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 triple 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. 192. Example 1. when fired with 5 each of the several powder, viz. If a ball of 1lb. acquire a velocity of 1600 feet per second, ounces of powder; it is required to find with what velocity kinds of shells will be discharged by the full charges of Nature of the shells in inches Their weight in lbs. Charge of powder in lbs. Ans. The velocities are 13 10 8 5 43 196 90 48 16 9 4 2 1 594 584 565 693 693 Ex. 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. Ex. 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'. Ex. 4. With what impetus, velocity, and charge of powder, 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, charge = = 2.95lbs. Ex. 5. A shell being found to range 3500 feet, when discharged at an elevation of 25° 12′; how far then will it range at an elevation of 30° 15′ with the same charge of powder ? Ans. 4332 feet. Ex. 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. 6 lb. of powder. Ex. 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. * In a series of experiments carried on at Woolwich, a few years ago, 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 Sir Thomas Blomfield and Dr. Hutton made their experiments. Putting B for the weight of the ball, and C for that of the charge, v = 1600, is now found a good approximative theorem for the initial velocity, and is the expression of the rule in the text. Ex. 8. In what time will a shell range 3250 feet, at an elevation of 32°? Ex. 9. How far will a shot range on a plane which ascends 8° 15′, and another which descends 8° 15'; the impetus being 3000 feet, and the elevation of the piece 32° 30′ ? Ans. 4244 feet on the ascent, and 6745 feet on the descent. Ex. 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. 4.92535lb. or 4lb. 15oz. nearly. Ex. 11. At what elevation must a 13-inch mortar be pointed, to range 6745 feet, on a plane which descends 8° 15'; the charge 41lb. of powder. Ans. 32° 18'. Ex. 12. Suppose, in Rirochet firing PO = 1200 feet (fig. in last note) OH = 10 feet, OR = 50 feet; required the elevation and the velocity, so that the ball shall just clear H and hit R. Ex. 13. Given the heights y, y', of a ball above a horizontal plane, at the horizontal distances x, x', to find the initial velocity and the angle of elevation of the ball. Ex. 14. The greatest horizontal range of a shell is 3990 feet: what is the greatest range with the same velocity on a plane elevated 9°, and on another depressed 9°? and what must be the elevation in each case? Ex. 15. Given the proportion between the range and greatest height of a ball projected with a given velocity to find the angle of elevation; and for a numerical example, take the range to the greatest height as 10 to 1. DESCENTS ON INCLINED PLANES. 193. WHEN We treated of the Inclined Plane as a mechanical power, and compared the forces which sustained a weight upon any such plane, we found that the sustaining force acting parallel to the plane, was to the absolute weight or gravity of the body sustained, as the height of the plane to its length, (art. 77.) that is to say, taking the initial letters, we found f g h : l. This, therefore, is the value of the accelerating force which nrges a body down a plane of height h, length 7; and which must be substituted for g in the expressions v = gt, s = gt2, v = √ 2gs. Which hence become, in the case of the inclined plane, h [In these equations we may for the plane they will thereby be simplified for practical uses.] substitute sin. i, i being the inclination of h If we deduce the value of the time from the equation s = 19t2 4, making s (space) = 1 (length of plane) we shall have Whence we learn that, abstracting from friction and the resistance of the air, the times of descent down planes of the same height are as the lengths. 194. Supposing, still, s = l, or the body to descend through the whole length C Whence we learn, that the velocity acquired by a body in falling down an inclined plane, is the same as the velocity acquired by falling vertically through the height h; and consequently if bodies descended freely over ever so many inclined planes, CP, CP', CP", &c. of the same altitude CA, the velocity acquired at P, P', P", P", &c. IN THE RESPECTIVE DIRECTION CP, is equal to the vertical velocity at A, after falling through CA. 195. Once more: all the cords CP, CP', CP", &c. of the same vertical circle, drawn from the same extremity C of the upright diameter CD, are run over in the same time by bodies falling down them. For, putting CD= d, calling CP, 1, and CM, h, PM being horizontal, the well known property of the circle CP2 = CD. CM, gives l2 = dh. Substituting this 212 value of 12 for it, in the equation t = comes t= it be gh' gether independent of CP, or 7, and indicates evidently the time of descent through CD. Therefore the time of descent down either of the chords is to the time through the vertical diameter. 