Q. 32. There are four vessels of equal capacity; and wine is poured into them, so that th of the first, th of the second, th of the third, and rd of the fourth are filled with wine. The first is then filled up with water, and from this mixture the second is filled up, again from this second mixture the third is filled up, and in like manner the fourth is filled up from the third. What is the ratio of wine to water in the fourth vessel ?

These two questions were less often tried than the others in this section, but occasionally they received good answers.

C.

Q. 33. (a) Define the terms logarithm, mantissa, characteristic.

Referring to the logarithms given below, write down the logarithm of 535 51, and also the logarithm of 0.053551.

(b) From logarithms given below find the logarithm of 27 and that of 125; also calculate the numerical value of

(c) If

(125) 0-3.

(27) 0-25
log (x2) =

find the value of x.

3

5'

Was on the whole satisfactorily answered, but many failed to make out (c).

Q. 34. (a) Define a degree, a minute, and a second of angle.

Explain what is meant by the number commonly denoted by the Greek letter, and write down its approximate value, true to four decimal places.

(b) If an angle of 90 is represented by 2'

find the number of de

grees, with the odd minutes and seconds, in the unit of angle. By what name is that unit commonly called?

(c) Calculate the numerical value of

π

log 3'

"the 360th part of a The definition of a degree was often faulty; e.g., circle"; on the other hand there were several careful and correct definitions.

Q. 35. (a) Draw an appropriate diagram, and, with reference to it, define the sine, the tangent, the cosine, and the cotangent of an angle less than 90°.

(b) Express the sine, the cosine, and the cotangent of an angle in terms of its tangent.

(c) Show that sin tan is greater than 2(1 − cos 0), when ℗ is a

positive angle less than 90°.

-

The last part, (c), was seldom well done; what was usually given was merely a verification for a particular value of 0.

Q. 36. (a) Construct the positive angle less than 180° whose tangent is 3 and find the sine and the cosine of the angle correct to

two places of decimals.

(b) Establish the following relations, in which A denotes an acute angle:

(i) cos (180° - A) tan (180° – A)

=

sin A.

(ii) sin (90° + A) cos (90° +A) + sin (90° – A) cos (90° — A) = 0. (c) Taking the tangent of 31° as 0'6, how many degrees are there in each of the positive angles (less than 360°) whose tangent is 0.6?

As

There was much vagueness and confusion in many of the answers. usual, angles greater than a right angle proved to be outside the range of knowledge of the candidates.

Q. 37. Write down the formula for the sine of the half of any one of the angles of a triangle, in terms of the sides, and explain the notation.

The sides of a triangle are 491, 682, and 827 feet long; employ the formula to calculate its largest angle.

Q. 38. Show how to find the remaining parts of a triangle when two angles and the side adjacent to both are given.

The distance between two stations P and Q on the sea shore is 1,000 yards; the line joining P with a buoy B subtends an angle of 38° at Q, and the line BQ subtends an angle of 65° at P. Find the distance of the buoy from P.

The problems were often worked correctly, but the explanation required in the first part of Q. 38 was not well given.

STAGE 3.

Results 1st Class, 26; 2nd Class, 47; Failed, 21; Total, 94.

The work in this Stage quite reaches the standard of last year.

A.

Q. 41. If two triangles have one angle of the one equal to one angle of the other, and the sides about the equal angles proportionals, show that the triangles are equiangular to one another, and that those angles are equal which are opposite to the homologous sides.

What is meant by "homologous sides"?

Perpendiculars AD, BE are drawn to the sides BC and CA of a triangle ABC. Let ED and AB be produced to meet in 0; show that OD is to OE as triangle ADB is to triangle AEB.

Q. 42. If a straight line be divided into any two parts, show that a rectilinear figure described on the whole line exceeds the sum of the two similar figures similarly described on the two parts by twice the similar figure similarly described on the mean proportional between the parts.

Show that either of the complements is a mean proportional between the parallelograms about a diagonal of a parallelogram.

Q. 43. Given two squares P and Q, and a straight line AB, show how, by a geometrical construction, to find a point E in AB such that the ratio of AE to EB may equal the ratio of P to Q.

Q. 44. Having given the sum of the squares on two straight lines, and also the rectangle contained by the lines, show how to construct them.

Draw a circle of radius two inches, and inscribe in it a rectangle having an area of 5 square inches.

Q. 45. A point O is taken within a triangle ABC, and from A, B, and C lines are drawn through O to cut the opposite sides in D, E, F ; show that

If the line drawn through O bisects the angle CAB, and if FE is parallel to BC, show that AB equals AC.

Q. 46. ABCD is a given quadrilateral; show how to construct another quadrilateral with its angular points P, Q, R, S on the four lines AB, BC, CD, DA respectively, and with its sides drawn parallel to four given directions.

9897.

B 2

In GEOMETRY, Qs. 41, 43 and 45 were those most often chosen; Qs. 41 and 45 were on the whole well answered; in Q. 43 several candidates assumed that the ratio a2: 62 was equal to a : 6, which showed to what little advantage they had studied the subject of ratio.

Of the remaining questions in this section, the first part of Q. 42 was made out by a fair number, and the second part by a few; there were five or six complete answers to Q. 44, and two or three to Q. 46.

B.

In ALGEBRA the work was very satisfactory, many good answers being made in the case of each of the questions.

C.

In TRIGONOMETRY all the questions were fully answered except Q. 56, which no one attempted.

Q. 56. If a, b are the adjacent sides of a parallelogram, and 0, & the acute angles between the sides and between the diagonals respectively, show that

Both the candidates in this Stage sent up good papers.

