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 τ=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 altitude-function is useful in direct fire, for giving immediately the angle of elevation φ 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 β.

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

(88) tan φ = C[I(V) - δA].
δS

Also

(89) tan φ - tan β = C [I(V) - L(v)]

so that

(90) tan β = C[δA- I(v)],
δS

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

(91) sin 2φ = 2C [I(V) - δA],
δS
(92) sin 2β = 2C [δA - I(v)].
δS

To simplify the work, so as to look out the value of sin 2φ without the intermediate calculation of the remaining velocity v, a double-entry table has been devised by Captain Braccialini Scipione (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 2φ = 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 φ and β, as calculated for the range tables above, are also given there for comparison.