12. When does one quantity vary as another ? If A a B, C, D, &c., when only one of the quantities is changed, show that A a BCD... when all change, Apply this principle to the following example:If 10 men do a piece of work in 12 days of 12 hours each, in what time will 23 men do three times as much, each working 9 hours per day? 13. Find (1) the number of permutations which can be formed from the letters of the word Sebastopol, taken all together. (2) the number of combinations when three letters are taken together. EUCLID, ALGEBRA, AND TRIGONOMETRY. Set to Candidates for the Office of the Committee of Council on Education. Note.- In this Examination Mathematics are not prescribed, but may be selected by any candidate who has made them his especial study," with the view of displaying his industry and intelligence. 1. If two triangles have two sides of the one equal to two sides of the other, each to each, but the angle contained by the two sides of one of them greater than the angle contained by the two sides equal to them, of the other; the base of that which has the greater angle shall be greater than the base of the other. Book III. 2. Prop. 20. — The angle at the centre of a circle is double of the angle at the circumference upon the same base, that is upon the same part of the circumference. BOOK IV. 3. Prop. 11.— To inscribe an equilateral and equiangular pentagon in a given circle. Book VI. 4. Prop. 18.— Upon a given straight line to describe a rectilineal figure similar, and similarly situated, to a given rectilineal figure. -5. A common tangent is drawn to two circles which touch externally; prove that if a circle be described on that part of it which lies between the points of contact, as diameter, it will pass through the point of contact of the two circles. 6. Inscribe a circle in a given quadrant of a circle. 7. Divide 4 b x3 + (4c – ab) x? — (4d + ac) x + ad by 4x 2 a vit x2 8. Find the value of when x = x + 1 + x2 a. + + 9. Reduce to its simplest form the expression 1 4 2:3 (x y) 2.2.2 (212 + y) 10. Solve the following equations : 12 5 x 24 (1) -(1-1) ** + 2 y + 32 = 17 (3) y + 2 z + 3x = 13 -2 + 2 x + 3y = 12 (4) 32 + xy + y2 = a? 24 + xy2 + y = 64 11. A and B have the same annual income, and occupy lodgings for 30 weeks in the year, the former at 14s., the latter at 21s. per week, all other expenses being exactly the same for both : B exceeds his income by as much as A comes short of his, and finds that he has spent one-tenth too much : Required the annual income and the whole expenditure of each. 1 3 2 to 14 terms. : + + . 10 COS 13. Write down the expansion of (3 x – 4y)”, and by means of the bi. nomial theorem approximate to 31/31. 14. Find the values of tan 30° and of sin 18°. 15. Prove A - B A + B (1) Sin A - sin B=2 sin 2 2 A 16. Having given the numerical value of sin A, find that of cos and show that there ought to be four corresponding values. Determine which is the proper value when A lies between 180° and 270°. 17. In a plane triangle, having given two sides and the included angle, obtain the formulæ for solving the triangle. Ex. Given a = 205, b = 195, C = 4°, 1102 = 30103, Lcot 20: 11.4569162, L cot 54° 20'= 1.8559376, Lcot 54° 21'= 9.8556708; find the remaining angles. 18. What are the advantages of employing the number 10 as the base for logarithms? Having given the logarithms of a number to the base e, show how to find the logarithms of the same number to the base 10. Given log 10 71968 = 4.8571394; diff. for l = 60: find the value of 8_•0719686 to seven places of decimals. -a. {Viva 2 5 ALGEBRA. Set to Candidates for the Admiralty, who selected Algebra as a Subject of Examination. 1. Divide 4 b 23 + (4c – ab) – (4 d + ac) x + ad by 4 2 a w1 + x2 2. Find the value of when x = x+wl + x2 ] 1 2x2 + y2) 4. Show that the product of two quantities equals that of their greatest common measure and least common multiple. Find the greatest common measure of 35 x3 + 47 23 + 13 x + 1 and 42 x4 + 41 203 - 92 - 9 x - 1. 5. Solve the following equations 12 5 (1) x 24 1 1 1 + 3 35 z + 2 x + 3y = 12 ** + xy2 + y = 64 6. A and B have the same annual income, and occupy lodgings for 30 weeks, the former at 14s., the latter at 21s. per week, all other expenses being exactly the same for both : B exceeds his income by as much as A comes short of his, and finds that he has spent one tenth too much : Required the annual income and the whole expenditure of each. 7. Find the sum of the following series : 1 3 2 (1) to 14 terms. 