5. In an obtuse-angled triangle, if a perpendicular be drawn from either of the acute angles to the opposite side produced, prove that the square of the side subtending the obtuse angle is greater than the squares of the sides containing the obtuse angle by twice the rectangle contained by the side upon which, when produced, the perpendicular falls, and the straight line intercepted without the triangle, between the perpendicular and the obtuse angle. If the triangle ACB be isosceles, C the obtuse angle, AD the perpendicular on BC produced, prove the square on AB is equal to twice the rectangle contained by BC and BD. 6. Prove that equal straight lines in a circle are equally distant from the centre. Shew that all equal straight lines in a circle may be touched by another circle. 7. When are segments of circles said to be similar? ABC is an acute-angled triangle, perpendiculars AP and BQ are drawn from A and B to the opposite sides meeting in O; prove that the angles POC and PQC are equal. 8. Describe a square about a given circle. Compare the areas of the squares described about and inscribed in the same circle. 9. An isosceles triangle, having each angle at the base double of the angle at the vertex, is inscribed in a given circle; complete the construction for inscribing a regular pentagon in the circle, and prove that it is equilateral. Shew generally that a pentagon may be equilateral without being equiangular, and equiangular without being equilateral. 10. When are rectilineal figures said to be similar, and when similarly situated? On a given straight line describe a trapezium similar and similarly situated to a given trapezium. What must be the ratio of the given line to the homologous side of the given figure, so that the figure described upon it shall be nine times the given figure in area? II. Prove that in any right-angled triangle any rectilineal figure described on the side subtending the right angle is equal to the similar and similarly described figures on the sides containing the right angle. Is this true if all the figures so described be semi-circles? II. ARITHMETIC. (Obligatory.) (Including the use of Common Logarithms.) [N.B.-Great importance will be attached to accuracy in numerical results.] 4. Find the quotient of divided by 1; divide the result by 0. 5. Add together 6375 of a gallon, 01283 of a peck, and '00412 of a bushel; subtract the result from 93 of a gallon, and give the answer in pints and the decimal of a pint. 6. Multiply 5'3129 by 46000. 7. Divide 1 by '00375. 8. Divide 3'81 by 2'227, and express the answer as a decimal correct to six places. 9. Reduce ti of half a mile to the decimal of 6 poles. 10. Express 2 lbs. 5 oz. 15 dwts. 12 grs. as the decimal of 7 lbs. Troy. II. 12. Divide 317 days 13 hrs. 31 mins. 57 secs. by 53. Find the dividend on £4146. 12s. 6d. at 11s. 8d. in the £. 13. What principal will amount to £2775. 195. in 6 years and 4 months at 7 per cent. per annum simple interest? 14. If 400 metres be equal to a quarter of a mile, find the number of square metres in a quarter of an acre. 15. A person sold a horse at a loss of 20 per cent.; if he had received £10 more for it, he would have gained 10 per cent. Find the cost of the horse. 16. A man rode a bicycle from A to B, 54 miles, at an average rate of 8 miles an hour; another man started from A on horseback half an hour after the bicyclist and arrived at B 15 minutes before him. Find the ratio of their speeds. 17. Divide the cube root of 240 17712 by the square root of 176400. 18. Find (by duodecimals) the cubical content of a block of stone 4 ft. 7 ins. long, 4 ft. 2 ins. 5 pts. wide, and 2 ft. 7 ins. 6 pts. high. Express the answer in cubic feet, cubic inches, and the fraction of a cubic inch. 19. A steamer A is in distress and stationary and fires a gun which is heard on another steamer B coming direct towards her. A fires another gun ten minutes after the first, and this is heard on B 9 mins. 48 secs. after the first. At this moment B is 4 miles from A. How soon will the two steamers be alongside of each other? Give the answer to the nearest second, supposing sound to travel at the rate of 1130 feet per second. 20. A man sold 150 one-hundred-pound shares of railway stock which were paying 5%, at 105. With the proceeds he purchased 4 per cents. at 90 and re-sold them at 96. He then re-invested in the railway stock, which was still at 105 and paying 5%. What was the change in his income? 21. Two sums of money amounting to £1946. 5s. were invested, the smaller at 4 per cent. and the larger at 4 per cent. per annum. At the end of 18 months the simple interest on the two sums amounted together to £125. 45. 10d. What were the two sums? 22. Find log *024, log (234), and log(15) × (51)3, and find the number whose log is 3'1187804. III. ALGEBRA. (Obligatory.) (Including Equations, Progressions, Permutations and Combinations, and the Binomial Theorem.) [N.B.-Great importance will be attached to accuracy in results.] 8. A grocer has two weights, one as much over a lb. as the other is under a lb., and he finds that on selling 511 lbs. 14 oz. of tea at 2s. 6d. a pound he gains £2 more by using the lighter weight than he would have done by using the heavier; what were the respective weights? 9. Find the sum of an arithmetic series whose first term is a and common difference d to n terms. IO. The first and third terms of an arithmetical progression are 22 and 14 respectively; how many terms must be taken that the sum may be 64? Explain the double answer. If x-a, y-a, and z-a be in geometrical progression, prove that twice y-a is the harmonic mean between y-x and y-z. II. If a b::c:d, and that 12. prove that ma+nb: ma- nb :: mc+nd: mc-nd, a2+c2: b2+ d2 :: ac: bd. Write down the number of permutations of n things r together, and deduce the number of combinations. Eight men sit down to two tables at whist; in how many different ways may the games be arranged? 13. Write down the term involving xr in the expansion of (1+x)" and in the expansion of (1-x)-". If p and q be two numbers whose difference q-p is very small compared with either of them, prove that |