Substitute in the above equation (1) and we obtain

Integrate this equation by parts in the usual way, and it becomes

here (ny) and (nz) denote the angles which the normal to the surface at the point (x, y, z) makes with the axes of y and z respectively; and ds is an element of the curve of intersection of the body by a plane at right angles to the axis of x.

If we equate to zero the term in brackets in the double integral we obtain the equation which must hold at every point of the interior; and if we equate to zero the term in brackets in the single integral we obtain the equation which must hold at every point of the surface.

But Saint-Venant does not explain why we must equate these terms separately to zero; that is, he does not explain why he breaks up equation (2) into two equations. Moreover the whole process borrows so much from the memoir on Torsion that it has not the merit of being an independent investigation.

Saint-Venant says:

Or la deuxième et la première parenthèse carrée, égalées séparément à zéro... :

by this he means the terms contained within the square brackets in (2). The English translation has very strangely "Now the squares of the second, and of the first parenthesis, each equated to zero,..."

[158.] A remark of Saint-Venant's on p. 809 may be cited:

Le calcul du potentiel de torsion a aussi, en lui-même, une valeur pratique; car les ressorts en hélice, qu'on oppose souvent à divers chocs, travaillent presque entièrement par la torsion de leurs fils, ainsi que je l'ai montré en 1843, et que l'ont ermarqué, au reste, Binet dès 1814, M. Giulio en 1840, et récemment des ingénieurs des chemins de fer.

See our Arts. 175*, 1220*, 1382* and 1593-5*. The 1814 and the récemment (1864) mark the wide interval which too often separates theory from practice!

[159.] Théorie de l'élasticité des corps, ou cinématique de leurs déformations. Les Mondes, Tome 6, 1864, pp. 607 and 608. If a body is deformed any small portion originally spherical becomes an ellipsoid: see our Art. 617*. In the present paper SaintVenant undertakes to establish this proposition by simple general reasoning; the process does not seem very satisfactory.

SECTION III.

Researches in Technical Elasticity.

[160.] Résumé des Leçons...sur l'application de la mécanique à l'établissement des constructions et des machines....Première section. De la Résistance des corps solides, par Navier....Troisième Édition avec des Notes et des Appendices par M. Barré de SaintVenant. The title-page bears the imprint, Paris, 1864. A footnote, however, on p. 1 tells us that pp. 1-224 appeared in 1857, pp. 225--336 in 1858, pp. 337-496 in 1859, pp. 497—688 in 1860, pp. 689-849 in 1863, while the Notices et l'Historique, pp. i-cccxi, were finally added in 1864. Thus the whole work of more than 1100 pages occupied some seven years in the production, and thus necessarily lacks somewhat of the unity which is to be met with in other treatises. Under the form of notes to a few sections of Navier's original work (see our Art. 279*), Saint-Venant has given us a complete text-book of elasticity from the practical standpoint. At the same time, by additional notes and appendices, he has rendered his text-book of surpassing historical value and physical suggestiveness. The leading characteristics of the book are simplicity of analysis and copiousness of reference. See Notice I., pp. 41-2 and Notice II., pp. 28—9.

[161.] The cccxi. pages of introductory matter are occupied with the following subjects: Table of Contents, pp. i-xxxviii; Notice biographique sur Navier by de Prony extracted from the Annales des ponts et chaussées (1837, 1er semestre, p. 1), pp. xxxix—

li; the funeral discourses on Navier by Emmery and Girard, pp. li-liv: a bibliography of the works of Navier with copious remarks due to Saint-Venant, pp. lv—lxxxiii; the original prefaces to the editions of Navier's Leçons published in 1826 and 1833; pp. lxxxiv -xc; and finally Saint-Venant's Historique abrégé des recherches sur la résistance et sur l'élasticité des corps solides, pp. xc—cccxi.

[162] The Historique abrégé is practically the only brief account of the chief stages of our science extant. Girard had written what was for his day a fair sketch of the incunabula (see our Art. 123*), but it remained for Saint-Venant, without entering into the analysis of the more important memoirs, to describe their purport and relationship. It fulfils a different purpose to our own history—for it makes no attempt to replace the more inaccessible memoirs-but as a model of how mathematical history should be written, we hold it to be unsurpassed, and can only regret that a recent French historian has not better profited by the example thus set1.

We would especially recommend to the student of SaintVenant's memoirs pp. clxxiii-cxcii, which treat of the relation of his own researches by means of the semi-inverse method to the work of his predecessors. The point we have referred to in our Arts. 3, 6, 8 and 9 is well brought out in relation to Lamé's problem of the right-six-face.

