If a parabola touch two given straight lines and the chord of contact passes through a fixed point, the focus will be on a fixed circle.

3. Find the condition that the line

la + mß + ny = 0

may touch the conic

√πa + √μß + √vy = 0.

Two triangles are inscribed in a conic; shew that their six sides touch another conic.

4. Find the shortest distance between two straight lines whose equations are given in the symmetrical form.

Find the shortest distances between the diagonal of a rectangular parallelopiped and the edges which it does not meet.

5. If a', b', c' be the lengths of three conjugate semidiameters of the ellipsoid

prove that

a'2 + b22 + c22 = a2 + b2 + c2

b'2 c'2 sin2a + c'2 a'2 sin2 ß + a'2 b′2 sin2 γ
= b2 c2 + c2 a2 + a2 b2

where a ẞy are the angles which a', b', c' make
with one another.

and that the planes containing two of three equal conjugate diameters touch the cone

6. Find the circular sections of an ellipsoid.

If the two circular sections which pass through a point on an ellipsoid have the same radii, prove that the point must lie on one of the principal planes through the mean axis.

7. Shew how to find the asymptotes of a curve referred to polar coordinates.

Find the asymptotes of the curves

r cos n o = a, r cos n o = a cos m 0.

8. Find the equation to that curve of the family
(r) A cos 0 + B sin 0

which touches the curve r = f (0) at the point
where a, A and B being parameters and
a given function.

Hence find the polar equations to the tangent and the circle of curvature of the curve r = f(0) at the point where 0 = a, and deduce the ordinary expressions for the radius of curvature in polar coordinates.

9. Investigate a formula for the volume of any solid referred to rectangular axes.

Find the volume enclosed by the surfaces x2 + y2 = 2cz, x2 + y2 = 2ax, ≈ = 0, illustrating by figures the process of summation.

10. Shew how to differentiate

If F(x) be an algebraical polynomial of less than n dimensions, prove that

PHYSICAL GEOLOGY AND MINERALOGY. SECOND PAPER.

The Board of Examiners.

1. Describe fully the various experiments by which the mean density of the earth has been ascertained, and discuss the bearing of this ascertained density upon the question of the structure and composition of the interior.

2. Give Mallet's definition of an earthquake, and exexplain the method of determining

(a) The point on the earth's surface vertically above the focus of an earthquake.

(b) The depth of the focus.

3. Explain clearly the difference between the "strike" and the "outcrop" of stratified rocks, mentioning the varying circumstances where they coincide, and where there is the greatest divergence between them.

4. Explain fully what is meant by False bedding, Thinning-out, Unconformity, and Overlap respectively, giving the exact geological inference you would draw from each.

5. What do you understand by Zeolites? What is their general chemical composition? Where are they found? And whence are they derived? Give examples.

6. Explain clearly the difference between Isomorphism and Heteromorphism with its two varieties Dimorphism and Trimorphism. Illustrate your answer by examples.

7. Define the terms "Holohedral" and "Hemihedral" as used in Crystallography, with illustrations from the Monometric System, and state what you understand by "Twin" Crystals, and "Macles" "Hemitropes," and how they are usually recognised?

or

DEDUCTIVE LOGIC.

The Board of Examiners.

PAPER No. 2.

1. Explain the following statements:-(a) Formal logic is the science of concepts of the second intention applied to first. (b) It is not an organon of science. (c) It is a canonic of thought and science.

2. What logical distinction may be drawn between Perception or Intuition and Thought? Illustrate.

3. Mention general characteristics of the Concept or Notion. Consider any alleged cases in which the concept entirely represents the individual.

4. Show the connexion of the laws of Identity, Contradiction, and Excluded Middle. Discuss the question whether there is required, for purposes of Formal Logic, the addition of any other law equally fundamental.

5. Consider whether immediate inference by way of Conversion can be justified by reference to logical principles.

6. Mention rules in accordance with which relatively simple propositions may be drawn from (a) complex universal propositions; (b) complex particular propositions.