Q. 3. ABC is a triangle having a right angle at C, and the side AC is twice as long as the side BC. A force of 20 units acts from B to A, one of 10 units from B to C, and one of 15 units from A to C. By means of a construction drawn to a fairly large scale, find the resultant completely.

The triangle and the force BC being as above, how must the forces along CA and AB be chosen so that the three forces may not have a single resultant?

A usual mistake was to assume that the forces act at a point.

Q. 4. Define the centre of gravity of a body, and state what is the position of the centre of gravity of a triangular board-putting the thickness of the board out of the question.

Draw a triangle ABC to a fairly large scale, and construct the position of its centre of gravity.

In the diagram, which you have drawn, take a point P in BC such that PC is one-fourth of BC; find the position of the centre of gravity of ACP, and show that its distance from the centre of gravity of ABC is one-fourth of BC.

Many omitted to give a proof in the last part of the question.

Q. 5. Define the moment of a force with reference to a point. If two forces act along intersecting lines, show how to find any one of the points about which the moments of the forces are equal but of opposite signs.

Two lines AB and AC are inclined at an angle BAC of 60°; a force of 7 units acts from A to B, and one of 10 units from A to C; show how to draw the straight line in which are situated all the points such that the moments of the forces about them may be equal and of opposite signs.

In most instances an insufficient explanation was given.

Q. 6. Sketch the beam of an equal-arm balance showing how it is itself supported and how it supports the pans for the goods and weights. Explain what the sensibility depends upon.

How can stability of equilibrium be ensured?

An article is found to weigh 438 grains, but on weighing again with the article in the weight-pan the weight is found to be 437 grains; what is the true weight of the article?

Explanations as to "stability" were in general not well given. In the rider it was a frequent assumption that the result must be the arithmetic mean.

Q. 7. AB is a rod loaded at A so that the distance of the centre of gravity (G) from A is one-tenth of AB; the rod can turn freely round an axis at C, which is halfway between A and G; the weight of the rod is 4 lbs. A weight of 10 lbs. is hung from A, and a weight (P) of 1 lb. is placed so as to balance the weight at A; find AP as a fraction of AB.

Q. 10. State what is meant by "acceleration" and by constant acceleration."

The units of distance and time being feet and seconds, illustrate your statement by explaining what is meant by an acceleration of 10.

Using the same units, two points A and B begin to move at the same instant in straight lines; A has an initial velocity of 100 feet per second, and the acceleration of its velocity is 10: B has no initial velocity, and its acceleration is 20. Find (a) how many seconds elapse before they are moving with equal velocities, (6) the distances that they describe respectively in that time.

These two questions were often well done.

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Q. 12. A simple pendulum 40'5 inches long was observed to make 59 beats in a minute, at a certain place; show that g = 32'209 at that place.

If the pendulum were lengthened by half an inch, how many fewer beats would it make in an hour?

Was seldom attempted, and not once with complete success. Arithmetical calculation by contracted methods is imperfectly understood.

STAGE 2.

Results 1st Class, 22; 2nd Class, 62; Failed, 89; Total, 173.

The number of candidates was 173, and there were fewer good papers than usual. The work compares unfavourably with that sent up in the Evening Examination.

Q. 21. State Newton's second law of motion, and show that, with a proper choice of units, it can be expressed by the equation. FMf.

If the units of length and time are a furlong and a minute, find the number of pounds in the unit of mass, when the unit of force equals the weight of 11 lbs. (g = 32).

Was generally attempted, but the majority did not understand how to change units.

Q. 23. A uniform rod AB rests on a rough horizontal table; state how the reaction of the table acts, and explain whether the roughness of the table affects your answer.

A thread AP is fastened to the end 4 and pulled by a force P in a direction inclined to the vertical. The force P is not sufficient to cause motion. Find the line along which the reaction now acts, and show that the rod will not slide, so long as P sin (0 + 4) < W sin ; where denotes the angle of friction, and the inclination of the thread to the vertical. A common fault was to take the reaction as acting at the centre of the rod.

