instantaneous axis and the axis of the fixed cone passes through the axis of the rolling cone. This plane turns round the axis of the fixed cone with an angular velocity, which must clearly bear a constant ratio to the angular velocity w of the rigid body about its instantaneous axis.

117. The motion of the plane containing these axes is called the precession in any such case. What we have denoted by is the angular velocity of the precession, or, as it is sometimes called, the rate of precession.

The angular motions w, are to one another inversely as the distances of a point in the axis of the rolling cone from the instantaneous axis and from the axis of the fixed cone.

I

For, let OA be the axis of the fixed cone, OB that of the rolling cone, and OI the instantaneous axis. From any point P in OB draw PN perpendicular to OI, and PQ perpendicular to OA. Then we perceive that P moves always in the circle whose centre is Q, radius PQ, and plane perpendicular to OA. Hence the actual velocity of the point P is Q.QP. But, by the principles explained above (§ 110) the velocity of P is the same as that of a point moving in a circle whose centre is N, plane perpendicular to ON, and radius NP, which, as this radius revolves with angular velocity w, is w.NP. Hence

A

N

Q

B

or

Q.QP=w.NP,
w::: QP: NP.

118. Suppose a rigid body bounded by any curved surface to be touched at any point by another such body. Any motion of one on the other must be of one or more of the forms sliding, rolling, or spinning. The consideration of the first is so simple as to require

no comment.

Any motion in which the bodies have no relative velocity at the point of contact, must be rolling or spinning, separately or combined.

Let one of the bodies rotate about successive instantaneous axes, all lying in the common tangent plane at the point of instantaneous contact, and each passing through this point-the other body being fixed. This motion is what we call rolling, or simple rolling, of the movable body on the fixed.

On the other hand, let the instantaneous axis of the moving body be the common normal at the point of contact. This is pure spinning, and does not change the point of contact.

Let it move, so that the instantaneous axis, still passing through the point of contact, is neither in, nor perpendicular to, the tangent plane. This motion is combined rolling and spinning.

119. As an example of pure rolling, we may take that of one cylinder on another, the axes being parallel.

Let P be the radius of curvature of the rolling, σ of the fixed,

cylinder; w the angular velocity of the former, V the linear velocity of the point of contact. We have

1

1= + V.

For, in the figure, suppose P to be at any time the point of contact, and Q and p the points which are to be in contact after a very small interval 7; 0, O the centres of curvature; POp=0, PO'Q=0.

Then PQ=Pp=space described by point of contact. In symbols

Also, before ƠQ and OP can coincide in direction, the former must evidently turn through an angle

θ + Φ.

Therefore

and by eliminating

get the above result.

ωτ:

· = 0 + $ ;

and p, and dividing by T, we

It is to be understood here, that as the radii of curvature have been considered positive when both surfaces are convex, the negative sign must be introduced for either radius when the corresponding surface is concave.

R

G

Hence the angular velocity of the rolling curve is in this case equal to the product of the linear velocity of the point of contact into the sum or difference of the curvatures, according as the curves are both convex, or one concave and the other convex.

120. We may now take up a few points connected with the curvature of surfaces, which are useful in various parts of our subject.

The tangent plane at any point of a surface may or may not cut it at that point. In the former case, the surface bends away from the tangent plane partly towards one side of it, and partly towards the other, and has thus, in some of its normal sections, curvatures oppositely directed to those in others. In the latter case, the surface on every side of the points bends away from the same side of its tangent plane, and the curvatures of all normal sections are similarly directed. Thus we may divide curved surfaces into Anticlastic and Synclastic. A saddle gives a good example of the former class; a ball of the latter. Curvatures in opposite directions, with reference to the tangent plane, have of course different signs. The outer portion of the surface of an anchor-ring is synclastic, the inner anticlastic.

121. Meunier's Theorem.-The curvature of an oblique section of a surface is equal to that of the normal section through the same tangent line multiplied by the secant of the inclination of the planes of the sections. This is evident from the most elementary considerations regarding projections.

122. Euler's Theorem.-There are at every point of a synclastic surface two normal sections, in one of which the curvature is a

maximum, in the other a minimum; and these are at right angles to each other.

In an anticlastic surface there is maximum curvature (but in opposite directions) in the two normal sections whose planes bisect the angles between the lines in which the surface cuts its tangent plane. On account of the difference of sign, these may be considered as a maximum and a minimum.

Generally the sum of the curvatures at a point, in any two normal planes at right angles to each other, is independent of the position of these planes.

If

1

P

1

and be the maximum and minimum curvatures at any

σ

point, the curvature of a normal section making an angle ✪ with the normal section of maximum curvature is

which includes the above statements as particular cases.

123. Let P, p be two points of a surface indefinitely near to each other, and let r be the radius of curvature of a normal section passing through them. Then the radius of curvature of an oblique section through the same points, inclined to the former at an angle a, is r cos a (§ 121). Also the length along the normal section, from P to p, is less than that along the oblique section-since a given chord cuts off an arc from a circle, longer the less is the radius of that circle.

