determined, with one extremity of a diameter coinciding with an extremity of the major axis of the ellipse. See the construction given in Chap. VII. converse of 2.

The straight line, through point Q, and the point in which the circle touches the plane, is the required line of contact of the cylinder and its given tangent plane.

Work the problem :

(1) When the given plane is horizontal,

X.

INTERSECTIONS OF SOLIDS WITH PLANE

SURFACES.

NOTE. Developments should be drawn to all the problems of this Chapter, for which refer to Chapter VII.

PROBLEM I.

Two right prisms, with square bases of 25 inches side, intersect. One is placed with its edges horizontal and parallel to the vertical plane, and with its lowest face inclined 30o and its lowest edge I inch above the horizontal plane. The other stands on its base with one face inclined 25° to the vertical plane and with its axis passing through the highest edge of the horizontal prism. Other dimensions at pleasure. Draw plan and elevation.

Commence by drawing an auxiliary elevation of the horizontal prism, so arranged, that the elevations of both of its bases coincide: that is, draw the square and a ground line inclined 30° to one side and 1 inch below the lowest corner or angle of square. From this determine plan.

The plan of the vertical prism can then be easily placed in position, and will be a square with one side inclined 25° to the ground line of the required elevation.

Then proceed with the required elevation. The points in which the plans of the edges of the horizontal prism meet the square, which is the plan of the vertical prism, will be the plans of the points of intersection between. those edges and the vertical faces; and their elevations can readily be found by perpendiculars from them to the ground line to meet the elevations of the same edges.

The points in which the vertical edges of the one prism intersect the faces of the other will not be evident in plan from the vertical position of those edges, but they can be marked off in elevation by transferring their heights, which can be readily seen in the first or auxiliary elevation, to the required one.

The proper points must then be joined, and this will best be ascertained by inspection from the auxiliary elevation.

PROBLEM II.

Take the same prism similarly arranged, except that the horizontal edges of the one are inclined 30° to the vertical plane, and determine plan and elevation.

PROBLEM III.

Let the horizontal prism be intersected by one with base a regular hexagon of 1.5 inches side standing on its base, and with one edge at least outside the other solid.

ments at pleasure.

Other arrange

PROBLEM IV.

Plan and elevation of a right prism, with bases equilateral triangles of 2 inches side, piercing horizontally a 2 inch square pyramid standing on its base. No face of prism to be horizontal or vertical. All horizontal edges to enter pyramid. One side of the base of the pyramid to be inclined 30° with the vertical plane. Other dimensions and arrangements at

pleasure.

(1) Make an auxiliary elevation at right angles to the horizontal edges of the prism. This elevation of the prism will be a triangle.

(2) The points where the elevations of the edges of the pyramid cross this triangle will be the elevations of the points of intersection of the edges of the pyramid with the faces of the prism. Perpendiculars to the xy from these points will be loci of their plans.

(3) To find the points in which the horizontal edges enter and leave the pyramid, draw in the auxiliary elevation lines from the vertex to the base passing through the elevations of the horizontal edges. These are really lines on the faces of the pyramid passing through the points. where the horizontal edges of prism enter and leave the pyramid. Determine these lines in plan. The points where they cross the plans of the horizontal edges of the prism are the plans of the points required.

(4) The proper points must then be joined from inspection.

PROBLEM V.

Take a pyramid and prism of four and five sides in each base respectively, and find intersection. Let the pyramid be placed as in the last problem, and the prism with horizontal edges, and with two entirely free from the pyramid.

PROBLEM VI.

To determine the penetration of a prism standing on its base with a pyramid, when the base of the pyramid is inclined 0°, has no side horizontal, and is not at right angles to the vertical plane.

Draw pyramid and prism, preparatory to finding their interpenetration.

The plans of the points at which the edges of the pyramid enter the prism are shown where these edges in plan cross the plan of the prism, and can thence be determined in elevation.

The intersection of the vertical edges of the prism with the faces of the pyramid can be found, either, by determining the traces of the planes of those faces and finding their intersection with the vertical edges passing through them, or, which is more simple, by drawing lines from the plan of the vertex of the pyramid through the plans of the edges of the prism to the base of the pyramid, and showing these lines, which are really lines on the surfaces of the faces of the pyramid, in elevation. The intersections of the elevations of these lines, with the ele