RANGE TABLE FOR 6INCH GUN.
Projectile Palliser shot, Shrapnel shell. Weight, loo lb.
Muzzle velocity, 2154 f/s. Nature of mounting, pedestal. Jump, nil.
To strike 5' elevation or Fuse 50 % of rounds
alters
an object depression
scale for should fall in.
Remain to ft. Slope of impact. T. and P. Time Penetra
point of
ing high De Later Eleva Range. middle of tion into
Velocity. range scent. ally or tion. No. 54 Flight. Wrought
must be Range. Verti Marks I., Length. Breadth. Height.
known to cally. II., or III.
fjs. yds. in. yds. yds. yds. yds. yds. yds. secs. in.
2154 ,. .. o•o0 0 0 0 .. .. .. 0•O0 13.6
2I22 1145 687 125 0.14 0 4 100 ; .. 0.4 o•16 13.4
2091 635 381 125 0.29 0 9 200 0.4 0'31 13.2
2061 408 245 125 0.43 0 13 300 1 .. 0.4 .. 0.47 13.0
2032 316 190 125 0.58 0 17 400 Iq .. 0.4 .. o•62 12.8
2003 260 156 125 0.72 0 21 500 If .. 0.5 0.2 0.78 12.6
1974 211 127 125 .0.87 0 26 600 2 .. 0.5 0'2 0.95 12.4
.1946 183 1I0 125 I.OI 0 30 700 21 .. 0.5 0.2 I•II 12.2
1909 163 98 125 1.16 0 34 800 2 .. 0.5 0.2 1.28 12.0
1883 143 85 125 1.31 0 39 900 3 .. o•6 0.3  1.44 II.8
1857 130 78 125 1.45 0 43 1000 311 .. o•6 0.3 1•6t II.6
1830 118 71 125 160 0 47 II00 3a o•6 0.3 I.78 II.4
1803 110 66 125 1.74 0 51 1200 4 o•6 0.3 I.95 I1.2
1776. IOI 6, 125 I•$9 0 55 1300 4± .. 0.7 0.4 2.12 11.0
1749 93 56 125 2.03 0 59 1400 41 .. 0.7 04 2.30 to.8
1722 86 52 125 2.18 I 3 1500 5 .. 0.7 0.4 2.47 Io•6
1695 8o 48 125 2.32 I 7 1600 5Z 25 o•8 0.5 2.65 10.5
1669 71 43 125 2.47 1 II 1700 51 25 0.9 0.5 2.84 10.3
1642 67 40. too 2.61 I t6 i800 64 '25 1•0 0.5 3.03 10•I
1616 61 37 100 2.76 I 22 1900 62 25 1.1 o•6 3.23 9.9
1591 57 34 100 2.91 127 2000 7 25 I.2 0•6 3.41 9.7
The last column in the Range Table giving the inches of penetration into wrought iron is calculated from the remaining velocity an empirical formula, as explained in the article ARMOUR PLATES. _ High Angle and Curved Fire." High angle fire," as defined and row
officially, " is fire at elevations greater than 15°," and "curved (53)
fire is fire from howitzers at all angles of elevation not exceeding 15°." In these cases the curvature of the trajectory becomes considerable, and the formulae employed in direct fire must be modified; the method generally employed is due to Colonel Siacci of the Italian artillery.
Starting with the exact equations of motion in a resisting medium,
also (56) dy_Cgsecitani.
dq f (q sec i) '
(57) di _ Cg
dq _q sec i . f (q sec i)'
(58) d tan i C g sec i
__
dq q •f(q sec i)'
from which the values of t, x, y, i, and tan i are given by integration with respect to q, when sec i is given as a function of q by means of (51).
Now these integrations are quite intractable, even for a very simple mathematical assumption of the function f(v), say the quadratic or cubic law, f(v) = v2/k or v'/k.
But, as originally pointed out by Euler, the difficulty can be turned if we notice that in the ordinary trajectory of practice the quantities i, cos i, and sec i vary so slowly that they may be replaced by their mean values, n, cos n, and sec n, especially if the trajectory, when considerable, is divided up in the calculation into arcs of small curvature, the curvature of an arc being defined as the angle between the tangents or normals at the ends of the arc.
Replacing then the angle i on the righthand side of equations (54) (56) by some mean value n, we introduce Siacci's pseudovelocity u defined by
(59) u = q sec n,
so that u is a quasicomponent parallel to the mean direction of the tangent, say the direction of the chord of the arc.
by
and eliminating r,
(45) dx d2ydy ex_ dx dt dt2 _ dt dt2 
gdt'
and this, in conjunction with
(46) tan i=dv_dy dx
dt~dc'
2 .di dx d2y  dy d2x dx sec idt= (at d2i dt di2) (dt) 2
(47)
reduces to (48)
di tan i _ g
dt =  v g COs . Or d dt  v cos i
the equation obtained, as in (18), by resolving normally in the trajectory, but di now denoting the increment of i in the increment of time dt.
