CONJUGATE DIAMETERS.

181

since any such point might be taken for the point p. Hence if we take any two conjugate diameters oq, or of the section oqr, the three lines op, oq, or are such that the tangent plane at the extremity of each is parallel to the other two. These lines are called a set of conjugate diameters of either of the two surfaces; one of them always meets the hyperboloid of two sheets, and the other two meet the hyperboloid of one sheet,

Now in the surface of revolution, any section through oa being a hyperbola whose semi-axes are equal to oa and

"om2 pm2

ob2

ob, we have

is unaltered by a homogeneous strain, the equation is equally true for any hyperboloid of two sheets, if now oa, ob, oc form a set of conjugate semi-diameters in the sense just explained. It may be shewn in the same way that in a hyperboloid of one sheet we should find

x, y, z being written for om, mn, np, as before.

It follows immediately that any plane parallel to aob cuts the surfaces in two hyperbole whose common centre is on oc, the asymptotes of all being parallel.

DISPLACEMENT-QUADRIC.

It shall now be proved that any homogeneous strain of a solid may be represented by means of a central quadric surface, namely, either an ellipsoid or a pair of conjugate hyperboloids, in the following manner. The displace

ment of any point p of the surface, relative to its centre o, will be at right angles to the diametral section conjugate to op, and will contain as many centimeters of length as that section contains square centimeters of area. For this purpose it is necessary to shew that the hyperboloids have the same property which we proved true for the ellipsoid; namely that if a section A and its conjugate diameter a be respectively projected upon a section B and its conjugate diameter B, by lines parallel to B and B respectively, the ratio of the projection of A to B is equal to the ratio of the projection of a to B. We shall prove this first for surfaces of revolution, and then extend it to the other surfaces by a homogeneous strain.

When the central section is a hyperbola, we cannot properly speak of its area at all. In this case we shall suppose it to be replaced by an ellipse having the same axes; so that in general, if the semi-axes of an ellipse or hyperbola are a, b, the area is always to be reckoned as Tab.

Let the figure represent two conjugate hyperbolæ, which, by revolving about the axis aa, are to generate a pair of conjugate hyperboloids of revolution. The diametral section conjugate to op is made by a plane through qq perpendicular to the paper. The semi-axes of this section are oq and a line oc perpendicular to the paper equal in length to ob. The projection of the section on ob is therefore π. nq. oc, and its ratio to the area of

a in

the section obc is π. nq.oc: π.ob.oc = nq: ob. But ng: obom: oa; thus the projection of section oqc on obc bears the same ratio to section obc that projection of op on oa bears to oa.

In the same way, the section conjugate to oq is made by a plane through op perpendicular to the paper, and its "area" is to be reckoned as T. op.oc. It follows at once

DISPLACEMENT-QUADRIC.

183

that its projection on the plane oac, namely T. om. oc, bears the same ratio to section oac that ng bears to ob.

Passing now to the case of hyperboloids not of revolution, we have proved that any pair of conjugate surfaces may be altered by homogeneous strain into surfaces of revolution, so that any given diameter aa' of the hyperboloid of two sheets becomes the axis of revolution. And since the ratios of parallel lengths and of parallel areas are unaltered by the strain, it follows that the property just proved for surfaces of revolution is true for all hyperboloids.

This being so, let e, f, g be principal elongations of a homogeneous strain, and let a, b, c be three lengths such that a√e=b√f=cNg=√(efg). If any of the quantities e, f, g be negative, we must in this formula consider it replaced by its absolute value. If e, f, g are all of the same sign, construct an ellipsoid with semi-axes a, b, c; but if one is of a sign different from that of the other two, construct a pair of conjugate hyperboloids with the same semi-axes, so that the axes whose elongations are of the same sign shall meet the one-sheeted surface, and the remaining axis the two-sheeted surface. The relation between x, y, z in the quadric surface or surfaces thus constructed, whether ellipsoid or hyperboloids, is

π (ex2 +ƒy2+gz3) = ± efg,

as may be seen by comparing the values just given for x2, y3 z2 ±

a, b, c with the equation ±

a2 = b2 = c2

=

1. The surface

for which π (ex2+fy2+gz2) = efg is called the displacementquadric. If e, f, g are all positive or all negative, the displacement-quadric is an ellipse; if two of them are positive and one negative, or if one is positive and two negative, it is a hyperboloid of two sheets; but in the latter case we must call in the assistance of the conjugate surface in order to represent the strain.

If then oa, ob, oc be semi-axes of the displacementquadric, the displacement of the point a is ea which is Tbc, the area of the conjugate section; and this displace

ment is along oa, and therefore normal to the area of that section. When the displacement-quadric is a hyperboloid, elliptic and hyperbolic areas must be regarded as having different signs; but which sign is to be attributed to each depends on the signs of e, f, g, and it will be found in fact that the elliptic area is always of the same sign as the product efg.

Since the displacements of a, b, c are vectors representing the conjugate areas, it follows that the displacement of any point p on the displacement-quadric or its conjugate surface is a vector representing the area of section conjugate to op. For we have shewn that the components of that area, namely its projections on the principal planes, bear the same ratio to the principal areas Tbc, Tсa, Tab, that the components of op, namely its projections on the axes, bear to those axes. Now if om be the projection of op on oa, the displacement of m is

om

om

να

x the displacement of a, that is, it is Xπbс. Con

оа

sequently the displacement of m is a vector representing the projection on obc of the area conjugate to op. Now the displacement of p is the resultant of the displacements of its projections on the axes; and therefore it represents the area which is the resultant of the three projections here considered, namely, the area of section conjugate to op.

The case of a point k lying on the asymptotic cone of the displacement-quadric requires some explanation. In that case the length of the line drawn in the direction ok to meet the surface is in

finite, and the displacement of its end is infinite also. The conjugate section is made by a plane through ok touching the asymptotic cone, which cuts the conjugate surface in two parallel straight lines. In the case of sur

α

LINEAR FUNCTION.

185

faces of revolution it is clear that the distance between these lines is bb'; they lie on either side of ok in a plane through it perpendicular to the paper. Thus the displacement of p (the infinitely distant point on ok) is .ob.op, perpendicular to ok in the plane of the paper. Hence the displacement of k is .ob.ok in the same direction. And generally the displacement is π. ok multiplied by half the breadth of the conjugate section.

π

In any other case if ot be the perpendicular on the tangent plane at p, the displacement of p is parallel to ot and equal to Tabc ot. For the perpendicular on a tangent plane, multiplied by the area of the parallel diametral section, is constant, and therefore equal to Tabc. This follows at once for surfaces of revolution from the corresponding property of the hyperbola; and it is extended to any hyperboloids by the consideration that all volumes are altered in the same ratio by a homogeneous strain. We shall write H for abc or efg: 72, so that displacement of p = H: ot.

LINEAR FUNCTION OF A VECTOR.

Just as in the case of a plane strain, the strained position of a vector or the displacement of its end is said. to be a linear function of the original vector when the strain is homogeneous. If the displacement of the end of p be denoted by $(p), the strained position of it is p+ $(p) =(1+) p. When the strain is pure, & is said to be a pure function.

Let i, j, k be three unit-vectors at right angles to one another, and let

pi = ai + hj + g'k,

øj = h'i + bj +fk,

pk=gi+fj+ck.