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2. Distinguish between the subjective and objective elements 7 in our idea of cause. Explain the various theories to account for the presence of the subjective element, and state whose theories they are. Does Hume's definition, invariable antecedent, fully express our idea of cause ?

3. Fowler, in various places, refers to (a) the process of 8 crystallisation, (b) the discovery of Neptune, (c) the epicyclical theory of the heavens, (d) the derivation of French words from the Latin. What positions are these references intended to illustrate ? Show how they do so.

4. Explain Dr. Whewell's views on the inductive methods, 7 and on the tests of a true hypothesis. State Fowler's criticisms thereon. Are they just ?

5. Explain the relation of induction to deduction. Show how 6 they may be combined in reasoning, and in what relation they stand to the process of verification,

6. Why are Analogy and Simple Enumeration not proper 5 inductive methods ? Under what conditions may those two methods be employed for inductive purposes ?

7. Examine the following opinions with regard (1) to the 6 inductive method employed in forming them, and (2) to their validity, stating the class of fallacies to which the invalid ones belong.

(a.) Men's fortunes in this world are caused by their acts in a previous state of existence.

(b.) Men's condition after death will depend on their conduct during their present life.

(c.) When you speak about an absent person, he is sure to appear on the scene. (d.) Where there is smoke there must be fire.

8. Give a tabular view of the fallacies incident to Observation 5 and to the Inductive Methods.

Moral Science.

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1. What is Moral Philosophy ? What is its relation to Mentat Philosophy? What is the amount of authority due to Sacred Books of Religion with regard to the study of the former ?

2. Distinguish between Will and Desire. Assign to them their respective places as springs of action. Explain the mutual relations subsisting between Will, Intellect, and Sensitivity.

3. Compare Instinct and Habit, with regard to their nature and origin. Is it possible to identify them in any way psychologi. cally and morally ? State briefly the various theories on the subject.

4. Define Conscience, and analyse its functions. Review the various theories that have been held regarding its nature and origin. Is it possible to act wrongly under its guidance ?

5. In what sense is the will free, and in what sense is it not free? What makes us responsible for vice, even though our understandings be depraved ?

6. What is meant by the System of Utility ? Describe it, and review the controversy regarding it.

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7. Define Justice and Veracity. Can it be right, under any circumstances, to violate the laws of either of them ? State your reasons ?

8. Define the place in Moral Philosophy of the qnestions regarding the Existence of God and the Immortality of the Soul. Explain briefly the arguments in favour of both doctrines.

Butler's Analogy. 1. What is the meaning of the title “ Analogy of Religion" ? Explain the nature, degree, and foundation of the kind of evidence that Butler adduces. Show the peculiar utility of his method.

2. What objections have been urged against the doctrines that we are under Divine Government, and that that Government is a moral one? How does Butler answer them?

3. What a priori objection founded on the divine attributes might, in the absence of experience, be urged against the supposition of our being subjected to hazard and danger ? Whence does that speculative difficulty arise ? Show that experience supplies a practical answer.

4. Butler says:

(a.) “ The principle of virtue, improved into habit, of which “improvement we are thus capable, will plainly be, in proportion " to the strength of it, a security against the danger which finite “ creatures are in, from the very nature of propension or particular affection."

(6.) “ It is ridiculous to assert that self-denial is essential to “ virtue and piety ; but it would have been nearer the truth,

though not strictly the truth itself, to have said, that it is “essential to discipline and improvement."

Explain fully these statements, clause by clause.

5. What is the doctrine of Necessity? Butler shows that even admitting that doctrine to be true, the conclusions deduced therefrom by the Atheist and Deist do not follow. Explain this.

6. Hobbes held (a) “that personal gratification was the sole “ end of every act," (6) " that every exercise of passion or faculty was equally authoritative," and (c) “ that man has no moral “ faculties of perception or action.” State Butler's arguments in refutation of these three positions. Where are they set forth?

7. Butler maintains that "there is a natural principle of “ benevolence in man, which is in some degree to society what “self-love is to the individual.” What does he mean by this ? how does he prove it? and what important conclusions does he deduce from it ?

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ALGEBRA AND TRIGONOMETRY

MORNING.

+

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+

Examiner-MR. W. GRIFFITHS, M. A.

