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]) A0 + 2B0 = 1, (14) A0 B0 1 - A0 1 [alpha] = ------, [beta] = ------ = ------ = ------------. (15) 1 - A0 1 - B0 1 + A0 1 + 2[alpha] The length of the shot being denoted by l and the calibre by d, and the length in calibres by x l/d = 2a/2b = x, (16) x 1 A0 = ----------- ch^(-1)x - ------, (17) (x²- 1)^3/2 x² - 1 -x x² 2B0 = ----------- ch^(-1)x + ------, (18) x² - 1)^3/2 x² + 1 x sh^(-1) [root](x²-1) x x²A0 + 2B0 = ---------------------- = ------------ log [x + [root](x² - 1)]. (19) [root](x²-1) [root](x²-1) If [sigma] denotes the density of the metal, and if the shell has a cavity homothetic with the external ellipsoidal shape, a fraction f of the linear scale; then the volume of a round shot being 1/6 [pi] d^3, and 1/6 [pi] d^3 x of a shot x calibres long W = 1/6 [pi] d^3 x (i - f^3) [sigma], (20) d² Wk1² = 1/6 [pi] d^3 x -- (1 - f^5) [sigma], (21) 10 l² + d² Wk2² = 1/6 [pi] d^3 x ------- (1 - f^5) [sigma]. (22) 20 If [rho] denotes the density of the air or medium W´ = 1/6 [pi] d^3 x [rho], (23) W´ 1 [rho] --- = ------ -------, (24) W 1 - f³ [sigma] k1² 1 1 - f^5 k2² x² + 1 --- = --- -------, ---- = ------, (25) d² 10 1 - f³ k1² 2