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46.

ranging from zero to 1000 ft/sec., 1000 to 3000 ft/sec., and from 3000 ft/sec. upward. The first curve represented the experimental results of A. Frank (3) obtained with prolate ellipsoids. The second curve represented the experimental results of A. Mallock (4), whereas the third curve represented an empirical formula by Mallock (5), which agrees well with experimental results up to 4,500 ft/sec. -- the highest velocity that has been attained by projectiles-- and hence may be used for still higher velocities with a fair degree of safety. Mallock's expression, reduced to the absolute ft. lb. sec. system and multiplied by 1/4, the coefficient for projectiles with pointed heads, becomes

P = 0.00006432 v[[superscript]] 2 [[/superscript]] (v[[superscript]] ' [[/superscript]]/a)[[superscript]] 0.375 [[/superscript]] + 480  (8)

where v = the velocity of the projectile,
v[[superscript]] ' [[/superscript]] = the velocity with which a wave is propagated in the air immediately in front of the projectile; which equals the velocity of the body when that velocity exceeds the velocity of sound in the undisturbed gas; and 
a = the velocity of sound in the undisturbed gas.

The constant, 480 poundals, must be added for velocities over 2,400 ft/sec. owing to the vacuum in the rear of the projectile.

[[underlined]] The Quantity, S [[/underlined]].

The above expression (8), for the resistance, holds only at atmospheric pressure. At high altitudes the pressure, of course, decreases greatly. If we call [[underlined]] S [[/underlined]] the mean density throughout any

[[line]]
(3) A. Frank, Zeitschr. Verein Deutsches Ing. 50 pp. 593-612, 1906.
(4) A. Mallock, Proc. Roy. Soc., 79 [[underlined]] A [[/underlined]], pp. 262-273, 1907.
(5) A. Mallock, Proc. Roy. Society, 79 [[underlined]] A [[/underlined]], pp. 267, 1907.