Very well answered in regard to the first two parts. The remarks under Q.1 apply to the last part of the question. The tracing paper method of finding the length of the arc was often successfully used, and with due care gave the most accurate result.

Q. 22. The figure is the plan of a field in which there is a pond P; scale inch to 10 yards. Find the total area of the field (including the pond), and the area of the pond in square yards. A favourite question and very well answered. The measurements were usually made in decimals. A good proportion showed a fair knowledge of some method of obtaining the area of the curved figure.

Q. 23. In arranging some elementary experiments in statics, cyclists' trouser clips are used as spring balances. In order to be able to measure pulls, a series of weights are hung on a clip, and the corresponding openings AA are measured. The results are as tabulated.

Pull-P ounces

0 4

8 12 16 20 24 28 32

Opening AA=x inches

0 0.51 1.15 1.83 2.53 3.24 3.89 4.44 4.96

Plot a curve showing the relation between P and x, the scale for P being " to 1 ounce. Use this curve to graduate a decimal scale of ounces, which being applied to the spring at AA shall measure any pull up to 32 ounces.

A fair number plotted the curve and did so well, but very few understood how to obtain the required scale.

Q. 24. The figure is the plan of a corner of a landing, scale 1" to 1'. Sis a stone on which a stove is to rest. Show the corner of the landing carpet folded over along the line LL. On this fold draw the hole which must be cut in the carpet, allowing a margin of 13" all round for turning under, so that when the carpet is turned back into position, the hole shall just fit the stone S.

The candidates were well able to answer this question, but sometimes the drawing was a little inaccurate.

Q. 25. Four pieces of sheet material are hinged together at A, B, C, D, these points forming the corners of a jointed parallelogram. One of the pieces is pinned or hinged to the drawing board at P, and the hinge point B is moved in a straight line from L to M. Find the locus of Q, that is the path that would be traced by a pencil moving with Q.

Frequently attempted and fairly well done. Owing to slight inaccuracies in working, a large number failed to obtain a straight line locus, and very few appeared to recognize the special properties of the mechanism.

Q. 26. A and B are two points in a body having plane motion. At a certain instant the positions of A and B are given by the vectors

O being a fixed point of reference, and angles being measured anti-clockwise from a line drawn along the tee-square to the right (eastwards). Plot the points O, A, B, to a scale of " to O'I'. What is the position of B relatively to A?

At the instant under consideration the velocity of A is 515 feet per second, and the angular speed of the body 10 radians per second. To a scale of inch to 1 foot per second, draw a vector triangle, or velocity image, showing the velocity of A, the velocity of B relatively to A, and the velocity of B. Read off the velocity of B.

Seldom attempted, and good answers were very rare indeed.

Q. 27. If my, my, my are the masses of three bodies in a plane, and A1, A2, A3, three vectors defining the positions of their centres of mass G1, G2, G3, then the position of the centre of mass G of the three bodies is given by the vector equation

Ոջ = 10, My = 2'5.

Determine G and measure a and a.

Rarely attempted. The plotting of the triangle was fairly satisfactory, but the vector equation was not well understood. A few obtained the answers by the vector method suggested, but the majority employed other and less direct methods.

Q. 28. A cardboard lever is pinned or hinged to a board at the point Forces P and Q are applied to the lever as shown.

The linear scale of the drawing being, and the magnitude of P being 185 lbs., what is the moment of P about 0 Find the magnitude of Q if the lever is balanced.

Show a force which, being applied to the lever near O, shall relieve the pin from all pressure, and allow of its being removed without disturbing the equilibrium.

A fair number of candidates gave good answers. Some failed to express the moment of P in proper units. In other cases an incorrect balancing force was shown which did not pass through O, or did not pass through the inter-section of P and Q.

Q. 29. A thin metal plate ABC resting on the ground is shown in plan. A piece of thin wire AD, 2.5" long, with one end soldered to the plate at A is also shown in plan at Ad.

(a) Find the distance of the end D of the wire from the plate, and the angle between wire and plate.

(b) Draw the elevation of the wire and plate on xy.

(c) From the elevation project a new plan on x11.

Generally well done. A considerable number, however, were unable to project the new plan properly.

