_High Angle and Curved Fire._--"High angle fire," as defined officially, "is fire at elevations greater than 15," and "curved fire is fire from howitzers at all angles of elevation not exceeding 15." In these cases the curvature of the trajectory becomes considerable, and the formulae employed in direct fire must be modified; the method generally employed is due to Colonel Siacci of the Italian artillery.
Starting with the exact equations of motion in a resisting medium,
(43) d^2x/dt^2 = -r cos i = -r dx/ds,
(44) d^2y/dt^2 = -r sin i - g = -r dy/ds - g,
and eliminating r,
(45) dx/dt d^2y/dt^2 - dy/dt d^2x/dt^2 = -g{dx/dt};
and this, in conjunction with
(46) tan i = dy/dx = {dy/dt}/{dx/dt},
(47) sec^2 i{di/dt} = ({dx/dt}{d^2y/dt^2} - {dy/dt}{d^2x/dt^2}) / (dx/dt)^2,
reduces to
(48) di/dt = -{g/v} cos i, or {d tan i}/dt = -g/{v cos i},
the equation obtained, as in (18), by resolving normally in the trajectory, but di now denoting the _increment_ of i in the increment of time dt.
Denoting dx/dt, the horizontal component of the velocity, by q, so that
(49) v cos i = q,
equation (43) becomes
(50) dq/dt = -r cos i,
and therefore by (48)
(51) dq/di = {dq/dt} {dt/di} = {rv}/g.
It is convenient to express r as a function of v in the previous notation
(52) Cr = f(v),
and now
(53) dq/di = {v f(v)}/{Cg},
an equation connecting q and i.
Now, since v = g sec i
(54) dt/dq = -C sec i / f(q sec i),
and multiplying by dx/dt or q,
(55) dx/dq = -C q sec i / f(q sec i),
and multiplying by dy/dx or tan i,
(56) dy/dq = -C q sec i tan i / f(q sec i);
also
(57) di/dq = Cg / {q sec i . f(q sec i)},
(58) d tan i/dq = C g sec i / {q . f(q sec i)},
from which the values of t, x, y, i, and tan i are given by integration with respect to q, when sec i is given as a function of q by means of (51).
Now these integrations are quite intractable, even for a very simple mathematical assumption of the function f(v), say the quadratic or cubic law, f(v) = v^2/k or v^3/k.
But, as originally pointed out by Euler, the difficulty can be turned if we notice that in the ordinary trajectory of practice the quantities i, cos i, and sec i vary so slowly that they may be replaced by their _mean_ values, [eta], cos [eta], and sec [eta], especially if the trajectory, when considerable, is divided up in 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] - [theta] = 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),