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*8. Determine the intersection between the planes VOH and VO,H, and also the angle between them. Write the latter down in degrees.

(10.) *9. f is the plan of a point F, lying in the given plane RST. Draw the projections of a line, lying in the plane, passing through F, and inclined at 20° to the vertical plane of projection.

(14.) *10. The projections are given of a ladder, leaning against a wall. Draw an elevation of the ladder and wall on the given xy line.

(12.) 11. Draw the plan of a regular octohedron of 2" edge, lying on a face. Show a horizontal line drawn along two adjacent faces at a height of 11′′ above the horizontal plane. (12.) *12. The parallelogram N and the circle M are the plans of two cylinders or discs, exactly alike, one resting on the other. Show the section of the two solids made by a vertical plane, of which AB is the horizontal trace.

(10.) 13. A right prism is 4" long, and its base is a regular pentagon of 1" side. A hole 1" in diameter is bored through its centre, parallel to its long edges. The prism rests with one edge of the base on the horizontal plane, the long face containing that edge being inclined at 22° to the horizontal plane.

Show the real shape of the section of the prism made by a horizontal plane, 2′′ above the horizontal plane of projection. (If the hole be omitted, 6 marks will be deducted.)


Advanced Stage.


Read the General Instructions on page 1.

Only eight questions are to be attempted.

Plane Geometry.

*21. C is one vertex of a triangle; Q is the centre of the circumscribed circle; O the centre of the inscribed circle. Draw

the triangle.



22. Draw an indefinite line BC. At any point in it, N, draw a line perpendicular to BC, and set off from N on the perpendicular, above and below BC, lengths NP, NP2, each equal to 2.4 inches. BC is the axis of a parabola, and P, P2 are points on the curve. Calling A the vertex of the parabola, draw the curve such, that the area bounded by the double ordinate PP, and the portion of the curve PAP2 shall be 4 square inches in area. (20.) *23. Two equal elliptic wheels, A and B, are in contact at 0. Their foci are those of A, F1 and F2; those of B, f1 and f2. The wheel A is driven by the wheel B; and the wheels rotate round fixed axes at the foci F1 and f1 respectively. A pin is attached to A at its focus F2. Draw a diagram representing the vertical heights of the pin above the line CD during a complete revolution of the wheels, the pin at the commencement of the revolution heing at the point F2 on the diagram. Take for abscissae inch to represent 1th of a complete revolution; and for ordinates the actual heights of the pin corresponding, above CD. The position should be shown for, at least, everyth of a complete revolution. (22.)


Solid Geometry.

*24. ab, a'b' are the projections of a line AB: l'h, hg are the traces of a plane. Draw the projections of a line in the plane, meeting AB, and making with it an angle of 35°.

(24.) 25. A cube, inscribed in a sphere of 11⁄2 inches radius, has two adjacent faces inclined at 30° and 70°, respectively, to the horizontal plane. Draw the solids in plan and elevation.


*26. Draw an equilateral triangle abc (see diagram, which is not drawn to scale) of 4.2 inches side, and set off lengths a▲, aB, cF, &c., 1 inch distant from each vertex, along the sides. The figure ABCDEF, so formed, is a horizontal section of a regular octohedron, lying with one face on the horizontal plane. Draw the octohedron, showing on it the outline of the section in plan, and its height above the horizontal plane in elevation.

(26.) *27. A and B are the scales of slope of two planes. Draw the plan of a sphere of 1 inch radius, resting on the horizontal plane, and touching the two planes, both of which pass over the sphere. Show the points of contact with the planes, and write down their figured heights. Unit =0.1 inch.


*28. aa', bb', cc' are the projections of three points A, B, and C. Find the projections of a fourth point D, distant 24 inches from each of the points A, B, and C. Complete the tetrahedron formed by joining the four points. (22.)

29. The vertex of a cone is 1.5 inches above the horizontal plane, and its axis is inclined at 45°. Its generating lines make an angle of 30° with the axis. Determine the scale of slope of a plane tangent to the cone, and inclined at 60°. Show the line of contact on the cone in plan. = 0.1 inch.

Unit (24.)

*30. The diagram represents a cone ABV, lying on a block DEFG whose thickness is DD2. Draw on the plan the outline of shadow thrown by the solids, one on the other, and on the horizontal plane of projection. Show also on the plan the limit of light and shade on the cone. The arrows indicate

the direction of the parallel rays of light, inclined at 45° to the xy line in plan and elevation.


