APPENDIX, Containing the QUESTIONS proposed in several years during the first three days of the SENATE-HOUSE EXAMINATION, in accordance with the Grace of the SENATE passed in May 1846. 1848. MODERATORS. GEORGE GABRIEL STOKES, M.A., Pembroke College. EXAMINERS. WILLIAM NATHANIEL GRIFFIN, M.A., St John's College. CHARLES OCTAVUS BUDD, M.A., Pembroke College. THURSDAY, Jan. 6. 9...12. 1. If two triangles have two sides of the one equal to two sides of the other, each to each, and have likewise their bases equal, the angle which is contained by the two sides of the one shall be equal to the angle contained by the two sides, equal to them, of the other. How does it appear that the two triangles are equiangular and equal to each other? 2. The opposite sides as well as the opposite angles of a parallelogram are equal to one another, and the diameter bisects it. If the two diameters be drawn, shew that a parallelogram will be divided into four equal parts. In what case will the diameter bisect the angle of a parallelogram? 3. If a straight line be divided into two equal parts, and also into two unequal parts, the rectangle contained by the unequal parts, together with the square of the line between the points of section, is equal to the square of half the line. t 4. 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. Shew that all equal straight lines in a circle will be touched by another circle. 5. 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. If two straight lines AEB, CED in a circle intersect in E, the angles subtended by AC and BD at the centre are together double of the angle AEC. 6. Describe an isosceles triangle having each angle at the base double of the third angle. 7. If the first of six magnitudes have to the second the same ratio which the third has to the fourth, but the third to the fourth a greater ratio than the fifth to the sixth, the first shall have to the second a greater ratio than the fifth has to the sixth. 8. Equal triangles which have one angle of the one equal to one angle of the other have the sides about the equal angles reciprocally proportional; and conversely. 9. If two planes which cut each other be each of them perpendicular to a third plane, their common section shall be perpendicular to the same plane. 10. Define a parabola, and its tangent. Assuming the tangent at any point P of a parabola to make equal angles with the focal distance SP and the diameter at that point, prove that SY, the perpendicular upon it from the focus, meets it in the tangent at the vertex. If PM be the ordinate at P, and T the intersection of the tangent at P with the axis, TP. TY = TM . TS. 11. Prove that in a parabola QV2 = 4 SP . PV. Define the parameter at any point of a parabola, and prove that it is proportional to the focal distance of the point. 12. Define an ellipse. If one of the focal distances. SP of a point P be produced to L, a straight line PT which bisects the exterior angle HPL is the tangent to the curve at P. For what position of P is the angle SPH greatest? 13. State when diameters of an ellipse or hyperbola are conjugate. Prove that all parallelograms whose sides touch an ellipse at the ends of conjugate diameters are equal. Prove that such parallelograms have the least area of all which circumscribe the ellipse. 14. In an ellipse the sum of the squares of any two conjugate diameters is invariable. When is the square of their sum least? 15. Define the asymptotes of an hyperbola. If any straight line Qq perpendicular to either axis of an hyperbola meet the asymptotes in Q, q, and the curve in P, the rectangle QP.Pq is invariable. 16. If a circular right cone be cut by a plane which meets both of its slant sides, the section is an ellipse. 17. The chord of curvature of an ellipse or hyperbola 2 CD2 through its centre is equal to CP The chords of curvature through the centre and focus are in the ratio of AC to CP. THURSDAY, Jan. 6. 11...4. 1. PROVE the rules for finding the greatest common measure and least common multiple of two integers. Find the least number of pounds which can be paid in either half-crowns or guineas. 2. Explain the meaning and use of fractions in a system of arithmetic, and shew that the value of a fraction remains unchanged when its two members are replaced by any equimultiples of their former values. 3. If each inmate of a workhouse cost per week, 28. 24d. for food, 51d. for clothes and washing, and 3d. for lodging, and the number of persons thus maintained be estimated at 100000, what is the whole yearly cost; and how many labourers at 18d. a day wages would earn the same sum in the same time? 4. If 25 men do a piece of work in 24 days, working 8 hours a day, in how many days would 30 men do the same piece of work working 10 hours a day? 5. Supposing arithmetical addition and subtraction represented by the signs +, respectively, prove the equivalence of the two sets of operations indicated by (a - b) (cd) and ac + bd - bc - ad, a being a number greater than b, c greater than d. What is the nature of the generalization with respect to the use of the signs +, in Symbolical Algebra ? - 6. Find the highest common divisor of x3- x2 - 2x + 2 and x − 3 x3 + 2x2 + x − 1. 7. Define what is meant by the nth root of a given numerical magnitude, and explain the principle according to which a root is represented by means of a fractional index. 8. Define a logarithm, and the base of a system of logarithms. Prove that log xy = log x + log y, log x" = n log x. Given log 7 8450980, log 587514-7690153, log 5875247690227, find ✔07 to 7 significant figures. Discuss the nature of the roots in (2) as affected by the value of m. 10. When is one directly or inversely? quantity said to vary as another, If A & B when C is constant, and A ∞ C when B is constant, prove that A ∞ BC when B and C both vary. Given that the area of an ellipse varies as either axis when the other is constant, and that the area of a circle of radius unity 3.14..., find the area of the = ellipse whose axes are 3 and 5. 11. Find the sum to n terms of a geometric series. What is meant by the sum of an infinite series? When can such a series be said to have a sum? If P be the sum of the series formed by taking the 1st and every pth term of an infinite geometric series whose first term is 1 and whose ratio <1, Q the sum of the series found by taking the 1st and every qth term, prove that |