normal sec planes tion by elastic flexure, in torsion and finitesimal. correspond flexure and 593. Considering now a wire of uniform constitution and Warping of figure throughout, and naturally straight; let any two of reference perpendicular to one another through its central line when straight, cut the normal section through P in the lines PK and PL. These two lines (supposed to belong to the substance, and move with it) will remain infinitely nearly at right angles to one another, and to the tangent, PT, to the central line, however the wire may be bent Rotations or twisted within the conditional limits. Let and λ be the ing to component curvatures (§ 590) in the two planes perpendicular torsion. to PK and PL through PT, and let T be the twist (§ 120) of the wire at P. We have just seen (§ 590) that if P be moved. at a unit rate along the curve, a rigid body with three rectangular axes of reference OK, OL, OT kept always parallel to PK, PL, PT, will have angular velocities K, X, 7 round those axes respectively. Hence if the point P and the lines PT, PK, PL be at rest while the wire is bent and twisted from its unstrained to its actual condition, the lines of reference P'K', P'L, P'T' through any point P'infinitely near P, will experience a rotation compounded of κ. PP' round P′K', λ. PP' round P'L', and T. PP' round P'T". energy of in bent and un- twisted wire. 594. Considering now the elastic forces called into action, Potential we see that if these constitute a conservative system, the work elastic force required to bend and twist any part of the wire from its strained to its actual condition, depends solely on its figure in these two conditions. Hence if w. PP' denote the amount of this work, for the infinitely small length PP' of the rod, w must be a function of x, λ, T; and therefore if K, L, T denote the components of the couple-resultant of all the forces which must act on the section through P' to hold the part PP' in its strained state, it follows, from §§ 240, 272, 274, that where Sw, Sw, 8,w denote the augmentations of w due respectively to infinitely small augmentations Ɛ, dλ, ST, of K, λ, T. 595. Now however much the shape of any finite length of the wire may be changed, the condition of § 588 requires energy of in bent and twisted wire. Potential clearly that the changes of shape in each infinitely small part, elastic force that is to say, the strain (§ 154) of the substance, shall be everywhere very small (infinitely small in order that the theory may be rigorously applicable). Hence the principle of superposition [§ 591, II.] shows that if x, λ, T be each increased or diminished in one ratio, K, L, T will be each increased or diminished in the same ratio: and consequently w in the duplicate ratio, since the angle through which each couple acts is altered in the same ratio as the amount of the couple; or, in algebraic language, w is a homogeneous quadratic function of κ, λ, τ. Components of restituent couple. Thus if A, B, C, a, b, c denote six constants, we have w = }(Ak2 + Bλ2 + Cr2 + 2adr + 2btk + 2ckλ) .......................... (2). Hence, by § 594 (1), Three principal or normal axes of torsion By the known reduction of the homogeneous quadratic function, these expressions may of course be reduced to the following simple forms: where, J., 9, are linear functions of к, λ, T. And if these Hence 596. There are in general three determinate rectangular directions, PQ,, PQ,, PQ, through any point P of the middle and flexure. line of a wire, such that if opposite couples be applied to any two parts of the wire in planes perpendicular to any one of them, every intermediate part will experience rotation in a plane parallel to those of the balanced couples. The moments Three principal flexure of the couples required to produce unit rate of rotation round torsionthese three axes are called the principal torsion-flexure rigidities rigidities. of the wire. They are the elements denoted by A,, A„, A ̧ in the preceding analysis. 597. If the rigid body imagined in § 593 have moments of inertia equal to A,, A,, A, round three principal axes through kept always parallel to the principal torsion-flexure axes through P, while P moves at unit rate along the wire, its moment of momentum round any axis (§§ 281, 236) will be equal to the moment of the component torsion-flexure couple round the parallel axis through P. 598. The form assumed by the wire when balanced under Three principal or northe influence of couples round one of the three principal axes mal spirals. is of course a uniform helix having a line parallel to it for axis, and lying on a cylinder whose radius is determined by the condition that the whole rotation of one end of the wire from its unstrained position, the other end being held fixed, is equal to the amount due to the couple applied. Let be the length of the wire from one end, E, held fixed, to the other end, E', where a couple, L, is applied in a plane perpendicular to the principal axis PQ, through any point of the wire. The rotation being [§ 595 (4)] at the rate of length, amounts on the whole to L ', per unit angular space occupied by the helix on the cylinder on which it Case in which tral line is a normal axis of torsion. 