(74) y/x = tan [eta].

so that [eta] is the inclination of the chord of the arc of the trajectory, as in Niven's method of calculating trajectories (_Proc. R.S._, 1877): but this method requires [eta] to be known with accuracy, as 1% variation in [eta] causes more than 1% variation in tan [eta].

The difficulty is avoided by the use of Siacci's alt.i.tude-function A or A(u), by which y/x can be calculated without introducing sin [eta] or tan [eta], but in which [eta] occurs only in the form cos [eta] or sec [eta], which varies very slowly for moderate values of [eta], so that [eta] need not be calculated with any great regard for accuracy, the arithmetic mean ([phi] + [theta]) of [phi] and [theta] being near enough for [eta] over any arc [phi] - [theta] of moderate extent.

Now taking equation (72), and replacing tan [theta], as a variable final tangent of an angle, by tan i or dy/dx,

(75) tan [phi] - dy/dx = C sec [eta] [I(U) - I(u)],

and integrating with respect to x over the arc considered,

(76) x tan [phi] - y = C sec [eta] [xI(U) - [Integral,0:x] I(u)dx],

But

(77) [Integral,0:x] I(u)dx = [Integral,U:u] I(u) dx/du du = C cos [eta] [Integral,x:U] I(u) {u du}/{g f(u)} = C cos [eta] [A(U) - A(u)]

in Siacci's notation; so that the alt.i.tude-function A must be calculated by summation from the finite difference [Delta]A, where

(78) [Delta]A = I(u) u[Delta]u / gp = I(u)[Delta]S,

or else by an integration when it is legitimate to a.s.sume that f(v)=v^m/k in an interval of velocity in which m may be supposed constant.

Dividing again by x, as given in (76),

(79) tan [phi] - y/x = C sec [eta] [I(U) - {A(U) - A(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 [phi] with velocity v_[phi], the curvature of the arc [phi] - [theta] is first settled upon, and now

(80) [eta] = ([phi] + [theta])

is a good first approximation for [eta].

Now calculate the pseudo-velocity u_[phi] from

(81) u_[phi] = v_[phi] cos [phi] sec [eta],

and then, from the given values of [phi] and [theta], calculate u_[theta]

from either of the formulae of (72) or (73):--

(82) I(u_[theta]) = I(u_[phi]) - {tan [phi] - tan [theta]}/{C sec [eta]}, (83) D(u_[theta]) = D(u_[phi]) - {[phi] - [theta]}/{C cos [eta]}.

Then with the suffix notation to denote the beginning and end of the arc [phi] - [theta],

(84) _[phi]t_[theta] = C[T(u_[phi]) - T(u_[theta])], (85) _[phi]x_[theta] = C cos [eta] [S(u_[phi]) - S(u_[theta])], (86) _[phi](y/x)_[theta] = tan [phi] - C sec [eta] [I(u_[phi]) - [Delta]A/[Delta]S];

[Delta] now denoting any finite tabular difference of the function between the initial and final (pseudo-) velocity.

[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.