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We have


w = q(t − t')


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= 9, con a' con a)


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= 9


=4(sina sin a)



sin a'




These three formulas are respectively equivalent to (9), (8), and (10) of § 111.


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on sur


125. In the same case, suppose the trace on one only of Surface rollthe surfaces to be given. We may evidently impose the con- face without dition of no spinning, and then the trace on the other is determinate. This case of motion is thoroughly examined in § 137, below.

The condition is that the projections of the curvatures of the two traces on the common tangent plane must coincide.


If /1/

and be the curvatures of the rolling and stationary surfaces in a normal section of each through the tangent line to the trace, and if a, a', p, p' have their meanings of § 124,

p'=r' cos a', p=r cos a (Meunier's Theorem, § 129, below).



sin a, hence tan a′



-tan a, the condition re


Surface rolling on surface; both traces given.


126. If a straight rod with a straight line marked on one Examples of side of it be bent along any curve on a spherical surface, so and twist. that the marked line is laid in contact with the spherical surface, it acquires no twist in the operation. For if it is laid so along any finite arc of a small circle there will clearly be no twist. And no twist is produced in continuing from any point along another small circle having a common tangent with the first at this point.

If a rod be bent round a cylinder so that a line marked along one side of it may lie in contact with the cylinder, or if, what presents somewhat more readily the view now de


Examples of sired, we wind a straight ribbon spirally on a cylinder, the and twist. axis of bending is parallel to that of the cylinder, and therefore oblique to the axis of the rod or ribbon. We may therefore resolve the instantaneous rotation which constitutes the bending at any instant into two components, one round a line perpendicular to the axis of the rod, which is pure bending, and the other round the axis of the rod, which is pure twist.

The twist at any point in a rod or ribbon, so wound on a circular cylinder, and constituting a uniform helix, is

cos a sin a

V cos a


if r be the radius of the cylinder and a the inclination of the
spiral. For if V be the velocity at which the bend proceeds
V cos a
along the previously straight wire or ribbon,
will be the

angular velocity of the instantaneous rotation round the line of
bending (parallel to the axis), and therefore

V cos a


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sin a and


are the angular velocities of twisting and of pure bending respectively.

From the latter component we may infer that the curvature of the helix is



a known result, which agrees with the expression used above (§ 13).

127. The hodograph in this case is a small circle of the sphere. If the specified condition as to the mode of laying on of the rod on the cylinder is fulfilled, the transverses of the spiral rod will be parallel at points along it separated by one or more whole turns. Hence the integral twist in a single turn is equal to the excess of four right angles. above the spherical area enclosed by the hodograph. If a be the inclination of the spiral, a will be the arc-radius of the hodograph, and therefore its area is 27 (1- sin a). integral twist in a turn of the spiral is 27 sin a, with the result previously obtained (§ 126).

Hence the which agrees

of surface.

128. As a preliminary to the further consideration of the Curvature rolling of one surface on another, and as useful in various parts of our subject, we may now take up a few points connected with the curvature of surfaces.

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 point 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 Synclastic a good example of the former class; a ball of the latter. Cur- clastic sur vatures in opposite directions, with reference to the tangent plane, have of course different signs. The outer portion of an anchor-ring is synclastic, the inner anticlastic.

and anti


of oblique

129. Meunier's Theorem.-The curvature of an oblique sec- Curvature tion of a surface is equal to that of the normal section through sections. 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.


130. Euler's Theorem.-There are at every point of a syn- Principal clastic 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.

vatures in

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

tions at right angles to each other.

Taking the tangent plane as that of x, y, and the origin at the point of contact, and putting

Principal normal






(d), -4, (ty),
(darty) = B, (d).

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or if

we have


The curvature of the normal section which passes through the point x, y, z is (in the limit)


that is



g x2 + y2

If the section be inclined at an angle to the plane of XZ, this




= 1⁄2 (Ax2 + 2Bxy + Cyo) + etc.


angles to each other,


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R =

x2 + y2



+- -=





be curvatures in normal sections at right


A+C constant.

(2) may be written

1 1

= = = {-4(1 + cos 20) + 2B sin 20 + C(1 − cos 20)}





= {4 + C + A − C' cos 20 + 2B sin 20},


¦ (4 – C) = R cos 2a, B = R sin 2a,

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we have

(4 + C) + √ √ { } ( 1 − C')2 + B3,


cos 2 (0 – a).

The maximum and minimum curvatures are therefore those in
normal places at right angles to each other for which ✪ = a and
0 = a+


and are respectively

1 (4 + 0) { (4-0)2 + B)

Hence their product is AC – B2.

If this be positive we have a synclastic, if negative an anticlastic, surface. If it be zero we have one curvature only, and the surface is cylindrical at the point considered. It is demonstrated

(§ 152, below) that if this condition is fulfilled at every point, the Principal surface is "developable" (§ 139, below).

normal sections.

By (1) a plane parallel to the tangent plane and very near it cuts the surface in an ellipse, hyperbola, or two parallel straight lines, in the three cases respectively. This section, whose nature informs us as to whether the curvature be synclastic, anticlastic, or cylindrical, at any point, was called by Dupin the Indicatrix.

of Line of

A line of curvature of a surface is a line which at every point Definition is cotangential with normal section of maximum or minimum Curvature.


line be

points on a

131. Let P, p be two points of a surface infinitely near to Shortest each other, and let r be the radius of curvature of a normal tween two section passing through them. Then the radius of curvature surface. of an oblique section through the same points, inclined to the former at an angle a, is (§ 129) r cos a. 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 the radius of that circle.

If a be the length of the chord Pp, we have

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


Such a curve is called a Geodetic line. And it is easy to see Geodetic 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.

133. If an infinitely narrow ribbon be laid on a surface along a geodetic line, its twist is equal to the tortuosity of its axis at each point. We have seen (§ 125) that when one body rolls on another without spinning, the projections of the traces on the common tangent plane agree in curvature at the point

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