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R' =




r = F'r' = 1036 × 3·15 — 3262 feet,



the actual range.


12.217; hence, '3-15,

Or, R' may be found thus:—

LR' = 2 Lv + L sin 2 e― (11·505150 + LF') = 5·908484 +9.698970—(11·505150 +3·015649) = 15.607454 - 14·520799 = 1·086655, and R′ = 12·21.


1. What is the actual range of a ball of 6 inches diameter, fired at an elevation of 25°, with a velocity of 1000 feet? Ans. 10570 feet.

2. What is the actual range of a shell 10 inches in diameter, its weight being of that of a ball of the same diameter, when discharged at an elevation of 40°, with a velocity of 400 feet? Ans. 3939.

238. Case II.-When the potential random exceeds 39000, or the impetus exceeds 19500, or the velocity ex

ceeds 1050.

" Find two mean proportionals between 39000 and the potential random, and take the less of them for the reduced potential random; then the true potential random is to the reduced potential random, as the potential range to the reduced potential range. This reduced potential range, being divided by the reduced terminal height F', will give the tabular potential range, from which the actual range is found as in the last case."

LR 3.139977+ LR" + Lh,


and adding 10 to both sides, and substituting for Lh its value 2 Lv-1·806180 (233); then

LR" LR-Lv +4.064143,


LR=L2h+L sin 2e-10 (234),
LR2 Lv + L sin 2e-11-505150.


Hence, LR" Lv + L sin 2 e-7·441007.


Find F and F' as in article 237.

EXAMPLE. Required the range of the bullet in the example of the first case, discharged at the same elevation, with a velocity of 2100 feet.

In this case, v

1050, or 2 h 39000.

As in the former example, F = 1057, and F' = 1036. And LR" Lv + Lsin 2e-7.4410072.214813+ 9.698970-7.44100711·913783-7.441007 4.472776; hence, R" 29701. LR'LR"-LF'-4-472776-3-0156491.457127, and R' 28-65; hence by table, r' =4·1734,


and r = F'r' = 1036 × 4·1734 = 4324 feet, the actual range.

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Although the velocity in this example is more than double that in the preceding, yet the range is only 1062 feet greater.


1. Find the actual range of a 42-pound ball, discharged with a velocity of 1800 feet, at an elevation of 36°.

Ans. 14587 feet. 2. What will be the actual range of a 24-pound ball, fired at an elevation of 35°, with a velocity of 1760 feet per second? Ans. 13695.

It can be shown, by dynamical principles, that balls of the same density projected at the same elevation, with velocities that are proportional to the square roots of their diameters, describe similar curves. The reason of this is, that the resistances are proportional to the masses or weights of the balls. Their velocities at their greatest height, which are horizontal, are proportional to their diameters; and any corresponding lines of their trajectories, that is, of the curves described by them, are proportional to their diameters. Their actual ranges are therefore proportional to their diameters, or to the squares of their initial velocities; but their potential ranges are in the same proportion; hence their actual and potential ranges are proportional. But the terminal heights, being 900 d, are proportional also to their diameters, or their terminal velocities are proportional to their initial velocities. The terminal heights are therefore also proportional to their ranges, both actual and potential. Hence, the quotients of the actual and potential ranges of one ball by its terminal height are respectively equal to the corresponding quotients for another ball, both being projected under the conditions stated above; that is, the tabular

ranges, both actual and potential, are the same for all balls of the same density, discharged at the same elevation, with velocities proportional to the square roots of their diameters. Thus, a comparatively limited set of experiments with a ball of given dimensions and density, would be sufficient to determine the data for the construction of the preceding table; by means of which the ranges of balls, of an unlimited variety of density and size, could be computed.

The weights of two balls being w, w', their diameters d, d', their velocities v, v', and the resistances to them r and ', then (228)

r: r' = d2v2: : d'2'2 nearly,

if the velocities are both greater or both less than 1050. And if v: v'd:d', then

r: r' = d2d: d'3d' = d3: d'3 = w: w',

so that in this case the resistances are as the weights. If and are the terminal velocities, then r=w, and r' = w' ; hence r:'w: w'.



d2x2: d'2v'2 = d3: d'3,

v2: v'2d: d', or v: v'd:d'.

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239. Fortification treats of the method of putting a military position in such a state of defence as will enable a body of men to withstand the assault of a greater body.

240. The attack of a military position is called a siege; the defenders are called the garrison; and their assailants the besiegers. A siege is said to be carried when the fortified place is taken; and it is said to be raised if it is abandoned.


241. Of all the works of a fortification, the most important is the rampart and ditch. The rampart is a mound of earth or a wall surrounding the fortified place; and before, that is, on the outside of the rampart, is the ditch, which is formed by the excavation of the earth used for building the rampart.

242. All that is contained within this first rampart is called the body of the place, or the enceinte; and all the works beyond the ditch are called outworks; the ground beyond the outworks is called the field.

243. On the top of the rampart next the field is erected another smaller mound or wall called a parapet. The rampart is elevated from 10 to 20 feet according to circumstances, and the parapet is 6 or 7 feet higher than the rampart. On the rampart, immediately behind the parapet, is a step called the banquette. The parapet covers the besieged from the fire of the enemy.

244. All works have a parapet of 3 fathoms thick, and a rampart of 8 or 10, besides their slopes.

245. The outside of the rampart is commonly lined with a wall of stone called the revêtement, which is strengthened within by buttresses, called counterforts.

246. Along the outside of the ditch a passage is cut called the covert way; and from it the ground gently declines towards the field, and is called the glacis.

247. The flat part on the top of the rampart is called its terreplein; the sloping side of any work that declines towards the field is called an escarpe; and when it inclines towards the place, the counterscarpe.

248. The openings in the parapet, through which the garrison fire their guns, are called embrasures. Across the covert way walls are erected called traverses.

Traverses serve for defending the covert way when the enemy has entered it at any place; the covert way is also covered with trees, thorns, cheveaux de frise, crow-feet, &c., and it has a line of palisades along its outer side, to entangle and annoy the enemy when they approach near the place.

Every work ought to be at least 6 feet higher than those before it, that the garrison may fire from all the works on the same side at the same time. The covert way ought to be lower than the level ground, otherwise the body of the place would be unnecessarily high.

249. The bases of all inward slopes ought to be at least equal to their heights; and the bases of all outward slopes two-thirds of their heights. The slopes of all walls and revêtements ought to be one-fifth of their height. The

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