Second case, Fig. 86. Let the cutting plane LMN be parallel to one generator, and one only. In the figure it is parallel to AV.

The method of determining the curve is the same as in the first case, and is evident from the figure. The curve ONL is a parabola—OnL is also a parabola.

Third case, Fig. 87. Let the cutting plane LMN be parallel to two generators DE and FG, or, what comes to the same thing, let it cut both sheets of the conical surface. The curve of intersection has then two branches. That curve is the hyperbola.

The horizontal projection of the curve might be found as in the other two cases, but as the straight generators cut LMN so obliquely a better solution is obtained by taking a series of horizontal circles on the cone. The rest of the construction will be evident from the figure. The curve is for convenience rabatted on the vertical plane.

The tangent to the plane section of a curved surface at any point is the line of intersection of the cutting plane and the tangent plane at that point. The tangent to any of these conic sections at a given point may be readily obtained in that way.

The asymptotes of the hyperbola, that it is to say the tangents at points infinitely distant, when the conical surface is extended indefinitely, are the lines of intersection of LMN with the tangent planes along DE and FG; LP, QR are the projections of the asymptotes, and L,O,, O,Q, their rabat

ments.

The general problem of finding the section of any conical surface by a given plane is solved by finding the points of intersection of a sufficient number of straight lines, which are the generators, with the given plane, by Prob. XIV. Chap. II.

The ellipse diminishes in size as the cutting plane approaches the vertex, and its limit is a point. The limit of the hyperbola is a pair of straight lines intersecting at the vertex; the limit of the parabola is a straight line.