2. A coach wheel turns 280 times in a mile; what is its radius? 3. A pipe 2 inches in diameter discharges 20 gallons of water in a minute; what is the uniform speed at which the water is issuing ? 4. What is the content of a tree whose length is 17ft., and which girts, in 5 equidistant places, as follows, viz. 9 • 43 ft., 7.92 ft., 6. 15 ft., 4.74 ft., 3:16 ft. ? Section 4. Give an example of a field book, and calculate the area of the corresponding field. Division II. will receive marks according to the merit of the answers; but no the first part be unsatisfactory. 1. Equal straight lines in a circle are equally distant from the centre ; and, conversely, those which are equally distant from the centre are equal to one another, 2. About a given circle, to describe a triangle equiangular to a given triangle. 3. If two triangles have one angle of the one equal to one angle of the other, and the sides about the equal angles proportionals, the triangles shall he equiangular, and shall have those angles equal which are opposite to the homologous sides. 4. Show that straight lines, which are perpendicular to parallel straight lines, are themselves also parallel. 5. If from any point in the diagonal of a parallelogram, straight lines be drawn to the angles, the parallelogram will be divided into two pairs of equal triangles. 6. If tangents be drawn at the extremities of any two diameters of a circle, and produced to intersect one another, the straight lines joining the opposite points of intersection will both pass through the centre. ALGEBRA. (Four Hours allowed for this paper, with that on Higher Mathematics.) DIVISION I. Section 1. 1. Add together, 2 (a + b) x y?, 3 (a − 2b) x yand 4 (2 a—b) xy. 2. From (a + b) x + (b + c) y, take (a - b) x — (b − c) y. 3. Prove that (a - b) (x a) (x – 5) + (6 — c) (x – b) (x − 6 +(c-a) (x–c)(x − a) is equal to (a - b)(b — c)(a - c). Section 2. 62 a+b+ a.roy 1. Reduce to their simplest forms and a+b+ a 3 3 Section 3. Solve one of the following equations — 1. 4(0–1) –*? = 32 (-) im - 1 2. 2x2 -2 -2x ✓ 1 - 2 = 11. 3. A farmer buys a flock of sheep at the rate of 71. for every 5 sheep, he afterwards loses 9, and sells the remainder at the rate of 161. for every 11, and the sum for which he sells the flock is 241, more than that which he gave for it; how many sheep were there ? DIVISION 2. receive marks according to the merit of the answers; but no first part be unsatisfactory. 1. A blank paper book containing 48 sheets is sold for 14s., and another containing 78 sheets of the same size for 198., the binding cost the same in both, and the paper was of the same quality. What was the price of the binding? 2. A rectangular field is to be enclosed from the waste with a fence 40 chains long, it is to be surrounded, within the fence, by a gravel walk 6.05 yards wide, and there is to be left in the middle, an area of 6.05 acres of grass. What must be the dimensions of the field ? 1 1 1 3. 0 x2 + 3x+ 5 + + 6 x+8 + una 9 x +7 4. Find what sum of money at r per cent. compound interest will amount in n years to £a? And find the number of years in which a sum of money will double itself at r per cent. compound interest. 5. What annuity, to commence at the end of two years and to continue two years certain, can be bought for 1,0001., allowing 5 per cent. compound interest ? 6. If the present population of Great Britain be thirty millions, and the annual emigration--supposed to take place at the end of the year-be one quarter of a million, by what proportion must the number of births in each of the next two years exceed the number of deaths that the population may have increased at the expiration of that time by one-twentieth? HIGHER MATHEMATICS. Division I. Section 1. 1. Find the common difference of a series in arithmetical progression, whose first.term is 8, and the sum of the eight first terms 28. 2. Find the present value of an annuity of £A to be paid for n years, allowing compound interest. 3. Required the vulgar fraction equivalent to the decimal PQQQ, &c.; where P contains p digits, and Q contains q digits recurring in inf. Section 2. 1. Prove generally the rule for pointing off in the product of two decimal numbers as many decimals as there are in the multiplier and multiplicand together, 10 2. Define the logarithm of a number, and show that - log n 3. What is meant by the radix, and what by the digits of a scale of notation? What is the number which is expressed by 2 4 3 1, in the scale of notation, where radix is 5? Prove the rule for expressing a number in any scale of notation. Section 3. 1. Prove one of the following formulæ of plane trigonometry :Cos (A + B) = Cos A Cos B Sin A Sin B. (S – b) (S – c) Sin · A= bc 2. How might the height of the top of St. Paul's above Wimbledon Common be determined by observations made on Wimbledon Common, from whence St. Paul's can be seen? 3. By what means can the distance from one another of two remote objects, C and D, be determined by observations taken from stations A and B, whose distance from one another is known, and whence the objects C and D are visible? Give the formulæ which must be used in this computation. Section 4. 