124. Hence, if the shortest possible line be drawn from one point of a surface to another, its osculating plane, or plane of curvature, is everywhere perpendicular to the surface.

Such a curve is called a Geodetic line. And it is easy to see that it is the line in which a flexible and inextensible string would touch the surface if stretched between those points, the surface being supposed smooth.

125. A perfectly flexible but inextensible surface is suggested, although not realized, by paper, thin sheet-metal, or cloth, when the surface is plane; and by sheaths of pods, seed-vessels, or the like, when not capable of being stretched flat without tearing. The process of changing the form of a surface by bending is called 'developing. But the term 'Developable Surface' is commonly restricted to such inextensible surfaces as can be developed into a plane, or, in common language, 'smoothed flat.'

126. The geometry or kinematics of this subject is a great contrast to that of the flexible line (§ 16), and, in its merest elements, presents ideas not very easily apprehended, and subjects of investigation that have exercised, and perhaps even overtasked, the powers of some of the greatest mathematicians.

127. Some care is required to form a correct conception of what

is a perfectly flexible inextensible surface. First let us consider a plane sheet of paper. It is very flexible, and we can easily form the conception from it of a sheet of ideal matter perfectly flexible. It is very inextensible; that is to say, it yields very little to any application of force tending to pull or stretch it in any direction, up to the strongest it can bear without tearing. It does, of course, stretch a little. It is easy to test that it stretches when under the influence of force, and that it contracts again when the force is removed, although not always to its original dimensions, as it may and generally does remain to some sensible extent permanently stretched. Also, flexure stretches one side and condenses the other temporarily; and, to a less extent, permanently. Under elasticity we may return to this. In the meantime, in considering illustrations of our kinematical propositions, it is necessary to anticipate such physical circumstances.

128. The flexure of an inextensible surface which can be plane, is a subject which has been well worked by geometrical investigators and writers, and, in its elements at least, presents little difficulty. The first elementary conception to be formed is, that such a surface (if perfectly flexible), taken plane in the first place, may be bent about any straight line ruled on it, so that the two plane parts may make any angle with one another.

Such a line is called a 'generating line' of the surface to be formed.

Next, we may bend one of these plane parts about any other line which does not (within the limits of the sheet) intersect the former; and so on. If these lines are infinite in number, and the angles of bending infinitely small, but such that their sum may be finite, we have our plane surface bent into a curved surface, which is of course 'developable' (§ 125).

129. Lift a square of paper, free from folds, creases, or ragged edges, gently by one corner, or otherwise, without crushing or forcing it, or very gently by two points. It will hang in a form which is very rigorously a developable surface; for although it is not absolutely inextensible, yet the forces which tend to stretch or tear it, when it is treated as above described, are small enough to produce absolutely no sensible stretching. Indeed the greatest stretching it can experience without tearing, in any direction, is not such as can affect the form of the surface much when sharp flexures, singular points, etc., are kept clear off.

130. Prisms and cylinders (when the lines of bending, § 128, are parallel, and finite in number with finite angles, or infinite in number with infinitely small angles), and pyramids and cones (the lines of bending meeting in a point if produced), are clearly included.

131. If the generating lines, or line-edges of the angles of bending, are not parallel, they must meet, since they are in a plane when the surface is plane. If they do not meet all in one point, they must

meet in several points: in general, let each one meet its predecessor and its successor in different points.

132. There is still no difficulty in understanding the form of, say a square, or circle, of the plane surface when bent as explained above,

provided it does not include any of these points of intersection. When the number is infinite, and the surface finitely curved, the developable lines will, in general, be tangents to a curve (the locus of the points of intersection when the number is infinite). This curve is called the edge of regression. The surface must clearly, when complete (according to mathematical ideas), consist of two sheets meeting in this edge of regression (just as a cone consists of two sheets meeting in the vertex), because each tangent may be produced beyond the point of contact, instead of stopping at it, as in the preceding diagram. 133. To construct a complete developable surface in two sheets from its edge of regression—

Lay one piece of perfectly flat, unwrinkled, smooth-cut paper on the top of another. Trace any curve on the other, and let it have no point of inflection, but everywhere finite curvature. Cut the paper quite away on the concave side. If the curve traced is closed, it must be cut open (see second diagram). The limits to the extent that may be left uncut away, are the tangents drawn outwards from the two ends, so that, in short, no portion of the paper through which a real tangent does not pass is to be left.

Attach the two sheets together by very slight paper or muslin clamps gummed to them along the common curved edge. These

must be so slight as not to interfere sensibly with the flexure of the two sheets. Take hold of one corner of one sheet and lift the whole. The two will open out into two sheets of a developable surface, of which the curve, bending into a curve of double curvature, is the edge of regression. The tangent to the curve drawn in one direction from the point of contact, will always lie in one of the sheets, and its continuation on the other side in the other sheet. Of course a double-sheeted developable polyhedron can be constructed by this process, by starting from a polygon instead of a curve.

134. A flexible but perfectly inextensible surface, altered in form in any way possible for it, must keep any line traced on it unchanged in length; and hence any two intersecting lines unchanged