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the fixed directors. Developable surfaces cannot therefore in general have more than two directors.

352.] On developable surfaces.

Since in developable surfaces every generating line and its consecutive line are in the same plane, this plane is the tangent plane to the surface at every point along the first line: for consider any point on the first generating line; the tangent plane at that point passes through the next consecutive point on the line, and therefore contains the line; and as the tangent plane also passes through an indefinite number of points infinitesimally near to the point at which it is drawn, it also passes through a point on the consecutive generating line; and this line is in the same plane with the first generating line; therefore the tangent plane which contains the first line also contains this latter line; the tangent plane therefore touches the surface along the whole length of the generating line. Hence also we have the following property of such surfaces:

Let any number of generating lines be represented by Oj,

02, Ga, ;then the surface is made up of the infinitesimal

plane areas contained between Gi and G3, between o2 and G3,

Now the plane area lying between oj and o2 may be brought into the same plane with that lying between G2 and G3, by being turned through a small angle about G2; and similarly, by turning this last area about Gs, may all the areas between Gi and Gs be brought into the same plane without any discontinuity. Let these operations be performed for all the elements, then all will be brought into the same plane; and if we suppose any thin flexible and inextensible film to be laid on such a surface, it will be unfolded into a plane without tearing, rumpling, or doubling. For this reason such surfaces have obtained the expressive title of Developable Surfaces.

If all the generating lines of such surfaces meet in one point the surface is called conical, and the point is called the vertex of the cone; and if that point is at an infinite distance, so that all the generators are parallel, the surface is called cylindrical. Of these surfaces also independent definitions are given in the following Articles.

353.] On conical surfaces.

A conical surface is generated by a straight line which passes through a given point, and through a given director curve. Price, Vol. 1. 3 x

Let a, b, c be the coordinates to the given point, which is

the vertex of the cone; then the equations to the generating

line are ,

x-a = y-b = z_-c


in which a, b, c are constant; and L, M N, which are constant for any one position of the generator, vary as the generator passes from one position to another. Let the equations to the director be

Fi y\ *') = 0, F2 (af, y\ z') =0; (12)

and therefore, as the generator has to pass through (x't y, z), its equations become,

whence x a = -—- (z c),

z — c

. \; (14)

y--b = *-±«-c)\


between which equations and (12), eliminating x', y\ z', we have a function of the form,

ix—a y b\

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sz c z c

which is the general functional equation of conical surfaces. Equation (15) may also be written in the explicit form

xra _ tiy-b\


and may also be put into the symmetrical form,

iv b z c x—a\

* I- , , 7) = 0. (16)

vz — c x—a y b'

If the origin is taken at the vertex of the cone, a = b = c 0, and the last three equations severally become

which are homogeneous functions of 0 dimensions; such functions therefore represent conical surfaces.

Two particular forms which the director takes require notice. Firstly, if the director is a curve in the plane of xy, let us suppose its equation to be

r(x0,y0) = O;

Again, suppose the cone to be circumscribed about a given surface; then each generating line touches the surface; and if the equation to the surface is F (a?', y, sf) = 0, we have

and this and the equation to the surface are the equations to the director curve of the conical surface.

Equation (17) is that of the first polar of the surface with reference to (a, b, c) the pole, and gives the line of contact of all the generating lines of the cone with the surface. So if an eye is placed at the point (a, b, c), the visible part of the surface is separated from the unseen part by this line of contact. In surfaces of the second order the line of contact is a plane curve.

354.] Equations (15) and (16) are the general equations to all conical surfaces. They contain an undetermined functional symbol because the director curve is undetermined; and the variables enter under the functional symbol in a particular combination. Now that function may be coutinuous or discontinuous. If it is continuous, partial derived functions may be found, and by means of them, as in Article 53, the functional symbol may be eliminated, and a differential expression will arise, which gives a general property of all conical surfaces. Thus, from (15) we have, see Art. 53, Ex. 2,

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so that the equation to the conical surface is laz cx bz cy\ _

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which is indeed identical with the preceding, as the theory of Art. 50 shews. Now the geometrical property which this equation expresses is this: nil tangent planes of a conical surfact pass through the vertex of the cone, and also contain a generating line of the cone. Sometimes this geometrical property i< assumed to be (see Monge, Application de l'Analyse &c.) the defining property of conical surfaces; in which case the mathematical translation is the preceding differential equation (19), and from it b}- integration the integral equation of the surface is found. This question is considered in Vol. II of our treatise.

355.] Examples of conical surfaces.

Ex. 1. To find the equation to a cone whose director is a circle in the plane of xy.

Let the equations to the circle and to the generator be

xj + y0l = *», (20)

, x ay bz c and = =; (21)

L If N

therefore, when z = 0, xv a c, y0 = b e;


squaring and adding which, by means of (20), and replacing the variable parameters by their values from (21), we have

(cx - azf + (cy - bzf = *»(*- c?;

which is the general equation to a cone of which the director is a circle in the plane of xy.

If the line joining the centre of the circular director and the vertex is at right angles to the circle, the cone is called right; in which case a = b = 0, and the equation is


where - is the tangent of the semi-vertical-angle.

Ex. 2. To find the equation to a cone circumscribing a givcu ellipsoid.

Let the vertex of the cone be (.«•„, y0, s0), and the equation to the ellipsoid be ^ . ^

Let the equations to the generator be

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whence, operating on the equality (23), and reducing by means of (22), (24) and (25), we have

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which is the equation to the circumscribing cone.

As (x, y, z) is the point on the ellipsoid common to it and to the cone, (25) is the equation to the plane of contact; and it, and equation (22), are those to the director-curve.

Ex. 3. As another example let us investigate the relations between the coefficients of the equation of the second degree when it represents a cone. The equations to all conical surfaces are subject to the condition (19). Let the general equation of the second degree be

A-z^-f By'* + cz* + 2a! yz + 2bj Zx + 2ci xy

+ 2A2a?+ 2Bay + 2c22 + K = 0;

... (£) = 2(A*+clSH- -,. + *>, {%)=-,

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