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 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 40deg; 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 [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 altitude-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 [[De