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ject to the action of a certain pressure distributed along all the porta of their length, and tending to produce a fracture. That they may be as light as possible, it is necessary that they resist equally the efruit acting on them; and in fact it is observed that the axis of all the feathers is a solid, of which the bases of fracture decrease, according to a certain law, from its origin to its extremity to which there is always correspondent a base of fracture equal (o), a characteristic property of the solids of equal resistance.'
The theory invented by Galileo was not advanced by any of his immediate disciples. The first experiments on the resiste ance of solids were made by Wurtzius, a Swede; and Bindel, a French architect, was the second author on the subject. In 1669, Alarchetti wrote on the resistance of solids; and in 1712, Pere Grandi, the adversary of Marchetti, and the critical examiner of his writings, published a treatise which has been inserted in the complete edition of the works of Galiles. In 1980, allaricit made several experiments; and, perceiving that their results did not accord with those given by the theory of Galiley, lie proposed to substitute a new hypothesis concerning the nature of the resistance of the fibres of bodies.
Mariett was followed by the great Leibnitz, a man destined to enlarge, illustrate, and inform every subject which he cons templated. Observing that a state of inflexion preceded the rupture of bodies, he supposed bodies to be composed of extensive fibres which, at the instant of their rupture, resist proportionally to their extension ; hence, by a simple calculation, he obtained results conformable to the observations of Maricii.
On the subject of the resistance of solids, Bernoulli also wrote; and he proved that the hypothesis of Leibnitz was not more generally admissible than that of Galilea. In the Memoirs of the Academy of Paris for 1705, he gave a paper on elastic furjes, but did not apply his demonstrations to the resistance of solids ; this application was however made by Euler, in his treatise de curvis elasticis; and he gave the method of determining the absolute elasticity or moment of elasticity, by virtue of which, solids resist their inflexion with more or less energy. This elasticity cannot be deduced in finite terms from the equa
on of the clastic curve which is not integrable: bus, since the inoment of elasticity is independent of the curvature of bodies, here is no reason why the curvature should not be deemed so Small, that the element of the curve may coincide with that of che abscissa; in which case, ihe expression for the radius of kurvature becoming more simple, an equation is obtained that is easily integrable; and thence the value sought, which is a Junction of the co-ordinates of the curve, and of the charge producing the intlexion. .
In 1769, La Grange found the same expression that Eulerhad used for the resistance of an elastic spring, pressed parallel to its length, when engaged in inquiring whether the practice of all architects, from the time of Vitruvius, (viz. of swelling the column near the third of its height,) was founded on the circumstance of this form contributing to its strength. The result of La Grange’s inquiries was, that, if the swell of pillars increases the elegance of their form, it adds nothing to their strength; and that, neglecting their own weight in the expres. sion for the charge that compresses them, the cylivdrical figurę is that which essentially suits them.
A curious and important memoir, by Coulomb, treats of the resistance of those solids which are not composed of flexible fibres, but of particles adherent to each other, as stones, minerals, &c.
The last labours of mathematicians on this subject were those. of the illustrious Euler ; who gave three memoirs in the Pe. tersburg Acts, to determine the height of a prismatic or cylin- . drical column, such as it is at the moment of its bending beneath its own weight.
Mariott, Belidor, Mussembrock, and Buffon, have made experiments on the resistance of solids. The experiments of the latter are most valuable for number and accuracy; and, had he known the theory of elastic curves, he would probably have determined the absolute elasticity: but the co-operation of calculation and experiment is rare.
The first section of the present work relates to the resistance of solids, and is purely theoretical.—The general expressions for the absolute and relative resistances of solids are cal. culated, and considered relatively to the hypothesis of Galileo and I.uibnitz. It is judiciously observed that experience alone can indicate the modifications to be made in these formulas, since no body is either perfectly hard or perfectly elastic. An illustration of the two hypotheses is exhibited by means of an indefinite-number of levers; and the expressions are calculated accordingly, and shewn to agree with those which arise from a fluxionary process. Galileo's theory is applied to several examples : it is manifested that, of an hollow cylinder, where the diameters of the two circles remain the same, the resistance is greatest when the interior circle touches the exterior in the lowest part. Leibnitz's theory is likewise applied to several examples. The resistance of solids is next considered in those cases in which the power of producing a fracture acts at their extremities; and it is demonstrated that the weight, capable of producing the rupture of a prism firmly inserted at its parts of support, is double that which is necessary to break the same
body, freely sustained on the same parts: a result which accords with the experiments of Mariott.
N. GIRARD then proceeds to shew that the formulas of the resistances, on the theories of Galiles and Leibnitz, agree in this, that the resistance of the rectangular bases of fracture are as the squares of the heights multiplied by their length. At the end of the section designed to determine the relation between weights which, compressing solids of given dimensions parallel to their length, are able to make them bend, we have a digression on elastic curves, extracted from Euler's treatise. The relation above mentioned was deduced by this geometrician, as a consequence from the theorems appertaining to the theory of elastic curvesa. M. Girard extracts only what is indispensably necessary to lead him to the proposition which he had in view, and which is thus announced; that the dimensions of the bases, and the elasticity of a series of columns, remaining the same, the weights which they can support before their fexure will be in the inverse ratio of the squares of their respective lengths.'
