By which it appears that, from the mean of 56 comparisons, the centre yard of the Aberdeen scale is 001385 inch (or about of an inch) shorter than the centre yard of the Royal Astronomical Society's scale, at the temperature of about 47° of Fahrenheit's thermometer.

I have also myself made 16 direct comparisons of the whole length of the same scales, namely 5 feet; and have obtained the following results, viz.

By which it appears that, from the mean of the 16 comparisons, the whole measure of the 5 feet Aberdeen scale is 001971 inch (or about 50 of an inch) shorter than the whole 5 feet of the Royal Astronomical Society's scale, at the temperature of 73° 6 of Fahrenheit.

I beg to add that the Aberdeen scale is in very good condition, and in excellent preservation: it appears to have been exceedingly well finished, and is by far the best of any that I have seen of Mr. Troughton's execution. And although the above results show a greater discordance from the correct measures than is desirable, yet as perfect accordance is seldom or never attainable, no inconvenience can arise from this circumstance, now the amount of the error is ascertained, and will consequently be known to those parties who may, at any future time, have occasion to make use of this scale.

Independent of the value of these experiments in thus determining the comparative length of the Aberdeen scale, they are important in a general point of view, in as much as, coupled with others that I have made with a similar object, they evidently show that the too prevalent notion “that standard scales, "made from one and the same prototype agreeably to Mr.

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Troughton's method, would accord with each other," is not strictly correct. Indeed, it is now too evident (as I shall at some future time show more in detail) that a great number of minute, yet important, circumstances have hitherto been neglected in the formation of such scales; and without an attention to which, they cannot be expected to accord with that degree of accuracy which the present state of science demands.

Impact upon Beams. By EATON HODGKINSON.

THE object of the present paper is an inquiry into some of the effects of impact upon beams when struck by bodies of different weight, hardness, and elastic force; and to compare theory with, and endeavour to adapt it better to, the results of experiment. The paper is a continuation of some experimental researches on the collision of imperfectly elastic bodies, which were published in the Fourth Report of the Association; and it is intended to contain proofs of the principal statements made in a short communication read at the Cambridge Meeting.

The preliminary conclusions, and some calculations with their results, will first be given; and afterwards the experiments, to which constant reference will be made for proofs and illustrations.

With the castings and every assistance in making the experiments I have been supplied, as on former occasions, through the liberality of Mr. Fairbairn, engineer, of Manchester.

Conclusions from Experiments, &c.

Conclusion 1.-If different bodies of equal weight, but differing considerably in hardness and elastic force, be made to strike horizontally with the same velocity against the middle of a heavy beam supported at its ends, all the bodies will recoil with velocities equal to one another.

This is shown by the experiments on the 3rd beam, in which two balls 8 lbs. each, one of lead and the other of cast iron, suspended as pendulums, were made to fall through equal arcs against the middle of the beam, 13 lbs. weight between the supports; and the recoil of the leaden ball was nearly the same as that of the iron ball, though the hardness and elasticity of the two balls were widely different. This equality of recoil in the two balls was likewise shown to exist whether both fell through a small or large arc.

To vary the weight and quality of the balls, two were used, half the weight of the former, 44 lbs. each, one of lead and the other of bell-metal. In these, as before, when both were let fall through equal arcs, whether great or small, the recoil of one ball was nearly equal to that of the other.

To vary further the experiment on this beam, three balls were

used, about of the weight of the first, 9 oz. 7 drs. each, one of lead, one of bell-metal, and the other of hardened steel. The recoils from equal impacts in all these, though very anomalous, tended toward equality as before.

The beam here used was of steel, but that did not affect the results; for the same equality in the recoils will be found in the experiments on the 2nd beam, which was of cast iron.

Conclusion 2.-If, as before, a beam supported at its ends be struck horizontally by bodies of the same weight, but different hardness and elastic force, the deflection of the beam will be the same whichever body be used.

case.

This conclusion is proved by the experiments upon the 2nd and 3rd beam, and with the same generality as in the former In those on the 3rd beam two balls 8 lbs. each, one of lead and the other of cast iron, were made to strike the beam with velocities varying from 1 to 5; and the deflections from equal impacts by the two balls are as below:

In the impacts with the 4 lbs. balls of lead and bell-metal, the deflections from equal impacts, and varying in velocity from 1 to 6, are as below, and nearly equal:

The same equality is shown too, though with greater anomalies than above, in the deflections from impacts with the balls of lead, bell-metal, and hardened steel, 9 oz. 7 drs. each.

Conclusion 3.-The quantity of recoil in a body, after striking against a beam as above, is nearly equal to (though somewhat below) what would arise from the full varying pressure of a perfectly elastic beam as it recovered its form after deflection.

