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plane. On account of the difference of sign, these may be conaidered as a maximum and a minimum.
Generally the sum of the curvatures at a point, in any two pormal planes at right angles to each other, is independent of the position of these planes. · If and be the maximum and minimum curvatures at any point the curvature of a normal section making an angle 0 with the normal section of maximum curvature is
- cosoe + sino e, which includes the above statements as particular cases.
123. Let P, o 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 an is r cos a (8 121). Also the length along the normal section, from P to 0, 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 sup posed 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 com. mon 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 Alexible, and we can easily form hhe 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’ (9*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 Do 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 musk meet in several points : in general, each one meets its predecessor and its successor in different points.
132. There is still no difficulty in understanding the form of, säý 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 in. flection, but everywhere finite curvature. Cut the paper quite away on the con. cave 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 muslip 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 dėvelopable surface, of which the curve, bending into a tor. tuous curve, is the edge of regression. The tan. gent to the curve drawn in one direction from
the point of contact, will always lie in one of the LUI
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 fexible 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 in mutual inclination. Hence, also, geodetic lines must remain geodetic lines.
135. We have now to consider the very important kinematical conditions presented by the changes of volume or figure experienced by a solid or liquid mass, or by a group of points whose positions with regard to each other are subject to known conditions.
Any such definite alteration of form or dimensions is called a Strain.
Thus a rod which becomes longer or shorter is strained. Water, when compressed, is strained. A stone, beam, or mass of metal, in a building or in a piece of framework, if condensed or dilated in any direction, or bent, twisted, or distorted in any way, is said to experience a strain. A ship is said to strain' if, in launching, or when working in a heavy sea, the different parts of it experience relative motions.
138: If, when the matter occupying any space is strained in any way, all pairs of points of its substance which are initially at equal distances from one another in parallel lines remain equidistant, it may be at an altered distance; and in parallel lines, altered, it may be, from their initial direction; the strain is said to be homogeneous.
137. Hence if any straight line be drawn through the body in its initial state, the portion of the body cut by it will continue to be a straight line when the body is homogeneously strained. For, if ABC be any such line, AB and BC, being parallel to one line in the initial, remain parallel to one line in the altered state; and therefore remain in the same straight line with one another. Thus it follows that a plane remains a plane, a parallelogram a parallelogram, and a parallelepiped a parallelepiped.
188. Hence, also, similar figures, whether constituted by actual portions of the substance, or mere geometrical surfaces, or straight or curved lines passing through or joining certain portions or points of the substance, similarly situated (i.e. having corresponding parameters parallel) when altered according to the altered condition of the body, remain similar and similarly situated among one another.
139. The lengths of parallel lines of the body remain in the same proportion to one another, and hence all are altered in the same proportion. Hence, and from § 137, we infer that any plane figure becomes altered to another plane figure which is a diminished or magnified orthographic projection of the first on some plane.
The elongation of the body along any line is the proportion which the addition to the distance between any two points in that line bears to their primitive distance.
140. Every orthogonal projection of an ellipse is an ellipse (the case of a circle being included). Hence, and from $ 139, we see that an ellipse remains an ellipse; and an ellipsoid remains a surface of which every plane section is an ellipse; that is, remains an ellipsoid.
141. The ellipsoid which any surface of the body initially spherical becomes in the altered condition, may, to avoid circumlocutions, be called the Strain Ellipsoid.
142. In any absolutely unrestricted homogeneous strain there are three directions (the three principal axes of the strain ellipsoid), at right angles to one another, which remain at right angles to one another in the altered condition of the body. Along one of these the elongation is greater, and along another less, than along any other direction in the body. Along the remaining one the elongation is less than in any other line in the plane of itself and the first men. Lioned, and greater than along any other line in the plane of itself and the second.
Note.-Contraction is to be reckoned as a negative elongation: the maximum elongation of the preceding enunciation may be a minimum contraction: the minimum elongation may be a maximum contraction.
143. The ellipsoid into which a sphere becomes altered may be an ellipsoid of revolution, or, as it is called, a spheroid, prolate, or oblate. There is thus a maximum or minimum elongation along the axis, and equal minimum or maximum elongation along all lines perpendicular to the axis.
Or it may be a sphere; in which case the elongations are equal in all directions. The effect is, in this case, merely an alteration of dimensions without change of figure of any part.
144. The principal axes of a strain are the principal axes of the ellipsoid into which it converts a sphere. The principal elongations of a strain are the elongations in the direction of its principal axes.
145. When the positions of the principal axes, and the magnitudes of the principal elongations of a strain are given, the elongation of any line of the body, and the alteration of angle between any two lines, may be obviously determined by a simple geometrical construc. tion.
146.' With the same data the alteration of angle between any two planes of the body may also be easily determined, geometrically.
147. Let the ellipse of the annexed diagram represent the section of the strain ellipsoid through the greatest and least principal axes.
Let S'OS, T'OT be the two diameters of this ellipse, which are equal to the mean principal axis of the ellipsoid. Every plane through 0, perpendicular to the plane of the diagram, cuts the ellipsoid in an ellipse of which one principal axis
is the diameter in which it cuts the ellipse of the diagram, and the other, the mean principal diameter of the ellipsoid. Hence a plane through either SS or TT', perpendicular