Q. 30. The roof of a house is rectangular in plan, and two adjacent surfaces are inclined to the horizontal at 30° and 40° respectively. Find the inclination to the horizontal of the "hip," or line in which the two surfaces meet.

What is the magnitude of the dihedral angle between the surfaces?

Answered by about 35 per cent. of the candidates, with, for the most part, satisfactory results.

Q. 31. The figure represents a hanging lamp shade and a tilted mirror. Draw the two elevations of the image of the conical shade in the mirror.

N.B.-If P is any point in space, and P' its image, then the plane of the mirror bisects PP' at right angles.

Not very frequently attempted. A common mistake was to give the projection of the shade on the mirror, instead of the image as specified in the question.

Q. 32. The figure shows a funnel made of sheet metal. Draw the developments of the cylindrical and conical portions, showing the shapes of the plates from which the funnel is bent. Omit all allowances for overlap at the joints.

The favourite of the optional questions. The answers were generally right in principle, but there was great inaccuracy in plotting the length of the outer circular arc in the development of the cone. The tracing paper method is not sufficiently practised.

Q. 33. A timber joint is shown. Represent, in pictorial or metric projection, the two portions of the joint, separated from each other and placed in any suitable positions. The projection is to be similar to that employed for the cube, where lines parallel to oy and oz are drawn horizontally and vertically, and to a scale of full size, and lines parallel to or are drawn by using the 45° setsquare, and to a scale of half size.

Fairly often attempted, and very well answered. A few did not follow the instructions in regard to angles and scales, but most of the answers were very good.

STAGE 3.

Results 1st Class, 26; 2nd Class, 81; Failed, 104; Total, 211.

:

The number of candidates was 211, which, compared with 161 last year, shows the large increase of 31 per cent. The proportion of Failures was about the same as last year, but there was a higher percentage of FirstClass passes, and the work of this year was superior. Candidates in this stage have now a good knowledge of vectors, and also of fundamental problems in Descriptive Geometry, as evidenced by the satisfactory answers to Question 48.

Q. 41. Four pieces of sheet material are hinged together at A, B, C, D, these points forming the corners of a jointed parallelogram. One of the pieces is pinned or hinged to the drawing board at P, and the hinge point B is moved round the quadrant M. Trace the locus of the point Q.

A favourite question. The majority understood "quadrant" as meaning the circular arc, and so omitted the two radii. The drawing was somewhat inaccurate, and few recognised the principle of the mechanism and the nature of the locus.

Q. 42. In the figure, LL represents a horizontal beam, and the diagram gives the moment of inertia, I, of any cross section about the neutral axis. You are required to divide LL into eight parts, such that the length of each part shall be approximately proportional to the mean moment of inertia of the part. That

is

I

is to be the same for all the segments. Give the value of this common ratio, l and I being measured on the same scale. Only ten attempts. Of these, five were worthless, four were poor, and one was very creditable. The method adopted in the latter case was first to draw an curve, next to plot an integration curve, and then to use the latter in projecting the required points of division after the manner of Question 4.

I

Q. 43. The horizontal scale of the given map, contoured in feet, is six inches to the mile. Find the horizontal area inclosed by the given figure FFFF. Find also what the area measures approximately on the actual sloping surface of the ground. Not often attempted. There were some very good answers. The most frequent error was to take the slope of the arête as the average slope of the ground.

Q. 44. A and B are two points in a body having plane motion. At a certain instant the positions of A and B are given by the

O being a fixed point of reference, and angles being measured anti-clockwise from the eastward direction.

Plot the points O, A, B to a scale of 1" to 0'1'. What is the position of B relatively to A?

At the instant under consideration the velocity of A is 515 feet per second, and the angular speed of the body is 10 radians per second. Draw a vector triangle, or velocity image, showing the velocity of 4, the velocity of B relatively to A, and the velocity of B. Read off the velocity of B.

Show the point I of the body which is instantaneously at

rest.

Attempted by 25 per cent. of the candidates. There were one or two full answers, but few could proceed beyond the finding of the position B relatively to A.