599. In the most important practical cases, as we shall elastic cen- see later, those namely in which the substance is either "isotropic," as is the case sensibly with common metallic wires, or, as in rods or beams of fibrous or crystalline structure, with an axis of elastic symmetry along the length of the piece, one of the three normal axes of torsion and flexure coincides with the length of the wire, and the two others are perpendicular to it; the first being an axis of pure torsion, and the two others axes of pure flexure. Thus opposing couples round the axis of the wire twist it simply without bending it; and opposing couples in either of the two principal planes of flexure, bend it into a circle. The unbent straight line of the wire, and the circular arcs into which it is bent by couples in the two principal planes of flexure, are what the three principal spirals of the general problem become in this case. Case of equal flexi directions. A simple proof that the twist must be uniform (§ 123) is found by supposing the whole wire to turn round its curved axis; and remarking that the work done by a couple at one end must be equal to that undone at the other. 600. In the more particular case in which two principal bility in all rigidities against flexure are equal, every plane through the length of the wire is a principal plane of flexure, and the rigidity against flexure is equal in all. This is clearly the case with a common round wire, or rod: or with one of square section. It will be shown later to be the case for a rod of isotropic material and of any form of normal section which is "kinetically symmetrical," § 285, round all axes in its plane through its centre of inertia. 601. In this case, if one end of the rod or wire be held fixed, and a couple be applied in any plane to the other end, a uniform spiral (or helical) form will be produced round an axis perpendicular to the plane of the couple. The lines of the substance parallel to the axis of the spiral are not, however, parallel to their original positions, as (§ 598) in each of the three principal spirals of the general problem: and lines traced along the surface of the wire parallel to its length when straight, become as it were secondary spirals, circling equal flexi directions. round the main spiral formed by the central line of the Case of deformed wire; instead of being all spirals of equal step, as in bility in all each one of the principal spirals of the general problem. Lastly, in the present case, if we suppose the normal section of the wire to be circular, and trace uniform spirals along its surface when deformed in the manner supposed (two of which, for instance, are the lines along which it is touched by the inscribed and the circumscribed cylinder), these lines do not become straight, but become spirals laid on as it were round the wire, when it is allowed to take its natural straight and untwisted condition. Let, in § 595, PQ, coincide with the central line of the wire, and let A1 = 4, and 4 ̧= à ̧ = B; so that A measures the rigidity of torsion and B that of flexure. One end of the wire being held fixed, let a couple G be applied to the other end, round an axis inclined at an angle to the length. The rates of twist and of flexure each per unit of length, according to (4) of § 595, will be respectively. The latter being (§ 9) the same thing as the follows (§ 126, or § 590, footnote) that B sin 0 is the radius of strained to spiral and curvature of its projection on a plane perpendicular to this line, that is to say, the radius of the cylinder on which the spiral lies. 602. A wire of equal flexibility in all directions may clearly Wire be held in any specified spiral form, and twisted to any stated any given degree, by a determinate force and couple applied at one end, twist. the other end being held fixed. The direction of the force must be parallel to the axis of the spiral, and, with the couple, must constitute a system of which this line is (§ 559) the central axis: since otherwise there could not be the same system of balancing forces in every normal section of the spiral. All this may be seen clearly by supposing the wire to be first brought by any means to the specified condition of strain; then to have rigid planes rigidly attached to its two ends perpendicular to its axis, and these planes to be rigidly |