If a, b, c be in geometrical progression prove that (Candidates must answer satisfactorily in each of the three divisions of this paper.) I.—1. If from the vertical angle of a triangle a straight line be drawn perpendicular to the base, the rectangle contained by the sides of the triangle is equal to the rectangle contained by the perpendicular and the diameter of the circle described about the triangle. 2. Prove that the tangent to a parabola at any point is the bisector of the angle which the focal distance of the point makes with the diameter produced. 3. Prove that the projections of the two foci upon any tangent to a central conic lie on the auxiliary circle. 4. Prove that the triangle contained by the asymptotes and any tangent to a hyperbola is equal to the rectangle contained by the semi-axes. 5. Prove that in any conic the semi-latus rectum is C a harmonic mean between the segments of any focal chord. II.-1. If ✪ be the circular measure of a positive angle less than a right angle, prove that sin 0, 0, tan ✪ are in ascending order of magnitude. 2. Prove that in general the change of the tangent of an angle is approximately proportional to the change of the angle. 3. Shew that q different values and no more can be found for the expression p {cos +(-1) sine 3. 4. Prove Gregory's series for the expansion of tanla in powers of x. 5. Sum to infinity the series cos a + c cos (a +ß) + c2 cos (a + 2ß) + &c. III.-1. If a1, A2, a, be the roots of an equation .... f(x) = 0 of the nth degree which is in its simplest form, shew that − ƒ (x) = (x — a1) (x — α2) .... (x — α„). 2. Prove that an infinite series is convergent if from and after some fixed term the ratio of each term to the preceding term is numerically less than some quantity which is itself numerically less than unity. Shew that the series 13 + 33 x + 53 x2 + 73 x3 + &c. is convergent if a be numerically less than unity. 3. State and prove the exponential theorem. 4. State and prove the rule for finding the sum of In terms of the series whose nth term is (a n + b) (a n + 1 + b) . . . . (a n + m −1 + b). which the denominators are all of the same sign, prove that lies in magnitude between the least and greatest of the aforesaid fractions. ADVANCED MATHEMATICS. Professor Nanson. 1. Explain and illustrate what is meant by the word "limit" in the differential calculus, and find the limit of for the value infinity of the independent variable x. 2. State and prove the rule for finding the differential coefficient of a function of a function. 4. State and prove Leibnitz's Theorem, and find the nth differential coefficient of 3 sin (ax + b). 6. Find the equation to the straight line passing through two given points. Deduce the equation to a straight line in terms of the intercepts it makes on the axes. 7. Find the equation to a circle whose centre and radius are given. Form the equation to the circle having the line joining the points h, k; h', k' for a diameter. 8. Shew that if the conic ax2 + 2hxy + by2 + c = 0 when referred to its axes takes the form then a, ẞ are the roots of the quadratic 9. Define the excentric angle of any point on an ellipse, and find the equation to the tangent at a point whose excentric angle is given. 10. Find the locus of the middle points of a set of parallel chords of a parabola. 11. Explain the process of integration by substitution, and shew how to find the limits of the transformed 13. Shew how to find the partial fraction corresponding to a single factor of the first degree when a rational fraction is decomposed into partial fractions. 14. Find the area of a loop of the curve a1y2 + b2x2 = a2b2x2. MIXED MATHEMATICS.-I. Professor Nanson. 1. Find the distance of a given point from a straight line whose equations are given in the symmetrical form, and find the equation to the plane containing the point and line. 2. Determine the circular sections of a central quadric |