um of all the values of [Delta]S up to any assigned
velocity v,

(15) S(v) = [Sum]([Delta]S) + a constant, by which S(v) is calculated
from [Delta]S, and then between two assigned velocities V and v,

(16) S(V) - S(v) = [Sum,v:V][Delta]T = [Sum]v[Delta]v/gp or
[Integral,v:V]vdv/gp,

and if s feet is the advance of a shot whose ballistic coefficient is C,

(17) s = C[S(V) - S(v)].

In an extended table of S, the value is interpolated for unit increment of
velocity.

A third table, due to Sir W. D. Niven, F.R.S., called the _degree_ table,
determines the change of direction of motion of the shot while the velocity
changes from V to v, the shot flying nearly horizontally.

To explain the theory of this table, suppose the tangent at the point of
the trajectory, where the velocity is v, to make an angle i radians with
the horizon.

Resolving normally in the trajectory, and supposing the resistance of the
air to act tangentially,

(18) v(di/dt) = g cos i,

where di denotes the infinitesimal _decrement_ of i in the infinitesimal
increment of time dt_.

[v.03 p.0272] In a problem of direct fire, where the trajectory is flat
enough for cos i to be undistinguishable from unity, equation (16) becomes

(19) v(di/dt) = g, or di/dt = g/v;

so that we can put

(20) [Delta]i/[Delta]t = g/v

if v denotes the mean velocity during the small finite interval of time
[Delta]t, during which the direction of motion of the shot changes through
[Delta]i radians.

If the inclination or change of inclination in degrees is denoted by
[delta] or [Delta][delta],

(21) [delta]/180 = i/[pi], so that

(22) [Delta][delta] = 180/[pi] [Delta]i = 180g/[pi] [Delta]t/v;

and if [delta] and i change to D and I for the standard projectile,

(23) [Delta]I = g [Delta]T/v = [Delta]v/vp,
[Delta]D = 180g/[pi] [Delta]T/v, and

(24) I(V) - I(v) = [Sum,v:V][Delta]v/vp or [Integral,v:V]dv/vp,
D(V) - D(v) = 180/[pi] [I(V) - I(v)].

The differences [Delta]D and [Delta]I are thus calculated, while the values
of D(v) and I(v) are obtained by summation with the arithmometer, and
entered in their respective columns.

For some purposes it is preferable to retain the circular measure, i
radians, as being undistinguishable from sin i and tan i when i is small as
in direct fire.

The last function A, called the _altitude function_, will be explained when
h