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 right-hand side of equations (54) - (56)
by some mean value [eta], we introduce Siacci's pseudo-velocity u defined
by

(59) u = q sec [eta],

so that u is a quasi-component parallel to the mean direction of the
tangent, say the direction of the chord of the arc.

[v.03 p.0274] Integrating from any initial pseudo-velocity U,

(60) t = C[Integral,u:U] du/f(u),
(61) x = C cos [eta] [Integral] u du/f(u),
(62) y = C sin [eta] [Integral] u du/f(u);

and supposing the inclination i to change from [phi] to [theta] radians
over the arc,

(63) [phi] - [theta] = Cg cos [eta] [Integral] du/{u f(u)},
(64) tan [phi] - tan [theta] = Cg sec [eta] [Integral] du/{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,

(65) [Integral,u:U] du/f(u) = [Integral] du/gp = T(U) - T(u),
(66) [Integral] u du/f(u) = S(U) - S(u),
(67) [Integral] g du/u f(u) = I(U) - I(u);

and therefore

(68) t = C[T(U) - T(u)],
(69) x = C cos [eta] [S(U) - S(u)],
(70) y = C sin [eta] [S(U) - S(u)],
(71) [phi] - [theta] = C cos [eta] [I(U) - I(u)],
(72) tan [phi] - tan [theta] = C sec [eta] [I(U) - I(u)],

while, expressed in degrees,

(73) [phi]deg - [theta]deg = C cos [eta] [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 [eta] cannot be exactly the same mean angle in all
these equations; but if [eta] is the same in (69) and (70),

(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 altitude-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],