gun, the time in seconds of the ball's flight through the air, and its range, or the distance where it fell on the horizontal plane. From which it is hoped that some aid may be derived towards ascertaining the resistance of the medium, and its effects on other elevations, &c. and so afford some means of obtaining easy rules for the cases of practical gunnery. Though the completion of this enquiry, for want of time at present, must be referred to another work, where we may have an opportunity of describing another more extended course of experiments on this subject, which have never yet been given to the public.

Another subject of enquiry in the foregoing experiments, was, how far the balls would penetrate into solid blocks of elm wood, fired in the direction of the fibres. The annexed tablet shows the results of a few of the trials that were made with the gun n° 2, with the most frequent charges. of 2, 4, and 8 ounces of powder; and the mediums of the penetrations, as placed in the

last line, are found to be 7, 15, and 20 inches, with those charges. These penetrations are nearly as the numbers

2, 4, 6, or 1, 2, 3; but the charges of powder are as

·

**

2, 4, 8, or 1, 2, 4; so that the penetrations are proportional to the charges as far as to 4 ounces, but in a less ratio at 8 ounces; whereas, by the theory of penetrations the depths ought to be proportional to the charges, or which is the same thing, as the squares of the velocities. So that it seems the resisting force of the wood is not uniformly or constantly the same but that it increases a little with the increased velocity of the ball. This may probably be occasioned by the greater quantity of fibres driven before the ball; which may thus increase the spring and resistance of the wood, and prevent the ball from penetrating so deep as it otherwise might do.

From a general inspection of this second course of these experiments, it appears that all the deductions and observations made on the former course, are here corroborated and strengthened, respecting the velocities and weights of the balls, and charges of powder, &c. It further appears also that the velocity of the ball increases with the increase of charge

A

charge only to a certain point, which is peculiar to each gun, where it is greatest; and that by further increasing the charge, the velocity gradually diminishes, till the bore is quite full of powder. That this charge for the greatest velocity is greater as the gun is longer, but yet not greater in so high a proportion as the length of the gun is; so that the part of the bore filled with powder, bears a less proportion to the whole bore in the long guns, than it does in the shorter ones: the part which is filled being indeed nearly in the inverse ratio of the square root of the empty part.

It appears that the velocity, with equal charges, always increases as the gun is longer; though the increase in velocity is but very small in comparison to the increase in length; the velocities being in a ratio somewhat less than that of the square roots of the length of the bore, but greater than that of the cube roots of the same, and is indeed nearly in the middle ratio between the two.

It appears from the table of ranges, that the range increases in a much lower ratio than the velocity, the gun and elevation being the same. And when this is compared with the proportion of the velocity and length of gun in the last paragraph, it is evident that we gain extremely little in the range by a great increase in the length of the gun, with the same charge of powder. In fact the range is nearly as the 5th root of the length of the bore: which is so small an increase, as to amount only to about a 7th part more range for a double length of gun.-From the same table it also appears, that the time of the ball's flight is nearly as the range; the gun and elevation being the same.

It has been found, by these experiments, that no difference. is caused in the velocity, or range, by varying the weight of the gun, nor by the use of wads, nor by different degrees of ramming, nor by firing the charge of powder in different parts of it. But that a very great difference in the velocity arises from a small degree in the windage: indeed with the usual established windage only, viz. about of the calibre, no less than between 1 and of the powder escapes and is lost and as the balls are often smaller than the regulated size, it frequently happens that half the powder is lost by unnecessary windage.

14

2 0

It appears too that the resisting force of wood, to balls fired into it, is not constant: and that the depths penetrated by balls," with different velocities or charges, are nearly as the logarithms of the charges, instead of being as the charges themselves, or, which is the same thing, as the square of the velocity.—Lastly, these and most other experiments, show, that balls are

greatly

!

greatly deflected from the direction in which they are projected; and that as much as 300 or 400 yards in a range of a mile, or almost 4th of the range.

