roulette is a curve which may be described by rolling C on A. Suppose B and C to roll simultaneously on A, so as always to have the same point of contact; then the motion of C relative to B is that which describes the roulette qp. Now cp is perpendicular to the tangent both of this roulette and of that which p describes by the rolling of C on A. Hence the two roulettes always touch one another, as was to be proved. Observe that the point p is not necessarily on the curve C.

Returning to the case of the circles, we observe that the extremities q, r of the moving diameter describe similar and equal cycloidal curves, such that a cusp of one and a vertex of the other are on the same diameter of the fixed circle. Hence if a straight line of constant length move with its ends on two such cycloidal curves, starting from a position in which one end is at a cusp and the other at a corresponding vertex, it will envelop a cycloidal

curve.

The following are cases of this theorem:

1. The chord of a cardioid through the cusp is of constant length. (A point is a special case of a cycloidal curve.)

2. A line of constant length with its ends moving in two fixed lines at right angles envelops a four-cusped hypocycloid.

3. The portion of the tangent to a three-cusped hypocycloid intercepted by the curve is of constant length.

=

The curvature of cycloidal curves may be calculated by means of the general theorem already given for the curvature of roulettes, or directly as follows. Let o be the centre of the fixed circle, take ce: dc do co, draw a circle through e with centre o, and a circle on ce as diameter. Produce pc to meet this in q. If this circle roll on the circle through e, so that q is brought to h, we shall have eq=eh, and since eq: pdec: cd = oe: oc, pd is equal to the corresponding arc of the circle kc. Hence the two small circles may roll together on the two large ones, so that ce always passes through o, and pcq is a straight line. Then

EVOLUTE OF CYCLOID.

157

pq is normal to the path of p and tangent to that of q, or the latter path is the evolute of the former.

d

h

It follows that the length of the arc kq is equal to pq, or s = de cosy. It is clear that is in a fixed ratio to the angle which pq makes with the normal at k, and consequently sa cos mp, if a=de and m is this fixed ratio.

BOOK III. STRAINS.

CHAPTER I. STRAIN-STEPS.

STRAIN IN STRAIGHT LINE.

WE have hitherto studied the motion of rigid bodies, which do not change in size or shape. We have now to take account of those strains, or changes in size and shape, which we have hitherto neglected.

The simplest kind of strain is the change of length of an elastic string when it is stretched or allowed to contract. When every portion of the string has its length altered in the same ratio, the

strain is called uniform or homogeneous. Thus if apb is changed into a'p'b' by a uniform strain, ap: a'p' = ab a'b'. The ratio

ar

αν

b

a'p' ap, or the quantity by which the original length must be multiplied to get the new length, is called the ratio of the strain. The ratio of the change of length to the original length, or a'p' - ap: ap, is called the elongation; it is reckoned negative when the length is diminished. A negative elongation is also called a compression.

Let e be the elongation, s the unstretched length ap, ☛ the stretched length a'p', then σs=es, or σ =s (1+e). Thus 1e is the ratio of the strain.

In general, a solid body undergoes a strain of simple elongation e, when all lines parallel to a certain direction are altered in the same ratio 1 : 1+e, and no lines perpendicular to them are altered in magnitude or direction.

STRAIN OF PLANE FIGURE.

159

The strain is then entirely described if we describe the strain of one of the parallel lines.

HOMOGENEOUS STRAIN IN PLANE.

The kind of strain next in simplicity is that of a flat membrane or sheet. Suppose this to be in the shape of a square; we may give it a uniform elongation e parallel to one side, and then another uniform elongation ƒ parallel to the other side. It is now converted into a rectangle, whose sides are proportional to 1+e, 1+f. By each of these operations two equal and parallel lines, drawn on the membrane, will be left equal and parallel; though, if not parallel to a side of the square, they will be altered in direction.

We may prove, conversely, that every strain which leaves straight lines straight, and parallel lines parallel, is a strain of this kind combined with a change of position of the membrane in its plane. Such a strain is called uniform or homogeneous.

Since a parallelogram remains a parallelogram, equal parallel lines remain equal. Then it is easy to shew, by the method of equi-multiples, that the ratio of any two parallel lines is unal

tered by the strain. Next, if we draw a circle on the unstrained membrane, this circle will be altered by the strain into an ellipse. For in the unstrained figure

L

B

B'

A'M. MA: CA2=MP2 : CB2,

a

C

and since these ratios of parallel lines are unaltered, it follows that in the strained figure also

a'm.ma: camp3: cb3.

Hence the strained figure is an ellipse, whose conjugate diameters are the strained positions of perpendicular diameters of the circle.

It follows that there are two directions at right angles to one another, which remain perpendicular after the

strain; namely those which become the axes of the ellipse into which a circle is converted. If these lines remain parallel to their original directions, the strain is produced by two simple elongations along them respectively; in that case it is called a pure strain. If they are not parallel to their original directions, the strain is compounded of a pure strain and a rotation.

Two lines drawn anywhere in the strained membrane parallel to the axes of the ellipse into which a circle is converted, or in the unstrained membrane parallel to the unstrained position of those axes, are called principal axes of the strain. The elongations along them are called principal elongations; the ratios in which they are altered are called principal ratios,

REPRESENTATION OF PURE STRAIN BY ELLIPSE.

When the strain is pure, the new position of any step may be conveniently represented by means of a certain ellipse. Let the principal ratios be p, q, so that every line parallel to oX is altered in the ratio 1: p, and every line parallel to oY in the ratio 1 q. Take two lengths oa, ob, along oX, oY respectively, such that oa2: ob2=q: p, and let m be the positive geometric mean of p, q, so that m2=pq. Then we shall have, so far as length is concerned, p. oam. ob, and q. ob =m.oa. Hence, taking account of direction, oa becomes im. ob', and ob becomes im. oa, in consequence of the strain.

Now construct an ellipse having oa, ob for semi-axes; then if p be any point on it and gg' the diameter conjugate