6. The third law of motion is sometimes enunciated with reference to "action" and "re-action," and sometimes with reference to "momentum"; give both enunciations, and show how they become consistent with each other.

Find the pressure which, acting horizontally for 5 seconds on a weight of 160 pounds, placed on a smooth table, will generate a velocity of 15 feet per second.

7. When a weight (P) lifts another weight (W) by means of a string passing over a fixed pulley, find the accelerating force, and the tension of the string.

If P be 45 pounds and W 36 pounds, what weight suspended from a string at rest will produce the same tension as is produced by P lifting W?

8. How is an impulsive force estimated? Give some account of the action of one smooth elastic ball on another during direct impact. State a general principle from which the motion of two balls after impact may be obtained.

A given smooth elastic ball impinges directly with a given velocity upon another at rest, find the velocities of each ball after impact, and show that the velocity of the centre of gravity of the two balls before and after impact is the same. 9. Explain generally what is meant by centrifugal force. weight at the end of a string be whirled round a fixed point in the string with a given angular velocity, show how to find the tension of the string.

If a

A string 2 feet long can just support five pounds without breaking; if a weight of one pound be attached to the end of the string, which is whirled round its other end uniformly in a horizontal plane, how many revolutions per minute must it make so as just to break the string?

10. What is a simple pendulum? Find the time of the oscillation of a simple pendulum in a small circular arc, and determine approximately the length of the seconds pendulum.

If a seconds pendulum be lengthened one inch, find the number of seconds it will lose in 12 hours.

SET II.

EUCLID.

1. Construct a triangle of which the sides shall be equal to three given straight lines, but any two whatever of these must be greater than the third.

How would the process fail if the last condition were not fulfilled?

2. Parallelograms on the same base and between the same parallels are equal to each other.

Show that if two triangles have two sides of the one equal to two sides of the other each to each, and the sum of the two included angles equal to two right angles, the triangles are equal.

3. In a right-angled triangle the square on the side subtending the right angle is equal to the sum of the squares on the sides containing the right angle.

Show how to construct a straight line, the square on which shall be any given multiple of a given square.

4. If a straight line be bisected and produced to any point the rectangle contained by the whole line thus produced, and the part produced, together with the square on half the line bisected is equal to the square on the straight line which is made up of the half and the part produced.

5. The angle at the centre of a circle is double of the angle at the circumference on the same base that is on the same arc.

6. If two straight lines cut one another within a circle the rectangle contained by the segments of the one shall be equal to the rectangle contained by the segments of the other.

In a given straight line AB, find a point O such that the rectangle contained by the segments AO and OB shall be equal to a given rectangle not greater than the square on half of AB.

7. Describe a circle about a given triangle.

Lines are drawn parallel to one side of a triangle cutting the other sides or the other sides produced. Show that if circles be described about the new triangles so formed, their centres all lie in a straight line, and find the angles which this line makes with the two sides of the original triangle.

8. If the vertical angle of a triangle be bisected by a straight line which also cuts the base, the segments of the base shall have the same ratio which the other sides of the triangle have to one another.

If ABC be the triangle, A the vertical angle, BE and CE the segments of the base, and if circles be described about the triangles ABE, ACE, the diameters of these circles are to each other in the same ratio as the segments of the base.

9. If four straight lines be proportionals, the rectangle contained by the extremes shall be equal to the rectangle contained by the

means.

On a given straight line describe an isosceles triangle equal to a given triangle.

10. Similar triangles are to each other in the duplicate ratio of their homologous sides.

11. Given a point O in the line AB, find two other points CD, such that a line OP given in direction shall bisect the angle CPD, and the segments CO, OD shall bear a given ratio to each other.

ALGEBRA.

1. "The product of two algebraical quantities having like signs is
positive, and having unlike signs negative." State briefly the
steps by which this rule is obtained, and show its arithmetical
correctness when in the product obtained for (a − b) × (c − d),
α = 12, b = 9, c
10, d = 4.

=

If x and y differ in value, explain why (xy)2 = (y — x)2. Show how the rule of signs stated above leads to the introduction of impossible or imaginary quantities in algebra.

2. Multiply-

-

(1 + x + 2x2)2 - (1
(1-x-2x) by (1+x- 2x2) − (1 − x + 2x2)3.
Find the product of

x2 - (a+b)x+ab by x2 + (a - b) x — ab,
and examine what the product becomes if in it either (a) or (b)
be substituted for x.

(x − y) (y − z) + (x − y) (z − x) + (y − z) (z − x).

x (z − x) + y (x − y) + z (y − z)

4. If (m) be a whole number, prove that (x + y) is divisible by

1

my, when (m) is odd.

Write down the three last terms of the quotient of (x + y)

divided by (2+1+y+1), and examine how many terms the
quotient will contain.

5. Find the fraction in its lowest terms which is the square root of

(1.) (x — 4) (x — 8) (x − 10) = (x − 6) (x — 7) (x — 9).
(2.) 5x® 14x = 3.

(3.) 2cx2

abx+2abd-4cdx.

(4.) (x + y)2 - 2o 65)

=

x2 - (y + z) = 13.

7. Form a quadratic equation whose roots are 561.

8. A bill of £63. 58. was paid in sovereigns and half crowns, and the number of coins used in the payment was 100; how many sovereigns were paid, and how many half crowns?

9. A cask (A) is filled with 50 gallons of water and a cask (B) with 40 gallons of brandy; (C) gallons are drawn from each cask, mixed, and replaced. The same operation is repeated. Find (C) when there are 81 gallons of brandy in (A) after the second replacement.

10. Find the sum of (n) terms of the series (a − b) + (a 3b) + (a- 5b) &c., without assuming a formula for the sum of an arithmetic series, and apply the result to find the sum of 10 terms of 76+70 +64 + &c.

11. The sum of the two first terms of a geometric series is 12 and of the three first terms is 39, find the series, and determine if the condition is satisfied by more than one series. Find the sum of the infinite series

1+r+(1+b) r2 + (1 + b + b2) gr3 + &c.,

(r) and (b) being proper fractions.

12. Show how to transfer a whole number from one scale of notation to another, and prove that if (r) be the radix of the scale, the number itself will be divisible by (r+1) if the difference between the sum of the digits in the odd places and the sum of the digits in the even places be divisible by (r + 1).

13. Find the number of different permutations of (n) letters taken altogether when one letter occurs (p) times and another (q) times in each permutation.

If the n letters contain only a and b, and n be even, show that

the number of permutations will be greatest when p = q =

n

2

14. Assuming the form of the binomial theorem, find the greatest term of (1x)" when (n) is a positive integer and x is also positive. Apply the result to the determination of the greatest term of

(1+5), and express the value of the greatest term.

Show by actual multiplication that the whole coefficient of x in the product of the expansions of (1+x)" and (1 − x") is equal to the coefficient of x in the expansion of (1- x2)".