b. Prove completely that an imaginary root in Lagrange's equation for k corresponds to a real continually-increasing movement. c. The sufficiency of the condition of stable equilibrium can be deduced from the equation of vis viva. d. If a solid sphere be attracted, according to the law of nature, by a number of other solid spheres, show that it has necessarily at least one position of unstable equilibrium. 2. Deduce the equations of motion of a material particle upon a smooth surface from the principle of least action. a. If a number of equal particles, acted on by the same forces, and situated on the same surface, be projected from the same point, with the same velocity, but in different directions, and if their orbits be all touched by the same curve, show that the principle of least action does not in general hold, for each orbit, beyond the point of contact. b. How should the enunciation of the principle be modified so as to include this case? 3. Determine the value of the tension in an elastic inextensible rod, and show that if no forces act except at the extremities, 4. If Tension C× (Elastic ƒ)2 + C′ r = a { 1 + a(Yo+ Y1+ &c.) } be the equation to the Earth's surface, show from the permanence of the rotation that a. If the coefficients in the general value of Y2 be functions of a which do not change sign from the centre to the circumference, the same theorem will hold for any permanently rotating body composed of nearly spherical strata. 5. Deduce the general equation of the surfaces de niveau of a revolving heterogeneous fluid acted on only by the attractions of its own particles, namely, a (2i + 1) aa Yi (“pda3 + 3a2i1ı ' ["pd ( 14 ) − 3 [" pd (airs Y1) = 0 unless when i = 2. 10 a. What is the right-hand number of this equation when i = 2? b. Show from this equation that Yo when i has any value different from 2. = 6. Prove Laplace's theorem that the centre of gravity of a spheroid partly solid and partly liquid, which is in equilibrium, coincides with the centre of gravity of the surface, and state the physical assumptions on which it depends. 7. Deduce Clapeyron's expression for the amount of work done by forces acting at the surface of an elastic body which has at least three fixed points. a. Show that the internal work done in an element of the body can never vanish so long as there is any elastic force. 8. Explain the method by which the supposition of a single wave in the form of Fresnel's wave surface is shown to be consistent with the laws which govern motion in an elastic medium. 9. Deduce the equation of the wave surface from Professor M'Cullagh's equations of motion. 10. Write down the equations derived by Professor M'Cullagh from the principles of vis viva, and the equivalence of vibrations, and show from them that sin (1+2) tan 01 − sin (i1 — 2) tan 03 = sin 2i1 tan 02 where 1, 2 are the angles of incidence and refraction, and 01, 02, 03 are the angles which the incident, refracted, and reflected vibrations make with the plane of incidence. Experimental Physics. LIGHT. PROFESSOR JELLETT. 1. In the ordinary method of exhibiting Fresnel's fringes of interference, the light being very oblique, one of the mirrors can be moved parallel to itself, while the other can be turned round on an axis. The first movement displaces the fringes rapidly, but alters their width slowly, the reverse being true of the other movement. Explain this. a. If the mirrors be originally in the same plane no fringes are produced by the first movement alone. Why? b. If the positions of the radiant and of the mirrors be given, determine the greatest distance of the screen which will exhibit any fringes. 2. Show that fringes similar to those of Fresnel may be obtained by interposing a crystalline plate of variable thickness between two Nicol's prisms. a. Prove that the tint can only be changed by changing the thickness of the plate. 3. Describe Jamin's method of determining the elliptic polarization produced by metallic reflexion. a. What is the principal source of inaccuracy to be apprehended in making this experiment ? 4. In examining with a double-plane analyzer, a beam of polarized light which has passed through a Babinet's compensator, it is not desirable to use a prism cut at so small an angle as in the ordinary case. 5. Describe Jamin's compensator, and show how to apply it to the determination of the index of refraction of a gas by means of the fringes of interference. a. Show that this method, though applicable to a gas, is not applicable to a liquid. SOUND. DR. HAUGHTON. 1. Give an account of the correction made by Laplace, in the velocity of sound determined theoretically by Newton. 2. Explain the mode of determining the velocity of sound in solid, liquid, and gaseous bodies, by ascertaining the note caused by their longitudinal vibrations. 