1. At what price per head must a farmer purchase a flock of 100 sheep, that expending £10. in feeding them, and losing nine, he may be able to sell the remainder at £2. each and gain £20 ?

2. I turn over the pages of a book by fours and find three odd ones. I then turn them over by fives and find two odd ones. The last time I do not turn them over so often by twenty times as I did the first. How many pages were there?

3. A passenger train and a luggage train, the one travelling at 10 mile per hour less speed than the other, set out at the same time, the one from London and the other from Carlisle, 210 miles apart, and pass one another at a certain station on the road. The passenger train sets out from Carlisle to return, two hours after the luggage train sets out to return from London; and it is observed that they pass one another at the same station. At what rate do they travel, and how far from London is the station?

Section 7.

1. The first term of an arithmetical progression is 7 the number of terms 8 and the sum 28. What is the common difference?

2. A person sowed a bushel of wheat and the next year he sowed again the whole produce of that bushel, and so on until at the end of the third year he had a bushels. How many grains of wheat must each grain of seed have yielded, supposing it to have yielded the same number every year?

3. What is the present value of an Annuity of £a which increases in geometrical progression whose ratio is r for n years' interest being assumed at p per cent. per annum?

Section 8.

1. Approximate by the method of continued fractions to the value of

587 1943

2. Show in the above example that the approximating fractions must be alternately greater and less than the true value.

230

3. How can the fraction nominators are 7 and 11. 77

be divided into two others whose de

Section 9.

1. Find expressions for the number of years on which £ a will amount to £ A. at per cent. simple interest, and at r per cent. compound

interest.

2. Investigate a general method for expressing a number N in the scale of Notation, whose radix is r: and show that the number, when so expressed, will leave the same remainder when divided by (r− 1), as the sum of its digits will.

3. Show, that if one solution of the indeterminate equation a x + by C be given, all the rest may be determined from it.

1. Expand

2x + 3
3x + 2

Section 10.

to four terms in a series ascending by powers of r,

by the method of indeterminate co-efficients.

2. Resolve

3x+2

x Tinto partial fractions.

3. Expand as in a series ascending by powers of x.

Section 11.

1. Investigate an expression for the number of permutations of n things taken r together, and find the number of signals which may be made with six different flags placed one above the other.

2. Prove the Binomial Theorem in the case in which the index is a positive integer.

3. Investigate the general term of the Multinomial Theorem in the case in which the index is a positive integer.

Section 12.

1. Show that every number, which is a perfect square, is of one of the forms 5 m or 5 m + 1.

2. Find the number of balls in a triangular pile, each side of the base of which contains 25 balls.

3. The plate of a looking-glass costs £a per square foot, and the frame £b per linear foot of the top and bottom, and £c per linear foot of the sides. What is the largest glass that can be bought for £ m.

Section 13.

1. Show that the perpendicular from the focus of a parabola upon any tangent, intersects that tangent in the same point in which a tangent to the vertex intersects it.

2. Show that the sum of the distances of any point in an ellipse from the foci, is equal to the axis major.

3. Show that the semi-axis major of an ellipse in a mean proportional between the distance, C N, from the centre to the foot of the ordinate to any point, and the distance C T from the centre to the intersection of the tangent, to that point with the axis.

Section 14.

1. Investigate the general equation to a straight line, and draw the line whose equation is

2. Find the perpendicular distance of a point whose co-ordinates are xy, from a line whose equation is ax + by = c.

3, Having given the co-ordinates of a front with reference to any given axis, to determine them with reference to an axis inclined to the former at any given angle, the origin being the same.

Section 15.

1. Determine the locus represented by the equation, x2 + y2 - 2 cx + 6 cy + 9 c2 = 0.

2. Determine the equation to the tangent to an ellipse.

3. Show that the second term in the general equation of the second degree may be made to disappear by turning the axis of the co-ordinates through a certain angle.

Section 16.

1. To inscribe a square in a given triangle, one side being on the side of the triangle.

2. To find the locus of the vertex of a triangle, whose base and the ratio

of its sides are given.

3. To find the locus of the middle points of all the chords passing

through a given point in an ellipse.

VOL. II.

3 M

Section 17.

1. Show that the pressures acting on the arms of a lever will balance when they are reciprocally proportional to their distances from the fulcrum.

2. If a point be kept at rest by three pressures, any two are to one another inversely as the sines of the angles they make with the third.

3. To find the centre of gravity of any number of heavy bodies, whose respective centres of gravity do not all lie in the same plane.

Section 18.

1. Investigate the relation between the power and the weight in a single moveable pulley, when the strings are not parallei.

2. Show that the centre of gravity of a triangle is situated in the line joining one of its angles with the bisection of the opposite side at twothirds the length of that line.

3. Show that a couple will produce the same statical effect upon a rigid body, when it is moved parallel to itself in its own place.

4. Investigate a relation between the space and time when the motion of a body is uniformly accelerated.

Section 19.

1. Describe the common steelyard, and give a formula for graduating it. 2. A body is projected vertically upwards from the top of a precipice with a velocity of 160 feet per second, and five seconds afterwards a body is let fall from the same place. Both bodies reach the foot of the precipice together. What is its height?

