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7. If z vary as y when x is constant, and z vary as x when У is constant, prove that z varies as xy when x, y both vary.

The volume of a cylinder varies directly as the square of its radius when its height is constant, and directly as its height when its radius is constant. If the volume of a cylinder whose radius is and height h be increased by the same amount by increasing either its radius or its height by x, find the value of x.

8. Define a harmonical progression, and insert a given number of harmonic means between two given quantities.

If a2, b2, c2 be in arithmetical progression, prove that b + c, c + a, a + b are in harmonical progression.

9. Find the number of permutations of n things r

at a time.

There are n different books and p copies of each; find the number of ways in which a selection can be made from them.

10. Prove the binomial theorem for a positive integral exponent.

If a, be the coefficient of x" in the expansion of (1x)", prove that

12a1 + 22a2 + 32α3 +
=2"-2n (n + 1).

+ n2an

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(ii) (ya) (za) bc, (z — b) (x — b) = ca

=

(x-c) (y-c) ab.

GEOMETRY AND TRIGONOMETRY.

The Board of Examiners.

In the first six questions the symbol - must not be used; and the only abbreviation admitted for "the square described on the straight line AB" is "sq. on AB," and for "the rectangle contained by the straight lines AB, CD" is "rect. AB, CD."

1. Enunciate the cases of equivalent triangles (triangles equal in all respects) considered in the first book of Euclid. What other case is indirectly considered in the third book? Is it completely treated? Give fully the reasons for your

answer.

2. Prove that the perpendiculars from the vertices of a triangle to the opposite sides pass through a point, and that the circle which passes through

the feet of these perpendiculars also bisects the sides of the triangle.

3. The sum of the squares on the diagonals of a quadrilateral is equal to twice the sum of the squares on the straight lines joining the middle points of its opposite sides.

4. If two circles touch each other, the chords of the two arcs intercepted between two straight lines drawn through the point of contact are parallel.

5. To find a fourth proportional to three given straight lines. Shew how to cut off from a given triangle another similar to it and having a given ratio to it.

6. In equal circles angles at the centres or at the circumferences have the same ratio as the arcs on which they stand.

If arcs of different circles have a common chord, straight lines drawn from one of its extremities will cut the arcs proportionally.

7. Define the unit of circular measurement of angles, and find a relation between the members expressing a given angle in terms of this unit and of a degree. Find the circular measure of an angle of a regular quindecagon.

8. Define the tangent of an angle.

If tan (0 + 4) =tan (0-4) prove from definition that must be a multiple of a right angle.

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9. Prove that cos (0 − ) = cos e cos + sin 0 sin p. If cos (A C) cos B cos (A B+ C) prove that tan A, tan B, tan Care in Harmonical Progression.

10. Define a common logarithm.

Adapt to logarithmic computation the equation a sin x + b cos x = c.

11. Two objects, A and B, were observed from a ship to have at the same instant the bearing N. 15° E. The ship then sailed N.W. for five miles, when it was found that A bore E. and B bore N.E. Find the distance between A and B.

12. If R and rare radii of the circumscribed and the inscribed circles of a triangle whose sides are

a, b, c, prove that

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The Board of Examiners.

1. In the following passages parse fully each word which is printed in italics :

66 And now I

pray you, sir,

For still 'tis beating in my mind, your reason

For raising this sea storm."

"In every cabin

I flamed amazement.”

"I will pay thy graces

Home, both in word and deed."

2. Make a full analysis of each of the following pas

sages:

(a) There was to have been a combat with cold steel, had it not struck Adrastus that his chiefs had better reserve their fury for the enemy than expend it on each other.

(b) On what foundation stands the warrior's pride, How just his hopes, let Swedish Charles decide. (c) You said it was me you cared for, not her.

3. Write a short note on the philology of each of the

following words :

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4. Write out the twelve lines from the "Progress of Poesy" commencing "Man's feeble race."

5. (a) Who was Witherington, and what was his fate?

(b) Explain

"Time has but half succeeded in his theft, Thyself removed, thy power to soothe me left."

(c). Explain

"Where wasted nations raise a single name, And mortgaged states their grandsires' wreaths regret,

From age to age in everlasting debt."

(d) Explain

"The sad historian of the pensive plain."

(e) What, according to Addison, are the qualifications required in the action of an epic poem?

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