FELLOWSHIP EXAMINATION. Examiners. ANDREW SEARLE HART, LL. D. THOMAS STACK, M. A. JOHN H. JELLETT, B. D. Mathematics, and Mathematical Physics. PURE MATHEMATICS. DR. HART. I. Prove that the tangential of any point on a cubic lies on the polar of that point with regard to the Hessian. 2. Find the conditions that the equation of a cubic curve may be capable of being resolved into two or three factors. 3. If a quadrilateral is inscribed in a cubic, so that tangents at its four angles shall meet on the curve, prove that every line which passes through their point of meeting is cut harmonically by the opposite sides of the quadrilateral and the points where it meets the curve again. 4. Give expressions for the co-ordinates of a point on a trinodal quartic, as functions of a single parameter. 5. Show how to arrange the bitangents of a quartic in sixty-three groups of twelve, so that the points of contact of each group may lie by eights on fifteen conics. 6. If seven bitangents of a quartic are given, show how the others may be found. 7. Given n2 + 3n 2 2 points on a curve of the nth degree, find the locus of the intersection of tangents at two of these points. 8. If three quadrics intersect in seven given points, find the co-ordinates of their eighth intersection. 9. State and prove Chasle's extension to quadrics of Pascal's theorem. = 10. Investigate the number of umbilics on a surface of given degree. 11. The co-ordinates of any point on a given surface being represented as functions of two parameters, show how to form the equation of the lines of curvature as functions of these parameters. 12. Express the measure of curvature and the directions of inflexional tangents as functions of the same parameters. MR. BURNSIDE. 1. F(w) is a monodrome function of w = u + iv, having the following properties: moreover it becomes twice infinite in the interior of the rectangle constructed on the sides 2K, 2K' for w = iK' and w = 2K + iK'. From the identical equation F (w) ew dw = o, the course of integration being the perimeter of the rectangle, deduce the formula and K < K < K" < 1, by integrals of the same type and o. 2 the limits being where f (x) is a rational and integral function of x of the degree 2m— 1, prove that may be reduced to the form II (0) de, where F(x) is a rational function of x. 7. State and prove the Jacobian theory of the last multiplier. 10. In order that SS V dx dy be reducible to single integrals, V involving no higher differential coefficients than the second: V is of the 11. Show how the seven invariants, the two fundamental linear covariants, and the six quadratic covariants of a system of two binary cubics, may be obtained. 12. If P and Q denote the skew invariants of the same system, express P2, PQ, Q2 in terms of the other invariants. 13. If U (a, b, c, d...) (x, y, z .. .)" be a quantic in any number of variables of which the discriminant A vanishes, prove when the singular roots of U are substituted for the variables x, y, z. (a). Prove the following form for A in general: A = Aa + (B, C, D, ) (b, c, d, )2 14. If a fixed curve, .... .... be intersected by a curve changing arbitrarily, of constant degree m, in the points x1, X2, X3, r=mn Xmn, prove 15. Describe a curve passing through two fixed points such that the area included by the curve, its evolute, and the radii of curvature at the fixed points, be a minimum. MATHEMATICAL PROBLEMS. DR. HART. 1. Find the condition that each pair of opposite sides of a quadrilateral inscribed in a binodal quartic may intersect at a node, and show that when this condition is fulfilled, an indefinite number of such quadrilaterals may be inscribed. 2. Find the lines of curvature on the quartic surface (x2 + y2 + z2)2 + Az2 + By2 + Cz2 + D = o. 3. Find the co-ordinates of a cusp of the Cayleyan of 23 + y3 + z3 + 6 mxyz, and of the points where the tangent at this cusp meets the Cayleyan again. MR. BURNSIDE. 4. Putting_ƒ(x) = ƒ, (x) ·ƒ, (x), where ƒ,(x), ƒ2(†) are quadratic functions of x, prove that √ƒ, (x) ƒ1⁄2 (4) − √ƒ, (y) ƒ2 (x) is an integral of the differential equation = C 5. If x ̧, x2, x ̧, ¤, be the roots of a quartic U, wanting the second term, and λ, μ, v the roots of its reducing cubic, prove that (P2 − $3) ($1 − $4) = (μ − v) ( A + Bλ), (P3 − 1) (02 − P1) = (v − λ) (A + Bμ), (P1 − 2) (P3 − s) = (λ − μ) (A + Bv), where ø (x) is a rational and integral function of (x), and A, B rational functions of the coefficients of the quartic, and of the coefficients of p. (a). Hence prove that the invariants of the quartic whose roots are P1 P2 P3 P4 are the same as those of the quartic AU + BH, where H is the Hessian of U. 6. Determine the value of cot am u. ▲ am (u + j1 (2m + 1) K + j1⁄2 (2m′ + 1) i K') |