3. One of the cannons of a ship is made to describe a circle on the deck. Show that the centre of gravity of the ship will thereby be made to describe a circle, and compare the diameters of the two circles.

4. At what angle must the double wedges which support the keel of a ship in dock be cut, that its weight may just cause them to slide apart, the co-efficient of friction being supposed to be given.

Section 20.

1. Explain the formation of artesian wells.

2. Determine the pressure of a fluid upon a portion of a plane immersed in it, and inclined at any angle to the surface.

3. Show that if a body be not at rest in a fluid, the moving force by which it ascends or descends is the difference between the weight of the solid and that of an equal bulk of the fluid.

Section 21.

1. Describe the common hydrometer, and show how it may be graduated. 2. The weight of a bottle full of water is found to be w; a body, whose weight is w', having afterwards been placed in it, the weight of the whole is found to be w11. What is the specific gravity of the body?

3. What are the laws in respect to elastic fluids known as those of Marriotte and Gay-Lussac? Express by a formula the relation between the elastic force, density, and temperature of a gas.

Section 22.

1. What is the chemical constitution of the air? In what way is it connected with the support of animal and vegetable life?

2. In what different forms does carbon exist? What are its combinations with oxygen? How may they be obtained?

3. Explain what is meant by the law of multiple proportions in chemistry.

Report on the Normal School for Training Regimental Schoolmasters, and on the Model School at the Royal Military Asylum, Chelsea; by Her Majesty's Inspector of Schools, the Rev. H. MOSELEY, M.A., F.R.S.

MY LORDS,

24 August 1849.

In compliance with your instructions, communicated to me in the letter of your Secretary of the 4th June, I devoted the week commencing with the 23rd July to an examination of the students of the Normal school for the instruction of regimental schoolmasters at the Royal Military Asylum, Chelsea, established by Royal Warrant, bearing date the 21st November, 1846. I find that it is placed under the general control of a Board of Commissioners, of whom the following are constituted a School Committee: The Secretary-at-War, the Bishop of London, the Paymaster-General, the Judge-Advocate-General, the Deputy Secretary-at-War, Viscount Hardinge, and the Right Hon. Sidney

Herbert.

The Inspector-General of Military Schools has, moreover, free access to the schools for the purpose of inspecting and reporting upon the same.

Under the authority of this Warrant there are also appointed1. A chaplain and head-master of the Normal school. 2. A second-master of the Normal school. 3. An upper-master of the Model school. 4. A first-master of the Model school. 5. A second-master of the Model school. 6. A master of the Infant school.

For the management of all that does not belong to the education of the children, and for their charge and supervision out of school-hours there is, moreover, appointed a staff of military officers, consisting of a commandant, an adjutant and secretary, a quarter-master and steward, a surgeon, a quarter-master sergeant, and one sergeant-assistant for every fifty boys.

The following are the officers in the educational department of

the institution :

Normal School.

The Rev. W. S. O. Dusautoy, head-master and chaplain.
Mr. Robert Fowler, assistant-master.

Mr. John Hullah, singing-master.

Mr. H. Gandee, drawing-master.

Model School.

Mr. W. McLeod, upper-master and master of method.
Mr. T. Baker, second-master.

Mr. G. Renwick, third-master.

Mr. T. Dexter, master of the Infant school.

The masters of the Model school are assisted by eight paid monitors, destined eventually to become students of the Normal school. This number is hereafter to be increased to 16.

The students of the Normal school are wholly maintained at the public expense. Such vacancies as occur are made public, and the candidates for admission being examined by the InspectorGeneral, the best qualified are appointed on his report to the Secretary-at-War. They enter into a bond to enlist in Her Majesty's service at the expiration of their course of instruction, of which the prescribed period is two years. The office they are then to hold in the army is designated in the Warrant as that of schoolmaster-sergeant; they are to rank with the sergeant-major, and have the same pay and appointments. They are to teach the children of soldiers from 9 to 11 o'clock daily, and the adult schools of soldiers and recruits at such hours as may be ordered in each regiment. Recruits are required to attend these adult schools two hours daily. With soldiers after they are discharged drill, attendance is voluntary.

The apartments assigned to the students in the Asylum are situated in one of its wings. They were formerly occupied by the boys, when the full number were resident. But few changes have been made in these apartments to adapt them to their present use. Such changes are obviously required, and I am informed that they are to be made. The number of students at present resident is 26; their ages vary from 19 to 25 years, and the period during which they have resided, from nine months to two years. Their previous attainments appear generally to have been superior to those of the students of other Normal schools which I have examined; they have for the most part been educated at private schools, many having followed very laborious callings, and the greater part affording in their manners and deportment the evidences of a respectable parentage. I have adverted to this fact thus particularly, because I consider it to be in some degree characteristic of this institution as compared with others. I find no advantages in the condition of a regimental schoolmaster, as compared with that of a national schoolmaster, to account for the difference, and can only attribute it to the fact that the career in life of the military schoolmaster is the more adventurous one,that the entire expense of the students' residence in the Military Normal school is borne by the Government, and that the appointments are thrown open to public competition. In the estimate which may be formed of the advantages of thus providing for the