12. Two ships sail from one port to another, both of which are in the same latitude, the difference of longitude of the two places being given; one ship sails along the parallel of latitude, the other along a great circle; prove that, if the difference between the lengths of the courses of the two ships be the greatest possible, then 2 being the given difference between the longitudes, and a being the latitude, MIXED MATHEMATICS. Examiner-MR. A. M. NASH, M. A. 1. Find the general conditions of equilibrium of a rigid body acted on by given forces; and show how to reduce a given system of forces to two forces at right angles. 2. Obtain general formulæ for determining the position of the C. of G. of a solid body. Find the C. of G. of the segment of a parabola cut off by any chord. 3. Find the general equations of epuilibrium of a flexible inextensible string. A heavy chain hangs in the curve y = log sec z; prove that the weight of each element of the curve is proportional to the secant of the angle which the tangent to that element makes with the horizon. 4. Find the attraction (1) of a circular ring upon a particle placed upon its axis; (2) of a uniform sphere upon a particle within it. 5. In an elastic fluid at rest, find the surfaces of equal density and of equal pressure and determine the condition that they should coincide. 6. A cube edge a, containing liquid, rotates with uniform angular velocity w about a vertical edge; if 2aw 3g, prove that the pressure on the base is double of the pressure upon either of the faces containing the axis, and equal to the pressure upon either of the other vertical faces. 7. A solid is floating in stable equilibrium in a homogeneous liquid, find the time of a small vertical oscillation. A cone, whose axis is vertical and vertex downwards, is filled with liquid whose density varies as the square of the depth: if the liquid be transferred to a cylinder whose base is equal to that of the cone, find the whole pressure on the curved surface. 8. A vessel formed of flexible and inextensible material in the form of a surface of revolution is held with its axis vertical and filled with homogeneous liquid; find the principal tensions at any point. 9. Find the illumination at any point of a surface illuminated by a given surface of uniform brightness. A series of surfaces is formed by the revolution about an axis through the pole of all the arcs of an equiangular spiral contained in a given quadrant: if a bright point be placed at the pole, find the whole illumination of one of these surfaces, and show that it is the same for all. 10. Determine the length of a caustic by reflexion. In a transparent sphere the index of refraction at a distance r k2 from the centre is : a ray of light falls upon the sphere, prove that the path of the ray within the sphere is an arc of a circle. 11. Find the angle of minimum deviation for a ray of light passing through a prism in a principal plane. 12. Describe the construction of Huyghens' eye-piece, and trace the course of a pencil of rays through it. What special advantages does it possess? PROBLEMS. Examiner-MR. A. M. NASH, M. A. 1. Through the centres of the inscribed and escribed circles of a triangle ABC an infinite number of rectangular hyperbolas can be drawn; the triangle ABC is self-conjugate with respect to each of these hyperbolas, and their centres all lie upon the circumscribed circle. 2. Prove by reciprocation the following theorems :- (1.) The tangent to a conic at any point bisects the angle be tween the focal distances of the point. (2.) Two hyperbolas have one focus common and a pair of paral. lel common tangents, the four asymptotes all touch the same circle. 3. The section of an ellipsoid by the polar plane of P has a given area; prove that the locus of P is another ellipsoid. 4. Find the plane circular sections of the surface (x + y2 + z3)? = a2 x2 + b2 y2 + c2 z2. 5. The envelope of a straight line which moves in such a manner that the sum of the squares of the perpendiculars on it from three given points is constant is a conic whose centre is the centroid of the triangle formed by the three points. 6. Find the maximum value of the angle included between the tangents to an ellipse and its auxiliary circle at the extremities of coincident ordinates. 7. Integrate (x vers-1x)a dx. 8. A perfectly rough oblate spheroid is placed with its axis vertical on the surface of a sphere of radius c; if the equilibrium be neutral, prove that a = c (a - b), where 2a, 2b are the axes of the generating ellipse. 9. An elastic band of natural length a passes round three small rough pegs A, B, C, forming an equilateral triangle in a vertical plane, the base, BC, of which is horizontal; determine the greatest weight which can be fastened at the middle point of BC without causing the band to slip round the pegs. 10. If a, B, y be the depths of the vertices of a triangular lamina immersed in a homogeneous liquid, the depth of the centre of presΣ(a3) + Σ(By) sure will be 11. A cylinder of radius a contains liquid, the density of which 2 at a depth z is 1 + ; a cone (height h and radius of base a) rests in equilibrium when just complete, immersed with its axis vertical, and vertex downwards: show that its density must be equal to that Th of the liquid at a depth 12' 12. A luminous point is placed at the vertex A of a regular hexagon ABCDEF, and images are formed by reflexion at the sides BC, CD, DE, EF, FA, AB in succession: prove that (1) any three consecutive images are equidistant from some one of the angular points, (2) the directions of the first and last images as seen from A make equal angles with AB, and (3) find the distance of the last image from A. MENTAL PHILOSOPHY. Examiner-REV. J. ROBERTSON, M. A. 1. Define mind, and explain how the notion of self or ego has been accounted for. State and criticise the various theories as to the connexion between mind and body. Point out the defects of current modes of expressing that connexion, and state what you regard as the best mode. 2. Does physiology facilitate the study of the human mind? Show how the principle of evolution has been applied to mental philosophy, and with what results. 3. Give an analysis of the intellectual phenomena, showing clearly the difference between the empirical and the critical philosophy. Discuss at length the grounds on which the latter rests and explain its method. 