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hence v = 2v, or v = of the wheel equal only to
3v, and v = 4V, or the velocity
To determine the Form and Dimensions of Gunpowder
In the practice of engineering, with respect to the erection of powder magazines, the exterior shape is usually made like the roof of a house, having two sloping sides, forming two inclined planes, to throw off the rain, and meeting in an angle or ridge at the top; while the interior represents a vault, more or less extended, as the occasion may require; and the shape, or transverse section, in the form of some arch, both for strength and commodious room, for placing the powder barrels. It has been usual to make this interior curve a semicircle. But, against this shape, for such a purpose, I must enter my decided protest; as it is an arch the farthest of any from being in equilibrium in itself, and the weakest of any, by being unavoidably much thinner in one part than in others. Besides, it is constantly found, that after the centering of semicircular arches is struck, and removed, they settle at the crown, and rise up at the flanks, even with a straight horizontal form at top, and still much more so in powder magazines with a sloping roof; which effects are exactly what might be expected from a contemplation of the true theory of arches. Now this shrinking of the arches must be attended with other additional bad effects, by breaking the texture of the cement, after it has been in some de gree dried, and also by opening the joints of the voussoirs at one end. Instead of the circular arch therefore, we shall in this place give an investigation, founded on the true, principles of equilibrium, of the only just form of the interior, which is properly adapted to the usual sloped roof.
For this purpose, put a DK the thickness of the arch at the top, x = any absciss DP of the required arch ADCM, U KR the corresponding absciss of the given exterior line KI, and y PC RI their equal ordinates. Then by the principles of arches, in my tracts on that subject, it is found that c1 or w = a xxx = a + x
W = Q X
yx-xy or Q X a constant > y3 ya quantity, and where a is some certain quantity to be determined hereafter. But KR or u is
ty, if t be put
the tangent of the given angle of elevation KIR, to radius 1;
and then the equation is w= a + x = ty =.
Now, the fluxion of the equation
ty, ; there
and the 2d fluxion is
fore the foregoing general equation L
and hence wri
w= a + x
becomes w = Qw
ww the fluent of which gives w2=
ty, is w = x
: but at D the value of w is a, and
= 0, the curve at D being parallel to KI; therefore the correct fluent is чес a2=
we √(w2 — a2); the correct fluent of which gives ya x hyp. log. of w+(x2-a2)
Now, to determine the value of a, we are to consider that when the vertical line cr is in the position AL or MN, then wc becomes AL or MN the given quantity c suppose, and y AQ or aмb suppose, in which position the Last equation becomes bax hyp. log. 2-40); and hence it is found that the value of the constant quantity √, is h. l.c+ √(e-¿2); which being substituted for it, in the above general value of y, that value becomes w+ √(w2- -a2) log. of y = 6 x
Hence then j2
= c + √(c2 - α2)
or j =
from which equation the value of the ordinate FC may always be found, to every given value of the vertical cı.
But if, on the other hand, rc be given, to find cr, which will be the more convenient way, it may be found in the following manner: Put A = log. of a, and c = xlog. of
+(-a); then the above equation gives cy + A = log.
of w + √ (w2 — a); again, put in the number whose log. is cy + A; then nw+(-a); and hence w =
Now, for an example in numbers, in a real case of this
CI = W
a2 + n2
mature, let the foregoing figure represent a transverse vertical section of a magazine arch balanced in all its parts, in which the span or width AM is 20 feet, the pitch or height na is 10 feet, thickness at the crown DK = 7 feet, and the angle of the ridge LKN 112° 37′, or the half of it LKD = 56° 18',` the complement of which, or the elevation KIR, is 33° 41′1⁄2, the tangent of which is, which will therefore be the value of t in the foregoing investigation. The values of the other letters will be as follows, viz, DK=α=7; AQ=b=10; BQ=h=10; AL=c=10; 4 = log. of 78450980; c+ √(c2 ~ Q2) 31+/520 C= x log. of =log. of log. of 2.562070408591; cy+A0408591y + 8450980 log. of n. From the general equation then, viz, +, by assuming y successively equal to 1, 2, 3, 4, &c, thence finding the corresponding values of cy+A or 0408591y8450980, and to these, as common logs. taking out the corresponding natural numbers, which will be the values of n; then the above theorem will give the several values of w or cI, as they are here arranged in the annexed table, from which the figure of the curve is to be constructed, by thus finding so many points in it. Otherwise. Instead of making n the number of the log. cy + A, if we put m the natural number of the log. 20+ √(2 w2 cy only then m 2, and am―w=√(w2 — a2), or by squaring, &c, aam2 — 2amw + w2 = w2-a2, and hence
m2 + 1
xa; to which the numbers being applied, the very same conclusions result as in the foregoing calculation and table.
