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From the above experiments, it is evident that the round-ended shot loses more than one-half its power of resistance to pressure in the direction of its length; and this may be accounted for by the hemispherical end concentrating the force on a single point, which, acting through the axis of the cylinder, splits off the sides by a given law of cleavage in every direction. On the other hand, the flat-ended specimens have the support of the whole base in a vertical direction; and from these we derive the following comparative results :—

Taking the resistance of the flat-ended shot at 54.82 tons per square inch, and that with hemispherical ends at 26-86, we have a reduction from the mean of the flat-ended columns of 27.96 tons, being in the ratio of 100: 49; or, in other words, a flat-ended shot will require more than double the force to crush it than one with one of its ends rounded. Now, as the same results were obtained at Shoeburyness, in the appearance of the fractured ends, when similar shot was fired from a gun, we arrive at the conclusion that the same law is in operation whether rupture is produced by impact or statical pressure. In the experiments on cast-iron shot, the mean compression per unit of length of the flat-ended specimen was '0665, and of the round-ended ⚫1305. The ratio of the compression of the round- to the flat-ended was therefore as 1.96: 1, or nearly in the inverse ratio of the statical crushing pressure. It has been correctly stated that it requires a considerable amount of force to break up shot when delivered with great velocity against an unyielding object, such as the side of an iron-cased ship, or a target representing a portion of that structure; and it may be thence inferred that the force expended in thus breaking up the shot must be deducted from that employed in doing work on the plate. This is confirmed by experiment, which shows that though the whole of the force contained in the ball, when discharged from a gun at a given velocity, must be delivered upon the target, the amount of work done, or damage done to the plate, will depend on the weight and the tenacity of the material of which the shot is composed.

If, for example, we take two balls of the same weight, one of cast iron and the other of wrought iron, and deliver each of them with the same velocity upon the target, it is obvious that both balls carry with them the same projectile force as if they were composed of identically the same material. The dynamic effect or work done is, however, widely different in the two cases, the one being brittle and the other tough: the result will be, that the cast iron is broken to pieces by the blow, whilst the other either penetrates the plate or, what is more probable, flattens its surface into a greatly increased area, and

inflicts greatly increased punishment upon it. In this instance the amount of work done is in favour of the wrought iron: but this does not alter the condition in which the force was first delivered upon the target; on the contrary, it is entirely due to the superior tenacity of wrought iron to that of cast iron, which yields to the blow, and is broken to pieces in consequence of its inferior powers of resistance. The same may be said of steel in a much higher degree, which delivers nearly the whole of its vis viva upon the plate. In the foregoing experiments it will be observed that the resistance of castiron flat-ended shot to a crushing force is about 55 tons per square inch, whilst in the two following we find that the round-ended specimens, of the same material, gave way and were crushed with a pressure of only 26 tons-rather less than one-half the force required to crush the flat-ended ones. is a curious but interesting fact (provided the same law governs the force of impact as dead pressure) that the round-ended projectile which strikes the target should lose, from shape alone, one-half its powers of resistance. This may be accounted for as under.

It

Take, for example, a cylinder of cast iron, a, with a rounded end forcibly pressed against the steel plate A, until it

is crushed by a fixed law of fracture observable in every description of crystalline structure; that is, the rounded end or part s forms itself into a cone, which, acting as a wedge, splits off the sides ec in every direction at the angle of least resistance, and these, sliding along the sides of the cone, are broken to pieces on the surface of the plate.

At Shoeburyness the same results were observable in all the experiments with spherical and round-ended shot, each of

them following precisely the same law. In every case where the shot was broken to pieces, the fractured parts took the same direction, forming a cone or central core similar to that shown at s, as exhibited in my own experiments on statical pressure with the round-ended cylindrical shot.