Q. 24. A uniform beam, of length 2a, is placed in a fixed smooth hemispherical bowl of radius r, r<a; show that in the position of equilibrium the angle which the beam makes with the vertical is given by

Several candidates ignored the fact that having drawn the diagram so that the three forces acted through one point, the result could be found by elementary geometry.

Q. 27. A weight rests on the lowest point of a fixed smooth vertical ring, and is pushed uniformly up the ring till it is level with the centre.

Draw a diagram of the work done, and show from it how the rate at which the force increases depends upon the angular distance traversed by the weight.

The subject of the diagrammatic representation of work when variable seems to have been neglected by teachers, as only one or two candidates understood what was required.

Q. 29. Prove that the free path of a projectile under gravity is a parabola.

Find equations for the determination of the initial conditions that the projectile may pass through a given point.

The answers were usually restricted to bookwork and that was frequently

cumbrous.

Q. 32. Prove that the centrifugal force at the equator due to the Earth's rotation is nearly the 289th part of the Earth's attraction at the same place.

Very well done by those candidates, few in number, who attempted it.

DIVISION II. (FLUIDS.)
STAGE 1.

Results 1st Class, 11; 2nd Class, 43; Failed, 14; Total, 68.

There were several good sets of answers sent up, but on the whole the results are not nearly so good as those of the Evening Examination. Most of the faults and errors were such as are due to imperfect knowledge; but in some cases they arose from the inability of the writer to express his meaning clearly.

There are two prevalent faults to which attention may be drawn.

Q. 8. Explain the distinction between stable and unstable equilibrium. If part of a wooden sphere be cut off by a plane, explain why the remainder will float with the flat surface upward.

Very few understood that in the case of a floating sphere, or segment of a sphere, the fluid pressure at each point acts at right angles to the surface, and consequently the resultant fluid pressure must act through the centre of the sphere.

Q. 10. The formula for a perfect gas is PV=RT; explain carefully the meaning of the letters, and state the two laws or properties of a perfect gas which are embodied in the formula.

If the temperature of a gas were 25° C., what would be the numerical value of 7' (in the formula)?

A certain quantity of gas is at a temperature of 17° C. under a certain pressure; another quantity which is twice the former is under the same pressure at a temperature of 27° C.; find the ratio of their volumes.

Very few realize that Boyle's Law applies to some definite quantity of a gas. The statement "The volume of a gas varies inversely as the pressure is inadequate, and in most cases implies a misconception. It should run thus: "The volume of any given quantity of a gas, &c."

STAGE 2.

Results 1st Class, 5: 2nd Class, 13; Failed, 20; Total, 38.

:

The general remarks on the work in Stage 1 are applicable to the work in Stage 2. Several good papers were sent up.

Q. 23. A pair of opposite vertical faces (or sides) of a cubical vessel are kept from rotating outwards about their lowest edges, which are hinged, by a taut string connecting their middle points.

Find the tension of the string when the cube is filled with water.

Evaluate the tension when the capacity of the vessel is 6'23 gallons.

Q. 24. A side, of length a, of a vertically immersed quadrilateral is in the surface of a fluid; the opposite side, of length 6, is parallel to it at a depth h prove that the depth of the centre of pressure is 1 a +36

2a + 26

h

These two questions were often well answered, but a good many mistakes were made in applying statical principles,

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Q. 26. Suppose that a rectangle ABCD represents a vertical face of a rectangular solid floating in stable equilibrium, and that AB is under water; let E and F be the middle points of BC and CD. Suppose that E is fastened to one end of a thread, the other end of which is fastened to a fixed point at the bottom of the water; suppose also that the body comes to rest with ADF above the surface of the water. If s denote the specific gravity of the solid, show that

Q. 27. A cylinder formed of a thin flexible substance is subjected to an internal fluid pressure; explain how the tension at any point is measured.

In the case of an upright cylinder filled with water, find the tension at any assigned point of the curved surface. Find also the whole resultant force which tends to tear the curved surface asunder along any vertical line.

Q. 28. Define the metacentre of a floating body. Assuming the general formula for its position (HM.VA), find its position in the case of a right cylinder whose axis is vertical.

