and equating this to the impressed couple (multiplied by g), that is,
to
c1
gN = (c1 - c2)-- u² tan [theta], (5)
c2
and dividing out sin[theta], which equated to zero would imply perfect
centring, we obtain
c1
C2[mu]² cos [theta] - C1p[mu] + (c2 - c1)-- u² sec [theta] = 0. (6)
c2
The least admissible value of p is that which makes the roots equal of
this quadratic in [mu], and then
C1
[mu] = ½ --p sec [theta], (7)
C2
the roots would be imaginary for a value of p smaller than given by
c1
C1²p² - 4(c2 - c1)-- C2u² = 0, (8)
c2
p² c1 C2
-- = 4 (c2 - c1) -- ---. (9)
u² c2 C1²
_Table of Rifling for Stability of an Elongated Projectile, x Calibres
long, giving [delta] the Angle of Rifling, and n the Pitch of Rifling
in Calibres._
+-----------------------+-----------------+-----------------+-----------------+-----------------+
| | Cast-iron Common| Palliser Shell | Solid Steel | Solid Lead |
| | Shell [f] = 2/3,| [f] = ½, S.G. 8.| Bullet | Bullet [f] = 0, |
| | S.G. 7.2. | | [f] = 0, S.G. 8.| S.G. 10.9. |
+--------+--------------+--------+--------+--------+--------+--------+--------+--------+--------+
| x |[beta]-[alpha]|[delta] | n |[delta] | n |[delta] | n |[delta] | n |
+--------+--------------+--------+--------+--------+--------+--------+--------+--------+--------+
| 1.0 | 0.0000 | 0° 0´|Infinity| 0° 0´|Infinity| 0° 0´|Infinity| 0° 0´|Infinity|
| 2.0 | 0.4942 | 2 49 | 63.87 | 2 32 | 71.08 | 2 29 | 72.21 | 2 08 | 84.29 |
| 2.5 | 0.6056 | 3 46 | 47.91 | 3 23 | 53.32 | 3 19 | 54.17 | 2 51 | 63.24 |
| 3.0 | 0.6819 | 4 41 | 38.45 | 4 13 | 42.79 | 4 09 | 43.47 | 3 38 | 50.74 |
| 3.5 | 0.7370 | 5 35 | 32.13 | 5 02 | 35.75 | 4 58 | 36.33 | 4 15 | 42.40 |
| 4.0 | 0.7782 | 6 30 | 27.60 | 5 51 | 30.72 | 5 45 | 31.21 | 4 56 | 36.43 |
| 4.5 | 0.8100 | 7 24 | 24.20 | 6 40 | 26.93 | 6 32 | 27.36 | 5 37 | 31.94 |
| 5.0 | 0.8351 | 8 16 | 21.56 | 7 28 | 23.98 | 7 21 | 24.36 | 6 18 | 28.44 |
| 6.0 | 0.8721 | 10 05 | 17.67 | 9 04 | 19.67 | 8 56 | 19.98 | 7 40 | 23.33 |
| 10.0 | 0.9395 | 16 57 | 10.31 | 15 19 | 11.47 | 15 05 | 11.65 | 13 00 | 13.60 |
|Infinity| 1.0000 | 90 00 | 0.00 | 90 00 | 0.00 | 90 00 | 0.00 | 90 00 | 0.00 |
+--------+--------------+--------+--------+--------+--------+--------+--------+--------+--------+
If the shot is moving as if fired from a gun of calibre d inches, in
which the rifling makes one turn in a pitch of n calibres or nd
inches, so that the angle [delta] of the rifling is given by
tan [delta] = [pi]d/nd = ½ dp/u, (10)
which is the ratio of the linear velocity of rotation ½dp to u, the
velocity of advance,
c1 C2d²
tan² [delta] = [pi]²/n² = d²p²/4u² = (c2 - c1) -- ----
c2 C1²
W´ / W´ \ / k2 \²
1 + --- [alpha] ( 1 + ---[epsilon] ) ( -- )
W´ W \ W / \ d /
= --- ([beta] - [alpha]) --------------- · -----------------------------. (11)
W W´ / k1 \^4
1 + ---[beta] ( -- )
W \ d /
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
[pi]² W´ / k2 \² / / k1 \^4
tan²[delta] = ----- = --- ([beta] - [alpha]) ( -- ) / ( -- ), (12)
n² W \ d / / \ d /
if then we can calculate [beta], [alpha], or [beta] - [alpha] for the
external shape of the shot, this equation will give the value of
[delta] and n required for stability of flight in the air.
The ellipsoid is the only shape for which [alpha] and [beta] 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,
_ _
/ [oo] ab² d[lambda] / [oo] ab² d[lambda]
A0 = | ------------------------------------------------------- = | -----------------------------------, (13)
_/ 0 (a² + [lambda])[root][4(a² + [lambda])(b² + [lambda])²] _/ 0 2(a² + [lambda])^3/2 (b² + [lambda])