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TP perpendicular to it, each amounting to np.PE per unit of area.

An application of force equal and opposite to the distribution thus found over the prismatic boundary, would of course alone

'produce in the prism, otherNAV

wise free, a state of strain which, compounded with that supposed above, would give the state of strain actually produced by the sole application of balancing

couples to the two ends. The result, it is easily seen (and it will be proved below), consists of an increased twist, together with a warping of naturally plane normal sections, by infinitesimal displacements perpendicular to themselves, into certain surfaces of anticlastic curvature, with equal opposite curvatures in the principal sections (122) through every point. This theory is due to St. Venant, who not only pointed out the falsity of the supposition admitted by several previous writers, that Coulomb's law holds for other forms of prism than the solid or hollow circular cylinder, but discovered fully the nature of the requisite correction, reduced the determination of it to a problem of pure mathematics, worked out the solution for a great variety of important and curious cases, compared the results with observation in a manner satisfactory and interesting to the naturalist, and gave conclusions of great value to the practical engineer.

674. We take advantage of the identity of mathematical conditions in St. Venant's torsion problem, and a hydrokinetic problem first solved a few years earlier by Stokes 1, to give the following statement, which will be found very useful in estimating deficiencies in torsional rigidity below the amount calculated from the fallacious extension of Coulomb's law:

675. Conceive a liquid of density n completely filling a closed infinitely light prismatic box of the same shape within as the given elastic prism and of length unity, and let a couple be applied to the box in a plane perpendicular to its length. The effective moment of inertia of the liquid 2 will be equal to the correction by which the torsional rigidity of the elastic prism calculated by the false extension of Coulomb's law, must be diminished to give the true torsional rigidity.

Further, the actual shear of the solid, in any infinitely thin plate of it between two normal sections, will at each point be, when reckoned as a differential sliding (§ 151) parallel to their planes, equal to and in the same direction as the velocity of the liquid relatively to the containing box.

1. On some cases of Fluid Motion,' Cambridge Philosophical Transactions, 1843.

2 That is the moment of inertia of a rigid solid which, as will be proved in Vol. 11., may be fixed within the box, if the liquid be removed, to make its motions the same as • they are with the liquid in it.

676. St. Venant's treatise abounds in beautiful and instructive graphical illustrations of his results, from which we select the following :

(1) Elliptic cylinder. The plain and dotted curvilineal arcs are 'contour lines' (coupes topographiques) of the section as warped by

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torsion; that is to say, lines in which it is cut by a series of parallel planes, each perpendicular to the axis. These lines are equilateral hyperbolas in this case. The arrows indicate the direction of rotation in the part of the prism above the plane of the diagram.

(2) Equilateral triangular prism. The contour lines are shown

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as in case (1); the dotted curves being those where the warped section falls below the plane of the diagram, the direction of rotation

of the part of the prism above the plane being indicated by the bent arrow.

(3) This diagram shows a series of lines given by St. Venant, and more or less resembling squares. Their common equation containing only one constant a. It is remarkable that the values

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a=0.5 and a=-{(V2-1) give similar but not equal curvilineal squares (hollow_sides and acute angles), one of them turned through half a right angle relatively to the other. Everything in the diagram outside the larger of these squares is to be cut away as irrelevant to the physical problem; the series of closed curves remaining exhibits figures of prisms, for any one of which the torsion problem is solved algebraically. These figures vary continuously from a circle, inwards to one of the acute-angled squares, and outwards to the other : each, except these extremes, being a continuous closed curve with no angles. The curves for a=0:4 and a=-0.2 approach remarkably near to the rectilineal squares, partially indicated in the diagram by dotted lines.

(4) This diagram shows the contour lines, in all respects as in the cases (1) and (2) for the case of a prism having for section the figure indicated. The portions of curve outside the continuous closed curve are merely indications of mathematical extensions irrelevant to the physical problem.

(5) This shows, as in the other cases, the contour lines for the warped section of a square prism under torsion.

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(6), (7), (8). These are shaded drawings, showing the appearances presented by elliptic, square, and flat rectangular bars under

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exaggerated torsion, as may be realized with such a substance as India rubber.

677. Inasmuch as the moment of inertia of a plane area about an axis through its centre of inertia perpendicular to its plane is obviously equal to the sum of its moments of inertia round any two axes through the same point, at right angles to one another in its plane, the fallacious extension of Coulomb's law, referred to in § 673, would make the torsional rigidity of a bar of any section equal to í (§ 665) multiplied into the sum of its flexural rigidities (see below, $ 679) in any two planes at right angles to one another through its length. The true theory, as we have seen (§ 675), always gives a torsional rigidity less than this. How great the deficiency may be expected to be in cases in which the figure of the section presents projecting angles, or considerable prominences (which may be imagined from the hydrokinetic analogy we have given in § 675), has been pointed out by M. de St. Venant, with the important practical application, that strengthening ribs, or projections (see, for instance, the fourth annexed diagram), such as are introduced in engineering to give stiffness to beams, have the reverse of a good effect when torsional rigidity or strength is an object, although they are truly of great value in increasing the flexural rigidity, and giving

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