Val. of y
Val. of wo
To construct Powder Magazines with a Parabolical Arch.
It has been shown, in my tract on the Principles of Arches of Bridges, that a parabolic arch is an arch of equilibration, when its extrados, or form of its exterior covering, is the very same parabola as the lower or inside curve. Hence then a parabolic arch, both for the inside and outer form, will be
very proper for the structure of a powder magazine. For, the inside parabolic shape will be very convenient as to room for stowage: 2dly, the exterior parabola, everywhere parallel to the inner one, will be proper enough to carry off the rain water: 3dly, the structure will be in perfect equilibrium : and 4thly the parabolic curve is easily constructed, and the structure erected.
constant. Then ci= × Q is =
a constant quan
ÿ3 tity ➡a, what it is at the vertex; that is, ci is everywhere equal to KD.
Consequently KR is = DP; and since RI is = PC, it is evident that KI is the same parabolic curve with DC, and may be placed any height above it, always producing an arch of equilibration, and very commodious for powder magazines.
THEORY AND PRACTICE OF GUNNERY.
In the 2d vol. of this course have been given several particulars relating to this subject. Thus, in props. 19, 20, 21, 22, p. 151 &c, is given all that relates to the parabolic theory of projectiles, that is, the mathematical principles which would take place and regulate such projects, if they were not impeded and disturbed in their motions by the air in which they move. But, from the enormous resistance of that medium, it happens, that many military projectiles, especially the smaller balls discharged with the higher velocities, do not range so far as a 20th part of what they would naturally do in empty space! That theory therefore can only be useful in some few cases, such as in the slower kind of motions, not above the velocities of 2, 3, or 400 feet per second, when the path of the projectile differs but little perhaps from the curve of a parabola.
Again, at pa. 160 &c, are given several other practical rules and calculations, depending partly on the foregoing parabolic
theory, and partly on the results of certain experiments performed with cannon balls.
Again, in prop. 58, pa. 219, are delivered the theory and calculations of a beautiful military experiment, invented by Mr. Robins, for determining the true degree of velocity with which balls are projected from guns, with any charges of powder. The idea of this experiment, is simply, that the ball is discharged into a very large but moveable block of wood, whose small velocity, in consequence of that blow, can be easily observed and accurately measured. Then, from this small velocity, thus obtained, the great one of the ball is immediately derived by this simple proportion, viz, as the weight of the ball, is to the sum of the weights of the ball. and the block, so is the observed velocity of the last, to a 4th proportional, which is the velocity of the ball sought.—It is evident that this simple mode of experiment will be the source of numerous useful principles, as results derived from the experiments thus made, with all lengths and sizes of guns, with all kinds and sizes of balls and other shot, and with all the various sorts and quantities of gunpowder; in short, the experiment will supply answers to all enquiries in projectiles, excepting the extent of their ranges; for it will even determine the resistance of the air, by causing the ball to strike the block of wood at different distances from the gun, thus showing the velocity lost by passing through those different spaces of air; all which circumstances are partly shown in my 4to vol. of Tracts published in 1786, and which will be completed in my new volumes of miscellaneous tracts now printing.
Lastly, in prob. 17 on Forces, near the end of volume 2, some results of the same kind of experiment are successfully applied to determine the curious circumstances of the first force or elasticity of the air resulting from fired gunpowder, and the velocity with which it expands itself. These are circumstances which have never before been determined with any precision. Mr. Robins, and other authors, it may be said, have only guessed at, rather than determined them. That ingenious philosopher, by a simple experiment, truly showed that by the firing of a parcel of gunpowder, a quantity of elastic air was disengaged, which, when confined in the space only occupied by the powder before it was fired, was found to be near 250 times stronger than the weight or elasticity of the common atmospheric air. He then heated the same parcel of air to the degree of red hot iron, and found it in that temperature to be about 4 times as strong as before; whence he inferred, that the first strength of the in