For a shot in air the ratio W′/W is so small that the square may be neglected, and formula (11) can be replaced for practical purpose in artillery by
| tan2 δ = | π2 | = | W′ | (β − α) ( | k2 | ) | 2 | / ( | k1 | ) | 4 | , |
| n2 | W | d | d |
(12)
if then we can calculate β, α, or β − α for the external shape of the shot, this equation will give the value of δ and n required for stability of flight in the air.
The ellipsoid is the only shape for which α and β have so far been determined analytically, as shown already in § 44, so we must restrict our calculation to an egg-shaped bullet, bounded by a prolate ellipsoid of revolution, in which, with b = c,
| A0 = ∫∞0 | ab2 dλ | = ∫∞0 | ab2 dλ | , |
| (a2 + λ) √ [ 4 (a2 + λ) (b2 + λ)2 ] | 2 (a2 + λ)3/2 (b2 + λ) |
(13)
A0 + 2B0 = 1,
(14)
| a = | A0 | , β = | B0 | = | 1 − A0 | = | 1 | . |
| 1 − A0 | 1 − B0 | 1 + A0 | 1 + 2α |