7. If a vessel, the bottom of which is horizontal and the sides vertical be filled with fluid, the pressure upon the bottom will be equal to the weight of the fluid.

If the sides incline inwards or outwards, how must the enunciation of this proposition be modified?

8. When a body of uniform density floats on a fluid the part immersed: the whole body :: the specific gravity of the body: the specific gravity of the fluid.

A triangular lamina of uniform thickness floats in a vertical position with its base horizontal and its sides half immersed in a fluid; compare its specific gravity with that of the fluid.

9. Air has weight.

If a body were floating on a fluid, with which the air was in contact, and the air were suddenly removed, would the body rise or sink in the fluid? State your reasons.

10. Explain the action of the common syphon.

If the ends of the syphon were immersed in two fluids of the same kind and the air were removed, describe what would take place.

11. Describe the construction of the common pump and its operation.

ARITHMETIC AND ALGEBRA.

WEDNESDAY, January 9, 1856. 12 to 31.

SECOND DIVISION.—(A.)

1. A POUND of silver is coined into 66 shillings, of which 62 only are issued. If nineteen half-crowns and fifteen sixpences are melted into bullion, and sent to the Mint to be recoined, what sum will be reissued?

2. If a decimal system of coinage were adopted, and the tenth, hundredth, and thousandth parts of a pound sterling called respectively florins, cents, and mils, what sum would 4 florins, 7 cents, 5 mils represent?

3. Reduce £65. 12s. 6d. to the decimal of £125.

4. Prove that (y − z)3 + z3 — y3 = 3yz (z − y).

5. Solve the equations

6. If one watch loses and another gains at the rate of a minute a day, and they are both set at noon on Monday, what time will be indicated by the latter, when the former points to 10h. 491 min. P.M. on the following Saturday?

7. Find the squares of 1039681 and 328776; and divide the greater result by the less, to the first significant digit in the decimal places.

8. A sets off from M to go to N at the same time that B sets off from N to go to M. When they meet, A returns to M, and then goes back again to N, which he reaches at the same time that B reaches M. Compare their rates of going.

10. In how many years will £1000 amount to £1123. 12s. at 6 per cent. compound interest? What would it amount to in 5 years at the same rate at simple interest?

11. Give the algebraical definition of proportion, and prove that if a: b :: c: d, and a is the greatest of the four quantities, d will be the least, and a + d> b + c.

12. Find the length of a solid whose thickness is one foot, breadth 18 inches, and solid content 3 cubic feet, 216 cubic inches.

13. Divide x + 64 by x2 + 4x + 8.

14. A person invests £1365 in the 3 per cents. at 91, he sells out £1000 stock when they have risen to 933, and the remainder when they have fallen to 85. How much does he gain or lose by the transaction? If he invests the produce in 43 per cents. at 102, what is the difference in his income?

SECOND DIVISION.-(B.)

1. A POUND of silver is coined into 66 shillings, of which 62 only are issued. What amount of silver coinage must be sent to the Mint, in order that 18 florins, 26 fourpenny-pieces, and 28 threepenny-pieces may be reissued?

2. If a decimal system of coinage were adopted, and the tenth, hundredth, and thousandth parts of a pound sterling called respectively florins, cents, and mils, what sum would 5 florins, 2 cents, 5 mils represent?

7. At what time between the hours of 3 and 4 will the hands of a watch be at right angles to each other?

8. Find the squares of 7380481 and 1905632, and divide the greater result by the less, to the first significant digit in the decimal places.

9. Two persons set off to walk from different points in opposite directions along the line joining them; when they meet, the quicker turns back, and on arriving at the point from which he started, again sets off; they arrive simultaneously, each at the point from which the other started. At what point do they first meet?

10. Solve the equation

(x + 9)+ + (x − 7) = 8.

11. If the sum of £1200 be put out at 10 per cent. per annum compound interest, and interest paid half yearly, to what will it amount in a year and a half?

What would it amount to in 4 years at the same rate at simple in

terest?

