they will all contain the line joining the vertices of the cones. Hence the horizontal traces of all the cutting planes will pass through the horizontal trace of the line joining the vertices of the cones. Also, the vertical traces of the cutting planes will pass through the vertical trace of the line joining the vertices of the cones.

EXAMPLE (Fig. 743). vv, is the vertex of a cone whose horizontal trace is a circle 3 inches in diameter. vv is the vertex of a cone whose vertical trace is a circle 2.5 inches in diameter. The other dimensions are given on the figure. It is required to show, in plan and elevation, the intersection of the surfaces of the two cones.

Seven cutting planes passing through the vertex of each cone will determine all the important points in this example, but in Fig. 743 only four of these planes, numbered 1, 2, 3, and 4, are shown. Both traces of each cutting plane are employed. All the horizontal traces of the cutting planes pass through v2, and all the vertical traces pass through v

Consider plane number 1. This plane touches one cone and cuts the other. The horizontal trace of the plane is drawn first, and if the point where this trace meets the ground line comes within the paper the vertical trace is obtained by joining v, to this point. If however the point where the horizontal trace meets the ground line is off the paper, then the construction given in Art. 11, p. 9, may be used for drawing the vertical trace. The intersections of the line in which plane number 1 touches the one cone with the lines in which it cuts the surface of the other cone determine the points numbered 1 on the intersection required. In like manner the other points are determined.

324. Intersections of Cylinders and Cones enveloping the same Sphere.-If two cylinders envelop the same sphere their

intersection will be plane sections of both and will therefore be ellipses. Fig. 744 shows a projection of two such cylinders on a plane parallel to their axes. The straight lines ab and cd form the projection of the intersection of the cylinders.

The same remarks apply to the cone and cylinder (Fig. 745), and to two cones (Fig. 746). Referring to Fig. 745, if the vertex of the

cone is placed on the curved surface of the cylinder, one of the ellipses will become a straight line. Referring to Fig. 746, it is obvious that if the cone vad be turned round so as to make ved more nearly parallel to v,a, the straight line cd will become more nearly parallel to v,a and vb, and when vd is parallel to v,a, cd will also be parallel to va and vb, and the intersection cd will then be a parabolic section of each cone. Continuing the motion of the cone vad in the same direction, the intersection cd will become a hyperbolic section of each cone.

325. Intersection of Cylinder and Sphere.-Since the surface of a sphere is a particular form of a surface of revolution this problem is a particular case of that discussed in Art. 326, p. 398. If

the cylinder is also a surface of revolution, that is, a right circular cylinder, then this problem is also a particular case of that considered in Art. 328, p. 401. But instead of using the method described in Art. 326, or the method of Art. 328 cutting planes parallel to the axis of the cylinder and perpendicular to one of the planes of projection may be employed, because such planes will cut the sphere in circles and the cylinder in straight lines.

If the axis of the cylinder is inclined to both planes of projection then an auxiliary elevation on a vertical plane parallel to the axis of

the cylinder should be drawn, and the cutting planes being vertical and parallel to the axis of the cylinder they will cut the sphere in circles which will appear as circles in the auxiliary elevation.

EXAMPLE (Fig. 747). The plan and elevation of a cylinder and sphere are given, the dimensions being marked on the figure. It is required to show, in plan and elevation, the intersection of the surfaces of the cylinder and sphere.

The method used here is that of cutting the surfaces by horizontal planes.

The upper right hand portion of Fig. 747 is a projection of the cylinder and sphere on a plane perpendicular to the axis of the cylinder and it is from this projection that the positions of the cutting planes which give the important points on the required intersection are determined.

326. Intersection of Cylinder and Surface of Revolution.— In all the problems hitherto considered on the intersection of surfaces, the auxiliary cutting surfaces which have been used, in order to find points on the required intersection, have been planes. The only simple plane sections of a surface of revolution are, in general, sections at right angles to its axis, which are always circles. Now it is evident that only in very particular cases would a plane which cuts the surface of revolution in a circle cut the surface of the cylinder in a circle or in straight lines. For instance, a plane which cuts the surface of revolution in a circle may cut the cylinder in an ellipse, and it would clearly be a laborious process to construct an ellipse for each cutting plane. This objection may however be got over in the present problem by using a tracing of the section of the cylinder by a plane which cuts the surface of revolution in a circle, in a manner to be explained at the end of this article.

