SECTION V.-1. Describe an isosceles triangle, each of the sides of which shall be double of the base. 2. Show that in a parallelogram the sum of the squares of the diagonals is equal to the sum of the squares of all the sides. 3. The arcs intercepted between any two parallel chords in a circle are equal. HIGHER MATHEMATICS. Section I.--). The sum of an arithmetic series is 440, the first term 3, and the common difference 2. What is the number of terms ? 2. Prove the rule for the number of permutations of n different things taken all together. In how many different ways may the letters forming the word Buckingham be arranged ? 3. Prove that if the sum of the digits of any number be divisible by 3, the number itself is divisible by 3. Section II.--1. Investigate a rule for finding the present value of £a, due n years hence, r being the rate of interest. 2. A person is entitled to an annuity of £100, commencing three years hence, and continuing for three years. He wishes to exchange it for an annuity to commence immediately, and terminate at the same time with the other annuity. The rate of interest being 5 per cent.; to what amount of annuity is he entitled ? 3. What is a logarithm? Find a logarithmic expression for the value of x in the equation a b + c = e. d Section III.-1. Define the secant of an angle, and find its value in terms of the sine. 2. tan. A + cot. A V3 Find the values of sin. A. and cos. A. 3. Given an angle (A) of a triangle, and the sides (b, c) containing it, in. vestigate formulæ for the values of the opposite side (a), and the remaining angles (B, C). Section IV.–1. Prove that the sides of a triangle are proportional to the sines of the opposite angles. 2. Find the value of sin. 18°. 3. The elevation of a tower is observed from a certain distance. At a station a feet nearer, the elevation is the complement of the former; b feet nearer still, it is double the first elevation. What is the height of the tower? SECTION V.-1. Explain what you mean by a system of coordinates, and by the equation to a line, and find the equation to a straight line which cuts the axis of x at a distance from the origin a, and the axis of y at a distance b, the system of coordinates being rectangular. 2. What do you mean by a conic section ? Find the equation to the. ellipse, the extremity of the axis major being the origin of coordinates, and the axis major being on the axis of x: and from it deduce the equation when the origin is at the centre of the ellipse. 3. What do you mean by maxima and minima of a function of a variable. Investigate a rule for their discovery, and find the largest possible area of a right-angled triangle, the hypothenuse of which = c. LIST OF MASTERS WHO HAVE RECEIVED CERTIFICATES. The Committee of Council on Education have awarded Certificates of Merit, after Examination before Her Majesty's Inspectors, at Easter, 1851, to the following Masters in National and other schools in connection with the Church of England, and Students in the Exeter Training School : . . THIRD DIVISION. John Austin Axminster N. S. William Branter Ingrow N. S. Skirbeck N. S. Aldeburgh, N.S. Market Weighton N. S. Henry William Humphreys . Bradford Ch. Ch. N. S. John Loam Tavistock Ch. of E. S. James Macfarland Oakhain N. S. William Moss . Manchester, All Souls N. S. George William Oliff Grantham N. S. John Oliver Brampton Brian N. S. (Salop.) William Pearson . Leeds, St. Peter's Ch. of E. S. Henry Pinder London, St. Sepulchre's Parochial. Edward Poore Cranley N. S. Henry Smith Seiston N. S., near Saxmundham. Wellow, Romsey Ch. of E. S. 180 . . . . ANSWERS TO THE MATHEMATICAL QUESTIONS. Ques. 99.—Proposed by J. H. A SPHERICAL balloon, 30 feet in diameter, constructed of silk, weighing 2 ounces 9 drams per square yard, is filled with hydrogen gas, the specific gravity of which is to that of atmospheric air' as .069 to 1, a cubic foot of atmospheric air weighing 14 ounce : what weight will just balance the balloon to prevent it from rising in the atmosphere? Answered by Mr. Rowlatt, Mr. Hill, Mr. Salter, Mr. Royds, and Mr. Righton. Area silk =302 x 3.1416 sq. ft. =314:16 sq. yds. = 216 x 314.16=805.035 oz. =15647•1315 oz. =8 cwt. 2 qr. 25 lb. 15.1315 oz. Ques, 105.