4. How do we obtain our knowledge of the material world, and what is the extent of our knowledge? Defend the conclusions you adopt. Explain and criticise the following:: (1.) "We are incapable of discussing the existence of an independent material world." (2.) "It is a principle generally assumed by philosophers that the relation of knowledge infers a correspondence between the subject knowing and the object known." 5. Explain fully the various uses of the term consciousness. Give a logical estimate of the theory of unconscious mental modifications, and its application to the phenomena of memory and reproduction. 6. Describe and analyse the phenomenon of belief, and determine its place in a classification of mental phenomena. State clearly the relation between belief and knowledge. 7. Discuss the value of self-evidence and necessity as tests of intuition. How does the empirical philosophy account for necessity of thought? How far is man's power of conception a test of truth? 8. Explain what you understand to be the province of mental philosophy, and define the limits of its various departments. Is a philosophy of being possible? MORAL PHILOSOPHY. Examiner-REV. J. ROBERTSON, M. A. 1. State fully the relation of Ethics to Psychology, Metaphysics, and Politics. 2. State concisely the Socratic, Aristotelian, and Stoic views as to what constitutes the moral character of an action. State and defend your own view. Can the material rightness of an action be determined apart from its formal rightness? Examine the following: As all schools teach that a person may mistake his duty and do what is wrong, sincerely believing it to be right, it results that an action may be wrong in one sense and right in another. 3. How is diversity of moral judgments accounted for by those who hold the intuitive theory of conscience ? 4. Sketch the ethical systems of Epicurus and Butler; and indicate clearly the merits and defects of the latter. Expound and discuss fully Butler's opinion as to the object of desire; and contrast his view of self-love with Kant's. 5. Explain and discuss the theory which asserts that "the characteristic of the moral faculty is an education under law through the instrumentality of punishment;" and show how it accounts for the element of authority in our moral sentiments. Discuss the applicability of the theory of hereditary transmission to our moral sentiments. 6. It is asserted that the free-will doctrine seeks to establish actions without any motives whatever: Examine this. State and examine the arguments against the freedom of the will founded on the existence of motives. Does the notion of disinterested motives necessarily involve freedom? The perplexity of the question of free-wili is mainly owing to the employment of unsuitable language to express the facts: Explain and illustrate this from the history of the controversy in England. 7. Hegel holds that good conduct consists in the conscious realisation of the free reasonable will, which is essentially the same in all rational beings: develop Hegel's ethical views, and show how, though starting from the same principle, they differ from Kant's. Show how Kant develops the connexion between duty and free-will. Hegel has no place for a doctrine of freedom of will in the individual: Explain this. 8. Examine the following: (1.) The common dislike to utility as the standard resolves itself into a sentimental preference amounting to the abrogation of reason in life. (2.) The common antithesis between Intuitionalists and Utilitarians must be discarded: since such abstract moral principles as we can admit to be really self-evident seem required to furnish a rational basis for a Utilitarian system. HISTORY OF PHILOSOPHY. Examiner-MR. A. E. GOUGH, M. A. 1. "King Nomos must be superseded by the scientific professor' Prove from history, and from the nature of the thing, that philosophy is always militant here on earth. Picture the life-work of Socrates from the Platonic Apology. Mark the attitude of Descartes towards the accredited learning of his day, as exhibited in the Discourse on Method. 2. Specify the elements incorporated into the philosophy of Plato from the pre-Socratic schools. Point out germs of Stoicism and of asceticism in the teachings of Plato. Explain the following positions: Verification consists in the appeal to ante-natal experience." "Reminiscence is kindled up in the mind of the philosopher by the aspect of visible beauty, which is the great link between the world of sense and the world of ideas." 3. "With Aristotle the conscious subject is the animated organism, body and soul in one." Prove this from the Aristotelian definition of the soul or vital principle (psyché), and define the several terms of which the definition is built up. Describe after Aristotle the successive gradations of psychic life. Contrast the Platonic and Aristotelian views of method. Compare the Platonic and Aristotelian doctrines of spiritual immortality. 4. Note the principal points in the reply of Leibnitz to the general empiricism of Locke. "pro 5. Hume treats the transcendental elements of cognition as pensions to feign." Distinguish the counterpositions taken up by the Kantian and Scottish schools. Explain the Kantian doctrine of the antinomies and of the dialectic of pure reason, and show how these limits to the philosophising process are really or apparently surmounted by Hegel. 6. Justify or refute the charge of atheism brought against the philosophy of Fichte. 7. Philosophy is the science of the formation and transformation of thought, the investigation of the successive phases of the intelligence of the human race." Refer this definition to its school, and indicate the place it assigns to the history of philosophy in a system of liberal education. Show that it vindicates the study of philosophy against the popular phonomenalism of the present day. LOGIC. Examiner-MR. A. E. GOUGH, M. A. 'Thought is 1. "The understanding has no power of intuition." operative only within the field of possible experience." State the Kantian doctrine of the relation of logic to metaphysic. Show that any philosophy to be complete must posit verities be yond the scope of the logical faculty. 2. "Logic treats both of the process of advancing from known truths to unknown, and of all other intellectual operations in so far as auxiliary to this." Examine this statement, and argue pro and contra the position that the Aristotelian Organon and the Baconian Organon form parts of one system. |