3. A circular elastic plate thrown into a condition of sonorous vibration, divides itself into sectors, which are in opposite phases, alternately; state the conclusive experiments of M. Lissajous, which demonstrate this fact experimentally. 4. State the law of Secondary notes emitted by a vibrating bell; and show which of them are Harmonics of the Primary note. TERRESTRIAL MAGNETISM. MR. GALBRAITH. 1. If a magnet bar lie in a right line perpendicular to another magnet, and passing through its centre, prove that its moment with regard to the other is M1=Sqrdr, M3 = √ qr3 dr, M5=§ qr3 dr, &c., and of my mз m5, &c., the corresponding quantities for the other bar is the magnetic intensity at the distance r from the centre, and a the distance of the centres of the magnets. 3. Describe the principle of Gauss's method of finding the absolute value of the horizontal component by observing the time of vibration of a horizontal magnet, and the deflection it produces in another horizontal magnet. State Lamont's modification of this method, and its advantages. 4. What were the disadvantages of the earlier method of determining the relative values of the intensity by the oscillations of a horizontal magnet. 5. On Humboldt's scale the intensity at London is 1.372; if the number of oscillations in 300 seconds at London is 775, find the number on his isodynamic line of unit intensity, the dip at London being taken at 70°. 6. Assuming the theory of two magnetic poles, each at a small distance a from the centre of the Earth, prove the following expressions : : Deduce the expressions for the horizontal and vertical components in terms of the magnetic colatitude. HEAT. DR. HAUGHTON. 1. Ten grains of charcoal burned in oxygen are found to give out 8080 units of heat; and ten grains burned in nitrous oxide are found to give out 11158 units of heat; find from these data the quantity of heat necessary to decompose an amount of nitrous oxide containing ten grains of oxygen. 2. Assuming that the quantity of heat lost by a radiating surface is proportional to the surface, to its emissive power, and to the time, jointly; deduce from this hypothesis Newton's Law of Cooling of heated bodies. 3. Prove the following formula for finding the weight, in grains, of a moist gas: 5.375 V (p −0.378ƒ); where V is the volume in cubic inches, t the temperature in Fahrenheit degrees, and p, f, the pressures of the atmosphere and vapour in inches of mercury. 4. Describe the methods of finding the Specific Heats of bodies, known as 1. The Ice Calorimeter. 2. The Method of Cooling. Mental and Moral Sciences. DR. TOLEKEN. 1. Explain and discuss Kant's assertion that "we ourselves make nature," and show that the paradox is rather apparent than real. Express in your own words, avoiding as far as possible Kant's technical language, his general explanation of our à priori concepts and judgments. 2. Point out the peculiarities of the analogies of experience as compared with the axioms of intuition and the anticipations of perception, and show that the distinction has been laid down by Hume and Hegel. The principle on which the analogies depend is stated differently in the first and second editions of the "Critick;" Kuno Fischer objects to the alteration? How may it be defended ? Kant's application of the word "analogy" seems to have been misapprehended by Fischer ? 3. In treating of the postulates of Modality, Kant alludes to certain questions the solution of which depends on the principle of the synthetic unity of apperception ? How is the application of the criterion of necessity in existence limited? The question of Idealism appears under the same topic in the Transcendental Logic and Dialectic; why? 4. State accurately the mode in which Kant traces the transition from Formal to Transcendental Reasoning; and show the connexion of the latter with Hegel's System. Mention any other leading points in which Hegel's System seems to have been anticipated by Kant. 5. How does Kant contrast the speculative and practical employment of Reason; and why, according to him, can we not found a system of Rational Psychology on the latter? How does Mr. Mahaffy state this objection? What does Kant observe respecting the application of our à priori conceptions in the Critick of the Practical Reason? 6. Compare the theories of Judgment given by Fichte and Hamilton with that of Mr. Mill. Give Hegel's explanation of what is meant by a judgment, and show its relation to his general method. There is a close approximation in language between Hegel and a celebrated English Nominalist? 7. How does Hamilton contrast the mode of action of the common and proper sensibles, and what reason does he allege "why the common are in propriety objects of sense per se" ? In his account of the ancient doctrine of "intelligible species," there appears a curious anticipation of Kant's description of the function of the understanding in perception ? |