- Proposed by Mr. E. Whittle. The expense of paving a court-yard with Dutch clinkers, at 5s. per hundred, was 401. ; required the dimensions of the court-yard, allowing 144 clinkers to a square yard, and that the length of the court-yard exceeds its breadth by 15 feet. Answered by Mr. J. Bolton, Mr. Hill, and Mr. Bolton. Let x=the breadth of the court-yard, 40 x 20 x 100 = 16000; 5 16000 .. area = x 9 144 1000 sq. ft. ; .'. x (x + 15) = 1000; and the length=25 +15=40 feet. . Ques, 101.- Proposed by Nemo. Given the base (6) of an isosceles triangle, to find its perpendicular height, when the area of the inscribed circle has a given ratio (p) to the area of the whole triangle. Answered by Mr. J. Sheppard, Mr. Morris, A. M., Gillingham, and Emma Royds. the radius of the circle r; the height of the triangle =or; and half the Let c; then each of the equal sides = N 02 72 + c2; (1) (3) + from this cubic equation v may be found; and then by substitution r may be determined from the equation r = pco LIST OF MATHEMATICAL ANSWERS, A. M. Gillingham, ans. 99, 100, 101; J. Sheppard, ans. 99, 100, 101; J. Royds, Belfield, ans. 99, 100, 101; J. Rowlatt, Evercruch, ans. 99, 100; H. Hill, Mortlake, ans. 99, 100; J. Herbert, Woolton, ans. 99, 100, 101; J. Salter, Durham, ans. 99, 100; W. Brown, Ightham, ans. 100; F. Dungate, Ightham, ans. 99; G. Scott, Low Moor, ans. 99, 100; J. Webb, ans. 100; J. Fox, Burton-on-Trent, ans. 100; J. Reid, Brendon, Cornwall, ans. 99, 100; T. Horsman, Chelsea, ans. 100; D. O'Sullivan, Preston, ans, 99, 100, 101; W. Righton, jun., ans. 99, 100, 101 ; W. Davis, Coniston Cold, ans. 99, 100; J. B. Bayley, 99, 100; G. Morris, Gosport, ans. 99, 100, 101; C. Smith, ans. 100 ; J. Bolton, Old Malton, ans. 99, 100; Sam. Dyer, ans. 99, 100; C. H. Kennion, Tamworth, ans. 100; E. Carthew, Roehampton, ans. 99, 100. . NEW QUESTIONS, Ques, 102.- Proposed by Mr. Abbott, Twickenham. If 3lbs. of tea be worth 7lbs. of coffee, and 12lbs. of coffee worth 30lbs. of (sugar, what quantity of sugar may be had for a chest of tea weighing 50lbs. ? Ques. 103.- Proposed by H. M. I. If the modulus of a hydraulic ram be •87, the water spent 18 gallons per minute, the fall of the water 14 feet; to what height will it raise a cubic foot of water in 80 seconds ? Ques. 104.--Proposed by Tom Tomkins. Required an expression for the side of an equilateral triangle inscribed in a given parabola ? Intelligence. NATIONAL SOCIETY.—The annual meeting of this Society was held on the 4th June, at the offices of the Society at Westminster, and was most numerously attended. No. tices of motion having been given, which seemed likely to lead to considerable discussion, and to interrupt the harmony with which the proceedings have been usually conducted, the following statement was offi. cially published through the Times the day before the meeting : " The committee of the National Society, to whom the management of its affairs is by the charter exclusively intrusted, earnestly deprecate the discussion which they have reason to expect at the general meeting of the society on Wednesday next. They consider each and all of the propositions of which individual members of the society have given notice alike unnecessary, and they deem the public discussion of them at the annual meeting calculated to embarrass the operations and to impair the efficiency of the society. They believe that there is in the public mind some misconception as to the position of the society in respect of its relations to the Committee of Privy Council on Education, and they had intended that a resolution should have been proposed to the general mee:'ng, declaratory of the views of the committee on this subject. But the committee, fearing that by that course they might appear to encourage such discussion, have on consideration resolved not to adopt this course. It seems to them necessary to announce this change of intention, and at the same time to declare that the committee consider it of the utmost importance to preserve harmonious co-operation between the National Society and the Committee of Council on Education; and while they regret the continued adherence of the Committee of Council to the resolution of excluding from all share of the Parliamentary grant for building school houses those Church schools the promoters of which decline to constitute their trust deeds on the model prescribed by their Lordships, thereby interfering with that prin. ciple of local freedom on which the society has ever voted its grants ac |