which is half the moment of ab in regard to cd. Now this parallelepiped is made up of the tetrahedra

abcd, abdd', abcc', cdaa', cdbb',

of which the last four are equal, and each of them (being one-third of height x base) is one-sixth of the parallelepiped. It follows that abcd is one-third of it.

We learn then that a twist may be resolved in an infinite number of ways into two spins, but that the tetrahedron, whose opposite edges are their representative lines, is always of the same volume, namely, one-sixth of the squared magnitude of the twist multiplied by its pitch.

INSTANTANEOUS MOTION OF A RIGID BODY.

We shall prove presently that when a plane is in motion, sliding on another plane, the system of velocities at any instant is that of a spin about a certain point in the plane, called the instantaneous centre. As the motion goes on, the instantaneous centre in general changes continuously, describing a curve in the fixed plane and a curve in the moving plane. These curves are called centrodes (Kévтpov ódós, path of the centre), and the motion is such that the centrode in the moving plane (the moving centrode) rolls upon the centrode in the fixed plane (the fixed centrode). Thus every motion of a plane sliding on a plane may be produced by the rolling of one curve on another; the point of contact being the instantaneous

centre.

Similar theorems hold good when a body moves about a fixed point, or, which is the same thing, when a spherical surface slides upon an equal sphere. In this case the velocity-system at any instant is that of a spin about a certain line through the fixed point, called the instantaneous axis; or, in describing the sliding of a sphere, we may say that at any instant it is rotating about a point on the spherical surface, called the instantaneous centre. As the motion goes on, the instantaneous axis moves, always

CONDITIONS OF RIGIDITY.

137

passing through the fixed point, so as to describe a certain cone, called the fixed axode. At the same time it traces out in the moving body another cone, called the moving axode. We may describe the same thing in other words by saying that the instantaneous centre on the spherical surface describes a fixed and a moving centrode on the fixed and moving spheres respectively. The motion is such that the moving cone rolls upon the fixed cone, and therefore the moving spherical curve rolls upon the fixed

curve.

The most general motion of a rigid body is that of a twist about a certain screw, called the instantaneous screw. The axis of this screw, in moving about, generates two surfaces, one fixed in space, and one moving with the body. These surfaces are called axodes; being generated by the motion of a straight line, they belong to the class of ruled surfaces or scrolls. The motion is such that one axode rolls and slides on the other, the line of contact being the axis of the instantaneous screw.

a

B

Returning to the motion of a plane on a plane, we may approximately represent it during a certain interval by considering a series of successive positions at certain instants during the interval. We know that the body may be moved from one of these to the next by turning it round a certain point. Let a, B, C, D, E... be the points round which the body must be turned in order to take it from the first position to the second, the second to the third, etc., and let b, c, d, e... be the points in the moving plane which successively come to coincide with B, C, D, E... Then we can move the body through this series of positions by rolling the polygon abcde on the polygon aBCDE, it being obvious that corresponding sides of them are equal. By taking the successive positions sufficiently near to one another, we can make this approximation as close as we like to the actual motion of the plane; and the nearer the successive positions are taken, the more closely do the polygons approximate to continuous curves which roll upon one another.

Precisely similar reasoning may be used in the case of a sliding sphere, and of the general motion of a rigid body.

There are some difficulties in this proof, which the following exact investigation may clear up. The question is, what velocity-systems are consistent with rigidity? We shall secure that the body does not change in size or shape, if we make sure that no straight line in the body is altered in length. Let a and b be two points in the body, then the motion of b relative to a must be at right angles to ab; for its component along ab is the flux of the length ab, which has to be zero. We shall find it convenient to denote the velocity of the point a by ȧ. This being so, it is necessary and sufficient for rigidity that b— å should be either zero, or perpendicular to ab, where a, b are any two points in the moving body. It follows at once that if two velocity-systems are consistent with rigidity, their resultant is consistent with rigidity.

Now suppose a plane to be sliding on a plane, and combine with its velocity-system a translation equal and opposite to the velocity of any point a. Then the new

motion is consistent with rigidity, and the point a is at rest. Consequently the new motion is a spin about the point a. The original motion, therefore, is the resultant of this spin and of a translation equal to the velocity of a; it is therefore a spin of the same magnitude w, about a point o situate on a line through a perpendicular to its direction of motion, at a distance such that ȧiw.oa.

To determine the motion of the instantaneous centre, we must find the acceleration of any point in the plane. The instantaneous centre shall be called c in the fixed plane, and c, in the moving plane; and at a certain instant of time it shall be supposed to be at a point o in the moving plane. Then at that instant c, c,, o are the same point; but ċ means the velocity of the instantaneous centre in the fixed plane, c, its velocity in the moving plane, and o the velocity of o in the moving plane, which we know to be zero.

ROLLING OF CENTRODES.

139

Now if p be any point in the moving plane, we know that at every instant piw.cp. To find the acceleration of p we must remember that the flux of cp is p-c. Therefore

p = iw.cp+iw (p − ċ) = (iw — w3) cp — iw .ċ.

Now let p coincide with o, that is (for the instant) with c. Then ö-iw.ċ, or the acceleration of o is at right angles to the velocity of c, and equal to the product of it by the angular velocity.

If we suppose the moving plane to be fixed, and the fixed plane to slide upon it so that the relative motion is the same, then if p, is the point of the fixed plane which at a given instant coincides with p in the moving plane, the velocity and accleration of p, on one supposition are equal and opposite to the velocity and acceleration of p on the other supposition; also a becomes - w. Hence we shall have ö,+iw.c,, but ö,ö. Therefore ċ, ċ, or the velocity of the instantaneous centre in the moving plane is the same in magnitude and direction as its velocity in the fixed plane.

=

Because these velocities are the same in direction, the two centrodes touch one another; and because they are the same in magnitude, the moving centrode rolls on the fixed one without sliding. For let s, s, be the arcs ac, bc

measured from points a, b which have been in contact; then s=8,, and therefore (since they vanish together) s=s1.

The angular velocity w is equal to s multiplied by the difference of the curvatures of the two centrodes. For suppose them to roll simultaneously on the tangent ct; then their angular velocities & and will be respectively equal to their curvatures multiplied by s, and the relative angular velocity will be the difference of these. When the curvatures are in opposite directions one of them must be con

sidered negative. The same result may be obtained by calculating the flux of the acceleration of o.

1

Thus if r, r, are radii of curvature of the fixed and rolling centrodes, we have

We may derive some important consequences from the expression just obtained for the the acceleration of a point in the moving plane, namely

p = (iww') cp-iw.ċ.

This consists of three parts; w. pc is the acceleration towards c due to rotation about it as a fixed point; i. cp is in the direction np perpendicular to cp, due to the change in the angular velocity; and i.ċ is in the

P

direction cn, due to the change in position of c as the centrode rolls. Hence the normal acceleration of p, that is, the component along pc, is in magnitude w2. It vanishes for those points p for •pc-w.ċ cose. which w.pc ċ cose, or for which cnc : w. These points lie on a circle having cn for diameter; the curvature

=

=