But this diversity in the value of the quantity n, or the first force of the inflamed gunpowder, is probably owing in some measure to the omission of a material datum in the calculation of the problem, namely, the weight of the charge of powder, which has not at all been brought into the computation. For it is manifest, that the elastic fluid has not only the ball to move and impel before it, but its own weight of matter also. The computation may therefore be renewed, in the ensuing problem, to take that datum into the account.

PROBLEM XVIII.

To determine the same as in the last Problem; taking both the Weight of Powder and the Ball into the Calculation.

BESIDES the notation used in the last problem, let 2p denote the weight of the powder in the charge, with the flannel bag in which it was inclosed.

Now, because the inflamed powder occupies at all times the part of the gun bore which is behind the ball, its centre of gravity, or the middle part of the same, will move with only half the velocity that the ball moves with; and this will require the same force as half the weight of the powder, &c, moved with the whole velocity of the ball. Therefore, in the conclusion derived in the last problem, we are now, instead of w, to substitute the quantity p+w; and when that is done, the last 2230nhd b. velocity will come out, v√( And from this equation is found the value of ", which is

22 =

v2=log. of

b p+w

=

a 8567b

x com. leg.-).

a

p + w v2=log. of -, by 2o‚by 2230hd substituting for d is value 1.96, the diameter of the ball. Now as to the ball, its medium weight was 16 oz. 13 dr. = 16'81 oz. And the weights of the bags containing the several charges of powder, viz. 4 oz, 8 oz, 16 oz, were 8 dr, 12 dr, and 1 oz. 5 dr; then, adding these to the respective contained weights of powder, the sums, 4.5 oz, 8.75 oz, 17.31 oz, are the values of 2p, or the weights of the powder and bags; the halves of which, or 2.25, and 4·38, and 8-66, are the values of the quantity p for those three charges; and these being added to 16.81, the constant weight of the ball, there are obtained the three values of p+w, for the three charges of powder, which values therefore are 19.06 oz, and 21.19 oz, and 25°47 oz. Then, by calculating the values of the first force n, by the last rule above, with these new data, the whole will be found as in the following table.

The

And here it appears that the values of n, the first force of the charge, are much more uniform and regular than by the former calculations in the preceding problem, at least in all excepting the smallest charge, 4 oz, in each gun; which it would seem must be owing to some general cause or causes. Nor have we long to search, to find out what those causes may be. For when it is considered that these numbers for the value of n, in the last column of the table, ought to exhibit the first force of the fired powder, when it is supposed to occupy the space only in which the bare powder itself lies; and that whereas it is manifest that the condensed fluid of the charge, in these experiments, occupies the whole space between the ball and the bottom of the gun bore, or the whole space taken up by the powder and the bag or cartridge together, which exceeds the former space, or that of the powder alone, at least in the proportion of the circle of the gun bore, to the same as diminished by the thickness of the surrounding flannel of the bag that contained the powder; it is manifest that the force was diminished on that account. Now by gently compressing a number of folds of the flannel together, it has been found that the thickness of the single flannel was equal to the 40th part of an inch; the double of which, or 05 of an inch, is therefore the

quantity

quantity by which the diameter of the circle of the powder within the bag, was less than that of the gun bore. But the diameter of the gun bores was 2.02 inches; therefore, deducting the *05, the remainder 1.97 is the diameter of the powder cylinder within the bag: and because the areas of circles are to each other as the squares of their diameters, and the squares of these numbers, 1.97 and 2.02, being to each other as 308 to 408, or as 97 to 102; therefore, on this account alone, the numbers before found, for the value of n, must be increased in the ratio of 97 to 102.

But there is yet another circumstance, which occasions the space at first occupied by the inflamed powder to be larger than that at which it has been taken in the foregoing calculations, and that is the difference between the content of a sphere and cylinder. For the space supposed to be occupied at first by the elastic fluid, was considered as the length of a cylinder measured to the hinder part of the curve surface of the ball, which is manifestly too little by the difference between the content of half the ball and a cylinder of the same length and diameter, that is, by a cylinder whose length is the semidiameter of the ball. Now that diameter was 1.96 inches; the half of which is 0.98, and of this is 0.33 nearly. Hence then it appears that the lengths of the cylinders, at first filled by the dense fluid, viz. 3.45, and 5′99, and 11·07, have been all taken too little by 0.33; and hence it follows that, on this account also, all the numbers before found for the value of the first force n, must be further increased in the ratios of 3·45 and 5·99 and 11.07, to the same numbers increased by 0.33, that is, te the numbers 3.78 and 6.32 and 11:40.

