Page images
[ocr errors]
[ocr errors][ocr errors][ocr errors]

fore has no effect to change it; but the former BG, being prep.
to the ship's motion, is the only part of the force to turn the
ship about and change her course. But BF BG: CF : CB,
the force upon the rudder

therefore CF : CB :: BF2 :

to turn the ship about.
Now put a

[ocr errors][merged small][merged small][merged small]


CF, x = BC; then Br2 = a2 x2, and the x(a2-x2

[ocr errors]



; and, to have this a

maximum, its flux. must be made to vanish, that is, a3-3x2
x=0; and hence xa
BC the natural sine of 35°
16' angle F; therefore the complemental angle c =
as before, for the obliquity of the rudder, when it is most

The case will be also the same with respect to the wind acting on the sails of a wind-mill, or of a ship, viz. that the sails must be set so as to make an angle of 54° 44′ with the direction of the wind; at least at the beginning of the motion, or nearly so when the velocity of the sail is but small in comparison with that of the wind; but when the former is pretty considerable in respect of the latter, then the angle ought to be proportionally greater, to have the best effect, as shown in Maclaurin’s Fluxions, pa. 734, &c.

A consideration somewhat related to the same also, is the greatest effect produced on a mill-wheel, by a stream of water striking upon its sails or float-boards. The proper way in this case seems to be, to consider the whole of the water as acting on the wheel but striking it only with the relative velocity, or the velocity with which the water overtakes andstrikes upon the wheel in motion, or the difference between the velocities of the wheel and the stream. This then is the power or force of the water; which multiplied by the velocity of the wheel, the product of the two, viz. of the relative velocity and the absolute velocity of the wheel, that is (v-v) v=vv—v2, will be the effect of the wheel; where v denotes the given velocity of the water, and v the required velocity of the wheel. Now, to make the effect vo v2 a maximum, or the greatest, its fluxion must vanish, that is vʊ — 2vv 2vv = 0, hence v = v; or the velocity of the wheel will be equal to half the velocity of the stream, when the effect is the greatest; and this agrees best with experiments.

A former way of resolving this problem was, to consider the water as striking the wheel with a force as the square of the relative velocity, and this multiplied by the velocity of the wheel, to give the effect; that is, (vv) the effect. Now the flux. of this product is (vv)2v-(vv) X 2vv=0;


[ocr errors]

hence v

2v, or v = 3v, and v = v, or the velocity of the wheel equal only to of the velocity of the water.


To determine the Form and Dimensions of Gunpowder Maga


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 degree 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 adopted to the usual sloped roof.

For this purpose, put & 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 ci or wα + x u =QX

[ocr errors]
[ocr errors]



[ocr errors]

or=QX supposing ý a constant quan



tity, and where a is some certain quantity to be determined hereafter. But KR or u is ty, if t be put to denote the tan

[merged small][merged small][ocr errors][merged small]

gent of the given angle of elevation KIR, to radius 1, and then

[merged small][merged small][graphic][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

at p the value of w is


Q w w


[ocr errors]

: but




a, and w=0, the curve at D being pa

rallel to KI; therefore the correct fluent is wa a2

[ocr errors]


[merged small][merged small][merged small][merged small][merged small][ocr errors][merged small]
[ocr errors][ocr errors][ocr errors]

fluent of which gives y=ax hyp. log, of

w + √ (w3 — a2)


Now, to determine the value of a, we are to consider that when the vertical line ci is in the position AL or MN then w➡CI becomes AL or MN = the given quantity c suppose, and yaq or qм=b suppose, in which position the last equation bec+✓ (c2-a3) comes b = √ a X hyp. log.

[ocr errors]


and hence it is

found that the value of the constant quantity, is

[merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small]

from which equation the value of the ordinate PC may always be found, to every given value of the vertical cI.

But if, on the other hand, Pc 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 = X log, of c+√(c2 -α2) ; then the above equation gives cya

[blocks in formation]

Now, for an example in numbers, in a real case of this nature, 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 DQ is 10 feet, thickness at the crown DK = 7 feet, and the angle of the ridge LKS 112o 87′, or the half of it LKD = 56° 18′1⁄2, the complement of which, or the elevation KIR, is 33° 411, 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; DQ = h = 10; AL = C 103; =log. of 78450980;

[blocks in formation]

× log. of c+√(c2 — a2) ·


log. of

[ocr errors]
[blocks in formation]

log. of 2.56207 =0408591; cy + ▲ = •·0408591y + •8450980=log. of n. From the general equation then, viz. n, by assuming y successively

[ocr errors]
[ocr errors]


[ocr errors]



equal to 1, 2, 3, 4, &c. thence finding
the corresponding values of cy + a or
·0408591y+8450980, and to these,
as common logs. taking out the corres-
ponding natural numbers, which will
be the values of n; then the above
theorem will give the several values
of wor ci, as they are here arranged
in the annexed table, from which the
figure of the curve is to be construct-
ed, by thus finding so many points in it.
Otherwise. Instead of making n the
number of the log. cy+A, if we put
the natural number of the log.
w + √ (w2 — a2)


cy only; then m =



[ocr errors]
[ocr errors]

and am

[ocr errors]
[ocr errors]

a2), or by squaring, &c. a2 m2 2amw + w2 = w2 W2 a2, and

[blocks in formation]

xa: to which the numbers being applied,

the very same conclusions result as in the foregoing calculation and table.


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, every where paral-
lel 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,

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][subsumed][merged small][subsumed][merged small][merged small][subsumed][subsumed][merged small][merged small][merged small][merged small]


[ocr errors]

X Q is =

a con


by making y constant. Then cr

stant quantity =a, what it is at the vertax; that is, cr is every
where equal to KD.

Consequently KR is DP; and since RI is PC, it is evi-
dent 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.

[ocr errors]


In the Doctrine of Motion, Forces, &c. have been given several particulars relating to this subject. Thus, in props. 19, 20, 21, 22, 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 art. 104, &c. of same doctrine, are given several other practical rules and calculations, depending partly on the


[ocr errors]
[ocr errors]
« PreviousContinue »