Denoting dx/dt, the horizontal component of the velocity, by q, so that
(49) v cos i=q,
equation (43) becomes
(5o) dq/dt= r cos i,
and therefore by;(48) (51)
It is convenient to notation
(a2)
(43) 2 _
 r cos i = rds'
(44) d2 = 
sin i
= rdsr
dqdq dt_ry di=dt dig'
express r as a function of v in the previous
Cr=f(v),
vf(v) di_ Cg '
an equation connecting q and i.
Now, since v=q sec .t
(54) dt _ sec i
dq Cf(q sec i)' and multiplying by dx/dt or q,
(55) dx C q sec i
dq f(q sec il' and multiplying by dy!dx or tan i,
Integrating from any initial pseudovelocity U,
u du t=(. f f~u)f
di 6=Crcosn fuf,
tan 47—tan 6= Cgsecn du J u.f(u)
But according to the definition of the functions T, S, I and D of the ballistic table, employed for direct fire, with u written for v,
Now calculate the pseudovelocity ins from
(81) uo=vi cos 4) sec n,
and then, from the given values of 4 and 0, calculate ue'from either of the formulae of (72) or (73) :
(82)
I(ue) I(u4) —tan ) —tang
c(
sec n
°—6°
D(ue)=D(us)— cos n.
Then with the suffix notation to denote the beginning and end of the arc 4—6,
(84) mte =C[T (u4)T{ue)I,
(85) 0+xe =C cos n [S(uo) —S(ue)],
y _ DA
(YCsecn I(u~)—S
A now denoting any finite tabular difference of the function between the initial and final (pseudo) velocity.
x=C cos n f f~u)
y=C sin rt f f(u);
and supposing inclination . to 0 radians the arc.
(63)
(6o)
(61) (62)
over
(64)
(83)
(86)
(65) fu
ufn(t =J gp=T(U)—T(u),
(66) J u du =S (U) —S (u),
(u)
(67) `g du
J =I(U)—I(u)>
uf(u)
and therefore
(68) t =C [T(U) —T(u)],
(69) x=C cos n [S(U) —S(u)],
(70) y=C sin n [S(U) —S(u)],
(71) 47—0=C cos n [I(U) —I (u)],
(72) tan 4—tan B=C sec n [I(U)—I (u)],
while, expressed in degrees,
(73) 0°—6°=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 n cannot be exactly the same mean angle in all these equations; but if n is the same in (69) and (70),
(74) y/x = tan n,
so that n is the inclination of the chord of the arc of the trajectory, as in Niven's method of calculating trajectories (Prot. R. S., 1877): but this method requires n to be known with accuracy, as i %, variation inn causes more than 1 % variation in tan n.
The difficulty is avoided by the use of Siacci's altitudefunction A or A(u), by which y/x can be calculated without introducing sin n or tan n, but in which n occurs only in the form cos n or sec n, which varies very slowly for moderate values of n, so that n need not be calculated with any great regard for accuracy, the arithmetic mean 2(47+6) of y and 0 being near enough for n over any arc ¢—B 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,
(75) tan 4—=C secn [I(U)—I(u)] and integrating with respect to x over the arc considered, (76)
But
(77) f o I (u)dx = fuI (u)dudu
=C cos n f UI(u)udu
9f (u)
=C cos n [A(U) —A(u)]
in Siacci's notation; so that the altitudefunction A must be calculated by summation from the finite difference AA, where
(78)' AA=I(u) 9uAu
P `I(u)AS,
or else by an integration when it is legitimate to assume that f(v) =v'0/k in an interval of velocity in which m may be supposed constant.
Dividing again by x, as given in (76),
(79) tan 4— =Csecn[I(U)—S(U)—S((u))]
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 v4), the curvature of the arc ¢—0 is first settled upon, and now
(8o) n=1(0+6)
is a good first approximation for n.
N, M'
Qz Ns
Also the velocity tie at the end of the arc is given by
(87) ve=u0 sec 0 cos n.
Treating this final velocity vo and angle 0 as the initial velocity vo and angle 47 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 4—0 of an arc should be so chosen that ¢ye, the height ascended, should be limited to about moo ft., equivalent to a fall of 1 inch in the barometer or 3 % diminution in the tenuity factor r.
