into another between the variables u, v which are given by the equations and determine in this manner the value of the definite integral 5. If a, ß, y, &c. are the roots of the equation n+px-1+q^2+3+ find the value of the determinant da da da dp dq dr αβ αβ αβ dp dq dr dy dy dy dp dq dr 6. Prove that the differential equation of the cuspidal edge of a developable circumscribed to the ellipsoid is a® (ydz – zdy)? + b ( d – xđã ) + c (xảy – đư) = MATHEMATICAL PHYSICS. DR. HART. 1. Prove generally that the differential coefficients, with regard to the time, of the elements of a disturbed planet's orbit can be expressed as functions of the differential coefficients of the disturbing function with regard to the elements, and not involving the time. 2. If the mutual action of two planets causes the apse of each to oscillate, determine the extents of oscillation, and find in what case they will be equal. 3. Prove that the secular variations of the eccentricities and inclinations of the planetary orbits are of an order higher than the third. 4. Calculate the mean annual motion of the Moon's nodes either by Newton's method, or by any other of equal accuracy. 5. State Newton's method of computing the variation of the Moon. 6. Compute the coefficients of the terms in P and T which give rise to the Moon's parallactic inequality, and deduce their effect on her longitude. 7. Find the form of the chief term in the disturbing function of the Moon resulting from the spheroidal form of the Earth, and show how to deduce its effect on the Moon's latitude. 8. The radius of the Earth being expressed by a series of La Place's coefficients, investigate the form of the coefficient of the second order so that the axis of rotation may be a principal axis. PROFESSOR JELLETT. 1. Assuming Laplace's equations for the rotation of the Earth, sc. show that the sensible change in the obliquity of the ecliptic depends wholly on P'. a. Show that it is not true that the precession depends wholly on P. b. Show that the following terms, if they existed, in P' would produce sensible inequalities in the obliquity : 1. Terms of long period. 2. Terms whose period is a little more than 24 hours. 3. Terms whose period is nearly 12 hours. 2. Show that the change in the precession produced by the variation of the obliquity of the ecliptic is expressed by adding to the coefficient of each of the inequalities arising from the motion of the solar node the quantity 3. Two perfectly elastic spheres, whose radii are equal, moving on an imperfectly rough horizontal plane, impinge against each other; if there be no friction between the spheres, find their velocities after impact. 4. A system of material particles, acted on by given forces, and connected by given relations, commences to move, with velocities which begin from zero; find the initial values of the internal forces. 1. Determine, by the method of Lagrange, the equations of equilibrium of a system of particles continuously arranged so as to form a thin lamina, and show that for this system Xdx + Ydy + Zdz is a perfect differential. a. How is this problem more general than that of the equilibrium of a flexible and inextensible surface? 2. Find the equations of equilibrium of a system composed of three particles connected by an inextensible rod, which is elastic at the point where the second particle is situated; and show that the force of elasticity, as defined by Lagrange, is equal to the moment round the second particle of either of the forces acting at the extreme particles. 3. Deduce, by the method of Lagrange, the equations for the velocities of rotation round fixed axes produced by the action of instantaneous forces. a. If the moments of these forces vanish, the rotations will vanish also; how does this appear from these equations ? b. Show from the same equations that, unless the velocities and moments vanished together, we might have infinite velocities with finite moments. c. How does Lagrange show that these quantities do vanish together? 4. If x=f(a, b, t) be the solution of the dynamical equation 5. If a material particle, acted on by two centres of force of equal absolute intensity, and each varying inversely as the square of the distance, be projected from a given point; find the direction and velocity of projection such that the particle may describe an ellipse of which the centres of force are the foci. 6. Define Lamé's surface of elasticity, and show that it is an ellipsoid. a. Prove from this the existence of three planes in an elastic body for each of which the elastic force is perpendicular to the plane. 7. From the general equations of equilibrium of a membrane, deduce the equations of equilibrium of a plane membrane, and show that they admit of being completely integrated, the form of the integral being, however, imaginary. 2 2 2 8. Prove, by Professor M'Cullagh's mechanical theory, that if at any point in a crystal an ellipsoid be described whose axes coincide with the axes of elasticity, and are equal to and if a', b' be the axes of a central section made by the wave polarized parallel to a', b' will be a plane, the wave velocities of light I I respectively. a. Hence deduce the equation of the wave surface. 9. In reflexion at the surface of a uniaxal crystal, the axis of the crystal lying in the plane of incidence, determine, according to Professor M'Cullagh's theory, the azimuth of the plane of polarization of the reflected ray in terms of the azimuth of that of the incident ray, and of the angles of incidence, refraction, and inclination of the extraordinary ray to the wave normal. 10. The potential of the attraction of a spheroid covered by a fluid in equilibrio, under the action of forces whose potential is represented by the series Zo+Z1+ Z2+ &c. where r is the distance of the attracted point from the origin, and the equation of the spheroid is R = 1 + aΣ Yi II. If any one of the equilibrium surfaces of an insulated body, electrified by induction, be wholly without the body, there will be no action on any point without this surface; prove the theorem upon which this depends. 12. Show that there is one (and only one) distribution of a given mass upon a given surface, which will cause the potential to take at each point of the surface a given value. Experimental Physics. LIGHT. PROFESSOR JELLETT. 1. A beam of plane-polarized light passes obliquely through a plate of glass; it is found, when examined by an analyzer, to be no longer perfectly plane-polarized a. How does this change show itself? b. Show that it may arise from a weak elliptic polarization of the light. c. Suggest an experiment by which it may be decided whether this is the true cause. d. If, on the side of the glass next to the incident light, an opaque band be placed perpendicular to the plane of incidence, there will appear on one side of this a fringe of plane-polarized light; explain this. 2. Explain the deviation and dispersion of a ray which passes through a fine grating, and account for the purity of the spectrum. a. Show that, in general, one system of fringes will disappear. 3. State the method of observing the phenomena of interference where the difference of phase is a large multiple of the length of a wave, and show how to calculate this difference. 4. Describe the construction of the double-plane analyzer, and show that its sensibility varies nearly inversely as the angle between the planes. 5. Explain Newton's rule for the composition of colours. a. Jamin points out that this rule leads to an erroneous result in the composition of red and blue; this may be avoided by a limitation which is necessary in applying Newton's rule? 6. A lens, placed on a flat glass, and illuminated by homogeneous light, is slowly raised; describe and explain the phenomena which are seen. HEAT AND ELECTRICITY. PROFESSOR GALBRAITH. 1. Deduce the following formula for the wet and dry bulb thermo from theoretical considerations; compare it with Dr. Apjohn's formula; show to what extent they differ, and the reason of the difference. and show how we may calculate from it the emissive power of the body. 3. Describe Leslie's experiment for proving Lambert's law of cosines; state the results of MM. Provostaye and Desains' repetition of this experiment; and show that if q be the heat emitted in the normal direction, and q' the heat obliquely through the same aperture, that in which R and R' are the portions internally reflected. 4. Describe Leslie's and Melloni's method of measuring the absorbing powers of bodies; and show how MM. Provostaye and Desains avoided the sources of error which vitiate both. 5. Describe Buff's method of measuring the quantity of electricity developed by a plate machine. At what result did he arrive? 6. Give the expression for the action of one element of a voltaic current on another, and state the principles on which it depends. |