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

be 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, sc 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.

It has

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. 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

the ellipse, so that it can never reach p, it describes the inverse of an ellipse. Intermediate between these is the case in which q is on the ellipse, when the curve which it describes has a cusp and is the inverse of a parabola, which only goes to infinity in one direction.

We have here considered the relative motion of the two short sides of a crossed rhomboid. That of the two long sides is equivalent to the rolling of two equal and similar hyperbolas. For in this case we have

so that the locus of o is a hyperbola having a, d for foci and ab for transverse axis. Remarks may be made about the path of a point in the moving plane entirely similar to those made on the other case.

a

In the general case of three-bar motion, the lengths of the three bars being arbitrary, an important theorem has been obtained by Mr S. Roberts. Any path described by a point in a plane moving with three-bar motion may also be described in two other ways by three-bar motion. Suppose (second figure) that ah, hk, kb are the three bars, o the moving point which is rigidly connected with hk by the triangle ohk. Then the theorem is that the path of o may

TRIPLE GENERATION.

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also be described by means of the bars ag, gf, fc, or the bars bd, de, ec. The triangles hko, gof, ode are similar to

one another, and the figures ahog, bdok, cfoe are parallelograms.

The theorem has been put by Prof. Cayley into the following elegant form. Take any triangle abc (first figure) and through any point o within it draw lines kf, eh, gd parallel to the sides. Let the triangles hko, gof, ode be

supposed rigid and jointed together at o, and let the other lines in the figure represent bars forming three jointed parallelograms. Then however the system is moved about in its plane (e.g. into the configuration of the second figure) the triangle abc will be always of the same shape. Now

the system is one which in shape (independently of its position) has two degrees of freedom; for if we fix one of the three triangles, the other two may be turned round independently. If therefore we impose a single condition, that the area abc shall be constant, the system will still have one degree of freedom. But this is equivalent to fixing the size of abc as well as its shape, so that we may fix the points a, b, c; and still o will be able to move. In so moving it will describe a path which is due at the same time to three different three-bar motions.

All that remains to be proved, therefore, is that the shape of abc is invariable. This can be made clear by very simple considerations. Let q be the operation (complex number) which converts hk into ho, so that ho=q.hk. Then the same operation will convert go into gf and od into oe, since the three triangles are similar. Consequently

ac = ag+gf+fc=ho+ gf + oe = q. hk + q . go
+q.od = q(hk+ ah + kb) = q . ab,

that is, ac is got from ab by the same operation which converts hk into ho; therefore the triangle abc is similar to hko. Or in words, the components ag, gf, fc of ac are got from the components hk, ah, kb of ab by altering all their lengths in the same constant ratio and turning them all through the same constant angle. Therefore the whole step ac is got from ab by altering its length in a constant ratio and turning it through a constant angle.

It is to be observed that the configuration in the first figure forms an apparent exception to the theorem. The area abc is then a maximum, and the path of o has shrunk up into a point, so that it is really not able to move.

We may use Mr Roberts' theorem to transform motion due to the crossed rhomboid into that due to a figure called a kite by Prof. Sylvester. It also is a quadrilateral having its sides equal two and two, but the equal sides are adjacent.