We will note one or two further points of the Historique in the following five articles.

[163.] On p. cxcviii in the footnote Saint-Venant gives the expres

sion for the work-function in terms of the stresses when there is an ellipsoidal distribution of elasticity: see our Art. 144. He finds

1 The essential feature of scientific history is the recognition of growth, the interdependence of successive stages of discovery. This evolution is excellently summarised in Saint-Venant's Historique. Our own history' is only a bibliographical repertorium of the mathematical processes and physical phenomena which form the science of elasticity, as a rule for the purpose of convenience chronologically grouped. M. Marie's Histoire des sciences mathématiques is a chronological biography, without completeness as bibliography or repertorium. Excellent fragments there are in it, but the conception of evolutionary dependence is wanting.

[164.] Pages cxcix-ccix deal with the history of the problem of rupture. According to Saint-Venant, two kinds of rupture may be distinguished: rupture prochaine and rupture éloignée. The former falls outside the theory of 'perfectly elastic' bodies, the latter he thinks may be deduced from the hypothesis that when the limit of mathematical elasticity is passed,―i.e. when the stretch is greater than the limit at which stretch remains wholly elastic and proportional to traction,-then the body will ultimately be ruptured if it has to sustain the same load. The reader who has followed our analysis of the state of ease and the defect in Hooke's Law given in the appendix to Vol. I. and also our Arts 4 (y) and 5 (a) in the present volume will recognise that this hypothesis has only a small field of application. What we have really obtained is a limit to linear elasticity. It is the more important to notice this because SaintVenant argues that we must take as our limit the maximum positive stretch, for, as Poncelet has asserted: "que le rapprochement moléculaire ne peut être une cause de désagrégation" (p. cci). It is probably true that rupture can only be produced by stretch, but squeeze can surely produce failure of linear elasticity when the body is so loaded that no transverse stretch is possible. Hence when Saint-Venant introduces the stretch and slide-moduli into his condition for safe loading and so makes it a question of linear elasticity, it seems to me that he ought at the same time to alter his statement as to the greatest positive stretch being the only quantity we are in search of. Indeed, his condition seems partly based upon an idea associated with rupture, and is then applied to constants and equations deduced from the principle of linear elasticity (see his p. ccviii, § XLVIII.). The limitations to which his theory is subjected were, however, partially recognised by Saint-Venant himself (see his pp. ccv-vii). Thus he writes:

Nous ne prétendons pas, au reste, qu'une théorie subordonnant uniquement le danger de rupture d'un solide à la grandeur qu'atteint une dilatation linéaire n'importe dans quelle de ses parties, et indépendamment des autres circonstances où il se trouve en même temps, soit le dernier mot de la science et de l'art.

He refers on this point to the experiments of Easton and Amos: see our Art. 1474*.

[165.] Pages ccxiv-xxiv deal with the problems of resilience and impact.

In the footnote p. ccxvii, there is an error in the integral of the equation

d2%

dr2 = 9 cos a

g

z there given. It should be

The error was noted by Saint-Venant himself in a letter to the Editor of this History, August, 1885.

On p. ccxxii and footnote there should have been a reference to Homersham Cox with regard to the factor k = 17/35. His memoir of 1849 (see our Art. 1434*) seems to have escaped SaintVenant's attention.

A further consideration of the effect of impact on bars when the vibrations are taken into account occurs on pp. ccxxxii-viii, and then follows (pp. ccxxxix-xlix) an account of Stokes' problem of the travelling load (see our Art. 1276*). Saint-Venant refers to the researches of Phillips and Renaudot, but his account wants bringing up to date by reference to more recent researches.

[166.] On pp. ccxlix-ccliii Saint-Venant refers to the rupture conditions given by Lamé and Clapeyron and again by Lamé for cylindrical and spherical vessels. It seems to me that he has not noticed here that these conditions are, on his own hypothesis of a stretch and not a traction limit, erroneous: see the footnotes to our Arts. 1013* and 1016*.

[167.] After an excellent and succinct account of the course of the investigations of Euler, Germain, Poisson, Kirchhoff &c. with regard to the vibrations of elastic plates (pp. ccliii-cclxxi) the Historique closes with two sections LXI. and LXII. (pp. cclxxi—cccxi) on the experiments made by technologists and physicists previously to 1864 on the elasticity and strength of materials. Good as these pages are, they are insufficient to-day in the light of the innumerable experiments of first-class importance made during the last twenty years.