, to the co-ordinate planes be required, proceed to treat the line as in Prob. 85, or by the cone method of Chapter XV. It is done, as shown in dotted lines, in connection with Case II., by the latter method. It should be observed that in Case II., if we assume any v.t., v'o', of one of the planes, we may not afterwards draw any line o'h indiscriminately for the h.t., as the two traces may not meet in xy when produced. We may, however, assume any point h, and find the correct h.t. by Prob. 52.

2nd. When only one pair of traces intersect. Cases III., IV., V. and VI. Fig. 210.

The intersection in each case is ab, a'b'.

Case III. Two planes perpendicular to the H.P., and inclined to the V.P.

The intersection is a vertical line. The v.t.'s are supposed to meet beyond b', at an infinite distance off.

Case IV. Two inclined planes. The intersection is a horizontal line.

Case V. An oblique plane v.t., h.t., and a horizontal plane v.t. The intersection is a horizontal line.

Case VI. Two oblique planes with their horizontal traces parallel. The intersection is a horizontal line.

The third kind-when the respective traces do not meet within the limits of the paper-cannot be here treated.

PR. GEO.

13

Exercises.

1. The v.t. of an oblique plane makes 45° with xy; draw the h.t., when the real angle between the traces is 522.

Hint.-See the note to Prob. 128.

Ans. h.t. at 30° to xy.

2. In Exercise 1, ascertain in one figure the angles of inclination of the plane to the co-ordinate planes.

Hint.-Combine Probs. 129 and 130.

Ans. 0 = 63; $:

= 39°.

3. The v.t. of a plane is parallel to and 2" from xy; determine the h.t. when the plane is inclined to the H.P. at 53°.

4. The common h.t. of two oblique planes makes 30° with xy. If each plane be inclined to the V.P. at 63°, draw the vertical traces.

ht2 (6)

(i)

(e)

(f)

75" from each of the given planes (a), (b), (c), (d), (e) and (ƒ).

6. Determine the intersection between the given planes in each of the cases (g), (h), (i), and (j), and write down the inclinations of the intersection to the co-ordinate planes in each case.

Hint.-In case (j) get a point s in the intersection, and join s, s' to the point where all the traces meet.

7. An irregular tetrahedron stands with its base in the H.P., and one face in the V.P. Its base is an equilateral triangle of 2" side, and the face in the V.P. is also an equilateral triangle of 2" side. Determine the angles which either of the inclined faces of the solid makes with the planes of projection, the angle between its two edges which are in the planes of projection, and the true length and inclinations to the co-ordinate planes of the intersection line between the two inclined faces.

Hint. The projections of the solid are two equilateral triangles, having a side coinciding in xy, and a line joining their vertices. Figs. 201 and 209 will assist the student in conceiving the form of the solid. Taking two of the edges as traces of a plane, proceed as in Probs. 127, 129, 130. The line joining the vertices is plan and elevation of the intersection of the two oblique planes, which is to be treated as in fig. 207.

8. Make a pictorial view to scale of the solution to Exercise 7.

9. Represent by their traces :-a. Two planes at right angles to each other and to the vertical plane of projection, and one of them inclined at 40° to the horizontal plane. b. Two parallel planes not at right angles to either plane of projection. (1886)

10. The vertical trace of a plane makes an angle of 48° with the xy line. The plane is inclined at 60° to the horizontal plane. Draw the horizontal trace.

Hint.-See Prob. 132.

(1896) Ans. h.t. at 40° to xy.

11. Determine the angles which the given plane (c'a, ab) makes with the vertical and horizontal planes of projection. (1893) N.B. For the figure, see Chapter XIX., Q. 11.

12. Draw two parallel planes, inclined at 52° to the horizontal plane. Their horizontal traces make an angle of 47° with the xy line, and the planes are " apart, the distance measured perpendicularly to their surfaces. (June 1900)

Hint.-Draw one h.t. at 47° to xy, and from the note to Prob. 132 complete the plane. Draw the second plane by Prob. 134.

13. Find the intersection of two planes whose horizontal traces are parallel and 1" apart, and which are inclined in the same direction at 25° and 50° respectively. (1881)

Hint.-Draw the h.t.'s parallel, 1" apart, and [by preference] perpendicular to xy. Draw the v.t.'s at 25° and 50° respectively to xy as inclined planes. Complete as in fig. 210, Case IV.

14. Draw the traces of two planes not at right angles to either plane of projection, and determine the angle which the intersection of these two planes makes with the vertical plane of projection.

(1889)

Hint.-See fig. 207, Case II.

CHAPTER XVIII.

OBLIQUE PLANES AND LINES.

Rules on Planes and Lines.

26. If a line be contained by a given fixed plane, its h.t. is in the h.t. of the plane, and its v.t. is in the v.t. of the plane.

If the line be horizontal, its plan will be parallel to the h.t. of the plane; if it be parallel to the V.P., its elevation will be parallel to the v.t. of the plane; in each case the trace of the plane and the corresponding projection of the line will be supposed to meet at infinity. The inclination of a line in a plane varies between zero and 0, where angle of inclination of the plane. Compare

=

figs. 200, 202, and 210.

27. If two lines be parallel, and a plane contain one, the remaining line is parallel to, or contained by, the plane.

PROBLEM 136.

Having given the plan, p, of a point, P, in a plane v'o'h, to determine the elevation of the point.

Fig. 211.

b

Through p draw any line ab, terminating in the h.t. of the plane and xy, at a and b respectively, and regard ab as the plan of a line AB in the given plane. The elevation a' is in xy, and b' is in the v.t. of the plane (Rule 26).

Join a'b'. The required elevation p' must be in a'b', and is determined by now projecting up from the plan. If the elevation p' were given, and not the plan, the same or similar construction lines would be drawn to determine the latter.

Such a remark will apply to the succeeding problems where the changes may be rung. It is often convenient to assume the line AB parallel to one of the planes of projection, as will be seen in Prob. 138.

PROBLEM 137.

Having given the projections, p,p', of a point, and the horizontal trace, oh, of a plane containing it, to determine the vertical trace of the plane.

Assume the plan ab of any line in the plane passing through p x and meeting the h.t. in a.

Find

a' in xy, join a'p', and ascertain its v.t., b'. The v.t. required is b'o'.

This construction is also based on Rule 26. If the traces do not meet on the paper assume a second line, cd, c'd' (shown dotted), passing through P, and contained by the

Fig. 212.

plane. Join its v.t., c', to b', for the required v.t. of the plane.

PROBLEM 138.

To determine a point, 1, in the given plane, v'o'h, when it is

Imagine a horizontal line, AB, "high, in the plane. Imagine also a line, CD, in the plane 1" from the V.P The intersection of the lines AB, CD, is the point required. First draw a'b',

above xy to meet v'o' in a'. The plan of A is a in xy. Draw ab parallel to o'h. Next draw cd 1"

below xy to meet o'h in c. The elevation of c is c' in xy. Draw c'd' parallel to v'o'. Where the plans ab, cd intersect in i is the plan, and the intersection of the elevations a'b'

c'd' is the elevation of the point required.