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 twothirds 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 twothirds 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 0.2in. gun to a shot weighing 38o 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. WolleyDod, R.A., in the Proceedings R.A. Institution, 1888, employing Siacci's method and about twenty arcs: and Captain Ingalls, by assuming a mean tenuityfactor r=o•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 altitudefunction is useful in direct fire, for giving, immediately the angle of elevation 4 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 ,B.
In direct fire the pseudovelocities U and u, and the real velocities V and v, are undistinguishable, and sec n may be replaced by unity so that, putting y=o in (79),
tan 4)=C [I (V)AA. S]•
Also
(89)
(90) tan /3=C [S—I(v)
or, as (88) and (90) may be written for smal angles,
(91) sin 24) 2C [I (V) —AA] AS'
(92) sin 2/3=2C [oS — I (v)]'
To simplify the work, so as to look out the value of sin 2¢ without the intermediate calculation of the remaining velocity v, a doubleentry table has been devised by Captain Braccialini Scipione
x tan 4'—y=C sec n [xI (U)—f I(u)dx].
(88)
tan 4—tan 13=C II(V) L (v)] so that 7
(Problemi del Tiro, Roma, 1883), and adapted to yd., ft., in. and lb units by A. G. Hadcock, late R.A., and published in the Proc. R.A. Institution, 1898, and in Gunnery Tables, 1898.
In this table
(93) sin 20=Ca,
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 /3, 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 5, 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 lefthanded 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 vacuum; 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 righthanded twist. The effect is as if there was a mean sidelong thrust w tan S on the shot from left to right in order to deflect the plane of the trajectory at angle S 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 S.
An effect similar to drift is observable at tennis, golf, baseball and cricket; but this effect is explainable by the inequality of pressure due to a vortex of air carried along by the rotatingball, and the deviation is in the opposite direction of the drift observed in artillery practice, so artillerists are still awaiting theory and crucial experiment.
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 eliminated, such as variations in the air and in the muzzlevelocity, 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 accordance with the laws of probability of the 5o% zones shown in the range table (see PROBABILITY.)
II. INTERIOR BALLISTICS
The investigation of the relations connecting the pressure, volume and temperature of the powdergas inside the bore of the gun, of the work realized by the expansion of the powder, of the
V
dynamics of the movement of the shot up the bore, and of the stress set up in the material of the gun, constitutes the branch of interior ballistics.
A gun may be considered a simple thermodynamic machine or heatengine which does its work in a single stroke, and does not act in a series of periodic cycles as an ordinary steam or gasengine.
An indicator diagram can be drawn for a gun (fig. 3) as for a
21 20
.u..ESMMMEiiiiiiiiiie?CC ENNEEMiMEMMEM~iiiiiMEINIM.'.MO
m~mmmussmommmmms sonommmmmmmsummummummammmm~
IMMMIRMEEMEEEMMMMMMEMMMOUNUMMMMMMENEEMENECdOMSM
UMMEEPMEMEEMPEEEMESEMEMEMMMMMUENWIMMMMOREAMIll
11 ENNUMmIROMEMEMEMMMMMMMENSIMMMEMMMUMMMMIN11'aERAMMMIME
111:MMIEMEMEMMMEEEMMEEIMMIUMEMMEBIMOW. MM 'A MM nAMEMIEC..M1011 ". 11SIMMMMM11MM&EEMOREMISMEMIMIMEMM111M/MlMSZWEIMMMIMOEMM 1M MMMEMIMIMMIMMEMINEEMMMMISIM2WaTelOOMMIENNUME14:: 1ECISEMMMEEMMEMMUSMEMMMIOMMAMERMPEEMMMMEMMMMIMMSMINE
11!..!!^^!!!!!!!!!!!li7Ca!!!!!.!!!!!!!!.!!!!!!!!!! .! .!!!_!!!MMMIM1,~.~ /!!!!!!..!!!!!! ^!!!!!!!!!. ^•^!!! !.^!!lS..a.S!!.!.!!l..S!!!!!iM^!.!!!l.ar3
w6 •so
0.3 Rai/lathe 20 lbs.
18
French 8.N 25
20 „
Amide Lot 232 32 „
R.L.G2 23
EXE 42
Plugs
O 346 8 7 8 9 10 tt 12 t9 —_i }5 ~~'~161I
0 22 24 26 28 30 32 34 36 38 40 42 44 46 Travel in feet __ Pressure Curves, from Chronoscope experiments in 6 inch gun of :oo calibres, with various Explosives.
' ;,n Observed Pressures.
1f ~m 20.1 TONS
19 TONS R.L.G2
18
End of Article: RANGE TABLE FOR 

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