y, and the curving
effect of gravity is ignored.

If [Delta]t seconds is the time during which the resistance of the air, R
lb, causes the velocity of the shot to fall [Delta]v (f/s), so that the
velocity drops from v+1/2[Delta]v to v-1/2[Delta]v in passing through the
mean velocity v, then

(3) R[Delta]t = loss of momentum in second-pounds,
= w(v+1/2[Delta]v)/g - w(v-1/2[Delta]v)/g = w[Delta]v/g

so that with the value of R in (1),

(4) [Delta]t = w[Delta]v/nd^2pg.

We put

(5) w/nd^2 = C,

and call C the ballistic coefficient (driving power) of the shot, so that

(6) [Delta]t = C[Delta]T, where
(7) [Delta]T = [Delta]v/gp,

and [Delta]T is the time in seconds for the velocity to drop [Delta]v of
the standard shot for which C=1, and for which the ballistic table is
calculated.

Since p is determined experimentally and tabulated as a function of v, the
velocity is taken as the argument of the ballistic table; and taking
[Delta]v = 10, the average value of p in the interval is used to determine
[Delta]T.

Denoting the value of T at any velocity v by T(v), then

(8) T(v) = sum of all the preceding values of [Delta]T plus an
arbitrary constant, expressed by the notation
(9) T(v) = [Sum]([Delta]v)/gp + a constant, or [Integral]dv/gp + a
constant, in which p is supposed known as a function of v.

The constant may be any arbitrary number, as in using the table the
difference only is required of two tabular values for an initial velocity V
and final velocity v and thus

(10) T(V) - T(v) = [Sum,v:V][Delta]v/gp or [Integral,v:V]dv/gp;

and for a shot whose ballistic coefficient is C

(11) t = C[T(V) - T(v)].

To save the trouble of proportional parts the value of T(v) for unit
increment of v is interpolated in a full-length extended ballistic table
for T.

Next, if the shot advances a distance [Delta]s ft. in the time [Delta]t,
during which the velocity falls from v+1/2[Delta]v to v-1/2[Delta]v, we
have

(12) R[Delta]s = loss of kinetic energy in foot-pounds
=w(v+1/2[Delta]v)^2/g - w(v-1/2[Delta]v)^2/g = wv[Delta]v/g,
so that
(13) [Delta]s = wv[Delta]v/nd^2pg = C[Delta]S, where
(14) [Delta]S = v[Delta]v/gp = v[Delta]T,

and [Delta]S is the advance in feet of a shot for which C=1, while the
velocity falls [Delta]v in passing through the average velocity v.

Denoting by S(v) the s