for the ellipse (26 × 2c) M= = or < T1π b2c2 Such results as these he has reproduced and considerably added to in his edition of Navier, pp. 52-60, pp. 122-126 and 128-136. Indeed, we may affirm that Saint-Venant was the first to insist on the practical importance of investigating the relation between the planes of flexure and of loading, when the latter plane is not one of inertial symmetry. [15.] The chapter concludes with the deduction of SaintVenant's all-important discovery that the cross-sections of a beam under flexure do not remain plane even within the limit of elasticity. There is also an investigation of the change in the cross-sectional contour (pp. 318-323). We shall return to these points later, but meanwhile may quote the concluding words of the chapter as some evidence of the satisfaction which Saint-Venant legitimately felt at the results of his new process: On voit, par ce chapitre IV, que la méthode mixte de solution des problèmes de l'équilibre des corps élastiques peut, non-seulement confirmer des résultats connus, en apprenant à quelles conditions ils sont exacts, mais encore les compléter, et donner sur les circonstances de la flexion des résultats nouveaux. [16.] Saint-Venant's fifth chapter defines torsion and deduces the general equations by the semi-inverse method; it occupies pp. 323-333. The definition of torsion which does not involve the maintenance of the primitive planeness of the cross-sections is contained in the following paragraph: Et nous nous donnerons une partie des déplacements ou de leurs rapports, en ce que nous supposerons que ces déplacements ont produit une torsion autour d'un axe parallèle à ses arêtes, torsion qui consiste en ce que les déplacements transversaux des divers points appartenant primitivement à une même section quelconque perpendiculaire à l'axe ne different de ceux des points homologues d'une autre section, que par une rotation d'un même angle pour tous, autour du même axe; en sorte que les points qui se correspondaient primitivement sur les droites parallèles à l'axe puissent être ramenés à se correspondre encore, en les faisant tourner d'un angle qui est le même pour les points des deux mêmes sections (p. 324). S.-V. 2 We will now sketch the method by which our author reaches the general equations of torsion. [17.] The axis of torsion will be taken as axis of x; the direction of torsion will be from the axis of y towards that of z. The area of a cross-section will be denoted by w, and we shall write = fy3dw, wk, fz3dw, these being the sectional moments of inertia. 2 = The torsion referred to unit of length will be 7; that is, if we draw the radius-vector of a displaced point in one section, and also that of the homologous point in a section at distance έ from the first, then the second radius-vector makes with a parallel to the first an angle of which the circular measure is Tέ; this angle is measured from the axis of y to that of z. This language implies that the torsion is constant, but the meaning of 7, when it is not constant, will be assigned in the same manner as before at any point, provided we consider έ as infinitesimally small. The above definition of torsion leads us at once to the results: dv/dx=-TZ, dw/dx=ry. (i). The consideration that there is no lateral load gives for every point of a sectional contour the equation On p. 329 Saint-Venant fixes a point, line and elementary plane as in our Art. 10, and remarks that the total torsion between the terminal sections may be considerable provided each short element into which we may divide the prism by two cross-sections receives only a small distortion relative to itself, the length of the prism being great as compared with the linear dimensions of the section. The total shifts can then be obtained by summation from the solutions of the above equations for each short element. Referring to the equations in our Art. 4 (0) we easily obtain = xy=f, (du/dy-TZ), xze, (du/dz +Ty) ...........(iii). Whence if M be the moment of all the stresses on a cross-section about the axis of x, dw [e, (du/dz +Ty) y − ƒ, (du/dy — T≈) ≈] ......................... (iv). − — 72) It will be seen that this agrees with the old theory-which gave ω ‚7 [" dw (y2 + 2"),—only when e1 = ƒ, and du/dz = du/dy. This, since du/dx is assumed constant, amounts to u = 0, or the old theory that the cross-sections remain plane and perpendicular to the axis. Substituting in the equation of our Art. 4 (k), and in (ii) above, we find for body and surface shift-equations: aau để tới đâu để ta đều để tfdu/ddy tedu do dã +(ey-fz) dr/dx = 0 (v). e(du/dz +ry)dy -ƒ, (du/dy-Tz) dz=0) Saint-Venant (p. 331) at once simplifies these equations by taking d3u/dx2=fd2u/dxdy = e d3u/dxdz = 0; these follow at once from the supposition that du/dx, or the longitudinal stretch, is constant or zero, or again from the second supposition that it is constant only along lines parallel to the axis of torsion and that a principal plane of elasticity is perpendicular to this axis (i.e. e=ƒ=0). In general we shall adopt the notation e1 =μ, ƒ1=μ1, so that our equations become Saint-Venant for the purpose of simplifying the form of his results takes μ1 =μ ̧=μ in the following four chapters. Further to avoid the complexity which would be initially introduced by treating at the same time the problem of flexure Saint-Venant takes We shall see in the sequel that Clebsch has combined the two problems of torsion and of flexure by preserving the general form of the equations. The next four chapters of the memoir VI.-IX. are occupied with the torsion of prisms of various cross-sections. I shall briefly give the results here for the purpose of reference; the reader will find little difficulty in deducing the proofs for himself, if the original memoir be not accessible. At the same time I shall draw attention to one or two important points involved in Saint-Venant's discussion. [18.] The sixth chapter occupies pp. 333-352, and is entitled: Torsion d'un prisme ou cylindre à base elliptique. The following results are obtained, the axes of the cross-section being 26 and 2c, and the notation being otherwise as before: We see at once from (i) that the primitively plane sections suffer distortion (gauchissement), and become hyperbolic paraboloids. In the accompanying figure the contour lines of these surfaces of distortion are marked; broken lines denoting depressions. The point dangereux or fail-point is obtained by making b+c'y a maximum, thus it is at the extremity of the minor axis, i.e. is the point nearest to the axis of torsion. From (iv) we obtain by means of our Art. 5 (ƒ), if S1 = S2 = S1: The general appearance of the prism under torsion is given in the figures on the next page, the torsion being diagrammatically exaggerated. [19.] There are one or two important points to be noticed in this chapter. In the first place Saint-Venant solves equation (vi) of Art. 17 by a series ascending in powers of y and z; one term (a',yz) suffices for the elliptic cross-section, he makes use of others later. Secondly he points out pp. 339-341 that his results agree with the theory of Coulomb only in the case of a circular section, |