STAGE 5.

Results 1st Class, 6; 2nd Class, 12; Failed, 10; Total, 28.

Of the 28 candidates, about half answered fairly well, but the work as a whole is less satisfactory than that of last year.

STAGE 6.

Results 1st Class, 1; 2nd Class, 1; Failed, 2; Total, 4.

Four papers were sent up; one was good, one fairly good, the two others weak.

The paper sent up was creditable, but did not reach the standard for a First Class.

Report on the Examinations in Theoretical Mechanics.

EVENING EXAMINATIONS.

DIVISION I. (SOLIDS).
STAGE 1.

Results 1st Class, 141; 2nd Class, 193; Failed, 117; Total, 451. The work in this Stage was much as usual, and does not call for any general remarks. The following may be worth the attention of the teachers. Q. 2. State how to find the centre of gravity (or centre of mass) of two particles of given masses.

ABC is a triangle formed by three uniform rods, whose thickness is not to be considered. AB is 5 ft. long and weighs 4 lbs., BC is 4 feet long and weighs 1 lb., CA is 3 ft. long and weighs 2 lbs. Find the centre of gravity of the rods, and show, in a carefully drawn diagram, the position taken by the triangle when hung up by the point A.

The centre of gravity was found correctly in a good many cases; but it seldom or never occurred that the triangle was rightly constructed with AG vertical.

Q. 4. If three forces of given magnitude are in equilibrium at a point, state the construction by which the angles between their directions can be found.

If the forces contain 5, 7, and 9 units, make the construction in a carefully drawn diagram.

It is proved in Geometry that any two sides of a triangle are together greater than the third side; what is the corresponding property of three forces in equilibrium at a point?

The actual arrangement of the forces was often not shown, even when the triangle had been correctly drawn. In only a few cases was the approximate number of degrees in the angles given.

Q. 6. A straight uniform bar 2 feet long is supported horizontally by pegs placed symmetrically 3 inches from either end. If the weight of the bar be 5 lbs., what weight must be placed on one end so that the pressure on one of the pegs may be wholly relieved?

The number of pounds in the weight (15) was often found, but there was seldom a good reason assigned for this weight wholly relieving the pressure on one of the pegs.

Q. 7. One of the supports of a rod is a smooth fixed point; in what direction is the reaction of that point exerted?

A uniform rod AB can turn freely round a hinge at A, and rests against a fixed point C, which is so placed that C is on a lower level than A, and that AB is inclined at an angle of 45° to the vertical; also AC is ths of AB. Draw a diagram to scale showing the forces which act on the rod; also draw a triangle for the forces.

This is an instructive question. Of course there were many failures; but there were several good answers, as many perhaps as could be fairly expected.

Q. 8. When is a force said to do work?

Two weights of 10 lbs. and 15 lbs. respectively are joined by a fine thread which passes over a smooth fixed pulley. The heavier weight is allowed to fall and to draw up the lighter weight. When it has fallen 6 ft. how much work has been done by the first weight, and against the second weight? What is there to show for the difference between these two quantities?

Many found the 90 and 60 foot-pounds and the excess of 30 foot-pounds, but very few seemed to understand that this excess is represented by the Kinetic energy of the weights and pulley.

Q. 10. Describe Atwood's machine and show how it may be employed to determine the numerical value of the acceleration due to gravity.

Weights of 4 and 5 lbs., connected by a string which passes over a smooth fixed pulley, are held at rest and then gently released so that motion ensues; how soon will the system be moving with unit velocity?

The determination, by Atwood's Machine, of the numerical value of the acceleration due to gravity was often omitted, and in other cases seldom well explained.

Q. 11. Describe a simple pendulum and state the property of its small oscillations.

What is the effect upon the number of oscillations performed in a given time

(i) of increasing its length,

(ii) of transferring it to a place where the force of the earth's attraction is less?

Of course, an oscillation is small when the arc through which the oscillating body moves is small. Now, very few seemed to understand that small arcs may have different lengths, e.g., one small arc may be two or three times as long as another small arc, and consequently only a few distinctly realised the property of the simple pendulum, viz., that its small oscillations are (sensibly) made in a given time, whatever be the length of the arc, provided it be small.

STAGE 2.

Results 1st Class, 157; 2nd Class, 338; Failed, 192; Total, 687.

Much of the work showed an adequate grasp of principles, and the examination may be considered to be satisfactory.

The most noticeable failure was in regard to the polygon of forces; nearly all the candidates failed to distinguish between "a force acting along a line" and " a force represented in direction by a line." The attention of teachers may be called to this matter.

Q. 21. Find the unit of acceleration when the units of length and time are 1 yard and 1 minute respectively.

Candidates often failed to understand that they should have compared the magnitude of the unit of acceleration with that which obtains when the units of length and time are 1 foot and 1 second respectively. Many gave as the answer 1,200 yards per minute.

Q. 22. Ox, Oy are two lines inclined at a given angle; one particle, whose mass is m, moves along Ox with a velocity u; another, whose mass is n, moves along Oy with a velocity v; they start together from 0. Show in a diagram the line along which the centre of gravity moves, and find its velocity along that line.

Few were able to find the velocity of the centre of gravity.

Q. 25. A labourer having to move roots, or earth, or stones a considerable distance (such as a hundred yards), puts them into a wheelbarrow, and thereby can transfer at one journey about 1 cwt. of material. Explain the mechanical principles on which he derives advantage from the use of the machine.

A wheelbarrow is sometimes cited as an instance of a lever; explain this. Explain also the use of the wheel.

The answers were generally very fair; few were entirely satisfactory.