d Fire._--"High angle fire," as defined officially,
"is fire at elevations greater than 15deg," and "curved fire is fire from
howitzers at all angles of elevation not exceeding 15deg." 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