stress on one side of an indefinitely small plane face in the interior of a solid or fluid body is the resultant of all the actions of the molecules on this side of the plane on the molecules on the other side, the directions of which cross the face: see our Art. 616 and Moigno's Statique, pages 619, 675. Cauchy makes some remarks on the phrase moment d'une force d'élasticité which is used in the Mécanique analytique; Lagrange to obtain this moment multiplies the force by the differential of the angle which it tends to diminish. Cauchy says: Il est clair que, pour obtenir le véritable sens des formules de Lagrange, on ne doit pas attribuer ici aux expressions qu'il a employées leur signification ordinaire. After some explanations Cauchy concludes thus: En conséquence, dans la Mécanique analytique de Lagrange, par ces mots force d'élasticité tendant à diminuer un angle, on doit toujours entendre le moment d'un couple appliqué à l'un des côtés de cet angle, c'est-à-dire la surface du parallelogramme construit sur les deux forces du couple. 679. Observations sur la pression que supporte un élément de surface plane dans un corps solide ou fluide. Comptes Rendus, Vol. XXI. 1845, pages 125–133. Cauchy as we saw in the preceding article adopted the definition of stress given by Saint-Venant, and in this article he investigates expressions for the resolved stresses on the definition. The results are of the same nature as those in Moigno's Statique pages 674 and 675. Cauchy's method is somewhat obscure, and does not present any obvious advantage. See Moigno 619. 680. Mécanique moléculaire. Comptes Rendus, Vol. xxviii. 1849, pages 2-6. 2-6. All that concerns us here is the announcement of a design which it is to be regretted was never accomplished. Des savants illustres, dont plusieurs sont membres de cette Académie, m'ayant m'engagé à réunir en un corps de doctrines les recherches que j'ai entreprises et poursuivies depuis une trentaine d'années, sur la mécanique moléculaire et sur la physique mathématique, j'ai cru qu'il était de mon devoir de répondre, autant que je le pouvais, à leur attente, et de réaliser prochainement le vœu qu'ils m'avaient exprimé. Il m'était d'autant moins permis de résister à leur désir, qu'en y accédant je remplis, en quelque sorte, un acte de piété filiale, puisque ce désir était aussi le vœu d'un tendre père, qui joignant, jusqu'en ses derniers jours, l'amour de l'étude et la culture des lettres à la pratique de toutes les vertus, s'est endormi du sommeil des justes, et s'est envolé vers une meilleure patrie. Pressé par tous ces motifs, je me propose de publier bientôt un Traité de mécanique moléculaire où, après avoir établi les principes généraux sur lesquels cette science me paraît devoir s'appuyer, j'appliquerai successivement ces principes aux diverses branches de la physique mathématique, surtout à la théorie de la lumière, à la théorie du son, des corps élastiques, de la chaleur, &c. This passage is condensed in the Répertoire d'optique moderne...par l'Abbé Moigno, page 1741; but the allusion to the father of M. Cauchy is rendered unintelligible, and even ungrammatical, by the omission of its last clause. 681. Note sur l'équilibre et les mouvements vibratoires des corps solides. Comptes Rendus, Vol. XxxII. 1851, pages 323–326. This consists merely of generalities, and is apparently of no importance. [One point in this memoir seems to me suggestive. Cauchy remarks that if we consider a homogeneous body as built up of a system of molecules, and each molecule be in itself a system of atoms, then les coefficients renfermés dans les équations des mouvements vibratoires de ce corps cesseront d'être des quantités constantes. The conception as Cauchy points out is by no means without a possible application in the Theory of Light.] 682. Rapport sur divers Mémoires de M. Wertheim (Commissaires, MM. Regnault, Duhamel, Despretz, Cauchy rapporteur). Comptes Rendus, Vol. XXXII. pages 326-330, 1851. This report speaks very highly of the investigations of Wertheim. The principal point noticed is the value assigned by Wertheim as the ratio of one constant to another; Wertheim holds, using our notation, that λ= 2p. Cauchy cites experiments in favour of Wertheim's view, and holds that there is no valid theoretical objection against it. 683. Saint-Venant alludes to the papers noticed in Arts. 540 and 682; see his Torsion page 262, and Moigno's Statique page 706 but the object of the allusions is not very clear to me. For instance at the last cited page we have, La possibilité que Cauchy s'efforce d'y établir, contrairement à ses beaux travaux de 1828 à 1845, de plus de quinze coefficients...: but I do not see to what passage of Cauchy's these words refer1. 