in backing off the top face of the rafter to bring it into the planes or plane of the roof surface.

Fig. 479 shows in detail a plan (1), a side elevation (2) and a front elevation (3) of the upper part of a jack rafter where it joins the hip rafter. All the rafters have their side faces vertical, and X,Y, is the plan of the vertical face of the hip rafter to which the jack rafter is

attached. This end of the jack rafter requires to be bevelled in two directions in order to fit against the face of the hip rafter. The angles for these bevels are marked a and B. The angle B is shown on the side elevation (2). To determine a the upper surface of the rafter is brought into a horizontal position by turning it about a horizontal axis through B at right angles to XY, as shown in the plan (1) and elevation (2). The true form of the end of the rafter is shown at (4).

Exercises XVII

1. The solid whose projections are given in Fig. 480 is cut in two by a vertical plane whose horizontal trace is HT. Draw an elevation of the part to the left of the plane of section on a ground line parallel to HT.

2. The solid whose projections are given in Fig. 481 is cut in two by a vertical plane whose horizontal trace is PQ. Draw an elevation of the larger portion on a plane parallel to the plane of section.

3. The solids whose projections are given in Fig. 482 are cut by a vertical plane

whose horizontal trace is RS. Draw an elevation of the portions whose plans lie between RS and XY on a ground line parallel to RS.

4. The plan of a right square prism with a rectangular hole through it is given in Fig. 483. A side of the square base is on the horizontal plane. Draw the

In reproducing the above diagrams take the small squares as of 03 inch side.

elevation on XY and show on this elevation the section of the solid by the vertical plane of which HT is the horizontal trace. Determine also the true form of the section.

5. A right square prism, side of base 1 inch, altitude 3 inches, is cut into two equal parts by a plane. The true form of the section is a rhombus of 1.5 inches side. Draw a plan of one part when it stands with its section face on the horizontal plane.

6. A pyramid has for its base a square of 2 inches side which is on the horizontal plane. The altitude of the pyramid is 2.5 inches and the plan of the vertex is at the middle point of one side of the plan of the base. This pyramid is cut into two portions by a horizontal plane which is 1 inch above the base. Draw the plan of the lower portion.

7. A right hexagonal pyramid, side of base 1 inch, altitude 2 inches is cut by a plane which contains one edge of the base and is perpendicular to the opposite Determine the true form of the

face. section.

8. The plan and elevation of one of the rectangular faces ABCD of a right hexagonal prism are given in Fig. 484, AB and CD being on the ends of the prism. Complete the plan and elevation of the prism and show on the elevation the section by the vertical plane whose horizontal trace is PQ.

9. The plan and elevation of one of the triangular faces VAB of a right square pyramid are given in Fig. 485, V being the vertex of the pyramid. Complete the plan and elevation of the pyramid and show on the elevation the section by the vertical plane whose horizontal trace is RS.

a' b'

10. A horizontal moulding has (e), Fig. 486, for its cross-section. Determine the true form of a vertical section which is inclined at 45° to the longitudinal lines of the moulding.

11. A moulding whose cross-section is given at (f) in Fig. 486 is placed with its back on the vertical plane of projection and its longitudinal lines inclined at 30° to the horizontal plane. Draw the plan and elevation and show the true form of a section by a vertical plane whose horizontal trace is inclined at 60° to XY.

12. Two mouldings of the same cross-section (c), Fig. 486, are fixed to the face

of a vertical wall. One moulding is horizontal and the other rakes at 30° to the horizontal. The two mouldings join correctly together. Determine the true form of the joint section.

In reproducing the above sections take the small squares as of inch side.

13. A horizontal moulding mitres correctly with a raking moulding. Inclination of raking moulding to the horizontal, 25°. The backs of the mouldings are in vertical planes which are at right angles to one another as shown in plan in Fig. 487. Select from Fig. 486 a crosssection for the horizontal moulding and then determine the cross-section of the raking moulding.

14. Same as exercise 13 except that the angle between the vertical planes is 120° instead of 90°.

15. Two raking mouldings on vertical walls which are at right angles to one another as in Fig. 475, p. 244, mitre correctly. Both mouldings are inclined at 20° to the horizontal. Select a cross-section for one of the mouldings from Fig. 486 and then determine the cross-section of the other.

FIG. 487.

