WEDNESDAY, January 7, 1885. 1° to 44. GROUP B. 1. Give in the form of a series the definition of Bessel's function J, (x) when n is not integral, and verify that it satisfies the differential equation d'y 1 dy +11 dx2 T x dx Assuming that the same form holds for functions of positive and negative orders, + (1-7y=0. prove that if n be positive. Prove also that do dt - رقة (A A-C = M2 2 2. Establish Euler's equations of motion in the form A (B-C) ww,= L............; and determine w, w, w, in terms of the time t by means of elliptic functions, when no forces act upon the body. If the impressed forces reduce to a couple parallel to and 1 times the resultant angular momentum, prove that W, = N, ed? cn je (edt - 1), Α. (含 1, ent snu (ent - 1), W, = 12 ede dn je (edt - 1); where 12,, 0, , are the initial values of W, W., wg; and (B - C)(A-C) 129 2 22 C B - C Ω, fixed plane, and that confocal ellipsoid will roll on a parallel plane which revolves with angular velocity proportional to edt about the axis of resultant angular momentum. 3. Prove that the potential outside a homogeneous oblate spheroid of mass M and small ellipticity é at a point which is distant from the centre and whose distance makes an angle A with the axis of revolution is Mr-- - } Mk* er-* (3 cos' 0 - 1) where k is the mean radius of the spheroid. Supposing this body to be Saturn and originally liquid, investigate the equation mk3 wok MC (c* – 63) determining é the ellipticity, due partly to its own rotation w and partly to the disturbance caused by its ring, supposed to be a flat concentric circular disc, of uniform thickness and density and lying in the plane of the equator, of mass and bounding radii c tb, where c is large compared with k. And then gravity at colatitude D is to equatorial gravity in the ratio 1 + e cos' 0:1, a m FF 8 WEDNESDAY, January 7, 1885. 1} to 47. GROUP C. 1. DETERMINE the conditions to be satisfied in order that a mass of homogeneous gravitating liquid in the form of an ellipsoid, rotating with given angular velocity about a principal axis, may have a free surface of equal pressure, the liquid having a given constant molecular rotation parallel to this axis ; and shew that the shape is the same when the axes are stationary (Dedekind's ellipsoid) as when there is no relative motion in the liquid (Jacobi's ellipsoid). Prove that if a rigid ellipsoidal shell be filled with two homogeneous gravitating liquids of different densities, the denser liquid will form a nucleus in the shape of an ellipsoid ; and that if the shell be made to revolve with constant angular velocity about any given fixed axis, a possible form of the nucleus when the liquids are in relative equilibrium will be an ellipsoid, not coaxial with the external surface. 2. Prove that the velocity of propagation of waves of small displacement of length x in liquid of density p and depth h is 27T gp 27TO ! " 2п λ supposing the surface of the liquid covered by a thin flexible membrane of tension T and superficial density o; and discuss the case of waves in ice on water of uniform depth. Supposing the liquid originally still, and plane aerial vibrations of wave-length a and velocity v in air of density R to impinge on the surface at an angle B; prove that, when the motion of the system has become periodic, we may represent the displacements of the incident and reflected waves of air and the displacement of the surface by (i) a sin {m (a sin B + z cos B) – nt – a} (iii) bcos (ma sin B – nt) respectively, where m=27/1, n=2710/; and then a, the change of phase, is given by 2πT 20h
sin B). ava 2πυ, 2710 + 912) cos B – p cot ß coth λ il 3. An approximately circular membrane whose radius in any direction is a + or is vibrating in a type approximating to w= AJ, (kr) cos no. Shew that the pitch is the same as for a circular membrane of radius and that there are two positions of the nodal diameter one of which renders this period a maximum and the other a minimum. A heterogeneous membrane in the shape of a circular annulus, whose edges are fixed and inner and outer radii are b and c and whose density is u/r where r is the distance from the centre, is stretched with tension T and is performing small symmetrical normal vibrations. Shew that a possible motion is given by waves 4. Investigate the differential equation of the propagation of waves of longitudinal displacement in a uniform cylindrical bar; and prove that these take the same time to traverse the length of the bar whatever be the permanenttension. Prove that, if an elastic bar of length 1 with flat ends impinges directly with velocity V on another longer bar, at rest, of length nl of the same material and cross section, also with flat ends, the first bar will be reduced to rest by the impact, and the second bar will appear to move by successive advances of the ends with velocity V for intervals of time 2 l/a, and intervals of rest of 2 (n − 1) l/a, a denoting the velocity of propagation of longitudinal vibrations. UNIVERSITY SCHOLARSHIPS AND CHANCELLOR'S MEDALS. MONDAY, January 19, 1885. 9—12. Any one of the following subjects may be chosen for LATIN HEROIC VERSE: 1. "Εσπερε, πάντα φέρεις. 2. Eva Paradiso depulsa lamentatur. 3. Furiis agitatus Orestes. 4. Honor est a Nilo. The Thesis chosen should be written outside the paper with the author's name, and at the head of his Exercise. MONDAY, January 19, 1885. 9-12. Any one of the following subjects may be chosen for an ENGLISH ESSAY: 1. "A δεί μαθόντας ποιείν, ταύτα ποιoύντες μανθάνομεν. ARISTOT. Eth. N. II. a'. 2. The greatest happiness of the greatest number” has been often laid down as "the End of Political Government." Consider the value of this definition ; and, if you think any other more valuable, propose and defend it. 3. Sketch the life, character, and historic influence of any one of the following men: Luther. |