PROBLEM VII.

Arrange a cone placed vertically, and a sphere, and find intersection. Let centre of sphere and axis of cone be free from one another.

PROBLEM VIII.

To determine the intersation of two cylinders, the axes of which are not parallel to either plane of projection.

(1) Draw two lines making any angle, say 60°, with each other. Assume an ellipse having its major axis on one of them as the horizontal trace of one cylinder. Complete 'the plan of this cylinder.

(2) To determine its elevation. Make a vertical section on the major axis to obtain the portion of the axis of the cylinder. This is done by describing a semicircle on the major axis, and setting off from one extremity as a chord the length of the minor. A line at right angles to this through the centre of the ellipse is the axis of the cylinder.

(3) Arrange a second cylinder similarly on the second line of the angle 60°, so that its convex surface may penetrate that of the former.

(4) To find their intersection. Assume a point in plan and elevation, and from it draw two lines parallel to the axes of the cylinders. Determine the horizontal traces

of these lines, and join them. The line thus drawn will be the horizontal trace of a plane containing these two lines.

(5) Points in the intersection of the cylinders must then be found by assuming auxiliary planes, cutting the bases of the cylinders and parallel to the plane before found, that is, with their horizontal traces parallel to the horizontal trace of this plane. As each of these cutting planes will contain lines on the surfaces of the cylinders springing from the points where it crosses their bases, and as these lines are all in the same plane, the points in which they actually cut one another will be points in the intersection required.

Any number of points can be found, by assuming a corresponding number of auxiliary cutting planes. The projections of these points must then be joined by hand to get the required curve of intersection.

axes.

PROBLEM IX.

To determine the intersection of two cones with oblique

Assume two intersecting cones by the converse of Problem 5, Chapter VII.

Planes assumed passing through the vertices of the two cones and cutting their surfaces, will do so in straight lines, the intersections of which lines will give points in the intersection of the two cones.

Draw a line through the vertices of the two cones and find its horizontal trace. The horizontal traces of the cutting planes must pass through this point and cross the two bases. The rest of the work will be similar to that of the last problem.

XII.

SPHERICAL TRIANGLES.

CASE I.

Given the three sides a, b and c; to determine the three angles A, B and C.

Let the three sides be developed on the horizontal plane of one of them, viz., b.

Take a plane of elevation xy at right angles to Ol, the common edge of the sides c and b; and let it meet the second edge of side c in point P2, and cut Ol in e and Om in f.

Make OP, on the second edge of a equal to OP; since OP, and OP, are really the same line, viz., the common edge of sides c and a.

Then eP, ef, fP, are the sections of the three sides by the plane xy.

Construct this section epf, in this plane; taking the centre e and radius eP, and centre ƒ and radius ƒP1, and describing arcs intersecting in p: join p'e and pf. Then p, the plan of P, is determined in xy from p, and Op is the plan of the edge OP on the plane of b. Angle pef is the angle A. The plane through at right angles to Om gives the angle B. And the third angle C is determined by a plane through P, in space, at right angles to the third edge OP, in space, by Problem 15, Chap. I. Im is the horizontal trace of this plane, which is that of the profile angle of C.

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CASE II.

Given the sides a, b, and the angle C: to determine A, B and c.

Let the side a be developed on the horizontal plane of b: and since the angle C between these sides is known, a plane xy at right angles to their intersection Om exhibits this angle pfp'; f being the vertical trace of the side a, the length f being made equal to ƒP,, p in xy determined from p', gives Op, which produced is the indefinite plan of the third edge OP on the plane of b. A plane through at right angles to the edge Ol gives the angle A, and enables us to 'construct' the third side c about Ol its horizontal trace. A plane through P in space, and at right angles to the edge OP in space, and having lm for its horizontal trace, will determine the profile angle of B.