and ΔS is the advance in feet of a shot for which C=1, while the velocity falls Δv in passing through the average velocity v.
Denoting by S(v) the sum of all the values of ΔS up to any assigned velocity v,
(15) S(v) = ∑(ΔS) + a constant, by which S(v) is calculated from ΔS, and then between two assigned velocities V and v,
| (16) S(V) - S(v) = | ∑ | V v | ΔT = | ∑ | vΔv | or | ∫ | V v | vdv | , |
| gp | 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,