10. The point, say V, of which the vector is p= 2a, is a double point of (a), near which that surface has a cusp, which coincides nearly with its tangent cone at that point; and the semi-angle of this cone is

AUXILIARY EQUATIONS.

π

2. Determine the nature of the loci represented by the equations

3. Find a curve always touched by the same diameter of a circle rolling along a straight line.

4. Find the differential equation of surfaces formed by the motion of a straight line, which is always parallel to a given plane, and always passes through two given curves; and show how to integrate it.

5. The ordinates of a plane curve (B) are equal to the corresponding arcs of another plane curve (4), counted from a given abscissa a; what must be the nature of the curve (4) in order that the area of (B), taken between given abscissæ a and b, may be a maximum or minimum?

6. Let p = f(w) be the polar equation of a curve, and let p1 =ƒ(w+π); required the general equation of the class of curves for which βρι is invariable.

7. If on a surface of the second degree we consider a finite rectangle formed by four lines of curvature, the distances of the vertices of this rectangle from the consecutive confocal surface form a proportion.

8. The sides of a given rectilineal angle touch two given curves in the points t and ť; and the bisector of this angle, as it moves about, envelopes a third curve, touching it at T. Prove that the arc described by Tbears a constant ratio to the sum or the difference of the arcs described by t and ť.

PROFESSOR JELLETT.

1. If ds be the element of the arc of a curve, p the radius of curvature, and p the perpendicular from a fixed point on the tangent; it is known that

pd

taken between an apse and any other point on the curve, is equal to the difference between the arc and the projection of the extreme radius vector on the tangent. Prove the analogous property for a surface, namely, that if ds be the element of the surface, R, R' the principal radii of curvature, and P the perpendicular from a fixed point on the tangent plane, the integral

taken through the space enclosed by any closed curve on the surface, is equal to the difference between the enclosed portion of the surface and the sum of the projections of the elements of the cone formed by the extreme radii vectores on the corresponding tangent planes.

2. If it be proposed to draw between two fixed points a curve, such that if a new curve be formed from it by producing in a given ratio the normal (terminated by the line joining the given points), the area of this curve shall be a minimum; the solution is plainly the right line joining the two points. What solution is given by the calculus of variations ? 3. Admitting that the solution of the equation

A, B, C, u, v, being functions of x and y, the quantities u and v must satisfy three partial differential equations. Determine these equations, and explain how the solution of the original equation is facilitated if it be known that its solution is of the form

5. Let it be required to determine a function u which shall satisfy the partial differential equation

du

the values of u and

dt

values of x and y.

being also given for a given value of t, and for all

Show that there can be but one such function.

6. Find the value of the definite integral

S

∞ cos ax dx

I + x2

7. Determine the arbitrary functions in the equation-

2=

= x¢ (x2 + y2) + y¥ (x2 + y2).

by the condition that the surface which it represents shall pass through the ellipses

2 = c,
ax2 + by2 = 1,

z = c',

8. Eliminate the arbitrary functions from the preceding equation.

EXAMINATION FOR PREMIUMS IN CIVIL LAW.

PROFESSOR ANSTER.

The books ordered for Examination are:-Institutes of Justinian, Schrader's edition; Gibbon's Decline and Fall of the Roman Empire, chap. xliv.; Arnold's History of Rome, vol. 1., chap. xiv.; Merivale's Romans under the Empire, vol. III., chap. xxxi.; Milman's Latin Christianity, Book III., chap. v. ; Story's Conflict of Laws, first three chapters; Grotius, De Jure Beli et Pacis, Prologomena; Vattel, Droit des Gens, Préface et Préliminaires; Creasy's Rise and Progress of the English Constitution; Guizot's Essais sur l'Histoire de France, Sixième Essai; Civil Law Class, Introductory Lecture, 1851, and Examination Papers printed in the University Calendars, from 1851 to 1858, inclusive; the Class Lectures of the year 1857-58.

1. State the definition of "liberty" given in the "Institutes.”

2. Do you regard this definition as adequate? If inadequate, state in what respect inadequate.

3. It has been said that, in the great Charter of Runnymede, the aristocracy of England only aimed at obtaining what would benefit themselves what is M. Guizot's answer to this?

4. The Charter was renewed at the accession of Henry the Third, and several times during his reign. State the principal provisions of "Magna Charta and its supplements," as given by Mr. Creasy.

r

5. State the securities to the nation for the observance, on the part of the king, of the Charters of Liberty.

6. State the means taken for their publication.

7. With respect to the administration of justice, Mr. Creasy tells you that two great principles are found in these Charters. State them.

