7. Find the area of the triangle formed by the three points (6, 7), (8, 9), (4, 5), and put down the equations of the lines drawn from each angle perpendicular to the opposite side. Verify that these lines meet in a point.

8. Find the equation to a circle which cuts the axis of y at right angles, and passes through the two points (3, 4), (-5, 6).

determine points on a hyperbola. Prove that the chord joining the two points t = p, t = q is

10. Find the locus of the middle points of chords of a parabola which pass through a fixed point.

SECOND PAPER.

PROFESSOR MCWEENEY.

GEOMETRY, Differential anD INTEGRAL CALCULUS.

1. Draw a straight line through a given point so that the portion of it intercepted by two given straight lines may be bisected at the point.

Show that this line makes, with the two given lines, a triangle of smaller area than that made by any other straight line drawn through the point and terminated on opposite sides of the point by the given straight lines.

2. Show that the rectangle under the perpendiculars on the diagonals of a cyclic quadrilateral let fall from any point on the circumscribing circle is equal to the rectangle under the perpendiculars from the same point on a pair of opposite sides.

3. Show how to describe a sphere passing through a given circle and having its centre in a given plane. Express the radius of this sphere in terms of the radius of the circle, the angle between the plane of the circle and the given plane, and the perpendicular distance of the centre of the circle from the given plane.

4. Find from first principles the differential coefficient of cot z.

Find the limiting value towards which y

=

(1 − x) tan

πX

2

tends as x approaches unity. Prove that for all values of x between 975 and 1.025, the value of y differs from the limiting value by less than 0005.

5. Find a maximum or minimum value for sin ̄1x+2/1 − x2, distinguishing which it is.

6. Find the equation of the tangent and normal to the curve x2 + y2 53 at the point (1, 2).

7. Find the radius of curvature of the curve x'y = a3 + 2x3 at the point where the tangent is parallel to the axis of x. 8. Find the following integrals :

10. Find the length of the arc of the curve y3 = x3 from the origin to the point (1, 1). Find also the volume of the solid formed by rotating this are round the axis of x.

MATHEMATICAL PHYSICS.

SECTION A.

PROFESSOR CONWAY.

1. Prove the theorem known as the parallelogram of displacements. Deduce similar theorems for velocities and accelerations.

2. A body rests on a horizontal plane whose coefficient of friction is μ

to move it.

find the least force which will be sufficient

3. A triangular lamina has one-ninth of its area cut off by means of a section parallel to one side: find the centre of mass of the remainder.

4. A body is projected at an angle to the horizon: find the least velocity in order to reach a certain range on a horizontal plane.

5. A body of mass m grammes is moving with a velocity v centimetres per second at any instant under the influence of a force which is working at the rate of H ergs per second find its acceleration.

SECTION B.

PROFESSOR MORTON.

6. An equilateral triangular plate is immersed in a liquid, with one corner in the surface and one side vertical: find the depth of the centre of pressure.

7. Establish the condition for the stable equilibrium of a floating body.

8. Show that when an image is formed by direct reflexion at a spherical surface, the positions of object and image divide the radius of the surface externally and internally in the same ratio.

9. A convex lens, of focal length f, is used as a magnifying glass, and the image is formed at distance D from the eye find the apparent magnification.

10. Describe the construction and use of a transit instrument.

PRACTICAL CHEMISTRY.

[Special stress will be laid upon the written record of your work, and your attention is directed to the following points:

(a) Give a concise account of all the steps of the processes you employ, and of all the tests you use in searching for the different substances.

(b) If you find a metal capable of forming two series of compounds, ascertain, if possible, to which of these series the metal present in the substance you are examining belongs.

(c) In testing a solid, dry way tests, in addition to wet way tests, must be employed.

(d) In testing a solution, dry way tests should be employed in all cases where it is advisable to do so-in addition to liquid tests.

(e) Use confirmatory tests where it is possible to do so.

(f) At the end of your paper, give a statement of the constituents found in each solid or solution given you for examination.]

1. Detect acid and base or metal and non-metal in the solid marked 1.

2. Detect acid and base or metal and non-metal in the solid marked 2.

3. Identify the metal composing the powder marked 8.

FIRST PAPER.

PRACTICAL ENGINEERING.

SECTION A.

PROFESSOR FITZGERALD.

[Copies of Bidder's Table of Earthwork and Logarithmic Tables will be supplied with this Paper.]

1. Put down as in a level-book the following staff readings taken with a level, reduce them to a datum 27.41 below the third reading, and prove the result. Changes of position of instrument are indicated by §:—

6-75; 6.20; 11-32 § 3-82; 12.64 § 2-15; 5.40;
5.10; 2.78 § 12.46; 10.22.

2. Plot the above assumed taken at stations one chain apart, on a scale of ten feet to the inch vertical, and two chains to an inch horizontal.

3. Describe with the manner of using them, two forms of levelling instruments other than the telescope with bubble tube.

4. The depths in a cutting at distances from its beginning of 0, 8, 11, 18 chains are 0, 22, 18, 0 feet respectively: find the volume of earthwork by Bidder's table, arranging your work neatly in columns so as to be easily checked.

5. A pipe 16 inches outside diameter is to be laid at the bottom of a trench excavated with vertical sides, to a depth of 3 feet, and 2 feet wide. The trench is to be refilled after the pipe is laid, and the waste earth removed. If the stuff to be put back is thrown out on the left side of the trench, and that to be carted away on the right, what depth will the excavation be when the men begin to throw out earth on the right-hand side?

SECTION B.

PROFESSOR Townsend.

6. Make a figured sketch of a framed floor, showing the joists, binders, and girders: if secret nailing be adopted, sketch the joint of the flooring boards.

7. Make a figured sketch of the cross-section of the platform of a railway station, showing height of platform over rail, the distance of the edge of platform from a vertical line at nearest point of rail, and the minimum distance of any fixture on the platform from its edge.

8. The radius of a railway curve is 5 furlongs and 3 chains; the angle between the extreme tangents is 128° 26'. Calculate the tangent and distance of middle point of the curve from the intersection of the tangents.

9. Sketch a figured cross-section of a street tramway, with detail enlargement of rail.

10. Sketch the mode of fastening the double-headed rail, and also the flange rail to the sleepers, and state the relative advantages of these two methods of fastening.