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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.)
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 inperfect 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.
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 conseqnently 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
which are embodied in the formula. If the temperature of a gas were 25° C., what would be the numerical value of T (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
2a + 26 These two questions were often well answered, but a good many mistakes were made in applying statical principles,
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
ody 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. V = Ak?), 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 0:6; if the radius of the base is given, find
under what circumstances 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.
Results : 1st Class, 2; 2nd Class, 7; Failed 8; Total 17.
Only seventeen papers were sent in for this Stage, and the work was nut 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 coostruct à 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.
; 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.
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 63 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 osition with the bearing of a known point of land.
NAVIGATION AND ASTRONOMY.
STAGE 3. Results : 1st Class, ; 2nd Class, 2; Failed, 3 ; Total, 5. Of five candidates examined, one took Spherical Astronomy, the remainder confioing their attention to Section A, which deals with Navigation and Nautical Astronomy. One candidate gained 53 per cent. of total marks, and a second 38 per cent. The other papers were failures. The candidate who answered questions in Stage 3, Spherical Astronomy, sent up one good answer, and attempted two other questions not wholly without
HONOURS. Results : 1st Class, — ; 2nd Class, 1 ; Failed, 1 ; Total, 2. Two papers were sent in, one of which obtained 65 per cent. of the maximum. The other was a failure. Neither candidate took Spherical Astronomy.
BOARD OF EXAMINERS.
I. Practical Plane and Solid
V.p. Practical Mathematics
Professor John Perry, M.E., D.Sc.,
LL.D., F.R.S., Chairman.
F.R.S.E., M. Inst. C.E.
Report on the Examinations in Practical Plane and
STAGE I. Results : 1st Class, 649; 2nd Class, 518 ; Failed, 658 ; Total, 1,825. The number of candidates in this stage was 1,825, being five less than in 1905. The high standard of last year was maintained, and there are many schools in which the teaching is extremely efficient. Not only are the principles of the subject well understood, but care and accuracy in drawing is secured, the amount of distinctly careless work in these schools being very small. On the other hand, there are some schools in which the work is almost wholly bad, and these schools account for nearly the whole of the failures.
In all schools there is evidence that some portions of the subject do not receive adequate treatment, and that improvements are possible, more especially with students of less than average ability. The following points may be specially mentioned :
(a) There is a general lack of exact knowledge of the simpler facts relating to the tangency of lines and circles. See the detailed remarks under Question 3.
(6) In dealing with the important subject of vectors, the teaching often lacks precision. The simple graphical rules for the addition and subtraction of vectors have not been assimilated, and confusion results. In specifying direction, many candidates seem unable to distinguish East from West, and clockwise angles from counter-clockwise. Direction arrows are often omitted, a serious defect in vector work.
(c) The evidence afforded by the answers to Question 10, relating to a line in space, indicates that this branch of the subject is either badly taught or comparatively neglected. In a large number of cases the candidates were evidently quite unable to form any mental picture of the conditions of the problem, and their answers were absurd and worthless. The only successful method of dealing with lengths and angles in space is to provide students with a few simple models, devoid of construction lines, on which these quantities are directly measured, and the results verified by drawing projections of the models, and then determining the same lengths and angles by geometrical construction. Suitable and inexpensive models will be suggested by a study of the questions set in the day and evening examination papers of the last few years.