Compounding now these last ratios with the foregoing one, viz. 97 to 102, it produces these three, viz. the ratios of 334 and 581 and 1074, respectively to 385 and 647 and 1163. Therefore increasing the last column of numbers, for the value of n, viz. those of the 4 oz. charge in the ratio of 334 to 385, and those of the 8 oz. charge in the ratio of 581 to 647, and those of the 16 oz. charge in the ratio of 1074 to 1163, with every gun, they will be reduced to the numbers in the annexed table: where the numbers are still larger and more regular than before.

Powder.

oz.

4

8

16

The Guns.

1 2 3 4

13721387 1438 1430 1637 1677 1766 1812

157716161782 1784

Thus

Thus then at length it appears that the first force of the inflamed gunpowder, when occupying only the space at first filled with the powder, is about 1800, that is 1800 times the elasticity of the natural air, or pressure of the atmosphere, in the charges with 8 oz. and 16 oz. of powder, in the two longer guns; but somewhat less in the two shorter, probably owing to the gradual firing of gunpowder in some degree; and also less in the lowest charge 4 oz, in all the guns, which may probably be owing to the less degree of heat in the small charge. But besides the foregoing circumstances that have been noticed, or used in the calculations, there are yet several others that might and ought to be taken into the account, in order to a strict and perfect solution of the problem; such as, the counter pressure of the atmosphere, and the resistance of the air on the fore part of the ball while moving along the bore of the gun; the loss of the elastic fluid by the vent and windage of the gun; the gradual firing of the powder; the unequal density of the elastic fluid in the different parts of the space it occupies between the ball and the bottom of the bore; the difference between pressure and percussion when the ball is not laid close to the powder; and perhaps some others: on all which accounts it is probable that, instead of 1800, the first force of the elastic fluid is not less than 2000 times the strength of natural air.

Corol. From the theorem last used for the velocity of the b

ball and elastic fluid, viz. v =

8567hn

p+w

2230bd2
p + w

n = log. )

÷ log. ---), we may find the velocity of the elas

a

=

tic fluid alone, viz. by taking w, or the weight of the ball, = 0 in the theorem, by which it becomes barely v 8567hn

b

÷log.), for that velocity. And by com

a

puting the several preceding examples by this theorem, supposing the value of n to be 2000, the conclusions come out a little various, being between 4000 and 5000, but most of them nearer to the latter number. So that it may be concluded that the velocity of the flame, or of the fired gunpowder, expands itself at the muzzle of the gun, at the rate of about 5000 feet per second nearly.

VOL. II.

2 A

ON

ON THE MOTION OF BODIES IN FLUIDS.

PROBLEM XIX.

To determine the Force of Fluids in Motion; and the Circumstances attending Bodies Moving in Fluids.

1. It is evident that the resistance to a plane, moving perpendicularly through an infinite fluid, at rest, is equal to the pressure or force of the fluid on the plane at rest, and the Auid moving with the same velocity, and in the contrary direction, to that of the plane in the former case. But the force of the fluid in motion, must be equal to the weight or pressure which, gencrates that motion; and which, it is known, is equal to the weight or pressure of a column of the fluid, whose base is equal to the plane, and its altitude equal to the height through which a body must fall, by the force of gravity, to acquire the velocity of the fluid: and that altitude is, for the sake of brevity, called the altitude due to the velocity. So that, if a denote the area of the plane, the velocity, and n the specific gravity of the fluid; then, the altitude due to the velocity being the whole resistance, or motive force m, will be 4g' 72

a x n x

4g

anyz
4g

;g being 16 feet. And hence, cateris paribus, the resistance is as the square of the velocity.

2. This ratio, of the square of the velocity, may be otherwise derived thus. The force of the fluid in motion, must be as the force of one particle multiplied by the number of them; but the force of a particle is as its velocity; and the number of them striking the plane in a given time, is also as the velocity; therefore the whole force is as vx vor v2, that is, as the square of the velocity.

3. If the direction of motion, instead of being perpendicular to the plane, as above supposed, be inclined to it in any angle, the sine of that angle being s, to the radius 1: then the resistance to the plane, or the force of the fluid

against