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it between two normal sections, will at each point be, when reckoned as a differential sliding (S 151) parallel to their planes, equal to and in the same direction as the velocity of the liquid relatively to the containing box.

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 prisin 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=-1(12-1) give similar but not equal curvi. lineal squarės (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=04 and a =-02 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.

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(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 Alat rectangular bars iinder

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


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

M ($ 665) multiplied into the sum of its flexural rigidities (see below, $ 679) in any two planes at right angles to one another through

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

strength to bear ordinary strains, which are always more or less flexural. With remarkable ingenuity and mathematical skill he has drawn beautiful illustrations of this important practical principle from his algebraic and transcendental solutions. Thus for an equilateral (2)


(4) Square with curved Square with acute Star with four (5) Rectilineal corners and hollow angles and hollow

rounded points,

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triangle, and for the rectilineal and three curvilineal squares shown in the annexed diagram, he finds for the torsional rigidities the values stated. The number immediately below the diagram indicates in each case the fraction which the true torsional rigidity is of the old fallacious estimate (8 673); the latter being the product of the rigidity of the substance into the moment of inertia of the cross section round an axis perpendicular to its plane through its centre of inertia. The second number indicates in each case the fraction which the torsional rigidity is of that of a solid circular cylinder of the same sectional area,

678. M. de St. Venant also calls attention to a conclusion from his solutions which to many may be startling, that in his simpler cases the places of greatest distortion are those points of the boundary which are nearest to the axis of the twisted prism in each case, and the places of least distortion those farthest from it. Thus in the elliptic cylinder the substance is most strained at the ends of the smaller principal diameter, and least at the ends of the greater. In the equilateral triangular and square prisms there are longitudinal lines of maximum strain through the middles of the sides. In the oblong rectangular prism there are two lines of greater maximum strain through the middles of the broader pair of sides, and two lines of less maximum strain through the middles of the narrow sides. The strain is, as we may judge from ($ 675) the hydrokinetic analogy, excessively small

, but not evanescent, in the projecting ribs of a prism of the figure shown in (4) $677. It is quite evanescent infinitely near the angle, in the triangular and rectangular prisms, and in each other case as (3) of 8 677, in which there is a finite angle, whether acute or obtuse, projecting outwards. This reminds us. of a general remark we have to make, although consideration of spacę may

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