called principal axes of the strain, and the elongations along them are called principal elongations.

If the axes remain parallel to their original directions, the strain is called pure; if they are turned round, it is accompanied by rotation.

REPRESENTATION OF PURE STRAIN BY ELLIPSOID.

We e may now represent (in the case of a pure strain) the strained position of any vector by means of an ellipsoid, in a way entirely analogous to our previous representation of a plane strain by means of an ellipse. Let the principal elongations be e, f, g, and let p=1+e, q=1+f, r=1+g, so that the principal axes of the strain are multiplied by p, q, r respectively. Now construct an ellipsoid with semiaxes a, b, c such that the strained length of a shall represent the area of the section by the plane of b, c, and so for the others; that is, so that ра = πbс, ql=πсα, гс= πаb. This will be effected if we make a√p=b√ q = c√√r = √(pqr): π. Thus the axes of the ellipsoid must be taken inversely proportional to the square roots of p, q, r, which agrees with the rule for the ellipse.

This being so, it follows that the strained position of any vector op represents the area of the section by the conjugate diametral plane; that is to say, it is at right angles to this area, and contains as many linear centimeters as the area contains square ones. For since the projection of that area on the plane obc is to bc as om to oa, it follows that the strained position of om represents that projection; and similarly the strained positions of mn and np represent the projections on coa, aob. The strained position of op is the vector-sum of these three lines, and therefore represents the area of which they represent the projections.

Thus the strained position of any radius of this ellipsoid is a vector representing the area of the conjugate section.

We may easily see that the volumes of all portions of the solid are altered in the same ratio by the strain. For we may suppose these volumes cut up into small cubes by systems of planes at right angles, so as to leave pieces over at the boundaries. These cubes will be changed into equal and similar parallelepipeds, and therefore a volume made up of any number of the cubes will be altered in the same ratio as any one cube. Now any volume may be made up of cubes with an approximation which can be made as close as we like by taking the cubes small enough. Hence the proposition follows.

Now the cylinder standing on any diametral section of a sphere, and bounded by the tangent planes parallel to that section, is evidently of constant volume, whatever diametral plane be taken. Hence, in the ellipsoid' also, if we draw through every point of a diametral section a line parallel to the conjugate diameter, these lines will constitute a cylinder such that the volume of it enclosed by the two tangent planes parallel to the diametral section is constant, and therefore equal to 2πabc, its value when the section is one of the principal planes. The volume of a cylinder being the product of its base and height, and the height of this one being the perpendicular distance between the parallel tangent planes, that is, twice the perpendicular on either from the centre; it follows that the perpendicular on a tangent plane, multiplied by the area of the parallel diametral section, is equal to a constant, h. Hence if ot be the perpendicular on the tangent plane at p, the strained position of op is along ot and equal in length to h: ot,

PROPERTIES OF HYPERBOLOID.

We have hitherto supposed p, q, r to be of the same sign, which, for reasons already mentioned, is the case in all actual strains. If, however, we wish to represent in this way, not the strained position of op, but the displacement of p, we must make the squared axes of our surface inversely proportional to e, f, g, the principal elongations.

So long as these are of the same sign, the displacement may be represented in the way just described; namely, we can construct an ellipsoid so that the displacement of any point p on it shall be a vector representing the area of section conjugate to op. But when one is of a sign different from the other two, we require other surfaces, which shall be now described.

If we make a hyperbola rotate about its transverse axis aa', we obtain a surface of two sheets, each sheet being generated by a branch of the hyperbola. This surface is called a hyperboloid of revolution of two sheets.

a

By the same revolution the conjugate hyperbola generates a surface of one sheet, the two branches changing places after a rotation through two right angles. This surface is called a hyperboloid of revolution of one sheet.

Now let the whole figure be subjected to a uniform strain of any kind; then the surfaces will no longer be surfaces of revolution. They are then called hyperboloids of one and two sheets respectively; and in this particular relation are called conjugate. To every hyperboloid of one sheet there is a conjugate hyperboloid of two sheets, and vice versa. The properties of these more general hyperboloids may be derived from the particular case of the surfaces of revolution, just as those of the ellipsoid are derived from the sphere.

Then, the asymptotes of the revolving hyperbola generate a right cone, called the asymptotic cone, towards which the surface approaches indefinitely as it gets fur

CONJUGATE HYPERBOLOIDS.

179

ther away from the centre. The strain will convert this cone into an oblique cone (a cone standing on a circle with the vertex not directly over the centre of the circle) which will still be asymptotic. The shape of this cone determines the shape of the two conjugate surfaces.

Every central section of a hyperboloid of revolution is a conic; an ellipse when only the one-sheeted surface is cut, a hyperbola when both of the conjugate surfaces are cut. Let the section be made by a plane through ok perpendicular to the plane of the paper. When the point q is brought by the rotation to the position p vertically above m, np2 = nm2 + mp3, or mp2 = nq2 - nm3. Now

mp2, om2 Therefore + ob2 ok2

=

= 1, or the point p lies on an ellipse having ok for its semi-major axis, and a line oc perpendicular to the plane, of length equal to ob, for its semiminor axis. In a precisely similar way it may be shewn that a central section of the two-sheeted surface is a hyperbola, whose conjugate hyperbola is the section of the conjugate surface by the same plane.

Since after the strain an ellipse remains an ellipse and a hyperbola a hyperbola, it follows that a central section of any hyperboloid is a conic, is an ellipse when only the one-sheeted surface is cut, and a hyperbola when both the conjugate surfaces are cut.

Now take any point p on a hyperboloid of two sheets, and draw through the centre a plane oqc parallel to the tangent plane at p; this will cut the conjugate hyperboloid in an ellipse. Let the whole figure receive a shear, by sliding over one another the planes parallel to the tangent plane at p, until op becomes perpendicular to them; then elongate all lines parallel to the shorter axis oc of the ellipse until the ellipse becomes a circle. Then every section by a plane through op will be a hyperbola of which op is the transverse semi-axis, because it is perpendicular to the tangent at p. Consequently the other axis is in the plane of the circle and equal to its diameter; that is to say, all these hyperbolas have the same axes, and are therefore equal and similar. Hence the conjugate surfaces have been converted by this strain into surfaces of revolution.

In this state of the figure it is clear (1) that the tangent planes at points on the section by obc are parallel to op; (2) that all sections parallel to obc are circles having their centres on op. Hence in general if p be any point on a hyperboloid of two sheets, and oqc a diametral plane parallel to the tangent plane at p, the tangent planes at all points of the section of the conjugate surface by oqc are parallel to op, and all sections parallel to oqr are similar and similarly situated ellipses having their centres on op. If we draw through o a plane opr parallel to the

tangent plane at any point q of the section oqr, this will cut the hyperboloid of two sheets in a hyperbola, the tangent plane at every point of which will be parallel to oq,