orae-defluit saxis agitatus humor-vile potabis modicis Sabinum cantharis-minuit furorum vix una sospes navis ab ignibus-prae ardore impetuque tantae rei sensus non pervenit ad militem -ubi nunc navalia sunt-majus flagitium in Algido, major etiam clades accepta. 2. If a, 6 be the roots of c) (b + c a)3 b)3 (a — b), and find ( +1)(k + 1) = nhe (hæ − 1), prove that (a2 + 1)(62 + 1) = maß (aß — 1). 4. Explain what is meant by a" where m is fractional or negative, and prove that {am}" = amn for all real values of m and n. 6. Shew how to find the greatest and least values, if any, of the expression for real values of x, and shew that they occur when x is a root of the quadratic (ab'a'b) x2 - 2 (ca'-c'a) x + be b'e 0. = 7. Describe the method of mathematical induction. and apply it to prove that the square of any multinomial consists of the square of each term, together with twice the product of every pair of terms. 8. Shew that in an infinite geometrical progression, in which the common ratio is less than unity,. each term bears a constant ratio to the sum of all which follow it. If a, b, c be the pth, 7th, and th terms respectively of both an arithmetical and geometrical progression, prove that abc bc-a cab.— 1. 9. Find the formula for the number of permutations of n things r at a time. Find the number of ways in which n books may be arranged on a shelf so that two particular books may not be together. 10. Investigate the formula for the product (x + a1) (x + α2) (x + α3) ...... (x + an) in a series according to ascending powers of x. If 4. be the coefficient of a" in the product, and In the first six questions the symbol - must not be used; and the only abbreviation admitted for "the square described on the straight line AB" is "sq. on AB," and for "the rectangle contained by the straight lines AB, CD" is "rect: AB, CD." 1. Shew that the locus of a point, such that the triangles formed by joining it to the extremities of two given straight lines are equal, is two straight lines. 2. If two sides of a quadrilateral are parallel, the sum of the squares on the diagonals exceeds the sum of the squares on the other two sides by twice the rectangle contained by the firstmentioned sides. 3. If from a given point two tangents be drawn to a circle, and from the extremities of any diameter lines be drawn to the points of contact, cutting each other within the circle, the straight line which is drawn through their intersection to the given point shall be perpendicular to the diameter. 4. A straight line drawn parallel to one side of a triangle and cutting the other two cuts them proportionally. ABCD, EFGH are parallelograms on equal bases BC, FG, and between the same parallels. BE, CH cut CD, EF in K and L respectively. Prove that if the points A, K, L, G be in a straight line DC is parallel to EF. 5. In a right-angled triangle, if a perpendicular be drawn from the right angle to the base, either side is a mean proportional between the base and the segment of the base adjacent to that side. The circumference of one circle passes through the centre of another; through A, one of the points of intersection, a diameter AB of the first is drawn, meeting the other in C; prove that rectangle AB, AC is twice the square on OC. 6. Prove that if straight lines be drawn from a fixed point to points on a circle and be divided in a fixed ratio, the dividing points lie on another circle. 7. Describe the three common methods of measuring angles. Shew that there are the same number of grades in an angle of a regular octagon as of degrees in that of a regular dodecagon. 8. Shew how to solve a right-angled triangle. Straight lines are drawn from the angles of a triangle bisecting them and cutting the opposite sides into segments a, a'; ß, B'; y, y', taken in order. Prove that aßy = a'B'y'. 9. If and sin A sin B = sin P sin Q, cos A. cos B = cos P cos Q, and cos2 A+ cos2 B = cos2 (P + Q) + 1, sin2 P+ sin2 Q = sin2 (P + Q). If 0 is so small that cos (0 — 4) may be taken as 1, prove that 0+¢ cot 0+cot 2 cot 2 |