variegated patchwork of many languages, without any unity or inner coherence at all, various societies were instituted among them, at the beginning and during the course of the seventeenth century, for the recovering of what was lost of their own, for the expelling of that which had intruded from abroad; and these with excellent effect.-TRENCH.

French Language.

DR. ABELTSHAUSER.

CHATEAUBRIAND.

I. Faites un résumé de sa vie, contenant :

a. Sa famille; b. Son pays; c. Les rapports de ses destinées avec l'histoire politique de la France; d. Ses voyages; e. Ses œuvres; f. Leur but; g. Leur influence sur la littérature et sur les opinions religieuses et politiques de son temps, avec les dates approximativement.

II.-Donnez deux mots Français dérivés de chacun des mots Latins suivants, avec leur signification en Anglais :-pastor, pensare, captivus, nativus, homo, sacramentum, factio, calculus, blasphemare.

III. Quelle est la différence entre: prouver et éprouver; moudre et émoudre; perdu et éperdu; puiser et épuiser; mouvoir et émouvoir; chasse et châsse; économe et économique; découvrir et découvrer; ils révèrent et ils rêvèrent; gué et guet; place, lieu, et endroit. Donnez-en aussi la dérivation.

O guerriers! je suis né dans le pays des Gaules.
Mes aïeux franchissaient le Rhin comme un ruisseau,
Ma mére me baigna dans la neige des pôles
Tout enfant, et mon pére, aux robustes épaules,
De trois grandes peaux d'ours décora mon berceau.

Car mon pére était fort! L'âge à présent l'enchaîne.
De son front tout ridé tombent ses cheveux blancs.
Il est faible; il est vieux. Sa fin est si prochaine,
Qu'à peine il peut encor déraciner un chêne

Pour soutenir ses pas tremblants!

VICTOR HUGO.

VI. Traduisez en Français :

The base and plebeian origin of Mahomet is an unskilful calumny of the Christians, who exalt instead of degrading the merit of their adversary. His descent from Ismael was a national privilege or fable; but if the first steps of the pedigree are dark and doubtful, he could produce many generations of pure and genuine nobility; he sprung from the tribe of Koreish and the family of Hashem, the most illustrious of the Arabs, the princes of Mecca, and the hereditary guardians of the Caaba.--GIBBON.

CHARLES GRAVES, D. D., Professor of Mathematics.
JOHN H. JELLETT, M. A., Professor of Natural Philosophy.
GEORGE SALMON, D. D.

GEORGE F. SHAW, LL. D.

JOHN LESLIE, M. A.

DR. SALMON.

1. In an ellipse find the locus of the intersection

a. Of the central perpendicular on tangent with the focal radius

vector.

b. Of the focal perpendicular on tangent with the central radius vector.

2. If in a curve the product of the perpendiculars from two fixed points on the tangent be constant, show by the method of infinitesimals that the radii vectores from these points make equal angles with the tangent.

3. Trace the curve p sin 30 = a, and find its asymptote.

If the equation of a curve be ax3 + by3 + cz3 + dw3 = 0, where + y + z+w=0,—

a. Show immediately that any of the points xy is a point on the Hessian.

b. Form the equation of the Hessian.

5. The polar with respect to the Hessian of any point on a cubic meets the tangent at that point on the cubic ?

6. The symmetrical function Ear Bay", &c., contains the coefficients of the equation in a degree equal to the highest of the indices p, q, r, &c. ?

7. Show clearly how by Lagrange's method the solution of an equation of the fifth degree is made to depend on that of an equation of the sixth degree.

1. If u

=

MR. LESLIE.

sin (z+xeu); expand logu in a series of powers of x.

2. Find a maximum or minimum value of x y z subject to the condition ax + by + cz = k, and determine whether the result is a maximum or a minimum.

3. How does Leroy investigate the conditions that the general equation of the second degree should represent a surface of revolution?

4. If F(xyz) = o is the equation of a curved surface, and if we put

5. If a curved surface is given by the equations

x=f1(u, v), y=f2 (u, v),

the element of a curve traced on it may be

z=

z = fs (u, v),

expressed by the formula {Edu2 + 2 Fdudv + Gdv2}},

and the element of the surface by the expression (EG – F2) dudv. Assuming this, deduce the ordinary formula for the element of a surface in polar co-ordinates,

6. Express the element of the surface of an ellipsoid in elliptic co-ordinates.

7. Prove that the angle between conjugate tangents at any point of an ellipsoid is given by the equation

i being the angle which the tangent makes with one of the lines of curvature at the point.

DR. LUBY.

dp

dw

1. Given the equation (n-1) (1 + p2) − n == = 0, and p=

2. From the extremity of the axis-minor of an ellipse draw a line of maximum length, and find the ratio of the parts into which this line, if drawn, would divide the area of the semi-ellipse.

3. An ellipsoid of revolution about the axis major, with this axis vertical, empties itself through a given orifice at the lowest point; compare the times of emptying the upper and lower halves. The velocity at the orifice varies as the square root of the height of the fluid.

4. A line of a given length is joined at its middle point to another line l'indefinite and fixed, but not in the same plane, by a line at right angles to both; find the solid contents of the solid generated by the rotation of about l'.

5. If portions be taken on two given right lines from two given points, and the product of these portions be given; find the envelope of the line joining their extremities.

DR. GRAVES.

1. The formulæ employed in integrating by parts may be obtained by the method of separating symbols.

2. Prove the following symbolic equations:

3. If u1 and 2 are particular integrals of the linear differential equation,

Transform this unsymmetrical double integral into a symmetrical pair of single integrals.

4. Prove that the multiple integral (n) f(x)dan is equal to the single integral