8. What is the essential characteristic of trial by jury-that which, among all its changes, has been preserved unchanged, and which may be described as its principle?

9. M. de Tocqueville states what he regards as the greatest advantage of the trial by jury. State what he says.

10. Mr. Creasy quotes a passage from Dr. Arnold, in which, in addition to what is said by M. de Tocqueville, he mentions another great advantage. State what he says, as given by Creasy.

II. "Titio hominem Stichum do lego: Seio eundem do lego." These words occur in a Roman will. What is the effect of this bequest?

12. Titius, instituted heir in the will of Seius, survives Seius, but dies "antequam adierit hereditatem." What is the effect of this state of facts? Is Seius to be considered as intestate, or does the inheritance form a part of the property of Titius transmissible to his representatives?

13. "Testamentum, ab initio jure factum, propter capitis deminutionem irritum fit." Is such will wholly ineffectual? The case is given in the "Institutes:" state fully what is said there on the subject.

14. In Titulus 9, 2nd book of the "Institutes," "Per quas personas nobis acquiritur," the following passage occurs :-"Non solum autem proprietas per eos quos in potestate habetis acquiritur vobis sed etiam possessio-cujuscunque enim rei possessionem adepti fuerint id vos possidere videmini." What led to the doubt on the subject of possession implied in the language of this passage, when on that of property there was none? And how is the passage explained consistently with there being a real distinction in this respect between property and possession?

15. One man has the usufruct of a slave, the property of another; such slave acquires property; to whom does it belong?

16. A legacy, if left to such slave, to whom does it belong?-to the usufructuary?— -or to the dominus?

17. Such slave is instituted hæres in a will; to whom does the inheritance belong?

18. The Prætorian Jurisdiction, which, in testamentary matters, dispensed with some of the requirements of the Quiritary Law, introduced a new solemnity in the execution of wills. What was it?

19. Things sold and delivered do not, without something more, become the property of the purchaser. State the Roman law on this subject. 20. A case is stated where the nuda voluntas of the owner is enough, without delivery, to transfer property. State such case.

21. Define derelict.

22. Are things thrown in a storm from a vessel, for the purpose of lightening it, derelict?

23. State the distinction, as given in the "Institutes," between the usufructus and the usus.

24. A person believes a thing to have been given him, or that he has purchased it. He continues in possession of it for a sufficient length of time to have made it his by usucapio. In point of fact, however, he is wrong in believing it ever to have been given or sold to him. Has it, in this state of facts, become his by usucapio. If not, why not?

25. What is meant by extranei hæredes?

26. Give Justinian's definition of a legacy?

27. In the Roman system a legacy might be given of a thing not belonging to the testator. State any limitation to this proposition that occurs to you with reference to the thing so bequeathed, and also with reference to the testator. State what is said in the "Institutes" with

reference to this subject.

28. In the definition of "slavery" given in the "Institutes," you find a statement embodied in the definition, which seems to prepare the way for the final extinction of slavery. What is the statement ?

29. A case (unconnected with the privileges of a military testament) is stated in the "Institutes," in which the principle that a person cannot die partly testate and partly intestate is, or seems to be, violated. State the case.

30. Explain what is meant by capitis deminutio.

31. State the distinction between tutor and curator.

32. In the change effected from the republican system of government to a monarchy, Mr. Merivale mentions the different offices filled by the Emperors; among them, those of princeps senatus and tribune. State the way in which the assumption of those particular offices was useful for the purpose.

33. The Emperors gave their administration of government the appearance of depending more on popular principles than the rule of a king or dictator. In what way was this chiefly effected?

34. Mr. Merivale states ancient authority, not merely for the fact on which this view is founded, but for the view itself. Mention the passage he quotes, and state, as far as you can, what is precisely said by the author whom he quotes?

35. Other authority than that of written documents is given for the fact on which the view referred to in the last two questions rests, and indeed for the view itself. State what you find in Merivale on the subject?

36. Merivale gives you a definition of Treason from Cicero, and another from Ulpian. State both or either. If you cannot give the definitions with accuracy, state the leading principle found in both, and state, as far as you can, the distinction between them.

37. Augustus declined the consulate; he, however, contrived to have all the advantages which being actual consul could have given. How was this effected?

38. What was the chief advantage thus obtained?