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Edge of regression.



tion of a developable from its 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 annexed diagram.

149. 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 upper, and let it have no point of inflection, but everywhere finite curvature. Cut the two papers along the curve and remove the convex portions. If the curve traced is closed, it must be cut open (see second diagram). 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 the 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.

General property of

150. A flexible but perfectly inextensible surface, altered inextensible in form in any way possible for it, must keep any line traced surface. on it unchanged in length; and hence any two intersecting lines unchanged in mutual inclination. Hence, also, geodetic lines must remain geodetic lines. Hence "the change of direction" in a surface, of a point going round any portion of it, must be the same, however this portion is bent. Hence (§ 136) the integral curvature remains the same in any and every portion however the surface is bent. Hence (§ 138,

Gauss's Theorem) the product of the principal radii of curvature General at each point remains unchanged.

property of inextensible surface.

151. The general statement of a converse proposition, expressing the condition that two given areas of curved surfaces may be bent one to fit the other, involves essentially some mode of specifying corresponding points on the two. A full investigation of the circumstances would be out of place here.



152. In one case, however, a statement in the simplest Surface of possible terms is applicable. Any two surfaces, in each of specific which the specific curvature is the same at all points, and equal to that of the other, may be bent one to fit the other. Thus any surface of uniform positive specific curvature (i.e., wholly convex one side, and concave the other) may be bent to fit a sphere whose radius is a mean proportional between its principal radii of curvature at any point. A surface of uniform negative, or anticlastic, curvature would fit an imaginary sphere, but the interpretation of this is not understood in the present condition of science. But practically, of any two surfaces of uniform anticlastic curvature, either may be bent to fit the other.

triangles on

153. It is to be remarked, that geodetic trigonometry on Geodetic any surface of uniform positive, or synclastic, curvature, is such a suridentical with spherical trigonometry.


If a=



8, t, u are the lengths


C =


Jpp'' JPP'
of three geodetic lines joining three points on the surface, and
if A, B, C denote the angles between the tangents to the geodetic
lines at these points; we have six quantities which agree perfectly
with the three sides and the three angles of a certain spherical
triangle. A corresponding anticlastic trigonometry exists, al-
though we are not aware that it has hitherto been noticed, for any
surface of uniform anticlastic curvature. In a geodetic triangle
on an anticlastic surface, the sum of the three angles is of course
less than three right angles, and the difference, or "anticlastic
defect" (like the "spherical excess"), is equal to the area divided
by px-p', where p and -p' are positive.


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154. We have now to consider the very important kinema- Strain. tical conditions presented by the changes of volume or figure


Definition of homogeneous strain.

Properties of homogeneous strain.

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.

155. 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.

156. 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.

157. 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.

158. 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 § 156, we infer that any plane figure becomes altered to another plane figure which

of homo

is a diminished or magnified orthographic projection of the first Properties on some plane. For example, if an ellipse be altered into a geneous circle, its principal axes become radii at right angles to one another.


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.

159. Every orthogonal projection of an ellipse is an ellipse (the case of a circle being included). Hence, and from § 158, 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.

A plane curve remains (§ 156) a plane curve. A system of two or of three straight lines of reference (Cartesian) remains a rectilineal system of lines of reference; but, in general, a rectangular system becomes oblique.


x2 y2
a b3


be the equation of an ellipse referred to any rectilineal conjugate
axes, in the substance, of the body in its initial state. Let a and
B be the proportions in which lines respectively parallel to OX
and OY are altered. Thus, if we call έ and n the altered values
of x and y, we have

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έ= ax, n = By.

¿ n2





which also is the equation of an ellipse, referred to oblique axes
at, it may be, a different angle to one another from that of the
given axes, in the initial condition of the body.

Or again, let

x2 y3 22
+ +
a2 b2



n2 (Bb)

be the equation of an ellipsoid referred to three conjugate dia-
metral planes, as oblique or rectangular planes of reference, in the
initial condition of the body. Let a, ẞ, y be the proportion
β, γ
in which lines parallel to OX, OY, OZ are altered; so that if
έ, 7, be the altered values of x, y, z, we have


= ax, n=By, C=7z.


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= 1,

Properties of homogeneous strain.

Strain ellipsoid.

Change of volume.

Axes of a

which is the equation of an ellipsoid, referred to conjugate diametral planes, altered it may be in mutual inclination from those of the given planes of reference in the initial condition of the body.

160. The ellipsoid which any surface of the body initially spherical becomes in the altered condition, may, to avoid circumlocutions, be called the strain ellipsoid.

161. 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 (§ 158). 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 mentioned, 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.

162. 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.

The original volume (sphere) is to the new (ellipsoid) evidently as 1 aẞy.

163. 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.

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