ENTRANCE EXAMINATION. MR. W. ROBERTS. 1. Prove that the triangles whose sides are, respectively, 19, 23, 37, and 323, 391, 629, are similar; and find the ratio of their areas. 2. Give the construction in Euclid, Book II., for dividing a line so that the rectangle under the whole line and one part may be equal to the square of the other part; prove that the rectangle under the segments is equal to that under the sum and difference of the segments. 3. Solve, by means of a quadratic equation, the problem of dividing a line so that the rectangle under the whole line and one part may be n times the square of the other part. 4. Prove that the bisector of the vertical angle of a triangle divides the base into segments proportional to the sides. 5. Prove that the rectangle under the sides of a triangle is equal to that under the diameter of the circumscribing circle, and the perpendicular let fall from the vertex on the base. 6. What is the logarithm of 1, and what that of o? 7. Find the value of the continued product (a+b+c) (a + b − c ) (a + c − b ) (b + c − a). 8. Being given that log 3=0.47712, log 5=0.69897, log 7=0.84510, find the logarithm of 43. 9. Raise 1+1 to the fourth power. 10. Find the area of a triangle whose sides are 31.2, 47.8, 39.6. 1. The sides of a right-angled triangle are 85 and 204; find the hypotenuse, and the perpendicular from the right angle let fall upon it. 2. Find the values of √128+√72, and of 4√3√ 3. Multiply 2+3/-2 by 3-2-1. 4. Solve the equation I. EXAMINATION FOR THE DEGREE OF BACHELOR OF ARTS. Moderatorships in Mathematics and Mathematical Physics. Examiners. ANDREW SEARLE HART, LL. D. JOHN H. JELLETT, B. D., Professor of Natural Philosophy. RICHARD TOWNSEND, M. A. GEORGE F. SHAW, LL. D. PLANE GEOMETRY. DR. HART. 1. Find, in terms of the invariants and covariants of the trilinear equations of two conics, the general equation of conics which touch their common tangents. 2. Find the locus of a point from which tangents to two given conics make a harmonic pencil; and determine the relation between the invariants that the locus may be two right lines. 3. If three conics have double contact with the same conic S, show how to find their points of contact with another conic which touches the first three, and has also double contact with S. 4. Find the equation of tangents from any point x'y'z' to the cubic Ax3+ By3 + Cz3 + 6Dxyz=0. 5. Determine the points of inflexion of this curve, and the equations of the tangents at these points. 6. The equation of the tangents to a curve drawn from a point x1yızı on the curve being expressed in the form kAU1+ (A2 U1)2 = 0 find the degree of 4 in x1yızı, and in the coefficients of the equation of the curve. 7. Investigate the condition that there should be a cusp on the evolute of a given curve. 8. If the tangents to a cubic from a point on the curve make a harmonic pencil, find the anharmonic ratio of the tangents to its Hessian. 9. Determine all the foci of a. The ovals of Cassini; b. The conchoid of Nicomedes; and point out the distinguishing peculiarities of the different foci. 10. If the sides of a polygon pass through fixed points, and all the vertices but one lie on a curve of the mth degree, find the degree and class of the locus of the remaining vertex. GEOMETRY OF THREE DIMENSIONS. MR. TOWNSEND. 1. Two lines pass through the points a1 bi ci and a2 b2 c2, in the directions a1 B1 y1 and a2 B2 y2; find the equation of the plane bisecting at right angles the shortest distance between them. 2. Three lines pass through the points a b1 c1, a2 b2 c2, a3 b3 c3, in the directions a1 B1 y1, a2 B2 y2, aз B3 yз; find the co-ordinates of the centre of the parallelopiped of which they are edges. 3. Investigate the conditions that the general equation of the second degree in xyz should represent a cone of revolution. 4. Investigate the conditions that the general equation of the second degree in xyz should represent a parabolic cylinder. 5. A variable quadric passes in every position through the four sides of a fixed quadrilateral in space; determine geometrically the locus of its centre. 6. A variable quadric touches in every position the eight faces of a fixed octahedron in space; determine by any method the locus of its centre. 7. Explain the general method of obtaining the equation of the osculating plane at any point of the curve of intersection of two surfaces whose equations are given; and apply it to the case of two confocal quadrics. 8. Explain the general method of obtaining the length of the radius of curvature at any point of the curve of intersection of two surfaces whose equations are given; and apply it to the case of two confocal quadrics. 9. Investigate the differential equation common to a geodesic line and to a line of curvature on a surface of any order; and give the geometrical interpretation of its first integral in the case of the ellipsoid. 10. State and prove the principal properties of the system of geodesic lines which pass through one of the four umbilici on the surface of the ellipsoid. 11. In the skew surface generated by the motion of a variable line intersecting at right angles a fixed helix and its axis, show, from the equation or otherwise, that the two principal curvatures at every point are equal and opposite. 12. A surface of any nature, variable in position, but invariable in magnitude and form, moves without rotation according to any law; investigate in partial differences the equation of its envelope. 2. Prove that any root of the following equation in y has x sin x, x cos x for particular integrals; find the complete solution. 5. Prove that the integration of the equation when the coefficients are constants, depends on the integration of a homogeneous equation. and determine the constants so that for t=0, x=3, u=1. dz dx' dz q= integrate the partial differential equation (2y2 − xz) px + (y2 − x2 z) qy = ( 1 − 2x) y2 z. |