The law of fracture of cast iron has been carefully investigated by the late Professor Hodgkinson in his paper on the strength of pillars, to which we have referred. It is there clearly shown that the resistance of columns when broken by compression is in the ratio of 1, 2, and 3; the middle one, with only one end rounded, being an arithmetical mean between the other two. Now these important facts, according to all appearance, bear directly upon the forms necessary to be observed in the manufacture of projectiles, as we find cylindrical shot with round ends loses one-half its powers of resistance to a pressure or a blow which tends to rupture or to break it in pieces. My own experiments given above do not exactly agree with those of Professor Hodgkinson-the ratio of resistance in a column with one end rounded, and that of a column with both ends flat, being as 3: 1.5, instead of as 3: 2 as in his experiments, a discovery probably explained by considering that he employed cast-iron pillars from 20 to 30 diameters in length, whereas my own were only two diameters long. Professor Hodgkinson has, indeed, expressed an opinion that the difference of the strengths of the three forms of pillars becomes less according as the number of times the length of the pillar exceeds the diameter decreases, which is the reverse of the results obtained in the foregoing experiments. But on this I may observe, that the conclusion

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is founded on a very limited number of experiments on wrought-iron columns of 15 to 30 diameters long as compared with others of 60 diameters, which, in my opinion, has been prematurely assumed as a general law. With wrought iron especially, the crushing-up of the rounded ends would soon bring pillars of that form into the condition of flat-ended pillars when the breaking weight approached the ultimate strength of the material-a conclusion confirmed by observing that the experiments in question are exactly those in Mr. Hodgkinson's table in which the breaking weights of the pillars are greatest. However this may be, the experiments I have given show that short cylinders with flat ends have twice the strength of similar cylinders with one end rounded. From this it would appear that the law for short cylinders is not the same, but altogether different from that obtained by Mr. Hodgkinson for long cylinders.

The discrepancies which appeared to exist between my own experiments and those of Professor Hodgkinson induced me still further to inquire into the law which seems to govern short bolts of columns of two diameters in length. To account for those discrepancies, the experiments were extended to columns with both ends rounded; and what renders them interesting is, that in short columns with both ends rounded the powers of resistance are nearly the same as those with one end flat and one end rounded, and moreover they appear to follow a different law from that of Professor Hodgkinson's long columns, which, in most cases, broke by flexure.

The difference in strength between short columns with both ends rounded and those with one end flat and one end rounded is almost inappreciable, as will be seen by comparing their values as under :---

Tons per square inch.

Columns of two diameters long with flat ends crushed with 54.82
Columns with one end rounded and one flat
Columns with both ends rounded .....

26.86

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23.88

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So that the difference between them may be taken as the numbers 55, 27, and 24, or, in other words, in the ratio of 1: 49 with one end rounded and one end flat-that with both ends flat representing unity-and as 1 : 44 with both ends rounded; a comparatively slight difference between those with one end flat and the others with both ends rounded.

With regard to the dynamic effect, or work done, by round-ended shot as compared with flat-ended ones, it has already been shown that with dead pressure the indentations produced on wrought-iron plates by a round-ended shot are nearly 3 times greater than by those with the flat ends, and that the work done is twice as great in the case of the round ends as compared with that by the flat ends. This may be accounted for by rounded shot striking the plate with its pointed end, and the force of the blow being given by a comparatively small area; the vis viva or

the whole force is thus concentrated and driven into the target to a depth considerably greater than if spread over the whole area of the projectile. The flat-ended cylindrical shot, which indicates such powerful resistance to pressure, is generally fractured by one or more of its sides being forced downwards in the direction of the line a, and hence its superior resistance when the whole area of the cylinder forms the base as the means of support..

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The difference of form does not, however, lessen the quantity of mechanical force (the weights being the same), as each ball has the same work stored in it when delivered from the gun at the same velocity, and the blow upon the target ought to be the same in effect but for the difference of shape in the case of the round ends, which break to pieces with one-half the pressure.

It is difficult to estimate the difference of force or work done upon the target by the two balls; it is certainly not in the ratio of their relative tenacities (the metal being the same), but arising from form, as the one would strike the target with its whole sectional area in the shape of a punch adapted for perforation, whilst the other, although much easier fractured, would effect a deeper indentation upon the plate.

