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(3) Let a tangent plane roll without spinning over the Curratura boundary of a given closed curve or geodetic polygon on any horograph. curved surface. Show that the points of the trace in the tangent plane which successively touch the same point of the given surface are at equal distances successively on the circumference of a circle, the angular values of the intermediate arcs being each 2- K if taken in the direction in which the trace is actually described, and K if taken in the contrary direction, K being the "integral curvature" of the portion of the curved surface enclosed by the given curve or geodetic polygon. Hence if K be commensurable with the trace on the tangent plane, however complicatedly autotomic it may be, is a finite closed curve or polygon.

(4) The trace by a tangent plane rolling successively over three principal quadrants bounding an eighth part of the circumference of an ellipsoid is represented in the accompanying diagram, the whole of which is traced when the tangent plane is

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rolled four times over the stated boundary. A, B, C; A', B, C',
&c. represent the points of the tangent plane touched in order
by ends of the mean principal axis (4), the greatest principal
axis (B), and least principal axis (C), and AB, BC, CA' are the
lengths of the three principal quadrants.

tween lines

138. It appears from what precedes, that the same equality Analogy be or identity subsists between "whole curvature" in a plane and surfaces arc and the excess of 7 above the angle between the terminal curvature.

as regards

Analogy be- tangents, as between "whole curvature" and excess of 2′′ above and surfaces change of direction along the bounding line in the surface for


as regards

curvature. any portion of a curved surface.


Or, according to Gauss, whereas the whole curvature in a plane arc is the angle between two lines parallel to the terminal normals, the whole curvature of a portion of curve surface is the solid angle of a cone formed by drawing lines from a point parallel to all normals through its boundary.

change of direction Again, average curvature in a plane curve is length and specific curvature, or, as it is commonly called, curvature, change of direction in infinitely small length at any point of it = length

Thus average curvature and specific curvature are for surfaces analogous to the corresponding terms for a plane curve.

Lastly, in a plane arc of uniform curvature, i.e., in a circular change of direction 1





And it is easily proved (as below) that, in a surface throughout which the specific curvature is 2-change of direction integral curvature uniform,






p and p' are the principal radii of curvature. Hence in a surface, whether of uniform or non-uniform specific curvature, the


specific curvature at any point is equal to
In geometry of
three dimensions, pp' (an area) is clearly analogous to p in a
curve and plane.

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= where


Consider a portion S, of a surface of any curvature, bounded
Let there be a spherical surface, radius

by a given closed curve.
r, and centre C. Draw a radius CQ, parallel to the normal at
any point P of S. If this be done for every point of the bound-
ary, the line so obtained encloses the spherical area used in
Gauss's definition. Now let there be an infinitely small rect-
angle on S, at P, having for its sides arcs of angles and ', on
the normal sections of greatest and least curvature, and let their
radii of curvature be denoted by p and p'. The lengths of these
sides will be p and p' respectively. Its area will therefore be
pp'. The corresponding figure at Q on the spherical surface
will be bounded by arcs of angles equal to those, and therefore of

Hence Area of the


lengths rċ and r' respectively, and its area will be r'.
if do denote this area, the area of the infinitely small portion of


the given surface will be

In a surface for which pp' is

constant, the area is therefore

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PP2 [fdo
[sito = pp'× integral curvature.


139. A perfectly flexible but inextensible surface is sug- Flexible and gested, although not realized, by paper, thin sheet metal, or surface. cloth, when the surface is plane; and by sheaths of pods, seed vessels, or the like, when it is 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."

140. The geometry or kinematics of this subject is a great contrast to that of the flexible line (§ 14), 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.

141. 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 (S$ 717, 718, 719) we shall return to this. In the meantime, in considering illustrations of our kinematical propositions, it is necessary to anticipate such physical circumstances.




142. Cloth woven in the simple common way, very fine in two di- muslin for instance, illustrates a surface perfectly inextensible in two directions (those of the warp and the woof), but susceptible of any amount of extension from 1 up to 2 along one diagonal, with contraction from 1 to 0 (each degree of extension along one diagonal having a corresponding determinate degree of contraction along the other, the relation being e2+ e22 = 2, where 1:e and 1: e' are the ratios of elongation, which will be contraction in the case in which e or e' is < 1) in the other.

"Elastic finish" of muslin goods.

143. The flexure of a surface fulfilling any case of the geometrical condition just stated, presents an interesting subject for investigation, which we are reluctantly obliged to forego. The moist paper drapery that Albert Dürer used on his little lay figures must hang very differently from cloth. Perhaps the stiffness of the drapery in his pictures may be to some extent owing to the fact that he used the moist paper in preference to cloth on account of its superior flexibility, while unaware of the great distinction between them as regards extensibility. Fine muslin, prepared with starch or gum, is, during the process of drying, kept moving by a machine, which, by producing a to-and-fro relative angular motion of warp and woof, stretches and contracts the diagonals of its structure alternately, and thus prevents the parallelograms from becoming stiffened into rectangles.

Flexure of 144. The flexure of an inextensible surface which can be


developable. 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 Flexure of sum may be finite, we have our plane surface bent into a developable. curved surface, which is of course "developable" (§ 139).

145. 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 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 of.

146. Prisms and cylinders (when the lines of bending, § 144, 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.

147. 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, each one meets its predecessor and its successor in different points.

148. 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


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Edge of regression.

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