th of a second short of its impact at B, the height of the
burst is 7.84, say 8 ft.

The line of sight Ox, considered horizontal in range table results, may be
inclined slightly to the horizon, as in shooting up or down a moderate
slope, without appreciable modification of (28) and (29), and y or PM is
still drawn vertically to meet OB in M.

Given the ballistic coefficient C, the initial velocity V, and a range of R
yds. or X = 3R ft., the final velocity v is first calculated from (29) by

(35) S(v) = S(V) - X/C,

and then the time of flight T by

(36) T = C{T(V) - T(v)}.

Denoting the angle of departure and descent, measured in degrees and from
the line of sight OB by [phi] and [beta], the total deviation in the range
OB is (fig. 1)

(37) [delta] = [phi] + [beta] = C{D(V) - D(v)}.

To share the [delta] between [phi] and [beta], the vertex A is taken as the
point of _half-time_ (and therefore beyond _half-range_, because of the
continual diminution of the velocity), and the velocity v_0 at A is
calculated from the formula

(38) T(v_0) = T(V) - 1/2T/C = 1/2{T(V) + T(v)};

and now the degree table for D(v) gives

(39) [phi] = C{D(V) - D(v_0)},
(40) [beta] = C{D(v_0) - D(v)}.

This value of [phi] is the tangent elevation (T.E.); the quadrant elevation
(Q.E.) is [phi] - S, where S is the angular depression of the line of sight
OB; and if O is h ft. vertical above B, the angle S at a range of R yds. is
given by

(41) sin S = h/3R,

or, for a small angle, expressed in minutes, taking the radian as 3438',

(42) S = 1146h/R.

So also the angle [beta] must be increased by S to obtain the angle at
which the shot strikes a horizontal plane--the water, for instance.

A systematic exercise is given here of the compilation of a range table by
calculation with the ballistic table; and it is to be compared with the
published official range table which follows.

A discrepancy between a calculated and tabulated result will serve to show
the influence of a slight change in the coefficient of reduction n, and the
muzzle velocity V.

_Example_ 3.--Determine by calculation with the abridged ballistic table
the remaining velocity v, the time of flight t, angle of elevation [phi],
and descent [beta] of this 6-in. gun at ranges 500, 1000, 1500, 2000 yds.,
taking the muzzle velocity V = 2150 f/s, and a coefficient of reduction n =
0.96. [For Table see p. 274.]

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