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scending till the depth of the water be x. And when x = 0, the whole time of exhausting is barely √.
Hence, if A be = 10000 square fecet, a 1 square foot, and d 10 feet; the time is 7885 seconds, or 2h.11' 25"
Again, if the vessel be a ditch, or canal, of 20 feet broad at the bottom, 22 at the top, 9 deep, and 1000 feet long; 90+ x then is 90 90 + x :: 20 : x 2 the breadth of the 9
surface of the water when its depth in the canal is x; and
the fluxion of the
x = 0, is 1000 ×
× 2000 is the surface at that time.
time; the correct fluent of which, when 180 + 3d d 1000 x 186 × 3
9 × 45%
15459" nearly, or 4h. 17′ 39", being the whole time of exhausting by a sluice of 1 foot square.
To determine the Velocity with which a Ball is discharged from a Given Piece of Ordnance, with a Given Charge of Gunpowder.
a = AB, the part at first filled with powder and the bag; b = AE, the whole length of the gunbore;
c = 7854, the area of a circle whose diameter is 1;
d =BD, the diameter of the ball;
e = the specific gravity of the ball, or weight of 1 cubic foot; g= 16 feet, descended by a body in I second;
=230 ounces, the pressure of the atmosphere on a sq. inch; n to 1 the ratio of the first force of the fired powder, to the pressure of the atmosphere;
the weight of the ball. Also, let
AC, be any variable distance of the ball from A, in moving along the gunbarrel.
First, cd2 is the area of the circle BD of the ball; theref. mcd is the pressure of the atmosphere on BD; conseq. mned' is the first force of the powder on BD.
But the force of the inflamed powder is proportional to its density, and the density is inversely as the space it fills; there→ fore the force of the powder on the ball at B, is to the force on the same at C, as Ac is to AB; that is,
x: a :: mncd
= F, the motive force at c:
=ƒ, the accelerating force there.
Hence, theor. 10 of forces gives vu⇒ 2g få =.
the fluent of which is 4gmnaed
x hyp. log. of x,
But when v = 0, then ra; theref. by correction, 4gmnacd
x hyp. log. is the correct fluent conseq.
× hyp.log.) is the vel. of the ball at c.
× hyp. log.) the velocity with which
the ball issues from the muzzle at E; where b denotes the length of the cylinder filled with powder, and a the length to the hinder part of the ball, which will be more than h when the powder does not touch the ball.
Or, by substituting the numbers for g and m, and changing the hyperbolic logarithms for the common ones, then 2230nhd2
x com.log.), the velocity at E, in feet.
But, the content of the ball being cd3, its weight is
v = 2713√( x com log. -), the velocity at E.
When the ball is of cast iron; takinge=7368, the rule becomes
v = 100√( x log. -) for the veloc. of the cast-iron ball.
× log.) for the veloc. of the leaden ball.
Corol. From the general expression for the velocity v, above given, may be derived what must be the length of the charge of powder a, in the gun-barrel, so as to produce the greatest possible velocity in the ball; namely, by making the value of a maximum, or, by squaring and omitting the b constant quantities, the expression a x hyp. log. of
a maximum, or its fluxion equal to nothing; that is, b
a x hyp. log.
a = 0, or hyp. log. of
= 1; hence
2.71828, the number whose hyp. log. is 1. So that a: b :: 1:2.71828, or as 4 to 11 nearly, or nearer as 7 to 19; that is, the length of the charge, to produce the greatest velocity, is the 4th part of the length of the bore, or nearer of it.
By actual experiment it is found, that the charge for the greatest velocity, is but little less than that which is here computed from theory; as may be seen by turning to page 269 of my volume of Tracts, where the corresponding parts are found to be, for four different lengths of gun, thus, ; the parts here varying, as the gun is longer, which allows time for the greater quantity of powder to be fired, before the ball is out of the bore.
In the calculation of the foregoing problem, the value of the constant quantity n remains to be determined. It denotes the first strength or force of the fired gunpowder, just before the ball is moved out of its place. This value is assumed, by Mr. Robins, equal to 1000, that is, 1000 times the pressure of the atmosphere, on any equal spaces.
But the value of the quantity n may be derived much more accurately, from the experiments related in my Tracts, by comparing the velocities there found by experiment, with the rule for the value of v, or the velocity, as above computed by theory, viz.
v = 100 √(110 x log. of 2),
or = 100 √(xlog.of-).
Now, supposing that v is a given quantity, as well as all the other quantities, excepting only the number 7, then by reducing this equation, the value of the letter n is found to be as follows, viz.
when his different from a.
Now, to apply this to the experiments. By page 257 of the Tracts, the velocity of the ball, of 1.96 inches diameter, with 4 ounces of powder, in the gun No. 1, was 1100 feet per second; and, by page 109, the length of the gun, when corrected for the spheroidal hollow in the bottom of the bore, was 28.53; also, by page 237, the length of the charge, when corrected in like manner, was 3:45 inches of powder and bag together, but 2:54 of powder only: so that the values of the quantities in the rule, are thus: a = 3·45; b 28.53; d= 1·96; b = 2:54; and v = 1100: then, by substituting these values instead of the letters, in the the
orem 2 =
when his considered as the same as a. other experiments there treated of.
And so on, for the
It is here to be noted however, that there is a circumstance in the experiments delivered in the Tracts, just mentioned, which will alter the value of the letter a in this theorem, which is this, viz. that a denotes the distance of the shot from the bottom of the bore; and the length of the charge of powder alone ought to be the same thing; but, in the experiments, that length included, besides the length of real powder, the substance of the thin flannel bag in which it was always contained, of which the neck at least extended a considerable length, being the part where the open end was wrapped and tied close round with a thread. This circumstance causes the value of n, as found by the theorem above, to come out less than it ought to be, for it shows the strength of the inflamed powder when just fired, and when the flame fills the whole space a before occupied both by the real powder and the bag, whereas it ought to show the first strength of the flame when it is supposed to be contained in the space only occupied by the powder alone, without the bag. The formula will therefore bring out the value of n too little, in proportion as the real space filled by the powder is less than the space filled both by the powder and its bag. In the same proportion therefore must we increase the formula, that is, in the proportion of h, the length of real powder, to a the length of powder and bag together. When the theorem is dvv 1000h
so corrected, it becomes ÷ com. log. of 2.
Now, by pa. 237 of the Tracts, there are given both the lengths of all the charges, or values of a, including the bag, and also the length of the neck and bottom of the bag, which is 0.91 of an inch, which therefore must be subtracted from
all the values of a, to give the corresponding values of h. This in the example above reduces 3.45 to 2.54.
Hence, by increasing the above result 750, in proportion of 2.54 to 3.45, it becomes 1018. And so on for the other experiments.
But it will be best to arrange the results in a table, with the several dimensions, when corrected, from which they are computed, as here below.
Table of Velocities of Balls and First Force of Powder, &c.
Where it may be observed, that the numbers in the column of velocities, 1430 and 2200, are a little increased, as, from a view of the table of experiments, they evidently required to be. Also the value of the letter d is constantly 1.96 inch.
Hence it appears, that the value of the letter n, used in the theorem, though not yet greatly different from the number 1000, assumed by Mr. Robins, is rather various, both for the different lengths of the gun, and for the different charges with the same gun.