Q. 45. A card is temporarily pinned to a board. Forces measuring 13 and 2.2 lbs. are applied to the card as shown. A clockwise couple of 11 lb. inches is also applied. The linear scale of the diagram being 1, find the line of action and the magnitude of the force which, acting on the card, will produce equilibrium, so that if the pins are removed the card will remain at rest.

Attempted eighty-one times. There were twenty-three absolute failures. Of the remaining answers the majority were good.

Q. 46. The figure indicates how the position of a point P in space may be defined by a vector OP. "Here O is an origin, OZ a vertical axis, ZX a reference plane, and ZN a plane containing OP. Then the vector

If A1, A2, A3 are vectors defining the positions of the vertices of a triangle in space, then the position of the centre of area G of the triangle is given by the vector equation

Draw the plan of the triangle and determine G, having given

Not a very popular question, but the answers were generally very good. The plan of the triangle was nearly always right. The determination of the centre of gravity was generally found by a method other than that suggested by the vector equation.

Q. 47. The figure shows a braced tie of a large girder. It is loaded with its own weight, and there is also a longitudinal pull of 220 tons. The weight is taken as equivalent to 11 tons acting at each joint, R1, R2 being the forces supporting the weight. Write down the magnitudes of R1, and R. Draw the force diagram or reciprocal figure for the tie, and measure the forces in the bars a, b, c, and d, distinguishing pulls from thrusts.

Adopt a force scale of inch to 10 tons.

There were 116 attempts, fairly successful. Many lost time by using a graphical construction to determine R and R; from the symmetry of the figure it should have been evident that these were each equal to half the weight of the tie.

Q. 48. A thin metal plate ABC resting on the ground is shown in plan. A piece of thin wire AD, 25" long, with one end soldered to the plate at A, is also shown in plan at Ad.

(a) Find the distance of the end D of the wire from the plate, and draw the elevation of the wire and plate on xy.

(b) Draw the plan of a sphere which touches the plate, has its centre O on AD, and which passes through D.

(c)

Represent by a scale of slope a plane which contains BC and touches the sphere.

The answers to (a) and (b) were quite satisfactory. Part (c) was badly answered. The scale of slope was often displaced. And when two points on it were correctly found, the decimal sub-divisions were omitted.

Q. 49. The figure represents a hanging lamp shade and a tilted mirror. Draw the two elevations and the plan of the image of the conical shade in the mirror. From the plan project on x'y' the elevation of the mirror, the shade, and as much of the image as would be seen within the boundary of the mirror.

N.B.-If P is any point or space, and P' its image, then the plane of the mirror bisects PP at right angles.

Fairly often attempted. As in the answers to Q. 31, the projection of the shade on the mirror was frequently confused with the required image. Q. 50. The figure represents a sheet metal pipe. Draw the plan of the section of the pipe made by the horizontal plane SS.

Obtain a development of the truncated oblique cone, showing the shape of the piece required to form the middle portion of the pipe. Omit all allowances for seams.

A favourite question, but its popularity was not justified by the results. There are many surfaces which can be approximately built up of portions of oblique cones, and the development of the oblique cone should be better understood.

Q. 51. The circular arc ABC rotates about its chord AC. Draw the elevation and the plan of the figure generated.

Determine the shadow cast on the horizontal plane by parallel rays of light, one of which R is given. Show the margin of light and shade on the surface of the solid.

Another favourite question, also badly answered. The majority of the attempts got each less than 10 per cent. of the maximum marks. The answers were wrong in principle, indicating superficial knowledge and fallacious reasoning.

Q. 52. A person using a theodolite at a station point A observes the angular elevations of two objects P and Q above the horizon to be 35° and 65° respectively. The horizontal angle (or azimuth) between their directions, that is between vertical planes containing them, measures 75°. Find the angle PAQ subtended by the two objects at the place A, as would be measured by a sextant. A very popular question and well answered.

HONOURS.

Results 1st Class, 7; 2nd Class, 14; Failed, 21; Total, 42.

There were 42 candidates who took Honours, an increase of four over the number last year. There was a marked improvement of the work in comparison with that of recent years, the answers to the vector portion of the subject being exceptionally good. The only unsatisfactory part was the general avoidance of Questions 61 and 62, seeming to indicate a lack of inventiveness and resource in dealing with new problems.