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AP be bisected by the perpendicular sto; then with the centre o describing a circle through A and P, the same will also pass through q, because the angle gal, formed by the tangent Ai and AG, is equal to the angle APq, which will therefore stand on the same arc Aq.
97. Corol. 2. If there be given the range Ar and tłie ve. locity, or the impetus, the direction will hence be easily found thus: Take Ak = 4A1, draw ko perp. to Ah, meeting the circle described with the radius ao in two points q and 9; then Aq or Aq will be the direction of the piece: And hence it appears that there are two directions, which, with the same impetus, give the very same range Al. And these two directions make equal angles with AI and AP, because the arc Pq is equal the arc At They also make equal angles with a line drawn from A through s, because the art sq is equal the arc sq.
98. Corol. 3. Or, if there be given the range Al, and the direction Aq; to find the velocity or impetus. Take Ak = AT, and erect ka perp. to Ah, meeting the line of direction in
9; then draw qp making the EAP = L Akq; so shall AP be the impetus, or the altitude due to the projectile velocity
99. Corol. 4. The range on an oblique plane, with a given elevation, is directly proportional to the rectangle of the cosine of the direction of.the piece above the horizon, and the sine of the direction above the oblique plane, and reciprocally to the square of the cosine of the angle of the plane above or below the horizon. For, put s = sin. Lqar or APQ,
c = cos. Lqah or sin. PAq,
c= cos. LIAH or sin. akd.or akq or Aqp.
100. The range is the greatest when ak is the greatest; that is, when kq touches the circle in the middle point s; and then the line of direction passes through s, and bisects the angle formed by the oblique plane and the vertex. Also, the ranges are equal at equal angles above and below this direction for the maximum. 101. Coril. 5. The greatest height ov or kg of the projec
g2 tile, above the plane, is equal to X AP. And therefore it
ca. is as the impetus and square of the sine of direction above the plane directly, and square of the cosine of the plane's inclination reciprocally. For c (sin. aqe): s (sin. Apg) :: Ap : Aq,
and € (sin. Akq): s (sin, kaq):: Aq: kq,
theref. by comp. c?: 52:: AP : kq. 102. Corol. 6. The time of flight in the curve Avi is =25
where g = 1615 feet. And therefore it is as the velocity atd sine of direction above the plane directly, and cosine of the plane's inclination reciprocally. For the time of describing the curve, is equal to the time of falling freely through Gi or 4kq or ä * Ap. Therefore, the time being as the square root of the distance,
25 ✓g: ŅAP :: 1"
the time of flight.
103. From the foregoing corollaries may be collected the following set of theoremis; relating to projects made on any given inclined planes, either above or below the horizontal plane. In which the letters denote as before, namely,
of direction above the horizon,
the range on the oblique plane,
the projectile velocity,
the impetus, or alt. due to the velocity v, g = 1611 feet. Then,
8 And from any cf these, the angle of direction may be found
104. THE two foregoing propositions contain the whole theory of projectiles, with theorems for all the cases, regularly arranged for use, both for oblique and horizontal planes. But, before they can be applied to use in resolving the several cases in the practice of gunnery, it is necessary that some more data, be laid down, as derived from good experiments made with balls or shells discharged from cannon or mortars, by gunpowder, under different circumstances. For, without such experiments and data, those theorems can be of very little use in real practice, on account of the imperfections and irregularities in the firing of gunpowder, and the expulsion of balls from guns, but more especially on account of the enormous resistance of the air to all projectiles made with any velocities that are considerable. As to the cases in which projectiles are made with small velocities, or such as do not exceed 200, or 300, or 400 feet per second of time, they may be resolved tolerably near the truth, especially for the larger shells, by the parabolic theory, laid down above. But, in cases of great projectile velocities, that theory is quite inadequate, without the aid of several data drawn from many and good experiments. For so great is the effect of the resistance of the air to projectiles of considerable velocity, that some of those which in the air range only between 2 and 3 miles at the most, would in vacuo range about ten times as far, or between 20 and 30 miles.
The effects of this resistance are also various, according to the velocity, the diameter, and the weight of the projectile. So that the experiments made with one size of ball or shell, will not serve for another size, though the velocity should be the same; neither will the experiments made with one velocity, serve for other velocities, though the ball be the same. And therefore it is plain that, to form proper rules for practical gunnery, we ought to have good experiments made with each size of mortar, and with every variety of charge, from the least to the greatest. And not only so, but these ought also to be repeated at many different angles of elevation, namely, for every single degree between 30° and 60°elevation, and at intervals of 50 above 60' and below 30°, froin the vertical direction to point blank. By such a course of experiments it will be found, that the greatest range, instead of being constantly that at an elevation of 45°, as in the parabolic theory, will be at all intermediate degrees between 45 and 30,
being more or less, both according to the velocity and the weight of the projectile; the smaller velocities and larger shells ranging farthest when projected almost at an elevation of 45°; while the greatest velocities, especially with the smaller shells, range farthest with an elevation of about 30°.
105. There have, at different times, been made certain small parts of such a course of experiments as is hinted at above. Such as the experiments or practice carried on in the year 1773, on Woolwich Common; in which all the sizes of mortars were used, and a variety of small charges of powder. But they were all at the elevation of 45° ; consequently these are defective in the higher charges, and in all the other angles of elevation.
Other experiments were also carried on in the same place in the years 1784 and 1786, with various angles of elevation indeed, but with only one size of mortar, and only one charge of powder, and that but a small one too : so that all those nearly agree with the parabolic theory. Other experiments have also been carried on with the ballistic pendulum, at different times; from which have been obtained some of the laws for the quantity of powder, the weight and velocity of the ball, the length of the gun, &c. Namely, that the velocity of the ball varies as the square root of the charge directly, and as the square root of the weight of ball reciprocally; and that, some rounds being fired with a medium length of one-pounder gun, at 15° and 45° elevation, and with 2, 4, 8, and 12 ounces of powder, gave nearly the velocities, ranges, and times of flight, as they are here set down in the following Table.
106. But as we are not yet provided with a sufficient number and variety of experiments, on which to establish true rules for practical gunnery, independent of the parabolic theory, we must at présent content ourselves with the data of VOL. II.
some one certain experimented range and time of fight, at a given angle of elevation; and then by help of these, and the rules in the parabolic theory, determine the like circumstances for other elevations that are not greatly different from the former, assisted by the following practical rules.
SOME PRACTICAL RULES IN GUNNERY.
1. To find the Velocity of any Shot or Shell. Rule. Divide double the weight of the charge of powder by the weight of the shot, both in lbs. Extract the square root of the quotient. Multiply that root by 1600, and the product will be the velocity in feet, or the number of feet the shot passes over per second.
Or say-As the root of the weight of the shot, is to the root of double the weight of the powder, so is 1600 feet, to the velocity
II. Given the range at One Elevation ; to find the Range at .
Another Elevation. RULE. As the sine of double the first elevation, is to its range; so is the sine of double another elevation, to its range. III. Given the Range for One Charge ; to find the Range for
Another Charge, or the Charge for Another Range. RULE. The ranges have the same proportion as the charges; that is, as one range is to its charge, so is any other range to its charge: the elevation of the piece being the same in both cases.
107. Example 1. If a ball of 1 lb. acquire a velocity of 1600 feet per second, when fired with 8 ounces of powder ; it is required to find with what velocity each of the several kinds of shells will be discharged by the full charges of powder, viz.
108. Exam. 2. If a shell be found to range 1000 yards, when discharged at an elevation of 45°; how far will it