8. Find the sum of n terms of the following series ;

9. Given two sides of a triangle and the angle opposite one of them : show how to solve the triangle, and point out when the case is ambiguous.

If a 5, b = 7, and A= sint, is there ambiguity?

10. Given log10 71968 = 4.8571394; diff for 1 = 60, find the value of 8 0719686 to seven places of decimals.

11. If the three sides of a triangle are x2 x + 1, 2x + 1, and x2 1, show that the greatest angle is 120°.

12. A right cone is cut by a plane which meets the cone on both sides of the vertex; show that the section is a hyperbola. Under what condition is it possible to cut an equilateral hyperbola from a given cone?

13. In an ellipse prove that the line drawn through the focus at right angles to the focal distance intersects the tangent and the directrix in the same point.

No. 2.

1. If the square described upon one of the sides of a triangle be equal to the squares described upon the other two sides of it; the angle contained by these two sides is a right angle.

2. Upon a given straight line describe a segment of a circle, which shall contain an angle equal to a given rectilineal angle.

3. Similar triangles are to one another in the duplicate ratio of their homologous sides.

4. Investigate a rule for finding the greatest common measure of two algebraical expressions.

5. Reduce the following expressions to their simplest forms:

7. Write down the first 5 and the 7th term of the binomial expansion

8. If the sum of the digits of any number be divisible by r

number itself is divisible by r

system of notation.

1, r being the radix of the

Extract the square root of 25400544 in the senary scale.

9. Define a logarithm, and having given log 23010300, log 3 =

4771213, find log 36, log 75, and log

0012.

10. Express cos A in terms of sin 2 A when A is between 90° and 135°; and explain fully why when cos A is expressed in terms of sin 2 A the general expression ought to include four values.

11. Explain the meaning and use of subsidiary angles. In a triangle if 2/ab C

tan =

a- b

sin then will c= (a — b) sec p.

2

12. Find the polar equation to a circle. Show that the polar equation to the chord of a circle that subtends an angle 2 8 at the centre is p = a cosẞ sec

- α.

13. The section of a cone made by a plane which cuts all the generating lines of the cone on the same side of the vertex is an ellipse. Give a geometrical construction for the position of the foci. 14. If two equal tangents be drawn to a parabola, the alternate segments of them made by a third tangent will be equal.

B.-MIXED MATHEMATICS.

1. Assuming the truth of the parallelogram of forces as to direction, prove its truth as to magnitude.

2. Given the sum of two forces and their resultant, and also the angle which one of them makes with the resultant. Determine the forces, and the angle at which they act.

3. State the result of any experiments made with reference to friction.

A body weighing 12,000 tons, placed on a plane whose inclination is 1 in 12, and acted on by two chains, (each capable of sustaining a strain of 200 tons,) in the direction of the plane, is just on the point of moving when the chains break. Find the coefficient of friction between the body and the plane.

4. An area is cut off from one angle of a triangle equal to half the area of the triangle by a line parallel to the base. Find the centre of gravity of the remainder.

5. Enunciate the First and Second laws of motion, and mention any experiments which seem to suggest their truth. How is their truth finally established?

6. A body moving uniformly in a straight line is suddenly acted on by a constant force always acting in a given direction. Determine the subsequent motion.

7. A body of given elasticity is projected vertically upwards with a given velocity, and strikes against a horizontal plane. Determine the velocity with which it reaches the ground.

8. Find the line of quickest descent from the focus of a parabola, whose axis is vertical and vertex upwards, to the curve.

9. Define "specific gravity," and show that when a solid is immersed in a fluid the weight lost is to the whole weight of the body as the specific gravity of the fluid is to that of the solid.

10. Explain the principle of the hydraulic press, and find the mechanical power in a machine of given dimensions.

11. A particle moves in a circle under the action of a central force resident in an external point. Find the law of force. Is the force attractive or repulsive?

12. A particle describes a parabola under the influence of a force always parallel to the axis. Determine the law of force, and the velocity at any point.

13. When rays diverging from a point are incident on a plane mirror find the focus of reflection.

Within what space must the eye be situated to see a given point by reflection at the mirror.

14. Describe Galileo's telescope. Find its magnifying power and field of view.

15. Explain by a figure how it is that the length of the day varies, and that the sun rises at different points of the horizon.

16. Explain the nature of refraction, and show how it affects the moon's apparent horizontal and vertical diameters.

17. Explain the effects of the sun and moon upon the tides. Can you mention any local tidal peculiarities?

18. Define "sidereal," "solar," and "mean solar" time.

Explain clearly how the longitude is ascertained by means of chronometers.

PHYSICAL SCIENCE,

Set to Candidates for the situation of Inspector of Schools (Ireland). 3 hours allowed.

1. Define statical force, accelerating force, moving force, velocity, momentum, the resultant of two or more forces, centrifugal force, and the moment of a force.

A force that can statically support 50 lbs. acts uniformly for one minute on a body the weight of which is 200 lbs. : find the velocity and momentum acquired by the body.

2. Find the relation of the power to the weight in that system of pulleys where each pulley hangs by a separate string, (1) neglecting the weight of the moveable pulleys, (2) supposing them all of the same weight x.