*31. Draw the solids of Question 30 in isometric projection. The vertical isometric planes to be taken parallel to the planes DE and DG, and the vertical through D nearest to the observer. An isometric scale must be employed. (24.)

Graphic Statics.

Alternative and Optional.

32. A right truncated prism has for base an equilateral triangle of 2 inches side. The three edges perpendicular to the base are, respectively, 2 inches, 1.75 inches, and 1.25 inches in length. Find, by graphic construction, a line representing the cubic contents of the solid, to a unit of 1 inch. (20.) *33. ABCD is a horizontal rigid bar, hinged at A, loaded at B with 40 lbs., at C with 50 lbs., and retained in place by a cord DE (passing over a pulley) attached to a pin at D. Find the stress in the cord DE. Employ the funicular polygon, using, for the scale of loads, inch to represent 10 lbs.




Read the General Instructions on page 1.

Only six questions are to be answered.

Plane Geometry.

*51. A and B are two points on a coastline BAC. A ship sailing at the rate of 10 miles an hour leaves A. Its course is a straight line lying within the angle BAC. Another ship sailing 20 miles an hour leaves Bat the same moment. Draw the course which it must take, in order to meet the first ship: (1) when the ship leaving A is known to be making for a point D; (2) when the direction which the ship leaving A may take is unknown. Scale, 50 miles to 1 inch.

(70.) *52. The lines OA, OB are tangents to an ellipse, of which F is one of the foci. If the minor axis be 24′′ long draw the ellipse.

Solid Geometry.


*53. Find a point on the given line a5 b25, such that the line drawn from it to the given point P20, equal to the perpendicular let fall from it on the given line c10 d. The given lines may be produced if necessary. height, 1".

Unit of


*54. The elevation of a rectangular right prism is given. Draw the true shape of a horizontal section through the point of which p' is the elevation.


55. Draw a square of 21′′ side and letter its angles in succession, like figures on the face of a clock, a, b, c, and d. Let this square represent, in plan, the top of a mound standing on ground which is a plane surface. Let the levels of the corners a, b, and c of the top of the mound be respectively 108, 120, and 135. The plan of the 110 contour of the ground, passes through the points b and d, and the 120 contour of the ground through c. Assuming the slope of each side to be 50°, draw the plan of the mound, which will be a truncated pyramid. Contours to be shown at 5 units vertical interval. Unit of height height='1". (65.) 56. The base of a right cone is 3" in diameter, and its height is 4". Draw its projections, when it is in such a position that rds of its curved surface can be seen in plan. (60.)

*57. The projections of three lines are given. These are the axes of three right cones, the vertices being the points, vv′, v1ví', and v2 v2, while the generating angle in each case is 20°. Draw the plan and the elevation (on the given xy line) of the solids, showing their interpenetration. (65.) *58. The elevation b' of a circular bath is given, half filled with water (surface s). P (elevation, p') is an aperture, assumed as being a point, through which sunlight enters into the room, the line joining p, (the plan of P) with the plan of the centre of the bath making an angle of 60° with the face of the wall W. Assuming that the angle of incidence of the rays is equal to that of their reflection, draw an elevation showing the true shape of the reflection of the sunlight off the water on the wall W.

(65.) *59. The diagram shows the perspective drawings of two rectangular huts, standing on level ground, vv being two vanishing points. Assuming the height of one of the doorways to be 6 feet, draw orthographically the plan, front and end elevations of one of the huts, to a scale of. Mark the principal dimensions.

(If the paper be crowded, construction lines may be rubbed out, or drawn through other figures.) (70.)

Graphic Statics.

The weight

*60. AB is the axle of a waggon, CD the wheels. (1,200 lbs.) of the waggon and its contents, is distributed uniformly along the length MN. The weight of the axle. need not be considered. Show graphically, the bending moments and shearing stresses on the axle, when, in helping to unload the waggon, a man rests on each end of the axle, i.e., an extra weight of 150 lbs. is applied at A and also at B. Write down the bending moments (in foot-pounds) and the shearing stresses (in pounds) at the middle point (P) of the axle and at a point l' from it.

Scale for length,


{forces, 600 lbs. = 1".


61. Ascertain graphically, and mark distinctly, the centre of gravity of a crescent formed by two circular arcs, the shorter one being of 24 radius, the longer of 21". The crescent is 2" wide at its broadest part, i.e., between the two middle points of the arcs.

Explain briefly the means adopted in finding the centre of gravity.


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