TO FIND INSTANTANEOUS CENTRE.

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If we interchange the fixed and moving axodes, keeping the relative motion the same, we alter the signs of ä and w; therefore is unaltered, or the velocity of c is the same on either supposition. Hence it follows, as in the case of the plane, that the moving axode rolls in contact with the fixed one.

In the general motion of a rigid body, combine with its velocity-system a translation equal and opposite to the velocity of any point a. Then the new velocity-system is a spin round some axis through a. Hence the actual motion is the resultant of a spin and a translation, that is to say, a twist.

DEGREES OF FREEDOM.

The special problems presented by the motion of a plane on a plane are of two kinds. In the first kind, the motion being determined in any way, it is required to find the centrodes. In the second kind, the centrodes being given, it is required to find the path of any point or the envelop of any line in the moving body.

a

The motion of a plane is determined when each of two curves in the moving plane is made always to touch one of two curves in the fixed plane. Thus the figure bounded by the two curves a, b can be made to move about so that a shall always touch the curve A, and b shall always touch the curve B; and it is clear that its motion is then determined, except as to the time in

B

which it is performed. In particular cases one of the curves A, a may shrink into a point; the condition of tangency then resolves itself into the condition that a point in the moving plane shall lie on a fixed curve, or a curve in the moving plane shall pass through a fixed

Since it requires three conditions to fix a plane figure in its plane, it is said to have three degrees of freedom. If it is subjected to one condition, e. g. that a certain curve must always touch a fixed curve, it has two degrees of freedom left. When it is subject to two conditions, it has one degree of freedom left, and can only move in a certain definite manner.

When one curve has to touch another, the instantaneous centre is situated on the common normal, since the point of contact can only move along the tangent. And as a particular case, when a point has to lie on a given curve, or a curve has to pass through a given point, the instantaneous centre lies on the normal to the curve at that point. In general, if we know the direction of motion of any point, the instantaneous centre is in the line through the point perpendicular to that direction.

INVOLUTE AND EVOLUTE.

For example, if two lines at right angles pt, pn are made to move as a rigid body, so that pt is tangent and pn normal to a given curve, the motion of p will always be in the direction pt, and therefore the instantaneous centre will always be in pn. Hence pn is the moving centrode; and the fixed centrode, which pn rolls upon, is called the evolute of the given curve. If a is a point where the evolute meets the curve, pn = arc an in length. The curvature at p is 1: : pn, by the formula already obtained; thus n is the centre of curvature, and the evolute may be described as the locus of

=

centres of curvature of the given curve. Moreover, since pn arcan, the curve ap may be described by unwinding a string from the curve an. On this account ap is called an involute of the curve an. It is clear that every other

ENVELOPS OF LINE OF CENTRES.

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centre remains in the line ab. It now has only one degree of freedom, and the line ab is the fixed centrode. The rolling centrode is a curve in the moving plane which shall be called the curve A. This curve is clearly the envelop of the line of centres in the moving plane.

Let us now fix the moving plane, and move the fixed plane, subject to the same condition of relative motion. Then as before, for each position there will be a line of centres, and by restricting the instantaneous centre to this, we shall make the motion such that a curve B in the plane formerly fixed will roll upon the line of centres. This curve is the envelop of the line of centres in the fixed plane.

Hence the relative motion of the two planes is such that the curves A and B roll on the same straight line. Or when a plane slides on a fixed plane, having two degrees of freedom, its motion is such that a curve A in the moving plane rolls on a straight line which rolls on a curve B in the fixed plane.

Let x be any point on the line of centres, and draw the involutes of A and B which pass through x. Then they

will cut the line at right angles and therefore touch one another at x. But if we make A roll on the line, having its involute fixed to it, this involute will always pass through x at right angles to the line; and similarly for B. Hence the relative motion of the two planes is such that these two involutes always touch. Thus the motion is such that a curve in the moving plane always touches a curve in the fixed plane; but we may substitute for these two curves any two curves parallel to them at equal distances on the same side.

CHAPTER III. SPECIAL PROBLEMS.

THREE-BAR MOTION.

IF three bars, ab, bc, cd are jointed together at b, c, while the remaining ends are fixed at points a, d about which the bars are free to turn, a plane rigidly attached to

bc is said to have three-bar motion. Properly speaking, we ought to consider the jointed quadrilateral abcd, and study the relative motion of two of its opposite sides.

We may also specify the motion by saying that the points b, c in the moving plane have to lie respectively on two circles in the fixed plane, viz. the circles whose centres are a, d, and radii ab, dc. The instantaneous centre o is at the intersection of ab and dc, since the motions of b and c are respectively perpendicular to those lines.

The centrodes of the three-bar motion have only been determined in particular cases. The most important of these is that of the crossed rhomboid, so called because its

CROSSED RHOMBOID.

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opposite sides are equal. The figure is symmetrical; and if the intersection of ab, cd is at o, we have

thus the point o describes relatively to ad an ellipse of which a, d are foci, and ab the major axis. Similarly we have bo+coba, or the locus of o in the moving plane is an equal and similar ellipse. These, therefore, are the centrodes. The relative motion is most clearly understood by supposing both ellipses to roll on the common tangent ot, so as to preserve the symmetrical aspect.

In this way we may see that the path of any point in the moving plane is similar to a pedal of the fixed ellipse. For let p, q be corresponding points in the two ellipses, then the line pq is always bisected at right angles by the tangent ot, and therefore the locus of q, when p is fixed, is similar to the locus of t, but of double the size. It has been proved that the reciprocal of a conic section is always a conic section; from which it follows that the pedal of a conic is also the inverse of a conic (generally a different one; but the same in the case of an equilateral hyperbola in regard to its centre). Hence we see that every point in the moving plane describes the inverse of a conic. The inverse of a hyperbola passes twice through the centre of inversion, since the hyperbola goes away to infinity in two directions; but the inverse of an ellipse does not. Hence if q is outside the ellipse, so that it can coincide with p in some position of the two curves, it describes the inverse of a hyperbola; but if q is inside