(d) Wie still, wie dämmerig, wie rein war Alles, was ihn umgab! Wunderbare Ruhe umfing ihn, süsze Mattigkeit beschwichtigte jede stürmische Regung seines Herzens. So oft er das Auge aufschlug, begegneten ihm zärtliche, sorgende Blicke. Selbst wenn der Schmerz sich erneute, genosz er stilles, tröstliches Seelenglück. Auch das fühlte sie und empfand es als einen Lohn sondergleichen.

-EBERS.

PURE MATHEMATICS.-PART I.

The Board of Examiners.

1. If two circles touch one another externally, the straight line which joins their centres shall pass through the point of contact.

If the distance between the centres of two circles is equal to the sum of their radii, then the circles must meet in one point but in no other.

2. Inscribe a regular hexagon in a given circle.

AB, BC, CD, DE, .... are consecutive sides of a regular polygon, and AD, BE intersect in X; shew that AB = BX.

3. If two triangles be equiangular to one another, the sides about the equal angles shall be proportionals, those sides which are opposite to equal angles being homologous.

Ꮓ

If one of the parallel sides of a trapezium is n times the other, shew that the diagonals intersect one another at one of the points in which each diagonal is divided into n + 1 equal parts.

4. If four straight lines be proportional, the rectangle contained by the extremes is equal to the rectangle contained by the means, and conversely.

On a given straight line construct a rectangle equal to a given rectangle.

5. Shew how to solve two simultaneous equations when all the terms which contain the two unknown quantities are of the second degree.

6. State the meanings given to a when mis fractional or negative, and explain why such meanings are given.

7. Define an arithmetical progression, and find the sum of any number of terms of an arithmetical progression.

The sum of three quantities in arithmetical progression is 3a, and the sum of their squares is 3a2+262; find the quantities.

8. State and prove the formula for the number of combinations of n things r at a time.

In how many ways can two elevens be chosen out of 23 players to play a match?

9. Define the secant, cosecant, and cotangent of an angle of any magnitude, and prove that sec (-A) =sec A.

10. Prove that in any triangle

a = b cos C + c cos B,

a(b2 + c2) cos A + b(c2 + a2) cos B + c(a2 + b2) cos C = 3abc.

11. Shew how to solve a triangle, having given two sides and the included angle.

If a = 135, b = 105, C = 60°, find A,

12. Find an expression for the radius of an escribed

circle of a triangle.

Prove that

PURE MATHEMATICS.-PART II.

The Board of Examiners.

1. Find the coordinates of the point which divides in a given ratio the straight line joining two given. points.

Find the ratio in which the line joining the points x1, Y1; X2, y2 is cut by the line

ax + by + c = 0.

2. Find the general polar equation of a circle.

Draw the curves

r = a cos 0, r = b sin 0,

2ra cos 0 + b sin 0.

3. Obtain the equation of the normal at any point of a parabola in the form

Shew that from a given point three normals can be drawn to a parabola.

4. Find the condition that the line

x cos a + y sin a = P

Find the locus of the intersection of perpen

dicular tangents to an ellipse.

5. State and prove the rule for differentiating a".

6. State and prove a formula for the nth differential coefficient of eax cos (bx + c).

Find the nth differential coefficient of

eax cos px cos qx cos rx.

7. Assuming that f(x + h) can be expanded in a series of positive integral powers of h, shew that the expansion must be

h2

.....

f(x + h) = f(x) + hf'(x) + 12ƒ'" (x) + · · · · ·

Expand secx in ascending powers of x as far

as the term in x6.

8. Investigate a rule for finding maxima and minima values of a function of one independent variable. Find the maxima and minima values of

9. State and prove the rule for integration by sub

stitution.

Integrate

10. Shew how to find the partial fractions corresponding to a repeated factor of the first degree in the decomposition of a rational fraction.