4. State and prove the two laws of Sorites.

5. What errors of Locke concerning Logic does Whately notice?

6. What are Whately's remarks on the names and meanings of Infinitives, and how their occurrence in a logical proposition is restricted?

7. What mistake has arisen from merely probable conclusions being deduced in a Syllogism; and how does Whately explain it ?

8. What are the cases in which an argument might naturally fall into other figures besides the first?

9. What are Whately's subdivisions of the kinds of "Definition;" and with which kind does he state that Logic is concerned? Is he right ?

10. In arguing in a circle, what is the best way to veil the fallacy? How does Whately illustrate this?

3. Show how to integrate the system of m linear partial differential equations of the first order,

6. Apply the formula for the quadrature of a surface

to the determination of the entire surface of an ellipsoid, and show that it may be reduced to the two integrals

and

sin Ꮎ ᏧᎾ

(a2 sin 20+ c2 cos 20) (b2 sin 20+ c2 cos 20)

sin 30 de

10 √(a2 sin 20 + c2 cos 20) (b2 sin 20 + c2 cos 20)

7. Show that the equation of the plane curve which makes

S$ (p) ds

a maximum or minimum (p being the radius of curvature) will be p2 p' (p) = ax+by+c.

a. If the curve be drawn between two given points, the extreme tangents not being given, show that p2 p' (p) must vanish at each extremity. 8. Let it be required to draw between two points a curve which would make

(x, y) ds

a maximum or minimum. Then, if it be possible to draw more than one curve satisfying the condition

84(x, y) ds = 0,

and if these curves be arranged according to the angles which their initial tangents make with a given line, no two consecutive curves will in general give real maxima or minima?

TRIGONOMETRY AND LOGARITHMS.

MR. TOWNSEND.

1. Express, by a logarithmic formula, the volume of a tetrahedron in terms of the lengths a, b, c of any three conterminous edges, and of the opposite angles a, ß, y they make with each other in pairs.

2. Express again the volume of the same in terms of the lengths a and a, b and b, c and c' of its three pairs of opposite edges.

3. Express, as a determinant, the relation connecting the cosines of the six arcs a and a', ß and B', y and y' which connect in opposite pairs any four points P, Q, R, S on the surface of a sphere.

4. Investigate the equation of the nth degree whose roots are the cosines of the n arcs a, ß, y, d, e, &c., connecting any point P on a sphere with the n vertices A, B, C, D, E, &c., of any regular polygon on it.

5. Prove Legendre's Theorem, that if the sides a, b, c of a very small spherical triangle be straightened into lines, the angles A, B, C will be diminished each by a third of the spherical excess.

π

2

6. If x be an angle differing very little from and dx any small increase or diminution of it, investigate the corresponding changes in the functions log sin x, log tan x, log sec x.

7. If the function (a2 - 2ab cos x + b2) be expanded in an infinite series of the form 40+ A1 cos x + A2 cos 2x + &c., required the relation connecting every three consecutive coefficients Ak-1, Ak, Ak+1.

8. In general, if any function F of x, which is such that F(x) = F(−x), be expanded in an infinite series of the same form, express, as definite integrals, the values of the several coefficients Ao, A1, A2, &c.

9. Determine the value of the definite integral Sex dx between the limits infinity and nothing.

required the values of c, k, and 0 in terms of a, b, and x.

HEBREW EXAMINATION, AND THE PRIMATE'S PRIZES.

SENIOR CLASS.

DR. TODD.

1. It is objected against the authenticity of the later prophecies of Isaiah, that in the passage (lxiv. 10), "Thy holy cities are a wilderness, Zion is a wilderness, Jerusalem a desolation, Our holy and beautiful house, where our fathers praised thee, burned up with fire," &c., the author speaks as a contemporary of the Babylonian exile; how is this answered?

2. The later prophecies, beginning at ch. xl., divide themselves into three parts; parts I. and II. conclude with the same words; and part III. with an expansion or paraphrase of the announcement made in those words?

3. Give an account of the distinguishing differences, as regards the subject matter, in these three parts or sections of the later prophecies. 4. Point, and translate :

כה אמר האל יהוה בורא השמים ונוטיהם

רקע הארץ וצאציה

נתן נשמה לעם עליה ורוח לחלכים בה :

(a). To what D of the Law does the m in which these words occur correspond?

(6). The word DT appears to be grammatically anomalous; how is this explained?

how would this affect the ; נטעים e). It has been proposed to read)

translation, and what gain would it be ?

may be supported by another נטעי השמים d). The unusual phrase)

passage in Isaiah?

are repeated at the end of the Book of והיה מדי חדש The words .5

Isaiah; explain the reason of this.

6. In what other Books of the O. T. is the same thing done? Give the Masoretic mnemonical word for the names of these Books.

7. Translate the Masoretic note at the end of the Book of Isaiah.

SENIOR AND MIDDLE CLASS.

DR. TODD.

1. Translate and explain the Masoretic note which occurs at the end of the Psalms.

2. Give some account of the Masorah; (a) meaning of the name; (b) by whom first printed; (c) the manner in which it is printed, and its division into parva, magna, and finalis.

3. Translate the following ; and explain the Masoretic notes :

הושיע יהוה כי גמר חסיד כי פסו אמונים מבני אדם :

שוא ידבדו איש את רעהו

שפת חלקות בלב ולב ידברו :

יכרת יהוה כל שפתי חלקות ג' רף ג' לָשׁוֹן מדברת גדולות :

The Masora magna on the word npn is as follows:

חלקות ג' וסימן דברו חלקות שפת חלקות בלב ולב ודברו יכרת י' כל שפתי חלקות :

יחוה מי יגור באהלך קדשך : מי ישכן בהר

4. Point and translate :

(a). What is the difference between 1 and p? Show this by examples.

is not inconsistent with the Psalm בהר קדשך The allusion in .(6)

having been written by David? Show this by other examples.