684. Sur la torsion des prismes. Comptes Rendus, Vol. XXXVIII. 1854, pages 326-332. These pages contain a brief introduction to the subject of the torsion of prisms. Cauchy alludes to his own researches on it in the fourth volume of his Exercices de mathématiques; these he admits were only approximative, and holding under certain conditions. He speaks very highly of the recent researches of Saint-Venant with respect to the subject, and says that a careful perusal of them had led him to some new reflections. There are only two points which require notice in this article. Cauchy establishes in a simple way expressions for the six stress-component, involving twelve constants; Saint-Venant reproduces this in his Torsion in establishing the equations (18) on page 265 of that work. Compare also equations (33) on page 655 of Moigno's Statique, where however only nine constants are preserved. Again Saint-Venant assumes that the angle of torsion 7, corresponding to a unit of length is constant; Cauchy proposes to generalise this by assuming to be a function of the distance of the point from the axis. Saint-Venant himself pursues this suggestion in a note on pages 341-343 of his Torsion, and shews that it does not lead to any results of practical value. Cauchy's article concludes thus: 1 [Saint-Venant sees in these papers an abandonment by Cauchy of what he himself holds to be the true basis of elasticity, namely that molecular theory which reduces the 36 constants to 15, and not merely to 21, in the case of an aeolotropic elastic solid. ED.] Il reste à montrer comment, à l'aide du calcul des résidus, on pourra obtenir immédiatement l'intégrale donnée par M. de Saint-Venant, et l'intégrale du même genre relative au cas où est facteur de r. C'est ce que je me propose d'expliquer dans un prochain article. The intention here expressed seems not to have been carried out'. Moigno's Statique 616, 625, 640, 664, 665. Saint-Venant on Torsion 266, 340. 685. The remarks with which an enthusiastic pupil and friend of Cauchy closes a survey of the contributions of this great mathematician to the theory of light are equally applicable in relation to the subject of elasticity. See Epiphonème. pages 1748-1749. Moigno's Répertoire d'optique moderne, 1 [It may be remarked that Cauchy in this memoir first employs Coriolis's double suffix notation. ED.] CHAPTER VI. MISCELLANEOUS RESEARCHES OF THE DECADE [686.] WE must here note an historical controversy, which arose on the effect of a uniform tractive load on the inside and outside surfaces of a glass or metal vessel in changing its volume. The problem as to this change of contents sprung from the renewed experiments on the compressibility of water and other liquids which were made in various countries during this and the preceding decade. The controversy is interesting as giving a practical example of the need for a theory of elasticity. We may note the following memoirs : [687.] I. Oersted. Sur la compressibilité de l'eau. Annales de Chimie, Tome 22, pp. 192-198, 1823. This is an account of Oersted's first apparatus and experiments. He appears to have neglected the compressibility of his containing vessel. A remark made by him (p. 196) seems to shew that the material of that vessel obtained a set, not that the water lost its compressibility after several trials: Je dois encore signaler une autre circonstance qu'on devrait peutêtre prendre en considération ici : c'est que l'eau semble perdre un peu de sa compressibilité après quelques compressions. Je n'oserai cependant assurer ce fait, ne l'ayant pas soumis à des épreuves rigoureuses. [688.] II. Colladon and Sturm. Mémoire sur la compression des liquides. Mémoires...par divers savants, sciences mathématiques et physiques. Tome v. pp. 267-347, Paris, 1838. The paper was read on June 11, 1827. The authors commence that portion of their subject which concerns us with the following statement: ...ce principe assez évident, qu'un corps solide homogène, plongé 1 This chapter contains a few experimental researches falling outside this decade. |