FELLOWSHIP EXAMINATION. Examiners, ANDREW S. HART, LL. D. JOHN TOLEKEN, M. D. JOSEPH CARSON, D. D. THOMAS STACK, M. A, JOHN H. JELLETT, B. D. GEORGE LONGFIELD, D. D., Regius Professor of Hebrew. MICHAEL ROBERTS, M. A., Erasmus Smith Professor of Mathematics. RICHARD TOWNSEND, M. A., Professor of Natural Philosophy. JOHN R. LESLIE, M. A., Erasmus Smith Professor of Experimental THOMAS K. ABBOTT, M. A., Professor of Moral Philosophy. Mathematics, Pure and Applied. GEOMETRY. DR. HART. 1. Given the asymptoles of a cubic of the third class; find the locus of its cusp. 2. Given the conjugate point, and the three inflexional tangents of a cubic of the fourth class, find the points of inflexion. 3. All multiple points on a curve are also multiple points on its Hessian; what is the relation between the orders of these points? 4. If a plane section of a quadric contains a directrix, prove that the corresponding focus is the vertex of a right cone standing on the section. 5. Find the equation (referred to the common self conjugate tetrahedron of two quadrics) of the developable generated by tangents to their intersection. 6. If planes, through four given points on the curve of intersection of two quadrics, cut one another in any chord of this curve; prove that their anharmonic ratio is constant. 7. Show how to form the differential equation of a family of surfaces involving three arbitrary functions, and apply the method to the case of tubular surfaces generated by the motion of a given ellipsoid without rotation. 8. Investigate the condition for a point of osculation on the intersection of two surfaces. Find the curve of contact of a cubic with the developable that circumscribes it and its Hessian. 10. Determine the number of double points on the Hessian of a cubic, and find the simplest form of the equation of the cubic referred to planes intersecting in these points as co-ordinate planes. 11. If a plane, which touches any surface, intersects the polar cubic of its point of contact, and the polar plane of the same point with respect to the Hessian; what relation is there between the two lines of intersection? 12. What curve is represented by the vector equation Vap=p Vẞp? MR. M. ROBERTS. 1. State the method of applying Jacobi's modular transformation for an odd number p to the case p= ∞. If h is the modulus derived by the first theorem from k, and if h', k' are the complements of these quantities, the transcendental equation BF (k', σ) = F (h', 7) 2. From the following definitions of the functions H and 0: AH (u): = 2 ✩ 9 sin σ (1 − 2q2 cos 2σ + q1) ( 1 − 29a cos 2σ + q3) . . . 2 Κσ u = π where A is a numerical multiplier, prove that (u + iK') can pressed by H (u) multiplied by an exponential factor. be ex 3. Adopting as the definition of the function Z (u) the equation 4. If Z (u + iK') = U+iV, find the values of U and V. 5. Deduce immediately Jacobi's modular equation for the case p = 3, viz., from the equation which gives the value of sin am K 3 iK' ; and 3 6. Determine the equation which gives the value of sin2 am write down the arguments whose trisection is given by the remaining roots of this equation. 7. y2 = (x − a) (x − ß3) (x − y) denotes a cubic curve; a, ß, y are real, and such that a >ß>y; prove that it can be represented by a system of two equations by which the co-ordinates depend on an elliptic function (u) (whose periods are multiples of K and iK') which is called the argument of the point. If u, u2, us denote the arguments of the points in which any right line meets the curve, prove that u1 + U2+ U3 = 2m K + (2m2 + 1) ¿K', m, m' being any positive or negative integers. Show also that the arguments of the real points of inflexion are and determine the arguments of the points of contact of the tangents drawn to the curve from (1). 8. Let X=x(1- x) ( 1 − k3 x ) ( 1 − k12x) (1 − к'12x), where «">«'>«<1 : transform the differential if x = '<" depend on F(m1, 01) and F(m2, 02), where sin 01 sin 0. = (I - K'K') sin 0 I - K'K' sin2 0 deduce other relations between к, к', '' such that the corresponding hyperelliptic functions depend on elliptic integrals. 10. If M = 3α5 (α1 â2- αoαз) + аş (α1 α3 — 2α0αş + 3α22) - 2a2a32, find the coefficients of the covariant of (do, α1, aз, αs, A5, A6) (x,y)6 of which is M the source. If we take for the fundamental sextinvariant of this binary form that which involves ao2 as the highest power of ao, express it by the coefficients of the covariant (M). Express M by means of the roots of the equation (ao, A1, A2, A3, As, A5, A6) (X, 1) = 0. 11. Give the general expression for functions of the differences of the roots of a biquadratic, which are of the sixth degree in the coefficients, and of the tenth degree in the roots. |