fire and passes through the center of gravity, in such a manner that if the projectile had no rotation it would "tumble." Tumbling actually takes place when the velocity of rotation is not great enough to secure stability in the flight of the bullet, as sometimes occurs in shooting a 22 L.R. through a barrel designed for a 22 short, or when the bullet is too small and does not grip the rifling sufficiently. The rapidly spinning top furnishes an identical illustration of a projectile. As soon as it has attained an upright position, seemingly motionless, if the upper pole of the top be touched on one side, the top loses its equilibrium. If its rotation has been decreased too much it will tumble; if its rotation has not been appreciably affected the upper pole will describe a circle as long as the extraneous force continues. Just as soon as the tangent of the curve is no longer parallel to the axis of the projectile, the air strikes the point of the projectile on one side, see CT', Fig. 1, and being a constant extraneous force, the point (pole) of the axis is caused to move in a circle in the same direction as the twist of the barrel. The actual effect of the air at this stage is to lift the point of the projectile as air lifts a kite, but just as soon as the point is inclined to the plane of fire, that is—just as soon as it begins to describe a circle, the projectile is forced bodily outward. It may be shown analytically that the more rapidly the projectile rotates the longer will be the time required for its point to describe this circle; and it may be similarly shown that the longer the time required for the point to describe this circle, the greater will be the outward deflection of the projectile; hence the more rapidly the projectile spins the greater the outward deflection or so-called drift of the projectile. Again, it may be shown that this deflection increases more rapidly than the range, and thus the line that represents this deflection is an increasing curve convex to the plane of fire. The effect of wind upon this deflection or socalled drift plays a paradoxical role'. The wind acts chiefly upon the head of the projectile, being greatly condensed there around, and is thrown off in stream lines which unite again in the wake of the projectile. Any sidewise motion of the air will therefore cause an unequal pressure upon the head, the effect of which is to produce a precession of the axis independent of that produced by the rotation of the projectile and the direct resistance of the air. For illustration, a left wind blowing upon the head of a projectile with a right hand rotation causes the point to fall or droop, and by so doing brings its axis more nearly in coincidence with the tangent, and thus diminishes, while a right wind causes the point to rise and thereby increases the deflection to the right. Baill gives the following excellent formula for calculating drift, and for direct fire agrees practically with the result obtained by Mayevski's formula. nk gl D= 2n R2 h a V t Cos- 1+0.0000184 (1+2+) { t2 -) t2 3 For. 40 in which equals 3.1416, n the length of twist in calibers, k the radius of gyration of the projectile with reference to its axis, R the radius, g equals 32.16, a quantity depending upon the length of the projectile, shape of head, angle of resistance, and the distance of center of pressure from the center of gravity, angle a is the angle of departure. Formulae have been given for determining the co-efficient of form or reduction, c, and these formulae are analytically correct when the projectile went "point on," that is, when the resistance of the air was parallel to the axis of the projectile. But owing to the fact that the air strikes the head obliquely and is continually varying, the value of c is thereby increased and thus the value of C decreased. Therefore, the true value of c must habitually be determined by experiment. Chapter V. formula 13. But the reader should not conclude that the formulae given for determining the value of c are useless. Before the projectile, and afterward the cartridge, can be determined, a theoretical value of c must be known, and this value will not deviate greatly from that determined by experiment. See Research Work, Vol. 11. CHAPTER XIV The Mean Vertical, Mean Horizontal and Mean Absolute Deviations. T HE Mean Vertical Deviation is the average ver tical divergence and the Mean Horizontal the average horizontal divergence of an aggregate of shots from the calculated course of flight. Problem 26. To determine the mean vertical and the mean horizontal deviations. Assume A, B, C, D, etc., to be points of impact of 10 shots. Fig. 11. Draw any horizontal line, XX, and construct any line, YY, perpendicular to XX. Tabulate the distance of each point of impact from the line, XX, as follows, (using arbitrary numbers simply for illustration). 64.2 sum of ten shots, then 6.4 is the average. |