The fact in this Conclusion was sought for, because it seemed doubtful whether a bent beam would straighten itself with any nearer approach to the velocity arising from perfect elasticity than that given by the defective elasticity of the material of which it is made. Thus, two solid bodies of cast iron, struck against each other recoil with only of their velocity of im

pact, as appears from our experiments on the collision of imperfectly elastic bodies (Fourth Report of the Association). But a cast iron beam throws back a ball with a velocity much more nearly approaching to what would arise from perfect elasticity. This will be seen by comparing the observed results with the calculated ones in the experiments upon the 1st and 2nd beams, the calculated results being obtained from problem 1 following, where the beam is assumed to be perfectly elastic.

Conclusion 4.-The effect of bodies of different natures striking against a hard flexible beam seems to be independent of the elasticities of the bodies, and may be calculated, with trifling error, on a supposition that they are inelastic.

If the calculation be formed on a supposition that the time of the collision, in the first approach of the impinging body, is small compared with the time of deflection of the beam, and that the beam and striking body both proceed together afterwards as one mass (as is done in our following problems); the calculated deflections are somewhat greater than the observed ones, the dif.. ference sometimes amounting to one fifth or one eighth, as will be seen from the experiments on the 1st, 2nd, and 3rd beams. But the observed deflections in our experiments, excepting, perhaps, those on the 3rd beam, must be rather too small, arising from the resistance of the clay, into which a peg used for measuring the deflections was driven by the impacts.

This fourth Conclusion must only be admitted when there is nothing struck upon but the beam. When there is any other heavy body intervening between the striking body and the beam, touching the latter, and which must be struck before the beam can be deflected, then the elasticities of the concurring bodies exhibit their influence, and the result is greater than that obtained by calculation as above. This might be expected; and it is shown by the experiments on the 4th beam.

Dr. Young, in his Natural Philosophy, and Mr. Tredgold, in his Treatise on the Strength of Cast Iron, reason on this subject as if the striking body were inelastic, and we have here shown that this may be assumed, whatever the hardness and elasticity of the striking body may be*; or, probably, its weight with respect to that of the beam.

Of this curious fact the Author would beg to suggest the following as a possible explanation. In the first moment of the impact upon the middle of the beam, each half of it, if its ends were not fastened, would have a tendency to turn round its centre of oscillation, or a point two thirds of the distance from the middle to the end, which is seen in experiment by the ends springing up after a blow. But it is probable that, besides this, the whole beam is thrown by the blow into a state of nodal vibrations, like as in a musical chord; there

In the elaborate paper on the "Measure of Moving Force,' by Mr. Ewart, (Manchester Memoirs, vol. ii., second series,) there are, among other important matters, some ingenious inquiries respecting impact and the force of springs. These, however, do not appear to be easily applicable to our present subject generally; and the clay used by Mr. Ewart, in his experiments with pendulous balls, was the resisting medium, while in our case it was employed only to indicate the deflections.

Conclusion 5.-The power of a uniform beam to resist a blow given horizontally is the same in whatever part it is struck.

From the experiments on the 5th, 6th, and 7th beams, it appears that the beams, when supported at the ends, required the same blow to break them, whether they were struck in the middle, or half-way between that and one support. From a future investigation, too, it appears that the same is the case wherever the beam is struck.

Conclusion 6.-The power of a heavy uniform beam to resist a horizontal impact is to the power of a very light one as half the weight of the beam, added to the weight of the striking body, is to the weight of the striking body alone.

This is shown by Cor. 1. Prob. 2; for, from Cor. 2, the inertia appears to be half the weight of the beam; and the greater resistance of a large mass than of a small one may be inferred from the experiments on beam 4, and others.

Conclusion 7.-The power of a uniform beam to resist fracture from a light body falling upon it (the strength and flexibility of the beam being the same,) is greater as its weight increases, and greatest when the weight of half the beam, added

being one, two, or more nodes on each side of the middle. This will be understood from the adjoining

figure, which represents the beam, when bent by an impact from the ball A, and the small excursions of the parts

between the nodes. The time of a vibration of one of these parts is very small compared with the time of a vibration of the whole beam. Chladni has shown that if a uniform rod have its ends supported and be put into a state of double vibration as above, the number of nodes being n, there will be (n + 1)2 of these secondary vibrations for one whole vibration of the rod (Biot, Traité de Physique, tom. ii., p. 77-8). Hence, after the first concussion of a ball upon a heavy beam, the ball and beam in proceeding together are not constantly in contact, or in a state of equal pressure if they are. Their connexion appears to be a series of small impacts, or of approaches and retreats, the intervals between each of which are the time of one of these secondary vibrations. And during these intervals it is presumed that the compressed surfaces of the ball and beam recover themselves after the first concussion, leaving the effect the same whether the ball be elastic or not.