to test the goodness of anything, probus, good)

proof-butts, K=1.8, say 2 on the average . For spherical shot K is not See also:

• CONSTANT, BENJAMIN (1845-1902)

constant, and a See also:

• SEPARATE

separate ballistic table must be constructed; but K may be taken as 1.7 on the average .

Lastly, to allow for the See also:

• SUPERIOR

superior centering of the shot obtainable with the See also:

• BREECH (common in early forms to Teutonic languages)

breech-loading See also:

• SYSTEM

system, Bashforth introduces a factor e, called the coefficient of steadiness . This steadiness may vary during the See also:

• FLIGHT

flight of the projectile, as the shot may be unsteady for some distance after leaving the muzzle, afterwards steadying down, like a See also:

• SPINNING (from O. Eng. spinnen, to spin, cf. Ger. spinnen, &c., the Teut. root is seen, to draw out, cf. span, spider)

spinning-See also:

• TOP (cf. Dan. top, Ger. Topf, also meaning pot)

top . Again, e may increase as the gun wears out, after firing a number of rounds . See also:

• COLLECTING

Collecting all the coefficients, r, e, a, into one, we put (2) R =nd2p=nd2f(v), where and n is called the coefficient of reduction . By means of a well-chosen value of n, determined by a few experiments, it is possible, pending further experiment, with the most See also:

• RECENT

recent See also:

• DESIGN (Fr. desiin, drawing; Lat. designare, to mark out)

design, to utilize Bashforth's experimental results carried out with old-fashioned projectiles fired from muzzle-loading guns . For instance, n=o•8 or even less is considered a See also:

• GOOD, JOHN MASON (1764-1827)

good average for the modern See also:

• RIFLE

rifle See also:

• BULLET (Fr. boulet, diminutive of boule, ball)

bullet . 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 See also:

• TIN (Lat.- stannum, whence the chemical symbol " Sn "; atomic weight =117.6, 0=16)

tin 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 At seconds is the time during which the resistance of the air, R lb, causes the velocity of the shot to fall Av(f/s), so that the velocity drops from v+iAv to v—ZAv in passing through the mean velocity v, then (3) Rot =loss of momentum in second-pounds, w(v -{- z Ov)/g —w(v — a 0v) /g = wLv/g so that with the value of R in (1), (4) At = wAv/nd2pg• We put (5) wind2 = C; and See also:

• CALL (from Anglo-Saxon ceallian, a common Teutonic word, cf. Dutch kallen, to talk or chatter)

call C the ballistic coefficient (See also:

• DRIVING (from " to drive," i.e. generally to propel, force along or in, a word common in various forms to the Teutonic languages)

driving power) of the shot, so that (6) At = See also:

• CAT
• CAT, CIVET
• CAT, HOUSE

CAT, where (7) AT = Ov/gp, and AT is the time in seconds for the velocity to drop Lv 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 See also:

• FUNCTION

function of v, the velocity is taken as the See also:

• ARGUMENT

argument of the ballistic table; and taking iv = io, the average value of p in the See also:

• INTERVAL

interval is used to determine AT . Denoting the value of T at any velocity v by T (v), then (8) T(v) =sum of all the preceding values of AT plus an arbitrary constant, expressed by the notation (9) T(v) =E(Av)/gp+ a constant, or fde/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 See also:

• TABULAR

tabular values for an initial velocity V and final velocity v; and thus (to) T(V)—T(v)=1vOv/gp or fvdv/gp; and for a shot whose ballistic coefficient is C (II) t= CfT(V) —T(v)] . To See also:

• SAVE, or SAVA (Ger. Sau; Hungarian Szdva; Lat. Savus)

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 As ft. in the time At, during which the velocity falls from v-I 44ov to v— yav, we have (12) See also:

• RAS

RAs =loss of kinetic See also:

• ENERGY (from the Gr. ivipyela; iv, in, ipyov, work)

energy in foot-pounds w(v+lAv)2/g—w(v—iAv)2/g=wvav/g, so that (13) As=wvOv/nd2pg=CAS, where (14) AS = vLv/gp = vLT, and AS is the advance in feet of a shot for which C = t, while the velocity falls Av in passing through the average velocity v . Denoting by S(v) the sum of all the values of AS up to any assigned velocity v, (15) S(v)=E(OS)+ a constant, by which S(v) is calculated from AS, and then between two assigned velocities V See also:

• RAND

rand v, vvd (16) S(V) -5(v) = vAT = E gvAv p orJ _', 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 See also:

• SIR

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 See also:

• HORIZON (Gr. 6pfTwv, dividing)

horizon . Resolving normally in the trajectory, and supposing the resistance of the air to See also:

• ACT (Lat. actus, actum)

act tangentially, (18) v(di/dt) =g See also:

• COS
• COS, or STANKO (Ital. Stanchio, Turk. Islan-keui, by corruption from Etc rav KW)

cos i, where di denotes the infinitesimal decrement of i in the infinitesimal increment of time dt . In a problem of See also:

• DIRECT

direct fire, where the trajectory is flat enough for cos i to be undistinguishable from unity, See also:

• EQUATION (from Lat. aequatio, aequare, to equalize)

equation (16) becomes (19) v(di/dt)=g, or di/di=g/v; so that we can put (20) Di/At = g/v, if v denotes the mean velocity during the small finite interval of time i t, during which the direction of motion of the shot changes through Ai radians . If the inclination or change of inclination in degrees is denoted by S or M, (21) 6/18o=i/ir, so that 180 1805 At M = 4i= 7 v and if 6 and i change to D and I for the standard projectile, 2 AT Av 18og AT (3) AI =g v =vp, AD = v , and (24) I(V)-I(v)=Evvpor 'L 7, d ,D(V)-D(v)=t8o]I(V)-I(v)] .

The differences AD and AI are thus calculated, while the valuesof 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 See also:

• SIN
• SIN (O. Eng. syn: a common Teutonic word, cf. Dutch zonde, Ger. Siinde)

sin i and tan a when i is small as in direct fire . The last function A, called the See also:

• ALTITUDE (Lat. altitudo, from altus, high)

altitude function, will be explained when high angle fire is considered . These functions, T, S, D, 1, 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 to fjs; 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 See also:

• FORMULA
• FORMULA (Lat. diminutive of forma, shape, pattern, &c., especially used of rules of judicial procedure)

formula composed of a single power of v, say the integration can be effected which replaces the summation in (to), (16), and (24); and from an See also:

• ANALYSIS (Gr. avci and Meta, to break up into parts)

analysis of the See also:

• KRUPP, ALFRED (1812—1887)

Krupp experiments See also:

• COLONEL (derived either from Lat. column, Fr. colonne, column, or Lat. corona, a crown)

Colonel Zabudski found the most appropriate See also:

• INDEX

index m in a region of velocity as given in the following table, and the corresponding value of gp, denoted by f (v)or v'"/k or its See also:

• EQUIVALENT

equivalent Cr, where r is the retardation .