2. Give the feminine form of each of the following adjectives: Sec, grec; malin, marin; aigu, perdu; net, complet; vif, royal, formel.

3. Translate and write in full: "16 July, 1878; 200 men; 220 men; 2,220 men; 80 pounds; 84 pounds; 2,000 pounds; thousands of pounds." Explain the rule about vingt, cent, and mille.

4. "Cherchait à atteindre"; "je le vis lever la tête, essayer de se soulever." Why are these verbs in the infinitive? State the rule, or rules.

5. "Le brancard que l'on avait placé”; “celui qui avait cherché"; "le pauvre soldat était mort". With what part of the sentence do the participles placé, cherché, mort, agree? State the rule of agreement of the participle conjugated with avoir.

NOTE F.

Apportionment of naval instructors, midshipmen, and naval cadets among sea-going ships and otherwise, January 1, 1879.

* Exclusive of the class that entered in the preceding November, whose admission to the Britannia dated from January 15, 1879.

NOTE G.

EXAMINATION PAPERS: ROYAL NAVAL COLLEGE, JUNE, 1878.
EXAMINATION for rank of LIEUTENANT, R. N.

2. Prove the rule for finding the L.C.M. of two algebraical expressions. Find the L.C.M. of

6a1 — a3b— 3a2b2 + 3ab3 — b1 and 9a1 — 3a3b — 2a2b2 + 3ab3 — b1.

4. (a) Explain the terms: Surds, surds of the same order.

Express as surds of the same order √5 and

7.

6. A railway train 66 yards long, traveling at the rate of 50 miles an hour, met another train traveling at the rate of 22 miles an hour, which it passed in 5 seconds. Find the length of the second train.

8. If a and B are the roots of the equation ar2 + bx + c = 0, find the value of

9. Explain the terms: Ratio, Ratio of less inequality.

If a: b be a ratio of less inequality, and a positive quantity, show that the ratio a-r: b―x is less than the ratio a: b.

10. (a) Prove that the ratio of the sum of the latter half of 2n terms of any arithmetical series is to the sum of 3n terms of the same series as 1 to 3.

(b) Sum the series-1+2-&c. to 7 terms.

11. Having given two numbers and the difference of their logarithms, show how the base of the system may be determined.

In what system does the logarithm of 40 exceed that of 5 by 3?

II.-GEOMETRY.

(Time allowed, 3 hours.)

1. Describe an equilateral triangle upon a given finite straight line.

If a second equilateral triangle be described on the other side of the given line, what figure will the two triangles form?

2. If two triangles have two angles of the one equal to two angles of the other, each to each, and one side equal to one side, namely, either the sides adjacent to the equal angles, or sides which are opposite to equal angles in each, then shall the other sides be equal, each to each, and also the third angle of the one equal to the third angle of the other.

3. The complements of the parallelograms, which are about the diameter of any parallelogram, are equal to one another.

4. On the sides AB, BC, CD of a parallelogram are described equilateral triangles ABE, CDF without the parallelogram and BCG within it; prove that EG, FG are equal to the two diagonals respectively.

5. Describe a square equal to a given rectilineal figure.

6. Define a circle.

If two circles touch one another internally, the straight line which joins their centers, being produced, shall pass through the point of contact.

7. Define the segment of a circle.

On a given straight line describe a segment of a circle containing an angle equal to a given rectilineal angle.

8. In a given circle, inscribe a triangle equiangular to a given triangle.

If the inscribed triangle be equilateral, show that the distance of any point in the circumference of the circle from the most remote angle of the triangle is equal to the sum of the distances from the other angles.

9. Enunciate the axioms of Euclid's Fifth Book.

Find a mean proportional between two given straight lines.

10. In equal circles, angles, whether at the centers or circumferences, have the same ratio which the arcs on which they stand have to one another; so also have the sectors.

III.-TRIGONOMETRY,

(Time allowed, 3 hours.)

1. What are the methods of measuring angles commonly made use of?

If the radius of a circle be 20 feet, find to four places of decimals the length of the arc subtending an angle of 7° at the center of the circle.

2. Trace the changes in the sign and magnitude of the fraction

(a) Sin A (1 + tan A) + cos A (1 + cot A) = sec A + cosec A.

6. Investigate a formula giving the value of the sine of half an angle of a plane triangle in terms of the sides.

7. The area of a quadrilateral figure ABCD is 9,688 square yards. AB is 110 yards, BC 91 yards, AC 125 yards, and CD) 82 yards. Find the side AD.

8. If in the plane triangle ABC the angle A be three times the angle B, prove that—

9. Find an expression for the value of the cosine of an angle of a spherical triangle in terms of functions of the sides, and thence the cosine of a side in terms of functions

of the angles.

10. If a, b, c, be the sides of a spherical triangle, and if the arc d be drawn from the angle A to bisect the side a, show that

a

COS cos cos

b + c cos b = 0.

2

2

11. In the spherical triangle ABC, given a =90°, b = 71° 39′, and A find the remaining parts."

IV.-MECHANICS AND HYDROSTATICS.