A cylinder, whose specific gravity is 0'4, floats in a liquid whose specific gravity is 06; if the radius of the base is given, find under what circumstances it will float with its axis horizontal.

These three questions are rather hard, and the attempts to answer them were not worth much. Rather better results might have been fairly expected, at least in regard to Questions 27 and 28.

Q. 30. When is a vapour

(i) superheated,

(ii) saturated,

(iii) at its critical temperature?

Under what circumstances does a vapour obey very closely the Gaseous Laws?

Was tried by most of the candidates. The answers in very many cases illustrated the remark made above as to the want of ability on the part of the writers to express their meaning clearly. Attention had been very generally given to the subject of the question, but in few if in any cases were all three points clearly and briefly defined. Sometimes one or two points came out clearly, but in most cases the result of many words was to leave the matter obscure.

Report on the Examinations in Navigation and in Spherical and Nautical Astronomy.

NAVIGATION.

STAGE 1.

Results 1st Class, 2; 2nd Class, 7; Failed 8; Total 17.

Only seventeen papers were sent in for this Stage, and the work was not good. Two candidates only succeeded in reaching 60 per cent. of the total, and eight were below 40 per cent. Very few had any real grasp of the subject, and the logarithmic calculations were unusually erratic.

STAGE 2.

Results 1st Class, 11; 2nd Class, 24; Failed, 11; Total, 46.

In this Stage there were forty-six candidates, whose work was of an average character. Eight obtained 70 per cent. of the marks and upwards, 27 from 40 to 69 per cent., and 11 fell below 40 per cent. of the total.

A few remarks as to the answering of particular questions are appended. Q. 21. Two ships sail from the same point. The first sails NE a miles, and then SE a miles. The second sails SE a miles, and then NE a miles. Will they reach the same position? Give a reason for your answer.

There were few satisfactory answers, and a very general tendency was noticeable to solve the question by simply drawing a parallelogram. Q. 23. Explain the method of finding deviation of compass by reciprocal bearings taken on board and ashore.

If the bearings observed on board are S. 85° E. and N. 87° 40′ E., and the corresponding bearings observed ashore N. 82° 40′ W. and N. 84° 20′ W., find the respective deviations.

As on former occasions many mistakes were made in the work of the practical example. These are to a great extent due to the failure to construct a satisfactory diagram shewing simply a single line of bearing upon which the compass ashore is situated and the directions of correct magnetic and compass meridians. In several cases also, in reversing bearing, N. 84° 20′ E. was given as the reverse bearing to N. 84° 20′ W. instead of S. 84° 20′ E.

Q. 27. Find the compass course and distance from Jamaica (Lat. 17° 56′ N., Long. 76° 11′ W.) to the Lizard (Lat. 49° 58′ N., Long. 5° 12′ W.); Variation 3° 40′ E., Deviation 2° 50′ E.

Great want of accuracy was shewn in the work of this simple straightforward question.

Q. 31. A ship sails from Cape Finisterre (Lat. 42° 53' N., Long. 9° 15' W. to Cape Race (Lat. 46° 40′ N., Long. 53° 5' W.). Find the distance and the latitude of the vertex.

Considerable improvement was exhibited in the logarithmic work involved in this question, as compared with former examinations.

Q. 32. Day's Work. The necessary calculations were well and accurately done.

SPHERICAL AND NAUTICAL ASTRONOMY.

STAGE 1.

Results: 1st Class, 8; 2nd Class, 5; Failed, —; Total, 13.

The results in this Stage, for which thirteen papers were sent in, were unusually good, more particularly in the case of the group of candidates indicated by Numbers 69687-69696.

STAGE 2.

Results 1st Class, 1; 2nd Class, 11; Failed, 3; Total, 15.

Fifteen papers were sent in for this Stage, and the results were distinctly below the average. One candidate gained 80 per cent. of total marks, but no other succeeded in reaching 65 per cent. The practical work was inaccurate, and there was but little knowledge of theory.

The Sumner Double-Chronometer question was fairly well done, but hardly any candidates attempted Question 25, which deals with the combination of a line of position with the bearing of a known point of land.