12. Find the thickness of a solid whose length is 2 yards, breadth a yard and a half, and solid content 1 cub. yd., 6 cub. ft. and 1296 cub. in. 13. Give the algebraical definition of proportion and prove that if

:

a b c d then a+b: a :: c + d : c.

14. What sum must a person invest in the 3 per cents. at 90, in order that by selling out £1000 stock, when they have risen to 93, and the remainder when they have fallen to 841, he may gain £6. 5s. by the transaction? If he invest the produce in 4 per cents. at par, what will be the difference in his income?

THURSDAY, January 10, 1856. 9 to 12.

FIRST DIVISION.-(A.)

1. TWELVE rupees, three florins, and six half-crowns amount to £2.8. What is the value of a rupee?

2. The tenth, hundredth, and thousandth parts of a pound sterling being called florins, cents, and mils respectively, subtract 1 florin 4 cents

from 7 florins 1 cent 5 mils, and shew that eight times the difference equals £4. 12s.

3. State the rules for the multiplication and division of vulgar fractions. Multiply and divide + by 1-13. Find which result is the greater, and express their difference in its lowest terms.

4. What fraction of 10s. is 2s. 9d.? Reduce the result to a decimal. 5. What sum must be invested in the 3 per cents. at 92, in order to derive an income of £150 per annum? What will be the amount of

stock?

6. Define present value. What sum will amount to £820. 16s. in 3 years at 4 per cent. simple interest?

7. The distance between two stations A and B is 65 miles. A train starts from A to go to B at the rate of 15 miles an hour, and is delayed 10 minutes on the way: another train starts from A two hours after the former at the rate of 25 miles an hour, find the interval between their times of arrival at B.

8. Remove the brackets from the expression,

Find the value of ( √x2 + y2 + z) ( √x2 + y2 - z), when x=4, y=5, 2 = 6.

9. Reduce to their simplest forms the expressions

11. When is one quantity said to vary directly and when inversely as another? One horse takes 6 strides while another takes 5, but 7 strides of the latter horse are equal to 8 strides of the former; which is the swifter horse?

12. Divide a1 — xy3 — x3y + y1 by x2 + xy + y2, and extract the square root of the quotient.

13. The expense of carpeting a room 18 feet long was £7. 4s, but if the breadth of the room had been 4 feet less than it was, the expense would have been £5. 8s, what was the breadth of the room?

14. A merchant has teas worth 4s. 6d. and 3s. 6d. per lb. respectively, which he mixes in the proportion of 2 lbs. of the former to 1lb. of the latter and sells the mixture at 4s. 4d. per lb.; what does he gain or lose per cent?

FIRST DIVISION.—(B.)

1. NINE rupees, six crown-pieces, and eleven threepenny-pieces amount to £2. 13s. What is the value of a rupee?

2. The tenth, hundredth, and thousandth parts of a pound sterling being called florins, cents, and mils respectively, subtract 5 florins 3 cents 5 mils from 9 florins 6 cents, and shew that eight times the difference equals £3. 8s.

3. State the rules for the multiplication and division of vulgar fractions. Multiply and divide by 1-4. Find which result is the

greater, and reduce their difference to a decimal.

4. What fraction of 5s. is 2s. 3d.? Reduce the result to a decimal. 5. What sum must be invested in the 3 per cents. at 96, in order to derive an income of £120 per annum? What will be the amount of

stock?

6. Define present value. Find the present value of £402. 12s. due 3 years hence, simple interest being at 43 per cent. per annum.

7. The distance between two stations A and B is 75 miles. A train starts from A to go to B at the rate of 20 miles an hour, and is delayed 6 minutes on the way; another train starts from A an hour after the former, at the rate of 24 miles an hour; find the interval between their times of arrival at B.

8. Prove that a - (b −c) = a − b + c, specifying the limitations to the values of a, b, c necessary to your proof.

Shew that (a2 + b2) (c2 + d2) = (ac + bd)2 + (ad − bc)2.

9. Reduce to their simplest forms the expressions,