18. Same as exercise 15 except that the angle between the walls is 120° instead of 90°.

17. A, C, and E (Fig. 488) are three pieces forming part of a timber frame, each piece being of rectangular cross-section. The projections a' and c are incomplete. Complete the projections indicated and draw an elevation on a ground line inclined at 45° to XY.

18. A piece of timber having plane faces is shown in plan and elevation in Fig. 489. Draw two other elevations, one on a ground line perpendicular to XY, and the other on a ground line parallel to the longer sides of the plan. Determine the true form of the cross-section of the piece.

In reproducing the above diagrams take the small squares as of half-inch side.

19. Fig. 490 shows the plan of a piece of timber A, of uniform square crosssection, which lies between the faces RT and ST of two vertical walls. The longitudinal edges of A are inclined at 20° to the horizontal, rising from R to S. B is another piece of timber of uniform square cross-section. The longitudinal edges of B are horizontal. The piece B fits into a notch on A, the greatest vertical depth of this notch being equal to half the thickness of A. Draw two elevations of A and B, one on a ground line parallel to rs, and the other on a ground line parallel to rt. Determine all the bevels for the ends and the notch of the piece A.

CHAPTER XVIII

THE SPHERE, CYLINDER, AND CONE

212. Surfaces and Solids.-When a surface encloses a space and that space is filled with solid matter the whole is called a solid, and when the surfaces and solids have certain definite shapes they have certain definite names. But in speaking of certain solids and their surfaces the same name is frequently used to denote the solid and its surface. For example, in speaking of the sphere, the cylinder, and the cone the solids bearing these names may be referred to or it may be that it is their surfaces that are referred to, although the words "solid" or "surface" may not be used. In general the omission of the words "surface of" when a solid is mentioned by name will not lead to any confusion when it is the surface that is referred to. For example, by "the curve of intersection of two cylinders" is obviously meant "the curve of intersection of the surfaces of two cylinders."

213. The Sphere.-The sphere may be generated by the revolution of a semicircle about its diameter which remains fixed. The surface of a sphere is also the locus of a point in space which is at a constant distance from a fixed point called the centre of the sphere. The constant distance of the surface of the sphere from its centre is the radius of the sphere. A straight line through the centre of the sphere and terminated by the surface is a diameter of the sphere.

The orthographic projection of a sphere is always a circle of the same diameter as the sphere.

214. Plane Sections of a Sphere.-All plane sections of a sphere are circles. When the plane of section contains the centre of the sphere the section is a great circle of the sphere. All sections of the sphere by planes which do not contain its centre are called small circles.

A portion of a sphere lying between two parallel planes is called a zone, and a portion lying between two planes containing the same diameter is called a lune. See Fig. 491.

The determination of a plane section of a sphere is illustrated by Figs. 492 and 493. In Fig. 492 the plane of section is perpendicular to the vertical plane of projection and has V.T. for its vertical trace. In Fig. 493 the plane of section is inclined to both planes of projection and has V.T. for its vertical trace and H.T. for its horizontal trace. In both cases points on the projections of the section may be found

as follows. PQ parallel to XY is taken as the vertical trace of a horizontal plane. This plane cuts the sphere in a circle whose plan is a circle concentric with the plan of the sphere and whose diameter is equal to the portion of PQ within the circle which is the elevation of the sphere. This same plane cuts the given plane of section in a straight line. The points rr in which the plan of this line intersects the plan of the circle are the plans of points on the required intersection, and r'r' the elevations of these points are on PQ. In like manner by taking other positions for PQ any number of points on the projections of the required intersection may be found.

Since the required intersection is a circle and the projections of a circle are ellipses the axes of these ellipses may be found and the ellipses then be drawn by the trammel method (Art. 45, p. 41).

Referring to Fig. 492 a'a' the intercept of V.T. on the elevation of

the sphere is the elevation of the circle which is the section of the sphere by the given plane. Projectors from a' and a' to meet a parallel to XY through o the plan of the centre of the sphere determines aa the minor axis of the ellipse which is the plan of the required section. The major axis bisects the minor axis at right angles and has a length equal to a'a'.

Referring to Fig. 493, draw aab through o at right angles to H.T. aab is the horizontal trace of a vertical plane which intersects the sphere in a great circle and the given plane of section in a straight line, and AA the portion of this line within the great circle has for its plan the minor axis of the ellipse which is the plan of the required section. Let this vertical plane with the line AAB and the great circle in it be turned round about a vertical axis through O until it is parallel to the vertical plane of projection. A,A,B, is the elevation of the line AAB in its new position and the elevation of the sphere is the