Instead of cutting the given surfaces by planes they may be cut by the surfaces of cylinders and points on the required intersection obtained by means of circles and straight lines. Let a circle which is a section of the surface of revolution be taken and let a straight line move in contact with this circle and also remain parallel to the axis of the given cylinder. This moving line will describe the surface of a cylinder which intersects the surface of revolution in a circle and the surface of the given cylinder in straight lines, and the points in which these straight lines cut the circle will be points on the intersection required.

In order that the projections of the circular sections of the surface of revolution may be circles and straight lines the axis of revolution must be arranged perpendicular to one of the planes of projection.

Referring to Fig. 748, mn, m'n' is the axis of a cylinder which intersects a surface of revolution whose axis is vertical. The construction for the outline of the elevation of the surface of revolution is shown at (c).

The ellipse which is the horizontal trace of the cylinder is determined as explained in Art. 219, p. 253.

ab, a'b' is a circular section of the surface of revolution, oo' being the centre of this circle. ot, o't' is a line through oo' parallel to mn, m'n'.

ot, o't' is the axis of an auxiliary cylinder of which the circle ab, a'b' is one horizontal section, and all horizontal sections of this cylinder will be circles of the same diameter. tt' being the horizontal trace of the axis of this auxiliary cylinder the horizontal trace of its surface will be a circle whose centre is t and radius equal to oa. The horizontal trace

of this auxiliary cylinder intersects the horizontal trace of the given cylinder at 8 and r. The auxiliary cylinder intersects the given cylinder in straight lines sp, s'p' and rq, r'q' which are parallel to mn, m'n'. pp' and qq' the intersection of the lines sp, s'p' and rq, r'q' with

the circle ab, a'b' are points on the line of intersection of the given cylinder and surface of revolution. By taking other auxiliary cylinders, any number of points on the required intersection may be found.

The student should notice that each line of intersection of an auxiliary cylinder and the given cylinder intersects the corresponding circle on the surface of revolution in one point only although its plan may cut the plan of the circle in two points.

A convenient and practical method of solving the problem which has just been considered is to take horizontal sections of both the given surfaces. The sections of the cylinder are ellipses but these ellipses are all of the same size and if one of them be drawn and a tracing of it made, it only remains to draw a sufficient number of circular sections of the surface of revolution and apply the tracing of the ellipse to each to find points in the required intersection. The position of the centre of the ellipse corresponding to a particular circular section of the surface of revolution is where the plane of that section cuts the axis of the cylinder. For the circular section ab, a'b' (Fig. 748) e is the position of the centre of the ellipse in the plan and the tracing of the ellipse is placed so that the centre is at e and the major axis on mn. The points where the ellipse cuts the circle ab are points on the plan of the intersection required and these points may be pricked through. The elevations of the points are of course perpendicularly over their plans and on the elevation of the corresponding circular section.

327. Intersection of Cone and Surface of Revolution.Placing the surface of revolution so that its axis is vertical, horizontal sections of it will be circles, but except in the special case where the cone is a right circular cone and its axis is vertical, horizontal sections of the cone will not be circles or straight lines and all the horizontal sections will be different. The tracing paper method which is applicable to the intersection of a cylinder and a surface of revolution and which was described in the latter part of the preceding Art., is therefore not suitable in the case of the intersection of a cone and a surface of revolution.

Auxiliary cones are taken which have their vertices at the vertex of the given cone and for their directrices they have circular sections of the surface of revolution. These auxiliary cones will intersect the given cone in straight lines and the intersection of these straight lines with the corresponding circles on the surface of revolution will determine points on the intersection required.

Referring to Fig. 749, vn, v'n' is the axis of a cone whose vertical angle is 30° and which intersects a surface of revolution whose axis is vertical. The dimensions are given on the figure.

The horizontal trace of the cone is determined as explained in Art. 226, p. 259. ab, a'b' is a circular section of the surface of revolution, oo' being the centre of this circle. vt, v't' is a line passing through oo', the centre of the circle, and the vertex of the cone. vt, 't' is the axis of an auxiliary cone of which the circle ab, a'b' is one horizontal section. All horizontal sections of this auxiliary cone will be circles but they will be of different diameters. The point tt' being the horizontal trace