The rigorous equation of the elastic curve is of this form : - (my* +n) y
which admits no integration : v (
np'-(my? + n)“) if, however, the solid be considered in its first degrees of inflexion, the element of the curve (zo) is nearly equal to the element of the abscissa (x); in which case, the differential equation may be integrated, and the absolute elasticity determined.
In discussing the absolute negative resistance of solids charged solely with their own weight, M. GIRARD notices what neglect of circumstances led Euler to the paradox, that a heavy solid, prismatic, or cylindrical, standing vertically on a fixed plane, cannot bend beneath its own weight, to whatever height it is raised *.
The Second Section considers the solids of equal resistance, and is purely theoretical : their general equation is zy'=F(x), z being the breadth of the base of any rectangular fracture, y its height, x the distance of the base of the lever from the extremity of the arın to which the weight is applied, and F ) denoting a function of x. An application is made of this equation to cases in which the weight is constant ; or ia which the weight or charge is some function of the arm of the lever to
* The absolute negative resistanee of solids, in contradistinction to the ahsolute positive, is that of the fibres of a body against the action of a power tending to cupress them, and in a direction farallel to their length.
which it is applied. Solids of equal resistance are considered relatively to their weight. Solids of equal absolute negative resistance are also discussed, and an application is made to the form of pillars. The result of the investigation concerning the form of pillars we have already stated : the method of M, Girard is, first, to determine the form of pillars having only their own weight to sustain, and then that of those having their own and an added weight. The pillars, concerning the form of which the inquiry is made, are those which resist equally in all their bases of rupture; their form is canoidal, engendered by the revolution of a logarithmic curve about its axis: but, by reason of the excessive magnitude of the subtangent of the curve relatively to the height, the generating arc nearly coincides with a line parallel to the axis, and the conoid nearly with the cylinder.
The Third Section contains experiments on the resistance of solids. The first and principal object is to determine the value of Ek’, which represents the absolute elasticity : this value cannot be deduced from the equation y=-P(*+cx+3)x.
taken rigorously: but, by supposing the solid to be in the first degrees of inflexion, the equation may be integrated, and the yalue Eéz deduced.
This section contains likewise the description of the appa. ratus by means of which the experiments were made. It is difficult to form a complete and precise idea of a complicated machine, even with plates and a verbal description : but it would be vain to expect thạt verbal description alone could convey an adequate notion of the machinery, its parts, and the whole; we must therefore refer our readers, for satisfaction on this point, to M. GIRARD's book.
The description of the machinery is followed by a detail of experiments, and an explanation of the tables constructed according to the new French system of weights and measures.
Section 4th and last relates to the circumstances which attend the inflexion of bodies supposed to be perfectly elastic. What had been determined in the preceding sections concerned merely the equilibrium between the resistance and charge : but the equilibrium does not take place instantaneously; the inflexion of solids has always a certain duration ; and, reasoning by analogy, its motion ought to be subject to some law of continuity. To obtain the time which must pass before the equilibrium is established,
M. GIRARD first deduces the accelerating force by means of tie principle invented by D'Alembert, and so successfully emplored in his Dynamique. When Force (?) is determineil, Velocity (v) may, in cases in which the inflexion is small, from the equation vu:- Fy, and the time () from the equation auto
yo: the duration of the inflexion cannot rigorously be assigned.
Towards the end of the section, we find a dissertation ca the laws of the compressibility of aerial fluids, and an application of what is deduced concerning these laws to the setturs and inflexion of bodies imperfectly elastic. • In this valuable treatise, the analytical operations are condueted with perspicuity, and judiciously adapted to the pure
poses of experiment: useful remarks are also frequenely interspersed ; and the spirit of temperate and informed criticism is prevalent throughout. The work affords an additional instance of the success with which theory and experiment may be made to co-operate. We have reason to be elated at the advancements which science has made within the present age; yet what remains to be done is vast in its extent, and arduous of investigation! While we attend to the means by which our knowlege is enlarged, we are sensible that the limits are close on every side of us: what the sage Baçon long ago observed is true even now, " how little is done, and how much remains to be done!" Sufficiently sensible (as every man is who know's with what difficulty the least addition to truth is made) of the obstacles which oppose the advancement of physico mathematics, M. GIRARD observes that
" It cannot be dissembled that there is a very considerable dificulty in assigning the laws of certain phenomena, with precision suficicat to satisfy all cases. In the application of calculus to physies, and especially in questions involving the organization of bodies, we trust be contented to approach towards the truth: but, as every observatia made with care is a step towards it, no occasion of making such an observation should be onitted. Thus, although the experimeci., which have been related, 'were made during many years with as much exactness as they scenied to requiic, yet we are of opinion that it will be at all times highly advantageous to repeat them, or to make new or the same subject; a labo'ir especially reserved for engineers who possess skill, and zeal for the advancement of their art.'
This work is enriched with a valuable table of contents, and with copious tables relative to ti.c experiments.