Volume 3, Part 1, Slice 2 Part 31 (2/2)

[Ill.u.s.tration: FIG. 2.]

Also the velocity v_{[theta]} at the end of the arc is given by

(87) v_[theta] = u_[theta] sec [theta] cos [eta].

Treating this final velocity v_[theta] and angle [theta] as the initial velocity v_[phi] and angle [phi] 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 [phi] - [theta] of an arc should be so chosen that _[phi]y_[theta] the height ascended, should be limited to about 1000 ft., equivalent to a fall of 1 inch in the barometer or 3% diminution in the tenuity factor [tau].

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 a.s.sumes 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 ill.u.s.trated 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. Inst.i.tution_, 1888, employing Siacci's method and about twenty arcs; and Captain Ingalls, by a.s.suming a mean tenuity-factor [tau]=0.68, 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 alt.i.tude-function is useful in direct fire, for giving immediately the angle of elevation [phi] 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 [beta].

In direct fire the pseudo-velocities U and u, and the real velocities V and v, are undistinguishable, and sec [eta] may be replaced by unity so that, putting y = 0 in (79),

(88) tan [phi] = C [I(V) - [Delta]A/[Delta]S].

Also

(89) tan [phi] - tan [beta] = C [I(V) - L(v)]

so that

(90) tan [beta] = C [[Delta]A/[Delta]S - I(v)],

or, as (88) and (90) may be written for small angles,

(91) sin 2[phi] = 2C [I(V) - [Delta]A/[Delta]S], (92) sin 2[beta] = 2C [[Delta]A/[Delta]S - I(v)].

To simplify the work, so as to look out the value of sin 2[phi] without the intermediate calculation of the remaining velocity v, a double-entry table has been devised by Captain Braccialini Scipione [v.03 p.0275] (_Problemi del Tiro_, Roma, 1883), and adapted to yd., ft., in. and lb units by A. G.

Hadc.o.c.k, late R.A., and published in the _Proc. R.A. Inst.i.tution_, 1898, and in _Gunnery Tables_, 1898.

In this table

(93) sin 2[phi] = 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 [phi] and [beta], 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 [delta], called the permanent angle of deflection (see SIGHTS).

This effect is called _drift_ and the reason of it is not yet understood very clearly.

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