Volume 3, Part 1, Slice 2 Part 30 (1/2)

_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,