3. State the laws of friction, and explain the use of friction wheels. When a body can just rest on an inclined plane, find an expression for the coefficient of friction.

4. How far must a body fall by the action of gravity, from rest, to acquire a velocity of 128-8 feet per second? How far must it fall, supposing it to have an initial downward velocity of 32-2 feet per second, in order to acquire the velocity of 160 feet per second?

5. What different kinds of toothed wheels are used? Explain and illustrate them by figures.

6. Define friction wheels. A piece of metal whose weight in water is 12 ounces is attached to a piece of wood which weighs 16 ounces in vacuo, and the weight of the two in water is 8 ounces: find the specific gravity of the wood.

7. Describe Bramah's press, showing the principle on which it is constructed.

8. Describe any one of the reflecting telescopes, and draw a figure showing the course of a pencil of rays by which an object is seen through it.

9. Explain the general principles on which photography depends. 10. Explain what is meant by polarization of light, and describe the polariscope.

11. Mention the uses and give a general explanation of the principles on which the following depend :-The aneroid, the camera obscura, the eccentric, the governor.

PRACTICAL GEOMETRY, BUILDERS' WORK, &c.

Set to Candidates for the situation of Clerk of the Works in the Engineering Branch of the War Department.

GEOMETRY.

No. 1.

Time allowed, 3 hours.

1. Find the area of a square whose side is 35.25 chains.

2. What is the number of square yards of painting in a rhomboid whose length is 37 feet and height 5 feet 3 inches?

3. How many acres are there in the triangle whose sides are 2569, 4900, and 5025 links ?

4. What is the area of a ring the diameters of whose bounding circles are 10 and 20?

5. What is the content of a spherical segment 2 feet in height cut from a sphere of 8 feet in diameter ?

6. Describe a hexagon upon the line A B.

7. The lines AB and CD will meet if produced in an inaccessible point. Through the point E draw a line that would if produced meet

the prolongations of AB and CD in one and the same point, and explain the method by which it is done.

8. Draw a circular segment-arch of 15 feet span and 4 feet 6 inches rise, and state how it is done.

9. What is the perpendicular height of a hill, if its angle of elevation taken at the bottom is 46°, and 200 yards farther off on a level with the bottom the angle is 31° ?

(By construction and calculation.)

No. 2.

The same, with the following alterations :-
In question 1, 27 75 for 35.25.

In question 2, 40 ft. 6 in. for 37 feet.

In question 4, 14 for 10.

In question 5, 9 for 8.

In question 6, octagon for hexagon.

In question 8, 18 for 15.

In question 9, 350 for 200.

No. 3.

The same as No. 1, with the omission of questions 4, 5, 8, and the addition of the following:

(a) What is the length of a circular arc of 26 degrees, the radius being 117 feet; and what is the area of the circular segment?

(b) Find the solid content and the superficial area of a cone, whose height is 60 feet, and the diameter at the base 17 feet 6 inches. (c) Construct a semi-ellipse for an arch of 30 feet span and 8 feet

rise.

EXCAVATOR.
No. 1.

Time allowed, 3 hours.

1. How would you ascertain the nature of soil for foundations? 2. If the earth is hard in some places and soft in others, what method would you adopt to form a good foundation so as to guard against partial settlement?

3. Describe fully the method of making good concrete, and the proportion of the different ingredients.

4. Make an analysis of 10 yards of digging and throwing out ground, including finding and fixing, shoring, and filling in, and ramming around foundations and drains.

5. Make an analysis for sinking and steining wells 20 feet deep, also for boring, also for digging in rock requiring blasting.

6. Make an analysis for forming embankments, parapets, slopes, or other earthworks to required slopes, revetting the same with sods; describe size and mode of building sod work.

7. Make an analysis for a 3 inch and 6 inch cofferdam, double and single.

8. What description of tools, implements, and plant does a navigator and excavator require ?

No. 2.

1. How many days will it take to excavate a tunnel in chalk 150 yards long, 5 yards wide, and 6 yards high, the top being semicircular? How many cubic yards of chalk will such an aperture contain?

2. Describe the method of forming a new road; the materials used in the same; the quantity and description of gravel for 10 square yards, the mode of forming the surface drainage; and, assuming the road and drains to be 8 yards wide, find the cost of such work complete per rod lineal.

Along with questions 1, 3, 5, & 8 of No. 1.

No. 3.

The same as No. 1, with the omission of questions 1, 3, 5, and the addition of the following

:

(a) Describe the usual method of quarrying stone, the mode of boring, the position of the line of least resistance, and the amount of powder required in proportion to its length.

(b) How many yards of excavating are contained in a circular well 7 feet in diameter and 80 feet deep, and how many bricks will it require to stein the same dry in half-brick thickness?

No. 4.

The same as No. 1, with the omission of questions 5, 8, and the addition of the following:

(a) How many cubic yards are there in a piece of ground to be excavated 120 feet long, 5 feet 9 inches deep, and 80 feet broad? (b) How many men would you employ in such a work, and what would be the time in which they would probably execute it?