We have before adverted to a third set of experiments, of still more importance, with respect to the resistance of the medium, than any of the former; but, till the publication of those experiments we cannot avail ourselves of all the discoveries they contain. In the mean time however we may extract from them the three following tables of resistances, for three different sizes of balls, and for velocities between 100 feet and 2000 feet per second of time.

To determine the Resistance of the Medium against a Ball of any other size, moving with any of the Velocities given in the foregoing Tables.

The analogy among the numbers in all these tables is very remarkable and uniform, the same general laws running

through

through them all. The same laws are also observable as in the table of resistances in page 412 of this volume, particularly the 1st and 2d remarks immediately following that table, viz. that the resistances increase in a higher proportion than the square of the velocities, with the same body; and that the resistances also increase in a rather higher ratio than the surfaces, with different bodies, but the same velocity. Yet this latter case, viz. the ratios of the resistances and of the surfaces, or of the squares of the diameters which is the same thing, are so nearly alike, that they may be considered as equal to each other in any calculations relating to artillery practice. For example, suppose it were required to determine what would be the resistance of the air against a 241b ball discharged with a velocity of 2000 feet per second of time. Now, by the first of the foregoing tables, the ball of 1·965 inches diameter, when moving with the velocity 2000, suffered a resistance of 981b: then since the resistances, with the same velocity, are as the surfaces; and the surfaces are as the squares of the diameters; and the diameters being 1·965 and 5-6, the squares of which are 3.86 and 31-36, therefore as 3.86: 31.36 :: 98lb: 796lb; that is, the 241b ball would suffer the enormous resistance of 796lb in its flight, in opposition to the direction of its motion!

And, in general, if the diameter of any proposed ball be denoted by d, and r denote the resistance in the 1st table due d2 r to the proposed velocity of the 1.965 ball; then will de

3.86

note the resistance with the same velocity against the ball whose diameter is d; or it is nearly d2r which is but the 28th part greater than the former.

PROBLEM II.

To assign a Rule for determining the Resistance due to any Indeterminate Velocity of a Given Ball.

This problem is very difficult to perform near the truth, on account of the variable ratio which the resistance bears to the velocity, increasing always more and more above that of the square of the velocity, at least to a certain extent; and indeed it appears that there is no single integral power whatever of the velocity, or no expression of the velocity in one term only, that can be proportional to the resistances throughout. It is true indeed that such an expression can be assigned by means of a fractional power of the velocity, or rather one whose index is a mixed number, viz. 21 or 2:1; thus

v2.1

5400

the

resistance, is a formula in one term only, which will answer to all the numbers in the first table of resistances very nearly, and consequently, by means of the ratio of the squares of the diameters of the balls, for any other balls whatever. This formula then, though serving quite well for some particular resistance, or even for constructing a complete series or table of resistances, is not proper for the use of problems in which fluxions and fluents are concerned, on account of the mixed number 2, in the index of the velocity v.

r

We must therefore have recourse to an expression in two terms, or a formula containing two integral powers of the velocity, as 2 and v, the first and 2d powers, affected with general coefficients m and n, as mv2 + nv = the resistance. Now, to determine the general numerical values of the coef ficients m and n, we must adapt this general expression my2 +nv➡r, to two particular cases of velocity at a convenient distance from each other, in one of the foregoing tables of resistances, as the first for instance. Now, after making several trials in this way, I have found that the two velocities of 500 and 1000 answer the general purpose better than any other that has been tried. Thus then, employing these two cases, we must first make v 500, and r 4 lb, its correspondent resistance, and then again v 1000, and r-21.88lb, the resistance belonging to it; this will give two equations, by which the general value of m and of n will be determined. Thus then the two equations being

5002m+ 50ûn=4·5,

and 1000m+1000n 21·88;

dividing the 1st by 500, and the 500m+n=·009,

Hence then the general formula will be 00002576v2 -00388v=r the resistance nearly in avoirdupois pounds, in all cases or all velocities whatever.

Now,