1. Investigate an expression for the area of a triangle in terms of its sides. 2. Prove the expressions for the circumference and area of a circle in terms of the radius. 3. Show that the solid content of a cone or pyramid is found by multiplying the area of its base by one-third its height. Division II. will receive marks according to the merit of the answers; but no first part be unsatisfactory. 1. Find the sum of the series 1? + 22 + 32 + &c. to n terms. 2. Find the number of combinations of n things taken r together, and show that is the same as the number of combinations of n things taken r together. 3. Prove the binominal theorem in the case in which m is a positive N integer, and expand (at_). * b. 4. Show by the method of indeterminate coefficients that a + 1 5 4 4 3 5. Solve the equation zmi— 441x + 2430 = 0 6. Expand a*, and deduce from the expansion an expression-in a converging series—for the logarithm of a number to the base a. PHYSICAL SCIENCE. DIVISION 1. Section 1. 1. A barometer. Section 2. 1. A locomotive engine. Section 3. 1. What must be the diameter of an iron wire to sustain ll cwt., the tenacity of the iron being 25 tons per square inch ? 2. How long will ten men be in pumping dry the hold of a ship which contains 3,000 cubic feet of water, the centre of gravity of the water being 14 feet below the deck, and each man being supposed to yield 1,500 units of effective work per minute ? 3. Describe the wheel and double axle, and show how the relation of the power to the weight may be determined in this machine. Section 4. 1. What experiments serve best to show the expansion of metals, and the currents produced in liquids and gases by the application of heat ? 2. Give some account of nitrogen and of its compounds. 3. By what different methods may the chemical constitution of water be ascertained ? a a Division II. receive marks according to the merit of the answers; but no answers be unsatisfactory. 1. Show how it may be determined, by geometrical construction, whether the wall of a reservoir will stand or not, and prove fully the rule you use. 2. Wanting to determine the quantity of water discharged per minute from a pipe, I place under it a vessel, in the bottom of which is a rectangular aperture which I can close by means of a slide. I find that when the dimensions of the aperture are 4 inches by 3 the water stands steadily in the reservoir at a height of 3 feet. What is the efflux per minute from the pipe, neglecting the effects of contraction ? 3. Given the number of units of work, u, which a man can do on each stroke of a pump, the depth, a, of the water in the well beneath the bottom of the barrel, and the section, k, of the barrel. It is required to find the length of the stroke, so that the whole work on each stroke may just be expended in raising the water. 4. What experiments show the influence of the state of the surface of a body on the radiation of its heat, and what experiments show the existence of latent heat ? 5. Explain, fully, what is meant by the law of chemical equivalents, and what by that of gaseous volumes. Give some account of chemical nomenclature and chemical notation. 6. What are the chemical properties of oils and fats? What application is made of those properties to the uses of life? QUESTIONS IN FARMING. 1. Write out a farmer's calendar from February to June. 2. How has the ground to be prepared for turnips? What is the time for sowing them? What rules are to be observed in hoeing turnips ? 3. What is meant by the four course system? What are the advantages of keeping up stock as compared with pasture? What rules are to be observed in applying liquid manure ? MUSIC. DIVISION I. Section 1. 1. Write out the substance of a lesson on beating time, with examples of all the ordinary cases. Section 2. 1. How is a very gradual increase or decrease of velocity or of loudness expressed? When a note is to be forced how is it distinguished ? 2. How many sounds are there in a tetrachord? How many tetrachords are there in a major scale? Give the names of the sounds of the tetrachord. 3. What rules ought to be observed with regard to the turn in singing ? Write out the following melody with a direct and with an inverted turn introduced. Section 3. 1. Supposing that in a musical composition the time were to change from to 6, what change would be required in the manner of beating? What 4 would be the value of each of the beats in each case ? 2. In what scale is the following passage written? State how you determine it. 3. What is meant by reading in time? Show how the following passage should be read in time. to SUPPLEMENTARY PAPER. DIVISION II. will receive marks according to the merit of the answers; but no first part be unsatisfactory. 1. Explain, as you would to a class, what is meant by the terms prime, harmonic chord. |