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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 30o.

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.

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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

some

some one certain experimented 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.

SOME PRACTICAL RULES IN GUNNERY.

I. 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 sirot 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 other any 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.

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108. Exam. 2. If a shell be found to range 1000 yards, when discharged at an elevation of 45°; how far will it

range

range when the elevation is 30° 16', the charge of powder being the same? Ans. 2612 feet, or 871 yards.

109. Exam. 3. The range of a shell, at 45o 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′. 110. Exam. 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 4lb. 7 Joz.

111. Exam. 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 36° 15′ with the same charge of powder? Ans. 4332 feet.

112. Exam. 6. If, with a charge of 91b. 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.

113. Exam. 7. What will be the time of flight for any given range, at the elevation of 45° ?

Ans. the time in secs. is the sq. root of the range in feet.

114. Exam. 8. In what time will a shell range 3250 feet, at an elevation of 32°? Ans. 11 sec. nearly.

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115. Exam. 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.

116. 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.

117. 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 7 lb. of powder? Ans. 32° 28'.

118. Exam. 12. In what time will a 18-inch shell strike a plane which rises 8o 30', when elevated 45o, and discharged with an impetus of 2304 feet? Ans. 143 seconds.

THE

THE DESCENT OF BODIES ON INCLINED PLANES AND CURVE SURFACES. THE MOTION OF PENDULUMS.

PROPOSITION XXIII.

119. If a weight w be Sustained on an Inclined Plane AB, by a Power Pacting in a Direction WP,Parallel to the Plane. Then The Weight of the Body, w The Length AB, The Sustaining Power P, and The Pressure on the Plane, p, are respectively as

The Height BC, and
The Base AC,

of the Plane.

A

B

P

FOR, draw CD perpendicular to the plane, Now here are three forces, keeping one another in equilibrio; namely, the weight, or force of gravity, acting perpendicular to Ac, or parallel to BC; the power acting parallel to DB; and the pressure perpendicular to AB, or parallel to DC but when three forces keep one another in equilibrio, they are proportional to the sides of the triangle CBD, made by lines in the direction of those forces, by prop. 8; therefore those forces are to one another as BC, BD, CD. But the two triangles ABC, CBD, are equiangular, and have their like sides proportional; therefore the three BC, BD, CD, are to one another respectively as the three AB, BC, AC; which therefore are as the three forces w, p, p.

120. Corol. 1. Hence the weight w, power P, and pressure j, are respectively as

radius,

sine,

of the plane's elevation BAC above the horizon.

and cosine,

For, since the sides of triangles are as the sines of their opposite angles, therefore the three AB, BC, AC,

are respectively as

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sin. c, sin. A, sin. B,

radius, sine, cosine,
of the angle A of elevation.

Or, the three forces are as AC, CD, AD; perpendicular to their directions.

121. Corol. 2. The power or relative weight that urges a body w down the inclined plane, is = Xw; or the force

BC
AB

with which it descends, or endeavours to descend, is as the sine of the angle A of inclination.

122. Corol. 3. Hence, if there be two planes of the same height, and two bodies be laid on them which are proportional to the lengths of the planes; they will have an equal tendency to descend down the planes.

And consequently they will mutually sustain each other if they be connected by a string acting parallel to the planes.

123. Corol. 4. In like manner,

when the power P acts in any other direction whatever, we; by drawing CDE perpendicular to the direction we, the three forces in equilibrio, namely, the weight w, the power P, and the pressure on the plane, will still be respectively as AC, CD, AD, drawn perpendicular to the direction of those forces.

PROPOSITION XXIV.

B

P

E

E

B

124. Ifa Weight w on an Inclined Plane AB, be in Equilibrio with another Weight x hanging freely; then if they be set a-moving, their Perpendicular Velocities, in that Place, will be Reciprocally as those Weights. LET the weight w descend a very small space, from w to A, along the plane, by which the string PFW will come into the position PFA. Draw WH perpendicular to the horizon AC, and wG perpendicular to AF: then WH will be the space perpendicularly descended by the weight w; and AG, or the difference between FA and FW, will be the space perpendicularly ascended by the weight P ; and their perpendicular velocities are as those spaces wн and AG passed over in those directions, in the same time. Draw CDE perpendicular to AF, and DI perpendicular to Ac. Then,

in the sim. figs. AGWH and AEDI,

and in the siin. tri. AEC, DIĆ,

but, by cor. 4, prop. 23,

therefore, by equality,

VOL. II.

U

AG: WH: AE: DI;
AC: CD :: AE: DI;
AC CD :: W
: P;
AG: WH :: w
: P;
That

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