196. Hence follows this curious property. : If rings starting together run freely down straight wires CP", CQ", CR", CS", all posited in one vertical plane at any one moment of time all the rings will be in the circumference of one and the same circle PQRS. At the end of any other equal interval, they will all be in the circumference of another circle P'Q'R'S', touching the former in C. After any other interval, all in the circumference of another circle P"Q"R"S", touching both the former in C. And so on. P P' P" P" P P M P" D D If the wires, instead of being conceived all in one vertical plane, be imagined directed different ways in space-then the rings that run down them, will at the end of successive intervals, be found simultaneously in the surfaces of so many successive spheres, all touching in the point C, the origin of the motions. SCHOLIUM. 197. We may here introduce some useful formulæ, relative to motions along inclined planes, analogous to those already given for bodies falling freely (art. 158.) I. Let g, as before,= 32 feet, s the space along an inclined plane whose inclination is it the time, v the velocity; then II. Suppose V to be the velocity with which a body is projected up or down the plane; then, we have Making v = 0, in equa. 4, and the latter member of equa. 5; the first will give the time at which the body will cease to rise, the latter the space. III. If R be a constant resistance to motion on a horizontal plane, then where, making v = 0, we find when the motion ceases. 198. The first eight of the following problems will serve to exemplify these theorems. 1. How far will a body descend from quiescence in 4 seconds, along an inclined plane whose length is 400 and height 300 feet? 2. What velocity will such a body have acquired when it has reached the bottom A of a plane? 3. Suppose AD = DB, in what time will the body pass over each of those portions? 4. How long would a body be in falling down 100 feet of a plane whose length AB is 150 feet, and height BC 60? 5. If AB = 90, and BC= 25 feet, what velocity would a body acquire in falling through 70 feet? 6. A body is projected up an inclined plane, whose length is 10 times its height, with a velocity of 30 feet per second; in what time will its velocity be destroyed, and it cease to ascend? 7. Suppose that at the moment a body is projected up AB with the velocity acquired by falling down it, another body begins to fall down it, where will they meet, the length of AB being given ? 8. Given AB = 90, BC = 60 feet. And suppose two bodies to be let fall the same moment, one vertically, the other down the plane BA; what distance BD will the latter have moved, when the former has descended to C? 9. Ascertain, geometrically, the position of the right line of quickest descent, from a given point to a given plane. 10. Find, geometrically, the slope of a roof, down which rain may descend quickest. 199. PROP. If a body descend down any number of contiguous planes, AB, BC, CD; it will at last acquire the same velocity, as a body falling perpendicularly through the same height ED, supposing the velocity not altered by changing from one plane to another. D Produce the planes DC, CB, to meet the horizontal line EA produced in F and G. Then, by cor. 1, last art. the velocity at B is the same, whether the body descend through AB or FB. And therefore the velocity at C will be the same, whether the body descend through ABC or through FC, which is also again the same as by descending through GC. Consequently it will have the same velocity at D, by descending through the planes AB, BC, CD, as by descending through the plane GD; supposing no obstruction to the motion by the body impinging on the planes at B and C : and this again, is the same velocity as by descending through the same perpendicular height ED. Corol. 1. If the lines ABCD, &c. be supposed indefinitely small, they will form a curve line, which will be the path of the body; from which it appears that a body acquires also the same velocity in descending along any curve, as in falling perpendicularly through the same height. Corol. 2. Hence also, bodies acquire the same velocity by descending from the same height, whether they descend perpendicularly, or down any planes, or down any curve or curves. And if their velocities be equal, at any one height, they will be equal at all other equal heights. Therefore the velocity acquired by descending down any lines or curves, are as the square roots of the perpendicular heights. Corol. 3. And a body, after its descent through any curve, will acquire a velocity which will carry it to the same height through an equal curve, or through any other curve, either by running up the smooth concave side, or by being retained in the curve by a string, and vibrating like a pendulum: Also, the velocities will be equal, at all equal altitudes; and the ascent and descent will be performed in the same time, if the curves be the same. PENDULOUS MOTION. 200. PROP. The times in which bodies descend through similar parts of similar curves, ABC, abc, placed alike, are as the square roots of their lengths. That is, the time in AC is to the time in ac, as AC to ac. For, as the curves are similar, they may be considered as made up of an equal number of corresponding parts, which are every where, each to each, proportional to the whole. And as they are placed alike, the corresponding small similar parts will also be parallel to each other. But the time of describing each of these pairs of corresponding parallel parts, by art 194, 195, are as the square roots of their lengths, which, by the supposition, are as AC to 7 a B C ac, |