51. An elongated shot is made to preserve its axial flight through the air by giving it the spin sufficient for stability, without which it would turn broadside to its advance; a top in the same way is made to stand upright on the point in the position of equilibrium, unstable statically but dynamically stable if the spin is sufficient; and the investigation proceeds in the same way for the two problems (see GYROSCOPE). The effective angular inertia of the body in the medium is now required; denote it by C1 about the axis of the figure, and by C2 about a diameter of the mean section. A rotation about the axis of a figure of revolution does not set the medium in motion, so that C1 is the moment of inertia of the body about the axis, denoted by Wk1². But if Wk2² is the moment of inertia of the body about a mean diameter, and [omega] the angular velocity about it generated by an impulse couple M, and M´ is the couple required to set the surrounding medium in motion, supposed of effective radius of gyration k´, Wk2²[omega] = M - M´, W´k´²[omega] = M´, (1) Wk2² + W´k´²[omega] = M, (2) C2 = Wk2² + W´k´² = (W + W´[epsilon])k2², (3) in which we have put k´² = [epsilon]k², where [epsilon] is a numerical factor depending on the shape. If the shot is spinning about its axis with angular velocity p, and is preceding steadily at a rate [mu] about a line parallel to the resultant momentum F at an angle [theta], the velocity of the vector of angular momentum, as in the case of a top, is C1p[mu] sin [theta] - C2[mu]² sin [theta] cos [theta]; (4) 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._