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two refracted rays respectively, z1, z1′, z2, z2', are in the direction of the respective wave normals; a1, a1, a2, a'2, are the angles which the axis of xo makes with the axes x1, x1, x2, x'2 respectively; r, r', are the ratios of the lengths of the refracted rays to that of the incident ray, and s, s', are the velocities of wave propagation. Assuming these, and the principle of the equivalence of vibrations, deduce the theorem given by Professor Mac Cullagh, for the solution of the problem of crystalline reflexion and refraction.
II. Investigate the expression for the intensity of light passing through a small hole of any shape, and show that it may always be reduced so as to involve single integrals only.
12. If a beam of polarized light pass through a plate of uniaxal crystal cut perpendicular to the axis, and then through an analyzer, find the intensity of the light which reaches the eye, in various parts of the image.
1. How is it proved à priori that the free electricity of a conducting body of spheroidal form resides on the surface?
2. Give the numbers which express the electrical conducting powers of silver, gold, and platinum, the conductibility of mercury being taken as unity.
3. Describe Fechner's method of comparing experimentally the electromotive forces of different couples.
4. What is Professor Wheatstone's method of obtaining the same measures?
5. How does M. Pouillet deduce the electromotive force of the thermoelectric couple in absolute units? Give the steps of the deduction, and the final result.
6. Describe M. Pouillet's experiments to confirm the formula of Ohm for the intensity of the current engendered by the voltaic pile.
7. Compare the intensities of the current in a pile composed of equal couples arranged in the usual manner, and arranged with the similar elements united.
8. Prove that the direction of the force, exerted by a magnet on a magnetic particle, is in the line drawn from the latter to a point X, on the axis of the magnet produced, given by the formula
7 being half the length of the magnet, and r and r' the distances of the particle from its two poles.
OX = 1
9. Prove that, in the magnetic curve, the difference of the cosines of the angles, made with the axis of the magnet by the vectors drawn from any point to the two poles, is constant.
10. Find the moment of the force exerted by one horizontal magnet upon another, their axes being at right angles, and their centres in the same vertical line.
11. Find the expression for the disturbing effect upon a freely-suspended horizontal magnet, produced by a row of parallel iron rails placed in the same horizontal plane.
12. Deduce the time of vibration of a prismatic magnet suspended horizontally, in terms of the Earth's magnetic force, the moment of inertia of the magnet, and the coefficient of resistance.
13. The bifilar magnet is slightly disturbed from its normal position at right angles to the magnetic meridian; find its time of vibration.
14. Show that the horizontal component of the Earth's magnetic force may be determined by means of the dip-circle, and deduce the formulæ.
15. A horizontal magnet moveable on a vertical axis is acted on by a vertical conducting wire of indefinite length; show that the magnet will rest indifferently in any position, when the distance of the wire from the centre of the magnet is equal to half the length of the latter.
16. Prove that the moment of the force of a magnet, to produce rotation in a conducting wire, is independent of the form of the circuit.
17. How is the repulsion of a diamagnetic body by a magnetic pole explained by Professor Weber? Describe his experiments in confirmation of this explanation.
1. Given a ray of elliptically-polarized light, it is required to determine by the use of a compensator and an analyzer the position and ratio of the axes of the ellipse. Deduce the equation for the determination of both these quantities.
a. In determining the second of these, what must be the position of the compensator in order that an error in the use of the analyzer may produce the least possible effect on the result?
2. Let r, 0, be the polar co-ordinates of a point on a screen. If a beam of light falling on the screen be represented by
sin me sin sin (vt - x);
what is the appearance?
3. The expression for the intensity of polarized light falling on a screen
πVx2 cos2 a + y2 sin2 a
where x, y are the co-ordinates of a point on the screen, and a is the azimuth of the plane of polarization.
a. Describe the appearance presented at different points of the screen. b. What effect is produced by turning the polarizer round ?
4. Admitting that the difference in phase produced between the components of a polarized ray parallel and perpendicular to the plane of incidence, by a single reflexion at the surface of a transparent medium, may have any value from o to ; show how to obtain circularly-polarized light by a single reflexion?
a. The azimuth of the incident light must always be very large.
5. Explain the rings, seen by Newton, produced by light passing through a small hole and incident on a spherical mirror, and find an expression for the tint.
a. Could these rings be produced by a plane mirror?
b. If the hole through which the light passes be indefinitely small, and the distance of the screen from the mirror be great as compared with the thickness of the mirror, state how far the rings agree with the transmitted Newtonian rings, and how far they differ from them.
1. A known volume of carbonic acid is suffered to remain in contact with water, and the pressure and temperature are noted at the beginning and end of the experiment; determine the amount of the gas which has been absorbed by the water.
2. Assuming that the amount of heat generated is proportional to the amount of work destroyed; find the change in temperature produced in a bubble of gas by its expansion in rising from the bottom to the top of a vessel of water.
3. Regnault found the coefficient of dilatation of a gas to augment with the pressure; how may this be explained; and what experiment is suggested by Peclet for deciding whether this explanation be true?
4. In determining the absolute conductibility of a metal, Peclet found it sensibly the same for plates of different thickness; how does he explain this, and how was his explanation verified?
5. In a condensing engine determine the quantity of water, at a given temperature, which is necessary to reduce the temperature of steam from t' to t'.
Mental and Moral Sciences.