(Time allowed, 3 hours.)

1. Assuming the Parallelogram of Forces, so far as the direction of the Resultant is concerned, prove it for the magnitude of the Resultant.

Show that if the angle at which two forces are inclined to each other be increased, their resulta nt is diminished.

2. If two forces be inclined to one another at an angle of 150°, find the ratio between them when the resultant is equal to the smaller force.

3. Explain the term: Center of Parallel Forces.

Prove that if three forces, acting on a rigid body, balance each other, the lines in which they act must either be parallel or must pass through a point.

4. A uniform lever of the first kind is 12 feet in length. A weight of 50 lbs. is suspended from each extremity in turn, and it is found that weights of 20 lbs. and 110 lbs. are required at the other extremities to preserve equilibrium. Determine the weight of the lever and the position of the fulcrum.

5. The ninth part of the area of a triangle is cut off by a line parallel to the base. Find the center of gravity of the remaining area.

6. Investigate the conditions of equilibrium for the single movable pulley when the parts of the string are not parallel.

A cord fastened at A passes under a movable pulley bearing a weight, P; it then passes over a fixed pulley at B, and under a second movable pulley bearing a weight, Q, and is fastened to a peg at C. Find the tension of the cord when the angle at one of the movable pulleys is double that at the other, P being greater than Q.

7. Define velocity, distinguishing between uniform and variable velocity.

If 2 seconds be the unit of time and an acre be represented by 40, what is the measure of a velocity of 30 miles an hour?

8. A body projected up a smooth plane, with a velocity of 320 feet per second, returned to the foot of the plane in 30 seconds. How high did it ascend, and what was the inclination of the plane to the horizon?

9. If a circle be placed in a vertical plane, determine the cord passing through the lowest point, down which a body must fall so that it may acquire the greatest horizontal velocity at the bottom.

10. Distinguish between the terms: "Fluid pressure at a point" and "Fluid pressure on a point."

An area of 14 square feet is subject to a uniform fluid pressure of 2,568 lbs. Determine the measure of the pressure at any point when the unit of length is inch.

11. Describe the Hydrostatic Balance, and show how it may be used to compare the specific gravities of a solid and a fluid, the solid being of less specific gravity than the fluid.

12. What must be the weight of a mass of silver (sp. gr. 10.5) which, when weighed in a fluid of specific gravity 4.5, appears to weigh the same as a mass of lead (sp. gr. 11.4) the weight of which in vacuo is 15 lbs.?

V. PHYSICS.

(Time allowed, 3 hours.)

1. Give a diagram of a common pump, and show how the height of the barometer influences the limit of its action.

2. Give an account of the phenomena of capillary action, and describe the different ways in which it manifests itself in a water and a mercury barometer.

3. Describe Nicholson's Hydrometer.

A liquid is known to consist of water and alcohol; explain how a Nicholson's Hydrometer could be used to determine the proportions in which they are mixed.

4. What is meant by magnetic induction? Explain the statement that repulsion is a surer test of magnetization than attraction, and show that it is possible for one end of an iron bar under certain circumstances neither to attract nor repel the red pole of a magnet brought near to it.

5. What are meant by the lines of force in a magnetic field? In what directions do the lines of force of the terrestrial magnetism pass through the magnetic equator? 6. Distinguish between the temperature of a body and the quantity of heat which it contains.

If m lbs. of one body and n lbs. of another rise respectively p degrees and q degrees in temperature on the addition of the same quantity of heat to each, compare their specific heats.

7. What is meant by the hygrometric state of the air?

Explain the significance of the indications of the wet and dry bulb thermometers. 8. A ray of light passes from one medium to another; explain by what law the change in its direction is governed.

A ray of light falls on one surface of a triangular glass prism, is totally reflected at the second, and emerges from the third face. Draw a diagram showing the course of

the ray.

9. Distinguish between a virtual and a real image formed by a lens.

A large convex lens is used as a reading glass in the ordinary way; explain with a diagram the formation of the magnified image.

VI.-STEAM ENGINE.

(Time allowed, 3 hours.)

1. Water of 98° F. is under the atmospheric pressure. How many units of heat will be required to raise one pound of it to the boiling point, and how many more units to evaporate nine-tenths of a pound?

If the pressure to which it is subjected is 2 atmospheres, of which the corresponding temperature is 249° F., how much heat will be required to make it boil, and how much more to evaporate the whole of it?

2. What is meant by an indicated horse-power?

If the volume of a pound of steam under a pressure of 30 lbs. per sq. in. is 134 cubic feet, how many foot-pounds of external work is done during its formation from water under this pressure; and how many pounds of steam will be required per hour to develop one indicated horse-power?

3. Give a short description of coal, stating what its great calorific value is due to. 4. Describe the indicator, and how it is used.

5. How much salt does ordinary sea-water contain? At what density should the water in the boiler be kept? Describe how the density of the boiler water is determined, and how it is kept from becoming too great.