(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 E.caminers.

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.

N

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 un-
known quantities are of the second degree.
Solve the equations

x2 + y2 = a2 + b2, xy = ab.

6. State the meanings given to all when mn is

fractional or negative, and explain why such
meanings are given.
Multiply

a3 + b3 + o brot - chat – atbt
by

at + b + ch.

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 + 25; 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.
cosec (- A) = cosec A.
Find the general solution of the equation

tan20

=

tan’a.

10. Prove that in any triangle

a = b cos C + c cos B,
a(b2 + c) cos A + b(c2 + a) cos B

+ c(a? + 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, having given

= .3010300 L tan 13° 12' = 9.3348711, log 3 = .4771213 L tan 12° 13' 9.3354823. 12. Find an expression for the radius of an escribed

circle of a triangle.
Prove that
1

1 1

+ ni

log 2 =

+

r

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, Yı; X2, Yz 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,
2r = a cos 0 + b sin 0.

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

y = mx 2am amu. 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 may touch the ellipse

5. State and prove the rule for differentiating x”. Differentiate

a? )

x2)* + x

n

6. State and prove a formula for the nih differential

coefficient of eax cos (bx + c).
Find the nth differential coefficient of

eau cos pu 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) + x

a

2.4"(a) +

Expand sec x in ascending powers of x as far as the term in 26.

8. Investigate a rule for finding maxima and minima

values of a function of one independent variable.
Find the maxima and minima values of

25
(a – x)4 (b − x)4"

+

9. State and prove the rule for integration by sub

stitution.
Integrate

1
waz wa,

ea + e

10. Shew how to find the partial fractions correspond

ing to a repeated factor of the first degree in the decomposition of a rational fraction.