Corol. From the general expression for the velocity vs 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 v a maximum, or, by squaring and omitting the b b a a =0, or hyp. log. of a a constant quantities, the expression a x hyp. log. of a a maximum, or its fluxion equal to nothing; that is b b a x hyp. log. -=1; hence2.71828, 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 theth 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. 3 3 SCHOLIUM. 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. na b nh -), or = 100√(od = 100√(xlog. of), a b a × log.of --). Now, supposing that v is a given quantity, as well as all the other quantities, excepting only the number n, then by reducing this equation, the value of the letter n is found to be as follows, viz. 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 theorem n = dvv 1000a com. log. of b - a it comes out n = 750, when his considered as the same as a. And so on, for the other experiments there treated of. 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 pow der 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 theoremis dvv so corrected, it becomes b com. log. of 1000b a 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 theoremi, 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. But 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 2230nhd Ъ last velocity will come out, v = √(x com.log. ). p + w And from this equation is found the value of n, which is n = b p + w a = b a a p + w v2 ÷ log. of —, v2 log. of by 2230bd2 substituting for d its 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 |