in the coefficients of the equation is known; apply the method to an equation of the third degree. II. Determine the covariant of a binary quintic whose leading term is αo2 (αa2 — α3α5) + 3a。α1 (α2α5− αзα1) + 4A0A2 (A32 — A2A4) + 2a12 (a32 — A1A5) +5α12α2α +3α24 — 8a1a22α3. 12. Find the invariant of the eighteenth degree of the form Ax5+ Bу5+ Cz5, where x + y + z = 0, adopting as its definition the eliminant of the quintic 1. Show that the equations of the shortest distance between the lines (x-x') (cos a' - cos e cos a)+(y-y') (cos B' - cos 0 cos ẞ) + ( − z′) (cos y'cos e cos y) = o, (x-x") (cosa - cos 0 cos a ́) + (y − y′′) (cos ß — cos 0 cos B′) + (z − z′′) (cos y cos e cos y') = 0, O being the angle between the directions of the given lines. 2. Find the surface represented by the equation tan Ax2+cot Ay2 + 2 (1 + sin B) yz + 2 tan A cos Bxz + 2 sec Bxy + 2px and show that it represents a cylinder if +2qy + 2rz = 0, 3. Find the locus of the centre of a sphere of given radius which is touched by the edges of a self-conjugate tetrahedron with respect to a paraboloid. 4. Deduce the invariant relation which expresses the condition that two quadrics shall be such that a tetrahedron may have two pairs of opposite edges on the surface of one, while its four faces touch the other. 5. From any semidiameter D of an ellipsoid another line d can be derived, whose direction cosines are co-ordinates of the extremity of D divided by the parallel semiaxes. In this way let three lines d, d', d" be derived from three semidiameters D, D', D", which are mutually at right angles; let dd' be the angle between the lines d, d, and so for the others; prove that a, b, c being the semiaxes of the surface. 6. Show that the equation of either sheet of the wave surface can be presented in a distinct form by means of a relation between the primary semiaxes of an ellipsoid and one of the hyperboloids of a certain confocal system. DR. HART. 1. Show how to determine the number of double tangents to a curve of the nth degree. 2. Find the number of points on a curve of the nth degree where it has contact of the third order with a circle. 3. Investigate the relation of the asymptotes of a cubic with the polar conic of the line at infinity. 4. If a system of conics pass through three fixed points, and touch a fixed right line, it is obvious that two of them will pass through any other given point, and that four will touch any other given line. Hence show that the locus of a point which has the same polar with regard to one of these conics, and with regard to any fixed conic, is a curve of the sixth degree. 5. Find the equation of a curve such that the integral fy. ds between two given points shall be a maximum or minimum. 6. Find the surface for which fuds is a maximum or minimum, ds being the element of the surface, and μ a homogeneous function of the co-ordinates. 7. Given eleven points of intersection of a biquadratic with a given cubic, find the twelfth point by a simple rectilinear construction. dy dx 9. Investigate the condition that d2y (a + bxn) x2 + (c + ex1) x 1/4 + (ƒ + gxn) y = f (x) dx2 may be integrable in finite terms. 10. Integrate and find singular solutions for the equation dy2 dx2 dy dx Mathematical Physics. DR. HART. 1. Prove the formula for the longitude of the node and the inclination of a planet's orbit, in terms of the differentials of the disturbing function. 2. Assuming the expression for the non-periodical part of the disturbing function, F= m {{Co+} D1aa' (e2+é'2 —¿? —¿12+2 iï'cos (Q-Q')) — ‡D2 aa'ee' cos (w—w') }, find the secular variations of the eccentricities and of the perihelia of two planets, and find the condition that the latter should oscillate. 3. Show how to calculate the principal periodic term in the longitude of Uranus caused by the disturbance of Neptune, the periodic times being nearly as 1 : 2. 4. Compute approximately the perturbation of the radius vector of a planet's orbit, and show how to proceed to a second approximation. P 5. In the Lunar Theory, the values of and h2 u2 the fourth order, whose argument is the Moon's elongation; find the effect of these terms on the Moon's longitude. 6. Compute the term in the Moon's parallax, the argument of which is twice the distance of the Sun from the lunar apse. 7. Assuming that the term in the disturbing function R produced by the spheroidal form of the Earth is compute its effect on the Moon's latitude,- being the ellipticity of the Earth, a the ratio of centrifugal force at the equator to gravity, and d the Moon's declination. 8. Assuming the expression for the radius of any stratum of the Earth in a series of Laplace's coefficients, r = a and for the potential of the Earth's mass, find the values of the coefficients Yi, Y2, &c., that will satisfy the condi tion of fluid equilibrium, a being the mean radius of the Earth's surface, and a having the same signification as in the last question. 9. Deduce the above value of the potential from the value of r. 10. Find an expression for the attraction of a homogeneous sphere on an external particle, the force being supposed to vary inversely as the cube of the distance. show how to integrate it, taking into account the variation in 0. a. What term in the result is due to the correction? b. The importance of this term depends on the ratio between two motions? 2. Define the mean day, and show how to determine its variations. 3. If a material particle be in equilibrium under the influence of a number of centres of force varying inversely as the square of the distance, the equilibrium is neutral ? 4. More generally, if the force be everywhere attractive, and proportional to the nth power of the distancesa. If n lie between ∞ and 2, the equilibrium cannot be unstable ? b. If n lie between - 2 and ∞, the equilibrium cannot be stable ? c. In every case of neutral equilibrium, the directions of stable displacement are situated in one determinate plane? N. B.-By neutral equilibrium is meant equilibrium which is neither absolutely stable nor absolutely unstable. 1. Two particles are connected with a third by a rod which is elastic at its junction with the third particle; determine by the method of Lagrange the equations of equilibrium, and show how far they agree with the equations which would be obtained if the rod were rigid throughout, and how far they differ from them. 2. Deduce the equations of equilibrium of an elastic spring according to the method of Lagrange-no forces being supposed to act except at the extremities. 3. Prove that if the curve be plane at either extremity it will be plane throughout, and show that in this case the problem may be reduced to quadratures. 4. If the symbols & and A denote different variations of the arbitrary constants which enter into the solution of the dynamical equations of a system of particles, T the vis viva, and %, n, &c., the independent variables, prove that the function 5. Prove that if the attracting force be proportional to the cube of the distance, the force of gravity at each point of the surface of a homogeneous revolving spheroid is constant, whatever be the rotation, and point out the limitation with which this theorem is to be received. 6. If a solid nucleus of a form nearly spherical be covered by a fluid in equilibrio, the centre of gravity of a thin stratum of constant thickness at the surface of the fluid is the same as the centre of gravity of the whole spheroid? a. Hence show that, in the case of a heterogeneous fluid in equilibrio, the several strata of which it is composed have the same centre of gravity. 7. Assuming the general values of the elastic forces to be of the form show that for an uncrystalline solid the normal forces have the form a. How might λ, μ be determined experimentally? 8. How does Lamé obtain the equations of equilibrium of a flexible membrane? a. Show that his definition excludes an elastic plate. b. What equation must be satisfied by the normal and tangential elastic forces, independently of the form of the membrane ? c. Show that if the membrane be plane, the force must be tangential. 9. Assuming Professor Mac Cullagh's theorem of the polar plane, show that at the polarizing angle of a crystal the reflected ray is perpendicular to the intersection of the polar planes of the two refracted rays. 10. The equations obtained by Professor Mac Cullagh from the terms outside the sign of triple integration are dní dzi dzı' rs cos a2 + dz2 r's' cos a 2, where ni, n'i, n2, n'2 are the displacements for the incident, reflected, and |