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Fig. 4. projectile force. And the experiment may be varied by having a number of balls prepared of the same weight, and varying the velo cities and the distances from the centre. The effects of gravity, how ever, and the difficulty of representing by a straight line what maybe considered the direction of the circle, have prevented me from deter mining geometrically the direction of the projectile, although in prac tice it may easily be ascertained. If the ball be discharged from the point A with one revolution in a second, its velocity in the circle would be 12-57 feet per second, and its centrifugal velocity would be = 39-44 feet per second, and the initial projectile velocity would be = V 12-57 5 +39-44 2 =41-40 feet per second, disregarding for the present atmospheric resistance. And if, in the w-ay of illustration, AF be considered as the direction of the force in the circle AD, the sides Ak and Am, of the parallelogram Amvk, being made proportionate to the two velo cities 12-57 and 39-50 respectively, the diagonal Av of the parallelo gram will represent in direction and proportional amount the velocity 41-45 or initial projectile velocity, if a billiard-ball, moving upon a table with a velocity equal to 124 feet per second in the direction EF, were to receive at A an impulse in the direction of cn, which alone would cause it to move with a velocity equal to 394 feet per second, no other direction and velocity could be assigned to it, than that de signated by the diagonal Av of the parallelogram. The revolving ball is supposed to move in the direction Ak with the velocity of 12-57 feet per second, represented by that side of the parallelogram, and at the same time to be acted upon by a force which would cause it to move with a velocity equal to 394 feet per second, in the di rection of the side Am, which indicates that velocity, consequently no other direction nor amount can be assigned to it, when projected, than the diagonal Av of the parallelogram Amvk. If the velocity of the ball be doubled, the centrifugal velocity increasing as the square of the increased velocity in the circle, it would be = 39-44x4= 157*76 feet per second, and the initial projectilevelocitv would be =V25-14'+158 2 = 160 feet per second; and the two first would be represented by the sides Ah and An, respectively, of the parallelogram Anyh, and the diagonal Ay would indicate the direction and relative proportion of the initial projectile velocity. With four revolutions in a second, the initial projectile velocity would be 635 feet per second, in the di rection of the line Az. At least such would be the directions for those three velocities at the instant the ball leaves the point from which it may be discharged. But with such low velocities a pound ball would not indicate those directions by its path, for the reasons given above. With very high increasing velocities, however, the experimenter w-ill find that a small leaden ball will move in directions approaching that of the radius, as shown in the diagram. In repeated experi ments made with a machine revolving vertically, and having a tube placed in the direction of a tangent to the circle in which leaden balls were revolved, it w-as found that with very high velocities they were forced through the tuhe with difficulty, and a portion of each was removed by the friction, and the upper part of the tube, on the inside, was worn smooth. But with much lower velocities the balls passed through the tube without any apparent friction. In performing the first experiment, the bar, (A, Fig. 1,) moving with uniform velocity in every part of the circle BD, has the same centri fugal force at v that it would have after revolving for a minute or more; for the amount of that force depends upon the curvature and the circular velocity, and consequently was excited to the amount of thirty-nine pounds instantaneously, and if it had been discharged at three inches from B it would have been projected with that force. If this were not the case with bodies moving in space, supposed to be thus deflected, they would fall to the centre of attraction. Now as this is the fact, the tangent B.r in the diagram only serves, as every mathe matician knows, to show geometrically the amount of defection in a unit of time, measured at right angles to that line, the space .vv repre senting that through which the centripetal force alone, acting uni formly, would cause the body to fall in the fiftieth part of a second; the tangent, therefore, represents the line from w-hich the body mould be defected in an instant of time, and not that in the direction of which it would move with all its projectile force. Again, if the segment of a fly-wheel disintegrated by centrifugal force would be projected “ in a straight line, whose direction is that of the tangent,” the pressure which produces the fracture must act upon each particle of iron in the direction of a tangent to the circle in which the particle is revolved, for the direction of a moving body is always that in which a single force, or the resultant of two or more forces, acts to cause the motion. And it is self-evident that no amount of force, applied in that direction upon the particles in the revolving rim, could overcome the attraction of cohesion. And it is equally evident that such cannot be the direction in which the pressure acts, for whilst it is stated that the tangent is the direction in which the dissevered fragment is projected, we are informed that the force which causes the fracture acts at right angles to the tangent. By the theory given above, however, which is founded upon obser vation and experiment, all the circumstances that attend this pheno menon are easily explained. And when we consider the immense increase of centrifugal force as the velocity of the rim is increased, and the direction m which the resultant of the two forces acts, we ought not to be surprised to find that such masses of iron can be broken and projected with so much destructive effect by this powerful agent. The operation of the sling may also, in this way, be explained in a few' words. For a man, with a thong three and a half feet long, has only to give to a stone at the final effort a velocity, in a very small segment of a circle, equal to 132 feet per second, which would be at the rate of 360 revolutions in a minute, and he will project it with a force equal to that given to a ball of the same weight by an ordinary charge of gunpowder, after deducting one-third of its initial velocity for atmos pheric resistance. But to “ accumulate” an equal force in the circle by the strength of his arm, he would have to revolve the stone at the rate of 6850 revolutions in a minute, which is impossible. Without intending to enter into any particulars as to the probable results of a practical application of this principle, I will close with a few remarks designed to show the amount of force excited by the ro tation of heavy bodies about fixed axes, and the extent to which we may reasonably conclude it might be employed, if it could be con trolled, by giving the relative proportions of the power necessary to revolve a body and the central force excited, considered abstractedly, apart from friction and atmospheric resistance. “ The arc which the revolving body describes in a given time is a mean proportional be tween the radius of the circle and double the space which its centri petal force alone, acting uniformly, would cause it to fall through in the same time.”* Consequently the diameter is to the circumference as the circumference is to the space which the centripetal force of the body would make it fall through in the time of one revolution. That space, therefore, is to the circumference as 3-141 is to unit, [3‘141 being the circumference of a circle whose diameter is unit,] and the central velocity or force for an entire revolution in a second is equal * Cavallo’s Nat. Philos, p. tin.