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32. Given two lengths, AB = 2.25", CD = 2.70". Find a line whose length x is such that AB2 = (x - CD) × x. Write down the value of x.

(18.)

33. Draw two lines, AB, AC, making an angle of 30° with each other at A. On AC set off a point P, so that AP=3′′. AB is the axis of a parabola; AC is a tangent to the curve at the point P. Find the focus and directrix and draw the curve. (22.)

Solid Geometry.

*34. e'a, ad are the traces of a plane; fa is the plan of a line and aa' its horizontal trace. Draw the elevation of the line so that it shall meet the given plane at a point C such, that the real distance from A to C shall be 14 How many solutions can be found?

(20.)

*35. Draw a line through the point c to meet the line ab at a point d such, that the real angle cda shall be equal to 50°. Unit = 0·1".

(24.)

*36. Draw the plan of a sphere such that the line ab is tangent to it, and that the centre of the sphere is on the line cd. Unit = 0.1".

(25.)

37. A cone, base 2.70" diameter, height 2:35", has its axis inclined at 40°. A curve is traced on the cone which, in development, would be a circle of 1" radius touching the base of the cone. Draw the plan of the cone, and of the curve traced on it, touching the base of the cone at its highest point. (25.)

*38. Draw a plane perpendicular to the plane of the triangle (abc, a'b'c') and bisecting the sides bc, b'c' and ab, a'b'. (20.) *39. A twisted surface of revolution is generated by a line ae, a'e', revolving round a vertical axis o, o'o', to which it is rigidly fixed by the horizontal line ob (in elevation b'). Draw in plan and elevation the position of the generating line when it has revolved round the axis so as to pass in plan through p. (25.)

40. A right pyramid has for base a regular pentagon of which the diagonals measure 2.5". The vertex is 2" above the base. Draw the plan and elevation of the pyramid, with its base in a plane inclined at 55° to the vertical plane and at 60° to the horizontal plane; one diagonal inclined at 30°, and one end of that diagonal in the vertical plane.

(25.)

*41. The figure represents in elevation, in interrupted lines, a hexagonal pyramid and a sphere intersecting each other; the centre of the sphere is on the axis of the pyramid, and one side BC of the base of the pyramid is parallel to the xy line. Draw the plan and elevation of the solids complete, find their intersection, and draw the shadows on the solids; also the shadows cast by them on the horizontal plane. The rays of light are in the direction indicated by arrows. Shade in the portions of the solids in shadow, as well as the shadow thrown by them on the horizontal plane. (28.) [NOTE.-The visible portions of the edges of the pyramid and of the outline of the sphere, as well as the intersections of the solids, should be carefully drawn in full lines on the elevation.]

*42. Draw the isometric projection of the solids of the last question, showing their intersection. Use the appended scale.

(26.)

*43. Reduce the figure to an equivalent triangle, its base on the line AH and its vertex at E; and find the length of a line representing the area of the figure, taking for unit one inch. (20.)

*44. Find (and write down) the moment in foot-tons of the resultant of the pairs of parallel forces, A and B, C, and D, with regard to the point E. Scale of forces, 0.25" per ton; scale of distances, 0.1" per foot.

(22.)

Honours Examination.

INSTRUCTIONS.

Read the General Instructions on page 1.

Only eight questions are to be attempted.

Plane Geometry.

51. Draw a rectangle, sides 3′′ and 1". Suppose this rectangle to represent a garden. It is required to trace round the garden a path of uniform width, leaving in the centre a rectangular plot one half the area of the garden. Draw this plot.

(40.)

52. A circle of 21" diameter rolls on a fixed circle of 13′′ diameter, the smaller circle being inside the larger one. Draw the curve traced by a point moving with the larger circle and 13" from its centre, during one complete revolution. (40.)

$53. The given figure represents an arrangement of jointed bars forming "Lazy tongs." Assuming the point o to remain fixed and the arms od, oe to be brought into coincidence on ox, moving with a uniform angular velocity :

a. Draw the curve traced by the point p.

b. Draw a curve representing the velocity with which
the point q moves.
(45.)

Solid Geometry.

*54. Determine a vertical plane cutting the given parallel planes in lines 11" apart. (40.)

*55. Through the given point pp' draw a line inclined at 20° and making 65° with the given plane. (40.)

*56. ab is the plan of an edge (2" long) of an octohedron. The plan of an adjacent edge falls on ac. Complete the plan of the octohedron. Unit (figured heights)=0.1". (45.)

*57. Three lines intersect at an inaccessible point. The plans of two lines ab, cd are given and also the plan p of a point on the third line. Describe a sphere in the pyramid of which the lines are edges. Unit = 0·1".

(50.)

$58. A rod ab rotates round a fixed vertical axis xx. The angular velocity diminishes uniformly and becomes zero after two revolutions; the angle 0, which the rod makes with the axis uniformly diminishes and also vanishes after two revolutions. Trace in plan and elevation the locus of the point a. (50.)

59. A right cylinder (diameter 1") touches the two given spheres whose centres are a and b. The axis of the cylinder passes through the given point p. Determine the position of the cylinder. Unit (figured heights) = 0·1′′.

(45.)

*60. The elevation of a right cylinder, whose axis is parallel to the vertical plane of projection, is given. The cylinder penetrates the given sphere and also touches its surface internally. Draw in plan and elevation the complete curve of intersection. (50.)

*61. The plan of an anchor ring resting on the horizontal plane
is given. A is a vertical right cylinder (height 1·6′′)
standing on the horizontal plane and closely fitting the
ring. Determine the shadow cast on the horizontal
plane and the unilluminated portion of the anchor ring,
taking the point v as the source of light. Unit (figured
heights) 0.1".
(50.)
*62. The plan and elevation of an obelisk are given. A photo-
graph of this obelisk is taken on a vertical plane of which
pp is the ground line. The lens is so placed that all the
rays pass through the given point l'. Draw the resulting
photograph.
(50.)

= =

*63. The plan and an end elevation of an open box are given lg is the ground line of the plane of a vertical looking glass. Taking the point v as the position of the eye, draw on the transfe picture plane pp a view of the box as it would appear in the looking glass. The angles of incidence and reflection of rays may be assumed equal. Unit = 0.1". (55.) [N.B.-In order to bring the figure within a small the eye has been placed in such a position that the box would interfere with part of the reflection. This interference is to be neglected and the complete reflection shown.]

compass,

Graphic Statics.

*64. Determine by means of a polygon of forces and a funicular polygon, the magnitude and direction of the resultant of the six given forces pi Pε (Pi = 165 lb.; P2 = 355 lb.; P3 = 410 lb.; p. 323 lb. ; p1 = 162 lb. ; Ps = 205 lb.) using a scale of 1" 100 lb.).

(50.)

*65. A roof truss of given form is loaded on one side ab only
with a total uniformly distributed normal pressure of
2 tons due to wind. The foot b of the truss is fixed;
c is only supported. Determine and write down the
stresses in the respective members of the truss, distin-
guishing tension and pressure.
(55.)

SUBJECT II. MACHINE CONSTRUCTION AND
DRAWING.

EXAMINERS, PROF. T. A. HEARSON, M. INST. C.E., AND
J. HARRISON, Esq., M. INST. M.E., Assoc. M. INST. C.E.

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GENERAL INSTRUCTIONS.

If the rules are not attended to, the paper will be cancelled.

You may take the Elementary, or the Advanced, or the Honours paper, but you must confine yourself to one of them.

Put the number of the question before your answer.

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