1. Give a historical sketch of the leading atomistic theories, with Schwegler's statement of their relative position.
The objections of the Monadists to the infinite divisibility of Matter are open to criticism, according to Kant? How have they attempted to avòid these difficulties?
2. Explain Kant's three fundamental Dynamical Relations, and prove that they are thoroughly dynamical."
General conclusion from the discussion of the Analogies?
Kant points out the superiority of his own mode of proof, and several errors and deficiencies in previous writers?
3. Proceeding analytically in the Topic of Rational Psychology, Kant shows the impossibility of a definition of the Ego, as a merely thinking subject, on the basis of either Materialism or Spiritualism?
Kant gives an elaborate explanation of the first proposition of this "Topic"?
4. The importance of a correct application of the word "absolute" appears in the discussion of several questions involving the conceptions of Possibility and Necessity?
In contrasting our Theoretical and Practical Knowledge, as well as on other occasions, Kant notices the importance of distinguishing between the Absolute and the Relative?
5. How does Mr. Mansel conclude that Berkeley held the representative theory of Perception? State what you consider to be the chief difficulties in attributing to Berkeley Arnauld's theory of Ideas; and show that the same objections do not lie against Fichte.
6. A passage quoted by Sir W. Hamilton to disprove Brown's defence of Locke has been employed by him on another occasion for an opposite and apparently inconsistent purpose?
The distinction on which Mr. Webb founds his defence of Locke is found in Hamilton's account of the Cartesian opinions on the subject?
7. Give an account of the principal theories which have been proposed to explain our Personality and Personal Identity, with Mr. Mansel's statement of the leading difficulties, and his proposed mode of solution.
In what form should the question respecting Identity be stated, according to Sir W. Hamilton? and what argument of Mr. Mansel proceeds from this point of view?
How does Hamilton criticize Kant's doctrine? and how far, according to Kant, can we infer the permanence of mind speculatively?
8. The Conception of Design has been introduced under various forms into modern philosophical systems?
Compare the theories of Taste as given by Kant, Hamilton, and Mr. Mansel.
The solution of the apparent contradictions in our æsthetic judgments is given somewhat differently by Kant and Mr. Mansel ?
9. Explain what was meant by Species Impressæ and Expressæ, and give an account of the functions usually attributed to the Active and Passive Intellects.
Hamilton quotes what he terms a profound observation of Occam in connexion with this subject?
10. Give a summary of Spinoza's doctrine respecting the Conception of Substance; and show that his Paradoxes as to the existence of finite beings, as well as those of Schelling and Hegel, are chiefly founded on the ambiguity of language.
Some of the most important doctrines of Modern Psychology are found in an exaggerated form in Schelling?
11. How, and on what principle, does Mr. Mill classify the fallacies which arise from the disposition to attribute Reality to our subjective laws? He accuses Des Cartes of inconsistency, but, in so doing, begs the question between himself and his opponents?
What does he consider to be the extreme stage in the development of this propensity?
12. What appear to have been the principal contributions of preceding writers to the Aristotelian logic?
Show that although the fundamental Laws of Thought were indicated at an early period, correct views respecting them were first due to Kant. Give some account of Aristotle's doctrine of Definitions.
1.-a. What is Aristotle's classification of the ȧperaí? b. He uses the word aperʼn in three significations?
2. Give a full statement of the reasoning of Aristotle to define and ascertain the chief good.
3. ̓Αλλὰ τοῦτο μὲν εἰσαῦθις ἐπισκεπτέον, τό δ ̓ αὔταρκες τίθεμεν ὅ μονούμενον αἱρετὸν ποιεῖ τὸν βίον καὶ μηδενὸς ἐνδεᾶ· τοιοῦτον δὲ τὴν εὐδαιμονίαν οἰόμεθα εἶναι. ἔτι δὲ πάντων αἱρετωτάτην μὴ συναριθμούμένην, συναριθμουμένην δὲ δῆλον ὡς αἱρετωτέραν μετὰ τοῦ ἐλαχίστου των ἀγαθῶν· ὑπεροχὴ γὰρ ἀγαθῶν γίνεται τὸ προστιθέμενον, ἀγαθῶν δὲ τὸ μεῖζον αἱρετῶτερον ἀεί. Τέλειον δή τι φαίνεται καὶ αὔταρχες ἡ εὐδαιμονία, τῶν πρακτῶν οὖσα τέλος.
Translate, and write a commentary on this passage; explaining the distinction drawn by Aristotle between the τέλειον and the αὔταρκες ἀγαθόν.
4. How does Aristotle correct the Socratic and Peripatetic doctrine which seems to make virtue purely intellectual?
5. How does he show (a) that the intellectual and moral virtues cannot be separated? And (b) in what part of his formal definition of virtue is the relation of the intellectual to the moral virtues expressed?
6. Three things are necessary to the perfection of a moral agent, according to Aristotle, one of which is not of the same importance to the virtue of actions as the other two?
7. Give an account of the opinions of Mackintosh on selfishness and self-love; and Bishop Fitzgerald's critique upon them.
8. State the objections made by Mackintosh to Butler's treatment of the moral sentiments; and to his moral system in general.
9. Give an abstract of Bishop Fitzgerald's answer to the objections of Mackintosh.
10. Explain the theory of conscience put forth by Mackintosh, and show its inadequacy.
11. Give Dr. Adam Smith's analysis and classification of the questions which ought to be examined in a theory of moral sentiments.
12. State in full the reasoning of Bishop Butler to show that the precept to love our neighbour as ourselves includes all other virtues.