square sectioned bars it was only 5,636,625,000 on the old theory. Saint-Venant's however brings it up to 6,682,750,000, which may be considered in fair agreement with the result obtained from bars of circular section; especially when we remember the non-isotropic character which was inevitable in the iron bars of Duleau's experiments (see table p. 383). At any rate Saint-Venant's theory accounts for the greater part of the inferior resistance to torsion of square as compared with circular bars of equal sectional moment of inertia. Some experiments on copper wires of square and circular cross-sections are tabulated on p. 386. Here the mean for the circular cross-section is μ = 4,174,825,000; the old and the new theory give for μ the values 3,384,121,000 and 4,012,180,000; again to the advantage of the latter. The isotropy of these wires is however very questionable. [32.] Saint-Venant deduces on pp. 387-391 the value of the numerical factor which occurs in M (see our Art. 30) by an algebraic expansion for u and a calculation after the manner of Fourier (Théorie de la chaleur, chap. III. art. 208, Eng. Trans. p. 137) of the indeterminate coefficients. It does not seem a very advantageous process. A remark on p. 397 as to the difference between résistance à la rupture éloignée and rupture immédiate is to the point. Saint-Venant remarks namely that experiments on the latter can throw little light on the mathematical theory of elasticity. At the same time it is regrettable that he should have retained the word rupture in reference even to the first limit. Some support, however, for his theory may even be derived, he thinks, from Vicat's experiments on rupture; see our Art. 731* and p. 398 of the memoir. For Vicat found that for pierre calcaire, brique crue and plâtre the moment of the forces required to break a prism of square cross-section and length at least twice the diameter was less than in the case of an infinitely short prism, i.e. a case where the plane section cannot be distorted. This result of Vicat is of great interest and would be well worth further experimental investigation. [33] We now come to the general case: Cas d'un rapport quelconque des deux dimensions de la base (pp. 398-413). SaintVenant has calculated numerically all the particulars of the special case when b/c = 2. We reproduce the contour lines for the distorted cross-section as given by Saint-Venant on p. 400 according to the table on p. 399. The reader will at once note the change that these lines present, and Saint-Venant on pp. 400-1 determines the value of b/c for which the change from tetra-axial to bi-axial congruency takes place. = In order to ascertain this we must find when du/dz = 0 at the point y=b, z 0. For, with the tetra-axial congruency of the contour lines u is positive as we pass from z = 0, y=b along the line y=b into the first quadrant, but in the case of biaxial symmetry du/dz is negative, for u decreases or becomes negative as we pass along the same line. Our author thus obtains the equation the numerical solution of which gives b/c 1.4513. = [34.] The following general results are obtained (b>c): and this maximum slide takes place at the centre of the longer side of the rectangular cross-section. (p. 410.) These complex analytical results are rendered practically of service by a table on pp. 559-60 of the memoir, the most serviceable portion of which we shall reproduce later. This table gives the values of ẞ and of Bly for magnitudes of the parameter b/c varying from 1 to 100, after which they become sensibly constant. We are thus able to determine M and its limit M. Saint-Venant, however, gives in footnotes empirical formulae which agree with less than 4 per cent. error with the above theoretical values. He appears to have reached them by purely tentative methods, but he holds that they satisfy all practical needs. They are = 0, our μ = G, {It should be noted that our σ = 9, our ß=μ, our 7= our ST of the memoir.} [35.] On pp. 403-6 we have a further discussion of experiments of Duleau and Savart on the torsion of rectangular bars of iron, oak, plâtre, and verre à vitre, the paucity of the experiments, and the large variation in the values of the slide-moduli as obtained from Saint-Venant's formula do not seem to me very satisfactory. A series of experiments directly intended to test the torsion of rectangular bars for variations of the parameter c/b would undoubtedly be of considerable value. [36.] We now reach Saint-Venant's ninth chapter which is entitled: Torsion de prismes ayant d'autres bases que l'ellipse ou le rectangle. It occupies pp. 414-454. The chapter opens with an enumeration of the various forms of contour for which it is easy to integrate the equations of Art. 17. We will tabulate them on the next page. 72 1⁄2 (y2 + xo) + ΣAm cosh my cos mz 3 2 (y2 + 2o) − a ̧≈ + b‚y − a ̧ 2yz +b ̧ (y2 — 2o) — a ̧(3y2z — 23) α 4 1⁄2 (y2+ z2) + √ − 1 p (y + z √ − 1) − √ − 1 4 (y − z √ − 1) 1⁄2 r2 + ≥ (− a„r” sin no + bmrTM cos mp) Edm sinhmy sin mã + a‚y + bx + α, (y2o − x2) + b, 2yz + a ̧ (y3 — 3yz2) |