2. What is meant by the Greatest Common Measure of two algebraical quantities? Find that of

4x-6x-6x2 + 4x − 6 + 4x3 and 8x1 − 4x3 – 28x2 + 24x.

3. Prove the rule for the division of a fraction by a whole number, and simplify the following expression

investigate a meaning for a2, being a proper fraction.

Assuming a meaning for such an expression as a2, prove that

5. Investigate a rule for finding the cube root of an algebraical expression, and find that of

8x - 36x5 + 66xa – 63x3 + 33x2 – 9x + 1.

6. Solve the equations

(1) x3- 2x2 + 2x − 1 = 0. (2) 6x2 + 6y2 = 13xy, x3 + y3 = 35.

7. Shew how to find the sum of a series of quantities in Geometrical Progression, explaining what is meant by the limit of an infinite series, and in what case such a limit exists.

Sum the series 2 + 5x + 8x2 + 113+... to n terms and, when possible, to infinity.

8. Find the greatest term in the expansion of (1 + x)", n being a negativé quantity.

What are

9. What is meant by a unit of measurement ? the three different units usually employed in measuring angles? Shew how to pass from one system of measurement to another. 10. What are the Trigonometrical Ratios? What is the advantage of using them?

=

Shew that cos (180° - A) cos A and trace the changes in sign and magnitude of cos A as A increases from 0° to 270°. 11. Having given two sides of a triangle and the included angle, shew how, without determining the remaining angles, to

find the remaining side in a form adapted to logarithmic computation, and explain fully the nature and use of subsidiary angles.

12. Find the sum of a series of cosines of angles in arith

metical progression and, if ø =

13'

shew that cos + cos 30+ cos 50 +...+ cos 110 = 1.

13. Find expressions for cos a and sin a in terms of a. At what point of the proof is the necessity of a

measure introduced?

being circular

From Μὴ τοίνυν ἐᾶτε ταῦτ ̓ αὐτὸν λέγειν,....
. φιλανθρωπίας πολλῆς ἠξίωσαν.

to

DEMOSTH., Meidias, c. 12.

to

From Jamdudum ausculto;...
....nusquam tibi sit potandum.

HOR., Sat., II., 7., 1-32.

From Sic distractis exercitibus ac provinciis, Vitellio.,.
to ....cujus tum odio, etiam bonæ leges peribant.
TAC., Hist., 1. 77.

III.

1. Give the names of the various books of the Old Testament in their chronological order.

2. Why was Moses excluded from the promised land? State the circumstances of his death.

3. Who were the Rechabites?

4. Give some account of David's children, marking the fortunes and end of each.

5. "Make them and their princes like Oreb and Zeb: yea, make all their princes like as Zeba and Salmana," Ps. lxxxiii., 11. Give the history to which the passage refers.

6. "Mine own familiar friend whom I trusted," Ps. xli., 9. Of whom was this primarily spoken, and to whom did it prophetically apply? Draw a parallel between the two cases.

7. Give instances, both from the Old and New Testaments, of the use of forms of public worship. Mention the advantages attending the use of forms of prayer in general.

8. What are the proper lessons for Trinity Sunday? Wherein consists their special fitness for the occasion?

9. How would you deduce the opinion of the English Church as to the observance of the Sabbath? What does the prophet Isaiah say as to the manner in which the Sabbath should be observed, and the blessing attending such observance ?

10. What is the authority for Infant Baptism?

11. "Whereof is one Christ, very God, and very Man; who truly suffered, was crucified, dead and buried, to reconcile his Father to us, and to be a sacrifice, not only for original guilt, but also for all actual sins of men." Art. II. What passages of Scripture are commonly adduced in support of the words in italics?

12. Under what circumstances is a special confession of sin to the minister recommended by the English Church (1) in health, (2) in sickness? What objections are there to a frequent use of this kind of confession?

IV. EUCLID AND TRIGONOMETRY.

1. At a given point in a given straight line make an angle equal to a given rectilineal angle.

Having given an angle, make another 3 times as great. Is it possible to make one 23 times as great?

2. Define parallel straight lines. Give also the 12th axiom, and state your opinion whether it might be advantageously superseded by any other, with your reasons for the same.

3. Triangles upon the same base, and between the same parallels are equal to one another.

The line joining the bisections of two sides of a triangle is parallel to the base.

The lines joining the bisections of the sides of any quadrilateral figure, together constitute a parallelogram.

4. Describe an isosceles triangle having each of the angles at the base double of the angle of the vertex.

Which of the regular polygons depend on this proposition for their construction?

5. Triangles of the same altitude are as their bases.

6. If from the angle of a triangle a straight line be drawn perpendicular to the base, the rectangle contained by the sides of the triangle is equal to the rectangle contained by the perpendicular and the diameter of the circumscribing circle.

If D be the diameter of the circumscribing circle, shew that the prism whose base is the given triangle and axis D, is equal to the rectangular solid whose three adjacent sides are equal to the three sides of the triangle.

7. Draw a straight line perpendicular to a given plane from a given point above it.

Shew that this is the least straight line which can be drawn from the given point to the plane.

8. Define the sine, cosine, and tangent of an angle, and trace, in a general way, their magnitudes from 0° to 360°.

Supposing a to be the least value of which satisfies the equation tana, find all the values of which satisfy the same condition.

9. Expand sin (A + B) and cos (A + B) in terms of the sines and cosines of A and B.

10. Find the sines and cosines of 45°, 30°, 18°, 48°.

11. Express sin A and cos B in terms of cos A.

Find sin 15° regarded as the half of 30°, and show that it is identical with sin 15° regarded as 45° - 30°.

In the mensuration of triangles mention the case to which this formula applies.

13. If Stan A+ tan B + tan C+ &c. = sum of the tangents, sum of their products taken 2 and 2,

S2=

S1 =
&c.

1 - S2+ S- &c.

14. Express the sine of any angle of a triangle in terms of its sides.

15. Prove Demoivre's Formula when the index is a whole number, and shew that

cos (A + B + C + ...) + √(− 1) . sin (A + B + C + &c ) = {cosA+√(-1) sinД} {cos B+√(-1) sin B}. {cos C+√(-1) sin C} &c. 16. Shew the use of a subsidiary angle in solving the equation cos x + a sin x = b.

17. Describe briefly the instruments by which angles in a vertical and those in a horizontal plane are commonly measured. There is a tower on the other side of a river; show how its height may be determined without crossing the river.

V. ARITHMETIC AND ALGEBRA.

8

1. Define Vulgar and Decimal Fractions.
Shew from first principles that is equal to 21.
2. Find the sum of

of a guinea, 0125 of £1, 45 of £5. 10s., and 2.25 of 1s. in terms of the decimal of a crown. Reduce the result to pounds, shillings and pence.

3. What is the amount paid for rent and poor-rates on 12 acres, 1 rood, and 32.5 poles, when the rent is £1 per acre and the poor-rates 4s. in the pound on the rent?

4. Define Discount and Present Worth.

A bill of £2241. 158. is due two years hence: find the present worth allowing 5 per cent. compound interest.

What is the discount per cent.?

5. a and b are the lengths of two lines drawn from a certain point and terminated by the concave part of the circumference of a circle; the first is bisected, the second trisected, by the convex part of the circumference, express the relation between a and b.

6. Prove the rule for finding the greatest common measure of two integers. Hence shew that if a is prime to b, and divides bc, it divides c. Hence shew that if N is a whole number, (N) is a whole number or incommensurable.