187. Geometrical constructions of the principal cases in projectiles in a non-resisting medium, flow readily from the properties of the parabola; and in many cases those constructions suggest simple modes of computation. The following problems will serve by way of exercise. 1. Given the impetus and elevation; to find, by construc. tion, the range, on a horizontal plane, the greatest height, and thence the time of flight. 2. Given the impetus, and the range, on a horizontal plane; to find, by construction, the elevation, and the greatest height. 3. Given the elevation, and the range on a horizontal plane; to find, by construction, the impetus, the greatest height, and thence by computation, the time. 4. Given the impetus, the point and direction of projection, to find the place where the ball will fall upon any plane given in position. 5. Given the impetus and the point of projection, to find the elevation necessary to hit any given point; and to show the limits of possibility. Both construction and mode of computation are required. PRACTICAL GUNNERY. 188. We have now given the whole theory of projectiles, with theorems for all the cases, regularly arranged for use, both for oblique and horizontai planes. But, before they can be applied 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 utility 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 inVOL. II. 29 adequate, 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 aboul 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 prac tical 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 repcated at many different angles of elevation, namely, for every single degree between 30 and 60° elevation, and at intervals of 5° above 650 and below 30, from the vertical direction to point blank. By such a course of experi ments 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°, or little more. 189. 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 direct ly, 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 15o and 45° elevation, and with 2. 4, 8, and 15 ounces of powder, gave nearly the velocities, ranges, and times of flight, as they are here set down in the following table. But good experiments are wanted with large balls and shells. 190. 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 present content ourselves with the data of some one certain experimental range and time of flight, 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.— 191. SOME PRACTICAL RULES IN GUNNERY. I. To find the Velocity of any Shot or Sheil. 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, nearly. 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*. In more recent experiments carried on at Woolwich, by the Editor of the present edition, in conjunction with the select committee of artillery officers, it has been found that a charge of a third of the weight of the ball, gives, at a medium, a velocity of 1600 feet; gunpowder being much improved in its manufacture since the time when 11. 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. 192. EXAMPLE 1. If a ball of 1lb. 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. EXAM. 2. If a shell be found to range 1000 yards when discharged at an elevation of 45°; how far will it range when the elevation is 30° 16', the charge of powder being the 'same? Ans. 2612 feet, or 871 yards. EXAM. 3. The range of a shell, at 45° elevation, being found to be 3750 feet; at what elevation must the piece be set, to strike an object at the distance of 2810 feet, with the same charge of powder? Ans. at 24° 16', or at 65° 44'. EXAM. 4. With what impetus, velocity, and charge of pow der, must a 13-inch shell be fired, at an elevation of 32° 12', to strike an object at the distance of 3250 feet? Ans. impetus 1802, veloc. 340, change 4lb. 7 oz. EXAM. 5. A shell being found to range 3500 feet, when 3c Bir Tho. Blomfield and Dr. Hutton made their experiments. Putting for the weight of the ball, and c for that of the charge, v = 1600√now found a good approximative theorem for the initial velocity. discharged at an elevation of 25° 12'; how far then will it range at an elevation of 36° 15′ with the same charge of powder ? Ans. 4332 feet. EXAM. 6. If, with a charge of 9lb. of powder, a shell range 4000 feet; what charge will suffice to throw it 3000 feet, the elevation being 45° in both cases? Ans. 63lb. of powder. EXAM. 7. What will be the time of flight for any given range, at the elevation of 45°, or for the greatest range? Ans. the time in secs. is the sq. root of the range in feet. ÈXAM. 8. In what time will a shell range 3250 feet, at an elevation of 32° ? Ans. 11 sec. nearly. EXAM. 9. How far will a shot range on a plane which ascends 8° 15', and another which descends 8° 15'; the im. petus being 3000 feet, and the elevation of the piece 32° 30' ? Ans. 4244 feet on the ascent, and 6745 feet on the descent. EXAM. 10. How much powder will throw a 13-inch shell 4244 feet on an inclined plane, which ascends 8° 15', the elevation of the mortar being 32° 30' ? Ans. 7-3765lb. or 7lb. 6oz. EXAM. 11. At what elevation must a 13-inch mortar be pointed, to range 6745 feet, on a plane which descends 8° 15'; the charge 73lb. of powder? Ans. 32° 41'. EXAM. 12. In what time will a 13-inch shell strike a plane which rises 8° 30', when elevated 45°, and discharged with an impetus of 2304 fect? Ans. 143 seconds. THE DESCENT OF BODIES ON INCLINED PLANES AND CURVE SURFACES. PENDULUMS. THE MOTION OF 193. PROP. IF a weight w be sustained on an inclined plane AB, by a power r, acting in a direction wr, parallel to the plane. Then The weight of the body, w, The length AB, The height BC, and |