of the piece, and AP the altitude due to the projectile velocity at A. P By the last proposition, find the horizontal range AH to the given velocity and direction; draw HE perpendicular to AH, meeting the oblique plane in E; draw EF parallel to AG, and FI parallel to HE; so shall the R K H projectile pass through I, and the range on the oblique plane will be AI. As is evident by theor. 15 of the Parabola, where it is proved, that if AH, AI be any two lines terminated at the curve, and IF, HE parallel to the axis; then is EF parallel to the tangent AG. 94. Otherwise, without the Horizontal Range. Draw PQ perp. to AG, and QD perp. to the horizontal plane AF, meeting the inclined plane in K; take AE = 4AK, draw EF parallel to AG, and FI parallel to AP or DQ; so shall AI be the range on the oblique plane. - For AH therefore EH is parallel to FI, and so on, as above. Otherwise. 4AD, 1 95. Draw Pq making the angle APQ the angle GAI; then take AG = 4Aq, and draw GI perp. to AH. Or, draw qk perp. to AH, and take AI = 4ak. Also kq will be equal to cu the greatest height above the plane. For, by cor. 2, prop. 20, AP : AG :: AG: 4GI; or AP AP AG :: Aq: GI, AG: 4Aq: 4GI; therefore AG = 44q; and by sim. triangles, AI = 4AK. Also. qk, or GI, is to co by theor. 13 of the Parabola. זי k H A. G 96. Corol. 1. If 40 be drawn perp. to the plane AI, and AP be 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 GAI, 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 AI and the velocity, or the impetus, the direction will hence be easily found thus: Take Ak A1, draw kq perp. to AH, meeting the circle described with the radius Ao in two points q and q; 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 AI. And these two directions make equal angles with AI and AP, because the arc rq is equal the arc aq. They also make equal angles with a line drawn from a through s, because the arc sq is equal the arc sq. 98. Corol. 3. Or, if there be given the range AI, and the direction Aq; to find the velocity or impetus. Take ak= AI, and erect kq perp. to AH, meeting the line of direction in q; then draw qp making the AqP 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, puts sin. 4qAI or APq, c=cos. 4qAH or sin. PAq, IAH or sin. Akd or Akq or aqp. Then, in the triangle APq, and in the triangle Akq, theref. by composition, cc Aq ak; c2 : cs:: AP : Ak = ZAI. CS So that the oblique range = X 4AP. 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. Corol. 5. The greatest height cu or kq of the projec ܐܨ e, above the plane, is equal to X AP. And therefore c2 it 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. Aqp): 5 (sin. APq):: AP: Aq, and c (sin. akq) s (sin. kaq):: Aq: kq, theref. by comp. c:2:: AP: kq. 25 AP وست د 102. Corol. 6. The time of flight in the curve AI is = where g 16 feet. And therefore it is as C g the velocity and 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 X AP. Therefore, the 45 c2 time being as the square root of the distance, 103. From the foregoing corollaries may be collected the following set of theorems, relating to projects made on any given inclined planes, either above or below the horizontal plane. In which the letters denote as before, namely, c=cos. of direction above the horizon, c = cos. of inclination of the plane, R T H the greatest height above the plane, a the impetus, or alt. due to the velocity v, g= 16 feet. Then, And from any of these, the angle of direction may be found. PRAC 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 that are 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 5° above 60° and below 30°, from 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 present content ourselves with the data of VOL. II. M some |