10 5 4 16 (2) + 3 - &c. to 10 terms, and to infinity. 15 (1-12.)= + + + 8. Write down the expansion of (3 x – 4y)”, and by means of the bi nomial theorem approximate to 3 131. 9. What are the advantages of employing the number 10 as the base for logarithms? Having given the logarithms of a number to the base e, show how to find the logarithms of the same number to the base 10. Given log 1071968 = 48571394; diff. for 1= 60: find the value of 8.0719086 to seven places of decimals. 10. Solve the following equations : (1) Va- 712 + x2 с 2.c * (19) 2—* (3) (2}) x 532–5 = (14)*+1 11. Insert four harmonic means between 2 and 12. 12. Find what number r out of n things must be taken together so that the number of combinations formed may be the greatest possible. 13. When does one quantity vary directly as another, and when inversely as another? Given that y varies as the sum of two quantities, one of which varies y as x directly, the other as x inversely; and that when x= 1, y = 4, when x = 2, y = 6: Find the relation between x and y. 14. In what scale of notation is sixteen-hundred-and-sixty-four ten-thou sandth's of unity represented by .0404 ? GEOLOGY. Prepared for an Examination of Candidates for the Colonial Office. 1. Define the terms anticlinal, synclinal, unconformable, strike, and dip. 2. State the reasons for the division of rocks into igneous and sediment ary. What are metamorphic rocks? 3. What are the constituent minerals of granite, basalt, greenstone, gneiss, trachyte? 4. Describe the divisions of the wealden formation, and give a sketch of its distribution in England, and the reasons for looking on it as a freshwater deposit. 5. Where are the points of division placed by geologists to separate the hypozoic, the palæozoic, the mesozoic, and the cainozoic strata ? Exemplify the principles on which these divisions have been founded. 6. Coal has been accounted for sometimes as the result of drift, by water, of masses of vegetable matter; sometimes as an accumu. lation of such matter by growth in situ. What hypotheses do these views involve, and what circumstances lend probability to each view? 7. Give an accurate description of the stigmaria and the sigillaria, and of the facts that prove their mutual relation, with the most cha racteristic mode of their occurrence in the strata. 8. Describe the mountain limestone formation. How is it distributed over the world ? 9. By what observations and arguments does the geologist seek to deter mine the period of elevation of a mountain chain? Illustrate this by some example. 10. Show that, by the amount and by the characters of the distribution of organic remains in one and the same rock in different localities, we may predicate facts concerning its oceanic and littoral deposition, pointing to the limits of the sea in which it was formed. Give illustrations of this. 11. Give a description of the most important characteristics common to the trilobites, and give the history of their distribution in time. 12. What are the usual characters of a mineral vein ? How far is its wealth found to depend on the rock it traverses ? Describe the methods adopted by the practical miner for the discovery of a lode. 13. Describe the structure of the ammonite, and give an account of the distribution of its species in time. 14. Trace the changes in the character of the zoology during the oolitic period, as illustrated by the reptilia and the cephalopoda. 15. Describe some of the fossils characteristic of the chalk. 16. Give a sketch of the geology of the Malvern Hills. CHEMISTRY. Prepared for an Examination of Candidates for the Colonial Office. 1. Define the term element. What elements are gascous, what are liquids, under the ordinary conditions of the globe? What changes do these undergo by considerable alteration of such conditions? 2. State the law of multiple proportions; and illustrate it ly means of the oxides (1) of nitrogen (2) of manganese. 3. Give the chemical names of, and write in formula', alum, common salt, green vitriol, calomel, corrosive sublimate, and chloride of lime. 4. The equivalent of aluminium is 13.7. How much per cent of oxygen, of sulphur, and of aluminium is contained in the anhydrous normal (or neutral) sulphate of alumina : 5. Of what gases does the atmosphere consist? Give any accurate method of effecting its analysis; and state the results of this ana lysis. 6. Explain the changes resulting from the action (1) of hydrochlorie acid, (2) of strong nitric acid, (3) of very dilute nitric acid, on gold, |