5. Write a phrase in four-pulse measure, a phrase in three-pulse measure in secondary form, and a phrase in six-pulse measure, with one or two silent pulses.

6. Write the scale of G, or Sol, minor, descending and ascending.

6. Write the upper part of the scale in the minor mode, ascending or descending, in all the forms with which you are acquainted.

ARITHMETIC.

Males.

Three hours allowed for this paper.

Candidates are not permitted to answer more than one question in each Section.

The solution must be given at such length as to be intelligible. to the Examiner, otherwise the answer will be considered of no value.

SECTION I. Arrange the digits 58967537, so that the number formed may be divisable by 19.

How many times can 692 be taken from 4676536 ?

A man buys 300 quarters of corn for 1,000 guineas: he sells one hundred at £3 10s. per quarter; another hundred at 10s. per bushel; and the remainder at 1s. 6d. per gallon. Find the total gain.

(These form one question.)

SECTION II.-1. If 50 gallons of water weigh 525 lb., and a cubic inch of water weighs 7 oz., find the number of gallons in a tank containing 1,840 cubic feet.

2. If the ratio of the diameter of a circle to the circumference, find the length of the earth's diameter, the length of a degree of latitude being taken as uniformly equal to 69 miles.

3. If the total annual yield of all the gold mines of the world is 4,788,000 oz., find the number of gold coins that could be coined from 5 of the yield, so that 260 such coins can be made from 51 lb. of gold.

SECTION III.-1. A certain fraction exceeds 7 by 5, another is less than 5 by 3 find the product of the sum and difference of the two fractions.

2. If the franc in silver coinage is equivalent to 9 d., and in depreciated paper money to 8 d., find the loss incurred by paying bills to the amount of £100 in the former currency instead of in the latter, reckoning 25} francs to the pound.

SECTION IV.-1. Reduce 5 of a guinea + 32% of 138. 4d.-41 of a crown to the decimal of of '06 of £1 78. 6d.

2. Arrange in order of magnitude the fractions '06, 06, 069, 069, 069, and express their average decimally. SECTION V.-Write out clearly and concisely the rules

for

(a) Converting circulating decimals into vulgar fractions.

(b) For the extraction of the square root of a perfect square consisting of four digits.

(c) For the multiplication of numbers mentally by 25 and 625.

(These form one question.)

SECTION VI.-Find (by Practice) the value of 504 articles at £5 13s. 4d. each, and of 17 acres 1 rood 25 poles at £250 10s. per acre.

(These form one question.)

SECTION VII.-1. An engine can pump out 1,600 gallons per hour; after working for 19 hours it has pumped out 36 of the contents of a reservoir: in what time will it pump out the remainder with the help of another engine, whose power is 33 of its own? and how many gallons will each have pumped out?

2. If 27 men and 60 boys earn £134 in 17 days, how much will 18 men and 24 boys earn in 23 days, a boy's earnings being of a man's ?

SECTION VIII.-1. Find the edge and longest diameter of a cubical tank, which contains 134,217,728 cubic inches.

2. Find the cost of desk accommodation for a school of 168 children at 3s. 6d. per linear foot, each child requiring 21.75 inches; find also the average space in

square feet per child, if the room contain 8 equal groups of desks, three deep, cach group occupying 152.25 square feet, with 8 gangways occupying 288 square feet, and a space of 10 feet is left clear in front of the desks.

SECTION IX.-1. Find the Compound Interest on £3,600 for 2 years at 5 per cent. per annum, the interest being payable half-yearly.

2. Part of a sum of £3,000 is invested to produce 3 per cent. per annum, and the remainder 4 per cent.: if the interest on the whole for a year amounts to £122 98. 10 d., find the sum invested at each rate.

SECTION X.-1. Divide the sum of £2,529 12s. between A, B, and C, so that A's share may be 3 times B's, and A's and B's together 5 times C's.

2. In a school which has on its books 250 children uniformly, 18 per cent. are absent during the former half of the year, 123 per cent. during the latter half; in the former there are 3 weeks of holidays, in the latter 5: reckoning the full weekly atter lances of a child at 10, find the average attendance for the year, and the average number of attendances made by each child, reckoning 52 weeks to the year.

EUCLID, ALGEBRA, AND MENSURATION. Males.

Three hours allowed for this paper.

Candidates in Scotland may answer two questions out of Section IV. if they omit Section IX. With this exception Candates are not permitted to answer more than one question in each section. (Marks are given for portions of questions.)

EUCLID.

N.B. Capital letters, not numbers, must be used in the diagrams.

The only signs allowed are + and The square on AB may be written " sq. on AB," and the rectangle contained by AB and CD, "rect. AB. CD." Other abbreviations (if employed) must not be ambiguous.

SECTION I.-Define a plane superficies, an isosceles triangle, and an oblong.

Explain the term Axiom; write out the eleventh axiom in which proposition is this axiom first required ?

Give examples of converse propositions from the first book of Euclid.

(These form one question.)

SECTION II.-1. If the equal sides of an isosceles triangle be produced, the angles made by these lines produced with the base will be equal.

Show briefly that this property may be proved by a method similar to that employed in the 4th proposition

of the first book.

2. The greater angle of every triangle is subtended by the greater side, or has the greater side opposite to it.

Write out the converse of this proposition, and show that one of the two propositions is proved directly and the other indirectly.

3. If two triangles have two angles of the one equal to two angles of the other, each to each, and one side opposite to one of the equal angles of the one equal to the side opposite to the corresponding equal angle of the other, the third angle of the one shall be equal to the third angle of the other.

Show that this property would follow as a direct corollary from the 32nd proposition of the first book.

SECTION III.-1. Straight lines which are parallel to the same straight line, are parallel to one another.

If the straight line, to which two others are parallel, lie between them, show that this property follows from Euclid's definition of parallel straight lines.

2. Triangles upon equal bases and between the same parallels, are equal.

Given the middle points of the sides of a triangle, construct the triangle.

3. If the square described on one of the sides of a triangle be equal to the squares described on the other two sides of it, the angle contained by those two sides is a right angle.

Write out the corresponding proposition for obtuseangled triangles.

SECTION IV.-1. If a straight line be divided into two equal parts and also into two unequal parts, the rectangle contained by the unequal parts, together with the

square on the line between the points of section, is equal to the square on half the line.

At what point must a given line be divided, so that the rectangle contained by the two parts shall be the greatest possible?

2. To draw a straight line which shall touch a given circle, from a given point without it.

Two of the common tangents to two equal circles which do not cut or touch one another intersect in a point equi-distant from their centres.

3. The angle in the segment of a circle less than a semicircle is greater than a right angle.

Show that only four equal triangles can be described on the same base having equal vertical angles.

ALGEBRA.

SECTION V.-Simplify 7a-4b-{5a-3 [b-2 (a−b)]} Resolve into factors 256b1-81a1, 4a3-9a2b-16ab2+ 3663.

If x2-ax- is divisible by x-2, what is the value of a?

(These form one question.)

SECTION VI.-1. If x2+ax+b, and x2+cx+d have a b-d common measure of the form x+e, show that e=

a-c

2. Show that a quadratic equation cannot have more than two roots.

Form the equation whose roots are 2 and -3. 3. If a+b+c=0, show that

a+b+c=2 (ab+be+ca)2.

SECTION VII.-Solve the equations :

1. 20+3 x-3 3x+7 5x+6

8

9

=

7

11.

2. (x+2) (x-2) (x+8)=x (x-3) (x+16).

SECTION VIII.-1. A square field contains 1 acre 2 roods 27 poles 234 square yards; find the length and