N. B.The figures in the margin indicate full marhs. 1. Prove the exponential theorem.

If log (1 + x)(1 + ax) (1 + a-x) (1 + aox) to infinity be 6 expanded in a series of ascending powers of w, shew that the

(-1]n coefficient of an is

n (an — 1): 2. When is an infinite series said to be convergent or diver. 6 gent ?

1 1 1 Shew that

is divergent if p is eqnal to 1P 2P Зр or less than unity, and convergent if p is greater than unity.

Determine whether the series whose general term is (n3 + 1)-1 is convergent and divergent.

3. Prove the law of formation of successive convergents to the value of a continued fraction. P 1 1 1

1
If
9 +

shew that
4 + 12 + 93 + an
P

1
1

1
= a +
P
n-1

991

In - 3+ 4. Find the sum of n terms of the recurring series

6 Uo + Upx + Uz x2 + Uz 23 + whose scale of relation is 1

рх

aad
prove

that

(-q}m (u“, pun mun m

"n - m - 1 — que, m-1). 5. Prove that the arithmetic mean of any number of positive 6 quantities is not less than their geometric mean.

1 If a, b, c be the sides of any triangle and s = (a + b + c),

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-a) + 5 (8 — 6) + ✓ 5 (8—c). 6. Find expressions for the radii of the inscribed and circumscribed circles of a triangle.

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6 ab +

2

+

A dodecagon, six of whose sides are equal to a, and six equal to b, is inscribed in a circle ; prove that the radius of the circle is ✓a? + b2 + 3 ab; and that the area of the dodecagon is 3v3

(a” + 6%). 7. Assuming De Moivre's theorem, deduce the formula

63 05 Sin e = 0

- &c. 13 15

6

e Shew that the limit of

is indefinitely

sin 0 diminished is e.

8. Find exponential values for sin x and cos x.
Given tan ( + 0) cos 2a = tan P, prove that
1

1
tan’a sin 20 + tanta sin 40 + tan a sin 6° + &c.
2

3 9. Sum the series

23 1 + x cos 0 + cos 2 O +

cos 3 0 + &c.

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ANALYTICAL CONIC SECTIONS-AFTERNOON.

Examiner-MR, JOHN ELIOT, M. A.

N. B.-The figures in the margin indicate full marks. 1. Find the length of the perpendicular drawn from the point a B to the straight line Ax + By + 0 = 0.

2. Find the polar equation to the straight line passing through the two points (1, 0) (Tor 0,).

Thence deduce the condition that three points may lie in a straight line, and interpret the result geometrically.

3. Obtain an expression for the area of a triangle in terms of
the co-ordinates of its angular points.
Find the area of the triangle included by the straight lines

x cos ay + y sin ay P. 0.
2 cos ang + y sin ag - P2 0.

x cos az + y sin az — P3 = 0.
4. Explain what is meant by the terms pole and polar.

Find the co-ordinates of the pole of the line Ax + By + C = 0 with respect to the circle (x – a)? + (y B) ? = y.

5. What is meant by the radical axis of two circles ? Prove that the radical axes of three circles intersect in a point.

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Find the co-ordinates of this point for the three circles whose equations are

22 + ya

gua, (x

a)2 + (y B2

(u — a)2 + (y 6) 6. Define a parabola, thence obtain its equation in the simplest form, and trace its figure from the equation.

7. Obtain the equations to the tangent and normal at any point of a parabola in the form

y
= mx + -.

m
Y = mx — -2 am - am3.
Common tangents are drawn to the circle x + y2

to, and the parabola ya = 4 ax. Find their equations.

8. Obtain the polar equation to an ellipse, a focus being the pole and an axis the initial line. Also find the equation to the tangent at any point.

Prove that tangents drawn from an external point subtend equal angles at either focus.

9. What is meant by conjugate diameters of an ellipse ? Find the co-ordinates of the extremities of one conjugate diameter in terms of the co-ordinates of the extremities of the other diameter. Shew that the area of the parallelogram formed by normals at

(a2 — 62 ) the extremities of conjugate diameters of an ellipse is

ab sin3 2 , where p is the eccentric angle of one of the extremities.

10. All chords of a conic section which subtend a right angle at a given point of the curve pass through the same point in the normal at the given point.

11. If the normals at two points of a parabola intersect on the curve shew that the line joining the two points intersects the axis at a fixed point.

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