The same law of defective resistance is observable in wrought iron and steel as is indicated in cast iron, but not to the same extent. On comparing the mean of twenty-six experiments on wrought iron with those on cast iron, it is evident that the difference between the two is considerable in their respective powers of resistance to compression. In the experiments on cast iron the specimens were invariably broken into fragments, and those of wrought iron, although severely crushed, were not destroyed. The same law, however, appears to be in operation in regard to the flat- and the roundended specimens, although less in that of wrought iron, as both forms were squeezed so as to be no longer useful, the ratios being as 75: 50 nearly, or 100: 67-4. The round-ended shot, as might be expected, supported considerably more than one-half the pressure applied to the flat-ended one before it was finally distorted, whilst the cast iron was broken with less than onehalf the pressure required to crush the flat-ended specimens. From these and the experiments on impact, there cannot exist a doubt as to the damaging effects of wrought-iron projectiles.

The experiments on steel indicate similar results to those on cast and wrought iron, as may be seen from the mean of nineteen experiments given in the following summary of results :

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Here the same law of defective resistance is present in the round-ended cylinders as in those of cast iron, and doubtless the same ratio would have been obtained, provided the apparatus had been sufficiently powerful to have fractured the flat-ended specimens; we may therefore conclude that, instead of the above ratio of 100: 75, it would have been 100: 50 or thereabouts. From these facts, and those on wrought iron, we are led to the conclusion that the power of resistance to fracture of a cylindrical shot with both ends flat is to that with its front end rounded as 2 : 1 nearly.

The experiments of which the above is an abstract were extended to lead, as well as cast and wrought iron, and steel; but those on lead were of little value, as the compression was the same whether the ends were rounded or flat. This is accounted for by the extreme ductility of the metal and the facility with which it is compressed. As regards the wrought-iron specimens it may be observed that no definite results were arrived at, excepting the enormous statical pressure they sustained, equivalent to 78 tons per square inch of

sectional area, and the large permanent set which they exhibit. These comparative values are as follows:-

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From the experiments on the wrought iron, the flat-ended steel specimens, and the lead, no definite conclusion was arrived at, the material being more or less compressed without the appearance of fracture. The mean resistance of the cast iron is 800 foot-pounds per square inch, whilst that of steel is 2515 foot-pounds, or more than three times as much. The conditions which appear to be derivable from these facts, in order that the greatest amount of force may be expended on the iron plate, are therefore:-Very high statical resistance to rupture by compression. In this respect wrought iron and steel are both superior to cast iron; in fact, the statical resistance of steel is more than three times that of cast iron, and more than two and a half times that of wrought iron. Lead is inferior to all the other materials experimented upon in this respect. Again, resistance to change of form under severe pressure and impact is an important element in the material of shot. In this respect hardened steel is infinitely superior to wrought iron. Cast iron is inferior to both. In fact, the shot which would produce the greatest damage on armour-plates would be one of adamant, incapable of change of form, and perfect in its powers of resistance to impact. Such a shot would yield up the whole of its vis viva on the plate struck, and, so far as experiment yet proves, those projectiles which approach nearest to that condition are the most effective.

Report on the Progress of the Solution of certain Special Problems of Dynamics. By A. CAYLEY, F.R.S., Correspondent of the Institute. MY "Report on the Recent Progress of Theoretical Dynamics' was published in the Report of the British Association for the year 1857. The present Report (which is in some measure supplemental thereto) relates to the Special Problems of Dynamics: to give a general idea of the contents, I will at once mention the heads under which these problems are considered; viz., relating to the motion of a particle or system of particles, we have

Rectilinear Motion;

Central Forces, and in particular

Elliptic Motion;

The Problem of two Centres;

The Spherical Pendulum;

Motion as affected by the Rotation of the Earth, and Relative Motion in

general;

Miscellaneous Problems;

The Problem of three bodies.

And relating to the motion of a solid body, we have

The Transformation of Coordinates;

Principal Axes, and Moments of Inertia ;

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