2. How far does the history of Philosophy support the theory that the correspondence, in the relative order of thought and things, between the Universe and Man, is 'strictly macrocosm and microcosm' ?

SECTION B.

3. Trace the influence of Neo-Platonism and of Saint Augustine through the history of Medieval Philosophy.

4. (a) What is really at issue in the controversy as to the relations between intellect and will in the hierarchy of the Faculties?

(b) Give a history of this controversy, noticing the leading disputants, and indicating how far their attitude towards this question is in harmony with their characteristic theories.

SECTION C.

5. Discuss historically

'Kant followed the example of a long and illustrious line of predecessors, when he entered upon the realm of philosophy from the side of mathematical science.'

6. Explain, and examine briefly :- The law of rational progress in knowledge of the dialectical movement of consciousness, or, in one word, of experience, is not simple movement in a straight line, but movement by negation and absorption of the premisses.'

REV. PROF. DARLINGTON: PROF. MAGENNIS; Rev. Dr. MAGILL; PROF. PARK; REV. PROF. WOODburn.

THESIS.

Methods of Metaphysical Investigation.

III.-MATHEMATICAL AND EXPERIMENTAL

SCIENCES.

MATHEMATICAL SCIENCE.

FIRST PAPER.

THEORY OF PROBABILITY.

PROFESSOR CROFTON.

[Full credit will be given for answering ONE-HALF of this Paper.]

1. In two tosses of a coin, the chances are even that the two results shall be alike, or shall be different.

If the coin be slightly bent or injured, show that it is more probable that the two results will be alike.

2. If n persons stand in a row, and two are chosen by lot, show (by help of the Calculus of finite differences, or otherwise) that the probability that not more than intervene between them in the row, is

persons

3. The facility of success on a single trial as to a certain event is p. Assuming it proved, that in a large number (N) of trials, the probability that the number of successes shall lie between the limits pN±r is nearly

show that by increasing the number of trials, we can make it a certainty that the proportion of successes shall differ from p by an infinitesimal.

4. A bag contains a great number of black and white balls. If 2n balls are drawn, of which n are found to be black and n white, prove that if two more drawings are made, the chance that they will give a black and a white ball is

5. A line a is divided into n segments by n - 1 points Show that the mean value of the

taken in it at random.

Show also that the mean value of the square of the least segment is

6. A semi-circle is described on the line AB. If a point X is chosen at random on the circumference, and a point Y at random in the area, find the chance that Y shall fall on the triangle AXB.

If AY, BY are produced to meet the circle in P, Q, find the mean value of the arc PQ.

...

7. If a large number r of observations a12 . a, are made of the same magnitude, assuming that the law of error of the observations is of the form

8. An observation is affected by two independent sources of error, the equations of frequency for which are

Show that the equation of frequency for the composite error is

9. Several measurements are made of the same magnitude: show that the method of least squares gives the arithmetical mean as the best value to take.

An unknown magnitude (X) depends on another (x) by the relation X = f(x): X is measured and found to be A; x is also measured and found to be a show that the best

supposing the two measurements of equal weight.

10. Show that, according to D. Bernouilli's hypothesis, the advantage or moral value of a sum a to a person whose fortune is a, is

If he is to gain a sum a if an event happens whose probability = p, and a sum ẞ if another happens whose probability =q, the two events being mutually exclusive, what is his moral expectation? If Xis the sum of money which, to him, is the equivalent of this expectation, prove that

==

If both events may happen, e.g. the safe arrivals of two ships, prove that

11. Two points X, Y are taken at random in a triangle ABC. Prove that the mean value of the triangle AXY 2 ABC.

=

XY produced divides ABC into two segments: show that the mean value of that segment which contains the vertex A =

ABC.

12. If M is the mean area of the triangle formed by taking three points anywhere within a sphere, and M. the mean area when one of the three points is taken on the surface, prove that M & M。.

13. A point is taken at random in each of two plane areas, Show that the mean square of the distance between the points is

M (D2) = a2 + k2 + k2,

where a is the distance of the two centres of gravity, and k, k' the radii of gyration of each area round its centre of gravity.

14. Two lengths a, b are taken anywhere on a line 7. Prove that the chance that they have no common part is

What is the chance that the shorter length a shall fall within the longer one b?

SECOND PAPER.

CALCULUS OF FINITE DIFFERENCES.

PROFESSOR CROFTON.

[Full credit will be given for answering ONE-HALF
of this Paper.]

1. Show that a" expanded in a series of factorials is

2. Prove that "x"= (x + n) ▲"xTM−1 + NA”-1μ3-1