be the equation of a conic whose centre is the origin, prove that the values of z, got from the equation, 4(≈ - A) (≈ — C') = B2 are constant, if the axes be rectangular, no matter what the direction of the axes be. 10. Construct a conic passing through five given points. 11. If a, ß, y, be the roots of the cubic x3 + рx2 + qx + r = 0 and 80, 81, 82, 83, 84, the sums of the powers of those roots, prove that 12. Prove the Binomial Theorem, from a consideration of two solutions of the following question in combinations: "A painter has (x + y) colours, x dark, y light. He has to paint n balls, each ball being of one colour, but any number of balls may be of the same colour; in how many ways can he paint the balls ?" MR. BURNSIDE. 13. Find the position of equilibrium of a uniform beam, one extremity of which rests against a vertical plane, and the other on the interior surface of a given hemisphere. 14. A rectangular table stands on a rough inclined plane, and has two sides horizontal, if the coefficient of friction of the two lower feet be μ, and of the two upper u', find the inclination of the plane when the table is on the point of slipping. 15. A double ladder is placed in a vertical plane, and resting on a rough horizontal plane; from observing the greatest angle of opening consistent with equilibrium, determine the coefficient of friction. 16. Determine the maxima and minima values of the function x2+ y2+z2, when x y z are connected by the equations 17. If U=o be the equation of a plane curve, and the angle be tween the radius vector from the origin, and the normal at any point, prove that tan Ly - Mx = Lx + My' and transform the expression for tan to polar co-ordinates by the substitutions = r cos 0 y = r sin 0. 18. Form the equation which gives the distances of the four points determined by the conics x2 y3 a focus of the former. C. MR. WILLIAMSON. I. Prove that the equation which represents the pair of asymptotes to an hyperbola only differs in its absolute term from the equation of the hyperbola. Apply this principle to find the equation of the asymptotes to the curve ax2 + 2hxy + by2 + 2gx + 2fy + c = 0. 2. In a spherical triangle, prove the relation 1+ cos A cos B cos C sin2C I cos a cos b cos c sinc 3. Prove that the remainder after n terms in Taylor's expansion of f(x+h) can be represented by hn ·ƒn (x+Oh), I.2.3 ...n where lies between the limits o and 1. 4. Find the locus of the centre of an equilateral hyperbola which has contact of the second order with a conic at a fixed point. 6. A homogeneous right cone is placed with its base upon a rough plane, the inclination of which is gradually increased; investigate the condition in order that the cone should commence to tumble and slide simultaneously. DR. TRAILL. 7. Find the equation of the osculating parabola of the third order to the ellipse x2 y2 8. If the determinant a-λ, h, [PAL] be multiplied by the determinant got from this by changing the sign of X, prove that the product is of the form and determine the constituents A, B, C, F, G, H. 9. Two weights support each other on a rough double-inclined plane, by means of a string passing over the vertex; show that the plane may be tilted about either extremity of the base through an angle equal to twice the angle of friction, without disturbing the equilibrium. Prove this, and state what limitations, if any, are necessary. 10. If a man support a weight equal to his own by means of a system of three moveable pulleys, each of which hangs from the block by a separate string, find his pressure on the floor on which he stands. 11. A uniform beam rests across the top of a smooth wall, and its lower end touches a rough horizontal plane; prove that its angle of inclination with the horizon in its position of equilibrium is given by the equation 12. Show that the conditions requisite to make a balance stable are at variance with those required to make it sensible; and show how best both these properties may be ensured. MR. BURNSIDE. 13. Prove that the area of a triangle formed by three tangents to a conic is given by the formula where p', p', p'" are the three "distinct" perpendiculars dropped from the vertices of the triangle on the focal vectores to the points of contact of the sides, and P the semiparameter. 14. A right-angled triangle is inscribed in a conic; if the vertex be fixed, prove that the hypotenuse passes through a fixed point. 15. From the equations a2= a2 cos20+ b2 sin20, B2 a2 cos2 (0 - C) + b2 sin2 (0 – C'), = y2 = a2 cos2 (0 + B) + b2 sin2 (0 + B), prove that a2 sin 2A + ß2 sin 2B + y2 sin 2C = 2 (a2 + b2) sin A sin B sin C, where A, B, C are the angles of a triangle. 16. The sides of a spherical triangle are connected by the relation b = tan2 1c, tan a. tan 1. State accurately how Locke limits and defines his doctrine that all our ideas are derived from sensation and reflection. 2. State accurately the essential distinction, according to him, between primary and secondary qualities. Why does he think that the ideas of the former are resemblances, and those of the latter not so ? 3 Wherein consists the identity of the man, the soul, and the person, respectively? State the difficulties raised about Locke's doctrine as to personal identity, and his solution of them. How, according to Locke, do we get the idea of infinity, and what are the characters of the idea? 5. What is the cause of the "disorder in our names of substances," and why does Locke charge it as an imperfection in words? The uncertainty had not always been regarded as an inconvenience, according to him? 6. It is a mistake, according to Locke, to think that our senses only show us material things. Explain this, and state how Locke applies the remark. 7. Discourses on morality might be made as clear as those in natural philosophy, according to Locke; show this. 8. Many a man," says Locke, "who was pretty well satisfied of the meaning of a text in Scripture, or of a clause in the Code, has, by consulting commentators, quite lost it." How does he explain this? 9. Explain the origin of our ideas of moral relations. How does Locke confirm his statement as to the meaning of the word "virtue"? and how, according to him, does the mistake arise of confounding "sin" and "vice"? DR. STUBBS. 1. Bacon divides the variation of an experiment into three kinds; what example does he give of each kind? 2. Specify the several reasons assigned by Bacon which tend to show that the part of Logic which treats of the invention of arts and sciences was wanting in his time. 3. Examine the following argument: "The Epistle to be attributed to Barnabas is not to be reckoned among the writings of the Apostolic fathers; because, if genuine, it is a part of Scripture, and, if spurious, it is the work of some forger of later age." 4. What are the propositions relating to the discovery of truth, from which it is inferred that there are processes of reasoning to which the syllogistic theory is inapplicable, and point out the fallacy in the argument? 5. What three examples does Bacon give of Idola tribus? 6. What is the fallacy in the following argument : "If the Deity foresees exactly what I shall do on any occasion, it must be impossible for me to act otherwise, therefore my actions cannot be free ?" 7. Mention the principal points in which Whately's Synthetical Compendium differs from the corresponding parts of Murray's Logic. 8. Prove, in general, that the mutual interchange of a conclusion and premiss can never be legitimate. DR. TARLETON. 1. What, according to Stewart, are the causes which have retarded the progress of empirical Psychology? What is the real cause of its slow progress, and of the uncertainty of its results? 2. In what sense did Descartes consider extension as the essence of matter, and thought as the essence of mind? 3. The word "cause," according to Stewart, is used in two very different senses both by philosophers and the vulgar. Mention some leading philosophers to whom this remark would not apply. How does Stewart account for the confusion which has arisen between the two kinds of causes, and what are the results to which he thinks it has led ? In Stewart's remarks here, one of his leading doctrines in reference to Causality is implied, without being expressly asserted. Stewart elsewhere accounts for the harmony between the anticipations of the mind and the course of nature? 4. How does Stewart endeavour to show that imagination is attended with a belief in the existence of its object ? What addition does he make to Reid's theory of Perception ? |